A Density Functional Theory for VaporâLiquid Interfaces of Mixtures

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A Density Functional Theory for Vapor−Liquid Interfaces of Mixtures Using the Perturbed-Chain Polar Statistical Associating Fluid Theory Equation of State Christoph Klink and Joachim Gross* Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany S Supporting Information *

ABSTRACT: A density functional theory based on the perturbed-chain polar statistical associating fluid theory (PCP-SAFT) is extended to mixtures. The Helmholtz energy functional is suitable for inhomogeneous fluid phases and is here applied to vapor− liquid systems. The attractive branch of the van der Waals attractions are treated with a perturbation theory of first order in a nonmean field approach. The radial distribution function is considered in the Percus−Yevick closure for chain fluids with a threefluid theory approach. This perturbation term is brought to consistency with the PCP-SAFT equation of state using the approach of Gloor et al. [Gloor, G. J.; Jackson, G.; Blas, F. J.; del Rio, E. M.; de Miguel, E. J. Chem. Phys. 2004, 121, 12740]. The approach is applied to vapor−liquid interfaces of mixtures and the resulting surface tension is compared to experimental data. Binary mixtures with two subcritical components as well as mixtures with one species above its critical point, for the considered temperature, are considered. The density functional theory is applied to these systems with unaltered pure component parameters of the PC-SAFT or PCP-SAFT model. The binary interaction parameter was determined to bulk-phase vapor−liquid data (temperature, pressure, composition) only. Excellent agreement was found for the surface tension of mixtures without any model parameter adjusted to interfacial properties.

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INTRODUCTION The properties of interfaces are essential for example for evaporation processes of vapor−liquid mixtures or for the coalescence behavior of liquid−liquid systems, but they also determine the distinct behavior of confined and nanostructured fluids and solids. Predicting the interfacial properties of mixtures, such as the surface tension, is the objective of this study. Significant progress was made in the correlation and even prediction of bulk phase properties using equations of state derived from statistical mechanical theories. One line of development led to the statistical associating fluid theory (SAFT). The SAFT models are based on Wertheim’s graph theoretical work on directional interactions,1−4 that lead to simple association terms and approaches to model chainlike fluids.5−7 The family of SAFT models has several members, such as soft-SAFT,8,9 SAFT-VR,10,11 SAFT-γ,12,13 or PC-SAFT,14,15 and the polar version, PCP-SAFT.16−19 Reviews of these and other variants as well as fields of application are given by Müller and Gubbins,20 Economou,21 Paricaud et al.,22 and Tan et al.23 The SAFT equations of state are formulated as the Helmholtz energy as a function of temperature and the densities of all species. A central property of an interface in equilibrium is the density of each species as a function of the space coordinate. The density profile makes a continuous transition from one side of the interface to the other. For argon, the relevant density is simply the particle number density. More generally, the relevant density is a probability for each species to be in a certain molecular orientation and configuration at a certain position. A theory then requires an expression for the Helmholtz energy as a functional of the relevant density profile. The square gradient theory (SGT) uses an expansion of the Helmholtz energy in terms of the © 2014 American Chemical Society

density profile truncated after the second order (squared gradient) term in density. The SGT was developed by van der Waals and Rayleigh and a comprehensive account is given in ref 24. The SGT is usually applied with a so-called influence parameter that is empirically fitted to pure component (or mixture) properties. Numerous early applications of the SGT to interfacial properties apply cubic equations of state. Kahl and Enders have applied the SGT with the statistical associating fluid theory (SAFT)25,26 and several subsequent studies employ the framework with SAFT models. We refer to review articles for an overview.22,23 The influence parameter can also be expressed more fundamentally in terms of a direct correlation function.24,27 A recent study of Breure and Peters has taken the route to approximate the direct correlation function and estimate an averaged influence parameter.28 The theory does require an empirical scaling function, which, however, can well be treated as a universal function and the theory is numerically convenient. The DFT can be formulated in the outset as an exact approach, which requires an expression for the Helmholtz energy as a functional of the relevant densities. Various levels of approximation are common for the Helmholtz energy functional. Most DFT studies apply a perturbational approach to the Helmholtz energy, in which the intermolecular potential is divided in a repulsive branch and the attractive contributions are considered as perturbations. This decomposition does not in itself constitute an approximation, but the model for the reference fluid and the truncated expansions accounting for the attractive parts of the Received: Revised: Accepted: Published: 6169

September 10, 2013 February 27, 2014 March 5, 2014 March 5, 2014 dx.doi.org/10.1021/ie4029895 | Ind. Eng. Chem. Res. 2014, 53, 6169−6178

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intermolecular potential are approximate. The hard-sphere fluid, which serves as a reference to the majority of perturbation approaches, is now reliably described with the fundamental measure theory (FMT) proposed by Rosenfeld.29,30 A modification31,32 of the original parametrization of the FMT leads to consistency of the Helmholtz energy functional to the bulk phase Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) model.33,34 Wertheim’s theory for highly directional (associating) interactions, which is used to model hydrogen bonding interactions, was formulated in density functional form. A number of studies applied and evaluated this theory for associating fluids.35−40 In the limit of complete association, a chain fluid of permanently bonded segments can be constructed. The definition of an accurate Helmholtz energy functional for the formation of chains has been subject to many investigations41−47 and a powerful expression was recently devised by Tripathi and Chapman.48−50 The dispersion term that accounts for the van der Waals attraction, requires an approximation for the radial distribution function for pairs of all species. Many studies of chain fluids simply adopt a mean field approach, by assuming the radial distribution function to be unity.38,40,51−53 For an engineering approach, the so-obtained bulk phase properties however will not be accurate enough and several investigations construct dispersion terms by taking the nonuniform structure into account.54−60 It is desirable, however, to extend established SAFT models to interfacial problems without the need to construct entirely new dispersion terms. Two main pathways exist to achieve this objective. Some studies simply use locally averaged densities (also referred to as weighted densities61−63) in the dispersion term of PC-SAFT and SAFT-VR in order to go beyond the local density approximation.64−66 Another elegant approach for the SAFT-VR model was suggested by Gloor et al.67,68 They simply define a new Helmholtz energy functional for the dispersive attraction and account for (subtract) the small difference between this so-defined new nonlocal dispersion term and the existing (bulk) dispersion term of the SAFT-VR model. The small difference between the new nonlocal and the existing dispersion term is treated in a simple local density approximation. This approach leads to consistency between the Helmholtz energy functional and the well-known bulk-phase SAFT-VR model. Llovell et al. extended this DFT approach to mixtures69 and found very good agreement to experimental data for reservoir mixtures.70 In previous work from our group, we applied the prescription of Gloor et al.67 with the PC-SAFT equation of state and showed that this approach is forgiving with respect to the level at which the nonlocal functional for dispersive interactions is defined.71 A second order theory leads to almost indistinguishable results compared to a first order theory, although in bulk phases the effect of a second order term is significant. It is noteworthy to review an alternative route to a Helmholtz energy functional for attractive interactions, that is, by approximating the fluids direct correlation function. The starting point of Tang72 and of Tang and Wu57 is a functional expansion of the Helmholtz energy around the bulk fluid, which is described by the first-order mean spherical approximation (FMSA).73 The functional derivatives of order n in this expansion are n-body direct correlation functions. The FMSA also provides the approximation for the attractive part of the direct pair distribution function, after which the expansion is truncated. A comparable conceptual path has been taken by Zhou, who uses a

polymer-reference interaction site model (PRISM)74,75 to approximate a direct correlation function of chain fluids.76 Generally, considerable progress has been made in the field of inhomogeneous fluids (and solids) and reviews on theoretical developments as well as on applications are given, for example, by Wu,77,78 by Löwen79 and by Emborsky et al.80 In a previous publication from our group, we used the approach of Gloor et al.67 to formulate a DFT that is consistent with the PCP-SAFT equation of state for bulk phases.71 The analysis showed that relaxing the mean-field approximation, by considering the radial distribution function g(r) in the first-order perturbation functional for dispersive interactions, improved the representation of experimental surface tension data. An extension to a nonlocal second order perturbation functional, however, is not necessary. A subsequent study proposed a simple DFT-implementation of the renormalization group theory (RGT).81 The RGT takes the long-range density fluctuations into account that leads to the universal scaling behavior around a critical point.82,83 This study extends the previously proposed PCP-SAFT Helmholtz energy functional to multicomponent mixtures. We examine a first order perturbation theory with a nonmean field three-fluid approach for the dispersive contribution. The theory is applied to both, mixtures of small, spherical components and elongated, more complex components.

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CLASSICAL DENSITY FUNCTIONAL THEORY Before introducing the basics of the proposed DFT framework, we mention the molecular model underlying the PCP-SAFT equation of state. The molecular model of PCP-SAFT is a coarsegrained representation of real molecules, where any real molecule i is modeled as a chain of mi tangentially bonded spherical segments, that interact with van der Waals interactions. For the van der Waals interactions between two segments, we adopt a Lennard-Jones potential, that is decomposed according to the perturbation scheme of Barker and Henderson84 into a repulsive ϕij0(r ) = 4εij[(σij/r )12 − (σij/r )6 ]Θ(σij − r )

(1)

and an attractive branch uijPT(r ) = 4εij[(σij/r )12 − (σij/r )6 ]Θ(r − σij)

(2)

where Θ is the Heaviside step function and r is the distance between two Lennard-Jones segments i and j. In addition to the segment number mi, which is allowed to take on noninteger values, σii and εii are the second and third pure component parameter of the PC-SAFT or PCP-SAFT equation of state. For mixtures, we use a binary parameter kij, that acts on the crossenergy parameter, as εij = (εiiεjj)0.5(1−kij) for i ≠ j. In addition, hydrogen-bonding can be accounted for through a short-ranged square-well potential that is treated through Wertheims TPT11−4 with an angle-averaged coefficient, as suggested by Chapman et al.5 Associating compounds require two additional pure component parameters. Dipolar and quadrupolar interactions can also be accounted for through a third-order perturbation theory that was parametrized to molecular simulation data for two-center Lennard-Jones fluids.16,17,19 Both, the dipole moment μi and the quadrupole moment |Qi| can directly be taken from experiments or from quantum mechanical calculations of molecules in vacuum.16,17,19 For more detail on the molecular model we refer to a previous study.71 6170

dx.doi.org/10.1021/ie4029895 | Ind. Eng. Chem. Res. 2014, 53, 6169−6178


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δF res[ρk ]/kT

The starting point for the classical DFT is the grand potential Ω, which for mixtures in the absence of an external field, is Ω = F[ρk (r)] ̃ −

∑ μi ∫ dr ̃ ρi (r)̃

δρi (r)

where μi is the chemical potential of component i. Further, ρi(r)̃ denotes the molecular density with r ̃ as the array that specifies the position as well as the configuration of molecules. The square brackets around ρk(r)̃ indicate the functional dependence on the densities of all substances; that is, the index k in the variable list is a generic index that represents all N substances, k = (1,..., N). At equilibrium the grand potential is minimal, leading to conditions for the equilibrium density profiles of all substances, as δF[ρk ] δρi (r)̃

− μi

δρi (r)

= ln[ρi (r)Λi3]

(5)

F hs[ρ] = kT

(6)

[ρk ]/kT

δρi (r)

+

δF assoc[ρk ]/kT δρi (r)

δF multipolar[ρk ]/kT δρi (r)

(8)

(9)

∫ dr

f hs (n) kT

(10)

According to the FMT, the residual Helmholtz energy density f hs is only a function of four local weighted densities nα with α ∈ {0 ..., 3} and of two weighted vector densities nα with α ∈ {V1, V2}. For a description of the modified FMT we refer to the primary literature.31,32 The adaption to planar interfaces is given in ref 71. The Chain Contribution. The chain formation in the SAFT equation of state was first described for bulk fluid phases. A full functional form was later developed by Tripathi and Chapman.48,49 The theory is based on Wertheim’s perturbation theory of first order. An advantage of the Tripathi−Chapman functional is given by the choice of the ideal gas reference fluid, which is the mixture of nonconnected spherical segments. The theory not only describes the residual contribution of chain fluids based on nonbonded spherical segments, but it also accounts for the connectivity of the segments in the ideal gas state. To appreciate this feature, we assume for a moment the theory would only capture the residual part and require a chain fluid in the ideal gas state as a reference. In this case, one would have to provide all bond-length constraints and angle-distributions for each constituent of a mixture in the ideal gas state. The Tripathi− Chapman functional obviates this requirement and allows us to ignore intramolecular constraints. For the general form of the theory, we refer to the original literature.48,49 We are here concerned with the special case of averaged segments of homosegmented chains. Averaging eliminates the difference between the density profile of a terminal site of a chain and the density profile of any other segment of the chain. Rather, the density profiles ρi(r) of all segment-types of a homosegmented chain i are assumed equal. The Tripathi−Chapman theory can then be cast in the form

In these equations we have made a transition from the molecular density ρi(r)̃ ≈ ρi(r)f i(ω) to the molecular density ρi(r) that only depends on the (say) center of mass position r and on the orientational and conformational distribution function f i(ω). The orientational and conformational degrees of freedom are in eq 5 and eq 6 absorbed in the de Broglie wavelengths Λi(T). The chemical potential is also divided into an ideal gas contribution and a residual part, as μi = kT ln(ρLi Λ3i ) + μres,L . The i ideal part and the residual part of the chemical potential are density-dependent and vary across an interface, whereas the total chemical potential is constant with space coordinate for a system in equilibrium and in the absence of external fields. A choice for a location, at which we consider the residual chemical potential, can be made. Here, the index “L” for the residual chemical potential μres,L and for the liquid density ρLi refers to the bulk i liquid phase. The equilibrium condition for the density distribution of fluid mixtures, eq 4, can now be reformulated as ⎛ δF res[ρk ]/kT ⎞ ⎟⎟ ρi (r) = ρi L exp⎜⎜μi res,L /kT − δρi (r) ⎠ ⎝

δρi (r)

disp

In the following, we treat all of the contributions in some detail. To best evaluate the newly proposed dispersion term, this study, however, considers only non-hydrogen bonding fluids where the association term (assoc) vanishes and it will not further be discussed. For the hard-sphere term (hs) and the contribution due to chain formation (chain) we review existing mixture theories, while for the dispersion term we suggest a new formulation consistent with the PC-SAFT equation of state. The Hard-Sphere Contribution. An accurate description of hard-sphere mixtures became possible with the development of the Fundamental Measure Theory (FMT) by Rosenfeld.29,31,32 Subsequently, Roth et al.31 and Yu and Wu32 independently modified the FMT so that for homogeneous hard-sphere mixtures the theory simplifies to the BMCSL model. The residual Helmholtz energy functional Fhs[ρk(r)] of a hard-sphere mixture is given as

with the functional derivative to component i as δF ig[ρk ]/kT

δF chain[ρk ]/kT

μi res = μi hs + μichain + μidisp + μiassoc + μi multipolar

(4)

i

δF

+

and in terms of the chemical potential, it is

∀i

∫ dr∑ ρi (r){ln[ρi (r)Λi3] − 1}

δρi (r)

+

At this point it is useful in most DFT approaches to decompose the Helmholtz energy functional as an ideal gas contribution and a residual part, F[ρ] = Fig[ρ] + Fres[ρ]. The ideal gas contribution of an isotropic mixture can be written as F ig[ρk ]/kT =

δF hs[ρk ]/kT

+

(3)

i

0=

=

∀i (7)

where the de Broglie wavelengths Λi(T) cancel out and will not concern us further. This form of the equilibrium condition is convenient, because it suggests a simple Picard iterative scheme for solving the density profiles ρi(r) of all components. The functional derivative of the Helmholtz energy appearing in eq 7 has several contributions for the PCP-SAFT model, according to 6171


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chain i are thereby treated in an average way, as ρi,α(r) = ρi(r). The first order term is constructed as a three-fluid theory

∑ (mi − 1)∫ dr ρi (r){ln(ρi (r)) − 1} i

−

∑ (mi − 1) ∫ dr ρi (r){ln[yiidd (ρk̅ (r))λi(r)] − 1}

F1PT[ρk ]/kT =

i

(11)

With the two terms, we have kept the structure of the theory apparent as a chain-contribution in the ideal gas limit and a residual chain contribution. Thereby, yiidd (ρk̅ (r)) denotes the cavity correlation function evaluated at contact distance di between two nonbonded segments. For contact distances, the value of ydd ii is equal to the value of the radial distribution function. A full nonlocal description of ydd ii (r − r′) is not trivially available, so that in our previous work, we followed Kierlik and Rosinberg43 and Tripathi and Chapman48 in approximating ydd ii (ρ̅k(r)) locally, but with at a weighted density ρ̅k(r) as given below. It is important, however, to point to a thorough study of Schulte et al.85 who propose an elegant symmetrically averaged expression for ydd ii based on the FMT. Their functional clearly showed improved results for inhomogeneous fluids. Here, we nonetheless use our previous expression, rather than adopting the newly proposed and improved functional of Schulte et al., only in order to keep consistency with our earlier results for pure components. The BMCSL theory evaluated at space-fixed average densities ρ̅k(r) is yiidd (ρk̅ (r)) =

1.5diζ2̅ 0.5(diζ2̅ )2 1 + + 2 1 − ζ3̅ (1 − ζ3̅ ) (1 − ζ3̅ )3

with mixture segment densities (n = (2,3)) π ζn̅ (r) = ∑ ρi̅ (r)midin 6 i

3 4πdi3

∫ dr′ρi (r′)Θ(di − |r − r′|)

1 4πdi2

∫ dr′ρi (r′)δ(di − |r − r′|)

δρi (r)

(12)

(13)

(14)

(15)

− (mi −

− ∑ [(mj − 1)

∫

j

∂ ln yjjdd (ρk̅ (r′)) 3 dr′ρj (r′) ∂ρi̅ (r′) 4πdi3

δF disp[ρk ]/kT δρi (r)

i

i

(17)

(18)

where the density in round brackets indicates that, according to the LDA, the Helmholtz energy at any position r is calculated for a quasi-homogeneous phase of densities ρk = ρk(r). The functional derivative to the density of component i is

− 1}

∫ dr′ρi (r′) λ (1r′) 4π1d 2 δ(di − |r′ − r|)

kT

+ (F disp,PC − SAFT(ρk ) − F1PT(ρk ))

Θ(di − |r′ − r|)] − (mi − 1)

ϕijPT(r )̂

F disp[ρk (r)] = F1PT[ρk (r)]

= (mi − 1) ln(ρi (r)))

1){ln[yiidd (ρk̅ (r))λi(r)]

j

where the distance between position r and r′ is abbreviated as r̂ = |r −r′|. The radial distribution function ghc(r̂,η̂,mij) is the average segment−segment pair correlation of the hard-chain reference fluid with mij segments. The radial distribution function of the inhomogeneous fluid is approximated54 as that of a homogeneous fluid with the average of the packing fractions at r and r′, as η̂ = 1/2 (η(r) + η(r′)) = 1/2π/6∑imid3i (ρ(r) + ρi(r′)). Further, we use the combination rule mij = 1/2(mii + mij). The segment− segment radial distribution function of hard-chain fluids is considered in the Percus−Yevick approximation, where an analytic expression was developed by Tang and Lu86 based on the work of Chiew.87 For a binary i−j mixture the theory involves the correlation integral of three fluids, i.e. fluids with mii, with mjj, and with mij segments, but all with the same packing fraction η̂. The three-fluid approach is convenient in a density functional treatment, because the radial distribution function for the three chain-lengths can be precomputed (for various radial distances r̂, and densities η̂) and used for any composition. A one-fluid theory, in contrast, would require ghc for any composition observed across the interface. The functional for the attractive (dispersive) interactions according to eq 17 is not yet compatible with the PC-SAFT equation of state. The difference between the dispersion term of the PC-SAFT model and eq 17, however, is fairly small and we treat the difference in a local density approximation as proposed by Gloor et al.67 We note that the attractive Helmholtz energy functional given by eq 17 is different to the one developed by Gloor et al.67,68 We do not introduce the contact-value averaged radial distribution function and we use a different sequence in the two perturbation approaches: the formation of hard-chains is considered first so that the attractive perturbation theory considers hard-chains as a reference. The full dispersion term, consistent with the PC-SAFT equation of state, for mixtures is then

with δ as the Dirac-function. The functional derivative of the Helmholtz energy eq 11 is δF chain[ρk ]/kT

i

g (r ̂, η ̂, mij)

Furthermore, eq 11 contains λi(r) as the average density at contact-distance around a segment of chain i with position r, given as λi(r) =

∑ ∑ mimj ∫ ∫ dr dr′ρi (r) ρj (r′)

hc

where ρ̅i(r) is the weighted density in the interpenetration volume of two segments around fixed position r given as ρi̅ (r) =

1 2

=

δF1PT[ρk ]/kT δρi (r) ⎛ μ disp,PC − SAFT μ 1PT ⎞ + ⎜⎜ i − i ⎟⎟ kT kT ⎠ ⎝

(16)

The dispersive Attraction. The dispersion term is based on a first order perturbation theory. The individual segments α of a

(19)

with 6172


Industrial & Engineering Chemistry Research δF1PT [ρk ]/kT δρi (r) +

1 2

=

∑ mimj ∫ dr′ρj (r′)g hc(r ̂, η ̂, mij) j

∑ ∑ mjmkρk (r) ∫ dr′ρj (r′) j

k

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ϕijPT(r )̂ kT

∂g hc (r ̂, η ̂, mjk ) π ∂η ̂

ϕPT(r )̂ 3 jk di

the capillary waves than the vapor phase. A more thorough study is advisable on this particular point. The critical point of the mixture is calculated with the approach of Heidemann and Khalil90 and for m̅ we set m̅ = ∑iximi.

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mi

NUMERICAL PROCEDURE The numerical procedure for solving the DFT problem is very similar to the one described in an earlier study for pure substances.71 We discretize the interface with 1000 grid points and a grid spacing of 0.25 Å. The interactions are evaluated up to a cutoff distance of rc = 9σi, where i is the species with largest σi parameter. Tail-corrections are applied for larger distances. For further details we refer to the appendix of our provious study.71 For separation distances r > 4σi, however, we assumed the radial distribution function, ghc(r), is unity, so that part of integration of the dispersion term can be performed analytically. We use a Picard iteration (damped direct substitution algorithm) of eq 7 in order to iterate the density profile of all species in the mixture. The empirical relation for the starting values of all densities proposed in ref 71 is here used with the critical temperature of the mixture Tmix c . For any state point, the calculation procedure is 1. Determine the critical temperature of the mixture Tmix c according to the approach of Heidemann and Khalil.90 2. Determine the bulk residual chemical potentials, the bulk densities of all components in the mixture for both coexisting phases, and the bulk pressure of the mixture. 3. Generate the starting density profiles for all components. 4. Tabulate values for the pair correlation function ghc(r̂,η̂,mij) for all i−j pairs, and for the derivative to density (∂ghc/∂η̂) in two-dimensional grids of {r̂,η̂} and parametrize a cubic spline interpolation. 5. Iterate the partial densities ρi(z) to desired accuracy, by using eq 7 in a damped direct substitution scheme. 6. Calculate the surface tension from eq 22 and eq 23. Results and intermediate results of a sample calculation are given as a Supporting Information.

(20)

kT

The integration in eq 20 is carried out numerically up to a cutoff radius rc = 9σ, and an analytic tail-correction is applied.71

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SURFACE TENSION The calculation of the surface tension in mixtures is unchanged, compared to that of pure substances. The surface tension of the planar system can be expressed as the difference of the grand potential Ω[ρk] for the equilibrium density profiles ρk(r) and of the bulk phase grand potential, according to 1 γ = (Ω[ρk ] − Ω bulk ) (21) A where A denotes the area of the planar surface. A convenient reformulation of this expression is given by γDFT =

∫ dz{f [ρ(z)] − ∑ μi ρi (z) + pbulk } i

(22)

with f [ρ(z)] as the Helmholtz energy density, which is related to Helmholtz energy according to F[ρ(z)] = ∫ dz f [ρ(z)].

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CAPILLARY WAVE CONTRIBUTION In the DFT we prescribe a flat interface, whereas real systems lower their grand potential through long-wavelength fluctuations of a vapor−liquid interface. In a previous study a simple model for the capillary wave contribution to the surface tension of a vapor−liquid interface was suggested, based on a first order expansion of the universal critical scaling behavior. The development is based on the mode-coupling theory88 assuming, that the DFT accounts for short-wavelength fluctuations of the interface equivalent to the Fisk−Widom theory.89 Without repeating the derivation of the capillary wave term we provide the final result, as ⎛ 3 T 1 γ = γDFT⎜1 + 8π Tc (2.55)2 ⎝

1⎞ ⎟ κ⎠

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RESULTS AND DISCUSSION To evaluate the proposed Helmholtz energy functional, we analyze three classes of binary systems. First we consider the mixture methane and carbon monoxide as a system of small, almost spherical fluids. Methane is modeled as a spherically symmetric fluid, and carbon monoxide, although elongated, has a relatively small chain length. As a result, the effect of the threefluid approach in eq 17 is relatively small. This system thus allows the evaluation of the first order perturbation approach, without the detail of the three-fluid approximation. Figure 1 presents the surface tension of the mixture at a temperature T = 90.67 K with varying mole fraction in the liquid phase, xCO, and compares results of the DFT calculation to experimental data.91 The pure component parameters of all substances considered in this work are summarized in Table 1. The binary interaction parameter kij = 0.018 was independently adjusted to vapor−liquid equilibrium data,92 and the result is illustrated as an inset to Figure 1. Considering that no parameter was adjusted to interfacial properties (neither to pure, nor to mixture properties), we assess the agreement of the density functional theory to the experimental data as very good over the whole range of concentrations. The averaged absolute deviation between the calculated and the experimental values is 0.244 mN/m. Carbon monoxide is modeled as a dipolar fluid, with the polar term treated in a local density approximation. The result suggests that

−1

(23)

where Tc is the critical point and κ is the so-called amplitude ratio. Generally, the capillary-wave contribution is fairly small, that is, the term in the brackets of eq 23 is close to unity, with γ always lower than γDFT. Since the theory is approximate in several aspects,71 it was reparametrized to experimental surface tensions of n-alkanes, with κ(m̅ ) = 0.0045 + 0.0674m̅ , so that the model has a certain degree of empiricism. The resulting approach leads to excellent results for the surface tension of many real pure substances. Equation 23 is applied to all calculations we look at throughout the paper. For mixtures, we need to make a choice on how to determine the critical temperature in eq 23 and the relevant segment number m̅ in the expression for κ(m̅ ). Both values are calculated for the liquid phase composition, with mole fraction xi of species i. The mole fraction uniquely determines the critical point of a binary mixture. This choice is somewhat arbitrary and is driven by the intuition that the liquid phase should be more relevant for 6173



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Figure 3. Calculated dimensionless density of carbon monoxide in the mixture with methane at T = 90.67 K for different pressures.

Figure 1. Surface tension of the mixture methane and carbon monoxide at T = 90.67 K. Comparison of the DFT calculation (solid line) to experimental data91,92 (circles). Inset: Experimental vapor−liquid equilibrium data of methane and carbon monoxide (symbols92) to which the parameter kij = 0.018 of the PCP-SAFT model (lines) was adjusted.

the LDA is sufficient, at least for fluids with moderate dipole moment. Figure 2 shows the density profiles of both species across the liquid−vapor interface for the same mixture. The accumulation of carbon monoxide at the interfacial region (over length-scales of a few angstrom) is clearly observed. Figure 3 presents the

enrichment of carbon monoxide for various pressures and correspondingly varying compositions. With increasing composition of carbon monoxide the accumulation of carbon monoxide at the interface decreases. Enrichment of one species at a fluid− fluid interface (also referred to as ‘surface adsorption’) was early anticipated using square gradient theory.94 According to Telo de Gama and Evans95 the accumulation occurs for the compound with the lower surface tension. For substances of similar size, this corresponds to the substance with the lower attractive energy parameters. More recently, Llovell et al.69 conducted a systematic study, showing that both a difference in chain length and difference in energy-parameters lead to interfacial enrichment. We suspect that this accumulation can have an interesting effect on the resistivity of an interface to heat and mass transport. Johannessen and Bedeaux developed a description of the interfacial resistivities based on nonequilibrium thermodynamics.96,97 An earlier study applied a DFT approach to model an appropriate interface of pure substances.98 Winkelmann99 also earlier mentioned his expectation, that the enrichment of one species can build “a ‘chemical’ barrier to other molecules, penetrating the interface”. The second class of mixtures we investigate is the mixture of nalkanes and tetrahydrofuran (THF) representing nonspherical fluids. The binary interaction parameter, kij = 0.012, was first adjusted to experimental vapor−liquid equilibrium (VLE) data.100 Figure 4 compares the azeotropic phase behavior at

Figure 2. DFT calculation of the dimensionless component densities for the mixture methane and carbon monoxide at T = 90.67 K and xCO = 0.6875.

Figure 4. Vapor−liquid equilibrium of the mixture n-hexane and tetrahydrofuran (THF) at T = 313.15 K. Comparison of the PCP-SAFT EoS calculations with kij = 0.012 (solid line) to experimental data (circles100).

Table 1. Pure Component Parameters of All Investigated Species species i

mi

σi/[Å]

εi/k/[K]

μi/[D]a

17

1.4358 1.5131 2.4740 1.000 3.0576 3.8176 4.6627

3.1356 3.1869 3.5137 3.7039 3.7983 3.8373 3.8384

87.719 163.333 274.182 150.034 236.769 242.776 243.866

0.1098

CO CO216 THFc CH415 C6H1415 C8H1815 C10H2215

|Qi|/[DÅ]b 4.4000

1.6310

D = 3.3356 × 10−30 C·m. bDÅ = 3.3356 × 10−40 C·m2. cParameters adjusted to vapor-pressure and liquid density data compiled in the DIPPR database.93 a

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T = 313.5 K as calculated from the PCP-SAFT equation of state to the experimental data. Application of the DFT formalism using the same kij parameter allows the prediction of the surface tension for this binary system, now at a T = 298.15 K. Figure 5

Figure 6. Surface tension for the mixtures tetrahydrofuran (THF) with n-octane and n-decane at T = 298.15 K. Comparison of DFT calculations (lines) to experimental data (symbols101). For both calculations the parameter kij = 0.012 was taken from the n-hexane/ THF mixture. Figure 5. Surface tension of the mixture n-hexane and tetrahydrofuran (THF) at T = 298.15 K. Comparison of DFT calculation (solid line) to experimental data (circles101). The parameter kij = 0.012 was independently adjusted to experimental data100 as shown in Figure 4. (The dotted line shows the DFT calculation omitting eq 23.)

confirms an accurate prediction of the surface tension over the entire range of the mole fraction xTHF compared to experimental data.101 The averaged absolute deviation of the calculated surface tension to experimental values is 0.101 mN/m. To assess the role of the capillary wave contribution, a calculation result omitting this term (eq 23) is also shown in Figure 5. The calculated values for the surface tension are then too high by about 1 mN/m. The capillary wave term is temperature dependent, but the results presented in Figure 5 are quite representative for other systems studied here. It is well established that the pure component parameters of the PC-SAFT equation of state and other SAFT models are well behaved, say, for the members of a homologous series and, moreover, that the kij parameter is decently transferable from one mixture to other mixtures of the same chemical family. In order to test the robustness of the DFT approach in this respect, we calculated the surface tension of n-octane/THF and n-decane/ THF using the same binary interaction parameter as for the mixture n-hexane/THF. The averaged absolute deviation between the calculated and the experimental values for the surface tension is 0.086 mN/m for the mixture n-octane/THF, and 0.172 mN/m for the mixture n-decane/THF. The very satisfactory agreement of the calculated surface tension compared to experimental data,101 as shown in Figure 6, suggests that the calculation of interfacial properties is also relatively robust toward transferring (or otherwise estimating) k ij parameters. As a last class of system we analyze the surface tension of an isothermal mixture where one species is above its critical temperature. We consider the system of n-decane and carbon dioxide as a fairly asymmetric mixture. The binary interaction parameter was adjusted (kij = 0.053) to vapor−liquid data.102 The surface tension was calculated using this parameter and without any other adjustable parameters. Figure 7 compares the predicted surface tension for varying mole fractions xCO2 to experimental data.102 At the left (xCO2 = 0) of the diagram, the surface tension is that of pure n-decane and the system pressure is

Figure 7. Surface tension for the mixture of n-decane and carbon dioxide at T = 344.3 K. Comparison of DFT calculation (solid line) to experimental data (circles102). The parameter kij = 0.053 was independently adjusted to vapor−liquid equilibrium data.102

the vapor pressure of n-decane. With increasing xCO2 the surface tension decreases, while the system pressure increases, until, at the critical point, the surface tension is zero and the system pressure is the critical pressure of the mixture at T = 344.3 K. The DFT model is in excellent agreement to the experimental data, especially considering that no model parameter was adjusted to any interfacial property. The averaged absolute deviation between the calculated and the experimental values is 0.048 mN/m. Figure 8 illustrates the integrand to surface tension, according to eq 22. The value of the integrand is equal to the difference of normal pressure on the interface to parallel pressure. Toward the critical point, the absolute values of the integrand get small and the integral vanishes. An evaluation of the three-fluid approach of eq 17 requires an asymmetric mixture, i.e. a mixture with appreciably different segment numbers mi of two substances. The good agreement of the theory to the experimental values of the surface tension for the asymmetric mixture of Figure 7 suggests that the three fluid theory is suitable for a density functional treatment in fluid phases. Generally, the surprisingly accurate agreement between theory and experimental data for both, pure components and mixtures, suggests that data for the interfacial tension can be used for parametrizing a physically based equation of state. This is an 6175



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(2) Wertheim, M. S. Fluids with highly directional attractive forces. 2. Thermodynamic perturbation-theory and integral-equations. J. Stat. Phys. 1984, 35, 35−47. (3) Wertheim, M. S. Fluids with highly directional attractive forces. 3. Multiple attraction sites. J. Stat. Phys. 1986, 42, 459−476. (4) Wertheim, M. S. Fluids with highly directional attractive forces. 4. Equilibrium polymerization. J. Stat. Phys. 1986, 42, 477−492. (5) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase-equilibria of associating fluids chain molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1057−1079. (6) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase equilibria of associating fluidsSpherical molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1−31. (7) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Saft equation-of-state solution model for associating fluids. Fluid Phase Equilib. 1989, 52, 31−38. (8) Blas, F. J.; Vega, L. F. Prediction of binary and ternary diagrams using the statistical associating fluid theory (SAFT) equation of state. Ind. Eng. Chem. Res. 1998, 37, 660−674. (9) Pamies, J. C.; Vega, L. F. Vapor−liquid equilibria and critical behavior of heavy n-alkanes using transferable parameters from the softSAFT equation of state. Ind. Eng. Chem. Res. 2001, 40, 2532−2543. (10) GilVillegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. Statistical associating fluid theory for chain molecules with attractive potentials of variable range. J. Chem. Phys. 1997, 106, 4168−4186. (11) Galindo, A.; Davies, L. A.; Gil-Villegas, A.; Jackson, G. The thermodynamics of mixtures and the corresponding mixing rules in the SAFT-VR approach for potentials of variable range. Mol. Phys. 1998, 93, 241−252. (12) Lymperiadis, A.; Adjiman, C. S.; Galindo, A.; Jackson, G. A group contribution method for associating chain molecules based on the statistical associating fluid theory (SAFT-Υ). J. Chem. Phys. 2007, 127, 234903. (13) Lymperiadis, A.; Adjiman, C. S.; Jackson, G.; Galindo, A. A generalisation of the SAFT-group contribution method for groups comprising multiple spherical segments. Fluid Phase Equilib. 2008, 274, 85−104. (14) Gross, J.; Sadowski, G. Application of perturbation theory to a hard-chain reference fluid: An equation of state for square-well chains. Fluid Phase Equilib. 2000, 168, 183−199. (15) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An equation of state based on a perturbation theory for chain molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (16) Gross, J. An equation-of-state contribution for polar components: Quadrupolar molecules. AIChE J. 2005, 51, 2556−2568. (17) Gross, J.; Vrabec, J. An equation-of-state contribution for polar components: Dipolar molecules. AIChE J. 2006, 52, 1194−1204. (18) Kleiner, M.; Gross, J. An equation of state contribution for polar components: Polarizable dipoles. AIChE J. 2006, 52, 1951−1961. (19) Vrabec, J.; Gross, J. Vapor−liquid equilibria simulation and an equation of state contribution for dipole−quadrupole interactions. J. Phys. Chem. B 2008, 112, 51−60. (20) Müller, E. A.; Gubbins, K. E. Molecular-based equations of state for associating fluids: A review of SAFT and related approaches. Ind. Eng. Chem. Res. 2001, 40, 2193−2211. (21) Economou, I. G. Statistical associating fluid theory: A successful model for the calculation of thermodynamic and phase equilibrium properties of complex fluid mixtures. Ind. Eng. Chem. Res. 2002, 41, 953−962. (22) Paricaud, P.; Galindo, A.; Jackson, G. Recent advances in the use of the SAFT approach in describing electrolytes, interfaces, liquid crystals, and polymers. Fluid Phase Equilib. 2002, 194, 87−96. (23) Tan, S. P.; Adidharma, H.; Radosz, M. Recent advances and applications of statistical associating fluid theory. Ind. Eng. Chem. Res. 2008, 47, 8063−8082. (24) Evans, R. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Dekker: New York, 1992; Chapter 3.

Figure 8. Integrand of the surface tension, according to eq 22, for the mixture of n-decane and carbon dioxide at T = 344.3 K and for two pressures.

appealing perspective for substances that are (due to the lack of reliable vapor pressure data) otherwise difficult to parametrize, for example for ionic species or polymers and mixtures with such components.

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CONCLUSION This work proposes a density functional theory for mixtures compatible with the PCP-SAFT equation of state. The Helmholtz energy functional of the dispersive interaction is based on a perturbation theory of first order, where the radial distribution of chain fluids is considered in a three-fluid theory. The three-fluid approach is numerically appealing because it allows precomputation of the radial distribution function and uses an interpolation scheme in density and radial distance only. For a one-fluid theory, in contrast, the radial distribution function would additionally depend on the composition of the mixture. The proposed Helmholtz energy functional is consistent with the PCP-SAFT model for bulk phases, so that the pure component parameters and kij-parameters are unchanged compared to their regular values of earlier studies. The theory predicts the interfacial tension of mixtures in excellent agreement to experimental data without any model parameter adjusted to interfacial properties (neither pure component nor mixtures properties).

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ASSOCIATED CONTENT

S Supporting Information *

This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +49 (0)711 685 66103. Fax: +49 (0)711 685 66140. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support of the German Research Council (Deutsche ForschungsgemeinschaftDFG) through the collaborative research center SFB-TRR 75 is gratefully acknowledged.

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REFERENCES

(1) Wertheim, M. S. Fluids with highly directional attractive forces. 1. Statistical thermodynamics. J. Stat. Phys. 1984, 35, 19−34. 6176



Article

(25) Kahl, H.; Enders, S. Calculation of surface properties of pure fluids using density gradient theory and SAFT-EOS. Fluid Phase Equilib. 2000, 172, 27−42. (26) Kahl, H.; Enders, S. Interfacial properties of binary mixtures. Phys. Chem. Chem. Phys. 2002, 4, 931−936. (27) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford University Press: Oxford, 1989. (28) Breure, B.; Peters, C. Modeling of the surface tension of pure components and mixtures using the density gradient theory combined with a theoretically derived influence parameter correlation. Fluid Phase Equilib. 2012, 334, 189−196. (29) Rosenfeld, Y. Free-energy model for the inhomogeneous hardsphere fluid mixture and density-funtional theory of freezing. Phys. Rev. Lett. 1989, 63, 980−983. (30) Phan, S.; Kierlik, E.; Rosinberg, M. L.; Bildstein, B.; Kahl, G. Equivalence of two free-energy models for the inhomogeneous hardsphere fluid. Phys. Rev. E 1993, 48, 618−620. (31) Roth, R.; Evans, R.; Lang, A.; Kahl, G. Fundamental measure theory for hard-sphere mixtures revisited: The White Bear version. J. Phy.: Condens. Matter 2002, 14, 12063−12078. (32) Yu, Y. X.; Wu, J. Z. Structures of hard-sphere fluids from a modified fundamental-measure theory. J. Chem. Phys. 2002, 117, 10156−10164. (33) Boublk, T. Hard-sphere equation of state. J. Chem. Phys. 1970, 53, 471−472. (34) Mansoori, G.; Carnahan, N.; Starling, K.; Leland, T., Jr. Equilibrium thermodynamic properties of the mixture of hard spheres. J. Chem. Phys. 1971, 54, 1523. (35) Segura, C. J.; Chapman, W. G.; Shukla, K. P. Associating fluids with four bonding sites against a hard wall: Density functional theory. Mol. Phys. 1997, 90, 759−771. (36) Segura, C. J.; Vakarin, E. V.; Chapman, W. G.; Holovko, M. F. A comparison of density functional and integral equation theories vs Monte Carlo simulations for hard sphere associating fluids near a hard wall. Mol. Phys. 1998, 108, 4837−4848. (37) Pizio, O.; Patrykiejew, A.; Sokolowski, S. Evaluation of liquid− vapor density profiles for associating fluids in pores from densityfunctional theory. J. Chem. Phys. 2000, 113, 10761−10767. (38) Blas, F. J.; del Rio, E. M.; de Miguel, E.; Jackson, G. An examination of the vapour−liquid interface of associating fluids using a SAFT-DFT approach. Mol. Phys. 2001, 99, 1851−1865. (39) Yu, Y. X.; Wu, J. Z. A fundamental-measure theory for inhomogeneous associating fluids. J. Chem. Phys. 2002, 116, 7094−7103. (40) Bryk, P.; Sokolowski, S.; Pizio, O. Density functional theory for inhomogeneous associating chain fluids. J. Chem. Phys. 2006, 125, 024909. (41) Wertheim, M. S. Thermodynamic perturbation-theory of polymerization. J. Chem. Phys. 1987, 87, 7323−7331. (42) Kierlik, E.; Rosinberg, M. L. A perturbation density-functional theory for polyatomic fluids. 1. Rigid molecules. J. Chem. Phys. 1992, 97, 9222−9239. (43) Kierlik, E.; Rosinberg, M. L. A perturbation density-functional theory for polyatomic fluids. 2. Flexible molecules. J. Chem. Phys. 1993, 99, 3950−3965. (44) Kierlik, E.; Rosinberg, M. L. Perturbation density-functional theory for polyatomic fluids. 3. Application to hard chain molecules in slitlike pores. J. Chem. Phys. 1994, 100, 1716−1730. (45) Muller, M.; MacDowell, L. G. Interface and surface properties of short polymers in solution: Monte Carlo simulations and self-consistent field theory. Macromolecules 2000, 33, 3902−3923. (46) Yu, Y. X.; Wu, J. Z. Density functional theory for inhomogeneous mixtures of polymeric fluids. J. Chem. Phys. 2002, 117, 2368−2376. (47) Malijevsky, A.; Bryk, P.; Sokolowski, S. Density functional approach for inhomogeneous star polymer fluids. Phys. Rev. E 2005, 72. (48) Tripathi, S.; Chapman, W. G. Microstructure of inhomogeneous polyatomic mixtures from a density functional formalism for atomic mixtures. J. Chem. Phys. 2005, 122, 094506.

(49) Tripathi, S.; Chapman, W. G. Microstructure and thermodynamics of inhomogeneous polymer blends and solutions. Phys. Rev. Lett. 2005, 94, 087801. (50) Jain, S.; Dominik, A.; Chapman, W. G. Modified interfacial statistical associating fluid theory: A perturbation density functional theory for inhomogeneous complex fluids. J. Chem. Phys. 2007, 127, 244904. (51) Gloor, G. J.; Blas, F. J.; del Rio, E. M.; de Miguel, E.; Jackson, G. A SAFT-DFT approach for the vapour−liquid interface of associating fluids. Fluid Phase Equilib. 2002, 194, 521−530. (52) Lu, J.-F.; Fu, D.; Liu, J.-C.; Li, Y.-G. Study on surface tension for non-polar and associating fluids based on density functional theory. Mol. Simul. 2003, 29, 809−815. (53) Dominik, A.; Tripathi, S.; Chapman, W. G. Bulk and interfacial properties of polymers from interfacial SAFT density functional theory. Ind. Eng. Chem. Res. 2006, 45, 6785−6792. (54) Sokolowski, S.; Fischer, J. The role of attractive intermolecular forces in the density functional theory of inhomogeneous fluids. J. Chem. Phys. 1992, 96, 5441−5447. (55) Wadewitz, T.; Winkelmann, J. Application of density functional perturbation theory to pure fluid liquid−vapor interfaces. J. Chem. Phys. 2000, 113, 2447. (56) Zhou, S. A method to incorporate the radial distribution function of bulk fluid into the density functional approximation. J. Chem. Phys. 2001, 115, 2212. (57) Tang, Y.; Wu, J. Modeling inhomogeneous van der Waals fluids using an analytical direct correlation function. Phys. Rev. E 2004, 70, 011201. (58) Li, Z.; Cao, D.; Wu, J. Density-functional theory and Monte Carlo simulation for the surface structure and correlation functions of freely jointed Lennard-Jones polymeric fluids. J. Chem. Phys. 2005, 122, 174708. (59) Kahl, H.; Winkelmann, J. Modified PT-LJ-SAFF density functional theory. I. Prediction of surface properties and phase equilibria of non-associating fluids. Fluid Phase Equilib. 2008, 270, 50−61. (60) Zhou, S. Going beyond the mean field approximation in classical density functional theory and application to one attractive core-softened model fluid. J. Stat. Mech.: Theory Exper. 2010, 2010, P11039. (61) Tarazona, P. A density functional theory of melting. Mol. Phys. 1984, 52, 81−96. (62) Tarazona, P. Free-energy density functional for hard spheres. Phys. Rev. A 1985, 31, 2672. (63) Tarazona, P. Erratum: Free-energy density functional for hard spheres [Phys. Rev. A 31, 2672 (1985)]. Phys. Rev. A 1985, 32, 3148. (64) Hu, J.; Chen, J.; Mi, J. Prediction of Interfacial Structure and Tension of Binary Mixtures Containing Carbon Dioxide. Ind. Eng. Chem. Res. 2012, 51, 1236−1243. (65) Shen, G.; Ji, X.; Lu, X. A hybrid perturbed-chain SAFT density functional theory for representing fluid behavior in nanopores. J. Chem. Phys. 2013, 138. (66) Hughes, J.; Krebs, E. J.; Roundy, D. A classical density-functional theory for describing water interfaces. J. Chem. Phys. 2013, 138. (67) Gloor, G. J.; Jackson, G.; Blas, F. J.; del Rio, E. M.; de Miguel, E. An accurate density functional theory for the vapor−liquid interface of associating chain molecules based on the statistical associating fluid theory for potentials of variable range. J. Chem. Phys. 2004, 121, 12740− 12759. (68) Gloor, G. J.; Jackson, G.; Blas, F. J.; del Rio, E. M.; de Miguel, E. Prediction of the vapor−liquid interfacial tension of nonassociating and associating fluids with the SAFT-VR density functional theory. J. Phys. Chem. C 2007, 111, 15513−15522. (69) Llovell, F.; Galindo, A.; Blas, F. J.; Jackson, G. Classical density functional theory for the prediction of the surface tension and interfacial properties of fluids mixtures of chain molecules based on the statistical associating fluid theory for potentials of variable range. J. Chem. Phys. 2010, 133. (70) Llovell, F.; Mac Dowell, N.; Bias, F. J.; Galindo, A.; Jackson, G. Application of the SAFT-VR density functional theory to the prediction 6177



Article

of the interfacial properties of mixtures of relevance to reservoir engineering. Fluid Phase Equilib. 2012, 336, 137−150. (71) Gross, J. A density functional theory for vapor−liquid interfaces using the PCP-SAFT equation of state. J. Chem. Phys. 2009, 131, 204705. (72) Tang, Y. First-order mean spherical approximation for inhomogeneous fluids. J. Chem. Phys. 2004, 121, 10605−10610. (73) Tang, Y. On the first-order mean spherical approximation. J. Chem. Phys. 2003, 118, 4140. (74) Chandler, D. Derivation of an integral equation for pair correlation functions in molecular fluids. J. Chem. Phys. 1973, 59, 2742. (75) Curro, J. G.; Schweizer, K. S. Equilibrium theory of polymer liquids: Linear chains. J. Chem. Phys. 1987, 87, 1842. (76) Zhou, S. Thermodynamic perturbation theory in fluid statistical mechanics. Phys. Rev. E 2006, 74. (77) Wu, J. Z. Density functional theory for chemical engineering: From capillarity to soft materials. AIChE J. 2006, 52, 1169−1193. (78) Wu, J. Z.; Li, Z. Density-functional theory for complex fluids. Annu. Rev. Phys. Chem. 2007, 58, 85−112. (79) Löwen, H. Density functional theory of inhomogeneous classical fluids: recent developments and new perspectives. J. Phys.: Condens. Matter 2002, 14, 11897−11905. (80) Emborsky, C. P.; Feng, Z.; Cox, K. R.; Chapman, W. G. Recent advances in classical density functional theory for associating and polyatomic molecules. Fluid Phase Equilib. 2011, 306, 15−30. (81) Tang, X.; Gross, J. Density functional theory for calculating surface tensions with a simple renormalization formalism for the critical point. J. Supercrit. Fluids 2010, 55, 735−742. (82) Wilson, K. G. Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior. Phys. Rev. B 1971, 4, 3184− 3205. (83) White, J. A. Contribution of fluctuations to thermal properties of fluids with attractive forces of limited range: theory compared with PϱT and Cv data for argon. Fluid Phase Equilib. 1992, 75, 53−64. (84) Barker, J. A.; Henderson, D. Perturbation theory and equation of state for fluids. II. A successful theory of liquids. J. Chem. Phys. 1967, 47, 4714−4721. (85) Schulte, J. B.; Kreitzberg, P. A.; Haglund, C. V.; Roundy, D. Using fundamental measure theory to treat the correlation function of the inhomogeneous hard-sphere fluid. Phys. Rev. E 2012, 86. (86) Tang, Y.; Lu, B. C.-Y. Direct calculations of radial distribution function for hard-sphere chains. J. Chem. Phys. 1996, 105, 8262. (87) Chiew, Y. C. Percus−Yevick integral equation theory for athermal hard-sphere chains. II. Average intermolecular correlation functions. Mol. Phys. 1991, 73, 359. (88) Meunier, J. Liquid interfacesRole of the fluctuations and analysis of ellipsometry and reflectivity measurements. J. Phys. (Paris) 1987, 48, 1819−1831. (89) Fisk, S.; Widom, B. Structure and free energy of the interface between fluid phases in equilibrium near the critical point. J. Chem. Phys. 1969, 50, 3219−3227. (90) Heidemann, R. A.; Khalil, A. M. The calculation of critical points. AlChE J. 1988, 26, 769−779. (91) Sprow, F. B.; Prausnitz, J. M. Surface tensions of simple liquid mixtures. Trans. Faraday Soc. 1966, 62, 1105−1111. (92) Christiansen, L.; Fredenslund, A.; Mollerup, J. Vapour−liquid equilibrium of the CH4-Ar, CH4-CO, and Ar-CO systems at elevated pressures. Cryogenics 1973, 13, 405−413. (93) Rowley, R. L.; Wilding, W. V.; Oscarson, J. L.; Yang, Y.; Zundel, N. A.; Daubert, T. E.; Danner, R. P. DIPPR Data Compilation of Pure Chemical Properties; Design Institute for Physical Properties, AIChE: New York, NY, 2009, (94) Carey, B. S.; Scriven, L. E.; Davis, H. T. Semiempirical theory of surface tension of binary systems. AIChE J. 1980, 26, 705−711. (95) Telo da Gama, M.; Evans, R. Theory of the liquid−vapour interface of a binary mixture of Lennard-Jones fluids. Mol. Phys. 1980, 41, 1091−1112.

(96) Johannessen, E.; Bedeaux, D. Integral relations for the heat and mass transfer resistivities of the liquid−vapor interface. Phys. A 2006, 370, 258−274. (97) Glavatskiy, K.; Bedeaux, D. Transport of heat and mass in a twophase mixture: From a continuous to a discontinuous description. J. Chem. Phys. 2010, 133, 144709. (98) Johannessen, E.; Gross, J.; Bedeaux, D. Nonequilibrium thermodynamics of interfaces using classical density functional theory. J. Chem. Phys. 2008, 129, 184703. (99) Winkelmann, J. The liquid−vapour interface of pure fluids and mixturesApplication of computer simulation and density functional theory. J. Phys.: Condens. Matter 2001, 13, 4739−4768. (100) Wu, H. S.; Sandler, S. I. Vapor−liquid equilibria of tetrahydrofuran systems. J. Chem. Eng. Data 1988, 33, 157−162. (101) Piñeiro, Á .; Brocos, P.; Amigo, A.; Pintos, M.; Bravo, R. Surface tensions and refractive indices of (tetrahydrofuran + n-alkanes) at T = 298.15 K. J. Chem. Thermodyn. 1999, 31, 931−942. (102) Nagarajan, N.; Robinson, R. L. Equilibrium phase compositions, phase densities, and interfacial tensions for carbon dioxide + hydrocarbon systems. 2. Carbon dioxide + n-decane. J. Chem. Eng. Data 1986, 31, 168−171.

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