Generalization of the Fundamental-Measure Theory Beyond Hard

Article pubs.acs.org/JPCC

Generalization of the Fundamental-Measure Theory Beyond Hard Potentials: The Square-Well Fluid Case T. Bernet,† M. M. Piñeiro,‡ F. Plantier,† and C. Miqueu*,† †

Laboratoire des Fluides Complexes et leurs RéservoirsIPRA, Université Pau & Pays Adour/CNRS/TOTAL, UMR 5150, 64600, Anglet, France ‡ Departamento de Física Aplicada, Facultad de Ciencias, Universidade de Vigo, 36310, Vigo, Spain ABSTRACT: The fundamental-measure theory (FMT) developed at the end of the 1980s for hard-sphere inhomogeneous fluids has revolutionized density functional theories (DFTs), allowing one to describe accurately for the first time highly confined fluids. However, in order to apply DFTs to real fluids, less rigorous coarse-graining approaches have always been used for perturbative contributions in addition to the FMT hard-sphere treatment. With the aim to propose a formally consistent description avoiding this duality, a possible route is the generalization of the fundamental-measure theory for potentials different from the purely hard repulsive term. This is the main purpose of this work, achieved considering the square-well fluid as initial case test. As a reference for the bulk fluid, the statistical associating fluid theory for the square-well monomer has been selected, and a suitable functional has been developed and validated through comparison against Monte Carlo molecular simulations. This opens up a new class of efficient non-local DFTs directly applicable to mixtures of complex fluids.

■

INTRODUCTION

use of different approaches (FMT and CG) to deal respectively with the repulsive and attractive interaction potential terms. The main goal of this work is to generalize the FMT to interaction potentials with both repulsive and attractive terms. The case study selected is the ubiquitous square-well (SW) potential. This is undoubtedly the most widely used discontinuous interaction potential, and cornerstone of the formulation of bulk fluid equations of state (EoS) as SAFT-VRSW11,12 (statistical associating fluid theory of variable range for the SW potential). An interesting feature is that the performing and worldwide applied SAFT-type EoS13 rest upon originally on the first order thermodynamic perturbation theory (TPT1) developed by Wertheim.14−17 It was actually an inhomogeneous fluid theory, that eventually evolved yielding this homogeneous fluid molecular EoS through its application to the bulk case. In this paper we come back to the original inhomogeneous version of the TPT1 and the proposed developments could be used as possible guidelines for SAFT improvement. The procedure proposed here for the SW potential may be generalized for any other step potentials, either attractive or repulsive. This represents the basis for the FMT development for any generalized interaction potential, bearing in mind the well-known mathematical discretization technique that can be applied to any continuous potential. It is worth to point out that this would bridge the gap toward rigorous NLDFTs suitable to describe any complex fluid inhomogeneous behavior. This eventually will allow obtaining a rigorous

Density functional theory (DFT) represents undoubtedly one of the most rigorous and versatile theoretical tools to describe inhomogeneous fluids. Its use is widespread nowadays and is the key to rationalize physical phenomena as relevant as adsorption, confinement, wetting or capillarity. In this context, Rosenfeld1 developed the fundamental-measure theory2 (FMT) to describe the behavior of the hard-sphere (HS) potential, a classical reference in the base of the development of modern fluid theories. Later, other developments of the same nature have been proposed, but always for the case of rigid repulsive potentials.3,4 The application of DFT to real fluids5 entails of course the consideration of an attractive term within the interaction potential. This objective has been dealt with so far only in a partially convincing manner, by considering a HS reference term represented by the rigorous FMT, but the attractive term has always been treated in a more simplified way using approximated approaches as coarse-graining6−10 (CG). This hybrid FMT-CG approach presents evident inconveniences. For instance, the CG technique is an approximation that performs fairly well for pure fluids, but its extension to mixtures is far from being straightforward. In fact, it requires the assumption of further hypothesis as the adoption of spurious mixing rules, while the elegant FMT formulation is originally formulated for mixtures. This might be the principal reason why non-local DFTs (NLDFTs) are not used to describe complex mixtures, and even very seldom binary mixtures, in spite of the great advantages of its rigor, and also negligible computing times if compared to molecular simulations. This practical limitation relies on the inconsistent © XXXX American Chemical Society

Received: January 25, 2017 Revised: February 23, 2017 Published: February 23, 2017 A

DOI: 10.1021/acs.jpcc.7b00797 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

This is feasible due to the Heaviside functions appearing explicitly in the theory, leading to the decomposition with convolution products, or deconvolution, in short. Let us consider as well that these Heaviside functions are dependent on a characteristic distance 9 i + 9 j , where 9 i and 9 j represent the radii of spheres centered in molecules i and j, but not necessarily corresponding to the physical interaction radii Ri and Rj. This way, the FMT formalism can be generalized to different intermolecular interaction potentials, provided that the latter can be written in terms of Heaviside functions Θ(9 i + 9 j − | ri ⃗ − rj⃗|). Moreover, it is no longer necessary to consider the excess chemical potential low density limit as starting point, which additionally cast serious doubts over the possible extrapolation to higher densities. This way, the formalism can be adapted to an equation of state, or to an inhomogeneous fluid functional containing suitable Heaviside functions in their formulation. This means for instance that the interaction potential uij can be directly considered for the deconvolution, instead of the Mayer function f ij. This deconvolution adopts then the following expression

NLDFT whose bulk limit is an efficient and reliable EoS, providing a complete thermodynamic treatment of both homogeneous and inhomogeneous fluid phases with a fully consistent approach. The paper is organized as follows: in the first section the new formalism developed for the FMT is presented for a SW potential, after a brief recall of the HS original case, and the methodology is proposed for any discretized potential. In the section The FMT Formalism, a new density functional of the Helmholtz free energy is derived for a SW fluid from this extended version of FMT and SAFT bulk limit. Finally, in the section Results and Discussion, density profiles computed with the new version of the theory are discussed and compared to equivalent molecular simulations for validation.

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THE FMT FORMALISM Definition of the Problem. The starting point of the FMT is the definition of the excess free energy density functional Φ as follows β - ex =

∫ Φ d3r

(1)

ex

where - [{ρk ( r ⃗)}] with k = 1, ..., NC is the general excess free energy functional for a mixture of NC compounds, β =

1 , kBT

Θ(9 i + 9 j − | ri ⃗ − rj⃗|)

and

ρk(r)⃗ is the local density associated with the k chemical species, determined at a spatial location r.⃗ Originally, the intermolecular potential considered in the FMT framework is the HS, which can be written as follows for molecules i and j, placed at the distance r uijHS(r ) =

3

=

α=0

if r > R i + R j

(2)

where Ri and Rj represent the i and j spherical molecule radii. This is probably the roughest but useful way to describe intermolecular interactions. Indeed, only short-range repulsive interactions are considered, while long-range attractive terms are neglected, and the molecular overlapping is also excluded. In the FMT original version, Rosenfeld1 adopted the excess chemical potential low density limit as the starting point of his development. The reason for this was to propose a rigorous framework where the FMT formalism might be defined, but this decision might eventually conceal a possible generalization of the theory. First, let us point out that the HS Mayer function is written as HS

fijHS (r ) = e−βuij

(r )

− 1 = −Θ(R i + R j − r )

(4)

where we consider convolution product of p(α) functions for α = 3, 2, 1, 0 (in a tridimensional space), denoted as weighting f unctions. We will consider the decomposition proposed by Kierlik and Rosinberg,18 that uses only the scalar weights, differing from the original Rosenfeld decomposition,1 that used also vectorial weights. In the scalar formulation, the four weights can be expressed as

+∞ if r < R i + R j 0

∑ ∫ p(α) (9 i; | ri ⃗ − r |⃗ )p(3 − α) (9 j; | rj⃗ − r |⃗ ) d3r

p(3) (9 i; r ) = Θ(9 i − r ) p(2) (9 i; r ) = δ(9 i − r ) 1 δ′(9 i − r ) 8π 1 1 p(0) (9 i; r ) = − δ″(9 i − r ) + δ′(9 i − r ) 8π 2πr p(1) (9 i; r ) =

(5)

Geometrical considerations show that the weight integrals equal the 9 i radius sphere f undamental measures, that are

(3)

∫ p(3) (9 i; r ) d3r = 43 π 9 i 3,

0 where Θ is the Heaviside function, Θ(x) = 10 ifif xx < > 0 . This function is taken into consideration as it appears explicitly in the excess chemical potential expression used by Rosenfeld.1 Several ensuing FMT formulations share the same starting point, considering the Mayer function, and subsequently its physical meaning, instead of the Heaviside function, and its mathematical meaning. This way, a risk of confusion exists associated with the formalism generalization, and this fact has to be taken into careful consideration. Indeed, consider two spherical molecules i and j placed at ri⃗ and rj⃗ at a distance r = |ri⃗ − rj⃗ | apart. We will show in the following that the FMT central interest is splitting the contributions arising from ri⃗ and rj⃗ in the integral calculations.

∫ p(2) (9 i; r ) d3r = 4π 9 i 2 ,

∫ p(1) (9 i; r ) d3r = 9 i , and ∫ p(0) (9 i; r ) d3r = 1. This decomposition process is the FMT cornerstone. The use of this mathematical trick allows to uncouple the variables related to the distances ri⃗ and rj⃗ . However, we will show that, depending on the interaction potential considered, the coupling between the variables i and j associated with the molecules can still be present, as it will be noted for the SW case. The Heaviside function decomposed before meets within the DFT formalism a product of local densities ρi and ρj for species i and j placed at ri⃗ and rj⃗ , integrated over those two variables. It is therefore possible to write in a rigorous way B


Article


dependent on two characteristic variables, such as 9 i = {R i , λiR i}. The first of them is that of HS, evaluated at Ri + Rj, and enables the use of the original FMT formalism. However, considering the SW potential, one more characteristic distance appears, requiring the subsequent adaptation of the FMT. This new distance is λiRi + λjRj, and we can define accordingly the weight

∬ ρi ( ri ⃗)ρj ( rj⃗)Θ(9 i + 9 j − | ri ⃗ − rj⃗|) d3ri d3rj 3

=

∫

∑ n(α)(ρi ; 9 i; r ⃗)n(3 − α)(ρj ; 9 j; r ⃗) d3r

(6)

α=0

where the weighted densities associated with the spheres of radii 9 i are defined as n(α)(ρi ; 9 i; r ⃗) =

∫ ρi ( ri⃗)p(α)(9 i; | ri ⃗ − r |⃗ ) d3ri

πi(α)(r ) = p(α) (λiR i ; r )

(7)

to write

It is important to notice that each weighted density n(α)(ρi ; 9 i; r ⃗) is only dependent on the real local density ρi, the radius 9 i and the geometry of the space occupied by the fluid. The Original HS Case. As noted before, the first potential considered within the FMT to describe intermolecular interactions was the HS. In that case, only one characteristic variable is involved, namely 9 i = {R i}, where Ri is the HS geometrical radius. The associated weights can be then defined with the specific notation pi(α) (r ) = p(α) (R i ; r )

Θ(λiR i + λjR j − | ri ⃗ − rj⃗|) 3

=

νi(α)( r ⃗) = n(α)(ρi ; λiR i ; r ⃗) =

(16)

(8)

∬ ρi ( ri ⃗)ρj ( rj⃗)Θ(λiR i + λjR j − | ri ⃗ − rj⃗|) d3ri d3rj 3

=

∑ ∫ pi(α) (| ri ⃗ − r |⃗ )pj(3 − α) (| rj⃗ − r |⃗ ) d3r

(10)

with the equation

∬ ρi ( ri ⃗)ρj ( rj⃗)Θ(R i + R j − | ri ⃗ − rj⃗|) d3ri d3rj 3

∑ ni(α)( r ⃗)n(3j − α)( r ⃗) d3r α=0

(11)

In the following, it will be sometimes convenient to use the total weighted densities, defined with a summation over all species nα(r)⃗ = ∑i n(α) (r)⃗ . These variables are usually i employed in the HS FMT versions, where the dependence on the specie i is systematically eclipsed behind a summation over all appearing species, which is for instance the case for HS EoS. Otherwise, we will show that it might be interesting to keep explicit the i dependence in the weighted densities expressions, with the aim to deal with the cases we are interested in. The SW Case. The SW potential is decomposed into two terms, as follows uijSW (r ) = uijHS(r ) + uijatt (r )

(17)

The FMT formulation presented here is then adapted to the SW potential, and will be denoted hereafter as FMT-SW, with contributions of Ri and λiRi in the weights, in opposition to the original HS version (FMT-HS), that considered only Ri. However, let us recall that the coupling between molecules i and j is still present through the interaction potential, which is constant within the intervals defined by the characteristic distances Ri + Rj and λiRi + λjRj. For the SW potential case the next step is the application of this new formalism to an adapted free energy functional. In order to do so, we have selected the functional used to develop the homogeneous fluid SAFT-VR-SW EoS. Arbitrary Discretized Potential. Following the same reasoning that for the SW case, it is now possible to apply the generalized FMT formalism to any discretized potential

leading to the corresponding weighted densities expression

∫ ρi ( ri⃗)pi(α)(| ri ⃗ − r |⃗ ) d3ri

∫

∑ νi(α)( r ⃗)νj(3 − α)( r ⃗) d3r α=0

(9)

α=0

∫

∫ ρi ( ri⃗)πi(α)(| ri ⃗ − r |⃗ ) d3ri

participating in the decomposition

3

=

(15)

As for the HS case, the weighted functions associated with the interaction range for each molecule type can be now defined as

Θ(R i + R j − | ri ⃗ − rj⃗|)

ni(α)( r ⃗) = n(α)(ρi ; R i ; r ⃗) =

∑ ∫ πi(α)(| ri ⃗ − r |⃗ )πj(3 − α)(| rj⃗ − r |⃗ ) d3r α=0

These weights enable the functions decomposition

=

(14)

+∞

uij(r ) =

∑ ua ,ij[Θ(Ra ,i + Ra ,j − r) − Θ(Ra − 1,i + Ra − 1,j a=1

− r )]

(18)

with R0,i = 0, ∀ i, with a constant step value ua, ij between Ra−1, i + Ra−1, j and Ra, i + Ra, j. The main difficulty of this formalism is that it is applicable only for EoS or functionals written for an explicitly discretized potential (see e.g. the references19,20), and not to continuous potentials, as SAFT-VR Mie21 in its current version. It is therefore within this framework that it would be feasible to treat functionals with discretized Lennard-Jones or Mie potentials.

(12)

defining three different spatial regions. The first term is a repulsive HS identical to the case analyzed before. The second one is an attractive term which can be written as

■

uijatt (r ) = −εij[Θ(λiR i + λjR j − r ) − Θ(R i + R j − r )]

NEW DENSITY FUNCTIONAL FOR THE SW FLUID In this work we have selected as a test case the SW potential to describe the interaction between two particles i and j, split into a repulsive (reference) term and an attractive (perturbation) term.

(13)

where λiRi and λjRj stand for the corresponding molecular interaction ranges and εij is the well depth. The interest of this particular expression is the use of Heaviside functions C


Article

The Journal of Physical Chemistry C This representation of the molecular interaction potential is the foundation of the TPT, describing excess free energy as the addition of a HS contribution - HS and an attractive term contribution - att , thus reading - ex = - HS + - att . In former FMT versions, only the HS contribution was considered, but in order to describe a SW fluid, the excess free energy functional Φ must be expressed as Φ = ΦHS + Φatt, in the same spirit than the functional - ex . To achieve this, the new term Φatt needs to be derived, following the same philosophy than for the ΦHS case. It is then necessary to combine the classical DFT general results with the new FMT-SW. With this new formalism, it is possible to uncouple the terms as ρi ( ri ⃗)ρj ( rj⃗)(uijatt (| ri ⃗ − rj⃗|))m

K HS β -2 = − bulk 4 ×

1 2

(19)

Φ1 = −

Φ2 = −

(20)

α=0

HS Kbulk 4

⎧ ⎪

⎡

⎪ ⎩

⎣

⎛ ∂g HS ⎞⎤ bulk, ij ⎟⎥ ⎟⎥ ∂ ⎝ V ⎠⎦

HS ∑ ⎨(βεij)2 ⎢⎢gbulk, − V⎜ ij ⎜ i,j

⎫ ⎪

∑ (νi(α)νj(3 − α) − ni(α)n(3j − α))⎬ ⎪ ⎭

(25)

The HS Term. Different EoS have been used as reference to describe ΦHS within the FMT-HS framework: • The Percus−Yevick EoS,23,24 with the scaled-particle theory25 for original FMT versions of Rosenfeld1 and Kierlik and Rosinberg18 • The Carnahan−Starling EoS26 generalized for mixtures, with • The Boublı ḱ −Mansoori−Carnahan−Starling EoS27,28 for the White-Bear I version of the FMT by Roth,29 and the Modified-FMT by Yu and Wu30 (evoked by Kierlik and Rosinberg18) • The Hansen-Goos and Roth EoS31 for the WhiteBear II32 version of the FMT We propose to use here the HS EoS by Hansen-Goos and Roth,31 associated with the most accurate FMT-HS version, i.e. White Bear II,32 where the HS excess free energy reads as

⎡ ⎛ ∂g HS ⎞⎤ 1 HS ⎢ HS bulk, ij ⎟⎥ − Kbulk ⎢gbulk, ij − V ⎜ ⎜ ⎟⎥ 2 ⎝ ∂V ⎠⎦ ⎣

1

Φ

HS

(21)

= −ln(1 − n3)n0 + 4

where KHS bulk is the homogeneous bulk HS compressibility. The first advantage of this expression is its compatibility with the Barker-Henderson development included within the SAFT-VRSW EoS,11,12 thus enabling a formulation of the reference term att gHS perturbative contribubulk, ij. Therefore, the functional β tions can be expanded to the second order in βuatt ij , as β - att = β -1 + β -2 , where 1 2

i,j

∑ (νi(α)νj(3 − α) − ni(α)n(3j − α))}

α=0

×βuijatt (| ri ⃗ − rj⃗|)

β -1 =

3 HS ∑ {(βεij)gbulk, ij

3

×

The different DFT versions at this stage rest upon the approximation for the distribution function gij(ri⃗ ; rj⃗ ). Tang, Scriven and Davis used a combination of the Barker− Henderson22 theory and a CG approach to deal with pure fluids, but as said earlier the generalization to mixtures of this scheme entails the assumption of further approximations. Instead, we propose a new formulation for the distribution function gij(ri⃗ ; rj⃗ ), compatible with the Barker-Henderson theory, and opening the possibility of a rigorous development with the FMT-SW formalism. For this function, we consider as a reference the homogeneous bulk HS contribution, gHS bulk, ij, and the general function gij(ri⃗ ; rj⃗ ) only depends on ri⃗ and rj⃗ through the perturbative potential uatt ij (|ri⃗ − rj⃗ |). Thus, we can write gij( ri ⃗ ; rj⃗) =

1 2

and

i,j

×βuijatt (| ri ⃗ − rj⃗|) d3ri d3rj

(23)

(24)

∑ ∬ ρi ( ri ⃗)ρj ( rj⃗)gij( ri ⃗ ; rj⃗)

HS gbulk, ij

∬ ρi ( ri ⃗)ρj ( rj⃗)(βuijatt(| ri ⃗ − rj⃗))2 d3ri d3rj

The main profit of this hypothesis is that now the derivation of β -1 and β -2 can be obtained in a rigorous mathematical way, by means of the FMT-SW described above. The perturbative development is then Φatt = Φ1 + Φ2, where Φ1 and Φ2 are obtained from eqs 22 and 23 by expliciting the potential uatt ij with eq 13, and using eqs 11 and 17 issued from the FMT-SW formalism. This yields the following results

m being an integer number, provided that the attractive term is described by the SW potential, and this may be included to derive the formulation of other generalized attractive terms, according to the method introduced in the previous section. The Attractive Term. Classical DFTs allow obtaining a general formulation for - att , as shown in eq (2.9) in Tang, Scriven and Davis,8 which in this case reads as β - att =

⎡ ⎛ ∂g HS ⎞⎤ ⎢ HS ∑ ⎢gbulk,ij − V ⎜⎜ bulk,ij ⎟⎟⎥⎥ i,j ⎣ ⎝ ∂V ⎠⎦

+

1 − 9 n3 + 24π (1 −

1 + 9 n3 2 +

1 n2 18 3 n3)2

1 n3 18 3

(1 − n3)

n1n2

n2 3

(26)

and depends only on the total weighted densities nα(r)⃗ . The SAFT-VR-SW Monomer Reference. It is necessary at this point to state in an explicit way the elements participating in these new derived free energy densities. To do this, the Barker−Henderson development can be applied, in the same spirit that within the single monomer of SAFT-VR-SW EoS, decomposing the homogeneous excess free energy as Fex = FHS + F1 + F2, for a given mixing rule.12 The usual Lorentz− Berthelot expressions can be used, defining the coupled radius

HS ρ ( r ⃗)ρ ( r ⃗)βuijatt (| ri ⃗ − rj⃗|) d3ri d3rj ∑ gbulk, ij ∬ i i j j i,j

(22)

R ij =

and D

Ri + Rj 2

, the coupled interaction range λij =

λi R i + λj R j Ri + Rj

and


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Figure 1. Normalized density profiles obtained from the theory, for the HS term alone (dotted line), adding the first order attractive contribution (dash-dotted line), and including also the second order attractive contribution (solid line), at η = 0.05 (a, left) and η = 0.25 (b, right). σ = 2R represents the sphere diameter.

■

RESULTS AND DISCUSSION With the aim to validate this approach, we have considered as benchmark the case of a SW pure fluid in contact with a planar hard-wall, with now z representing, instead of r,⃗ the perpendicular distance to the wall. The SW potential well depth has been fixed to kBT/ϵ = 2, with an interaction range λ = 1.5. We analyze first the influence of the inclusion of attractive contributions to the model, if compared with the HS alone. The normalized density profiles have been thus plotted in Figure 1, for the case of HS alone ΦHS, then with the first order attractive contributions ΦHS + Φ1 and finally with both first and second order terms ΦHS + Φ1 + Φ2. The plot evidence the inhomogeneous layering structure of a fluid in contact with a solid planar wall. In Figure 1a, at low density 4π (η = 3 ρbulk R3 = 0.05), the profiles considering the attractive terms show a distinctive trend if compared with the classical equivalent HS reference term. In particular, at z = 0, ρ/ρbulk < 1, because for lower densities the attractive term among fluid molecules adopts a predominant role on density profiles, due to the lack of wall−fluid attractive interactions. In Figure 1b, at higher densities (η = 0.25), the difference in the behavior between HS and the HS with added attractive terms is still relevant in the immediacy of the wall, even if the higher density value occurs at z = 0, contrarily to the low density case. We can underline also that in the latter case the first order attractive contribution has a remarkable influence, while the second order term is a perturbation. With this setup for the SW fluid (at second perturbation order), the normalized density profiles are presented in Figure 2, for three different packing fraction values, η = 0.05, 0.15, and 0.25. As density increases, fluid layering becomes more evident due to the growing influence of the HS contribution over the attractive term. These density profiles are compared for validation to new NVT Monte Carlo simulations results performed for the equivalent molecular model, wall interaction and bulk densities. The excellent agreement found between both approaches clearly represents an encouraging initial outcome for the theoretical development proposed.

the coupled well depth εij = εiεj . With this mixing rule, it is necessary to formulate properly the reference distribution function gHS bulk, ij, depending on the homogeneous fluid limit of (α) the weighted densities n(α) i (r)⃗ , noted as ξi , with α = 3, 2, 1, 0. HS The distribution function gbulk, ij is related to the HansenGoos and Roth31 EoS pressure, PHS HG.R., by the expression HS βPHG . R . = ξ0 +

16π 3

HS ∑ ρbulk ,i ρbulk ,j R ij3gbulk , ij i,j

(see, e.g., Lebowitz and Boublı ́k27), and within the selected mixing rule framework, this leads to the new expression 33

HS gbulk , ij

⎡ R iR j ξ ⎤ 3ξijeff + (ξijeff )2 1 2 ⎥ = +⎢ eff 2 1 − ξijeff ⎣⎢ R i + R j 3ξ3 ⎥⎦ (1 − ξij ) ⎡ R iR j ξ ⎤2 2(ξijeff )3 2 ⎥ +⎢ ⎢⎣ R i + R j 3ξ3 ⎥⎦ (1 − ξijeff )3

with

ξefij f

= c1(λij)ξ3 +

c2(λij)ξ32

(27)

3

+ c3(λij)ξ3 and

⎛ ⎞ ⎛ c1 ⎞ ⎛ 2.25855 −1.50349 0.249434 ⎞⎜ 1 ⎟ ⎟⎜ λij ⎟ ⎜c ⎟ ⎜ ⎜⎜ 2 ⎟⎟ = ⎜⎜−0.669270 1.40049 −0.827739 ⎟⎟⎜ ⎟ ⎝ c3 ⎠ ⎝ 10.1576 −15.0427 5.30827 ⎠⎜ λij 2 ⎟ ⎝ ⎠

The latter coefficients are those of SAFT-VR-SW EoS.11,12 Following the same coherent logic with the reference selected, the new compressibility equation can be derived from ξ

HS Kbulk = − V0

(

∂V HS ∂βPHG .R.

), associated with the Hansen-Goos and

Roth EoS: ⎡ HS Kbulk = ⎡⎣(1 − ξ3)4 ξ0⎤⎦/⎢(1 − ξ3)2 ξ0 ⎣ ⎛ 4 2 ⎞ + ⎜2 + ξ32 − ξ33⎟(1 − ξ3)ξ1ξ2 ⎝ 3 3 ⎠ + (9 − 8ξ3 + 7ξ32 − 2ξ33)

1 3⎤ ξ2 ⎥ 36π ⎦

■

(28)

CONCLUSION Summarizing, we have proposed a generalization of the FMT for potentials including an attractive term. The SW fluid has

This way, we have detailed all terms within the Φ functional. It is therefore now possible to apply the DFT treatment to compute the density profile ρi(r)⃗ of an inhomogeneous fluid. E



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REFERENCES

(1) Rosenfeld, Y. Free-Energy Model for the Inhomogeneous HardSphere Fluid Mixture and Density-Functional Theory of Freezing. Phys. Rev. Lett. 1989, 63, 980−983. (2) Roth, R. Fundamental Measure Theory for Hard-Sphere Mixtures: A Review. J. Phys.: Condens. Matter 2010, 22, 063102. (3) Cuesta, J. A.; Martínez-Ratón, Y. Fundamental Measure Theory for Mixtures of Parallel Hard Cubes. I. General Formalism. J. Chem. Phys. 1997, 107, 6379−6389. (4) Hansen-Goos, H.; Mecke, K. Fundamental Measure Theory for Inhomogeneous Fluids of Nonspherical Hard Particles. Phys. Rev. Lett. 2009, 102, 018302. (5) Davis, H. T. Statistical Mechanics of Phases, Interfaces, and Thin Film; VCH Publishers: New York, 1996. (6) Tarazona, P. Free-Energy Density Functional for Hard Spheres. Phys. Rev. A: At., Mol., Opt. Phys. 1985, 31, 2672−2679. (7) Gross, J. A Density Functional Theory for Vapor-Liquid Interfaces using the PCP-SAFT Equation of State. J. Chem. Phys. 2009, 131, 204705. (8) Tang, Z.; Scriven, L. E.; Davis, H. T. Density-Functional Perturbation Theory of Inhomogeneous Simple Fluids. J. Chem. Phys. 1991, 95, 2659−2668. (9) Malheiro, C.; Mendiboure, B.; Plantier, F.; Blas, F. J.; Miqueu, C. Density Functional Theory for the Description of Spherical NonAssociating Monomers in Confined Media using the SAFT-VR Equation of State and Weighted Density Approximations. J. Chem. Phys. 2014, 140, 134707. (10) Malheiro, C.; Mendiboure, B.; Míguez, J.-M.; Piñeiro, M. M.; Miqueu, C. Nonlocal Density Functional Theory and Grand Canonical Monte Carlo Molecular Simulations of Water Adsorption in Confined Media. J. Phys. Chem. C 2014, 118, 24905−24914. (11) Gil-Villegas, 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. (12) 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. (13) 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. (14) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19−34. (15) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys. 1984, 35, 35−47. (16) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1986, 42, 459−476. (17) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 1986, 42, 477− 492. (18) Kierlik, E.; Rosinberg, M. L. Free-Energy Density Functional for the Inhomogeneous Hard-Sphere Fluid: Application to Interfacial Adsorption. Phys. Rev. A: At., Mol., Opt. Phys. 1990, 42, 3382−3387. (19) Benavides, A. L.; Gil-Villegas, A. The Thermodynamics of Molecules with Discrete Potentials. Mol. Phys. 1999, 97, 1225−1232. (20) Vidales, A.; Benavides, A. L.; Gil-Villegas, A. Perturbation Theory for Mixtures of Discrete Potential Fluids. Mol. Phys. 2001, 99, 703−710. (21) Lafitte, T.; Apostolakou, A.; Avendaño, C.; Galindo, A.; Adjiman, C. S.; Müller, E. A.; Jackson, G. Accurate Statistical Associating Fluid Theory for Chain Molecules formed from Mie Segments. J. Chem. Phys. 2013, 139, 154504. (22) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State for Fluids: The Square-Well Potential. J. Chem. Phys. 1967, 47, 2856−2861. (23) Wertheim, M. S. Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres. Phys. Rev. Lett. 1963, 10, 321−323.

Figure 2. Normalized density profiles obtained from the theory (solid lines) and Monte Carlo simulations (circles), at (from top to bottom) η = 0.05, 0.15, and 0.25. σ = 2R represents the sphere diameter.

been considered as an initial application, with very accurate quantitative results. In particular, we have developed a functional having as bulk limit the SAFT-VR-SW monomer fluid. In this context, and respecting the coherence between the different terms, we have derived new expressions for the HS distribution function gHS bulk and the compressibility Kbulk issued from the Hansen-Goos and Roth EoS. Additionally, these equations could be used as well to refine SAFT-type EoS formulations. It must be emphasized that the methodology presented here for the development of the FMT-SW is fully transferable to any other choice of discretizable potential and reference fluid. This unveils the perspective of obtaining new versions of fully consistent NLDFTs for mixtures in a quite straightforward and natural manner.

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AUTHOR INFORMATION

Corresponding Author

*(C.M.) E-mail: [email protected]. ORCID

M. M. Piñeiro: 0000-0002-3955-3564 C. Miqueu: 0000-0002-0924-4437 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.M.P. acknowledges CESGA for providing access to computing facilities, and financial support by Ministerio de Economı ́a y Competitividad, in Spain (Grant Ref. FIS201568910-P). T.B. acknowledges the Ministère de l’Éducation, de l’Enseignement Supérieur et de la Recherche, in France, for his MENRT Ph.D. Grant and Total SA for additional funding. We acknowledge P. Paricaud for the useful discussion on discretized potentials. F


Article

The Journal of Physical Chemistry C (24) Thiele, E. Equation of State for Hard Spheres. J. Chem. Phys. 1963, 39, 474−479. (25) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. Statistical Mechanics of Rigid Spheres. J. Chem. Phys. 1959, 31, 369. (26) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969, 51, 635−636. (27) Boublík, T. Hard-Sphere Equation of State. J. Chem. Phys. 1970, 53, 471−472. (28) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W., Jr Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres. J. Chem. Phys. 1971, 54, 1523−1525. (29) Roth, R.; Evans, R.; Lang, A.; Kahl, G. Fundamental Measure Theory for Hard-Sphere Mixtures Revisited: The White Bear Version. J. Phys.: Condens. Matter 2002, 14, 12063. (30) Yu, Y.-X.; Wu, J. Structures of Hard-Sphere Fluids from a Modified Fundamental-Measure Theory. J. Chem. Phys. 2002, 117, 10156−10164. (31) Hansen-Goos, H.; Roth, R. A New Generalization of the Carnahan-Starling Equation of State to Additive Mixtures of Hard Spheres. J. Chem. Phys. 2006, 124, 154506. (32) Hansen-Goos, H.; Roth, R. Density Functional Theory for Hard-Sphere Mixtures: The White Bear Version Mark II. J. Phys.: Condens. Matter 2006, 18, 8413. (33) Lebowitz, J. L. Exact Solution of Generalized Percus-Yevick Equation for a Mixture of Hard Spheres. Phys. Rev. 1964, 133, A895− A899.

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Generalization of the Fundamental-Measure Theory Beyond Hard

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