Nonlocal Density Functional Theory and Grand Canonical Monte

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Non-Local Density Functional Theory and Grand Canonical Monte Carlo Molecular Simulations of Water Adsorption in Confined Media Carine Malheiro, Bruno Mendiboure, Jose-Manuel Miguez, Manuel M. Piñeiro, and Christelle Miqueu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp505239e • Publication Date (Web): 06 Oct 2014 Downloaded from http://pubs.acs.org on October 12, 2014

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The Journal of Physical Chemistry

Non-Local Density Functional Theory And Grand Canonical Monte Carlo Molecular Simulations Of Water Adsorption In Confined Media Carine Malheiro†, Bruno Mendiboure†, José-Manuel Míguez†, ‡, Manuel M. Piñeiro‡, Christelle Miqueu*,† †

Univ Pau & Pays Adour, CNRS, TOTAL - UMR 5150 – LFC-R – Laboratoire des Fluides

Complexes et leurs Réservoirs, BP 1155 – PAU, F-64013, France ‡

Departamento de Física Aplicada, Universidade de Vigo, E36310 Vigo, Spain

KEYWORDS Slit-like micropores, mesopores, capillary condensation, capillary evaporation, activated surfaces.

ACS Paragon Plus Environment

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ABSTRACT

We present in this paper a study of water adsorption behavior in confined media by using on the one hand a Non-Local Density Functional Theory (NLDFT) coupled with the SAFT-VR equation of state and, on the other hand, Grand Canonical Monte-Carlo (GCMC) molecular simulations. The present work is a second step in an ongoing NLDFT/SAFT-VR coupling. The first step has focused on the monomer contribution and especially in the way to extend the dispersive terms of the monomer contribution of SAFT-VR in the NLDFT formalism. In the present work, the theory has been extended by introducing the associative contribution due to hydrogen bonding and is applied to water, which is modeled as one sphere with four identical associating sites placed in a tetrahedral geometry with the same interaction parameters for both theory and simulations. NLDFT/SAFT-VR and GCMC results for density distributions of water in graphitic slit-like micropores and mesopores are shown to be in good agreement. Moreover, the capillary condensation and evaporation were investigated with the theoretical model in micro and mesopores, and also in the case of activated surfaces.


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1. Introduction The properties of inhomogeneous and confined fluids are a subject of intensive study nowadays, as they are closely involved in many practical applications of remarkable relevance, and the detailed knowledge of complex fluids interfacial properties and their connection with their detailed molecular descriptions are still incomplete. An overview of the wide variety of processes where interfacial phenomena of fluids and interactions of fluids with solid substrates play a relevant role demonstrates the relevance of this topic. As a representative example, in the field of Petrophysics there are many processes where the precise characterization of interfacial properties and confinement effects is essential1. This includes enhanced natural gas recovery from non conventional sources as tight gas reservoirs (TGRs), shale gas or coal bed methane2,3, carbon dioxide capture and storage4–6, reuse of depleted oil wells for natural gas storage7,8, etc. In these cases, fluids are often adsorbed on porous substrates with variable pore size distribution, situations where extreme confinement is found. The modifications of the phase equilibria and thermophysical properties of the fluid in these cases if compared with the bulk reference behavior are closely connected to the details of the mutual intermolecular interaction potentials, and its precise description from a theoretical point of view remains a challenge. Without this basic knowledge, estimations of magnitudes as adsorption, interfacial tension, wetting transitions, etc., over wide ranges of temperature and pressure are far from being reliable. In this situation, typical problems as the determination of the optimal extraction yield of a given natural gas deposit upon the injection of an external aqueous fluid, or the situation once the process has finished, cannot be estimated with enough confidence with the currently available estimating tools.


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There are several different theoretical approaches to describe interfacial properties of a fluid. Skipping empirical approaches, which are widely used in practical applications but lack physical grounds, two approaches may be pointed out for their relevance. The first consists in the combination of inhomogeneous fluid theories (Density Gradient Theory9,10, DGT, or Density Functional Theory, DFT) with bulk equations of state (EoS), and the second is Molecular Simulation (MS). These two different strategies are complementary and offer information of the fluid behavior at different scales. Individually they are versatile and quantitative in many cases, but if used in close combination their mutual synergy is extremely useful to analyze critically the soundness of a given theoretical approach from different perspectives, taking advantage of their specific strengths. The recent developments in DFT application to interfaces of pure fluids and mixtures deserve special attention. The basic foundations of DFT and their applications are described in a series of excellent monographic works11–16. The basis of the method relies in the description of the single atom (or particle) density in an inhomogeneous fluid through a mathematical functional of the Helmholtz free energy, which provides a complete description of the system. The mathematical complexity of the description of an inhomogeneous fluid using DFT must be underlined, specially if compared with other existing approaches, but the purely predictive nature of the method, not dependent of any correlated or empirical parameter, and the quantitative performance in situations where for instance DGT is unable to capture the physics of the system is to be emphasized. One of the most successful treatments of DFT applied to molecular fluids is the perturbative approach around a hard sphere repulsive fluid reference term, combined with an attractive interaction term. Lately, several authors have proposed DFT approaches combined with the well-known Statistical Association Fluid Theory17,18 (SAFT) molecular equation of state


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(EoS). Based on the seminal works on thermodynamic perturbation theory (TPT1) by Wertheim, the original SAFT EoS has evolved due to many contributions, and has obtained a remarkable success in countless applications describing thermodynamics of complex fluids and mixtures. Nowadays, SAFT has undoubtedly become a widely used and remarkably popular estimation tool, as reported in several recent reviews19,20. Among the different SAFT-DFT schemes, the works by the group of Jackson deserve special mention21–26, and they have treated the case of interfaces of pure fluids, including chain and associating fluids, and their mixtures, including cases of real fluids involved in reservoir engineering. In this case, the authors considered the Local Density Approximation (LDA) approach, which performs very well when smooth density profiles across the interfaces are involved. Nevertheless, this approach finds its operative limitation when extreme density gradients are involved, as for instance in the cases of highly confined fluids, (the comprehensive review of Evans27 describes in detail phase equilibria and structural effects of fluids confined in narrow pores), where LDA is unable to describe the very sharp adsorption peaks and fluid layering appearing inside pores of molecular scale size. Here, a Non Local (NL) approximation becomes compulsory, including the consideration of Weighted Density Approximations (WDA), for instance using the Fundamental Measure Theory (FMT) as it was previously done by several authors16,28–39. In a previous work40, we have also adopted this strategy, and a NLDFT/SAFT-VR model was developed, considering a coarse graining approach for the non local density term. For the case of monomer molecules interacting through Square-Well (SW) dispersive interactions, the model was tested quantitatively with a twofold technique. First, the microscopic structure of the fluid within nanoscale carbon slit pores was compared against Grand Canonical Monte Carlo (GCMC) molecular simulations (MS) obtained considering a simple United Atom SW methane


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model. This model, albeit simple, has proved to be very efficient to describe properties of confined methane41. This first test pointed out the need of a proper NL treatment within the DFT to describe the microscopic fluid structure, which is not accessible experimentally. Second, experimental adsorption isotherms and capillary condensation of methane on a real carbon substrate were compared with NLDFT/SAFT-VR estimations obtaining excellent accuracy. This way, the model was tested against a reference MS molecular model to check the accuracy of the approximations performed, and proved to be reliable in the estimation of experimental data for this type of molecules. In the present work, the model is extended to associative molecules, focusing on the case of water as case study. With this aim, an associative term has been added to the previously reported formulation. Water has been modeled in the framework of SAFT following the description first proposed by Bol42 using the characteristic parameters determined by Clark et al.43, who provided a parameterization for the tetrahedral SAFT water model consisting of a single SW interacting segment with four off-center associative sites. The estimations of the model were again benchmarked against GCMC simulations, using the same water molecular model, so both MS and DFT are using the same underlying physical molecular model and the comparison can be established on fair terms. In this framework the fluid structure within graphite-like micro and mesopores is analyzed, together with capillary condensation and its trend with pore size. Finally, the change in the confined fluid behavior due to the presence of interacting sites on the substrate surface, reproducing the behavior of an activated surface, is discussed.


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2. Theoretical section We consider a system at a temperature T, chemical potential µ in a volume V. In the presence of an external potential Vext, the grand potential Ω of the inhomogeneous fluid, in the framework of the DFT11, is assumed to be a functional of the molecular density ρ(r):

[ ] [ ] ∫ drρ(r){V (r) − µ}

Ω ρ (r ) = A ρ (r ) +

€

ext

(1)

where A[ρ(r)] is the Helmholtz free energy of the fluid. The equilibrium density distribution is calculated through the minimization of the grandpotential by solving the corresponding Euler-Lagrange equation11:

[ ] = δA[ρ(r)] + V (r) − µ = 0 δρ(r) δρ(r)

δΩ ρ ( r )

ext

(2)

In a previous work40, we have developed a Helmholtz free energy functional that describes

€ spherical non-associative monomers using a non-local DFT associated to the SAFT-VR equation of state (NLDFT/SAFT-VR) with a square-well (SW) intermolecular potential defined by its molecular diameter σ, potential depth ε and interaction range λ. Here, we extend this model to describe fluids of spherical associative monomers of diameter σ, and more specifically water, that is modeled within the SAFT-VR formalism by one sphere and four associative sites in a tetrahedral geometry43, a model originally proposed by Bol42 that has become very popular and has found many successful applications31,35,43–48. Thus, we have followed our previous work40 for the dispersive contributions -i.e. a mean-field (MF) approximation, the use of weighted densities and a separation of the short and long range corrections as initially proposed by Gloor et al.23-


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and we have added the association contribution. The Helmholtz free energy of the system is given by

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

A ρ(r) = Aideal ρ(r) + AHS ρ (r) + A1sr ρ(r) + A2 ρ(r) + Aatt ρ (r) + Aassoc ρ(r) (3) where Aideal[ρ(r)] is the ideal free energy11,

[ ]

Aideal ρ(r) = kB T

∫ drρ(r)[ln(ρ(r)Λ ) −1] 3

(4)

with kB the Boltzmann constant and Λ the de Broglie thermal wavelength of the molecules. The interaction between monomers is described by the hard-sphere contribution AHS[ρ(r)] and a Barker-Henderson high-temperature perturbation expansion49,50 at second order A1[ρ(r)] and A2[ρ(r)]. We have chosen a mean-field (MF) approximation in which the correlations due to the attractive interactions are neglected. In that case, the first order term A1[ρ(r)] is partitioned into the “short-range” contribution A1sr[ρ(r)] and the “long-range” contribution that becomes the classical attractive term Aatt[ρ(r)] in the DFT formalism11. Each term of the monomer free energy contributions is only briefly recalled here. The hard-sphere term is written using the weighted densities n (r) of the FMT but following the Modified Fundamental Measure Theory (MFMT)51– α

53

to get the Carnahan-Starling equation of state54 for hard-sphere in the bulk limit in agreement

with the SAFT-VR bulk formalism. The weighted densities are defined by:

nα (r) =

(α )

∫ dr' ρ(r')w (r - r')

(5)

where α=0, 1, 2, 3, v1 or v2.


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The weighted functions w( )(r) are given by α

&σ ) w (2) (r) = πσ 2 w (0) (r) = 2πσw (1) (r) = δ ( − r + '2 *

(6)

€

%σ ( w (3) (r) = Θ' − r * &2 )

(7)

€

# r & #σ & w (v2) (r) = 2πw (v1) (r) = % (δ % − r ( $r' $ 2 '

(8)

€

where δ(r) is the Dirac delta function and Θ(r) is the Heaviside step function. The free-energy due to the interaction between hard-spheres is then given by

[ ]

AHS ρ(r) = kB T

∫ dr [Φ

HS (s)

{n (r)} + Φ

HS (v )

α

{n (r)}]

(9)

α

where

€ 3

Φ

€

HS (s)

n ln (1 − n ) nn n23 {nα (r)} = −n0 ln(1 − n3 ) + 1 −1 n2 + 2 36πn2 3 + 2 3 3 36πn3 (1 − n3 )

(10)

is the scalar part, and

{

}

ΦHS (v ) nα (r) = −

nv1 • nv2 n2nv2 • nv2 ln (1 − n3 ) n2nv2 • nv2 − − 2 1 − n3 12πn3 12πn ln (1 − n 3

3

)

2

(11)

is the vector part.

€


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The dispersive contributions A1sr[ρ(r)] and A2[ρ(r)] are treated non-locally by using a coarsegrained approach. It uses a weighted density defined by

ρ (r ) =

3 4πσ 3

∫ dr' ρ(r')Θ(σ − r − r' )

(12)

The first-order perturbation term is:

[ ] ∫ drρ(r)a (r)

A1sr ρ (r) =

sr 1

(13)

with

' * a1sr (r) = −α VdW ρ(r) )g HS σ ;ηeff (r) −1, ( +

(

)

(14)

where αVdW is given by

α VdW =

2π 3 3 σ ε λ −1 3

(

)

(15)

The radial distribution function gHS is calculated at contact length σ for an effective packing fraction ηeff, which is related to the real packing fraction η and whose expression can be found in literature55, and evaluated non locally through the weighted density described previously. Thus, we obtain

η(r) =

π ρ (r)σ 3 6

(16)

and


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(

)

g HS σ;ηeff (r) =

1−

ηeff (r) 2

(1 − η (r)) eff

(17)

3

The second-order perturbation term is:

€

[ ]

A2 ρ(r) =

€

1 ε 2 kB T

∫ drρ(r) K

HS

(η(r))ρ(r) ∂ρ((r)) ∂a1 r

where eff ' * a1 (r) = −α VdW ρ (r) g HS )σ , η (r), ( +

€

(18)

(19)

and

( )

K HS η(r) =

(1 − η(r))

4

(20)

1+ 4η(r) + 4η(r)2

The attractive contribution in the MF approximation is given by

€

[ ]

Aatt ρ (r) =

€

1 2

∫ drρ(r) ∫ dr' ρ(r')φ ( r − r' ) att

(21)

where φatt(|r-r’|) is, in this work, the attractive part of the square-well potential uSW used in the SAFT-VR equation of state:

u SW

' ∞ if r < σ ) = ( −ε if σ < r < λσ ) * 0 if r > λσ

(22)

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The extension of the previous work consists in considering associating contributions for the hydrogen bonds in water. For the associating functional, we have followed the work of Yu et al.56 who have introduced the scalar and vector weighted densities of the FMT into the expression of the Helmholtz free energy due to association. Here, as recommended in the SAFTVR framework43, water is modeled as a single sphere with four identical associating sites interacting through a SW potential defined by its attraction energy εHB and its bonding volume κHB. Hence, given the equivalence of the sites, the associating free energy may be written as

[

%

]

Aassoc ρ (r) = 4kB T

∫ drn (r)ζ(r)'& ln X (r) − 0

X (r) 1 ( + * 2 2)

(23)

where n0(r) is one of the weighted densities described previously for the hard-sphere term in the FMT formalism, ζ(r) is a proportional factor proposed by Yu et al.56 to take into account the vector-weighted densities defined by

ζ(r) = 1 −

nv2 (r) • nv2 (r) n2 (r)

(24)

2

and X(r) is the fraction of not bonded sites as introduced in Wertheim’s theory57–60. For a nonlocal description, X(r) is modified from the bulk version18 using the weighted densities of the FMT introduced by Yu et al.56 for the general expression of the fraction of not bonded sites and then transposed by Hughes et al.35 for the case of the model considered:

X (r ) =

−1+ 1+ 8n0 (r)ζ(r) Δ (r)

(25)

4n0 (r)ζ(r) Δ (r)


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Δ(r) = κHBfHBgSW(σ) is a function that characterizes the association between two sites where fHB=exp[εHB/(kBT)]-1 is the Mayer function and gSW(σ) is the radial distribution function of the molecules evaluated at contact length. Unlike Yu et al.’s work56 that uses the hard-sphere radial distribution function in terms of the weighted densities of the FMT, we have considered the expression of Gloor et al.23 for square-well radial distribution function from the hard-sphere reference and a high-temperature expansion at first-order and we have modified it to get a new expression of this function with WDA. Hughes et al.35 have previously developed a similar treatment to describe water interfaces but the weighted density used for dispersive contributions needed an adjustable parameter. In our work, we obtain

g

€

SW

' 1 ) ∂a1 (r) λ ∂a1 (r) *, (σ) = g (σ) + 4k T ) ∂η r − 3η r ∂λ , () () B ( + HS

(26)

where gHS(σ), according to Yu et al.’s work56, is written using the weighted densities from the FMT and the factor ζ(r):

% σ (2 1 σ n2ζ n22ζ g (σ) = + +' * , 1 − n3 2 2(1 − n )2 & 2 ) 18(1 − n )3 3 3 HS

€

(27)

and a1(r) andη(r) are respectively given by eq. 14 and eq. 16. Hence, the Helmoltz free energy of the system has been obtained with eqs. 3 to 27 and we have solved eq. 2 using a Picard iterative method to compute the density distributions of water in slitlike pores. We have used the water interaction parameters that can be found in Clark et al.43 i.e. σ = 0.3303 nm, ε/kB = 300.43 K, λ = 1.718, εHB/kB = 1336.9 K and κHB = 0.89369 10-3 nm3. The pore was discretized with n points along the direction perpendicular to the planar surfaces with a


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constant grid spacing of 0.005 nm and the calculation proceeded until the error (1/n)∑i=1,n(ρioldρinew)2 is equal or lower than 10-5, with 5 percent of the new solution mixed with the old solution at each iteration.

3. Molecular simulations GCMC simulations have also been performed for water in slit pores. For a pertinent comparison with the theoretical results, the molecular model selected for water and its interaction parameters are the same as in the NLDFT/SAFT-VR model. Thus, water molecules are modeled as single spheres interacting through the SW potential (eq. 22). Associative sites are modeled by four embedded off-center SW bonding sites placed in a tetrahedral geometry. Two of these sites represent the hydrogen atoms, and the two others the electron lone pairs of the oxygen atom. The SW association interaction is defined by:

% −ε if rAB < rc φ HB = & HB ' 0 otherwise

(28)

where rAB is the distance between two sites and rc is the cut-off distance of the hydrogen bonding. In the model, the sites are fixed at rd = 0.25 σ from the center of the sphere43. The cutoff distance rc is calculated using the relation61

κ

* HB

% ln r * + 2r * 6r *3 +18r *2r * − 24r *3 + ( c d c c d d ' * = 4π ' r * + 2r * −1 22r *2 − 5r *r * − 7r * − 8r *2 + r * +1 * d d c d d c c & c )

(

(

)( )(

)

)

(72r ) *2 d

(29)


14

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where κHB* = κHB/σ3, rc* = rc/σ and rd*=rd/σ. For κHB = 0.89373 10-3 nm3, we have found rc = 0.684333 σ. The equilibrium properties of the fluid in the slit pore are obtained by the classical GCMC simulation at fixed (µ, V, T)62. The simulation box is parallelepipedic, its size is (10σ, 10σ, H), where H is the pore width. For the slit pore, periodic boundary conditions and minimum image convention are applied on the two dimensions parallel to the walls. A corresponding bulk fluid GCMC simulation is performed in a (10σ, 10σ, 10σ) parallelepipedic box at the same chemical potential, and with periodic boundary conditions and minimum image convention in the three directions. A simulation at the same chemical potential allows obtaining the bulk density that is the input parameter in the NLDFT/SAFT-VR modeling, assuring a relevant comparison. GCMC simulations proceed as follows: each of the four types of movements that are molecule displacement, rotation, insertion and removal are chosen at random at every step with equal probability. For the insertion/removal attempt, the Configurational-Bias Monte Carlo (CBMC) method is used62. In order to apply the CBMC method, the interaction between molecules is separated into two contributions, the SW interaction between spherical cores, uSW, and the hydrogen bonding interaction φHB. For the particle insertion, the CBMC attempt is a process with three steps. First, a position for the molecule is selected by evaluating uSW for kI different random trial positions. One of them is selected according to the probability

p(i) =

exp( −βu SW (i))

(30)

WI

where β=1/(kBT) and W is the corresponding Rosenbluth factor. For the second step, kR different

€

orientations trials are randomly generated for the selected position of the first step and only φHB is


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calculated. Again, the orientation of the four embedded off-center SW bonding sites is chosen according to the probability

p(i) =

exp( −βφ HB (i)) WR

,

(31)

Finally, for the last step, the insertion of water molecules for the selected position and orientation is accepted or rejected according to a rule modified to remove the introduced bias62. For the particle removal step, the scheme is the same as for the insertion with the traditional modifications, where the actual position and orientation of the water molecule are kept in the calculus of the probabilities, and where only (kI – 1) randomly trial positions and (kR – 1) randomly trial orientations are generated. Each Monte Carlo simulation is made of 106 to 5 x 108 configurations for equilibration and 106 to 5 x 108 for production of averages.

4. Results and discussion In this section, we apply both the NLDFT/SAFT-VR model and GCMC molecular simulations to determine the adsorption behavior of water in slit-like pores. We consider the pore as two parallel graphitic surfaces (walls) at a distance H from carbon-center to carbon-center. Thus, the density distribution of the fluid only varies in one direction (z). The interaction parameters of the solid63 are σss = 0.34 nm and εss/kB = 28 K . The external potential applied to the fluid, due to the presence of the solid, is modeled by the Steele’s “10-4-3” potential64 from both surfaces: Vext ( z ) = Vsf ( z ) + Vsf ( H − z)

(32)

with


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Vsf ( z) = 2πρsε sf σ sf

2

10 4 . ' 1 ' σ sf * 4 σ sf 2 σ sf * 0 3 Δ ) , −) , − 3 05( z + z ( + 3Δ ( z + 0.61Δ ) 32 /

(33)

where ρs = 114 molecules/m3 is the density of graphite63, Δ = 0.335 nm is the spacing between two parallel layers of graphite molecules constituting the wall63, σsf and εsf are the interaction parameters determined by the Lorentz-Berthelot rules: σsf=(σss+σff)/2 and εsf=(εssεff)1/2 where the subscript ss refers to the solid while the subscript ff refers to the fluid. A. Adsorption in micropores In this section, we study the adsorption of water in slit-like graphitic micropores. Figure 1 shows the density distribution of water in the pores at T = 425 K and P = 0.05 MPa for several pore widths determined with both the NLDFT/SAFT-VR model and GCMC molecular simulations. The pore widths were selected in order to study different structures of the fluid with one or two distinct layers and two layers that overlap. It can be seen that both theory and simulations predict the same number of peaks and their positions are coincident. In the pores 0.8 nm, 0.9 nm and 1.4 nm wide, where two overlapped or separated peaks can be accommodated, both theory and simulations predict the same maximum density at adsorption peaks. However, in the narrower pore of 0.7 nm, theoretical results estimate higher maximum density than molecular simulations. Trying to understand these discrepancies, we have computed the density profile in this pore for a purely repulsive associating hard-sphere fluid, whose theory was developed by Yu and Wu56, using the same fluid interaction parameters defined previously. Figure 2 shows that for an associating repulsive fluid, the NLDFT/SAFT-VR results agree with GCMC simulations. Thus, it seems that dispersive interactions are responsible for the differences observed when the fluid is highly confined, especially in the limiting case where there is a single confined fluid layer within


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the pore. However, the results in larger micropores allow to confirm that the NLDFT/SAFT-VR model and the GCMC molecular simulations based on the same fluid model are fairly consistent in the description of the behavior of water in micropores.

400

50

H=0.7nm ρ (kg/m3)

ρ (kg/m3)

200 100 0 0.0

30 20 10

0.2

25

0.4 z (nm)

0 0.0

0.6

0.2

12

H=0.9nm

0.4 z (nm)

0.6

0.8

H=1.4nm

10 ρ (kg/m3)

20 15 10 5 0 0.0

H=0.8nm

40

300

ρ (kg/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 41

8 6 4 2

0.2

0.4 0.6 z (nm)

0.8

0 0.0

0.4

0.8 z (nm)

1.2

Figure 1. Density profiles of water in micropores at T = 425 K and P = 0.05 MPa. Solid lines are NLDFT/SAFT-VR results and symbols are GCMC molecular simulations.


18

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150 ρ (kg/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 € 57 58 59 60


100 50 0 0.0

0.2

0.4 z (nm)

0.6

Figure 2. Density profile of an associating hard-sphere fluid at T = 425 K, P = 0.05 MPa in a pore of width H = 0.7 nm. Solid line is NLDFT/SAFT-VR results and symbols are GCMC molecular simulations. Unlike GCMC molecular simulations, calculations using NLDFT/SAFT-VR are performed very quickly. Hence, we have only used the theory to explore the capillary condensation and evaporation of water that occur in micropores. We have obtained the water adsorption-desorption hysteresis at T = 425K (at this temperature, the saturation pressure is P0 = 0.502 MPa) for the same pore widths used previously (0.7 nm, 0.8 nm, 0.9 nm and 1.4 nm). To do this, at each pressure, the initial density distribution in the pore was the equilibrium density distribution obtained at the previous pressure. Figure 3 represents the average adsorbed quantity per surface area N as a function of the reduced pressure P/P0. N is defined by " 1 %1 N =$ ' #NA&2

H

∫ ρ( z)dz

(34)

0

where NA is the Avogadro number.


19


It is noticeable that in the narrowest pore (0.7 nm), the capillary condensation occurs remarkably soon (around P/P0 = 0.1) and this may explain that no hysteresis appears when desorption is described. It can also be seen that as the pore width increases, the pressure corresponding to the capillary condensation increases, and beyond a certain pore width there is no more capillary condensation but only condensation at the saturation pressure (see also Figures 4 and 5). On the contrary, once water has condensed in the pore, the pressure corresponding to the capillary evaporation (desorption) is fairly identical, therefore the hysteresis of water adsorptiondesorption grows as the pore width increases and then stabilizes.

2.5 2 N(µmol/m2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 41

1.5 1 0.5 0 0.0

0.2

0.4 0.6 P/P0

0.8

1.0

Figure 3. Water adsorption-desorption hysteresis determined with the NLDFT/SAFT-VR model at T = 425 K for H = 0.7 nm (crosses), H = 0.8 nm (circles), H = 0.9 nm (triangles) and H = 1.4 nm (squares). Closed symbols are for adsorption, open for desorption. Figure 4 shows the pressure corresponding to a condensation of water as a function of pore width for several temperatures in a range from 350 K to 500 K where the SAFT-VR EoS is very accurate. It can be seen that as the temperature increases, the range of pore widths where capillary condensation occurs also increases.


20

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1 0.8 Pcond/P0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 0.4 0.2 0 0.5

1.0

1.5 H(nm)

2.0

Figure 4. Reduced condensation pressure Pcond/P0 (P0 being the saturation pressure) of water as a function of the micropore size at T = 350 K (crosses), T = 400 K (circles), T = 425 K (triangles), T = 475 K (squares) and T = 500 K (diamonds) determined with the NLDFT/SAFT-VR model. We have also plotted the largest pore width where capillary condensation occurs as a function of temperature (Figure 5). This represents an interesting and relevant information because one can see from Figures 4 and 5 that the behavior of water, and particularly the pore filling, is highly dependent on both pore width and temperature, which can make water, at least at high temperature, an interesting probe fluid for the characterization of microporous materials. Another conclusion to be highlighted from these calculations is that for usual temperatures, capillary condensation will not occur for the majority of accessible micropores (H > 1nm).


21


2 1.6 Hmax (nm)

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Page 22 of 41

1.2 0.8 0.4 0 350

400 450 Temperature (K)

500

Figure 5. Largest pore size for which capillary condensation of water occurs as a function of temperature in graphitic slit-like micropores. B. Adsorption in mesopores To go further, in this section, we have studied water adsorption on graphitic slit-like mesopores. As expected, given what was previously observed for the largest micropores, the NLDFT/SAFTVR model does not predict any capillary condensation in mesopores. Figure 6 shows the hysteresis of water adsorption-desorption in a pore of width 3.033 nm (10 σ) at T = 425 K determined with the NLDFT/SAFT-VR model and confirms this prediction. As it can be seen in Figure 6, the condensation of water occurs only at the saturation pressure. We can also notice that the pressure corresponding to the capillary evaporation during the desorption process is the same as the one observed for the largest micropores.


22

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6 N(µmol/m2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 2 0 0.0

0.2

0.4

0.6 P/P0

0.8

1.0

Figure 6. Adsorption and desorption of water at T = 425 K in a pore of width H = 3.033 nm computed with the NLDFT/SAFT-VR model. Closed symbols are for adsorption, open for desorption. With the aim to test the adequacy and mutual consistency of both NLDFT/SAFT-VR model and GCMC molecular simulations to describe water adsorption in mesopores, we have determined the density distribution of water in the pore of width 3.033 nm, at T = 425 K and for pressures before condensation (P/P0 = 0.56, 0.68 and 0.84). It can be seen in Figure 7 that for the three pressures considered, the results given by the NLDFT/SAFT-VR model and by GCMC molecular simulations are in very good agreement. Indeed, when comparing these profiles, one has to keep in mind that the curves represent slightly different densities for theory and simulations. On the one hand, NLDFT symbols represent the exact density computed by the theory at each point. But, on the other hand, each GCMC symbol represents the average density value computed in a slab centered in that point; hence, due to the sharp variations of density, this averaged value is always lower that the real maximum density computed in the slab.


23


P/P0=0.56

ρ (kg/m3)

60

40

20

0 0.0

0.5

80

1.0

1.5 2.0 z (nm)

2.5

3.0

2.5

3.0

2.5

3.0

P/P0=0.68

ρ (kg/m3)

60 40 20 0 0.0

0.5

100

1.0

1.5 2.0 z (nm)

P/P0=0.84

80 ρ (kg/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 41

60 40 20 0 0.0

0.5

1.0

1.5 2.0 z (nm)

Figure 7. Density profiles of water before determined with NLDFT/SAFT-VR model (solid line) and GCMC molecular simulations (symbols) at T = 425 K in a mesopore of width H = 3.033 nm.


24

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€


C. Adsorption on activated surfaces Finally, we have explored the influence of activation sites placed on the graphitic surfaces using the NLDFT/SAFT-VR model in order to be closer to real porous activated carbons. Activated surfaces have been already investigated by several GCMC molecular simulations45,47,65–69 and DFT studies70. In MC simulations, oxygen atoms can be placed on the graphitic surface (with a certain density of sites) and form hydrogen bonds with water in order to mimic an activated carbon. By using the theoretical model in which the density is treated continuously, it is not possible to deal with discrete sites distributed on 2D surfaces interacting with water. Instead, as a first approximation, we propose to add an integrated potential to the external potential, following the same philosophy that transforms the Lennard-Jones intermolecular potential to become the Steele’s potential when integrated over the carbon planes. Thus, we consider a homogenous surface S with a number of sites per area ρsite placed at a distance zsite=0.5 σff from each graphitic surfaces inside the pore. The potential φHB resulting from this surface is given by

φ HB =

∫ρ

UO − H dS

(35)

site

S

where φHB is the same SW intermolecular potential that describes hydrogen bond between water molecules in the fluid: % −ε HB if r < σ HB UO − H = & ' 0 otherwise

(36)

where we have defined σHB by

σ HB

% 3 (1 3 =' κ * & 4 π HB )

(37)


25


Integrating eq. 35 we obtain ') −πε σ 2 − z − z HB HB site φ HB = ( )* 0 if z − zsite > σ HB

(

) if

z − zsite < σ HB

(38)

In order to test this model, we have compared the desorption isotherm of water in an activated €

carbon with GCMC simulation results from Müller et al.45. Figure 8 shows the average adsorbed quantity in the pore per surface area at T = 300 K (P0 = 3.536 kPa) for an activated carbon, with a density of sites ρsite = 1 nm-2. It can be seen that the model we have proposed to take into account the activated sites on the graphitic surface gives the same prediction of adsorbed quantities as GCMC simulation results and the same pressure at which the capillary evaporation occurs during the desorption process.

40 N(µmol/m2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 41

30 20 10 0 0

0.05 P/P0

0.1

Figure 8. Desorption isotherm of water at T = 300 K in a slit pore of width H = 2 nm with a density of activated site ρsite = 1 nm-2 on the graphitic surface. Circles are NLDFT/SAFT-VR modeling and triangles are GCMC molecular simulation data from Müller et al.45. Solid line is a guide to the eye.


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The model being validated in that case, the influence of site density on the adsorption behaviour of water was studied on both activated microporous and mesoporous carbons. According to Müller et al.45, it has been experimentally observed that the site density in activated carbons varies typically from 0.2 nm-2 to 1.4 nm-2. Figure 9 shows the adsorption isotherms of water in a micropore of width H = 0.9 nm at T = 425 K for different site densities: 0 (non-activated carbon), 0.4, 0.8 and 1.2 nm-2. It can be seen that increasing the site density, the capillary condensation transition occurs at lower pressure. This is also seen in Figure 10 in which we have represented the capillary condensation and evaporation pressures of water in a micropore of width H = 0.9 nm, at T = 425 K as a function of the site density from 0.2 nm-2 to 1.4 nm-2, adding the case for non-activated surfaces (ρsite = 0 nm-2). The presence of sites increases the affinity of water with the solid by creating hydrogen bonds and therefore, adsorption of water is enhanced leading to a shift of the condensation to lowest pressures. It can also be seen in Figure 10 that, compared to the case of non-activated surfaces for water desorption, the pressure at which we observe a capillary evaporation decreases as the site density increases. Moreover, the difference between capillary condensation and evaporation pressures is decreasing when the activation of the surface increases.


27


N(µmol/m2)

15

10

5

0 0.0

0.2

0.4

0.6 P/P0

0.8

1.0

Figure 9. Adsorption isotherms of water in a pore of width H = 0.9 nm at T = 425 K for nonactivated surfaces (crosses), and activated surfaces with a site density of 0.4 nm-2 (circles), 0.8 nm-2 (triangles) and 1.2 nm-2 (squares)

0.6

Ptransition/P0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 41

0.4

0.2

0 0.0

0.5 1.0 −2 ρsite (nm )

1.5

Figure 10. Reduced capillary condensation (circles) or evaporation (triangles) pressure of water as a function of the site density of in a pore of width H = 0.9 nm at T = 425 K. We have previously observed that no capillary condensation occurs in graphitic slit mesopores with the NLDFT/SAFT-VR model. Hence, we have studied the effect of the site density on this behavior. Figure 11 shows the adsorption isotherms of water in a pore of width H = 3.033 nm at


28

Page 29 of 41

T = 425 K when the site density varies between 0 and 1.2 nm-2. We can notice that increasing the site density, there is still no capillary condensation but an enhancement of the quantity of adsorbed water can be observed.

3

2

N(µmol/m )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

1

0 0.0

0.2

0.4

0.6 P/P0

0.8

1.0

Figure 11. Adsorption isotherms of water in a pore of width H = 3.033 nm at T = 425 K for nonactivated surfaces (crosses), and activated surface with site density of 0.4 nm-2 (circles), 0.8 nm-2 (triangles) and 1.2 nm-2 (squares)

5. Conclusions In this work, we have studied the adsorption behavior of water in confined media. On the one hand, we have developed a NLDFT/SAFT-VR coupling for the description of SW associating monomers, extending a previous work for the monomer contribution with the associating term. On the other hand, we have performed GCMC molecular simulations on the same fluid model (i.e. monomers interacting through SW potential). Both theory and simulations were used to generate density distributions of water in graphitic slit pores that were most of the time in very


29


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Page 30 of 41

good to excellent agreement. Only in the case of extreme confinement (i. e. for ultramicropores), a discrepancy can be seen, theoretical results giving in this case higher density than GCMC simulations, this difference being assigned to the MF level of the NLDFT/SAFT-VR theory. The theoretical model was also used to study the capillary condensation and evaporation of water in micropores. It has been shown that the pressure at which a capillary condensation may occur depends not only on the pore width but also on the temperature studied. Besides, the capillary evaporation appears at a nearly constant reduced pressure in micropores and mesopores. Finally, we have proposed, in a first approximation for the NLDFT/SAFT-VR model, to add an external potential to mimic activated sites on the porous solid. The model was first validated with GCMC simulations and then used to study the effect of site density on the adsorption behavior of water on activated surfaces. It has been shown that as the site density increases, capillary condensation in micropores is shifted to lower pressures, and also the adsorption is enhanced in micropores and mesopores.


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AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]. Phone: +33 559574415. Fax: +33 559574409 Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. ‡These authors contributed equally. (match statement to author names with a symbol)

ACKNOWLEDGMENT This work was sponsored by the ERC advanced grant Failflow (27769). This financial support is gratefully acknowledged. MMP acknowledges financial support from project FIS2012-33621 (cofinanced with EU FEDER funds), from Ministerio de Economía y Competitividad (Spain).


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TOC Graphic

25

water density (kg/m3)

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___ NLDFT/SAFT-VR +++ GCMC

20 15 10 5 0 0.0

0.2

0.4 0.6 z (nm)

0.8


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Nonlocal Density Functional Theory and Grand Canonical Monte

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