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Diffusion of supercritical fluids through single-layer nanoporous solids: theory and molecular simulations Fouad Oulebsir, Romain Vermorel, and Guillaume Galliero Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03486 • Publication Date (Web): 15 Dec 2017 Downloaded from http://pubs.acs.org on December 17, 2017

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Diffusion of supercritical fluids through single-layer nanoporous solids: theory and molecular simulations Fouad Oulebsir, Romain Vermorel,∗ and Guillaume Galliero Laboratoire des Fluides Complexes et leurs Réservoirs-IPRA, E2S, UMR5150, University of Pau and Pays de l’Adour/CNRS/TOTAL, 64000 Pau, France E-mail: [email protected]

Abstract With the advent of graphene material, membranes based on single-layer nanoporous solids appear as promising devices for fluid separation, be it liquid or gaseous mixtures. The design of such architectured porous materials would greatly benefit from accurate models that can predict their transport and separation properties. More specifically, there is no universal understanding of how parameters such as temperature, fluid loading conditions or the ratio of the pore size to the fluid molecular diameter influence the permeation process. In this study, we address the problem of pure supercritical fluids diffusing through simplified models of single-layer porous materials. Basically, we investigate a toy model that consists of a single-layer lattice of Lennard-Jones interaction sites with a slit gap of controllable width. We performed extensive Equilibrium and Biased Molecular Dynamics simulations to document the physical mechanisms involved at the molecular scale. We propose a general constitutive equation for the diffusional ∗

To whom correspondence should be addressed

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ACS Paragon Plus Environment

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transport coefficient derived from classical statistical mechanics and kinetic theory, which can be further simplified in the ideal gas limit. This transport coefficient relates the molecular flux to the fluid density jump across the single-layer membrane. It is found to be proportional to the accessible surface porosity of the single-layer porous solid and to a thermodynamic factor accounting for the inhomogeneity of the fluid close to the pore entrance. Both quantities directly depend on the Potential of Mean Force that results from molecular interactions between solid and fluid atoms. Comparisons with the simulations data show that the kinetic model captures how narrowing the pore size below the fluid molecular diameter lowers dramatically the value of the transport coefficient. Furthermore, we demonstrate that our general constitutive equation allows for a consistent interpretation of the intricate effects of temperature and fluid loading conditions on the permeation process.

Introduction Nanoporous materials are at the forefront of a variety of applications, including industrial processes such as gas treatment, purification and storage. 1–3 By nanoporous media, we refer to materials with a characteristic pore size comparable to that of fluid molecules susceptible to flow through them. Zeolites, Metallic Organic Frameworks (MOF), carbon nanotubes or manufactured amorphous carbons are typical examples of such materials and represent very active areas of research. With the advent of graphene material, new types of nanofluidic separation devices are soon to be added to this list. Namely, membranes based on nanoporous graphene 4 are on the verge of mass production. If their potential for water desalination retains most attention, 5 this promising new technology will also benefit many other applications thanks to their controllable pore size, high strength, 6 chemical stability 7 and their single-layer structure prone to minimize the energy required to maintain flow. More specifically, several experimental and numerical studies have investigated their efficiency at separating gaseous mixtures. 8–16 In this regard, the design of these architectured

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porous materials would greatly benefit from accurate theories that can predict their transport properties. The influence of parameters such as the pore to fluid molecules size ratio, or the effect of thermodynamical conditions on the diffusion coefficient first come to mind. Numerous authors have put a tremendous amount of work in the documentation of gas and supercritical fluids transport through a broad range of nanoporous materials. 17 Phenomenological diffusion equations based on the Maxwell-Stefan or Onsager formalisms seem to be well-suited so as to describe the flux of molecular species permeating through tight nanoporous systems. 18–20 These general theoretical frameworks should also apply to singlelayer nanoporous membranes. However, to the best of our knowledge, there is no precise and universal understanding of how important parameters such as fluid loading, temperature or pore size impact the value of the transport coefficients. Classically, diffusion in nanoporous materials is considered as an activated process 21 and several studies on gas permeation report fluid-dependent activation energies to characterize the membrane material, be it cristalline 22,23 or amorphous. 24–26 Since the pioneering work of Ford and Glandt, 27 the same assumption has been at the core of several theories addressing gas permeation through single-layer nanoporous solids. For instance, Drahushuk and Strano 28 have proposed a model based on activated mechanisms and Transition State Theory (TST) to describe gas permeation through nanoporous graphene. They assumed that fluid molecules might pass through the pore by means of two parallel processes: one part of the flux is due to gas molecules that directly cross the pore mouth from the bulk of the gas phase; the other part results from gas molecules that first adsorb at the surface of the solid wall before reaching the pore entrance. In their model, the description of the successive rate-limiting steps that lead fluid molecules from the bulk of the gas phase to the other side of the membrane requires the definition of several activation energies, supposedly independent of temperature. More recently, Sun et al. 29 used Non Equilibrium Molecular Dynamics (NEMD) simulations to document the permeation of several gas species through nanoporous graphene. They proposed a model based on kinetic theory in which they considered gas

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molecules and carbon atoms as hard spheres to predict the so-called direct flux incoming from the bulk gas phase. When this approach departs from Arrhenius-based models, their calculations based exclusively on geometric arguments are likely to underestimate the molar flux. From a fundamental point of view, the definition of the activation energy a fluid molecule is required to pass through a pore mouth is ambiguous. Let us for instance consider a spherical gas molecule that interacts with the solid atoms delimiting the pore space. If this gas molecule does not behave as a hard sphere, it can accommodate increasingly stronger repulsive interactions with the solid atoms as temperature increases. As a consequence, rising the temperature does not only boost the frequency of permeation events, it also widens the potential energy landscape accessible to the diffusing molecules. A change in temperature should therefore simultaneously affect the area of the pore accessible to the fluid and the effective free energy barrier opposed to the flow. Nguyen and Bhatia 30,31 accounted for this effect in their work on pore accessibility in disordered nanoporous materials. Since this subtle mechanism is likely to occur in single-layer nanoporous membranes, Arrhenius-based and geometric models are questionable. In this work we address the problem of pure supercritical fluids permeating through single-layer nanoporous solids. As a first approach, we consider the diffusion of a LennardJones (LJ) fluid through a simplified model of single-layer membrane that consists in a lattice of LJ interaction sites with a slit gap of controllable width. The purpose of this study is to lay the groundwork for a general theory of gas diffusion through single-layer nanoporous solids. We used MD simulations to document the diffusional mechanisms at the atomistic scale. More specifically, we report the values of the transport coefficients across the single-layer membrane computed from Equilibrium Molecular Dynamics (EMD) 32 for a range of temperatures, fluid densities and pore sizes. This transport coefficient relates the molecular flux density to the difference in number density across the thin membrane in the linear regime. To predict its value, we propose a constitutive equation based on kinetic

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theory and classical statistical mechanics. This theoretical model accounts for the actual fluid/solid potential of mean force and the complex effect of temperature via the definition of an accessible surface porosity. Furthermore, we account for the inhomogeneity of the fluid close to the pore entrance by introducing a corrective thermodynamic factor, which depends only on the fluid/solid interaction in the ideal gas limit. In the range of applicability of our model, we observe a good agreement between theory and simulations data obtained in a broad range of conditions. In addition, we document thoroughly how increasing fluid loadings affect the values of the transport coefficients.

Molecular models and simulation details (a)

(b)

Solid walls

(c)

W l

x y

z

L

z

L

H

l+h

x

x y

y

h

z

Figure 1: Basic features of the system under study. (a) Snapshot of the simulation box in the xz-plane. Periodic boundary conditions are applied in all directions. (b) Snapshot of one solid wall in the xy-plane. (c) Zoomed in view of the slit gap in the xz-plane.

Fluid and Solid Molecular Models The system used in this study is shown in fig. 1. It consists of three reservoirs of moving spherical fluid molecules [white spheres in fig. 1(a)] separated by two porous surfaces made of immobile spherical solid molecules [grey spheres in fig 1(a)]. In our simulations, we applied

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periodic boundary conditions in all directions so that the two lateral reservoirs are actually connected through their periodic images along the z axis. The central and lateral reservoirs, on the other hand, can only exchange fluid molecules through the two solid porous walls separated by a distance 2L. The solid walls structure is shown in fig. 1(b). It is comparable to the system found in the work of Ford and Glandt. 27 The porous surface is made of two single-layered sheets of spherical molecules with a horizontal slit gap centered around x = 0. In each solid walls the molecules are arranged in a square pattern with lattice spacing l. The width h of the slit gap is defined as the vertical distance between the edges of the opposing solid atoms [see fig. 1(c)]. The height and width of the solid walls, noted H and W respectively, coincide with the dimensions of the simulation box in the xy-plane. Unless otherwise specified, fluid and solid molecules are described as simple LennardJones (LJ) spheres with a truncation of the interactions at rij = rc = 2.5σ, " ULJ (rij ) = 4

σ rij

12

−

σ rij

6 # rij ≤ rc (1)

=0

rij > rc ,

where the same potential parameters and σ were used to describe both fluid/fluid and fluid/solid interactions. For model validation purposes, we used the repulsive component of the Weeks-Chandler-Andersen potential 33 (WCA) in some of our simulations to investigate the case of purely repulsive interactions between fluid and solid molecules. The solid phase in these specific simulations is referred to as the soft repulsive wall, for which the fluid/solid interaction is given by Urep (rij ) = ULJ (rij ) +

rij ≤ 21/6 σ (2) 1/6

=0

rij > 2

σ,

In the following of this article, our simulations data are expressed in standard LJ reduced units 34 and dimensionless variables are written with an asterisk in superscript (see Supple6


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mentary Information for the definition of the reduced units). The lattice spacing was set to l∗ = 21/6 and we investigated pore widths ranging from h∗ = 0.55 to h∗ = 10. We defined the pore width h∗ = h/σ in such a way that the two solid sheets form a perfect cristalline surface with lattice spacing l for h∗ = 0 [see fig. 1(c)]. The temperatures imposed in our simulations ranged from T ∗ = 1.5 to T ∗ = 4.5, which are above the critical temperature of the LJ fluid (Tc∗ = 1.1875 for rc∗ = 2.5 35 ). We therefore dealt with supercritical fluids and we observed no phase transition in our systems. Fluid densities ranged from ρ∗ = 0.1 to ρ∗ = 0.5. We also investigated what we refer to as the ideal gas limit upon switching the interactions between fluid molecules off, so that each fluid molecule interacted only with the solid phase. Typically, the porous solid wall was made of two single-layer sheets, each of them consisting of 8 and 14 layers of particles in x- and y-directions respectively, resulting in minimal transverse dimensions greater than H ∗ = 18 and W ∗ = 16. The half-distance between the solid walls was set to L∗ = 12. The number of layers and the distance L∗ were held constant for the range of pore widths h∗ under investigation.

Molecular Dynamics Simulations All simulations presented in this study were performed with the LAMMPS molecular dynamics simulations package. 36 We employed a method developed recently in our group to compute diffusional transport coefficients through single-layered porous solids from EMD simulations. 32 Contrary to NEMD methods, this technique guarantees that molecular diffusion is the only permeation mechanism occurring in our simulations. In the linear regime, the diffusional transport coefficient, Λ, relates the molecular flux density, J, to the number density difference across the thin membrane, ∆ρ, via the Fick’s law:

J = Λ∆ρ .

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(3)

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The computation of Λ from EMD simulations is based on a post-treatment algorithm that calculates the characteristic relaxation time, τ , of fluid concentration fluctuations between the reservoirs at equilibrium. The transport coefficient derives from the characteristic relaxation time as Λ=

L . 2τ

(4)

For a sufficiently large distance L, permeation events through the two porous walls are uncorrelated and τ is thus simply proportional to L. As a result, Λ does not depend on the distance between the interfaces and only accounts for the diffusion of the fluid particles across the solid walls. For an in-depth description of the computation of the transport coefficient Λ from EMD simulations, we refer the reader to the work of Vermorel et al.. 32 Furthermore, one can relate the transport coefficient Λ to the permeance of single-layer membranes, Π, usually defined in units of mol.Pa−1 .s−1 as 13,15,16,29,37

J=

NA Π∆P , Sw

(5)

where Sw = HW is the surface area of the single-layer porous solid, NA is the Avogadro’s number and ∆P is the pressure jump across the membrane. Equating the molecular flux densities in eq. 3 and eq. 5 yields the contribution of molecular diffusion to the permeance:

Π=

Sw ∆ρ Λ. NA ∆P

(6)

When the pore width h becomes sufficiently small, fluid molecules cannot permeate through the porous surface without overlapping, at least slightly, with molecules from the solid phase. For our system, this approximately happens for pore widths h∗ . 0.82. Under such conditions, fluid molecules therefore need to overcome free energy barriers to pass through the pore. In the ideal gas limit, computation of such free energy barriers is straightforward as it only requires knowledge of the fluid/solid interaction potential [from eq. 1 or 2]

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and of the solid wall structure. When it comes to systems with finite fluid densities however, the free energy landscape is also influenced by fluid/fluid interactions and its computation usually requires biased MD techniques able to compute free energy profiles, or so-called Potentials of Mean Force (PMF). 38,39 In this work we employed the Adaptive Biasing Force (ABF) method 40,41 implemented in the LAMMPS/colvars package. 42 We defined the reaction coordinate as the distance between the center of a pore and a selected fluid molecule, whose motion was restrained to a line perpendicular to the xy-plane of the solid wall [see the red sphere in fig. 2]. Here we refer to a pore as the pattern formed by four adjacent solid molecules located at the edges of the slit gap [see orange spheres in fig. 2]. Details about the simulation protocols, as well as the exhaustive list of simulations parameters can be found in the Supporting Information. X

Y

Z

Figure 2: Close view of the slit gap. White and grey spheres stand for the fluid and solid molecules, respectively. Orange spheres emphasize four solid molecules that form the pore pattern, materialized by the dashed black lines. During ABF simulations, the motion of a selected fluid molecule (red sphere) is restrained to the z-axis (dashed red line) perpendicular to the xy-plane and passing through the center of the pore.

Theoretical approach of gas diffusion through single-layered porous solids Hereafter we derive a theoretical model to predict the value of the transport coefficient, Λ, of the single-layered porous solid. Our approach is based on the kinetic theory of gas and 9


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classical statistical mechanics, it should therefore be valid in the ideal gas limit.

Large pore widths limit Let us first consider the classical problem of an ideal gas enclosed in a container. More specifically, we are interested in the motion of gas molecules in the vicinity of the container wall under thermodynamic equilibrium conditions. If we consider the solid wall as a planar surface, the Maxwell-Boltzmann distribution of molecular speeds yields the number of gas molecules incoming from one side of the wall and striking its surface per unit area per unit p time as n = ρ¯ v /4, where v¯ = 8kT /πm is the thermal velocity. 43 From now on, let us consider that the container wall is a porous solid surface, such as that of fig 1, so that some of the gas molecules hitting the surface can pass through it. If the pore width h is sufficiently large compared to the characteristic distance of interaction σ, most of the gas molecules passing through the solid surface do not interact with it. In this limit of large pore widths (i.e. h∗ >> 1), we recover the classical problem of effusion of gas molecules through a small hole. 44 In a fair approximation, the number of gas molecules per unit area per unit time incoming from one side of the wall and passing through it may thus be simply expressed as j = ρ¯ v h/4H, where the ratio h/H corresponds to the geometric surface porosity of the solid wall. We can then obtain the net flux density of gas molecules crossing the surface by finite differentiation of the local fluxes densities through a vanishingly small distance δ across the planar interface as follows J = j(z − δ/2) − j(z + δ/2) h¯ v [ρ(z − δ/2) − ρ(z + δ/2)] 4H h¯ v ∂ρ ≈− δ. 4H ∂z =

(7)

When fluid molecules pass through the interface they travel across a distance δ that ranges from the mean free path, when dealing with an ideal gas, to the molecular size when dealing

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with an incompressible liquid. 45 The determination of this characteristic length for any arbitrary thermodynamic state of the diffusing fluid is therefore ambiguous. The contribution of the undetermined distance δ is nonetheless lumped together with the density gradient to obtain the well-defined density difference across the membrane ∆ρ = | − ∂ρ/∂z|δ. Direct comparison with eq. 3 then provides the following expression for the transport coefficient:

Λ=

h¯ v . 4H

(8)

Let us emphasize that eq. 8 should be valid only in the limit of h∗ >> 1, for which interactions between diffusing gas molecules and the solid wall are negligible. In the following of this article, we refer to eq. 8 as the geometric model.

Accounting for solid/fluid interactions For pore widths of the same order of magnitude as σ, interactions with the solid molecules strongly impact the permeation mechanism. The first reason is the steric repulsion with the solid molecules located at the edges of the pore that becomes prominent as the pore width decreases. The surface porosity accessible to gas molecules is thus smaller than the simple geometric approximation h/H. In addition, because of their proximity with the solid atoms, the gas molecules that are prone to pass through the pore are subject to an external Potential of Mean Force (PMF). As a result, the density of the gas in the vicinity of the pore entrance cannot be considered as uniform anymore. The typical PMF felt by a fluid molecule in the ideal gas limit is shown in fig. 3(a). This corresponds to the free energy profile along the straight line orthogonal to the wall and passing through the center of the pore, computed from the LJ interaction potential of eq. 1. We define herein U (x, y, z), the free energy difference measured at the point of coordinate (x, y, z) with respect to the PMF minimum at (0, 0, zmin ) [see fig. 3(a)]. The surface porosity accessible to gas molecules may

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Figure 3: (a) PMF computed from solid/fluid interactions only for a pore width h∗ = 0.6. The PMF is computed along the straight line (0, 0, z) perpendicular to the solid surface and passing through the center of the pore. Only solid molecules within the cut-off distance were accounted for in the calculation. However, the effect of truncation was corrected to smoothen the PMF curve. (b) Arrhenius plots of the accessible porosity ϕ normalized by its maximum value ϕmax for pores of width ranging from 0.55 to 0.80.

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be obtained by integration of the Boltzmann factor over the surface of the solid wall: 1 ϕ= Sw

W

Z

Z

0

0

H

U (x, y, 0) dxdy exp − . kT

(9)

To compute the accessible surface porosity, we performed a dual numerical integration of the Boltzmann factors accordingly to the previous equation. Fig. 4 shows the surface maps of the Boltzmann factor in the integrand of eq. 9 for two different pore widths and a range of temperatures. The quantity e−U (x,y,0)/kT dxdy/Sw can be interpreted as the probability of a gas molecule passing through points located at (x±dx, y±dy, 0). The computed surface maps clearly show that the accessible porosity becomes significantly smaller than the geometric porosity as the pore width is reduced. The maximum of the Boltzmann factor distribution is observed at the center of the pore pattern, where U (0, 0, 0) = Uw [see fig. 3(a)] and which corresponds to the most favorable path. In the case of narrow pores the distribution features a sharp peak [see fig. 4 (a)], which widens significantly with increasing pore widths (see fig. 4 (b)). The effect of temperature appears clearly in fig. 4: rising the temperature does not only increase the value of the Boltzmann factor, it also spreads the bounds of the distribution towards the edges of the pore. The resulting Arrhenius plots of the accessible porosity thus exhibit significant curvature, as shown in fig. 3(b). As a consequence, one would need to define a temperature dependent activation energy to fit eq. 9 with an Arrhenius relation. 30,31 When dealing with a non homogeneous gas, the gas density involved in the expression of the local flux of molecules passing through the wall is not the reservoir bulk density ρ. In the following we assume that the local density of gas molecules prone to pass through the pore is ρzmin , measured at the minimum of the PMF. Similarly to eq. 7, we can thus express the net flux density of gas molecules crossing the solid wall in response to a bulk density gradient as ϕ¯ v ∂ρzmin δ 4 ∂z ∂ρ ϕ¯ v ∂ρzmin ≈− δ, 4 ∂ρ T ∂z

J ≈−

13


(10)

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Figure 4: Surface maps of Boltzmann factors in the plane of the solid wall z = 0, (a) pore size h∗ = 0.60, (b) pore size h∗ = 0.80. Graphics from left to right show the effect of increasing temperatures ranging from T ∗ = 1.5 to T ∗ = 4.5. Dashed lines stand for the edges of the four solid molecules forming the pore pattern. where the isothermal thermodynamic factor (∂ρzmin /∂ρ)T is introduced to correct for the inhomogeneity of the gas phase. This additional term is analogous to the Darken factor classically used in the context of membrane science to account for the effect of adsorption. 17,46 There again, comparison with eq. 3 yields the general expression of the transport coefficient that accounts for fluid/solid interactions: ϕ¯ v Λ= 4

∂ρzmin ∂ρ

.

(11)

T

In the ideal gas limit, the isothermal thermodynamic factor must be independent of the bulk fluid density ρ. Under such conditions, the thermodynamic factor equals the ratio of local to bulk density, i.e. (∂ρzmin /∂ρ)T = ρzmin /ρ. This density ratio is simply the ratio between the partition functions of a gas molecule located at (0, 0, zmin ) and that of a molecule located in

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the bulk. If we define U∞ = U (x, y, z → ∞) [see fig. 3(a)], then the density ratio writes ρzmin = exp ρ

U∞ kT

.

(12)

Substitution of the thermodynamic factor in eq. 11 leads to the expression of the transport coefficient in the ideal gas limit as follows, ϕ¯ v Λ= exp 4

U∞ kT

.

(13)

In the following we assume that the minimum of the PMF is unique and located at (0, 0, zmin ), and U∞ is thus defined as shown in fig. 3(a). This approximations should be correct in the limit of narrow pore widths h∗ . 1. Let us stress that eq. 13 is not equivalent to classical constitutive equations based on the Arrhenius law 23 because of the intricate effect of temperature on the accessible porosity ϕ. Thanks to eq. 9 and eq. 13, we can compute directly an approximation of the transport coefficient as the parameters ϕ and U∞ only depend on the fluid/solid interaction potential, which is known a priori. This constitutive equation stands for the asymptotic ideal gas limit and should therefore be all the more accurate as the fluid density is low. Furthermore, in the case of an ideal gas permeating through a soft repulsive wall [e.g. eq. 2], there is no potential well close to the wall and U∞ therefore vanishes. The computation of Λ should therefore come down to that of ϕ, as the transport coefficient, in the absence of attractive interactions between fluid and solid molecules, becomes

Λ=

ϕ¯ v . 4

(14)

We will use this interesting feature in our simulations in order to validate our computation of the accessible porosity.

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Λ

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

Λ

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Figure 5: Simulation results obtained for soft repulsive walls in the ideal gas limit. Main figure: plot of the quantity Λ/¯ v as a function of pore width, for the temperatures T ∗ = 1.5 ∗ (magenta circles), T = 3.0 (blue losanges) and T ∗ = 4.5 (yellow squares). Inset: Arrhenius plots of the quantity Λ/¯ v for pores of widths h∗ = 0.6 (magenta circles), h∗ = 0.9 (blue ∗ losanges) and h = 2.0 (yellow squares). In both figures, solid lines stand for the model predictions [see eq. 9 and eq. 14] with no adjustable parameter.

Results and discussion Hereafter we report and compare the results of our simulations to the theoretical model. In the following figures, we report the computed transport coefficients with error bars smaller than the size of symbols.

Computation of the accessible porosity We first investigated the case of soft repulsive walls combined to a fluid phase in the ideal gas limit. As previously discussed, we used the fluid/solid interaction potential of eq. 2 and we therefore expect the transport coefficient to satisfy eq. 14. Interactions between fluid molecules were turned off in order to mimic the conditions of infinite dilution. Fig. 5 shows the evolution of the diffusion coefficient with pore width and the Arrhenius plots of the quantity Λ/¯ v , supposed to be proportional to the accessible surface porosity ϕ according to eq. 14. Simulation results are in perfect agreement with the predictions of the model. This validates the expression of the accessible surface porosity in eq. 9 and the numerical

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Figure 6: Transport coefficient as a function of pore width in logarithmic scale. We obtained these simulation results for adsorptive walls and fluid densities ranging from ρ∗ = 0.1 to ρ∗ = 0.5. (a), (b), (c) and (d) stand for temperatures T ∗ = 1.5, T ∗ = 2.0, T ∗ = 3.0 and T ∗ = 4.5, respectively. Dashed lines stand for the predictions of the geometric model of eq. 8 and solid lines stand for the prediction of eq. 13, with no adjustable parameter. computation of this parameter. Let us point out that the Arrhenius plot for the pore width h∗ = 0.6 in the inset of fig. 5 exhibits significant curvature over the whole temperature range, which confirms that the Arrhenius relation is questionable in that case. Our results also show how the Arrhenius plots flatten for increasingly larger pore widths as the steric hindrance tends to decrease rapidly.

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Effect of pore width Fig. 6 shows results obtained in the case of adsorptive walls, corresponding to the fluid/solid interaction potential of eq. 1. For all the investigated temperatures and fluid densities, the evolution of the diffusion coefficient with the pore width follows two distinct regimes. In the limit of large pore widths (h∗ & 2), the effect of fluid/solid interactions becomes negligible and the geometric model of eq. 8 is in agreement with our simulations data. In such conditions, data obtained for different densities collapse on the same curve. This indeed suggests that most of the fluid molecules crossing the wall interact poorly with the solid molecules. As a result, crossing molecules are in the same thermodynamic state as the bulk of the fluid and the thermodynamic factor (∂ρzmin /∂ρ)T approaches unity. When dealing with smaller pores (h∗ . 1), the diffusion coefficient dramatically drops as a result of the steric interactions between permeating fluid molecules and the solid molecules located at the edges of the pore. The theoretical prediction of eq. 13 pertaining to the ideal gas limit captures this strong decrease of the diffusion coefficient and exhibits an overall good agreement with the simulation data. Obviously, the lower the fluid density, the more accurate this simplified model. In the crossover region between the two regimes (h∗ ≈ 1), the simulation data exhibits a bump. This is particularly noticeable at lower temperatures [see fig. 6 (a) and (b)], for which the simulation data might even exceed values predicted by the geometric model. We attribute this behavior to the fact that the minimum free energy barrier at the pore mouth Uw decreases more rapidly with the pore width than the free energy difference U∞ between the bulk fluid and the PMF minimum, which boosts the frequency of permeation events. Furthermore, the discrepancies between eq. 13 and the simulation results observed in the crossover region (h∗ ≈ 1) and for larger pore widths (h∗ & 2), can be explained quite simply. We indeed inferred eq. 13, and more specifically the free energy difference U∞ , from the projection of the PMF along a straight line perpendicular to the solid wall and passing through the center of the pore, which corresponds to the most favorable path. Upon increasing the pore size though, the probability of crossing the wall gradually changes from 18


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a sharp peak located at the center of the pore to a wider distribution spreading towards the edges of the pore [see fig. 4(b) for instance]. Consecutively, the 1D assumption becomes inaccurate and one would need to account for the three-dimensional free energy landscape to predict precisely the transport coefficients over the whole range of pore widths. As expected, lowering the temperature accentuates these inaccuracies as the Boltzmann factor in eq. 13 becomes more sensible to the value of U∞ . Calculations based on a 3D representation of the PMF are however out of the scope of the present work.

Effect of temperature In fig. 7 (a) and (b) we report the Arrhenius plots of the quantity Λ/¯ v for two porous walls, with pore widths of 0.60 and 0.80, representing the small pore and crossover regimes respectively. If the permeation process were a simple activated mechanism, the curves would follow a linear trend with a negative slope. In the case of small pore widths [see fig. 7 (a)], the Arrhenius plots exhibits an almost linear decreasing trend for a limited range of highly supercritical temperatures. However, when the temperature decreases the curves exhibit significant curvatures and might even pass through a minimum for lower fluid loadings. The simplified model of eq. 13 captures this typical trend that results from the competition between the opposed variations of the accessible porosity and that of the thermodynamic factor. Rising the temperature help fluid molecules to access increasingly larger areas of the pore leading to the increase of the accessible porosity. On the contrary, it also causes desorption and thus the probability of finding molecules in the vicinity of the pore entrance decreases. When it comes to pores of intermediate widths, the permeation process is dominated by adsorption as Uw decreases more rapidly with the pore width than U∞ . As a consequence, the corresponding Arrhenius plots [see fig. 7 (b)] show a monotonically increasing trend. A similar mechanism has been described by Bot, an et al. 20 in the context of gas permeation through amorphous porous carbons.

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Figure 7: Arrhenius plots of the quantity Λ/¯ v obtained from EMD simulations for pore ∗ ∗ widths h = 0.6 [subfigure (a)] and h = 0.8 [subfigure (b)] respectively. Symbols represent simulations data for fluid densities ranging from 0.1 to 0.5. The black solid line stands for the simplified model of eq. 13.

Effect of fluid loading Another interesting feature of the transport mechanism documented in our simulations lies in the evolution of the diffusion coefficient with bulk fluid density. In the case of small pore widths, increasing the bulk density of the fluid enhances the diffusion process. This effect is clearly visible on fig. 8 (a): for a pore width of h∗ = 0.60 at the reduced temperature of T ∗ = 4.5, the transport coefficient is roughly multiplied by a factor 3 when the density is increased from ρ∗ = 0.1 to ρ∗ = 0.5. The smaller the pore width, the stronger the effect. On the other hand, for pore widths in the crossover range, the transport coefficient may exhibit

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Figure 8: Transport coefficient as a function of bulk fluid density for pore widths h∗ = 0.6 [subfigure (a)] and h∗ = 0.8 [subfigure (b)] respectively. Symbols represent simulations data for reduced temperatures ranging from 1.5 to 4.5. a non monotonic evolution with fluid loading depending on the imposed temperature. For instance, fig. 8 (b) shows the results obtained for a pore of width h∗ = 0.8. At a temperature of T ∗ = 1.5, the diffusion coefficient decreases for fluid densities ranging from 0.1 to 0.4, and then increases for subsequent values of fluid densities. Interestingly, other MD studies on zeolites report comparable evolution of the diffusion coefficient with fluid loading. 47,48 In the large pore width limit however, the results we obtained for different fluid loadings converge on a single curve [see fig. 6]. This suggests that the effect of fluid loading on diffusion lies in the coupling between fluid/solid and fluid/fluid interactions that occurs near the pore entrance.

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Figure 9: (a) and (b): PMF (top) and reduced density profiles (bottom) obtained at the reduced temperature T ∗ = 3.0 for the pore widths h∗ = 0.60 and h∗ = 0.80 respectively. PMF and densities were computed along the straight line perpendicular to the solid wall and passing through the center of the pores. Colored dashed lines stand for the simulations data obtained for bulk fluid densities ρ∗ = 0.1 , 0.3 and 0.5. The solid black lines stand for the ideal gas limit predictions computed from the fluid/solid interaction potential. (c) and (d): evolution of the energy barriers U∞ (top) and Uw (bottom) with fluid loading for the pore widths h∗ = 0.6 and h∗ = 0.8 respectively. Symbols stand for the simulation data obtained for temperatures ranging from 1.5 to 4.5. Careful examination of the PMF under various thermodynamic conditions provides insights on how fluid loading influences the diffusion mechanism. We report in fig. 9 (a) and (b) the PMF and density profiles obtained at T ∗ = 3.0 for two distinct pore widths, respectively h∗ = 0.60 and h∗ = 0.80, and bulk fluid densities ranging from the ideal gas limit (ρ∗ → 0) up to ρ∗ = 0.5. The local density histograms were computed from unbiased MD by counting fluid molecules in cubic bins centered around the same straight line as the PMF, as shown in fig 2. We obtained convergence of the density values for bin widths smaller than 0.1σ. Our results show the clear structuration of the fluid phase in the vicinity of the pore entrance where the density reaches a maximum. As expected, the position of the density peak coincides with the minimum of PMF, even if we notice slight discrepancies most certainly due to the bias introduced during the computation of the PMF by means of ABF simulations.

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Fig. 9 (c) and (d) focus on the evolution of the energy barriers and thus illustrate how fluid loading impacts the global shape of the free energy profiles at different temperatures. The first noticeable effect is the decrease of the free energy barrier Uw [as defined in fig. 3(a)] for finite values of the fluid density. Starting from the ideal gas limit, the first consequence of introducing a finite fluid density lies in the contribution of the repulsive fluid/fluid interactions, which provokes additional steric hindrance at the pore mouth. As a result, the value of Uw obtained from the simulations performed at lower fluid loadings exceeds that of the ideal gas limit prediction. By rising the fluid density, this steric repulsion is compensated by the attractive interactions resulting from the adsorbed layers building up at the surface of the solid wall, which causes the free energy barrier to recess. The second striking effect lies in the modification of the energy difference U∞ [as defined in fig. 3(a)] concomitantly with the shift of the peak of density at zmin . The evolution of U∞ with fluid loading is almost antisymmetric to that of Uw and results from the same adsorption mechanism. U∞ indeed quantifies the reduction in free energy a fluid molecule coming from the bulk experiences when it reaches the position (0, 0, zmin ). There again, starting from the ideal gas limit, the first seeable consequence of accounting for a finite bulk fluid density lies in the steric fluid/fluid repulsion. This leads to a decrease of U∞ that translates in a lowering of the peak of density, visible on fig. 9 (a) and (b) between ρ∗ → 0 and ρ∗ = 0.1. For sufficiently high fluid loading conditions, the attractive interactions due to the adsorbed layers dominate leading to the increase of the free energy barrier. Consequently, both U∞ and the fluid density at the peak exhibit a minimum. The position of the latter depends on temperature, which is clearly visible in the case of a pore width of h∗ = 0.80 [see fig. 9 (d)]. We indeed observe that the minimum occurs at a density that exceeds ρ∗ = 0.5 at a temperature of T ∗ = 1.5, while it shifts to lower bulk densities when the temperature increases. As discussed in the previous paragraphs, the two characteristic energy differences Uw and U∞ drive the diffusion process. The first relates to the accessible porosity, while the second is closely related to the thermodynamic factor in eq. 11. First, let us treat the

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example of the pore of width h∗ = 0.60. Fig. 9 (c) shows that for temperatures T ∗ ≥ 2.0 and densities in the range 0.1 − 0.5 the free energy barrier U∞ is an increasing function of density. Moreover, in spite of being non monotonic, the evolution of U∞ with density at the temperature T ∗ = 1.5 is almost flat. On the other hand, the repulsive barrier Uw decreases with bulk fluid densities under the same thermodynamic conditions. The evolutions of both energy barriers are therefore favorable to the increase of the transport coefficient with fluid loading in the investigated range of temperatures, which is consistent with our simulations results [see fig. 8 (a)]. When dealing with pores of widths that fall in the range of the crossover regime (h∗ ≈ 1), the situation differs significantly. Under such conditions, the repulsive energy barrier Uw becomes negligible compared to U∞ , so that the latter dominates the diffusion process. The results reported in fig. 9 (d) for a pore of width h∗ = 0.80 exemplifies this mechanism. For bulk fluid densities in the range ρ∗ = 0.1 − 0.5, the free energy barrier U∞ increases with fluid loading for temperature T ∗ > 2.0, but steadily decreases with fluid density when the temperature is held at T ∗ = 1.5. This suggests that the transport coefficient should increase with fluid loading for T ∗ > 2.0, while it should decrease for a temperature of T ∗ = 1.5, which is in agreement with simulations results reported in fig. 8 (b). Since Uw U∞ in the crossover regime, the accessible porosity ϕ in eq. 11 is less sensitive to variations of fluid loading than the thermodynamic factor (∂ρzmin /∂ρ)T . To test the validity of the proposed constitutive equation, we inferred the values of the local density at the peak ρzmin upon substituting in eq. 11 the transport coefficients computed from simulations. Fig. 10 shows the local density isotherms for a pore width of h∗ = 0.80. The isotherms reconstructed from the simulated transport coefficients and eq. 11 are in good agreement with local densities computed in our simulations. This result demonstrates that the general form of eq. 11 is well-suited for the description of diffusion through single-layer nanoporous solids.

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Figure 10: Fluid density at the minimum of PMF as a function of the bulk fluid density, obtained from EMD simulations for a pore width of h∗ = 0.8. Symbols represent simulations data for reduced temperatures ranging from 1.5 to 4.5. Solid lines stand for the isotherms reconstructed from the simulations data shown in fig. 7 (b) by means of eq. 11. The only adjustable parameter is the value of ρ∗zmin at ρ∗ = 0.1 and we used spline interpolation to smoothen the fitting curves.

Conclusion In this study, we have used Equilibrium Molecular Dynamics to investigate the diffusion of supercritical Lennard-Jones fluids through models of single-layer nanoporous solids under thermodynamical equilibrium conditions. We observed different behaviors with respect to the ratio of the pore width to the fluid molecular diameter. For pores significantly larger than the molecular size, we observed a linear relation between the transport coefficient and the pore width. Under such conditions, most of the permeating fluid molecules weakly interact with the atoms of the single-layer membrane and the paradigm is thus equivalent to the classical problem of effusion through a small hole. 44 The situation is clearly different for pores of widths comparable to the molecular size and below, as the interaction between diffusing fluid molecules and solid atoms becomes important. To address this specific regime, we have proposed a theoretical model to predict the value of the diffusional transport coefficient for spherical molecules. Aside from the thermal velocity of diffusing molecules, our model requires two key inputs to compute transport coefficients: the accessible surface porosity of the single-layer membrane and a corrective thermodynamic factor, which accounts for the 25


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local effect of adsorption in the vicinity of the pore entrance. Both quantities depend on the Potential of Mean Force (PMF) between permeating fluid molecules and the porous solid. In the ideal gas limit, this PMF only depends on the fluid/solid interaction potential and a simplification of the model is therefore possible. Confrontation between the predictions of our theoretical model and simulations data yields satisfactory results for a range of pore widths, temperature and loading conditions. Importantly, our results suggest that the classical Arrhenius equation is not sufficient to capture the influence of temperature on the transport coefficients. This is due to several factors. The first lies in the intricate dependency of the accessible porosity on temperature. The second reason is the competition between the accessible porosity and the thermodynamic factor, which exhibits opposed trends with respect to temperature. As a result, one cannot define an activation energy over a wide range of temperatures unambiguously. In addition we have investigated the effect of fluid loading on the diffusional mechanism. Our simulations illustrate how fluid loading affects the PMF felt by permeating fluid molecules, which leads to non trivial evolutions of the transport coefficient with bulk fluid density. We could interpret the phenomenology documented in the simulations consistently with the proposed theoretical framework. Furthermore, we shall point out that the theoretical model proposed in this study does not split the diffusive flux in several contributions according to the trajectories of permeating molecules, as opposed to other works found in the literature. 27–29 Aside from the specific case of transient flows, we believe that the description of physical mechanisms involved at the pore mouth is sufficient to predict the diffusive transport coefficients correctly, as proven by other models pertaining to liquid water diffusion through nanoporous graphene. 45 Overall, the results from our simulations bring evidence that our approach is consistent. We emphasize that our simulations did not rely on the application of any external driving forces and, for this very reason, our results account for the sole contribution of molecular diffusion arising from thermal fluctuations. We therefore disregarded additional permeation

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mechanisms that might arise from the application of pressure gradients such as hydrodynamic flows, which are possible for pore sizes down to the nanometer scale. 49 The investigation of the diffusion of fluid mixtures, non equilibrium conditions and the extension of this study to the case of realistic nanoporous graphene/fluid systems are left for future work.

Acknowledgement This work was supported by Cellule Energie du CNRS (TESMMA project) and ESA (SCCO project). Fouad Oulebsir thanks the CNRS and TOTAL SA for financial support.

Supporting Information Available The following files are available free of charge. “Equilibrium and Biased Molecular Dynamics Parameters” This material is available free of charge via the Internet at http://pubs.acs. org/.

References (1) Ismail, A. F.; Rana, D.; Matsuura, T.; Foley, H. C. Carbon-based membranes for separation processes; Springer Science & Business Media, 2011. (2) Morris, R.; Wheatley, P. Gas Storage in Nanoporous Materials. Angewandte Chemie International Edition 2008, 47, 4966–4981. (3) Czaja, A. U.; Trukhan, N.; Muller, U. Industrial applications of metal-organic frameworks. Chem. Soc. Rev. 2009, 38, 1284–1293. (4) Yuan, W.; Chen, J.; Shi, G. Nanoporous graphene materials. Materials Today 2014, 17, 77 – 85. (5) Cohen-Tanugi, D.; Grossman, J. C. Water Desalination across Nanoporous Graphene. Nano Letters 2012, 12, 3602–3608, PMID: 22668008. 27


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(44) Graham, T. A short account of experimental researches on the diffusion of gases through each other, and their separation by mechanical means. Quart. J. Sci. 1829, 2, 74–83. (45) Strong, S. E.; Eaves, J. D. Atomistic Hydrodynamics and the Dynamical Hydrophobic Effect in Porous Graphene. The Journal of Physical Chemistry Letters 2016, 7, 1907– 1912. (46) Gubbins, K.; Travis, K. Adsorption and Transport at the Nanoscale; CRC Press, 2005. (47) Beerdsen, E.; Smit, B.; Dubbeldam, D. Molecular simulation of loading dependent slow diffusion in confined systems. Physical Review Letters 2004, 93, 248301. (48) Dubbeldam, D.; Beerdsen, E.; Vlugt, T.; Smit, B. d. Molecular simulation of loadingdependent diffusion in nanoporous materials using extended dynamically corrected transition state theory. The Journal of Chemical Physics 2005, 122, 224712. (49) Bot, an, A.; Rotenberg, B.; Marry, V.; Turq, P.; Noetinger, B. Hydrodynamics in Clay Nanopores. The Journal of Physical Chemistry C 2011, 115, 16109–16115.

Supercritical fluid reservoirs

Increasing Time / Diffusion Process

Single-layer porous solid

Figure 11: For Table of Contents Only

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