Molecular Simulation of Adsorption and Transport in Hierarchical

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Molecular Simulation of Adsorption and Transport in Hierarchical Porous Materials Benoit Coasne, Anne Galarneau, Corine Gerardin, François Fajula, and Francois Villemot Langmuir, Just Accepted Manuscript • DOI: 10.1021/la401228k • Publication Date (Web): 29 May 2013 Downloaded from http://pubs.acs.org on May 31, 2013

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Molecular Simulation of Adsorption and Transport in Hierarchical Porous Materials

Benoit Coasne,1,2,3,* Anne Galarneau,1 Corine Gerardin,1 François Fajula,1 and François Villemot1

1

Institut Charles Gerhardt Montpellier, CNRS (UMR 5253), Université Montpellier 2, ENSCM, Université Montpellier 1, 8 rue de l’Ecole Normale, 34296 Montpellier Cedex 05, France.

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MultiScale Material Science for Energy and Environment, CNRS/MIT (UMI 3466), 77 Massachusetts Avenue, Cambridge, MA 02139, USA. 3

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

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To whom correspondence should be sent. E-mail: [email protected]

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Abstract. Adsorption and transport in hierarchical porous solids with micro (~ 1 nm) and mesoporosities (> 2 nm) are investigated by molecular simulation. Two models of hierarchical solids are considered: microporous materials in which mesopores are carved out (model A) and mesoporous materials in which microporous nanoparticles are inserted (model B). Adsorption isotherms for model A can be described as a linear combination of the adsorption isotherms for pure mesoporous and microporous solids. In contrast, adsorption in model B departs from adsorption in pure microporous and mesoporous solids; the inserted microporous particles act as defects, which help nucleate the liquid phase within the mesopore and shift capillary condensation towards lower pressures. As far as transport under a pressure gradient is concerned, the flux in hierarchical materials consisting of microporous solids in which mesopores are carved out obeys Navier-Stokes so that Darcy’s law is verified within the mesopore. Moreover, the flow in such materials is larger than in a single mesopore, due to the transfer between the micropores and mesopores. This non-zero velocity at the mesopore surface implies that transport in such hierarchical materials involves slippage at the mesopore surface although the adsorbate has a strong affinity for the surface. In contrast to model A, flux in model B is smaller than in a single mesopore as the nanoparticles act as constrictions which hinder transport. By a subtle effect arising from fast transport in the mesopores, the presence of mesopores increases the number of molecules in the microporosity in hierarchical materials and, hence, decreases the flow in the micropores (due to mass conservation). As a result, we do not observe faster diffusion in the micropores of hierarchical materials upon flow but slower diffusion which increases the contact time between the adsorbate and the surface of the microporosity.


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1. Introduction Owing to their large surface area and tunable pore size, microporous materials (pores < 2 nm) such as zeolites1,2,3,4 and active porous carbons5,6,7,8 are widely used in industry for applications such as phase separation, catalysis, adsorption, etc. Nevertheless, because of their very small pore sizes, their use is often limited due to low permeability to the adsorbates and restricted access to the active sites located in the microporosity. To overcome such diffusion issues, significant efforts have been devoted to design and synthesize hierarchical porous materials combining different porosity scales usually in the range of the nm to several hundreds of nm.9,10,11,12,13 These solids combine the following advantages: (1) a large surface area of “active sites” for a desired application in the smallest porosity scale and (2) a high permeability and access to these sites which arise from the efficient (less restricted) transport in the largest porosity scale. Hierarchical solids include (1) mesoporous structures whose walls are composed of microporous materials such as zeolites or Metal Organic Frameworks (MOF) or (2) as microporous materials in which mesopores are introduced.14,15,16,17,18,19 Another class of hierarchical solids is obtained by incorporating a guest microporous material as nanoparticles, which cannot be easily shaped, entrapped into the larger porosity of a host material. In so doing, the resulting solids combine the interesting properties of the “guest material” for catalysis, phase separation, adsorption applications and those of the host material which is usually chosen for its ease of use or shaping. Examples falling in this second category of hierarchical solids are Metal Organic Frameworks (MOF) or zeolite nanocrystals incorporated in mesoporous silicas with well-ordered and tunable porosity.20,21 While the benefits of combining several porosity scales have been demonstrated in various applications relevant to the fields of catalysis and phase separation, some uncertainties remain regarding the behavior of adsorbates in such hierarchical porous solids. Shedding light on the mechanisms ruling adsorption and diffusion of adsorbates in hierarchical solids would help design in a comprehensive and predictive way specific


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hierarchical porous materials with optimal properties towards a desired application. Among open questions in the field of hierarchical porous solids, the exact role played by (1) each porosity scale and (2) the connections between the two porosity networks on the structure and dynamics of the confined phase remains to be clarified. Another crucial question is to determine to which extent the two porosities can be considered as independent subsystems: i.e., can the adsorption or dynamics in a hierarchical solid be described as the sum of the data in the smallest and largest porosity scales?

In this paper, we report molecular simulations of the adsorption and transport of a simple adsorbate in hierarchical solids exhibiting interconnected microporosity (~ 1 nm) and mesoporosity (> 2 nm). To do that, two realistic models of hierarchical porous materials are built. These two systems will be referred to as model A and model B throughout this article. Model A consists of a zeolitic material in which a large mesopore is introduced while model B consists of a mesoporous silica material with a cylindrical mesopore in which a zeolite spherical nanoparticle is included. In both cases, the mesopore consists of an hydroxylated silica mesopore of a diameter D = 4.2 nm while the zeolite material is modeled upon protonated faujasite (FAU) with a Si/Al ratio equal to 15. For both types of hierarchical porous solids, the system is opened at both ends towards external bulk reservoirs which mimic the macropores that are also present in real materials (the macroporosity in between the grains of the porous material for instance). While the models reported in the present paper cannot be referred to as true multiscale materials (in the sense that the porosity scales would span over several orders of magnitude), they include at least three porosity scales: micropores, mesopores and large mesopores (which can be considered as a reasonable approximation of the macropores). While many molecular simulation studies have been reported on the adsorption and dynamics in pure microporous or pure mesoporous materials, only a very few atomistic simulation studies of such phenomena in realistic models of hierarchical


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porous materials can be found.22,23,24 We first investigate the adsorption and confinement of nitrogen at 77 K in the hierarchical porous solids using Grand Canonical Monte Carlo simulations. The choice of nitrogen at 77 K was motivated by the fact that it consists of a simple probe representative of many other simple adsorbates. In addition, low temperature nitrogen adsorption is a routine characterization technique of porous solids including hierarchical porous materials.25,26,27 Starting from well-equilibrated Monte Carlo configurations, we then investigate the dynamics and transport of nitrogen in the hierarchical nanoporous solids by means of Molecular Dynamics. Both at equilibrium dynamics and dynamics under Poiseuille flow are considered; the latter case is used to simulate operating conditions encountered in industrial applications. The molecular simulations in this paper allow gaining insights into the mechanisms ruling the behavior of adsorbates in hierarchical materials; molecular modeling offers a theoretical basis for the interpretation of experimental data while experiments can be used to verify the theoretical predictions provided by molecular simulation. Of particular importance, it must be emphasized that the interpretation of experimental adsorption and diffusion data is often ambiguous as it is unclear whether they can be analyzed using models that consider independently each porosity scale (which is implicitly assumed in most of the experiments in this field).

2. Computational Details 2.1. Atomistic Models of Hierarchical Nanoporous Materials Two models of hierarchical porous materials were built. Model A consists of a zeolitic material in which a large mesopore is introduced while model B consists of a mesoporous silica material with a cylindrical mesopore in which a zeolite spherical particle is inserted. In both cases, the mesopore consists of a hydroxylated silica pore of a diameter D = 4.2 nm while the zeolite material is modeled upon protonated faujasite (FAU) with a Si/Al ratio equal to 15 as it corresponds to a typical value for protonated FAU


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used in catalysis. For both hierarchical porous solids, the system is opened at both ends towards large external bulk reservoirs, which mimic the macropores that are present in real materials. Model A: a zeolite in which a mesopore is carved out (Figure 1). FAU is an aluminosilicate zeolite of general chemical formula Mx/m, Alx, Si192-x, O384, where M is a cation or a hydrogen atom in the case of protonated materials. m is the charge of the compensating cation. The FAU structure belongs to the Fd3m space group of symmetry with a unit cell of 24.8536 Å that contains 192 TO4 tetrahedra (with T = Si or Al). Following our previous work on aluminosilicate zeolites, 28,29 the Al atoms were placed randomly among the 192 possible T-sites but we checked that the atoms obey the Lowenstein’s rule, which states that two Al atoms cannot be connected to the same O atom.30 The charge defect induced by the substitution of Si atoms with Al atoms was compensated by adding to the closest O atom to the Al atom a H atom (protonated FAU). A FAU crystal consisting of 3 × 3 × 5 cells along the x, y, and z directions was built (the size of the supercell is thus 7.45608 nm × 7.45608 nm × 12.4268 nm). Then, the first hierarchical porous solid was obtained by carving out of the FAU crystal a cylindrical mesopore of a diameter D = 4.2 nm. In addition, we added to the system external bulk reservoirs at both ends of the material along the z direction (in order to mimic the external surface of the hierarchical porous solid). In order to ensure the electroneutrality of the simulation box, the dangling bonds of the oxygen atoms at the external surface and mesopore surface were terminated with H atoms. The latter were placed perpendicularly to the surface at a distance of 1 Å from the closest unsaturated O atom. Model B: a zeolite nanoparticle inserted in a silica mesopore (Figure 2). Immobilization of nanocrystals of microporous materials such as zeolites of MOFs by coating or encapsulation on mesoporous bodies is a very promising method for their implementation as membranes or reactors for continuous flow separation or catalytic processes (see Refs. 21 and 31 for instance). An important example illustrating the potential – in terms of stability and productivity - of encapsulated nanocrystals of a copper based


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MOF for the continuous flow liquid phase Friedlander reaction has been published recently.21 In this paper, nanocrystals of the Cu-BTC (HKUST-1) MOF have been synthesized in situ in the mesopores of macroporous silica monoliths, leading to a composite microporous/mesoporous system corresponding to that depicted by model B. In the present work, we considered a system (model B) that consists of a zeolite nanoparticle inserted in a silica mesopore. A silica mesopore of a diameter D = 4.2 nm was first prepared according to the following method. 32 The porosity of the material can be defined using a mathematical function η(x,y,z) that equals 1 if (x, y, z) belongs to the silica wall and 0 if (x, y, z) belongs to the void.33 The silica mesopore used in this work was obtained by carving out of an atomistic block of cristobalite (cristalline silica) the void corresponding to η(x,y,z) = 0. The pore is of a finite length L = 12.4268 nm and is connected to bulk reservoirs so that it mimics real materials for which the confined fluid is always in contact with the external phase or macropores (this departs from infinitely long pores for which the evaporation of the confined liquid necessarily occurs through cavitation as there is no interface with the external phase). In a second step, in order to mimic the silica surface in a realistic way, we removed the Si atoms that are in an incomplete tetrahedral environment. We then removed all oxygen atoms that are non-bonded. This procedure ensures that (i) the remaining silicon atoms have no dangling bonds and (ii) the remaining oxygen atoms have at least one saturated bond with a Si atom. The electroneutrality of the simulation box was ensured by saturating all oxygen dangling bonds with hydrogen atoms. The latter are placed in the pore void at a distance of 1 Å from the unsaturated oxygen atom along the direction perpendicular to the silica surface. Then, a spherical nanoparticle of FAU zeolite with a diameter D = 3.2 nm was inserted in the silica mesopore. The latter was placed along the y axis at a position such that z = 0 (in the middle of the pore length) and that it is in contact with the pore surface. As in the case of the first type of hierarchical porous solids, the spherical particle was modeled upon protonated FAU zeolite with Si/Al = 15. The O dangling bonds at the external surface of the


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protonated zeolite were also saturated with H atoms placed perpendicularly to the surface at a distance of 1 Å.

2.2. Intermolecular Interactions N2 was described using the model of Potoff and Siepmann.34 In this model (Table S1 of the supporting information), each nitrogen atom of the rigid nitrogen molecule is a center of repulsion/dispersion interactions which interacts through a Lennard – Jones potential with the following parameter: σ = 0.331 nm and ε = 36 K. In addition, each nitrogen atom possesses a partial charge with q = -0.482 that interacts through Coulombic forces. At the center of the nitrogen – nitrogen bond (the N – N interatomic distance is 1.1 Å), a partial charge q = +0.964 compensates the negative charge on the nitrogen atoms. As pointed out by Herdes et al.,35 such a charge distribution reproduces the measured quadrupole moment of the nitrogen molecule. Interactions between the sites on the nitrogen molecule and the Si, O, Al, and H atoms of the silica nanopore and zeolite were calculated using the PN-TraZ potential as reported for rare gas adsorption in zeolite36 or in porous silica glass.37 The intermolecular energy is written as the sum of the Coulombic and dispersion interactions with a repulsive short-range contribution. The choice of the PN-TrAZ model to describe the benzene/silica interaction was motivated by the good transferability of this model in the case of nitrogen,38 water,39 and benzene in porous silica.40 The adsorbate – surface energy Uk(rk) of the site k of an adsorbate molecule at a position rk is given by in atomic units: U k ( rk ) =

5  C 2kjn qi qk  ( ) ( ) A exp − b r − f r +  ∑  kj ∑ kj kj 2 n kj 2n r rkj  j ={O , Si , H , Al }  n = 3 kj 

(1)

where rkj is the distance between the matrix atom j (O, Si, Al, or H) and the site k of the adsorbate molecule. The first term in equation (1) is a Born–Mayer term corresponding to the short range repulsive energy due to finite compressibility of electron clouds when approaching the adsorbate to very short


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distances from the pore surface. The repulsive parameters Akj and bkj are obtained from mixing rules of like-atom pairs (see below). The second term in the above equation is a multipolar expansion series of the dispersion interaction that can be obtained from the quantum mechanical perturbation theory applied to intermolecular forces.41 The multipolar expansion series was truncated after the first dispersion term C6/r6 in the present work. It has been shown that two-body dispersion C 2kjn coefficients for isolated or incondensed phase species can be derived from the dipole polarizability α and the effective number of polarizable electrons Neff of all interacting species, which are closely related to partial charges that can be obtained from ab initio calculations. f2n are damping functions that depend on the distance rkj and the repulsive parameter bkj: 2n b r  kj kj  f 2 n (rkj ) = 1 − ∑   exp(− bkj rkj ) m = 0  m! 

(2)

The role of these damping functions is to avoid divergence of the dispersion interaction at short distances where the wave functions of the two species overlap (i.e., when the interacting species are in contact).42 Each pair of interacting species is parameterized with the single bkj repulsive parameter. The third term in equation (1) is the Coulombic interaction between the charge of site k and that of the substrate atom j. The Coulomb energy was computed using the Ewald summation technique. The atomic parameters and coefficients for the nitrogen/silica and nitrogen/zeolite interactions are given in Table S2, Table S3, and Table S4 in the supporting information file. Following previous works on zeolites, the charges on the Si, Al, and O atoms of FAU zeolites were assumed to be qSi = +2.4e, qAl = +1.4e, qO = 1.2e.43 In order to ensure electroneutrality of the zeolite, qH = +1.0e. In contrast, following previous works on adsorption in MCM-41 mesopores, the charges on the Si, O, and H atoms of the silica mesopore were assumed to be qSi = +2.0e, qO = -1.0e, and qH = +0.5e.44 We note that the latter choices imply that the zeolite is more acidic than the surface of mesoporous silica. Finally, in order to keep


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things as simple as possible, we assume that the repulsion-dispersion parameters for Si and Al atoms are identical (the latter approximation is reasonable as Al and Si have similar number of electrons and polarizabilities). The repulsive parameters for like pairs are taken from the previous work by Pellenq and Nicholson for silicon36 and from the work by Filippini and Gavezzotti 45 for oxygen, hydrogen, and nitrogen. The repulsive cross-parameters were determined using the Bohm and Ahlrichs combination rules.46

2.3. Adsorption and Diffusion of N2 in Hierarchical Porous Materials N2 adsorption at 77 K in the hierarchical porous materials corresponding to model A and model B was simulated using Grand Canonical Monte Carlo simulations (GCMC). GCMC is a stochastic method that simulates a system having a constant volume V (the pore with the adsorbed phase), in equilibrium with an infinite reservoir of particles imposing its chemical potential µ and temperature T.47 The density of nitrogen in the pores is given by the ensemble average of the number of adsorbed atoms as a function of the pressure of the gas reservoir P (the latter is obtained from the chemical potential µ according to the bulk equation of state for an ideal gas). The use of the Grand Canonical ensemble allows estimating the number of molecules adsorbed in the system in equilibrium with a fictitious reservoir of molecules that imposes its chemical potential and temperature (so that the external pressure is also fixed). This allows determining adsorption isotherms and isosteric heat of adsorption curves. Monte Carlo simulations also provide crucial information regarding the position and structure of the confined adsorbate. Among structural properties that can be calculated, density maps, pair correlation functions, orientational bondorder parameters are key data that allow characterizing the heterogeneity of adsorbed or confined molecules.


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Starting from well-equilibrated Monte Carlo configurations, the dynamics of nitrogen confined in the hierarchical porous solids was investigated using Molecular Dynamics (MD) simulations. MD simulations provide key data about the dynamics of the adsorbed or confined phase (mean square displacement, self-diffusivity, time residence distribution of the adsorbate at the surface of the host material, fluxes, etc.). As in the GCMC simulations, the Si, Al, O, and H atoms of the zeolite and mesoporous silica were not allowed to move during the course of the simulation (frozen atoms). We modeled the nitrogen/nitrogen, nitrogen/silica, and nitrogen/zeolite interactions with the same interaction potentials as those used in the GCMC simulations. The PN-TrAZ interaction potentials for dispersion-repulsion interactions in Molecular Dynamics runs were fitted with a Lennard-Jones potential that is available in DL_POLY. The energy difference between the two potentials is less than a few %. As in the GCMC simulations, the Coulomb energy in the Molecular Dynamics simulations was computed using the Ewald summation technique. The equations of motion were integrated with a timestep of 1 fs using the Verlet algorithm47 implemented in the program DLPOLY.48 The temperature was maintained constant using a Nosé-Hoover thermostat with a coupling time to the thermal bath equal to 0.1 ps. The properties and configurations of the system were stored every 1 ps. In both the GCMC and MD simulations, periodic boundary conditions were used in order to avoid finite size effect.

3. Results 3.1. Adsorption in Hierarchical Porous Materials Figure 3 shows the N2 adsorption isotherms at 77 K in the two hierarchical porous materials considered in this work: (model A) a FAU zeolite in which a mesopore of a diameter D = 4.2 nm is carved out and (model B) a mesoporous silica material with a mesopore of a diameter D = 4.2 nm in which a spherical nanoparticle of FAU zeolite is inserted. Adsorbed amounts in mmol of adsorbed nitrogen molecules per


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g of material are plotted as a function of pressure in reduced unit with respect to the bulk saturating

vapor pressure, P0. We also show in Figure 3 the N2 adsorption isotherms at77 K in FAU zeolite with a Si/Al ratio = 15 and a MCM-41 mesopore of a diameter D = 4.2 nm. These data for pure microporous and mesoporous solids serve as reference data throughout this paper as the micro and mesoporosity of the hierarchical porous solids considered in this work were modeled upon these solids. Figure S1 in the supporting information shows the same adsorption isotherms in a log-log plot so that the effect of the different geometries on the low pressure region can be observed more clearly. We recall that all the porous solids considered in this work are connected at both ends to bulk external reservoirs in order to mimic explicitly their external surface and connections to larger pores (macropores). As expected, the reference N2 adsorption isotherms for the pure FAU zeolite and pure MCM-41 mesoporous silica are characteristic of microporous and mesoporous solids, respectively. The N2 adsorbed amount in the FAU zeolite increases rapidly at low pressure upon filling of the micropores and then reaches a plateau once the porosity is completely filled. Such a process is continuous and reversible as capillary condensation is suppressed because of drastic confinement (the pores in FAU are micropores as they are roughly about ~1 nm, i.e. cages of a diameter 1.3 nm with an opening of 0.78 nm). The adsorption isotherm for MCM41 mesoporous silica conforms the classical picture of capillary condensation in mesopores. The adsorbed amount increases continuously in the multilayer adsorption regime until a jump occurs at P = 0.5P0 due to capillary condensation within the pore. At low pressures, the adsorbed amount increases rapidly with increasing pressure which reveals the strong interaction between nitrogen and the silica surface. The evaporation pressure for the MCM-41 mesoporous silica, P = 0.4P0, is lower than the condensation pressure, so that a hysteresis loop is observed. While capillary condensation consists of a metastable transition which occurs as the cylindrical adsorbed film becomes unstable, evaporation occurs through the displacement at equilibrium of a hemispherical meniscus along the pore axis.44 The


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mesoporous volume of our MCM-41 model is lower than for real MCM-41 due to the thicker wall (3.2 nm) used in the model (real materials have wall thickness about1.0 nm).

The adsorption isotherm for the hierarchical porous solid consisting of a FAU zeolite crystal in which a mesopore is carved out (model A) possesses the features of both the adsorption isotherms for the pure microporous and pure mesoporous solids (Figure 3). As shown in the molecular configuration in Figure

4(a1), at low pressures, the adsorbed amount in this hierarchical porous solid increases very rapidly with pressures as filling of the zeolite cages (micropores) occurs. As a result, at a given pressure, the adsorbed amount in this hierarchical porous solid is larger than that for the pure silica mesopore. Once the zeolite porosity is filled, the adsorbed amount for the hierarchical material keeps increasing with increasing the pressure due to adsorption at the mesopore surface (i.e. multilayer adsorption regime, see molecular configuration in Figure 4(a2)). As is observed for the pure mesoporous solid, at a pressure much lower than the bulk saturating vapor pressure, the adsorbed isotherm for model A exhibits a discontinuous and irreversible increase which corresponds to capillary condensation/evaporation within the mesopore (Figure 4(a3)). Interestingly, the capillary condensation and evaporation pressures for this hierarchical porous material are very similar to those observed for the pure mesoporous solid. This result indicates that the presence of zeolite micropores at the mesoporous surface does not lead to significant changes in the adsorbed amount corresponding to the adsorbed film. The results above suggest that the data for hierarchical porous solids falling in the category corresponding to model A can be described as a combination of data for pure microporous and pure mesoporous solids, i.e. as a mechanical mixture of zeolite and mesoporous silica. To verify this conclusion, we estimated the adsorption isotherm

P N ads   (in mol per g of sample), expected for hierarchical porous solid A, as a linear combination of  P0 


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P the adsorption isotherm for the pure microporous solid, f micro   , and that for the pure mesoporous  P0  P solid, f meso   :  P0  P  P  P  N ads   = Vmicro f micro   + Vmeso f meso   ρ  P0    P0   P0 

(3)

where ρ = 0.0289 mol/cm3 is the bulk number density of nitrogen at 77 K and the weighing parameters Vmicro and Vmeso are the microporous and mesoporous volumes expressed in cm3 per g of sample of

P P sample, respectively. f micro   and f meso   , which are the reference adsorption isotherms for the  P0   P0  microporous and mesoporous solids, are expressed as a fraction of the porous volume being filled at a reduced pressure

P (i.e. N/N0 where N0 is the maximum adsorbed amount). Vmicro and Vmeso for the P0

model A can be easily determined as Vmicro = Lz(LxLy – πR2)Φmicro/M and Vmeso = ΦmesoπR2Lz/M where M is the sample mass, Lz and R the mesopore length and radius while Lx and Ly are the sizes of the hierarchical sample along the x and y directions perpendicular to the mesopore axis. Φmicro and Φmeso are the porosities of microporous and mesoporous phases, respectively. Φmicro = 0.48 for Faujasite (the latter value can be estimated from the amount of nitrogen molecules in a pure zeolite sample when saturated with the adsorbate) and Φmeso = 1 for MCM-41 (MCM-41 pores in a hierarchical porous material are 100% empty so that the porosity is 1). Vmicro and Vmeso, which correpond to the microporous and mesoporous volumes in the hierarchical porous solid, must not be confused with the volumes occupied by the microporous and mesoporous phases in the sample (in fact, these phase volumes are readily obtained by dividing Vmicro or Vmeso by the corresponding porosity, Φmicro or Φmeso). To make the


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comparison as accurate as possible, we corrected the adsorbed amounts for the adsorption on the external surface of the hierarchical porous solid and of the pure microporous and mesoporous solids; only data within the microporosity and/or mesoporosity, i.e. for molecules located at a position such that |z| < 4 nm, were taken into account in order to avoid any effect related to the presence of the external surface which is necessarily overestimated given the size of our samples (the pore length is about ~7 nm).

Figure 5 compares the simulated adsorption isotherm for model A with the linear combination obtained using Eq. (3). The linear combination is in very good agreement with the data for the hierarchical porous solid corresponding to model A. It should be emphasized that no fitting parameter was needed to obtain such a good agreement. This result shows that it is reasonable to assume that fillings of the microporosity and mesoporosity are independent in hierarchical solids falling in the category described by model A. The fact that the adsorption isotherm for model A can be described as a linear combination lies in the fact that the microporosity gets filled at a pressure much lower than the mesoporosity. In other words, capillary condensation in the smaller mesopores must occur at a pressure where all the micropores are already filled to have independent pore filling of the different porosity scales. Moreover, the fact that the capillary condensation pressure in the mesopore with microporous walls is close to that for the regular mesopore indicates that the interaction field felt by a molecule in the mesopore with micropores is equivalent to that for the regular mesopore. This is confirmed by the fact that the isosteric heats of adsorption for model A and for the regular MCM-41 mesopore are very similar at the onset of capillary condensation (about 15 kJ/mol, see the discussion below). In particular, this result shows that the difference in terms of adsorption energy between adsorption on hydroxylated silica surface (pure mesoporous solid) and adsorption on a hybrid surface (made up of a hydroxylated silica surface and zeolite cavities filled with nitrogen molecules) does not affect the adsorption of a film and capillary


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condensation/evaporation within mesopores.We note that the reasoning above would not be valid if the fluid molecule did not adsorb in small pores such as water in narrow hydrophobic carbons.

In contrast to the results above, the shape of the adsorption isotherm in Figure 3 for the hierarchical porous solid consisting of a FAU zeolite particle inserted in the porosity of a single mesopore (model B) drastically departs from what is expected on the basis of adsorption isotherms for pure microporous and mesoporous solids. As shown in Figure 4(b1), adsorption in this hierarchical porous material starts with the adsorption in the zeolite particle and at the mesopore surface at low pressures. Given its small volume, adsorption in the zeolite particle increases only slightly the adsorbed amout at a given pressure compared to the adsorbed amount for a regular MCM-41 pore having the same pore diameter. However, the adsorption around the zeolite nanoparticle increases slightly the adsorbed amount at a given pressure compared to the adsorbed amount predicted from the linear combination of the two porosities. The formation of a film around the nanoparticles once the zeolite is filled (see Figure 4 b2) leads to a higher adsorbed amount in the low pressure range (0.4 < P/P0), which may lead to an overestimation of the surface area and microporous volume of the hierarchical samples if not properly corrected. Moreover, the presence of the inserted particle drastically affects capillary condensation in the main mesopore of the hierarchical porous material model B. Such a particle, which acts as a defect, helps nucleate the liquid phase within the mesopore at a pressure much lower than capillary condensation for the regular mesopore without particle (Figure 4(b2)). This prevents the system from being trapped in a metastable state so that capillary condensation is triggered at the equilibrium pressure which corresponds to the evaporation pressure observed for the regular mesopore. Consequently, both capillary condensation and evaporation occur at the same pressure so that no hysteresis loop is observed for this hierarchical porous material (Figure 4(b3)). Consequently, the adsorption isotherm for model B cannot be described as a linear combination of the adsorption isotherms for the pure microporous and mesoporous solids (i.e.,


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Eq. (3)). Only the total pore volume and pore diameter from the desorption branch can be determined accurately.

The results above are important as they show that characterization of hierarchical porous materials based on data for reference solids can only be achieved for a given type of materials. In particular, this implies that the type of porous material (i.e. the type of hierarchy involved) must be known independently prior to any attempt to characterize it. For instance, our results show that the hierarchy factor (HF),49,50 i.e. the fraction of mesoporous surface multiplied by the fraction of microporous volume (Smeso/S ×Vmicro/V), which has been proposed to characterize hierarchical porous materials, must be used with caution. Indeed, since both the surface and volume measurements are affected by the type of hierarchy, the values estimated from such methods can be erroneous and misleading. Moreover, while the hierarchy factor constitutes a useful order parameter to describe true hierarchical porous materials as it describes continuously the change from a pure microporous or mesoporous material (HF = 0) to a hierarchical porous material (HF ≠ 0), it cannot be used to distinguish true hierarchical porous materials with interconnected micro and mesoporosity from a powder consisting of a mechanical mixture of a microporous and mesoporous solids.

In order to gain deeper insights into the adsorption in hierarchical porous materials, Figure 6 shows the isosteric heat of adsorption, Qst, for N2 adsorbed in the two types of hierarchical porous solids. We also report the data for reference solids which consist of a MCM-41 mesoporous material with a mesopore of a diameter D = 4.2 nm and a protonated FAU zeolite. Qst for the reference solids are typical of pure mesoporous and microporous solids. For the zeolite, Qst ~20 kJ/mol at low loading (when strongly adsorbing sites are being filled) and then decreases in a continuous way to ~12 kJ/mol when the


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microporosity is filled. The latter value is larger than the heat of liquefaction of nitrogen at 77 K (7 kJ/mol) as molecules within the microporosity of the zeolite always feels the interaction potential with the host material. Beyond 15 mmol/g, Qst for the zeolite decreases down to the heat of liquefaction of nitrogen as further adsorption takes place on the external surface of the material where an adsorbed film is already formed. For the MCM-41 pore, Qst is characteristic of adsorption of simple gases on heterogeneous surfaces; Qst ~ 18 kJ/mol at low loading (again when strongly adsorbing sites are being filled) and then decreases in a continuous way to a value close to the heat of liquefaction of nitrogen (7 kJ/mol) as further adsorption takes place. In contrast to the zeolite, due to its large size, Qst for the MCM-41 material decreases down to the heat of liquefaction of nitrogen because adsorbed molecules in the pore center do not feel the interaction potential when the pore gets filled. Interestingly, when plotted as a function of the adsorbed amount, Qst for the hierarchical porous materials possess the features of the reference data for pure microporous and mesoporous solids. The isosteric heat of adsorption for model A follows the data for the microporous solid up to ~10 mmol/g and then decreases to ~6-8 kJ/mol to follow the data for the MCM-41 mesopore for larger adsorbed amounts. Similarly, the isosteric heat of adsorption for model B first follows the data for the mesoporous solid up to ~1 mmol/g and then follows the values for the microporous solid for adsorbed amounts in between 1 and 3 mmol/g. We note that the range of adsorbed amounts for which Qst for model B follows the reference data for zeolite is small as the size of the zeolite particle is very small. Finally, Qst follows the isosteric heat of adsorption for the regular MCM-41 mesopore as condensation in the main mesopore of model B occurs (> 6 mmol/g). The results above show that calorimetry measurements combined with adsorption experiments can be very helpful to characterize hierarchical porous materials (while, as shown above, adsorption measurements may not be sufficient depending on the type of hierarchical porous materials).


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3.2. Dynamics and Transport in Hierarchical Porous Materials We first determined the self-diffusivity D of nitrogen confined at 77 K and a pressure P0 (the bulk saturating vapor pressure) in the two models of hierarchical porous solids. The latter were estimated by integrating the velocity autocorrelation function v(t )v(0) : t

1 D =lim ∫ v(t )v(0) dt t →∞ 3 0

(4)

where v(t ) is the molecule velocity at a time t and ... denotes an average over all the molecules.

Figure 7 shows v(t )v(0) and D(t) as a function of the upper boundary t of the integral in Eq. (4) for the two hierarchical porous materials. Data for each material have been separated into a contribution coming from nitrogen in the microporosity (dashed lines) and a contribution coming from nitrogen molecules in the mesoporosity (solid lines). For all hierarchical materials, calculations were restricted to molecules located in the region z < 4 nm (z is the mesopore axis) in order to avoid possible effects arising from the external surface (given the small diameter to length ratio in our materials, the latter effects would necessarily be overestimated compared to real materials). For the hierarchical material A, we considered that a molecule is located in the microporosity of the sample if its distance r to the mesopore axis is such that r > 2.3 nm (otherwise the molecule is considered as being located in the mesoporosity). For model B, we considered that a molecule is located in the mesoporosity unless it its distance to the particle center is less than 2.4 nm. While these criteria are somewhat arbitrary, they allow distinguishing molecules in different regions of the hierarchical materials and using different criteria would not change the qualitative conclusions below. While in theory D is obtained for t → ∞, Figure 7 shows that the integrals in Eq. (4) converge for t > 20 ps. We also show in Figure 7 the data for the pure microporous (zeolite) and mesoporous (a single MCM-41 mesopore) solids and for bulk nitrogen. As


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expected, due to confinement and the strong interaction with the surface of the host materials, the selfdiffusivity for confined nitrogen is always smaller than that for bulk nitrogen

(~3.1×10-9 m2/s).

Moreover, for a given material, the self-diffusivity in the microporosity (D < 10-9 m2/s) is much smaller than in the mesoporosity (D ~ 1.5 – 1.6×10-9 m2/s). For both hierarchical porous materials, nitrogen confined in the micropores has a self-diffusivity (< 10-9 m2/s) close to that for a pure microporous solid of the same nature (zeolite). Similarly, nitrogen in the mesoporosity of the hierarchical materials has a self-diffusivity that is close to that for the regular MCM-41 mesopore (although a slightly higher selfdiffusivity is observed in mesopores of model A which can be explained by the lower attraction of the zeolite walls in comparison to hydroxylated MCM-41 surface). We note that the self-diffusivity in the microporosity of model B is slightly larger than in the microporosity of model A; this result is due to the fact that nitrogen molecules in the small zeolite particle (model B) are less confined as they are close to its external surface so that their dynamics tends to be faster.

In addition to calculating the self-diffusivity D, which provides information about the translational dynamics for confined nitrogen, we also looked at the rotational dynamics of nitrogen confined in the micro and mesoporosity of hierarchical porous materials. Figure 8 shows the orientational pair correlation function C(t) = where θ(t) is the angle between the nitrogen molecular axis and a reference (arbitrary) axis and ... denotes an average over all the molecules in the system. C(t) provides information about the typical time needed for a molecule to relax its orientation. As in the case of the self-diffusivity calculations above, we calculated both C(t) for nitrogen in the microporosity (dashed lines) and for nitrogen in the mesoporosity (solid lines) of the hierarchical porous materials. Again, for all hierarchical materials, calculations were restricted to molecules located in the region z < 4 nm to avoid possible effects arising from the external surface. Figure 8 also shows the data for the


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pure microporous (zeolite) and mesoporous (a single MCM-41 mesopore) solids and for bulk nitrogen. All C(t) correlation functions were fitted in the short time range against a simple decaying function in order to estimate the typical relaxation time τ:

t C (t ) ∝ exp  τ 

(5)

In agreement with previous works on benzene confined in nanopores, 51 the rotational dynamics for confined nitrogen is slower than for bulk nitrogen (τbulk ~ 0.4 ps) due to confinement and the strong interaction with the surface of the host materials. For a given material, the relaxation time in the microporosity, τmicro ~ 3-4 ps, is longer than in the mesoporosity, τmeso ~ 1-2 ps. For both hierarchical porous materials, nitrogen molecules confined in micropores (mesopores) have a rotational relaxation time close to that for a pure microporous solid (a single MCM-41 mesopore).

The results above provide insights about the single dynamics of molecules confined in the microporosity and mesoporosity of hierarchical porous materials. On the other hand, they do not provide information about the collective dynamics and transport properties of adsorbates in this type of materials. To further investigate the dynamics of nitrogen confined in hierarchical porous materials, we report in Table 1 the fluxes in mol/cm2/s between micropores, mesopores, and macropores in models A and B. These data were estimated from at equilibrium dynamics so that the fluxes do not correspond to transport, i.e. mass transfer, between different porosity scales (they rather provide information about the transfer efficiency between two porosity scales). Given the uncertainty about how one should define the true surface area between microporosity and mesoporosity in the different porous materials under study, the flux from mesopores to micropores can be considered similar for all materials. This result was expected based on the data for the self-dynamics which were discussed above. Similarly, for all materials, the flux from


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macropores to mesopores does not depend on the type of materials being considered. Due to the fact that the self-dynamics becomes faster as molecules get confined in larger pores, the incoming flux from macropores is larger than that from mesopores. The results above have important implications for practical applications as they suggest that enhanced diffusion in hierarchical materials containing mesopores does not arise from an intrinsic faster diffusion from mesopores to micropores (in contrast, flux coming from the mesopores is slower than from the macropores) but from the large surface area generated upon creating mesoporosity in the material. In other words, adding mesoporosity to a microporous material does not make the overall diffusion faster but improves the efficiency of a given process (separation, catalysis, etc.) by increasing the amount of adsorbate accessing the microporosity per unit of time.

Table 1. Fluxes in mol/cm2/s between micropores, mesopores, and macropores for a microporous solid (protonated FAU zeolite), a mesoporous solid (a single MCM-41 mesopore of a diameter D = 4.2 nm), and two hierarchical porous solids (models A and B).

Material

Meso → Micro

Macro → Micro

Macro → Meso

Zeolite FAU

---

44.2

---

MCM-41

---

---

84.1

Hierarchical A

33.9

46.7

72.5

Hierarchical B

51.6

---

88.8

We also simulated transport in the two hierarchical porous materials using non-equilibrium Molecular Dynamics (NEMD). In NEMD, one simulates the collective dynamics of confined molecules by


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calculating the Poiseuille hydrodynamics flow resulting from an external force f.38,52 The latter external force that mimics the influence of a chemical potential gradient or a pressure is added to the force that acts on each molecule in the system. The system responds to this external perturbation by developing a flux. The steady-state is reached when the temperature and energy fluctuate around their mean value and that the flux and velocity profile remain constant over time. In these non equilibrium Molecular Dynamics simulations, a Berendsen thermostat with a timestep of 10 fs is used to ensure that the temperature of the system remains constant and equal on average to room temperature. In this work, we imposed a pressure gradient constant force f =

∇P along the mesopore axis which we modeled as a homogeneous

− ∇P . The hydrodynamic flow can be described using the Navier-Stokes equation

for incompressible newtonian fluids:

  ∂v + v ⋅ ∇v  = −∇P + η∇ 2 v + f grav  ∂t 

ρ

(6)

where v is the flow velocity, ρ the density, η the dynamic viscosity of nitrogen at 77 K, and fgrav = ρg

the gravitational force (g is the gravitational field).

∂v + v ⋅ ∇v is the fluid acceleration in which ∂t

v ⋅ ∇v is the convective term. In what follows, we neglect the effect of the gravitational force on molecules, which is necessarily very small at the microscopic scale compared to intermolecular forces such as dispersion and electrostatic interactions. We also assume that convection in nanopores cannot occur as the small pore size prevents a velocity gradient to appear. Under these assumptions and considering a cylindrical geometrical (

obtained for

∂v z   = 0 ), the stationary solution of Eq. (6), which is ∂r  r =0

∂v = 0 , is simply given by: ∂t


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vz (r ) =

∇zP 2 r +C 4η

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(7)

where C is a constant that can be determined from the boundary conditions at r = R0 (i.e., the pore radius). Moreover, given that the velocity is maximum in the pore center r = 0,

v z (0) = v zmax = C .

The velocity profiles obtained for nitrogen in the two hierarchical porous materials are shown in Figure

9. We also report the velocity profile for nitrogen in the single MCM-41 mesopore of a diameter D = 4.2 nm and a Faujasite zeolite. Instead of a Poiseuille flow, we observe that the flow in the pure microporous

sample is constant throughout the system. The latter result shows that the concept of Poiseuille flow and the related Darcy’s law, which rely on Navier-Stokes equation, does not apply for small pores such as micropores in zeolites. Interestingly, the flux for the pure microporous solid is larger than the flux in the microporosity of model A. This result is due to the fact that, because of the larg flow in the mesopore of model A, molecules in this region have larger velocities than in the microporosity. Consequently, in order to ensure mass conservation (ρv is conserved throughout the system), the density of molecules in the mesopore is smaller. On the other hand, global mass conservation imposes that the smaller density of molecules in the mesopore of model A is compensated by an increased density in the microporosity and hence a lower velocity (again because ρv is conserved throughout the system). As a result, the velocity in the micropores of model A is lower than in a pure microporous solid having the same type of microporosity. In contrast to the flux for the pure microporous solid, the velocity profiles for the single mesopore and for model A of hierachical materials obey the Navier-Stokes equation given in Eq. (9); the latter equation in which we impose the pressure gradient considered in our simulations quantitatively describes the velocity profiles observed for these materials. The viscosity found by adjusting the two sets of data against Eq. (9) is equal to 1.42×10−4 Pa.s, which is only 10% smaller than the experimental value for N2 at 77 K (1.58×10−4 Pa.s). Given that the model used in this work to describe nitrogen was


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not developed to reproduce its dynamic properties such as dynamic viscosity, the agreement between the simulated data and the viscosity found by adjusting Eq. (9) can be considered as very good. The slightly larger flow for the hierarchical porous material (model A) compared to that for the single mesopore material arises from the molecule transfer between the micropores and mesopores; molecules at the mesopore surface have a non zero velocity which contributes to enhance the flow within the mesopore. In contrast, due to the strong interaction with silica, molecules at the pore surface in the single mesopore have a zero velocity (sticky boundary conditions) so that the flux in this material is smaller. These results, together with the data in Figure 9, show that the flow and flux in hierarchical materials falling in the family of model A are larger than in a single mesopore due to the lower interaction of adsorbates with the surface made up of zeolite (in contrast to dense silica surfaces). This provides a theoretical frame to explain the fact that hierachical porous materials are very efficient for practical applications involved in catalysis, phase separation, etc. Interestingly, while the data for the single mesopore corresponds to sticky boundary conditions, the non-zero velocity at the mesopore surface for model A, which could be misinterpreted as an indication that the adsorbate does not wet the porous material, can be analyzed in terms of slippage with a slip length b. b is defined as the position from the mesopore

surface where the extrapolated velocity becomes zero, i.e.

dv v 53 = . We found b = 0.2-0.4 nm for dr b

model A. While the latter value is small compared to very hydrophobic materials such as carbon nanotubes,54,55,56 it is comparable to what was obtained for dehydroxylated (hydrophobic) silica.38 This result is important as even very small slip lengths can lead to very large flux enhancements, in agreement with the larger flow observed for model A compared to that for the single MCM-41 mesopore. In order to estimate the effect of such a slip length on the overall flux J, we integrate the flow over the mesopore surface v z (r )


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R0

J = ∫ 2πρv z ( r )dr

(8)

0

where ρ = 0.808 g/cm3 is the density of liquid nitrogen at 77 K and A = πR02 is the circular section of the mesopore surface. The calculated fluxes are 4.3×102 mol/cm2/s for the single MCM-41 mesopore and 5.2×102 mol/cm2/s for the mesopore in the hierachical mesoporous material (model A). Such a ~20 % flux enhancement confirms the important effect of slippage on transport in porous materials (in contrast to the results for the microscopic flux between micropores and mesopores discussed earlier, here the flux difference between model A and the MCM-41 pore is relevant as these two models were generated to exhibit the same microporosity/mesoporosity surface area). In contrast to the results obtained for model A, the velocity profile for model B shows that inserting a microporous nanoparticle in the mesoporosity of hierarchical porous samples has a detrimental effect (Figure 9). Indeed, as in the case of adsorption, the inserted particle acts as a constriction which obstructs transport within the main mesopore.57 This is confirmed by the fact that the velocity profile for this material is asymetrical with respect to r = 0; the particle being located at a position along the y axis in the region x < 0 the flow in this region r < 0 is smaller than for r > 0. The latter result shows that, although hierarchical porous materials of type B tend to be efficient towards a given application as they allow enhanced access to their microporosity, they might lead to reduced fluxes if the particles become too large and, hence, block the porosity of the material.

4. Conclusion


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This paper reports a molecular simulation study of the adsorption and transport of a simple adsorbate in hierarchical porous solids exhibiting interconnected microporosity (~ 1 nm) and mesoporosity (> 2 nm). Two types of hierarchical porous materials were considered as they correspond to porous adsorbents being envisaged for practical industrial applications. Model A consists of a zeolitic material in which a mesopore is introduced while model B consists of a silica mesoporous material with a mesopore in which a zeolite particle is inserted. For both types of hierarchical porous solids, the system is opened at both ends towards large external bulk reservoirs which mimic the macropores that are also present in real materials (the macroporosity located in between the grains of the porous materials for instance). Our findings are both of practical and fundamental interest as they show that hierarchy in porous materials, depending on its nature, can affect adsorption and transport of adsorbates in such materials.

Adsorption isotherms for hierarchical porous solids consisting of a microporous material in which a mesopore is carved out (model A) exhibit the features of both the adsorption isotherms for pure microporous and pure mesoporous solids. At low pressures, the adsorbed amount increases very rapidly with pressures as filling of the micropores occurs. Then, the adsorbed amount keeps increasing with increasing the pressure due to adsorption at the mesopore surface (i.e. multilayer adsorption regime). Moreover,

adsorption

isotherms

for

such

hierarchical

porous

solids

exhibit

capillary

condensation/evaporation within the mesopore. We have further shown in this paper that the adsorption isotherms for hierarchical porous solids consisting of microporous solids in which mesopores have been carved out can be described using a linear combination of the adsorption isotherms for the pure mesoporous solid and pure microporous solid. In contrast, we have found that adsorption in hierarchical porous solids consisting of microporous particles inserted in a mesoporous sample (model B) drastically departs from what is expected on the basis of adsorption isotherms for pure microporous and


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mesoporous solids. The inserted particles act as defects, which help nucleate the liquid phase within the mesopore, shift capillary condensation; depending on the particle size, the hysteresis loop can be suppressed as nucleation of the liquid phase (thanks to the particle) prevents the system from being trapped in a metastable state.

The latter findings are of practical interest for characterization purpose and industrial applications as they help determine accurately the amount of microporous and mesoporous phases in a given sample (by using the linear combination of two reference adsorption isotherms corresponding to the microporous and mesoporous phases). On the other hand, the latter results show the drawback of the hierarchy factor,49,50 i.e. the fraction of mesoporous surface multiplied by the fraction of mesoporous volume (Smeso/S ×Vmicro/V), which has been proposed to characterize hierarchical porous materials. The hierarchy factor (HF) constitutes a useful order parameter to describe hierarchically porous materials as it changes continuously from a pure microporous or mesoporous material HF = 0 to a hierarchical material HF ≠ 0. However, since both the surface and volume measurements as determined by gas adsorption are affected by the type of hierarchy involved, the values estimated from such methods can be erroneous and misleading. Moreover, the hierarchy factor does not allow distinguishing true hierarchical porous materials with interconnected micro and mesoporosity from a powder consisting of a mechanical mixture of a microporous phase and a mesoporous phase.

In this work, we also investigated the dynamics of molecules confined in hierarchical porous materials. For the two types of hierarchical porous materials, both the translational and rotational dynamics of confined molecules are slower than the bulk. Moreover, as expected, the dynamics of molecules located in the microporosity is much slower than in the mesoporosity. Our results suggest that the dynamics of


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molecules in hierarchical porous materials is similar to that in pure microporous and mesoporous solids as their rotational and translational dynamics only depend on their location (i.e. in the micro or mesoporosity) and are similar to that for pure microporous or mesoporous solids. We also looked at the collective dynamics and transport of adsorbates in the two types of hierarchical materials. We have shown that adding mesoporosity in a hierarchical material does not seem to make the overall diffusion faster but improves the efficiency of a given process (separation, catalysis, etc.) by increasing the amount of material accessing the microporosity per unit of time. Interestingly, we found that the type of hierarchical materials considered significantly affect the resulting molecular flow and flux. By simulating a Poiseuille flow, we oberved that the flow for hierarchical porous materials consisting of mesopores carved out of microporous solids obeys Navier-Stokes (i.e. Darcy’s law). Moreover, the flow for such hierarchical samples is larger than for a single mesopore, due to the important transfer between the micropores and mesopores. This result is important as it shows that the presence of microporous walls leads to important flux enhancements (20% for the specific materials considered in this work). From a fundamental point of view, the latter result shows that the transfer of molecules from the microporosity to the mesoporosity leads to a non zero velocity at the mesopore surface. In turn, such a departure from the usual sticky boundary conditions (which are assumed to hold in most applications dealing with transport in porous materials except for water in hydrophobic materials) has deep implications; this suggests that transport in hierarchical porous materials can involve slippage at the mesopore surface although the latter has strong affinity for the adsorbate. In contrast to the results obtained for model A, flux for hierarchical materials corresponding to model B is smaller than for a single mesopore as the inserted microporous particles act as constrictions which obstruct transport. Furthermore, we have shown that, in hierarchical materials used in flow processes, by a subtle effect related to the fast transport in the mesopores, the presence of mesopores increases the number of


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molecules present in the microporosity and, hence, decreases the flow in the micropores (due to mass conservation). In contrast to what is often suggested in the literature, we do not observe faster diffusion in the micropores of hierarchical materials in flow processes but slower diffusion which increases the contact time between the confined adsorbate and the surface of the microporosity.

Supporting Information Available: Interaction parameters. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgments: We thank Julien Nigon for his help and stimulating discussion.

References.

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(49) Verboekend, D.; Groen, J. C.; Perez-Ramirez, J. Interplay of Properties and Functions upon Introduction of Mesoporosity in ITQ-4 Zeolite. Adv. Funct. Mater. 2010, 20, 1441. (50) Perez-Ramirez, J.; Verboekend, D.; Bonilla, A.; Abello, S. Zeolite Catalysts with Tunable Hierarchy Factor by Pore-Growth Moderators. Adv. Funct. Mater. 2009, 19, 3972. ( 51 ) Coasne, B.; Fourkas, J.T. Structure and Dynamics of Benzene Confined in Silica Nanopores. J. Phys. Chem. C 2011, 115, 15471–15479. (52) Arya, G.; Chang, H.C.; Maginn, E. J. A critical comparison of equilibrium, non-equilibrium and boundary-driven molecular dynamics techniques for studying transport in microporous materials. J. Chem. Phys. 2001, 115, 8112–8124. (53) Cottin-Bizonne, C.; Barrat, J. L.; Bocquet, L.; Charlaix, E. Low-friction flows of liquid at nanopatterned interfaces. Nature Mater. 2003, 2, 237. (54) Holt, J. K.; Park, H. G.; Wang, Y.; Stadermann, M.; Artyukhin, A. B.; Grigoropoulos, C. P.; Noy, A.; Bakajin, O. Fast mass transport through sub-2-nanometer carbon nanotubes. Science

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Figure 1. (color online) Plane and transverse views of a hierarchical porous solid consisting of a cylindrical mesopore D = 4.2 nm carved out of protonated FAU with a Si/Al ratio = 15 (model A). The black segments indicate the simulation box boundaries. The hierarchical porous solid is connected at both ends to bulk external reservoirs which mimic the macroporosity present in real hierarchical materials. The orange, red, and blue spheres are the silicon, oxygen, and aluminum atoms of the FAU zeolite. The white spheres are the hydrogen atoms that saturate the O dangling bonds at the mesopore surface or compensate the Si/Al substitution in the zeolite framework (in both cases, the H atom is connected to a O atom of the hierarchical porous solid).


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Figure 2. (color online) Plane and transverse views of a hierarchical nanoporous solid consisting of a cylindrical mesopore D = 4.2 nm (carved out of bulk silica) which hosts a zeolite spherical particle having a diameter D = 3.2 nm (model B). The zeolite particle was cut out of protonated FAU with a Si/Al ratio = 15. The black segments indicate the simulation box boundaries. The hierarchical porous solid is connected at both ends to bulk external reservoirs which mimic the macroporosity present in real hierarchical materials. The orange, red, and blue spheres are the silicon, oxygen, and aluminum atoms of the FAU zeolite. The white spheres are the hydrogen atoms that saturate the O dangling bonds at the mesopore surface or compensate the Si/Al substitution in the zeolite framework (in both cases, the H atom is connected to a O atom of the hierarchical porous solid).


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24.0

Nads (mmol/g)

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

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18.0

12.0

6.0

0.0 0.0

0.2

0.4

0.6

0.8

P/P0 Figure 3. (color online) (top) N2 adsorption isotherm at 77 K in hierachical porous solids: (red squares) model A, a cylindrical mesopore of a diameter D = 4.2 nm carved out of FAU zeolite with a Si/Al ratio = 15 and (blue squares) model B, a MCM-41 mesopore of a diameter D = 4.2 nm which hosts a FAU spherical particle (Si/Al = 15) of a diameter D = 3.2 nm. The black symbols are N2 adsorption isotherms at 77 K for pure microporous and mesoporous solids: (squares) FAU zeolite with a Si/Al ratio = 15 and (circles) a MCM-41 mesopore of a diameter D = 4.2 nm. (bottom) same data as in top but with a log-log scale.


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(a1)

(b1)

(a2)

(b2)

(a3)

(b3)

Figure 4. (color online) Typical molecular configurations for N2 adsorbed at 77 K and different relative pressures in two different hierarchical porous solids: (a) model A and (b) model B. The pressure from top to bottom is 10-5 P0, 0.34 P0, 0.69 P0 (P0 is the bulk saturating vapor pressure for N2 at 77 K). The blue spheres are the N atoms of the N2 adsorbed molecules.


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6000.0

4000.0

Nads

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

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2000.0

0.0 0.0

0.2

0.4

0.6

0.8

1.0

P/P0 Figure 5. (color online) N2 adsorption isotherms at 77 K in hierachical porous solids: (red squares) model A, a cylindrical mesopore of a diameter D = 4.2 nm carved out of FAU zeolite with a Si/Al ratio = 15 and (blue squares) model B, a MCM-41 mesopore of a diameter D = 4.2 nm which hosts a FAU spherical particle (Si/Al = 15) of a diameter D = 3.2 nm. For each hierarchical porous solid, the line represents the adsorption isotherm which would be expected if adsorption in the microporosity and adsorption in the mesoporosity were independent; i.e. a linear combination of the adsorbed amounts in the microporosity and in the mesoporosity (see text and Eq. (3)).


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24.0

Qst (kJ/mol)

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18.0 12.0 6.0 0.0 0.0

6.0

12.0

18.0

24.0

Nads (mmol/g) Figure 6. (color online) Isosteric heat of adsorption for N2 at 77 K as a function of the adsorbed amount for hierarchical porous solids: (red squares) model A, a cylindrical mesopore of a diameter D = 4.2 nm carved out of FAU zeolite with a Si/Al ratio = 15 and (blue squares) model B, a MCM-41 mesopore of a diameter D = 4.2 nm which hosts a FAU spherical particle (Si/Al = 15) of a diameter D = 3.2 nm. The black squares and circles correspond to the isosteric heat of adsorption for N2 at 77 K in protonated FAU and in a single MCM-41 mesopore of a diameter D = 4.2 nm. The latter serve as reference data to compare/analyze the data for hierarchical porous solids.


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4.00

D (10-9 m2/s)

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3.00

8 3 -2 0

2.00

1 t (ps)

2

30

40

1.00 0.00 0

10

20

t (ps) Figure 7. (color online) Running value of the self-diffusivity for N2 at 77 K in hierachical porous solids as determined from the integral over time t of the velocity autocorrelation function (see text): (red data) model A and (blue data) model B. For each hierarchical solid, the solid and dashed lines are for N2 in the mesoporosity and microporosity of the solid, respectively. The black line is the data for bulk N2 at 77 K while the dashed and solid grey lines correspond to the data for N2 in protonated FAU and in a single MCM-41 mesopore of a diameter D = 4.2 nm, respectively. The latter serve as reference data to compare/analyze the data for hierarchical porous solids. The insert shows the velocity autocorrelation functions which were integrated from 0 to t to determine the running value of the self-diffusivity shown in the main figure. The legend is the same as in the main figure.


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1.0 0.8

C(t)

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0.6 0.4 0.2 0.0 0

2

4

6

8

t (ps) Figure 8. (color online) Orientational pair correlation function C(t) = for N2 at 77 K in hierachical porous solids: (red data) model A and (blue data) model B. For each hierarchical solid, the solid and dashed lines are for N2 in the mesoporosity and microporosity of the solid, respectively. The black line is the data for bulk N2 at 77 K while the dashed and solid grey lines correspond to the data for N2 in protonated FAU and a single MCM-41 mesopore of a diameter D = 4.2 nm, respectively. The latter serve as reference data to compare/analyze the data for hierarchical porous solids.


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0.4 Fits using Poiseuille flow

vz (nm/ps)

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

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0.3

0.2

0.1

0.0 -3.2

-1.6

0.0

1.6

3.2

r (nm) Figure 9. (color online) Velocity vz(r) for N2 confined in hierarchical porous solids: (red circles) model A and (blue circles) model B. The black squares are the data for a single silica cylindrical mesopore MCM-41 of a diameter D = 4.2 nm. The dashed line is the average flow for a zeolite sample. The black and red lines are fits using Poiseuille flow derived in the frame of Navier-Stokes equation (see Eq. (9)).


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TOC

Molecular Simulation of Adsorption and Transport in Hierarchical Porous Materials

Benoit Coasne, Anne Galarneau, Corine Gerardin, François Fajula, and François Villemot

Molecular simulation of adsorption and transport in hierarchical porous materials consisting of mesoporous materials with microporous walls (or mesopores carved out of a microporous material) or microporous particles inserted in a mesoporous material.


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Molecular Simulation of Adsorption and Transport in Hierarchical

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