Understanding Adsorption and Transport of Light Gases in

Article pubs.acs.org/JPCC

Understanding Adsorption and Transport of Light Gases in Hierarchical Materials Using Molecular Simulation and Effective Medium Theory Mauricio Rincon Bonilla,† Rustem Valiullin,‡ Jörg Kar̈ ger,*,‡ and Suresh K. Bhatia*,† †

School of Chemical Engineering, University of Queensland, Brisbane, QLD 4072, Australia Faculty of Physics and Earth Science, University of Leipzig, Linnéstr. 5, D-04103 Leipzig, Germany

‡

ABSTRACT: Ordered structures such as zeolites play an important role in a myriad of technological applications, from catalysis to adsorptive separations. Moreover, in recent years, a new class of zeolites with a “hierarchical” structure has been the subject of intense research, due to its potential to dramatically improve yield in those processes where the critical dimensions of the pore and the fluid are similar. In these hierarchical structures, a mesoporous network is purposefully embedded into the microporous framework; the presence of mesopores in the interior of microporous particles may significantly improve their transport properties. We analyze here our recent experiments on mass transfer in hierarchical CHA-like zeolite, microporous SAPO-34, having embedded mesopores as transport pathways providing access to intracrystalline micropores. We use effective medium theory (EMT), where no topological configuration of the phases is assumed other than they mix in an isotropic manner. Our results suggests that, while hierarchical materials yield significant improvement of mass transfer in nanoprorous crystals, the magnitude of such improvement is heavily influenced by the connectivity and availability of the transport pores. If the mesopore volume fraction is below the percolation threshold, there will exist microporous regions in the crystals where access of adsorbate cannot occur through an adjacent mesopore, slowing transport to a point where the difference between the diffusivity in the purely microporous and the mesopores-enhanced crystal will differ by no more than one or 2 orders of magnitude. Molecular simulation is used to predict the diffusion coefficients of ethane and propene in the microporous zeolite, providing results that compare well with the experimental data as long as the correct combination of framework/force field is chosen. In this sense, molecular simulation methods and EMT can potentially be combined to theoretically investigate hierarchical materials.

1. INTRODUCTION The use of nanoporous materials for mass separation and heterogeneous catalysis relies on the similarity of their pore size to the critical dimensions of the molecules under consideration.1,2 While close similarity enhances the equilibrium performance of the process, it also leads to dramatic retardation of molecular transport, turning mass transfer into the limiting factor for their technological application.3 Among the different strategies devised for overcoming this problem, the synthesis of nanoporous materials with hierarchical pore spaces has been under particular consideration over the past few years.4−6 In hierarchical materials, the microporous regions ensuring their functionality for mass separation and conversion are complemented by a network of mesopores intentionally introduced for accelerating mass transfer between the micropores as the “active sites” and the surrounding atmosphere. The success of this approach has been proven in adsorbate uptake and release experiments with remarkably superior exchange rates4,7−10 as well as in model reactions where the use of catalysts with a hierarchical pore structure was shown to lead to significantly enhanced conversion rates.10−13 Besides its obvious technological importance, mesopores-enhanced trans© 2014 American Chemical Society

port in hierarchical nanoporous material is also interesting from a fundamental perspective; history-dependent molecular diffusion in materials with hierarchical pore architecture forms an attractive tool for studying the equilibration process in complex pore systems.14,15 It is also relevant to draw attention here to the valuable insights molecular modeling and simulations have offered over the last two decades, given the experimental inaccessibility of the adsorbate configuration inside single pore spaces.16−18 One of the most important findings as far as molecular transport is concerned is that, even in the presence of a chemical potential gradient, the local equilibrium density profile is preserved;19,20 this conclusion is currently unattainable through experiments. Molecular transport in nanoporous materials with hierarchical pore topology, notably mesoporous zeolites, has been studied for the most part using the pulsed field gradient (PFG) mode of NMR.21,22 This technique allows studying the rate of the redistribution of the guest molecules within the system Received: March 21, 2014 Revised: June 2, 2014 Published: June 6, 2014 14355

dx.doi.org/10.1021/jp5028228 | J. Phys. Chem. C 2014, 118, 14355−14370

The Journal of Physical Chemistry C

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under study at equilibrium.23,24 Through this technique, it is even possible to record molecular diffusivities separately in the meso- and microporous pore spaces and to check not only overall diffusion enhancement by the presence of the mesopores, but also the opposite effect due to mesopore blocking.24 NMR studies of the rate of molecular uptake and release can also be performed by following the evolution of the NMR signal.15 The option of following uptake and release profiles in single crystals has recently been provided by the introduction of the techniques of microimaging.25 By exploiting the potential of interference microscopy (IFM)26,27 and/or infrared microscopy (IRM),25,28 it is possible to record transient concentration profiles, notably the variation of the cross-sectionally averaged concentration in the sample in the observation direction. To exploit the spatial resolution inherent to these techniques, about 0.5 μm × 0.5 μm in IFM and 3 μm × 3 μm in IRM, the particles under study must be transparent and of well-defined external shape. By using a single-element detector,25 IRM can also be applied in the pure integral mode. In this way, it is possible to follow molecular uptake or release on an individual particle. It is this mode of analysis that was exploited in our previous work investigating the transport properties of hierarchical CHA and LTA zeolites.29 In a previous publication, mass transfer in a hierarchical CHA-like zeolite, SAPO-34, was studied using the integral mode of IRM to determine the uptake curves of small adsorbates.29 The results were interpreted through the fastexchange model, according to which, for microporous particles traversed by a network of mesopores, the intraparticle diffusivity is given by the relation15,24 D = Dmicro + pmeso Dmeso

Following the Kirkpatrick effective medium theory,33 it can be shown that, for a network of channels with a coordination number (number of pores meeting at an intersection) N, the equivalent conductance ke is the solution to34−36 k(r , l) − ke k(r , l) +

(

N 2

)

=0

− 1 ke

(2)

where k(r,l) is the conductance of a pore with characteristic size r and length l, and the average ⟨...⟩ is taken over the number distribution of conductances. Assuming vanishing aspect ratios for the pores, uniform pore length and negligible volume of pore intersections, the pore conductance at finite density is given by k(r,l) = A(r) ρ(r) D(r)/l,38 where ρ(r) and D(r) are adsorbate density and diffusivity in a pore of characteristic size r, respectively, and A(r) is the pore cross section. With the effective conductance, the effective diffusivity can be extracted as34 (ρD)eff =

εkel 2 γ ⟨v(r )⟩

(3)

where v(r) = A(r)l is pore volume, ε is porosity, and γ is the tortuosity factor. If w1 is the number fraction of pores of size r1 and w2 = 1 − w1 the number fraction of pores of size r2 one has, for a thoroughly connected network (N → ∞) that ke = (w1ρ1D1v1 + w2ρ2D2v2)l−2, where vi, Di, and ρi represent v(ri), D(ri), and ρ(ri), respectively. Substituting this expression for ke in eq 3 yields, upon some rearrangement, (ρD)eff = ε1ρ1D1* + ε2ρ2 D2*

(4)

where εi = εwivi/[w1v1 + w2v2] is the porosity of pores with size ri and Di* = Di/τ. Now, if the effective diffusivity is defined with respect to the total adsorbate density ρa (amount adsorbed per unit volume of material), that is, Deff = (ρD)eff/ρa, eq 4 takes the familiar form of the fast-exchange model

(1)

in which Dmicro(meso) denotes the diffusivities in the micro(meso)porous spaces, with the relative amount pmeso of molecules in the mesopores (which is assumed to be negligibly small in comparison with the amount of molecules in the micropores). The fast exchange model has been used for interpretation of the variation of diffusivity with loading in pulse field gradient NMR experiments on the diffusion of cyclohexane in bimodal silica pores,30 and history dependent diffusion of cyclohexane in Vycor porous glass as a consequence of the variation of the relative amounts of capillary-condensed, adsorbed, and gaseous phases during adsorption and desorption.31 However, transport enhancement by just about 2 orders of magnitude was found for propene uptake in hierarchical SAPO-34 crystals,29 though eq 1 predicts D would be larger by about 5 orders of magnitude. In the calculations, Dmicro was taken from interpretation of IRM experiments in the fully microporous sample, whereas pmeso and Dmeso where estimated by assuming the adsorbate in the mesopores to behave as an ideal gas reflecting diffusely from the pore walls (thereby following Knudsen diffusion formula). Equation 1 implies the limiting case of “fast exchange”,30,32 where molecular exchange between the two pore spaces is sufficiently fast so that the molecular mean lifetimes in either of these spaces is negligibly small in comparison with the characteristic time constants of molecular uptake and release. Equation 1 can be readily derived in the context of Kirkpatrick’s effective medium theory,33 which has been previously used to interpret single34−36 and multicomponent transport in porous media.37 The derivation provides a clue on the topological conditions over which the fast exchange model is more accurate.

Deff = pD1* + (1 − p)D2*

(5)

where p = ε1ρ1/ρa is the mass/mole fraction of adsorbate in pores of size r1. If r1 and r2 correspond to micro and mesoporous sizes, one recovers eq 1 when the pressure is below the condensation point in the mesopores. A similar result could be obtained from a different starting point: a bundle of straight capillaries in parallel, where ρaDeff = ε1ρ1D1 + ε2ρ2D2. However, the current starting point better reveals the topological feature necessary for the fast exchange model to be accurate: at any given point, the particle must have an available path to move from one type of medium to the other in a single diffusion step (N → ∞), with a diffusion step understood as the transition from one pore to an adjacent one. For finite N, the probability that multiple steps must be performed for a particle to move from a micro- to a mesopore is nonzero, which in essence means that exchange between the two domains is either topologically or diffusionally restricted. In this paper, the results from our previous IRM experiments on ethane and propene uptake29 as well as those on butane uptake, measured gravimetrically by Schmidt et al.,10 are analyzed in the light of effective medium theory (EMT), while relaxing the assumption of infinite connectivity relating to the topological configuration of the two phases inherent to the fast exchange model. Moreover, molecular simulation methods are used to predict the diffusivity of ethane and propene in the 14356



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Figure 1. Schematic of the synthesis routes and final products of the different specimens of SAPO-34 considered in this study. Reprinted from Schmidt et al.10. Copyright (2012) with permission from Elsevier.

Table 1. Structure Data and Composition of the SAPO-34 Samples under Study10 sample

SBET (m2 g−1)

Sext (m2 g−1)

Vmicro (cm3 g−1)

Vmeso (cm3 g−1)

Al:Si:P(mol mol−1)

SAPO-34-micro SAPO-34-meso-1 SAPO-34-meso-2

531 517 501

0 98 184

0.285 0.203 0.153

0 0.456 0.364

45:12:43 45:15:40 48:14:38

carbon nanotubes. Figure 1 schematically depicts the pore spaces of the three samples of SAPO-34 considered in this study and their synthesis. The relevant porosity data as deduced from N 2 physiosorption and EDX measurements are summarized in Table 1. The physisorption experiments were interpreted through the multipoint BET model along with the t-plot model.10 The three SAPO-34 samples under study are specified according to the following convention:29 (1) SAPO-34-micro: This is the purely microporous sample, comprising cubic crystals with a mean edge size of about 20−40 μm. However, considerable variability could be observed throughout the sample, with crystals ranging from ∼10 to 60 μm. For the IRM experiments, nearly defect-free crystals of ∼30 μm edge size were chosen,29 as depicted in Figure 2a. (2) SAPO-34-meso-1: The carbon-nanoparticle-templated sample comprises cubic crystals with similar dimensions as those of microprous SAPO-34. This sample contains spherical insertions of 20 nm diameter within the microporous framework, constituting a collection of disconnected mesopores. For our tests, crystals of ∼30 μm edge size were chosen, similar to that depicted in Figure 2b. (3) SAPO-34-meso-2: The carbon-nanotube-templated sample comprises irregular particles with a wide range of sizes, ranging from about 10 to 100 μm. The microporous framework is traversed by a network of mesopores forming a spanning cluster. The mesopore diameters and lengths are expected to exhibit a broad distribution around mean values of about 10 nm and 1.5 μm, respectively. Figure 2c depicts examples of this type of crystal. The focus of this work will be on SAPO-34-micro and SAPO-34-meso-2, since SAPO-34-meso-1 was observed to be dynamically identical to SAPO-34-micro for all guest molecules. 2.2. IR Microimaging. In IR spectroscopy, information about the adsorbate loading may be obtained from the area under a characteristic IR band of the guest molecule under

microporous samples. Although quantitative agreement between simulation and experiment is not exact, the right order of magnitude for the diffusivity is always obtained, showing that molecular simulation and EMT can be successfully combined to study hierarchical materials from a theoretical perspective. In particular, analysis of propene and butane uptake curves suggests that percolation effects play a role on the quality of mass transfer enhancement in hierarchical samples. It must also be considered that the size of the microporous domains is crucial in determining the rate of mass transfer between the micro- and mesoporous spaces, since large microporous subparticles would clearly decrease the interfacial exchange area. It is likely that a combination of these two factors, namely, the significant surface barriers arising from the presence of large microporous subparticles and a reduced mesopore connectivity producing percolation effects, is responsible for the smaller than predicted enhancement of mass transfer in the hierarchical samples; while our previous work focuses on the former factor,29 this paper deals with the latter by assuming microscopic isotropy in the mixture of the two phases.

2. EXPERIMENTAL SECTION The experimental procedure is the same as described in our previous publication.29 However, we summarize the key points here for the sake of completeness. 2.1. Silicoalumophosphate SAPO-34. The material of structure type CHA considered in this paper is a crystalline silicoalumophosphate (SAPO-34).39,40 The synthesis and characterization of the material under study as well as its performance in adsorption kinetics and MTO (methane-toolefin) reaction is described in detail in Schmidt et al.10 By purposefully adding carbonaceous materials as secondary templates during the hydrothermal synthesis, samples with two types of mesopores were fabricated, namely, with mesopores separated from each other by the use of carbon nanoparticles and with a network of mesopores by the use of 14357



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3. MODELING AND SIMULATION Several molecular simulation techniques (MD, GCMC, and TST) were used to determine the adsorption isotherms and diffusion coefficients of methane, ethane, and propene in the microporous zeolite framework. Since SAPO-34 and AlPO4-34 are CHA-like structures, ideally differing in a single atom per unit cell (one Al is replaced by one Si), AlPO4-34 was taken as the host material for our simulations, taking advantage of the fact that it is a stable and well-characterized structure in its dehydrated form.43 The morphological details of the structure and the simulation box are summarized in Table 2. The critical Table 2. Lattice Parameters for AlPO4-34 Figure 2. Example of individual crystals chosen for the uptake experiments as observed through the optical element of the IRM. (a) SAPO-34-micro, ∼30 × 30 × 30 μm3, (b) SAPO-34-meso-1, ∼30 × 30 × 30 μm3, and (c) SAPO-34-meso-2, ∼50 × 40 × 40 μm3. The thickness of the crystal has been assumed to be equal to the smallest dimension in the plane perpendicular to the light beam. While SAPO34-micro and SAPO-34-meso-1 are regularly shaped, SAPO-34-meso-2 particles have an undefined geometry which makes uptake analysis approximate.

a=b=c (Å)

α=β=γ

sim. cell

no. of atoms

Al:P:O

window diameter (Å)

9.414

94.61

3×3×3

972

1:1:6

3.82

window size was established by taking the diameter of the O atom as 0.27 nm. The ratio Al:P in SAPO-34-micro is slightly higher than that reported in Table 2, which is likely to produce some small deviations in the cavity volume and window diameter. However, such deviation would expectedly be more pronounced if the all-silica CHA was used instead. 3.1. Grand Canonical Monte Carlo (GCMC) Simulations. The GCMC simulations used the standard Adams scheme,44 comprising trials of four types: moving a molecule, rotating a molecule, creating a molecule, and deleting a molecule. The probability of a move being accepted was evaluated by means of the Metropolis sampling scheme.45 The cutoff distance for potential energy calculations was 1.5 nm. Because no configurational bias was introduced, a large number of configurations were probed, sampling 5 × 107 configurations for ethane and 1 × 108 configurations for propene and butane, of which the first 10% were rejected in calculating averages. To estimate ethane and butane adsorption, the force field of Dubbeldam et al.46 for alkanes in zeolites was used, while propene adsorption was studied through the force field of Liu et al.47 for alkenes in zeolites. 3.2. Molecular Dynamics (MD) Simulation. MD simulations were performed using the parallel general purpose code DL_POLY_2.20.48 The Verlet algorithm was used for time integration of the Newton’s equations of motion, using a time step of 1 fs in all simulations. GCMC moves were first performed to place the molecules in the simulation domain, minimizing the energy. Next, an MD equilibration stage ranging from 500 to 800 ps was run to remove any initial large disturbances in the system; during equilibration, the velocity was scaled to match the temperature for each 100 time steps. After this, MD simulation production cycles were started, with the Nosé−Hoover thermostat applied to all the diffusing particles. The simulation times are 10 ns for methane, 30 ns for ethane, and 0.5 ns for determination of the dynamic correction in propene diffusion. Analysis of the trajectories is carried out from the history files generated. Intracrystalline self-diffusivity constants are obtained from the Einstein relation

study. A comprehensive overview of this technique can be found elsewhere.25,41,42 While good resolution of the spatial distribution of guest concentration can be achieved in some cases, the adsorbent particles in this study were neither sufficiently large nor homogeneous enough for allowing this type of measurement by use of a focal plane array detector. Instead, the IR measurements have been performed by using a single-element detector. The information thus extracted is the time dependence of total uptake or release, which is related to conventional transient sorption experiments.3 However, rather than a bed of particles or crystallites, it is now the individual host particle or crystallite that is actually probed. In addition to ensuring the absence of any bed resistance, uptake and release measurements with individual crystallites or particles may be also considered to be essentially unaffected by influences due to the finite rate of sorption heat release, as a consequence of the large (in fact, the largest possible) surface-to-volume ratio of the host system in such studies.41 Example of individual crystals chosen for the uptake experiments as observed through the optical element of the IRM are shown in Figure 2. While SAPO-34-micro and SAPO-34-meso-1 are regularly shaped, SAPO-34-meso-2 particles have an undefined geometry that makes uptake analysis somewhat approximate. The experiments were performed by use of a Fourier transform IR microscope (Bruker Hyperion 3000) composed of a spectrometer (Bruker vertex 80v) and an optical microscope. The optical cell in this device is connected to a vacuum system and mounted on a movable platform under the microscope. Such an arrangement facilitates the selection of a reasonably shaped particle/crystallite for subsequent uptake and release studies. Sample activation was accomplished by heating under vacuum at a rate of 1 K/min up to 300 °C for SAPO-34. The samples were kept under continued evacuation for 24 h at the maximum temperature. The subsequent uptake and release measurements have been performed after cooling down to 298 K. The transient sorption curves shown in this study are normalized; that is, they represent the fractional uptake or release as a function of time, following a pressure step in the surrounding atmosphere.

N

Ds =

⟨∑i = 1 (ri(t + Δt ) − ri(t ))2 ⟩ 1 lim 6N Δt →∞ Δt

(6)

where N is the number of particles. The transport (also termed collective) diffusivity is obtained from 14358



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N

⟨[∑i = 1 (ri(t + Δt ) − ri(t ))]2 ⟩ 1 lim (7) 6N Δt →∞ Δt Figure 3 depicts the self-diffusivity of methane as a function of loading in AlPO4-34 at 300 K. Four different interaction

molecule to diffuse through at its average thermal energy, leading to the decreasing self-diffusivity-loading profile commonly found in transport of small molecules through straight channels.58 If instead of AlPO4-34, the all-silica CHA from the IZA database59 is used, the characteristic entropic barrier appears for the all atom description. All-silica CHA has a window diameter of 0.386 if the LJ diameter of the oxygen is assumed to be 0.26 nm, which makes it evident that, at the present level of confinement, small variations in the framework composition may have significant consequences on the diffusivity profile. When quantitative comparison with experiment is sought, using a framework as close as possible to the actual one is clearly critical. There is some controversy as to whether framework flexibility per se plays a significant role on the dynamics of the adsorbates. While considerable impact on the short- and long-range dynamics of adsorbates in nanopores is theoretically predicted as a consequence of violation of the linear momentum conservation principle,60,61 the most important contribution seems to come from fluctuations in the critical diameter of the window, with instantaneous diameters above the average window size having a disproportionate weight on the diffusion constant.62,63 On the other hand, empirical potentials such as the BKS are found to be in general disagreement with crystallographic data, predicting considerable swelling of the structure in the presence of adsorbates.50,64 Figure 4 depicts the distributions of window sizes obtained for

Dc =

Figure 3. Self-diffusion coefficient of methane in AlPO4-34 at 298 K. The results from four different interaction potentials are depicted: allatom CH4 model in a rigid zeolite structure (AA-Rigid), all-atom CH4 but considering framework flexibility (AA-Flexible),49,52 united atom model of methane in a rigid zeolite structure (UA-Rigid),53 and united atom model in a flexible framework (UA-Rigid).52,53 The open circles represent the self-diffusivity of all-atom model methane in all-silica chabazite.

potentials were used to construct the curves: (1) An all-atom model of methane using the forced field of Catlow et al.49 and a rigid framework. This force field has previously been used in the estimation of diffusivities of small hydrocarbons in zeolites.50,51 (2) An all-atom model of methane in a flexible framework, with the Catlow et al.49 force field describing guest−guest and host−guest interactions and the van Beest− Kramer−Santen (BKS) potential52 for O−Al, O−O, and O−P interactions. (3) A united atom model of methane in a rigid framework, using the force field of Dubbeldam et al.46 for alkanes in zeolites. (4) A united atom model of methane in a flexible framework, with the Dubbeldam et al.46 force field accounting for guest−guest and host−guest interactions, and the BKS forced field describing intraframework interactions. Not only do these force fields differ quantitatively in their predictions, but, when comparing the united and all atom models, there is considerable difference in the diffusivity variation with loading. The all-atom model of methane produces a monotonically decreasing self-diffusivity, while the united atom model produces the typical maximum expected when transport is controlled by entropic barriers.53−55 This means the window does not represent a significant constriction for the motion of the all-atom methane molecules, which behave as if they were traversing a straight narrow channel. The Catlow et al.49 force field produces a maximum effective LJ diameter of 0.386 nm for methane (0.5σC−C + 0.5σH−H + rC−H, with σ being the LJ collision diameter and r being the equilibrium bond length), slightly below that produced by the OPLS/AMBER force fields of 0.41 nm56 or the ReaxFF force field of 0.48 nm.57 Moreover, the LJ diameter of the surface oxygen in the Catlow et al. potential is ∼0.26 nm, which means the critical diameter of the window becomes 0.392 nm for a rigid AlPO4-34 framework. This is wide enough for a methane

Figure 4. Distribution of window dimension obtained for flexible framework dynamics of AlPO4-34 following the BKS potential, at a loading of 1 molecule per cage.

flexible AlPO4-34 following the BKS potential at a loading of 1 molecule per cage. The average (crystallographic) window size (0.382 nm) is enlarged by 1.5 Å in the flexible structure, leading to a significant increase in methane diffusivity when flexibility is considered. Krishna and Van Baten64,65 have predicted through MD simulations that, if the lattice dynamics is suppressed and the window diameter is forced to coincide with the timeaveraged diameter of the flexible framework, there is no difference in the self-diffusivities produced by the rigid and flexible framework models in 8-ring zeolites. For the case of ethane, it will be shown that the rigid AlPO4-34 framework along with the united atom description of the adsorbate (Dubbledam et al.46 force field) produces good agreement with experiment. Given these results, the transition state theory 14359



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kmicro(meso) = Kmicro(meso)Dmicro(meso), leading to the EMT formulation

analysis of propene diffusion is performed exclusively with the Dubbledam et al.46 force field in a rigid framework. While more accurate force fields of the gas and the solid are available (with the best description of framework flexibility coming from the spatial fitting of the interatomic potential arising from ab initio simulations), the Dubbledam et al.46 force field is extraordinatily simple for its level of accuracy and, given that framework defects add an extra layer of uncertainty, we do not find further benefit in increasing the complexity of the force field. For the case of ethane, it will be shown that the rigid AlPO434 framework along with the united atom description of the adsorbate produces good agreement with experiment. 3.3. Effective Medium Theory. The inadequacy of the fast-exchange model to describe the experimental results29 indicates that the topological complexity of the material and/or the differences in time scales might not be properly captured by this model. An alternative treatment is offered by effective medium theory (EMT), in which the meso−micro framework (SAPO-34-meso-2) is modeled as an isotropic mixture of two phases with different mass transfer conductivities.66 For a threedimensional mixture of n phases in which phase i has a conductivity ki, the effective conductivity ke follows the solution of66 n

∑ ϕi i=1

ki − ke =0 ki + 2ke

ϕ

where ϕ is the volume fraction of the microporous phase and (KD)mix represents the effective conductance of the two-phase framework, which clearly contains the effects of both adsorption and molecular mobility. Solving for (KD)mix produces (KD)mix = −a/2 ± [(a2 − 8b)]1/2/4, with a = (1− 3ϕ)K m i c r o D m i c r o − (2 − 3ϕ)K m e s o D m e s o and b = KmicroKmesoDmicroDmeso; the positive root is taken as the answer. ϕ can be extracted from the micro- and mesoporous volumes Vmicro and Vmeso reported in Table 1 through ϕ=

ρs (Vmicro + Vmeso) + 1

(12)

4. RESULTS AND DISCUSSION 4.1. Ethane Uptake. Figure 5 compares the experimental uptake curves of ethane in purely microporous SAPO-34 and in

(8)

Figure 5. Uptake curves of ethane in purely microporous SAPO-34 and in SAPO-34-meso-2 at 298 K, following an ethane pressure step from 0 to 117 mbar. The symbols represent the experimental data obtained through IRM, and the lines the corresponding fitting through eqs 13 and 14.

(9)

where Kmicro is the equilibrium constant and cbulk is the pseudobulk density of fluid in equilibrium with the adsorbed molecules at a concentration c. For hydrocarbons, the equilibrium constant in window-cage type zeolites can be very large due to strong adsorption.67,68 On the other hand, adsorption in the mesopores is generally negligible at low pressures, so that Kmeso ≈ 1 and one has jmicro = Dmeso( −∇cbulk )

ρs Vmicro + 1

where ρs is the skeletal density of the framework. For AlPO4-34 with an Al:P ratio similar to that in the current samples, ρs = 2.29−2.48 g/cm,69 yielding a mean value of ϕ = 0.61.

where ϕi is the volume fraction of phase i. Notice that eq 8 corresponds to the particular case in which the coordination number in eq 2 is N = 6 and the conductance distribution is given by f(k) = Σiϕiδ(k − ki). The equivalence between the two treatments is complete if the “single channel” conductance in eq 2 is defined as ki = ρiDi⟨vi⟩/l2, with ⟨vi⟩ being a characteristic volume for domain i and l being a characteristic length that is assumed to be equal for all domains. With this definition, the volume average in eq 3 yields (ρD)eff = ke if ε/τ is assumed to be embedded in Di. The fact that eq 8 implicitly involves a finite coordination number means it does not require the thorough connectedness demanded by the fast exchange model, and hence can in theory predict more accurately mass transfer in heterogeneous porous media when topological restrictions exist in the exchange between the different domains. Low density transport within the microporous phase can be written as jmicro = Dmicro( −∇c) = K microDmicro( −∇cbulk )

K microDmicro − (KD)mix D − (KD)mix + (1 − ϕ) meso =0 K microDmicro + 2(KD)mix Dmeso + 2(KD)mix (11)

SAPO-34-meso-2 following a pressure step from 0 to 117 mbar. To fit the experimental results, we assume the diffusion coefficient to be constant throughout the concentration range of the experiment. In that case, for combined internal diffusion and external mass transfer, it can be shown that70

(10)

∞

Here, there is no assumption regarding the arrangement of the phases (i.e., whether they are in parallel or in series) other than they form an isotropic mixture at a sufficiently small lengthscale. It will be seen that this assumption works well with ethane adsorption and provides valuable insights on propene and butane uptake. The single phase conductivity is given by

m(t )/m(∞) = 1 − 6 ∑ n=1

(βn cos βn − sin βn)2 3

βn (βn − sin βn cos βn)

2

2

e−βn Dt / R

(13)

where {βn} are the nonnegative roots of the characteristic equation 14360


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= tan(βn)

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10−3 mmol cm−3, with the latter concentration corresponding to that of an ideal gas at 117 mbar. Thus, Kmicro does not exactly correspond to the infinite dilution Henry’s constant, but to the slope of the best linear approximation of the isotherm in the pressure interval of interest. Since the fitting produces r2 = 0.98, this procedure is not expected to lead to significant inaccuracies. It is straightforward to prove that

(14)

Here D is the effective diffusivity, α is the surface permeability for transport from the surrounding fluid through the geometric surface of the crystal, and R is the effective radius of the crystal particle, defined as the radius of a sphere with the same surface to volume ratio as the crystal particle. For SAPO-34-micro, the fitted parameters are α = 2.2 × 10−4 m s−1 and D = 4.6 × 10−12 m2 s−1, while for SAPO-34-meso-2, α = 2.0 × 10−5 m s−1 and D = 3.3 × 10−10 m2 s−1. The corresponding Biot numbers (Bi = αR/D) for SAPO-34-micro and SAPO-34-meso-2 are 866 and 0.015, respectively, indicating that the uptake rate in SAPO-34micro is fully diffusionally controlled and in SAPO-34-meso-2 is mainly controlled by surface resistance. It is important to consider that SAPO-34-meso-2 crystals are highly irregular, and the surface permeability likely nonuniform. Moreover, the variation of diffusivity with loading adds further uncertainty to the fitting procedure, which means the current values offer a semiquantitative characterization of the transport process. In our previous work,29 only the two limiting cases of surface barrier-controlled and diffusion-controlled transport are considered, with most curves displaying a tendency toward one limit or another. The current parameters are in good agreement with those found previously for SAPO-34-micro, with the sharp difference in α (83 × 10−7 m s−1 in our previous work) simply due to the high uncertainty arising from the comparatively small contribution of surface barriers to the total transport resistance. For SAPO-34-meso-2, it was shown29 that the barrier-controlled model predicted the experimental data slightly better, with the resulting parameters obtained here having similar order of magnitude as those estimated before (α = 1.0 × 10−5 m s−1 and D = 8.4 × 10−11 m2 s−1).29 To determine Kmicro, GCMC simulations were performed to obtain the corresponding adsorption isotherm, with ethane modeled as a two-sites rigid molecule and the fluid−fluid and fluid−solid interactions accounted for through the Lennard− Jones potential with Dubbeldam et al.46 parameters. The adsorption isotherm depicted in Figure 6 leads to Kmicro = 339, taken from the linear fitting of the data from cb = 0 to 4.7 ×

K mix = K microϕ + (1 − ϕ)

(15)

where Kmix is the equilibrium constant in hierarchical SAPO-34meso-2. For ethane, eq 15 yields Kmix = 207. For the mesoporous space, the diffusivity is expected to be close to the Knudsen value DKnudsen = (dmeso/3)[(8RgT/πM)1/2] = 1.5 × 10−6 m2 s, where dmeso (∼10 nm) is the mesopores diameter, T is temperature, M is the gas molar mass, and R is the universal gas constant. However, diffusion occurs also perpendicular to the pore axis within the mesopores, and mass transfer in this direction will clearly play an important role in defining the effective diffusivity. Neha et al.71 have theoretically studied the second frequency sum rule (Einstein frequency) of the velocity autocorrelation function in the presence of a density profile for spherical molecules under confinement in cylindrical and slitlike pores. They have shown that the sole existence of a density maximum near the surface tends to slightly increase the fluid diffusivity in the direction perpendicular to the wall, D⊥, with respect to the axial diffusivity D∥. On the other hand, molecular dynamics simulations of benzene in carbon nanotubes seem to indicate the opposite,72 with D⊥ being 0.8D∥ for a (12,12) nanotube. The permeable nature of the boundary of the microporous phase, which constitutes the wall surface of the mesoporous channels, is a factor that will certainly promote such diffusion anisotropy. Furthermore, the current material does not exactly correspond to the ideal isotropic mixture of two phases for which eq 11 is correct. Therefore, it is more convenient to take Dmeso as a fitting parameter, corresponding to the mesopore average three-dimensional diffusivity that would provide the effective diffusivity Dmix for an ideal isotropic mixture of two phases in which one of them has conductivity KmicroDmicro. Inserting the values of Kmicro, Kmix, Dmicro, and the experimental mixture diffusivity, based on the fit of eq 13, Dmix = 3.3 × 10−10 m2 s−1, for the pressure step 0−117 mbar into eq 11, and solving for Dmeso provides Dmeso = 7.02 × 10−7 m2 s−1. This value of Dmeso is half the Knudsen prediction, suggesting that D⊥ < D∥. However, it must be recalled that Dmix is based on a single crystal experiment, for which ϕ might not coincide exactly with the macroscopic average volume fraction of microporous phase reported in Table 1. If the content of microporous phase in the present crystal is 5% above the average (i.e., ϕ = 0.65), Dmeso becomes equal to DKnudsen. This is indeed reasonable since, for the IRM experiments, a single crystal must be selected. While the SAPO-34-micro and SAPO34-meso-1 crystals are generally uniform, this is not the case for the SAPO-34-meso-2 crystals which comprise a wide variety of shapes and sizes.73 It will be shown that a mesoporous volume just 15% below the macroscopic average (i.e., ϕ = 0.7) can explain intriguing results obtained in propene diffusion. It is interesting to compare the current result obtained through the estimated experimental value of Dmicro with that obtained through MD simulations. Figure 7 depicts the variation of self and transport diffusivity with loading for ethane in AlPO4-34 using the Dubbeldam et al.46 interaction parameters with both a rigid and a flexible framework structure.

Figure 6. Adsorption isotherm of ethane in AlPO4-34 at 298 K calculated through GCMC simulations. cb represents the bulk phase concentration, obtained through the ideal gas law cb = f/RT. The validity of the ideal gas law in the bulk phase was tested through bulk MC simulations (data not shown). 14361



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manifest earlier for a larger molecule like ethane. Again, the diffusivities for the flexible structure are larger than those obtained for the rigid one, mostly as a consequence of the larger average window size for the flexible framework. At 117 bar (cb = 4.7 × 10−3mmol cm−3), the adsorption isotherm shows c = 1.5 mmol cm−3, corresponding to ∼2.1 molecules/ cavity. At this loading, Dc = 1.0 × 10−10 m2 s−1 for the flexible framework and 4.0 × 10−12 m2 s−1 for the rigid one. The rigid framework provides a diffusivity very close to that experimentally determined, suggesting that, for this structure at room temperature, thermal vibrations might have little effect on the diffusivity of small hydrocarbons. A similar conclusion was reached by Krishna and Van Baten,64 although without direct comparison with experiment. The advantage of IRM microscopy with respect to bulk uptake measurements for determination of diffusivity in regular crystals is evident here: since it is possible to select the best crystals among a myriad of crystals where some are expectedly highly irregular and defective (very opaque), it is possible to minimize error to the point where quantitative comparison between simulation and experiment is possible. Using 4.0 × 10−12 m2 s−1 as Dmicro in eq 11 obviously produces essentially the same Dmeso = 1.7 × 10−7 m2 s−1 as before. 4.2. Propene Uptake. Figure 8 depicts the uptake curves of propene in the purely microporous SAPO-34-micro and in

Figure 7. Variation of self and transport diffusivity with loading for ethane in AlPO4-34 at 298 K calculated through MD simulations.

The curves display the typical maximum in self and transport diffusivity for hydrocarbons in window-cage zeolites,53 although at a slightly lower loading (in molecules per cage) than observed for methane. This is natural given that the packing transitions considered responsible for this behavior53,54 would

Figure 8. Uptake curves of propene in purely microporous SAPO-34 and in SAPO-34-meso-2 at 298 K, following an ethane pressure step from (a) 0 to 3 mbar, (b) 3 to 8 mbar, (c) 8 to 15 mbar, and (d) 15 to 30 mbar. The symbols represent the experimental data obtained through IRM, and the lines the corresponding fitting through eqs 13 and 14. 14362



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already in B) or may be caused by a nonoptimal selection of the reaction coordinate; in either case, a phenomenological correction to transition state theory can be done by introducing a transmission coefficient κ, in which a great deal of the dynamical information is condensed.77,78

SAPO-34-meso-2 at 300 K following a pressure step from 0 to 3 mbar, 3 to 8 mbar, 8 to 15 mbar, and 15 to 30 mbar. The uptake curves obtained from SAPO-34-meso-2 are very similar to those extracted from SAPO-34-micro and therefore not shown. The intracrystalline diffusivity and diffusion activation energy of propene in SAPO-34 obtained through the zero length column (ZLC) and frequency response methods67,74 are D(323 K) = 9.0 × 10−14 m2 s−1 and 23.4 kcal/mol, respectively, leading to D(300 K) = 5.5 × 10−15 m2 s−1. Fitting the experimental data through eqs 13 and 14 produces the diffusivities and surface permeabilities reported in Table 3,

k = κk TST

where k is the true diffusion rate constant. The transmission coefficient κ can be interpreted as the probability of acceptance in state B of a molecule at the top of the barrier that is moving in the direction of B. As such, it can be determined through MD simulations by repeatedly locating the molecule (center-ofmass) in the transition state with its initial velocity mapped from the Maxwell−Boltzmann distribution, and computing the fraction of trajectories for which the molecule effectively ends up in B. This fraction is time dependent, but reaches a plateau after a period of time that, for small adsorbates, is significantly smaller than the hopping period for sufficiently high barriers.78 In the present case, eq 17 can be rewritten as

Table 3. Fitting Parameters for Propene Uptake in SAPO34-micro, SAPO-34-meso-1, and SAPO-34-meso-2 at 298 K for Different Pressure Stepsa pressure step (mbar) −1

D × 10 (m s ) 14

2

0−3

SAPO-34-microporous (18 μm) SAPO-34-meso-1 (18 μm) SAPO-34-meso-2 (42 μm) α × 1010 (m s−1) SAPO-34-microporous (18 μm) SAPO-34-meso-1 (18 μm) SAPO-34-meso-2 (42 μm)

8−15

15−30

0.98 1.02 0.95 0.88 0.90 0.79 4.99 11.6 55.8 pressure step (mbar)

3−8

1.20 0.82 42.6

3−8

8−15

15−30

7.8

5.9

8.5

6.9 8.2 × 10−3

7.1 0.11

10.8 0.28

0−3 9.5 10.6 6.2 × 10−3

k = κv

which are of the same order of magnitude as those obtained through ZLC. These diffusivities tend to be slightly higher than those in our previous work, but within the same order of magnitude and following the same variation in D with pressure for SAPO-34-meso-2. 4.2.1. Diffusion Modeling. Considering a distance of 0.8 nm between the centers of connected cavities, the experimental diffusivities suggest an intercavity hopping period of ∼10 μs, beyond what can reliably be estimated through traditional MD methods. To obtain a theoretical estimate of the diffusion coefficient, we make use of transition state theory (TST), which has been found to predict satisfactorily the diffusion coefficient of slowly diffusing molecules in zeolites.54,75 In TST, diffusion is considered an activated process in which the molecule hops between the valleys in the free energy landscape through a barrier that is significantly higher than the thermal energy kBT. The thermally activated transit of the particle from state A to state B through the potential barrier can be characterized through the rate expression76 kBT Q * −E / kBT e h QA

e−βF(q*)

∫cageA e−βF(q) dq

(18)

where v is the average molecular velocity at the top of the barrier, corresponding to (kBT/2πm)1/2 for a Maxwell− Boltzmann distribution of velocities. F represents free energy, and q the selected reaction coordinate, with q* denoting the location of the transition state. We have used the Widom particle insertion method (WPIM)79 to obtain F(q). The WPIM uses a probe particle that is inserted at random positions and orientations to measure the energy released or required for insertion of a particle into the system. The particle is inserted during a canonical Monte Carlo simulation at the desired loading, and the corresponding energy ΔU is mapped onto the reaction coordinate, which in this case represents the axis between cavity centers as depicted in Figure 9.

a The number in parentheses corresponds to the effective particle radius used in the calculations.

k TST =

(17)

(16)

Here, the top of the potential barrier separating states A and B constitutes the transition state. E denotes the height of the barrier above the potential well; QA and Q* are the partition functions in the well and the transition state, respectively, and their ratio determines the probability for a particle to be found on top of the barrier. This equation ignores the nonreactive recrossings of the transition state; that is, it ignores trajectories that recross the potential energy barrier immediately after crossing it. Such recrossing events occur as a consequence of intermolecular interactions (e.g., collisions with particles

Figure 9. Lattice spanned by the cage centers of CHA-type zeolite. q* is the position of the barrier, perpendicular to the reaction coordinate. In the computation of κ, the particle starts in this plane. Reprinted with permission from Beerdsen et al.55 Copyright 2006 American Chemical Society. 14363



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Figure 10. Free energy profile of propene in AlPO4-34 at 298 K and loadings of 1 and 2 molecules per cage (mpc). Inset: variation of transmission coefficient with time. The plateau is reached at κ = 0.1 and 0.06 for 1 and 2 molecules per cage, respectively.

who showed that changes of 10% in the value of σgas−zeolite can produce variations of nearly an order of magnitude in Dc and Ds for argon-CHA at room temperature. Since CHA is isomorphic with SAPO-34, the fact that a small perturbation in σgas−zeolite can yield such a significant deviation in the diffusivity of a small molecule like argon makes the degree of accuracy on the predicted diffusivity of ethane quite remarkable, and the error in the estimation of diffusivity for propene acceptable. 4.2.2. EMT Analysis. Table 3 shows that propene uptake in SAPO-34-microporous and SAPO-34-meso-1 are esentially equivalent, with Biot numbers ranging between 0.5 and 0.67 throughout the different pressure steps, which indicates that mass transfer at the particle surface tends to be the controlling step during uptake, a finding already known for mass transfer in zeolite beds.83 However, the contribution of diffusion resistance is non-negligible (such assumption is commonly made for Bi < 0.1),70 which means the present values, determined by considering simultaneous surface permeation and diffusion, are somewhat more representative than those in our previous work.29 In a recent study on surface permeation and intracrystalline diffusion by microimaging in LTA-type zeolites84,85 and MOFs of type Zn(tbip),86,87 the formation of surface barriers was shown to be due to the total blockage of an overwhelming part of the micropores connecting the intraparticle pore space with the surroundings, rather than to the existence of a quasi-homogeneous layer of extremely reduced permeability on the external surface. In the case of single file diffusion, an additional boundary effect characterized by deviation of the concentration with respect to that expected in normal diffusion near the channel boundary has been observed, as a consequence of parallel occurrence of collective molecular motion and fast, uncorrelated displacement of individual molecules.88 The transport resistance in SAPO-34meso-2 is fully diffusional, with 105 < Bi < 107. Interestingly, despite the mesopores having a diffusivity that is 6 orders of magnitude higher than that in the micropores (Dmeso ∼ DK ∼ 10−8 m2 s and Dmicro ∼ 10−14 m2 s), the increase in propene diffusivity in SAPO-34-meso-2 with respect to SAPO-34-micro

The average of the Boltzmann factors yields the free energy at a given q according to βF(q) = −ln⟨e−β ΔU (q)⟩

(19)

The self-diffusion coefficient can be determined from Ds =

1 2 kλ 6

(20)

where λ is the distance between adjacent cage centers. At high loadings the WPIM, is known to give erroneous results.80 For this reason, we will restrict ourselves to low and moderate loadings in this analysis (1−2 molecules per cage). The free energy profile for concentrations of 1 and 2 molecules per cage is depicted in Figure 10. q = 0 corresponds to the location of the transition state. Equations 18 and 20 produce Ds,TST = 1.32 × 10−13 and 3.70 × 10−13 m2 s−1 with κ = 1 for loadings of 1 and 2 molecules per cage, respectively. Therefore, the transmission coefficient κ must be considerably smaller than unity at these loadings, which means that having one only molecule in the receiving cavity generates a hindrance significant enough for a molecule ready to make the jump from the transition state to backtrack its path. The transmission coefficient estimated as a function of time is depicted in the inset of Figure 10. The plateau is reached at κ = 0.10 and κ = 0.06 for 1 and 2 molecules per cavity, respectively, leading to Ds = 1.32 × 10−14 m2 s−1 for one molecule per cage and Ds,2 = 2.22 × 10−14 m2 s−1 for two molecule per cage. Since the transport diffusivity Dc at a given loading is always greater or equal than Ds because of the smaller effect of back correlations,53 it is clear that the method produces some overestimation of the diffusion coefficient. For a molecule that is large compared to the cavity diameter, it is reasonable for very small defects in the crystalline structure to yield significant variations in diffusivity. In particular, a slightly smaller cavity would greatly increase hindrance effects, reducing κ even further. The sensitivity of the diffusion coefficient to the solid−fluid interaction parameters in gas/zeolite systems was investigated by Fritzsche et al.81 and Krishna and Van Baten,82 14364



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is less than 2 orders of magnitude. It should also be noted that, from 0−3 mbar to 15−30 mbar, the diffusivity in SAPO-34meso-2 increases by an order of magnitude, consistent with the model of a constant corrected diffusivity with loading dependence determined by the Darken thermodynamic factor; for example, a Langmiurian isotherm provides D(c) = D0

1 1 − c /cmax

(21)

where cmax corresponds to the loading at saturation. A similar large increase in diffusivity is not observed in SAPO-34-micro, which might be due to the fact that the slightly dominant surface barrier does not allow for an accurate determination of D, while in SAPO-34-meso-2 surface effects are entirely negligible. If Dmicro is taken as 5.5 × 10−15 m2 s−1 as reported from early ZLC experiments67,74 and, instead of setting the effective particle radius equal to the geometric effective radius, its value is taken as a fitting parameter, then, for the interval 0−3 mbar, eq 13 yields R = 16 μm, which is about 40% of the geometric radius (42 μm) of the SAPO-34-meso-2 particle. This suggests that, instead of having a collection of small microporous particles embedded in a mesoporous phase, the microporous phase comprises relatively large particles for which a small, and perhaps not well connected, network of mesopores could not significantly enhance diffusion but can dramatically improve access to the microporous phase interior. In other words, the volume fraction of the mesoporous phase, 1 − ϕ, is likely below its percolation threshold, that is, the value of 1 − ϕ below which a spanning cluster of mesopores ceases to exist. Taking Dmicro = 0 and Dmeso = 1 in eq 11, it is straightforward to see that the percolation threshold according to Bruggeman’s EMT is ϕp is 0.67, relatively close to the macroscopic average volume fraction ϕ = 0.61. Therefore, it is reasonable to assume that, for a sizable fraction of crystals, the mesopore phase is generally disconnected, such as in the present case. Figure 11 depicts the adsorption isotherm of propene in the isomorphic AlPO4-34 as determined from GCMC simulations using the interaction parameters for alkenes in zeolites determined by Liu et al.89 From here, Kmicro = 2.5 × 103. Using this value of Kmicro, EMT

Figure 12. Variation of propene diffusivity in SAPO-34-meso-2 with volume fraction of micropores, following the Bruggemann EMT with Dmicro = 5.5 × 10−15 m2 s−1.

produces the variation of Dmix with ϕ depicted in Figure 12. To establish Dmeso, a reasonable approximation is obtained by taking Dmeso(prop) = Dmeso(eth)

Meth M prop

(22)

where M is molar mass and the subscripts eth and prop indicate ethane and propene, respectively. The above equation is a simple rescaling of the diffusivity to account for the effect of mass on velocity, following the usual kinetic theory result that velocity ∼ M−0.5. This leads to Dmeso = 1.3 × 10−7 m2 s−1. It is clear from Figure 12 that, for ϕ > ϕp, diffusion within the structure is dominated by the microporous phase. Therefore, a value of ϕ above 0.7 can potentially produce the experimentally observed Dmix for propene in SAPO-34-meso-2. Of course, for a scarcely connected mesoporous network, it is reasonable to argue that the isotropy demanded by EMT is not complied by the material. This, however, confirms the hypothesis that poor enhancement of intraparticle diffusion is caused by insufficient mesopore networking,90 leaving the overall transport enhancement to improvement in surface permeability for this particular crystal. To determine what fraction of particles fall below the percolation threshold and how their distribution affects macroscopic uptake at relatively low pressures, it is necessary to analyze macroscopic uptake over a sizable number of crystals. To do this, we study the adsorption of butane in the next section, experimentally investigated by Schmidt et al.10 on the same samples. 4.2.3. Butane Uptake. A very different perspective of the material can be extracted from the macroscopic uptake curves for a large molecule, butane, obtained gravimetrically by Smith et al.73 In the experiment, 1 mL/min n-butane (99.95% pure) was premixed with 39 mL/min nitrogen (99.999% pure) and passed over the sample at 298 K and atmospheric pressure. The mass change due to the adsorption of n-butane was recorded versus the time. Figure 13 depicts the uptake curve of butane in SAPO-34-microporous along with the fitting curve obtained through eq 13. The curve parameters are τ = R2/D = 4.4 × 105s,

Figure 11. Adsorption isotherm of propene in AlPO4-34 at 298 K determined through GCMC using the united-atom parameters from Liu et al.47 The symbols represent the simulation results, and the line the best fit with a two−site Langmuir isotherm. 14365



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Figure 13. Butane uptake in SAPO-34-micro. The symbols represent points in the uptake curve as obtained through macroscopic gravimetric methods.73 The curve is the model fitting following eqs 13 and 14.

Figure 14. Butane uptake in SAPO-34-meso-2. The symbols represent points in the uptake curve as obtained through macroscopic gravimetric methods.73 The solid line represents the fitting from case 1, the dotted line the fitting from case 2, and the dashed line the fitting from case 3.

Bi = 20.85, and m∞ = 100 mg butane/g sample. The magnitude of Bi shows that intracrystalline diffusion constitutes the dominant resistance during uptake. The attenuation of surface resistance relative to propene, for which surface barriers tend to dominate, is possibly due to the higher flow rates reached in this experiment. Adsorption of a small molecule like N2 is negligible at the temperature and pressure under examination, and its effect on butane diffusion within the structure is neglected. If the effective particle radius employed in the analysis of ethane adsorption (18 μm) is taken as the mean particle radius, we have Dmicro = 7.3 × 10−16 m2 s−1, which is in the expected order of magnitude considering that, for propene, Dmicro ∼ 0.5 × 10−14 m2 s−1. To analyze uptake in SAPO-34-meso-2, this value will be used as the diffusivity of butane in the microporous phase. The corresponding surface permeability is α = 8.4 × 10−11 m2 s−1, similar to that found for propene in this material. However, relatively weak surface barriers were found for propene in SAPO-34-meso-2, which makes it reasonable to assume that only diffusion resistance is important for butane in SAPO-34-meso-2. Figure 14 depicts the uptake curve for butane in SAPO-34meso-2 as extracted from macroscopic uptake curves obtained gravimetrically by Schmidt et al.73 Figure 15 depicts the adsorption isotherm of butane in AlPO4-34 at 298 K calculated through GCMC with the united atom parameters of Dubbeldam et al.;46 the corresponding Henry constant is Kmicro = 1.3 × 105. To analyze the influence of crystal size and microporous volume fraction variability on the overall uptake, the process is modeled according to the following three cases: (1) Uptake can be predicted solely by eqs 13 and 14, using the mean microporous volume fraction (0.61) and assuming a single mean particle size. In this case, the only fitting parameters would be the Biot number Bi = αR/D and the diffusion time constant τ = R2/D. (2) Uptake can be predicted through the diffusion limitation model (Bi → ∞), m(t )/m(∞) = 1 −

6 π2

∞

∑ i=1

2 2

Figure 15. Adsorption isotherm of butane in AlPO4-34 at 298 K calculated through GCMC simulations. cb represents the bulk phase concentration, obtained through the ideal gas law cb = f/RT.

the particles is assumed to distribute according to a Rayleigh distribution, fr (r ) =

(R − r0)2

⎛ r − r0 ⎞2 ⎟ −1/2⎜ ⎝ R − r0 ⎠ e

(24)

where R and r0 represent the mean and minimum particle radius. The normalized uptake is given in this case by, ∞

M(t )/M(∞) =

2

exp( −i π Dt /R ) i2

r − r0

∫0 m∞(r ) m*(t , r ) fr (r ) dr ∞

∫0 m∞(r ) fr (r ) dr

(25)

Here m*(t,r) = m(t,r)/m∞(r), where m(t,r) is the amount adsorbed in a particle of radius r at time t and m∞(r) = m∞(∞,r). For isotropic particles, m∞(r) ∝ r3, and therefore,

(23)

using the mean microporous volume fraction (0.61), but considering variability on particle sizes. For this, the radius r of 14366



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M(t )/M(∞) r − r0 2 ⎡

−1/2 ∫0 r 3(r − r0) e ( R−r0 ) ⎢1 − ∞

⎣

= ∞ 3 r (r 0

∫

−

6

∞

∑i = 0

⎛ −i2π 2Deff (ϕ0)t ⎞⎤ ⎟⎥ dr exp⎜ r2 ⎝ ⎠⎦

1 i

2

−1/2 R − r0 0 r0)e

)

π

2

(

r−r

2

dr

(26)

Deff(ϕ0) is the effective diffusivity obtained through the EMT, eq 11, for ϕ0 = 0.61, taking Dmicro = 7.3 × 10−16 m2 s−1 as derived from Butane uptake in SAPO-34-microporous. The fitting parameters are in this case r0 and R. (3) Uptake can be represented through the diffusion limitation case, eq 23, using a mean particle size (taken as a fitting parameter), but considering variability in micropore volume fraction. For this, ϕ is assumed to be distributed according to a normal distribution with mean ϕ0 = 0.61. While it seems more appropriate to use a truncated distribution given that 0 < ϕ < 1, and ϕ = 1 is not physically meaningful, it will be shown that a normal distribution provides very coherent results without further restrictions. Under these assumptions, ⎧ ⎪ M(t )/M∞ = ⎨ ⎪ ⎩

∫0

1

that is confined within reasonable limits for ϕ. This result certainly shows that nonuniformity in the micropore volume fraction should be taken into account to interpret macroscopic uptake experiments, and its effect is significantly more pronounced than that of particle size variability for the current sample. Furthermore, from the distribution, it can be estimated that 35% of the particles have a mesopore volume below the percolation threshold; that is, the region where the enhancing power of the hierarchical structure is severely reduced, as suggested by the single crystal uptake of propene. Finally, the current result shows that, in characterizing hierarchical structures, not only the mean fraction of conductive pores but also their distribution is of significant importance to explain the observed enhancement at a macroscopic level.

⎛ (ϕ − ϕ )2 ⎞ 0 ⎟ [(K micro − 1)ϕ + 1]exp⎜⎜− ⎟ 2σ 2 ⎠ ⎝

⎡ 6 × ⎢1 − 2 ⎢⎣ π ⎧ ⎪ ⎨ ⎪ ⎩

Figure 16. Volume fraction distribution as obtained from case 3.

∫0

1

∞

∑ i=0

⎛ − i 2π 2D (ϕ )(ϕ)t ⎞⎤ ⎫ ⎪ 1 eff 0 ⎟⎟⎥ dϕ⎬ exp⎜⎜ 2 2 ⎪ ⎥ i R ⎝ ⎠⎦ ⎭

⎛ (ϕ − ϕ )2 ⎞ ⎫ ⎪ 0 ⎟ [(K micro − 1)ϕ + 1] exp⎜⎜− ⎟ dϕ ⎬ 2 ⎪ 2 σ ⎝ ⎠ ⎭

(27)

As for case 3, Deff(ϕ0) is the effective diffusivity obtained through EMT, eq 11, with Dmicro = 7.3 × 10−16m2s−1. The fitting parameters are the mean particle size R and standard deviation σ. Figure 14 depicts the experimental and predicted uptake of butane in SAPO-34-meso-2. The symbols represent points in the uptake curve as obtained from the experiments,73 while the three model cases outlined above are represented by the lines as indicated in the legend. The resulting fitting parameters are shown in Table 4. Case 1 yields poor fitting of the experimental

5. CONCLUDING REMARKS Effective medium theory (EMT) has been employed to interpret uptake experiments in hierarchical SAPO-34. The current analysis suggests that, while hierarchical materials can improve mass transfer in microporous crystals, the magnitude of such improvement is heavily influenced by the connectivity and availability of the transport pores. While an automatic improvement comes from reduction of surface barriers, a significant increase of the effective diffusivity can only be reached if the microporous volume fraction is below the percolation threshold, that is, the volume fraction of micropores above which a spanning cluster of transport pores ceases to exists. The Bruggemann version of EMT66 assumes that the micro- and mesoporous phases are isotropically mixed, which is not necessarily correct for some crystals where considerable clustering of same type pores could potentially exist. In this regard, a more accurate analysis could be performed by numerical solution of the diffusion equations in a topologically similar pore network, as has been done in the past for fully microporous materials.37,91,92 More complex models accounting for topological correlations in the pore network may also be constructed, following the correlated random walk theories of Bhatia.93−95 Of particular interest would be the effect of a pore size distribution in the micro and/or meso space. This type of treatment on hierarchical structures is still awaited. The fact that MD simulations provide reasonable values for the diffusivities allows for meaningful theoretical investigation of hierarchical zeolites, which means prediction of optimal

Table 4. Fitting Parameters for Three Different Models of Butane Uptake in SAPO-34-meso-2 pressure interval (mbar)

parameter 1

parameter 2

case 1 case 2 case 3

Bi = 2.1 × 105 r0 = 2.7 × 10−4 μm σ = 0.104

τ = 73 757 s R = 20.5 μm R = 33.0 μm

data. However, the resulting parameter value, Bi → ∞, suggests that mass transfer is dominated by intraparticle diffusion, as assumed for cases 2 and 3. More interestingly, case 3 produces excellent fitting of the experimental data while case 2, having the same number of adjustable parameters, yields a prediction that is just marginally better than that in case 1. The mean particle sizes obtained from cases 2 and 3 are very reasonable, well within the observed range of particle sizes for SAPO-34-meso-2 (∼10 μm < 2R < 100 μm). Case 2 was also implemented with a normal rather than a Rayleigh probability distribution, but the resulting fitting error was even more pronounced. The pore volume fraction distribution obtained in case 3 is depicted in Figure 16. It is remarkable that the only fitting parameter for the unbounded normal distribution, σ, produces a distribution 14367



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mesopore geometry, volume fraction, and configuration can be potentially undertaken in the search for a systematic approach for the design of such materials.

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AUTHOR INFORMATION

Corresponding Authors

*(S.K.B.) E-mail: [email protected]. Tel: +61 7 3365 4263. Fax.: +61 7 3365 4199. *(J.K.) E-mail: [email protected]. Tel: +49 341 97 32502. Fax: +49 341 97 32549 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research has been supported by a grant from the Australian Research Council, under the Discovery Scheme. One of us (S.K.B.) acknowledges an Australian Professorial Fellowship from the Australian Researches Council. The authors gratefully acknowledge Franz Schmidt and Professor Stefan Kaskel from the Department of Inorganic Chemistry at Dresden University of Technology (Dresden, Germany) for providing the samples and much of the experimental data analyzed in this work. We thank the High Performance Computing Unit of the University of Queensland for access to supercomputing facilities.

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REFERENCES

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