Hindered Diffusion in Ordered Mesoporous Silicas: Insights from Pore

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Hindered Diffusion in Ordered Mesoporous Silicas: Insights from PoreScale Simulations in Physical Reconstructions of SBA-15 and KIT-6 Silica Stefan-Johannes Reich, Artur Svidrytski, Alexandra Hoeltzel, Justyna Florek, Freddy Kleitz, Wu Wang, Christian Kübel, Dzmitry Hlushkou, and Ulrich Tallarek J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b03630 • Publication Date (Web): 23 May 2018 Downloaded from http://pubs.acs.org on May 23, 2018

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The Journal of Physical Chemistry

Hindered Diffusion in Ordered Mesoporous Silicas: Insights from Pore-Scale Simulations in Physical Reconstructions of SBA-15 and KIT-6 Silica Stefan-Johannes Reich,† Artur Svidrytski,† Alexandra Höltzel,† Justyna Florek,‡ Freddy Kleitz,‡ Wu Wang,§ Christian Kübel,§ Dzmitry Hlushkou,† and Ulrich Tallarek†,*

†

Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse 4, 35032 Marburg, Germany

‡

Department of Inorganic Chemistry – Functional Materials, University of Vienna, Währinger Strasse 42, 1090 Vienna, Austria §

Institute of Nanotechnology and Karlsruhe Nano Micro Facility, Karlsruhe Institute of

Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany

* Corresponding author. E-mail: [email protected]

1 Environment ACS Paragon Plus

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ABSTRACT. The performance of SBA-15 and KIT-6 silicas as ordered mesoporous supports for heterogeneous catalysis, selective adsorption, and controlled drug release relies on their properties for diffusive transport of finite-size solutes or particles. We investigate this issue through a reconstruction–simulation approach, applied to purely mesoporous SBA-15 and KIT-6 silica samples with a mean pore size of 9.4 nm. Physical reconstruction by electron tomography confirms a highly interconnected, 3D mesopore network for both samples, but also reveals constrictions (both samples) and dead-ends (SBA-15 only) in the supposedly uniform cylinders of the primary pore system. Pore-scale simulations of the diffusion of point-like and finite-size tracers in the reconstructions show that a small number of bottlenecks suffices to observe a dramatic decline in the effective diffusion coefficient, accessible porosity, and pore interconnectivity with increasing tracer size. When the tracer size approaches one third of the mean pore size, diffusive transport becomes impossible. Below this limit, KIT-6 performs better than SBA-15, but both samples are decidedly inferior to random mesoporous silicas. Our results suggest that diffusive transport in SBA-15 and KIT-6 silicas could be improved by widening intrawall pores towards the mean pore size to provide bypasses around bottlenecks in the primary pore system.


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INTRODUCTION SBA-15 and KIT-6 are large-pore ordered mesoporous silicas1–6 that enjoy widespread popularity as support structures for catalysis, adsorption, separation, and drug delivery, and as nanocasting molds. The term “ordered” refers to the primary pore system, which consists of regularly arranged mesopores of defined shape and quasi-uniform size. The SBA-15 structure is based on hexagonally arranged, cylindrical pores, whereas the cubic structure of KIT-6 is based on a pair of interpenetrating, bicontinuous, cylindrical pore networks. These primary, ordered pores, whose size can be tightly controlled, are separated by amorphous silica walls, which contain a secondary, random pore system. Intrawall pores are usually in the micropore and smaller mesopore range, whereby details strongly depend on the preparation history7,8 (and references therein). Modulation of this complementary porosity allows to fine-tune the geometry (pore size distribution) and topology (pore network connectivity) of SBA-15 and KIT-6 silicas. The adjustable pore space morphology is a key feature, because it allows a variation of the material properties that goes beyond the possibilities offered by choosing arrangement and size of the primary pores. Knowledge about the relation between synthesis conditions and the porosity (and pore structure) of the resulting materials has mostly come from nitrogen physisorption analysis.7,8 After it was shown that pore condensation and hysteresis behavior depend sensitively on the synthesis parameters of a material, preparation conditions could be linked to particular structural properties through advanced analysis of physisorption hysteresis combined with non-local density functional theory (NLDFT) models.9 SBA-15 silicas, for example, can be prepared with structures ranging from corrugated and/or distorted (pseudo-1D) pore systems to highly interconnected, 3D pore networks.8 The knowledge established through gas sorption



analysis about the pore structure of SBA-15 and KIT-6 silicas, however, does not extend to their transport properties. The consequences of a given pore size distribution and network connectivity for the relevant mass transport properties (typically diffusion) of a material are difficult to predict quantitatively, unless structural descriptors with a proven strong correlation to the key transport phenomena are available.10 Up to now, structural descriptors for ordered mesoporous silicas have not been discovered, so that quantitative morphology–transport relationships between hysteresis behavior, pore structure, and effective diffusivity are not (yet) available.11 Compared with the familiar characterization by gas sorption analysis, detailed studies of the transport properties of SBA-15 and KIT-6 silicas are scarce. The few investigations of diffusion processes used the chromatographic zero-length column (ZLC) technique12–14 or pulsed fieldgradient nuclear magnetic resonance (PFG-NMR).15–19 Results of Kaliaguine and co-workers12 obtained with the ZLC technique indicate that diffusion of n-heptane in SBA-15 samples with different micropore volumes at low sorbate gas-phase concentrations (corresponding to the linear domain of the adsorption isotherm) occurs inside the silica walls and depends on their micropore content. With a larger amount of micropores, transport of n-heptane is characterized by relatively low diffusivities and high activation energies, resembling diffusion in typical microporous adsorbents like zeolites. With decreasing amount of micropores, diffusion becomes increasingly controlled by secondary mesopores, that is, smaller mesopores in the silica walls (as distinct from the primary, hexagonally arranged, larger mesopores), rendering diffusion faster and activation energies lower. Similar observations were made when the ZLC technique was used to study the diffusion of cyclohexane and 1-methylnaphthalene in SBA-15.14 It was concluded that mass transfer is dominated (rate-limited) by configurational diffusion in the micropores, and that


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transport differences originate from the varying amounts of microporosity in the studied samples, with less micropores resulting in faster diffusion. Very recently, NMR diffusometry has been applied to characterize effective tortuosities of SBA-15 and KIT-6 silicas through the transport behavior of carboxylic acids,19 with a focus on the use of sulfonic-acid functionalized SBA-15 and KIT-6 silicas as catalytic supports in the esterification of free fatty acids for biodiesel production.20 The effective tortuosity of both materials was found to increase with the alkyl chain length of an acid, with diffusion of longer chained acids to be dominated by the geometry of the support structure. The data moreover indicated that pore diffusion is unlikely to become rate-limiting in carboxylic acid esterification over sulfonated SBA-15 and KIT-6 silicas. Further advances toward quantitative morphology–transport relationships for SBA-15 and KIT-6 silicas (or mesoporous materials in general) are expected from a reconstruction‒ simulation approach.10 The pore space morphologies of mesoporous materials are physically reconstructed by electron tomography to be used as models in pore-scale simulations of diffusion, sorption, and reaction. This approach provides a comprehensive picture on the porescale, spatiotemporal dynamics of physicochemical phenomena under explicit consideration of the 3D morphology of a material, reflecting its preparation history. A unique feature of simulations is the possibility to analyse diffusion independently from other processes which are coupled in experiments, such as adsorption and reaction, by using inert (passive) tracers that neither interact with the material surface nor with one another.21 In this way, key aspects of diffusive transport in porous media, such as the effect of increasing solute or particle size on the effective diffusion coefficient, are quantitatively resolved. In addition, the transport properties can be linked to the accompanying morphological characteristics, for example, to the actual porosity and connectivity of the pore space accessible to the (finite-sized) diffusing species,



leading in a straightforward manner to the targeted morphology–transport relationships. We have recently used the reconstruction‒simulation approach to study hindered diffusion in random mesoporous silicas prepared by classical sol–gel processing.22 The degree to which diffusion through a porous material is hindered compared with diffusion in the bulk liquid depends on λ, the ratio of solute size to mean pore size, and is quantified by the overall hindrance factor H(λ). We simulated the diffusion of inert tracers of variable size in the reconstructed morphologies of three samples with different mean mesopore size. From the large data set we derived a quantitative hindrance factor expression that allows to predict the extent of hindered diffusion in random mesoporous silicas for 0 ≤ λ ≤ 0.9 from knowledge of the material porosity and tortuosity. Quantitative data on hindered diffusion in SBA-15 and KIT-6 silicas would be very welcome, as most applications rely on effective and/or selective transport of finite-size species. Hindered diffusion concerns the immobilization of enzymes,23–25 polyhedra,26 and clusters27,28 on the internal surface of these ordered mesoporous materials as well as the transport of bulky educts and products to and from the active surface sites,29,30 or the controlled release of drugs from the pore space.31–33 Partial exclusion of finite-size species reduces the active surface area of a support, and when interfacial kinetics are fast, diffusive transport may become rate-limiting to the process performance. Detailed 3D electron tomography data for SBA-15 and KIT-6 silicas are still scarce in the literature. We are unaware of any reconstruction of KIT-6, and the few studies conducted on SBA-15 typically focused on size, shape, and relative spatial location of catalytically active nanoparticles on the support,34–38 a key aspect in the design and assembly of catalysts with high durability. Two electron tomography studies of SBA-15 investigated pore wall corrugation in the primary mesopores39 and the merging of mesopores depending on the


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hydrothermal treatment (aging) temperature.40 So far, physical reconstructions of SBA-15 or KIT-6 have not been used in direct numerical simulations to quantitatively address the consequences of their morphological features for their transport properties. In this work, we apply the previously documented reconstruction‒simulation approach22 to SBA-15 and KIT-6 silica. Because micropores, if present, dominate overall diffusion behavior, we prepared purely mesoporous SBA-15 and KIT-6 silica samples by choosing appropriate conditions for hydrothermal treatment.7,8 An in-depth morphological analysis of the reconstructed samples complements the data obtained from the diffusion simulations. Our goal is to link the diffusive transport properties of the ordered mesoporous silica samples to their pore space morphology and to compare their diffusive transport behavior to that of random mesoporous silicas.

EXPERIMENTAL SECTION Synthesis of the Ordered Mesoporous Silica Samples. SBA-15 silica was synthesized according to the method of Choi et al.41 Briefly, 13.9 g of Pluronic triblock copolymer (P123, EO20PO70EO20, MW = 5800, Sigma-Aldrich) were dissolved in 252 g of distilled water and 7.7 g of hydrochloric acid (HCl, Fisher Scientific, 37%). After complete dissolution, 25.0 g of tetraethoxysilane (TEOS, Acros, 99%) were added at once. The mixture was left under stirring at 35 °C for 24 h, followed by hydrothermal treatment at 135 °C for 24 h under static conditions. The resulting solid product was filtered hot and dried for 24 h at 100 °C. For template removal, the as-synthesized silica powder was slurred in an ethanol‒HCl mixture for a short time, then filtered and dried for 24 h at 100 °C, and subsequently calcined at 550 °C in air.



KIT-6 silica was obtained by following the method reported previously.2,4 Briefly, 9.0 g of Pluronic P123 were dissolved in 325 g of distilled water and 17.4 g of HCl (37%) under vigorous stirring. After complete dissolution, 9.0 g of 1-butanol (BuOH, Sigma-Aldrich, 99%) were added at once. The mixture was left under stirring at 35 °C for 1 h, after which 19.35 g of TEOS (99%) were added at once to the homogeneous, clear solution. The synthesis was carried out in a closed polypropylene bottle. The molar ratio of the starting reaction mixture was 1/0.017/1.9/195/1.31 TEOS/P123/HCl/H2O/BuOH. This mixture was left under stirring at 35 °C for 24 h, followed by hydrothermal treatment at 130 °C for 24 h under static conditions. The resulting solid product was filtered hot and dried for 24 h at 100 °C. For template removal, the as-synthesized silica powder was first slurried in an ethanol‒HCl mixture for a short time, then filtered and dried for 24 h at 100 °C, and subsequently calcined at 550 °C in air. Sample Characterization. The calcined and powdered SBA-15 and KIT-6 samples were first characterized by X-ray diffraction (XRD) and nitrogen physisorption. Low-angle XRD patterns were recorded in the range of 2θ = 0.5–6° (with 0.01° step size) on a Rigaku SmartLab X-ray diffractometer using Cu Kα radiation (λ = 0.15406 nm) at 30 kV and 40 mA (KAIST, Daejeon, Republic of Korea). Nitrogen physisorption measurements were run at 77 K using an Autosorb-iQ2 sorption analyser (Quantachrome Instruments, Boynton Beach, FL). Prior to the measurements, the samples were outgassed under vacuum at 200 °C for 10 h. The specific surface area, SBET, was determined using the Brunauer‒Emmet‒Teller (BET) equation in the range of 0.05 ≤ p/p0 ≤ 0.20; the total pore volume was obtained at p/p0 = 0.95 with the Gurvitch rule. Pore size distributions were derived from the desorption branch of the isotherms by applying the kernel of equilibrium NLDFT desorption isotherms,9,42,43 considering an amorphous silica surface and a cylindrical


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pore model. The NLDFT calculations were carried out using the Autosorb 1.55 software provided by Quantachrome Instruments. Electron Tomography. For the 3D reconstruction by electron tomography,44‒46 powder samples of SBA-15 and KIT-6 were ground in a mortar. Received crumbs were dusted over a holey Cu grid (Quantifoil Micro Tools, Jena, Germany) on which Au fiducial markers (6.5 nm diameter) were deposited from an aqueous suspension (CMC, University Medical Center, Utrecht, The Netherlands). Electron tomography was performed using the image-corrected Titan 80–300 TEM (FEI, Hillsboro, OR) at the Karlsruhe Nano Micro Facility, operated at an acceleration voltage of 300 kV in scanning transmission electron microscopy (STEM) mode with a nominal beam diameter of 0.2 nm. STEM images were collected using a high-angle annular dark-field (HAADF) collector over the tilt-range of –76° to 76° for SBA-15 and –76° to 74° for KIT-6 (in steps of 2°). Image alignment was performed using IMOD 4.747 with fiducial markers yielding an average residual alignment error of ~1.5 pixels. The subsequent 3D reconstruction was performed using TomoJ48 with 10 iterations of the implemented DART algorithm, after applying the SIRT algorithm49 with 25 iterations. Images were denoised using the nonlinear anisotropic diffusion filter implemented in IMOD. The final stack of aligned, reconstructed, and denoised images had x × y × z dimensions of 110.4 × 110.4 × 46.5 nm3 with 0.463 nm3 voxels for SBA-15 and of 140.4 × 184.0 × 115.0 nm3 with 0.463 nm3 voxels for KIT-6. Medial Axis Analysis. An iterative-thinning algorithm, available as ImageJ plug-in bundle50 (Skeletonize3D and AnalyzeSkeleton), was applied to reduce the void space in the reconstructions of SBA-15 and KIT-6 to a medial axis of one-voxel thickness under conservation of the topological properties. The average pore connectivity Z of the resulting topological skeleton was calculated as the average number of branches of the medial axis meeting at a



junction according to Z = 3nt/nj + 4nq/nj + 5nx/nj, with nx/nj = 1 – nt/nj – nq/nj, where nj is the total number of junctions, nt is the number of triple-point junctions (connecting three branches), nq the number of quadruple-point junctions (connecting four branches), and nx the number of higherorder junctions (connecting five or more branches).51 Therefore, nt/nj, nq/nj, and nx/nj provide the fraction of nodes in the network connecting 3, 4, or >4 branches, respectively. Simulation of Diffusion. Details behind the simulation of hindered diffusion by a randomwalk particle-tracking (RWPT) technique have been reported previously.22,52 Thus, the approach is only briefly described here. Initially, a number of N = 106 passive tracer particles, which do not adsorb and react at the surface or interact with each other, was distributed randomly and uniformly in the void space of a reconstruction. The elimination of mutual interactions ("collisions") between tracer particles corresponds to the absence of any correlations between their current positions. It means that the use of N tracers in a random-walk simulation is equivalent to running N independent simulations with a single tracer. This is a quite accurate approach to analyze diffusion of guest molecules in a dilute limit typical, for example, in analytical chromatography. Therefore, the value of N in our simulations should not be associated with tracer concentration. The main purpose of using a large number of tracers is to reduce the statistical noise inherent in random processes like diffusion. During each time step δt of the simulation, adjusted such that the mean diffusive displacement did not exceed ∆h/10 = 0.046 nm (with ∆h, the spatial resolution of the reconstruction), the displacement of every tracer particle due to random diffusive motion was calculated from a Gaussian distribution with zero mean and standard deviation (2Dmδt)1/2 along each Cartesian coordinate. The passive interaction of the tracer particles with the pore walls was handled through a multiple-rejection boundary condition at the surface,53 that is, when a tracer particle hit the impermeable wall during an iteration, the


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displacement was rejected and recalculated until the position of the tracer particle was in the void space. After each time step, the positions of all tracer particles were recorded and a timedependent diffusion coefficient D(t) was calculated according to

() = ∑ Δ () − 〈Δ()〉 ,

(1)

where Δ () ≡ () – (0) and 〈Δ()〉 denote the displacement of the ith tracer particle and the average displacement of the tracer ensemble after time t, respectively. Effective diffusion coefficients Deff were determined from the asymptotes of the monitored transient diffusion curves D(t), observed in the long-time limit. In contrast to point-like tracers, which have access to the entire void space of a reconstruction, the void space accessible to finite-size tracers is smaller due to their steric interaction with the hard (nonpliable), solid (impermeable) pore walls. According to an approach originally developed by Torquato,54,55 the void space accessible to the center of a finite-size tracer becomes identical to the void space accessible to a point-like tracer if the pore diameter is reduced by the tracer size dtracer. The reduction of the accessible pore space for finite-size tracers can therefore be accounted for by eroding the pore space available to point-like tracers with a structuring element of size dtracer. We applied this mathematical morphology operation to generate the accessible pore space in the reconstructions for four dtracer-values: 0.92 nm, 1.84 nm, 2.76 nm, and 3.68 nm. This information was imported into the RWPT simulations to model the diffusion of passive tracers (of size dtracer) in the reconstructions. For each reconstruction and tracer size, the simulations provided the effective diffusion coefficient Deff(dtracer) and the hindrance factor Deff(λ)/Dm, where Dm is the diffusion coefficient in the bulk liquid and λ = dtracer/dmeso is the ratio of tracer size to mean mesopore size. The accessible porosity ε at a given value of λ was available as the void volume fraction of an eroded pore space at this λ.



The program realization of the RWPT algorithm used for the simulation of hindered diffusion in the SBA-15 and KIT-6 reconstructions was implemented as a parallel code in C language using the Message Passing Interface (MPI) standard on a supercomputing platform. All numerical codes and their description can be found in the Supporting Information of ref 52. The accuracy of the presented modeling approach has been confirmed by simulations of effective diffusion coefficients in regular (simple cubic and face-centered cubic) arrays of spheres56,57 and their comparison with analytically obtained values58 (cf. Figure 5 in ref 56 and Figure 3 in ref 57).

RESULTS AND DISCUSSION Structural Characterization of the Ordered Mesoporous Silica Samples. We begin with a short characterization of the studied silica samples by standard material chemistry techniques. Figures S1 and S2 in the Supporting Information inform us about the bulk characterization of the two powder samples by low-angle XRD and nitrogen physisorption. The XRD pattern of SBA15 shows the expected four diffraction peaks, indexable as (100), (110), (200), and (210) reflections with a cell parameter a = 11.3 nm, indicating an ordered 2D mesostructure (hexagonal p6mm symmetry);1,59 the XRD pattern of KIT-6 silica reflects an ordered 3D mesostructure (cubic Ia3d symmetry) with a = 24.0 nm.2,4 The nitrogen adsorption–desorption isotherms of both samples are type IV with a H1 hysteresis loop.9 Steep capillary condensation and evaporation steps at high relative pressure indicate large, channel-like pores in a narrow size range and are characteristic of high-quality SBA-15 and KIT-6 silicas.7,8 The pore size distributions derived from these isotherms are shown in Figure 1. They reveal a narrow range of


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mesopore sizes for both materials with a sharp peak, indicating a mean pore size of dmeso= 9.4 nm, and a low shoulder left of the main peak, ranging from ~7 to ~3.5 nm (insets in Figure 1). From the values of dmeso and a, the silica wall thickness t is calculated as t = a ‒ dmeso = 1.9 nm for SBA-151 and as t = a/2 ‒ dmeso = 2.6 nm for KIT-6.7 BET surface area, total pore volume, mean pore size, unit cell parameter, and silica wall thickness of the two materials are summarized in Table 1. The shoulder in the pore size distributions of the two materials indicates the presence of smaller mesopore sizes, which are usually interpreted as belonging to the intrawall pores, based on the expectation that the primary pores are larger and of uniform size.60–69 Because the pore size distribution does not indicate where certain pore sizes are located in a structure, this interpretation must be regarded as tentative. The reconstruction of the two silica samples by electron tomography (Figure 2) brought clarity in this respect. From the reconstructions (panel C of Figure 2), sections were enlarged to visualize structural details of the two samples (Figures 3‒ 5). The section taken from the SBA-15 reconstruction shows a primary pore system of hexagonally arranged mesopores with a diameter of ~9 nm (green circles in Figure 3) and smaller mesopores in the framework walls (red circle in Figure 3), as expected from XRD and nitrogen physisorption analysis. Another view of the same section (Figure 4) shows an intrawall mesopore (circle C in Figure 4) that penetrates the silica wall and thus forms a lateral connection between the primary pores. The presence of interconnections between the primary mesopores is consistent with an earlier, high-resolution 3D SEM study of SBA-15.70 For a quantitative analysis of the pore interconnectivity (Table 2), we used medial axis analysis, which traces the pore network by a topological skeleton (panel D of Figure 2). According to medial axis analysis, SBA-15 has an average pore coordination number (which corresponds to the average



connectivity of the green topological skeleton in panel D of Figure 2) of Z = 3.42, higher than the values previously found for random mesoporous silica samples (Z = 3.14–3.21).22,71 The SBA-15 silica sample thus contains a highly interconnected, 3D network of primary and secondary mesopores. Figure 4 also reveals dead-ends and constrictions in the primary pore system (circles A and B, respectively) caused by so-called plugs, well-known for this type of material.34,72–75 Figures 3 and 4 show that the primary mesopores of the studied SBA-15 sample are not the ideal, uniformly-sized cylinders expected on the basis of XRD and nitrogen physisorption data. Although it could be argued that the studied sample is a singular rather than a typical case, our observations agree very well with earlier studies of SBA-15 silicas by low-voltage, highresolution SEM.74,75 Because the latter technique does not penetrate into the sample interior, there remained doubt whether the detected structural defects were only present on the surface. Our data corroborate that the plugs are present in the interior of the material. The section taken from the KIT-6 reconstruction (Figure 5) shows the double-gyroid system of primary mesopores (with a diameter of ~9 nm), which are laterally connected by smaller mesopores that penetrate the framework walls (circle B in Figure 5). Similar to the SBA-15 structure, the primary pore system contains constrictions (circle A in Figure 5), though dead-ends were not observed in the KIT-6 reconstruction. The presence of connecting intrawall pores is consistent with low-voltage, high-resolution SEM images, which revealed many small openings connecting the two main pore systems in the rather open pore structure of KIT-6 silica.74 These intrawall pores enhance the pore network connectivity of KIT-6 silica, whose primary pore system is already intrinsically three-dimensional. In SBA-15, the presence of connecting intrawall pores is essential to turn the 1D primary pore system into a 3D pore network. With Z = 3.33, the average pore coordination number in KIT-6 (which corresponds to the average


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connectivity of the red topological skeleton in panel D of Figure 2) is lower than in SBA-15, but still higher than in random mesoporous silicas. According to medial axis analysis (Table 2), the ordered mesoporous silicas contain a significantly higher percentage of quadruple-point (17‒ 18%) and higher-order junctions (13% and 7% for SBA-15 and KIT-6, respectively) than the previously studied random mesoporous silicas, for which 11% quadruple-point and 2% higherorder junctions have been observed.22 Hindered Diffusion in Reconstructions of SBA-15 and KIT-6. We employed the reconstructions of SBA-15 and KIT-6 silica (panel C of Figure 2) as realistic models of the mesopore morphology to simulate effective diffusion coefficients and simultaneously characterize the pore space (porosity, connectivity) that is accessible to diffusing tracers of increasing size. Through the mathematical morphology operation described in the Experimental Section we generated accessible mesopore space in the SBA-15 and KIT-6 reconstructions for the following dtracer-values: 0.92 nm, 1.84 nm, 2.76 nm, and 3.68 nm. In addition, we simulated the diffusion of point-like tracers as the limiting case, with λ = dtracer/dmeso = 0. The chosen dtracerrange covers passive-tracer dynamics in the mesopore networks of SBA-15 and KIT-6, ranging from small-to-medium sized molecules – addressing metathesis, acylation reactions, C–C coupling reactions, base-catalyzed reactions, and the immobilization of organometallic complexes for catalytic applications including chiral catalysts76 – to large molecular species (triglycerides, proteins, and enzymes),23–25,30 as well as metal–organic polyhedra,26 metal clusters,27,28 and metal nanoparticles.36 RWPT simulations provided the effective diffusion coefficient Deff(dtracer) for each tracer size and reconstruction. Figure 6 shows typical transient diffusion curves D(t) for point-like and selected finite-sized tracers (dtracer = 1.84 nm), normalized by the diffusion coefficient in the bulk



. Here, = 1 liquid Dm, as a function of the dimensionless diffusion time = 2 ⁄

corresponds to the average time needed for a tracer to diffuse a distance of dmeso. The quick drop from the initial value of D(t)/Dm = 1 reflects the proximity of the tracers to the solid (impermeable) pore walls. At sufficiently long times, the D(t)/Dm-curves reach their individual asymptotic values, which correspond to the effective diffusivities Deff/Dm. The behavior seen in Figure 6 is very typical for diffusion in porous media and can be characterized experimentally by PFG-NMR.77 The asymptotic values in Figure 6 depend on the tracer size: An increase of dtracer results in a decrease of Deff, because the extent to which diffusion is hindered grows with the tracer size. In addition, the time required to reach asymptotic values also increases with dtracer, because the accessible pore space becomes more heterogeneous (as shown below in Figure 9). Figure 7 shows the results of the diffusion simulations in the reconstructed SBA-15 and KIT-6 silica samples and compares the results to data previously obtained for three random mesoporous silica samples, which are denoted according to their mean pore size as Si12 (dmeso = 12.3 nm), Si21 (dmeso = 21.3 nm), and Si26 (dmeso = 25.7 nm).22 The curves shown in Figure 7A convey immediately that effective diffusivities decline much faster with increasing tracer size in the ordered mesoporous silicas than in the random mesoporous silicas. At a tracer size of dtracer = 1.84 nm, the effective diffusivity in SBA-15 and KIT-6 is only about one-tenth and one-fifth of its value for λ = 0, respectively, and at dtracer = 2.76 nm, which is less than a third of the mean pore size (dmeso = 9.4 nm), diffusive transport is at a standstill. The random mesoporous silicas exhibit a much shallower mobility decay, and this advantage is not owed to their larger mean pore sizes, as Figure 7B proves. Here, the data were plotted as the hindrance to diffusion, Deff(λ)/Deff(λ = 0), that evolves with increasing ratio of tracer size to mean mesopore size, λ = dtracer/dmeso. The three data sets for the random meosporous silicas (open circles in Figure 7A)


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now collapse neatly onto a single curve, which is excellently fitted by the following expression (grey line in Figure 7B)22 #$%% (&) #$%% (&')

= (1 − 1.216λ − 0.582λ − 5.199λ. + 13.350λ0 − 7.455λ3 )

(2)

This equation describes hindered diffusion in the random mesoporous silicas remarkably well, but overestimates tracer mobility in the ordered mesoporous silicas substantially (Figure 7B). Hindered diffusive transport through the random mesoporous silicas remains finite even at λ = 0.9, whereas the SBA-15 and KIT-6 samples reach the diffusion threshold already at λ < 0.3. This striking difference between ordered and random mesoporous silicas is surprising against the background of their respective pore size distributions. The narrow main peaks in the pore size distributions of SBA-15 and KIT-6 silica (Figure 1) contrast with the uniform, but wider pore size distributions of the random mesoporous silicas.22 The preceding morphological analysis of the reconstructions links the seemingly unexpected behavior of the ordered mesoporous silicas to the intrawall mesopores and the constrictions in the primary pore system (Figures 3–5), that is, to the smaller mesopore sizes represented by the low shoulder in the pore size distributions (Figure 1). When the tracer size approaches the size of the smaller mesopores in the system, the diffusion threshold is quickly reached. While this is unsurprising per se, it is remarkable that a modest amount of smaller mesopores in the structures suffices to freeze the entire transport dynamics. This would not happen if small mesopores were confined to the silica framework walls. In this case, diffusive transport would be relegated to the primary pore system and not be slowed down.78 The fast decline of diffusive transport occurs, because the presence of plugs in the primary pores obliterates the available diffusion paths in the structure. The decline of effective diffusivities with increasing tracer size seen in Figures 6 and 7 shows qualitative agreement with the behavior revealed by diffusion NMR of liquid alkanes (heptane



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and cylcohexane) in SBA-15 and KIT-6 samples possessing different mean mesopore sizes, ranging from 7.0 nm down to 3.6 nm (SBA-15) or 3.9 nm (KIT-6).19 Assuming a molecular size of 0.58 nm for cyclohexane,79 for example, the corresponding λ-range is calculated as λ = 0.08– 0.16. The data in Figure 7 show that hindered diffusion is relevant in this λ-range, which could explain the decrease of alkane diffusivities with decreasing pore size illustrated in Figure 4 of ref 19. Further morphological details of the SBA-15 and KIT-6 samples studied by Rottreau et al.19 would be needed for a more quantitative comparison, thereby highlighting also possible differences between noninteracting-tracer and liquid diffusion data with regard to molecular size effects. The combined effect of small intrawall mesopores and plugs in the primary pore system is also reflected in the properties of the pore space (porosity, connectivity) accessible to finite-size tracers in the SBA-15 and KIT-6 samples (Figures 8 and 9). The value of ε(λ), the accessible porosity at a given value of λ = dtracer/dmeso, is available as the void volume fraction of the eroded pore space at the respective dtracer-value. Figure 8 summarizes the accessible porosity data, normalized by the porosity for point-like tracers ε(λ = 0), for the reconstructed mesopore space of SBA-15 and KIT-6 silica, together with the collapsed data for the three random mesoporous silica samples. The latter data are excellently fitted by the following expression (grey line in Figure 8)22 4(&) 4(&')

= 1 − 2.200λ + 1.245λ

(3)

The pore space in the ordered mesoporous silicas closes off much faster than in the random mesoporous silicas. Figure 9 visualizes the accessible pore network in SBA-15 and KIT-6 for three different λ-values. With increasing λ, whole pathways are closed off, the pore connectivity declines, and the network thins out substantially. At λ = 0.21, the fraction of triple-point


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junctions, the dominating junction type in these materials, has dropped to ~14% (SBA-15) and ~27% (KIT-6) of its original value (at λ = 0, Figure S3 in the Supporting Information). The decrease of effective diffusivities in Figure 7 reflects the removal of entire branches from the pore network and thus complex topological changes in the accessible pore space of both materials. These changes, that is, the increasing heterogeneity of the available pore space, result in longer times for larger tracers to approach asymptotic diffusion behavior (Figure 6). The dramatic decline of diffusive mobility, accessible porosity, and pore connectivity with increasing λ, illustrated by Figures 7–9 and S3, reflects that the ordered mesoporous silica samples respond far more sensitively to changes in the size of the transported species than random mesoporous silicas. This has consequences for the applications of SBA-15 and KIT-6 silicas. In catalysis, for example, a product that significantly exceeds the size of the educts will be enriched (or even trapped) in the pore space of the material, instead of being quickly removed. If, for the sake of simplicity, the space available to the educt is represented by the green or red network at λ = 0.1 (Figure 9) and the space available to the product by the corresponding one at λ = 0.21, the product will be trapped in the pores that were green or red at λ = 0.1 and have disappeared for λ = 0.21. Although this is a drastic example, it represents the dramatic decline of mobility and pore space accessibility observed for these materials. Furthermore, the finite-size educts have only limited access to the catalytically active surface and therefore utilize only part of it. At λ = 0.1, the educts would have access to only 50–60% of the pore space and its associated surface (Figure 8). Local entrapment of species in a pore network could also be advantageous, for example, by preventing metal clusters or nanoparticles loaded onto a support structure from getting into contact and aggregate or crystallize, an important consideration in the design and assembly of



catalysts with high durability. But nanoparticles in the pore systems could also cause constrictions and obstructions similar to those formed by silica plugs (Figure 4) and thus further deteriorate the transport properties of the support structure.80 Pore space morphologies of catalytic supports should therefore be characterized twice, before loading of the precursor and after formation of the catalytic species.34,36

CONCLUSIONS The derivation of quantitative morphology–transport relationships is a major challenge for porous materials, such as the SBA-15 and KIT-6 silicas studied in this work, but worthwhile to pursue in view of understanding and improving material properties for targeted applications. Hindered diffusion plays a central role for the process performance in heterogeneous catalysis, selective gas adsorption, and controlled drug release, the major application fields of SBA-15 and KIT-6 silicas. This study has shown that predicting the extent of hindered diffusion in ordered mesoporous silicas is even more complex than in random porous media. Our reconstruction‒simulation approach revealed that structural imperfections in the primary pore system of SBA-15 and KIT-6 silica have drastic consequences for the transport properties of these materials. Constrictions (SBA-15 and KIT-6) and even dead-ends (SBA-15) obstruct the diffusion of finite-size tracers through the primary pores, so that diffusive transport comes to a standstill at a relatively modest tracer size of dtracer = 2.76 nm, less than a third of the mean pore size of dmeso = 9.4 nm. The connecting pores in the silica walls are mostly too small to provide bypasses for tracers that cannot pass the constriction in a primary pore. The problem is worse for


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SBA-15 than for KIT-6 silica, because the primary pore system of SBA-15 is one-dimensional and any connections between primary pores must come from the secondary pore system. According to their pore size distributions, the studied SBA-15 and KIT-6 silica samples contain only a low fraction of small pore sizes, which suggests that a few bottlenecks in the primary pore system suffice for a dramatic decline of the transport dynamics with increasing tracer size. Interestingly, this situation – especially for SBA-15 – reflects recent observations on guest diffusion in nanoporous silicon (tubular pores in a silicon matrix) prepared by electrochemical etching.81 That work combined molecular simulations in a pore space model based on electron tomography data with quasi-elastic neutron scattering and PFG-NMR to analyze transport resistances and diffusion anisotropy over a hierarchy of length scales. Most notably, diffusion also occurred perpendicular to the main channels via connecting bridges, resembling transport through the intrawall mesopores in SBA-15, and was impeded by constrictions in the main channel system, similar to the bottlenecks in the primary mesopores of SBA-15. The hindered diffusion characteristics of SBA-15 and KIT-6 silicas may be advantageous or detrimental to the process performance. The presence of plugs could help to prevent the leaking of guest molecules or catalytic particles from the support, but slowed-down diffusion is a decided disadvantage when fast transport to and from the active surface sites is required. Fast, unimpeded diffusion would require perfecting the primary pore system through elimination of all plugs. Whether this may or may not be achievable, the outcome is certainly difficult to monitor by standard material characterization techniques. An alternative solution, which appears easier to implement as well as to monitor, is to widen the intrawall mesopores towards the main mesopore size, so that they become bypasses for tracers around any bottlenecks in the primary pore system. This would bring the transport dynamics of the ordered mesoporous silicas closer to that of



random mesoporous silicas. Future studies should be directed at realising the benefits of ordered structures (very narrow pore size distribution, highly defined pore shape, and an ordered arrangement), which remain currently obscured by structural imperfections. These studies may be complemented by molecular dynamics simulations and PFG-NMR measurements to address important and specific surface‒solvent‒solute interactions,82 on one hand, and transport over larger scales within and between particles (including the associated exchange through the external surface of the particles),83 on the other.

ASSOCIATED CONTENT Supporting Information. Bulk characterization of the two mesoporous silica powder samples by low-angle XRD (Figure S1) and nitrogen physisorption (Figure S2). Normalized fraction of triple-point junctions in the mesopore networks of SBA-15 and KIT-6 silica as a function of the ratio of tracer size to mean mesopore size (Figure S3). This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author * Corresponding author. Phone: +49-(0)6421-28-25727; Fax: +49-(0)6421-28-27065; E-mail: [email protected] (U.T.) Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


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ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft DFG (Bonn, Germany) under grant TA 268/9–1 and by the Karlsruhe Nano Micro Facility (KNMF) at the Karlsruhe Institute of Technology (Karlsruhe, Germany) under KNMF long-term user proposal 2017-019020749. F. K. and J. F. acknowledge support of the University of Vienna (Austria). We thank Professor Ryong Ryoo and his group (KAIST, Daejeon, Republic of Korea) for the powder XRD patterns of the two investigated samples.

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(69) Zhang, Z.; Melián-Cabrera, I. Modifying the Hierarchical Porosity of SBA-15 via MildDetemplation Followed by Secondary Treatments. J. Phys. Chem. C 2014, 118, 28689–28698. (70) Che, S.; Lund, K.; Tatsumi, T.; Iijima, S.; Joo, S. H.; Ryoo, R.; Terasaki, O. Direct Observation of 3D Mesoporous Structure by Scanning Electron Microscopy (SEM): SBA-15 Silica and CMK-5 Carbon. Angew. Chem., Int. Ed. 2003, 42, 2182–2185. (71) Stoeckel, D.; Kübel, C.; Hormann, K.; Höltzel, A.; Smarsly, B. M.; Tallarek, U. Morphological Analysis of Disordered Macroporous–Mesoporous Solids Based on Physical Reconstruction by Nanoscale Tomography. Langmuir 2014, 30, 9022–9027. (72) Van Der Voort, P.; Ravikovitch, P. I.; de Jong, K. P.; Neimark, A. V.; Janssen, A. H.; Benjelloun, M.; Van Bavel, E.; Cool, P.; Weckhuysen, B. M.; Vansant, E. F. Plugged Hexagonal Templated Silica: A Unique Micro- and Mesoporous Composite Material with Internal Silica Nanocapsules. Chem. Commun. 2002, 9, 1010–1011. (73) Kruk, M.; Jaroniec, M.; Joo, S. H.; Ryoo, R. Characterization of Regular and Plugged SBA-15 Silicas by Using Adsorption and Inverse Carbon Replication and Explanation of the Plug Formation Mechanism. J. Phys. Chem. B 2003, 107, 2205–2213. (74) Tüysüz, H.; Lehmann, C. W.; Bongard, H.; Tesche, B.; Schmidt, R.; Schüth, F. Direct Imaging of Surface Topology and Pore System of Ordered Mesoporous Silica (MCM-41, SBA15, and KIT-6) and Nanocast Metal Oxides by High Resolution Scanning Electron Microscopy. J. Am. Chem. Soc. 2008, 130, 11510–11517. (75) Kjellman, T.; Asahina, S.; Schmitt, J.; Impéror-Clerc, M.; Terasaki, O.; Alfredsson, V. Direct Observation of Plugs and Intrawall Pores in SBA-15 Using Low Voltage High Resolution



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Table 1. Structural Properties of the Mesoporous Silica Powder Samples.a

SBA-15

KIT-6

SBET [m2 g–1]

554

614

Vt [cm3 g‒1]

1.11

1.20

dmeso [nm]

9.4

9.4

a [nm]

11.3

24.0

t [nm]

1.9

2.6

a

Specific surface area (SBET), total pore volume (Vt), and mean mesopore size (dmeso) from nitrogen physisorption analysis (Figure S2 in the Supporting Information); unit cell parameter (a) from XRD analysis (Figure S1); silica wall thickness (t) calculated from cell parameter and mean mesopore size.

Table 2. Connectivities of the Topological Skeleton of the Reconstructed Mesopore Spaces.a

a

SBA-15

KIT-6

nt/nj [%]

70.3

74.5

nq/nj [%]

17.1

17.8

nx/nj [%]

12.6

7.7

Z

3.42

3.33

Percentage of nodes connecting 3, 4, or >4 branches (nt/nj, nq/nj, and nx/nj, respectively) as well as the resulting average pore connectivity (Z).


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Figure 1. Pore size distributions derived for bulk powder samples of SBA-15 (A) and KIT-6 silica (B). The maximum is at 9.4 nm for both materials. Insets enlarge the shoulder left of the main peak.



Figure 2. Reconstruction of the mesopore space of SBA-15 (top row) and KIT-6 silica (bottom row) by STEM tomography. Raw images (A) were segmented (B) and assembled into 3D structures (image stacks, C) with final dimensions of 110.4 × 110.4 × 46.5 nm3 (x × y × z, SBA-15) and 140.4 × 184.0 × 115.0 nm3 (KIT-6). (D) The void space of the reconstruction (available to pointlike tracers for diffusion) visualized by the topological skeleton; pores are represented by lines of one-voxel thickness to show the connectivity of the mesopore network.


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Figure 3. 2D slice (left) and selected 3D section (front and side view) from the reconstruction of SBA-15 silica, highlighting the hexagonal arrangment of primary mesopores (green circles) and a smaller mesopore in the silica wall (red circle).



Figure 4. Section from the reconstruction of SBA-15 silica, highlighting a dead-end (A) and a constriction (B) in the primary pore system as well as a smaller throughpore in the silica wall (C).


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Figure 5. 2D slice (left) and selected 3D sections from the reconstruction of KIT-6 silica. The center panel shows the primary mesopore system (solid is in grey) with a highlighted constriction (red circle). In the right panel, the void space is colored red to visualize an intrawall mesopore (red circle).



Figure 6. Diffusion of point-like and selected finite-size tracers in the reconstructed mesopore spaces of SBA-15 and KIT-6 silica. Transient diffusion coefficients D(t), normalized by the diffusion coefficient in the bulk liquid Dm, are shown as a function of the dimensionless diffusion time = 2 ⁄ .


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Figure 7. Diffusion of point-like and finite-size tracers in ordered and random mesoporous silicas. (A) Effective diffusion coefficient Deff, normalized by the diffusion coefficient in the bulk liquid Dm, as a function of the tracer size dtracer. (B) Effective diffusion coefficient Deff(λ), normalized by the corresponding value for point-like tracers Deff(λ = 0), as a function of λ = dtracer/dmeso, the ratio of tracer size to mean mesopore size. The grey curve represents the unifying expression (eq 2) obtained for random mesoporous silicas.22



Figure 8. Tracer-size dependent evolution of the accessible void volume fraction in ordered and random mesoporous silicas. Shown is the accessible porosity ε(λ), normalized by the corresponding value for point-like tracers ε(λ = 0), as a function of λ = dtracer/dmeso, the ratio of tracer size to mean mesopore size. The grey curve represents the unifying expression (eq 3) obtained for random mesoporous silicas.22


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Figure 9. Tracer-size dependent evolution of the accessible void space, represented as the topological skeleton, in SBA-15 and KIT-6 silica. At λ = 0.21 (i.e., for a tracer size of dtracer = 1.84 nm), the accessible porosity ε(λ) has dropped to 15–20% of its original value ε(λ = 0) (cf. Figure 8).



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Hindered Diffusion in Ordered Mesoporous Silicas: Insights from Pore

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