Tracer-Size-Dependent Pore Space Accessibility and Long-Time

Article pubs.acs.org/JPCC

Tracer-Size-Dependent Pore Space Accessibility and Long-Time Diffusion Coefficient in Amorphous, Mesoporous Silica Dzmitry Hlushkou, Artur Svidrytski, and Ulrich Tallarek* Department of Chemistry, Philipps-Universität Marburg, Hans-Meerwein-Strasse 4, 35032 Marburg, Germany S Supporting Information *

ABSTRACT: Heterogeneous porous materials are widely used as fixed beds in adsorption, separation, and catalysis. An important aspect of mass transport in these materials is the hindrance to diffusion of molecules with dimensions approaching the pore size. We numerically study hindered diffusion of finite-size, passive (i.e., nonadsorbing, nonreacting) tracer particles in physically reconstructed amorphous, mesoporous silica from a hierarchical, macroporous−mesoporous silica monolith. The long-time, effective diffusion coefficient Deff is determined as a function of λ, the ratio of the tracer diameter to the average size of the mesopores. Results demonstrate a strong reduction of Deff with increasing tracer size. Comparison with theoretical models of hindered diffusion in a single uniform, cylindrical pore reveals that these models significantly overestimate Deff for λ > 0.2. Morphological analysis of the mesopore pore space accessible for finite-size tracers shows that its effective geometrical and topological properties are a function of λ. The nonuniform distribution of mesopore size results in an increased fraction of pores that become impermeable for larger tracers. As a consequence, not only is the accessible mesopore space (porosity) reduced for larger values of λ but also the connectivity of the accessible pore network decreases. Thus, in contrast to a single uniform pore, an increase of tracer size leads to two complementary effects in a heterogeneous porous medium: a constriction of the accessible pore space in individual pores and a reduction of the number of available diffusion paths in the pore network.

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nm ≤ dpore ≤ 50 nm) and/or micropores (dpore < 2 nm). In contrast to the macropore domain, diffusion-limited transport prevails in the mesopores and micropores, where the tailored surface chemistry and the required surface area are critical to specific molecule−surface interactions. Therefore, the knowledge of relevant transport properties along with eventual transport limitations is essential for the targeted performance optimization of functional porous materials.7−10 Many theoretical, experimental, and numerical studies reveal that diffusive transport properties of a heterogeneous material depend strictly on its pore space morphology.1,11−15 However, the derivation of quantitative relationships between morphological characteristics of porous media and the effective diffusivity of transported species is still an outstanding scientific problem. A major challenge is the identification of morphological descriptors that adequately characterize diffusive mass transport.16,17 This problem becomes even more complex for porous media with different porosity scales.6,18 For instance, it was recently shown that diffusion in materials with hierarchical structure is strongly affected not only by the accessibility of the pore space but also by the interconnectivity of pore networks in

INTRODUCTION Characterization of the mass transport properties of heterogeneous disordered materials is one of the major challenges in industrial, biological, and environmental processes, including chemical separations, catalysis, transport through cell membranes and diatom frustules, confinement of highly radioactive wastes, and groundwater remediation.1−4 A prominent class of porous media, hierarchically structured materials, typically consist of spatial domains with different morphology and physical characteristics of the pore space.5 This results in specific transport properties for the different domains, e.g., domination of diffusion in small pores, in contrast to composite advective−diffusive mass transport in large pores.6 Examples of such hierarchically structured materials are silica and organicpolymer monoliths or packings of porous particles used as fixed-bed adsorbers, separators, and reactors. The hierarchical structure of the void space in these materials is typically characterized by two or three scales in pore size, dpore. Interconnected macropores (dpore > 50 nm) form a continuous, highly permeable network of flow-through channels in the interskeleton and interparticle void space of monoliths and particle-packed beds, respectively. These flow-through pores are responsible for advection-dominated transport through the material and toward the pore networks inside the monolith skeleton or the porous particles, which contain mesopores (2 © XXXX American Chemical Society

Received: January 10, 2017 Revised: March 30, 2017 Published: April 4, 2017 A

DOI: 10.1021/acs.jpcc.7b00264 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. Hierarchical pore space morphology of a macroporous−mesoporous silica monolith (adapted from ref 45). The macropores form a network of flow-through channels in the interskeleton void space, which is dedicated to advection-dominated transport through the material and toward the intraskeleton network of mesopores, in which transport remains diffusion-limited.

different spatial domains.16,19−21 Another important aspect in the development of quantitative relationships between morphology and mass transport properties of porous media is to account for the hindrance to transport of finite-size molecules, relevant when the molecular size becomes comparable with the pore size. In a liquid-filled pore, this hindrance is attributed to a combination of molecule−wall hydrodynamic interactions and steric resistance. However, in porous media with nonuniform pore size, local molecular sieving effects occur if the size of the molecules is larger than dpore. Therefore, the precise determination of the void space structure in porous media is a prerequisite for the accurate characterization of their transport properties. Several imaging techniques are nowadays available to receive information on the geometrical structure of porous media for three-dimensional (3D) physical reconstruction.22 The capability of these experimental approaches range, e.g., from nanometer resolution with techniques based on electron microscopy,23−27 through micron and submicron scale with X-ray tomography28−30 and confocal laser scanning microscopy,18,31−33 to several tens of micrometers with nuclear magnetic resonance imaging.34−36 The gained 3D information on the heterogeneous material structure can be directly used for pore-scale simulations of diffusive transport. However, it has been shown that the spatial resolution of the 3D reconstruction has to be much finer than the morphological feature size to ensure that possible effects of medium structure on the transport characteristics are adequately captured.37 Only in a few studies, information on the void space, obtained by physical reconstruction of porous materials with sufficiently fine resolution, was used to investigate numerically the diffusion of finite-size tracer particles. Langford et al.15 employed Brownian dynamics simulations to model diffusion of spherical probes of different sizes (varying from 0 up to 70 nm) in chromatographic adsorbents physically reconstructed using electron tomography with a resolution of about 3 × 3 × 5 nm. Average pore sizes determined from emulating inverse sizeexclusion chromatography on three reconstructed adsorbents were 28.8, 60.6, and 100.0 nm. Simulations revealed that effective probe diffusivity decreased with increasing probe size. In addition, the diffusive transport characteristics of the three studied adsorbents could be directly related to different connectivities of the reconstructed pore networks in these samples. However, changes in connectivity of the pore network accessible for tracer particles of different size were not analyzed.

A similar trade-off between diffusivity and tracer size was reported by Müter et al.38 They modeled the diffusion of finitesize particles in physically reconstructed pore networks of chalk samples, combining 3D X-ray imaging with a resolution of 25 nm and a dissipative particle dynamics approach. Their simulations showed that an increase in the tracer particle diameter from 1% to 35% of the average pore size can reduce the effective diffusion coefficient by as much as 60%, but no accompanying analysis of the accessible pore networks in the reconstructed samples has been conducted. In this study, we investigate numerically the impact of the diameter of passive tracer particles on their effective, i.e., longtime asymptotic, diffusion coefficient in the intraskeleton mesopore space of a hierarchical macroporous−mesoporous silica monolith together with an analysis of the morphological properties of the accessible void space. These silica monoliths, attractive supports for use in chemical separations and heterogeneous catalysis,39−42 have a hierarchical structure of the pore space realized by the perforation of a continuous silica block with intersecting networks of macropores and mesopores. Although advective−diffusive transport in the physically reconstructed interskeleton macropore space of these monoliths has been studied numerically in a number of studies,43,44 the analysis of diffusion-limited transport in the intraskeleton mesopore space is still an outstanding problem, especially for molecular species with dimensions approaching the mean mesopore size. The main challenge is the acquisition of comprehensive, high-resolution spatial information on the mesoporous void space. Here, we adapt information about the 3D structure of a mesoporous skeleton from a hierarchical silica monolith (Figure 1), previously obtained using scanning transmission electron microscopy (STEM) with a spatial resolution of Δh = 0.47 nm (resulting in a ratio between mean mesopore size and Δh of ∼34).45 The effective diffusion coefficient (Deff) in the reconstructed mesopore space of the monolith is determined as a function of the tracer diameter by numerical simulations using a random-walk particle-tracking approach. For the first time, we demonstrate that actual constriction of the accessible pore space, occurring with increasing tracer size, is accompanied by a substantial reduction in connectivity of the accessible pore network. We compare simulated values of Deff with those obtained according to theoretical hydrodynamic models for hindered diffusion of a finite-size particle in a single cylindrical pore. Our results show that these models fail to describe B


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Then, the change of the above two parameters in dependence of the domain size was analyzed (Figure 3). The calculated center of mass in the reconstructed crumb was placed in the void space of a mesopore. As a result, the porosity and the specific surface area of the initial (1 × 1 × 1 voxel) domain are unity and infinity, respectively. With increasing domain size, these parameters approach virtually constant values, indicating that the domain becomes statistically representative and its averaged morphological characteristics approach those of the reconstructed crumb. On the basis of the analysis of the data in Figure 3, a cubic domain with edge length of 100 nm (dashed lines in Figure 3A,B) was chosen for our diffusion simulations in the amorphous, mesoporous silica skeleton of the monolith. The 3D structure of this cubic domain is provided in the Supporting Information as a set of TIFF files with images of 2D slices.

adequately hindered diffusion in a heterogeneous pore network, because they do not account for the dependence of the accessible pore space topology on tracer size.

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EXPERIMENTAL SECTION To analyze the influence of tracer size on the effective diffusion coefficient in the mesoporous skeleton of the silica monolith by numerical simulations, we used information on its geometrical structure obtained previously by 3D physical reconstruction.45 The silica monolith was prepared in-house following a known synthesis route. Briefly, monolith synthesis comprised formation of a macroporous−microporous silica rod through phase separation, gelation, and aging, followed by widening the micropores to mesopores through hydrothermal treatment. Further details on the synthesis procedure can be found elsewhere.45 The prepared macroporous−mesoporous monolith rod (cf. Figure 1, left panel) had a diameter of 1.1 cm. Standard nitrogen physisorption measurements have been used to already assess the mesopore size distribution in the prepared monolith (see the Supporting Information for ref 45). The mean mesopore size ⟨dpore⟩ obtained by analysis of the physisorption data is 15.87 nm, but along with relatively large mesopores (dpore > 20 nm), there also exists a fraction of pores with dpore < 5 nm. We will return to this point in our discussion of the simulation results for diffusion of finite-size tracer particles in the reconstructed mesoporous skeleton of the monolith. The void space of the amorphous, mesoporous silica from the monolith skeleton (cf. Figure 1, right panel) was physically reconstructed by STEM tomography. The reconstructed mesoporous silica crumb occupying a volume of ca. 245 × 430 × 240 nm is shown in the left panel of Figure 2. A cubic

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NUMERICAL APPROACHES Diffusive mass transport in the cubic domain shown in Figure 2 was simulated by a random-walk particle-tracking (RWPT) technique.46,47 Initially, a large number (N) of passive (nonadsorbing, nonreacting) tracers was randomly distributed within the void space. In this work, we used N = 5 × 106, which is approximately equal to the number of voxels in the void space of the reconstructed cubic domain. Then, during each time step (δt), the displacement of every tracer due to random diffusive motion was calculated from a Gaussian distribution with a mean of zero and a standard deviation of (2Dmδt)1/2 along each Cartesian coordinate, where Dm is the molecular diffusion coefficient in free space. The time step δt was defined such that the mean diffusive displacement did not exceed Δh/ 10, where Δh = 0.47 nm is the spatial resolution of the physical reconstruction. If at the current time iteration a tracer hits the solid wall, this displacement is rejected and recalculated until the tracer position is in the void space, realizing the multiplerejection boundary condition at the solid surface.48 Positions of tracers were monitored after each time step, and timedependent diffusion coefficients D(t) were calculated from tracer displacements as D(t ) =

1 d 6N dt

N

∑ [Δri(t ) − ⟨Δr(t )⟩]2 i=1

(1)

where Δri(t) and ⟨Δr(t)⟩ denote the displacement of the ith tracer and the average displacement of the tracer ensemble after time t, respectively. The effective diffusion coefficients Deff were determined from the long-time asymptotes of the D(t) curves. The accuracy of this modeling approach was confirmed by comparing simulated values of Deff in regular arrays of spheres11,49 with values calculated using the analytical approach50 (cf. Figure 5 in ref 11 and Figure 3 in ref 49). Among the advantages of the employed RWPT approach are conservation of mass, absence of numerical dispersion, simplicity of program realization, and straightforward parallelization, allowing an implementation of the numerical model in parallel high-performance computational systems (supercomputers). Whereas the entire void space of the porous material can be sampled by pointlike tracers, the accessible void space for finitesize tracers is smaller due to their steric interaction with the solid pore walls. For instance, the center of a hard spherical tracer with diameter dtracer, diffusing in a cylindrical pore, cannot be found in an annular region adjoining the pore wall with a

Figure 2. An irregularly shaped reconstructed mesoporous silica crumb (left panel) and the cut cubic domain (right panel) used for the simulation of diffusion of finite-size tracer particles.

domain used for simulation of diffusion (right panel in Figure 2) was virtually cut from the reconstruction according to the following numerical procedure. First, the mass center of the crumb and the corresponding voxel, i.e., the initial 1 × 1 × 1 voxel domain, were determined. (The mass center does not coincide with the geometrical center because of the irregular shape of the reconstruction.) Next, the dimension of the domain along each direction was iteratively increased by one voxel, resulting in a set of domains with size of (3 × 3 × 3), (5 × 5 × 5), etc. voxels. At each iteration, the average porosity (void volume fraction) and specific surface area (ratio of solid surface area to mass) of the domain were calculated. These parameters belong to the constitutive morphological characteristics determining mass transport properties of porous media. C


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Figure 3. Average porosity (A) and specific surface area (B) of the cubic domain cut from the reconstructed mesoporous skeleton of the silica monolith (cf. Figure 2) as a function of the domain edge length. The dashed lines correspond to the size of the domain used in this study with an average porosity of 0.583 and a specific surface area of 373 m2/g.

thickness of dtracer/2 (Figure 4). Thus, the void space accessible for the center of a finite-size tracer becomes identical to that for

a pointlike tracer, if the pore diameter is reduced by the value of dtracer. A similar effect is observed in heterogeneous porous media. Figure 5A shows an individual two-dimensional slice of the reconstructed amorphous, mesoporous silica. The entire void space (white) is accessible for pointlike tracers, i.e., when dtracer = 0. By contrast, the accessible void space is significantly reduced for finite-size tracers. The red-colored regions in Figure 5B, for example, highlight the fraction of the pore space that is actually inaccessible for the centers of tracer particles with dtracer = 0.94 nm. This reduction of the pore space accessible for finite-size tracers can be equivalently determined by eroding the pore space (accessible for pointlike tracers) with a structuring element of the size of dtracer.14,24,27,51 We employed this mathematical morphology operation to determine the accessible mesopore space in the spatial domain shown in the right panel of Figure 2 for eight values of dtracer monotonically varied from 0.94 nm up to 7.52 nm. Then, this information was imported to the RWPT-simulations to model diffusion of passive tracers with a size equal to dtracer. The program realization of the RWPT algorithm was implemented as parallel code in C language using the Message Passing Interface (MPI) standard. The total computational time took about 7000 core-hours on the JUQUEEN super-

Figure 4. Illustration of the steric interaction between a spherical tracer particle with diameter dtracer and the solid wall of a cylindrical pore with diameter dpore. The red-colored annular area corresponds to the region that is inaccessible for the center of the tracer.

Figure 5. (A) Two-dimensional slice from the reconstructed mesoporous skeleton of the silica monolith (black is solid, white is void space). For a pointlike tracer (dtracer = 0), the entire void space is accessible. (B) Same slice, but with a fraction of the void space (red) inaccessible for the centers of spherical tracers with diameter dtracer = 0.94 nm. D


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correspond to effective diffusion coefficients Deff, which depend on λ: An increase of λ results in a decreased value of Deff, indicating more hindered diffusive transport for larger tracer particles. The results presented in Figure 6 were obtained for tracers randomly distributed at t = 0 in the entire void space of the reconstructed domain and with the mirror boundary condition imposed at the external faces of that domain. Implementation of the mirror boundary condition assumes that a tracer crossing an external face of the domain continues its motion in the mirror image of the original domain. An advantage of this approach is that, at any time, the tracer ensemble samples the complete accessible pore space and therefore reflects the spatially averaged (macroscopic) morphological properties of a structure along with their effect on the overall diffusive transport. However, the realization of the mirror boundary condition results in artificially symmetrical edge cavities, which may affect the value of Deff.15 To analyze eventual effects of mirror boundaries, we conducted additional simulations with the “annihilation” boundary condition imposed at the domain edges. In this case, all N tracers are initially placed in the center of the reconstruction (more properly, in the void voxel closest to the geometrical center of the cubic domain). Then, in contrast to eq 1, the transient diffusion coefficient Di(t) is determined for each individual tracer particle as

computer at the Jü l ich Supercomputing Center (JSC, Forschungszentrum Jülich, Jülich, Germany). The programs adapted for the calculation of the effective diffusion coefficients reported in this paper are available in the Supporting Information.

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RESULTS AND DISCUSSION With the approaches described above, we simulated diffusion of spherical tracer particles for nine different values of dtracer (varied from 0 up to 7.52 nm) in the reconstructed mesoporous domain of the silica skeleton (cf. Figures 1 and 2). In this study, the parameter λ is defined as the ratio of dtracer and the mean pore size ⟨dpore⟩ = 15.87 nm, determined from the N2physisorption-based pore size distribution. In dependence of dtracer, corresponding values of λ varied from 0.0 up to 0.474. The simulated transient diffusion coefficients D(t) normalized by Dm are shown in Figure 6 as a function of the dimensionless

ri2(t ) 1≤i≤N (2) 6t where ri(t) is the displacement of the ith tracer after time t from its initial position. Averaging Di(t) over the tracer ensemble provides the mean D(t): Di(t ) =

Figure 6. Transient diffusion coefficients D(t) in the reconstructed mesoporous skeleton of the silica monolith, normalized by the molecular diffusion coefficient in free space Dm, as a function of the dimensionless diffusion time tD = 6Defft/(L/2)2. The color-coding corresponds to the nine different values of λ, i.e., the ratio of the tracer diameter dtracer and mean pore size ⟨dpore⟩ = 15.87 nm. Deff is the asymptotic (long-time) value of D(t) determined for the respective value of λ, and L = 100 nm is the dimension of the adapted cubic simulation domain (cf. Figure 2). The column to the right summarizes the Deff/Dm data, color-coded in correspondence to the λ values.

D(t ) =

1 N

N

∑ Di(t ) i=1

(3)

If a tracer crosses an external face of the domain, it is excluded from further simulation. Figure 7 shows the time

diffusion time tD = 6Defft/(L/2)2; L = 100 nm is the dimension of the reconstructed cubic domain used for the simulations (cf. Figure 2, right panel). Here, a value of tD = 1.0 corresponds to the average time needed for a tracer to diffuse the distance of L/2 in an isotropic and macroscopically homogeneous porous medium, where the dimension of the local (microscopic) structural heterogeneities is much smaller than L. For all values of λ, D(t) decreases with time from the initial Dm value due to the resistance to tracer motion arising from interactions with the pore walls.15,38,52 This transient behavior of the diffusion coefficient, specific for porous media, was also observed in numerous experiments performed using the pulsed field gradient technique of NMR, which allows one to determine the evolution of D(t) in inhomogeneous media with a characteristic length of structural heterogeneities down to hundreds of nanometers.52 At long times, the transient diffusion coefficients shown in Figure 6 approach asymptotic values, with superimposed stochastic noise inherent for random processes. The amplitude of this noise decreases approximately with the square root of N (the number of tracers used for the RWPT simulations). However, the long-time asymptotic values of D(t) are not affected by N. These asymptotic values

Figure 7. Time-evolution of normalized transient diffusion coefficients D(t)/Dm following different initial tracer distribution for two selected values of λ = dtracer/⟨dpore⟩.

evolution of normalized transient diffusion coefficients D(t)/ Dm obtained for λ = 0.0 and 0.237 using these two different conditions regarding initial tracer distribution. The black curves correspond to the condition that tracers are initially distributed randomly in the entire pore space, the mirror boundary condition is imposed, and D(t) is calculated according to eq 1; E


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cylindrical pore by resolving the problem of enhanced drag due to the hydrodynamic Stokes friction effect, i.e., the increased molecular friction coefficient in constrained space compared to its value for an unbounded solution. Assuming that the drag coefficient is constant over the cross-section and equal to the drag coefficient at the pore centerline (centerline approximation), Renkin53 obtained the following expression for Deff(λ)/Deff(λ=0):

the red curves were obtained under the condition that all tracers are initially placed in the center of the domain, the “annihilation” boundary condition is employed, and D(t) is calculated using eqs 2 and 3. For tD < 0.4, the two types of initial tracer distribution result in different behavior of D(t). If all tracers are initially placed at the same position (red curves in Figure 7), the diffusivity at short times is determined only by the local pore structure close to that initial position. In the course of time, tracers diffuse from their initial position and gradually sample a more extended region of the mesopore space. Then, the actual value of D(t) becomes less sensitive to local pore structure near the starting point and instead reflects the spatially averaged diffusive properties of the whole structure. As a consequence, the long-time asymptotic values of D(t) obtained with the different approaches are indeed very close and do not reveal a noticeable effect of the boundary condition imposed at the domain edges and the initial distribution of tracers. On the basis of the results of this case study, we adapted the random initial distribution of tracer particles and the mirror boundary condition for our simulations. The normalized asymptotic values of Deff obtained with the simulations for different values of λ are presented in Figure 8.

Deff (λ) = (1 − λ)2 (1 − 2.104λ + 2.09λ 2 − 0.95λ 3) Deff (λ=0) (4)

Though eq 4 was originally developed to describe hindered diffusion of a finite-size particle in a single cylindrical pore, it is widely used to characterize hindrance to diffusion in heterogeneous mesoporous media.55−62 Later, Dechadilok and Deen54 improved eq 4, using the cross-sectional averaged drag coefficient, and derived the following expression: Deff (λ) 9 = 1 + λ ln λ − 1.56034λ + 0.528155λ 2 Deff (λ=0) 8 + 1.91521λ 3 − 2.81903λ 4 + 0.270788λ 5 + 1.10115λ 6 − 0.435933λ 7

(5)

In Figure 8, good agreement is observed for λ < 0.2 between the simulated data and the results obtained with eqs 4 and 5. However, for larger λ the simulated values of Deff(λ)/Deff(λ=0) are significantly smaller than those predicted with the above theoretical models. This discrepancy is explainable by the fact that eqs 4 and 5 have been developed for hindered diffusive transport in a single, uniform, cylindrical pore. Indeed, the mesopore network in the reconstructed amorphous silica skeleton is morphologically complex and cannot be reduced to an equivalent single pore. Figures 2 and 5 demonstrate that the void space in the reconstruction consists of interconnected, heterogeneous pores of varying shape and dimension. Inspection of the pore size distribution45 reveals the existence of a significant fraction of small pores with dpore < 5 nm, though the average pore size ⟨dpore⟩ used to define λ in this study is 15.87 nm. Figure 5B illustrates that already for relatively small tracer particles with dtracer = 0.94 nm (λ = 0.059), not only is the accessible void space reduced but also some of the pore throats become inaccessible for the diffusing particles. Thus, both the geometry and topology of the accessible void space in a porous medium change with the tracer diameter. Therefore, the overall effect of tracer size on the effective diffusivity in a porous medium cannot be reduced to exclusively the hydrodynamic effect characterized by the value of λ, which is based on the average pore dimension. Even in the same porous medium, tracer particles of different size diffuse in diverse accessible void spaces with different morphological properties. Figure 9 shows the accessible porosity (εa ) in the reconstruction as a function of λ. The value of εa is defined as the ratio of void space accessible for the centers of the finitesize tracers and the total volume of the reconstructed domain. With increasing tracer diameter, accessible pore space decreases, reflecting its constriction due to the steric effect illustrated by Figure 4. In contrast to a cylindrical pore, where this effect can be represented as isomorphic transformation of the pore space, this transformation is far more complex for the accessible void space in a heterogeneous material. In pores which are relatively large compared to the tracer diameter, an

Figure 8. Effective diffusion coefficient Deff as a function of λ = dtracer/ ⟨dpore⟩, normalized by its value for λ = 0. Black circles correspond to numerical simulation of diffusion in the reconstructed mesoporous skeleton of the silica monolith [the following values of Deff(λ)/ Deff(λ=0) are obtained with increasing λ: 1.0, 0.788, 0.563, 0.321, 0.0974, 0.0103, 3.21 × 10−3, 7.93 × 10−4, and 4.72 × 10−4]. The blue and red curves correspond to Deff(λ) relationships derived from the theoretical models for hindered diffusion in a single cylindrical pore by Renkin53 (eq 4) and Dechadilok and Deen54 (eq 5).

Here, the black circles correspond to values of Deff(λ) determined by averaging the D(t) curves in Figure 6 over the long-time range 0.9 ≤ tD < 1.0, where the transient diffusion coefficients do not exhibit any monotonic or systematic changes with time. This time-averaging procedure also eliminates the statistical noise observed for the simulated D(t) curves. In Figure 8, the values of Deff(λ) are normalized by Deff(λ=0), the effective diffusion coefficient obtained for pointlike tracer particles, which allows one to evaluate the hindrance to diffusion due to the finite size of the tracers. Blue and red curves in Figure 8 represent the values of Deff(λ)/Deff(λ=0) calculated according to relationships derived from models by Renkin53 and Dechadilok and Deen,54 respectively. These relationships were derived for hindered diffusion of finite-size spherical particles in a single uniform, F


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either belonging to a certain pore or shared by several pores. (iv) The pore boundaries are propagated to reduce shared pore volumes to single-voxel-thick boundary layers by watershed segmentation. The delineated boundaries are the watershed between individual pores at whose centers the imagined water sources are located. Figure 10 shows a single slice of the 3D reconstruction with the compartmentalized accessible mesopore space for λ = 0 (left panel) and λ = 0.237 (right panel), as obtained with the above procedure. Here, white color represents the pore space accessible for the centers of the tracers. It is composed of irregularly shaped and differently sized allotments, bordered by the pore throats colored red. The compartmentalization of the pore space was performed in 3D, while Figure 10 shows twodimensional sections of the compartmentalized pore networks for better visualization (pore throats are represented by segments of curved 3D surfaces). Figure 10 demonstrates that with increased tracer diameter (and correspondingly larger value of λ), not only the accessible pore space is strongly reduced but also the entire mesopore network becomes less connected. This effect is further illustrated with Figure 11, showing the mean diffusion paths obtained by skeletonization of the accessible pore space for tracer particles with diameters corresponding to λ = 0 and 0.237. The skeletonization procedure is based on an iterative-thinning algorithm, which reduces the pore space to a continuous medial axis of one voxel thickness under conservation of its topological properties.63,64 An implementation of this algorithm is available as a plug-in (Skeletonize3D) of the open-source image-processing program Fiji.65,66 The results presented in Figure 11 demonstrate that the topology of the accessible pore network is completely different for pointlike and finite-size tracers. The inability of the finite-size tracers to penetrate a pore for dpore < dtracer causes a strong reduction of the accessible-pore network connectivity and decreases the number of possible diffusion paths. Since pore throats are the connecting element in the compartmentalized representation of the pore space, we can evaluate the connectivity of the accessible pore network also from the perspective of the pore throats. By counting the number of pores that share a particular throat (cf. Figure 10), we receive the distribution of the pore throat coordination number.63 We applied the compartmentalization procedure to analyze the pore throat coordination number distribution for all nine values of λ used in this study. Results of this analysis are

Figure 9. Accessible porosity in the reconstructed mesoporous skeleton of the silica monolith determined for different values of λ = dtracer/⟨dpore⟩.

increase in dtracer leads to a reduction of the geometrical locus accessible for the centers of the tracers (similar to a single uniform pore). However, the same increase in dtracer also results in an impermeability of small pores with dpore < dtracer for finitesize tracer transport. Furthermore, the presence of such virtual dead-end and inkbottle pores results in a reduced connectivity of the accessible pore network. To quantify the changes in pore connectivity with varied tracer size, we analyzed the throat coordination numbers of the pores.63 For this purpose, the intrinsic borders made by the solid phase (solid−void borders) were supplemented with calculated boundaries in the mesopore network (void−void borders), which are referred to as pore throats. The compartmentalization of the reconstructed mesopore space into a set of individual pores delimited by pore throats proceeds along the following steps.63 (i) The largest sphere that can be inscribed around each voxel into the pore space is determined by calculation of the smallest distance to the pore wall through Euclidean distance transform. (ii) The complete set of inscribed spheres is reduced to a set of containing spheres by assigning each void voxel the radius of the largest sphere in which it is contained. The resulting field of containing sphere radii holds local maxima. (iii) The centers of containing spheres situated at local maxima are used as seed points for pore propagation. All void voxels are assigned as

Figure 10. Two-dimensional view of compartmentalized mesopore space accessible for pointlike tracers (λ = 0.0) and for tracers with dtracer = 3.84 nm (λ = 0.237). Black is the solid, white is the accessible void space, and red corresponds to the pore throats. G


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Figure 11. Mean diffusion paths accessible in the reconstructed mesoporous skeleton of the silica monolith for pointlike tracers (λ = 0.0) and for tracers with dtracer = 3.84 nm (λ = 0.237).

summarized in Figure 12. With increasing λ, the total number of compartmentalized pores is strongly reduced. For instance,

coefficient requires precise knowledge of the porous medium morphology. This aspect can be important also in heterogeneous catalysis applications, when hindered diffusive transport engenders a transport limitation to a process that was expected to be reaction-limited.14,19

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CONCLUSIONS We have reported a comprehensive modeling approach that combines three-dimensional physical reconstruction of the mesoporous skeleton from a silica monolith with a randomwalk particle-tracking method to simulate diffusion of finitesize, passive tracer particles in the reconstructed mesopore space. Up until now, rigorous analytical models of hindered diffusion of finite-size tracer particles have been developed only for a single uniform pore. These models account for the hydrodynamic Stokes friction effect, i.e., the increased molecular friction coefficient in constrained space compared to its value for an unbounded solution. Expressions obtained by numerical solution for the above problem53,54 describe the dependence of the effective diffusion coefficient Deff on λ = dtracer/⟨dpore⟩. Though these relationships were developed for a single uniform pore, they are widely used to characterize hindered diffusion of finite-size molecules in the heterogeneous pore space of random porous media. We demonstrated that these theoretical approaches fail to provide an adequate and accurate characterization of the relation between Deff and λ in heterogeneous porous media. We showed that the geometry and topology of the accessible pore network in the reconstructed mesopore space of the monolith skeleton change significantly with dtracer, and therefore, hindered diffusion cannot be attributed exclusively to the hydrodynamic drag effect in a single pore. A precise characterization of the diffusivity for finite-size molecules in a heterogeneous porous material requires detailed information on its geometrical structure. It cannot be virtually reduced to a single pore with a diameter determined as the average pore size in the material. Nonuniform pore size commonly observed in real porous media results in a complex relation between the morphological parameters of the pore network accessible for the molecules and molecular size. A key parameter of the pore network is its connectivity characterizing the number of possibilities for tracer molecules to diffuse from one point to another via different paths. Our morphological analysis of accessible pore space in the reconstructed

Figure 12. Distribution of pore throat coordination numbers in the accessible void space. The color-coding corresponds to the nine different values of λ = dtracer/⟨dpore⟩.

this number is 5327, 1057, and only 50 for λ = 0.0, 0.237, and 0.474, respectively. However, more relevant with increasing λ becomes the drastic decrease in the number of throats in the accessible pore space shared by more than two pores. While for λ = 0.0 this number is 1985, it drops to 150 for λ = 0.237 (cf. Figure 11) and to just 4 with λ = 0.474. The results confirm that, with increasing tracer diameter, not only is the accessible pore space strongly reduced but also the pore network becomes far less interconnected. The latter aspect is not accounted for by the adapted theoretical models for hindered diffusion of finite-size particles in a single cylindrical pore (cf. eqs 4 and 5), while it affects greatly the effective diffusivity in heterogeneous porous media. A similar impact of pore interconnectivity on diffusive transport is observed in porous media with different porosity scales, e.g., in zeolites comprising micro- and mesopores.6,19−21 The results presented in this study demonstrate that this effect is principal also for heterogeneous materials with pores of a single-porosity scale. As a consequence, theoretical approaches to hindered diffusion of finite-size particles that account only for hydrodynamic drag in a single uniform pore are not able to provide an adequate description of the diffusion process in heterogeneous porous materials. Accurate determination of the effective diffusion H



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mesoporous skeleton of the silica monolith revealed that this number drops strongly with increasing tracer diameter. Therefore, in contrast to a single uniform pore, the Stokes friction effect in heterogeneous porous media is complemented by changes in the connectivity of the accessible pore network, which decreases with increasing tracer size. Moreover, to prepare a porous material with enhanced diffusivity, it is not sufficient to increase ⟨dpore⟩. Independent from the actual ⟨dpore⟩ value, the presence of a fraction of small pores can markedly reduce the effective diffusivity (compared to its value estimated using just ⟨dpore⟩) due to a low connectivity of the accessible pore network. This reduction can be even aggravated by interaction (adsorption, binding, etc.) of the transported species with the pore surface.6,18,38 The approach presented in this study promises also potential for the simulation of transport coupled with adsorption and reactions in hierarchically structured porous media and the derivation of quantitative morphology−transport relationships for heterogeneous materials, in general. In particular, the proposed simulation approach was already applied to study advective− diffusive transport of pointlike tracers, coupled with adsorption/desorption at the solid−liquid interface, in an open channel and a random packing of nonporous spherical particles.67,68 Its extension to finite-size tracers and porous media with a hierarchical structure is straightforward and just requires detailed knowledge on the morphology of the void space in the investigated material. Then, the proposed numerical model can be applied to analyze the interplay between transport and morphological properties (Deff, the porosity and connectivity of the accessible pore space, specific surface area, etc.) in the macropore and mesopore domains dependent on molecular size. Finally, the presented approach can be combined with molecular dynamics simulations to account additionally for the local solvent composition and structural organization in the liquid interface region that develops between a solid surface and a bulk liquid phase.69,70 A recent study revealed that the pore-level solute diffusion coefficient can be strongly affected by the physiochemical properties of this formed interface region.70

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Article

AUTHOR INFORMATION

Corresponding Author

*Phone: +49-(0)6421-28-25727. Fax: +49-(0)6421-28-27065. E-mail: tallarek@staff.uni-marburg.de. ORCID

Ulrich Tallarek: 0000-0002-2826-2833 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Deutsche Forschungsgemeinschaft DFG (Bonn, Germany) under grant TA 268/9−1. We thank the John von Neumann Institute for Computing (NIC) and the Jülich Supercomputing Center (JSC), Forschungszentrum Jülich (FZJ, Jülich, Germany), for the allocation of a special CPU-time grant (NIC project number 10214, JSC project ID HMR 10).

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REFERENCES

(1) Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties; Springer: New York, 2002. (2) Sahimi, M. Heterogeneous Materials: Vol. I. Linear Transport and Optical Properties; Springer: New York, 2003. (3) Kachanov, M.; Sevostianov, I. Effective Properties of Heterogeneous Materials; Springer: New York, 2013. (4) Gin, S.; Jollivet, P.; Fournier, M.; Angeli, F.; Frugier, P.; Charpentier, T. Origin and Consequences of Silicate Glass Passivation by Surface Layers. Nat. Commun. 2015, 6, 6360. (5) Schwieger, W.; Machoke, A. G.; Weissenberger, T.; Inayat, A.; Selvam, T.; Klumpp, M.; Inayat, A. Hierarchy Concepts: Classification and Preparation Strategies for Zeolite Containing Materials with Hierarchical Porosity. Chem. Soc. Rev. 2016, 45, 3353−3376. (6) Coasne, B. Multiscale Adsorption and Transport in Hierarchical Porous Materials. New J. Chem. 2016, 40, 4078−4094. (7) Deria, P.; Gómez-Gualdrón, D. A.; Hod, I.; Snurr, R. Q.; Hupp, J. T.; Farha, O. K. Framework-Topology-Dependent Catalytic Activity of Zirconium-Based (Porphinato)zinc(II) MOFs. J. Am. Chem. Soc. 2016, 138, 14449−14457. (8) Vargas L, E.; Snurr, R. Q. Heterogeneous Diffusion of Alkanes in the Hierarchical Metal-Organic Framework NU-1000. Langmuir 2015, 31, 10056−10065. (9) Zeng, Y.; Moghadam, P. Z.; Snurr, R. Q. Pore Size Dependence of Adsorption and Separation of Thiophene/Benzene Mixtures in Zeolites. J. Phys. Chem. C 2015, 119, 15263−15273. (10) Chakraborty, S.; Colón, Y. J.; Snurr, R. Q.; Nguyen, S. T. Hierarchically Porous Organic Polymers: Highly Enchanced Gas Uptake and Transport through Templated Synthesis. Chem. Sci. 2015, 6, 384−389. (11) Liasneuski, H.; Hlushkou, D.; Khirevich, S.; Höltzel, A.; Tallarek, U.; Torquato, S. Impact of Microstructure on the Effective Diffusivity in Random Packings of Hard Spheres. J. Appl. Phys. 2014, 116, 034904. (12) Hlushkou, D.; Liasneuski, H.; Tallarek, U.; Torquato, S. Effective Diffusion Coefficients in Random Packings of Polydisperse Hard Spheres from Two-Point and Three-Point Correlation Functions. J. Appl. Phys. 2015, 118, 124901. (13) Burada, P. S.; Hänggi, P.; Marchesoni, F.; Schmid, G.; Talkner, P. Diffusion in Confined Geometries. ChemPhysChem 2009, 10, 45− 54.

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b00264. Modeling programs used for the calculation of effective diffusion coefficients of finite-size tracers in reconstructed mesoporous silica (PDF) PDF file “eroding.r” (PDF) Text file “eroding.r” (TXT) PDF file “tracers.cpp” (PDF) Text file “tracers.cpp” (TXT) PDF document containing the library files required to run the code “tracers.cpp” (RWPT simulation of diffusion) (PDF) Text document containing the library files required to run the code “tracers.cpp” (RWPT simulation of diffusion) (TXT) 3D structure of the cubic domain from the reconstruction of the mesoporous silica employed for these simulations (ZIP) I


Article

The Journal of Physical Chemistry C (14) Skaug, M. J.; Wang, L.; Ding, Y.; Schwartz, D. K. Hindered Nanoparticle Diffusion and Void Accessibility in a Three-Dimensional Porous Medium. ACS Nano 2015, 9, 2148−2156. (15) Langford, J. F.; Schure, M. R.; Yao, Y.; Maloney, S. F.; Lenhoff, A. M. Effect of Pore Structure and Molecular Size on Diffusion in Chromatographic Adsorbents. J. Chromatogr. A 2006, 1126, 95−106. (16) Kärger, J.; Ruthven, D. M. Diffusion in Nanoporous Materials: Fundamental Principles, Insights and Challenges. New J. Chem. 2016, 40, 4027−4048. (17) Müllner, T.; Unger, K. K.; Tallarek, U. Characterization of Microscopic Disorder in Reconstructed Porous Materials and Assessment of Mass Transport-Relevant Structural Descriptors. New J. Chem. 2016, 40, 3993−4015. (18) Skaug, M. J.; Schwartz, D. K. Tracking Nanoparticle Diffusion in Porous Filtration Media. Ind. Eng. Chem. Res. 2015, 54, 4414−4419. (19) Bonilla, M. R.; Valiullin, R.; Kärger, J.; Bhatia, S. K. Understanding Adsorption and Transport of Light Gases in Hierarchical Materials Using Molecular Simulation and Effective Medium Theory. J. Phys. Chem. C 2014, 118, 14355−14370. (20) Schneider, D.; Mehlhorn, D.; Zeigermann, P.; Kärger, J.; Valiullin, R. Transport Properties of Hierarchical Micro−Mesoporous Materials. Chem. Soc. Rev. 2016, 45, 3439−3467. (21) Galarneau, A.; Guenneau, F.; Gedeon, A.; Mereib, D.; Rodriguez, J.; Fajula, F.; Coasne, B. Probing Interconnectivity in Hierarchical Microporous/Mesoporous Materials Using Adsorption and Nuclear Magnetic Resonance Diffusion. J. Phys. Chem. C 2016, 120, 1562−1569. (22) Cocco, A. P.; Nelson, G. J.; Harris, W. M.; Nakajo, A.; Myles, T. D.; Kiss, A. M.; Lombardo, J. J.; Chiu, W. K. S. Three-Dimensional Microstructural Imaging Methods for Energy Materials. Phys. Chem. Chem. Phys. 2013, 15, 16377−16407. (23) Möbus, G.; Inkson, B. J. Nanoscale Tomography in Materials Science. Mater. Today 2007, 10, 18−25. (24) Koku, H.; Maier, R. S.; Czymmek, K. J.; Schure, M. R.; Lenhoff, A. M. Modelling of Flow in a Polymeric Chromatographic Monolith. J. Chromatogr. A 2011, 1218, 3466−3475. (25) Müllner, T.; Zankel, A.; Mayrhofer, C.; Reingruber, H.; Höltzel, A.; Lv, Y.; Svec, F.; Tallarek, U. Reconstruction and Characterization of a Polymer-Based Monolithic Stationary Phase Using Serial BlockFace Scanning Electron Microscopy. Langmuir 2012, 28, 16733− 16737. (26) Leary, R.; Saghi, Z.; Armbruster, M.; Wowsnick, G.; Schlögl, R.; Thomas, J. M.; Midgley, P. A. Quantitative High-Angle Annular DarkField Scanning Transmission Electron Microscope (HAADF-STEM) Tomography and High-Resolution Electron Microscopy of Unsupported Intermetallic GsPd2 Catalysts. J. Phys. Chem. C 2012, 116, 13343−13352. (27) Yao, Y.; Czymmek, K. J.; Pazhianur, R.; Lenhoff, A. M. ThreeDimensional Pore Structure of Chromatographic Adsorbents from Electron Tomography. Langmuir 2006, 22, 11148−11157. (28) Aste, T.; Saadatfar, M.; Senden, T. J. Geometrical Structure of Disodered Sphere Packings. Phys. Rev. E 2005, 71, 061302. (29) Dong, H.; Blunt, M. J. Pore-Network Extraction from MicroComputerized-Tomography Images. Phys. Rev. E 2009, 80, 036307. (30) Gommes, C. J.; Bons, A.-J.; Blacher, S.; Dunsmuir, J. H.; Tsou, A. H. Practical Methods for Measuring the Tortuosity of Porous Materials from Binary or Gray-Tone Tomographic Reconstructions. AIChE J. 2009, 55, 2000−2012. (31) Fredrich, J. T.; Menendez, B.; Wong, T. F. Imaging the Pore Structure of Geomaterials. Science 1995, 268, 276−279. (32) Bruns, S.; Müllner, T.; Kollmann, M.; Schachtner, J.; Höltzel, A.; Tallarek, U. Confocal Laser Scanning Microscopy Method for Quantitative Characterization of Silica Monolith Morphology. Anal. Chem. 2010, 82, 6569−6575. (33) Bruns, S.; Grinias, J. P.; Blue, L. E.; Jorgenson, J. W.; Tallarek, U. Morphology and Separation Efficiency of Low-Aspect-Ratio Capillary Ultrahigh Pressure Liquid Chromatography Columns. Anal. Chem. 2012, 84, 4496−4503.

(34) Seymour, J. D.; Callaghan, P. T. Generalized Approach to NMR Analysis of Flow and Dispersion in Porous Media. AIChE J. 1997, 43, 2096−2111. (35) Tallarek, U.; Bayer, E.; van Dusschoten, D.; Scheenen, T.; Van As, H.; Guiochon, G.; Neue, U. D. Dynamic NMR Microscopy of Chromatographic Columns. AIChE J. 1998, 44, 1962−1975. (36) Manz, B.; Gladden, L. F.; Warren, P. B. Flow and Dispersion in Porous Media: Lattice-Boltzmann and NMR Studies. AIChE J. 1999, 45, 1845−1854. (37) Gooya, R.; Bruns, S.; Müter, D.; Moaddel, A.; Harti, R. P.; Stipp, S. L. S.; Sørensen, H. O. Effect of Tomography Resolution on the Calculated Microscopic Properties of Porous Materials: Comparison of Sandstone and Carbonate Rocks. Appl. Phys. Lett. 2016, 109, 104102. (38) Müter, D.; Sørensen, H. O.; Bock, H.; Stipp, S. L. S. Particle Diffusion in Complex Nanoscale Pore Network. J. Phys. Chem. C 2015, 119, 10329−10335. (39) Guiochon, G. Monolithic Columns in High-Performance Liquid Chromatography. J. Chromatogr. A 2007, 1168, 101−168. (40) Unger, K. K.; Skudas, R.; Schulte, M. M. Particle Packed Columns and Monolithic Columns in High-Performance Liquid Chromatography − Comparison and Critical Appraisal. J. Chromatogr. A 2008, 1184, 393−415. (41) Galarneau, A.; Sachse, A.; Said, B.; Pelisson, C.-H.; Boscaro, P.; Brun, N.; Courtheoux, L.; Olivi-Tran, N.; Coasne, B.; Fajula, F. Hierarchical Porous Silica Monoliths: A Novel Class of Microreactors for Process Intensification in Catalysis and Adsorption. C. R. Chim. 2016, 19, 231−247. (42) Enke, D.; Gläser, R.; Tallarek, U. Sol-Gel and Porous GlassBased Silica Monoliths with Hierarchical Pore Structure for SolidLiquid Catalysis. Chem. Ing. Tech. 2016, 88, 1561−1585. (43) Hlushkou, D.; Bruns, S.; Höltzel, A.; Tallarek, U. From Pore Scale to Column Scale Dispersion in Capillary Silica Monoliths. Anal. Chem. 2010, 82, 7150−7159. (44) Hlushkou, D.; Hormann, K.; Höltzel, A.; Khirevich, S.; SeidelMorgenstern, A.; Tallarek, U. Comparison of First and Second Generation Analytical Silica Monoliths by Pore-Scale Simulations of Eddy Dispersion in the Bulk Region. J. Chromatogr. A 2013, 1303, 28− 38. (45) 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. (46) Salles, J.; Thovert, J.-F.; Delannay, L.; Prevors, L.; Auriault, J.-L.; Adler, P. M. Taylor Dispersion in Porous Media. Determination of the Dispersion Tensor. Phys. Fluids A 1993, 5, 2348−2376. (47) Delay, F.; Ackerer, P.; Danquigny, C. Simulating Solute Transport in Porous or Fractured Formations Using Random Walk Particle Tracking. Vadose Zone J. 2005, 4, 360−379. (48) Szymczak, P.; Ladd, A. J. C. Boundary Conditions for Stochastic Solutions of the Convection−Diffusion Equation. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 036704. (49) Daneyko, A.; Hlushkou, D.; Baranau, V.; Khirevich, S.; SeidelMorgenstern, A.; Tallarek, U. Computational Investigation of Longitudinal Diffusion, Eddy Dispersion, and Trans-Particle Mass Transfer in Bulk, Random Packings of Core-Shell Particles with Varied Shell Thickness and Shell Diffusion Coefficient. J. Chromatogr. A 2015, 1407, 139−156. (50) Blees, M. H.; Leyte, J. C. The Effective Translational SelfDiffusion Coefficient of Small Molecules in Colloidal Crystals of Spherical Particles. J. Colloid Interface Sci. 1994, 166, 118−127. (51) Serra, J. Image Analysis and Mathematical Morphology; Academic Press: London, 1982. (52) Kärger, J. Transport Phenomena in Nanoporous Materials. ChemPhysChem 2015, 16, 24−51. (53) Renkin, E. M. Filtration, Diffusion, and Molecular Sieving through Porous Cellulose Membranes. J. Gen. Physiol. 1954, 38, 225− 234. J


Article

The Journal of Physical Chemistry C (54) Dechadilok, P.; Deen, W. M. Hindrance Factors for Diffusion and Convection in Pores. Ind. Eng. Chem. Res. 2006, 45, 6953−6959. (55) Adiga, S. P.; Curtiss, L. A.; Elam, J. W.; Pellin, M. J.; Shih, C. C.; Shih, C. M.; Lin, S. J.; Su, Y. Y.; Gittard, S. D.; Zhang, J.; et al. Nanoporous Materials for Biomedical Devices. JOM 2008, 60, 26−32. (56) Wernert, V.; Bouchet, R.; Denoyel, R. Influence of Molecule Size on Its Transport Properties through a Porous Medium. Anal. Chem. 2010, 82, 2668−2679. (57) Sharma, K.; Kim, Y.-H.; Gabitto, J.; Mayes, R. T.; Yiacoumi, S.; Bilheux, H. Z.; Walker, L. M. H.; Dai, S.; Tsouris, C. Transport of Ions in Mesoporous Carbon Electrodes during Capacitive Deionization of High-Salinity Solutions. Langmuir 2015, 31, 1038−1047. (58) Lin, X.; Yang, Q.; Ding, L.; Su, B. Ultrathin Silica Membranes with Highly Ordered and Perpendicular Nanochannels for Precise and Fast Molecular Separation. ACS Nano 2015, 9, 11266−11277. (59) Gritti, F.; Guiochon, G. The Quantitative Impact of the Mesopore Size on the Mass Transfer Mechanism of the New 1.9 μm Porous Titan-C18 Particles. I: Analysis of Small Molecules. J. Chromatogr. A 2015, 1384, 76−87. (60) Notario, B.; Pinto, J.; Rodriguez-Perez, M. A. Nanoporous Polymeric Materials: A New Class of Materials with Enhanced Properties. Prog. Mater. Sci. 2016, 78−79, 93−139. (61) Babayekhorasani, F.; Dunstan, D. E.; Krishnamoorti, R.; Conrad, J. C. Nanoparticle Diffusion in Crowded and Confined Media. Soft Matter 2016, 12, 8407−8416. (62) Hormann, K.; Tallarek, U. Mass Transport Properties of Second-Generation Silica Monoliths with Mean Mesopore Size from 5 to 25 nm. J. Chromatogr. A 2014, 1365, 94−105. (63) Hormann, K.; Baranau, V.; Hlushkou, D.; Höltzel, A.; Tallarek, U. Topological Analysis of Non-Granular, Disordered Porous Media: Determination of Pore Connectivity, Pore Coordination, and Geometric Tortuosity in Physically Reconstructed Silica Monoliths. New J. Chem. 2016, 40, 4187−4199. (64) Lee, T.-C.; Kashyap, R. L.; Chu, C.-N. Building Skeleton Models via 3-D Medial Surface/Axis Thinning Algorithm. CVGIP: Computer Vision, Graphics, and Image Processing 1994, 56, 462−478. (65) Schindelin, J.; Arganda-Carreras, I.; Frise, E.; Kaynig, V.; Longair, M.; Pietzsch, S.; Rueden, C.; Saalfeld, S.; Schmid, B.; Tinevez, J.-Y.; et al. Fiji: An Open-Source Platform for Biological-Image Analysis. Nat. Methods 2012, 9, 676−682. (66) Arganda-Carreras, I. Software/Image Analysis. https://sites. google.com/site/iargandacarreras/software/ (accessed Mar 20, 2017). (67) Hlushkou, D.; Gritti, F.; Guiochon, G.; Seidel-Morgenstern, A.; Tallarek, U. Effect of Adsorption on Solute Dispersion: A Microscopic Stochastic Approach. Anal. Chem. 2014, 86, 4463−4470. (68) Hlushkou, D.; Gritti, F.; Daneyko, A.; Guiochon, G.; Tallarek, U. How Microscopic Characteristics of the Adsorption Kinetics Impact Macroscale Transport in Chromatographic Beds. J. Phys. Chem. C 2013, 117, 22974−22985. (69) Melnikov, S. M.; Höltzel, A.; Seidel-Morgenstern, A.; Tallarek, U. A Molecular Dynamics View on Hydrophilic Interaction Chromatography with Polar-Bonded Phases: Properties of the Water-Rich Layer at a Silica Surface Modified with Diol-Functionalized Alkyl Chains. J. Phys. Chem. C 2016, 120, 13126−13138. (70) Rybka, J.; Höltzel, A.; Melnikov, S. M.; Seidel-Morgenstern, A.; Tallarek, U. A New View on Surface Diffusion from Molecular Dynamics Simulations of Solute Mobility at Chromatographic Interfaces. Fluid Phase Equilib. 2016, 407, 177−187.

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Tracer-Size-Dependent Pore Space Accessibility and Long-Time

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