Hindrance diffusion factor in random mesoporous adsorbents


Diffusion of solute molecules and particulate matter through random network pores is a common phenomenon that occurs naturally in most cases. In various processes where the random porous media provide reactive and adsorptive interfaces, hindered diffusion may control them. Hindered diffusion results in a number of consequences which vary depending on the kind of process carried out. For example, it may lead to the partial exclusion of solute molecules which decreases the active surface area and its mass loadability.

Hindered diffusion-controlled processes can be efficiently optimized and improved by understanding the relationship between the diffusive hindrance factor and the morphology of the material. The relationship determines the extent to which diffusion of solute molecules and particles through a material is hindered. At first, this depends on the size of the diffusing species or tracer with respect to the pore size, commonly parameterized by the ratio (λ) of tracer size to mean pore size. The global diffusive hindrance factor is expressed as the ratio of the effective diffusion coefficient of the (finite-size) tracer through the confining pore space to the diffusion coefficient of the same tracer in the bulk liquid.

Despite hindered diffusion being a critical factor in the performance of random mesoporous silica materials in various applications such as storage, separation, and catalysis, little information is available to its effect on these processes.

Researchers at Philipps-Universität Marburg in Germany led by Professor Ulrich Tallarek derived an expression for the overall diffusive hindrance in random mesoporous silica through a global diffusive hindrance factor H(λ) with a λ-range from 0 to 0.9. The mesoporous silica samples were carefully selected, characterized, and physically reconstructed using scanning transmission electron microscopy (STEM) tomography. The researchers subsequently performed pore-scale simulations of diffusion of finite-size tracers in representative reconstructions of the porous media. Eventually, they predicted the extent of hindered diffusion from the various material properties and the available and reliable material parameter values, e.g., mean pore size and porosity. The research work is published in Industrial and Engineering Chemistry Research.

The authors observed that the diffusive hindrance factor can be precisely and accurately predicted only by taking into consideration the relationship between the evolving (λ-dependent) morphology and transport in the materials.

“Diffusive tortuosity–porosity correlations valid for point-like tracers in self-similar materials do not necessarily account for the topological evolution of a given material that is experienced by different-sized tracers”, Professor Tallarek says. “We have shown that the actual pore networks evolving from a given pore space at different λ-values are not self-similar, so their evolution cannot be predicted by a generic diffusive tortuosity‒porosity relationship.”

While smaller mesopores generally have the capacity of increasing the surface area and mass loadability, the derived diffusion hindrance expression will allow the determination of the optimum mesopore sizes for various solute sizes. The concept therefore can be used to identify, actually quantify, the compromising factors in the transport and mass loadability as well as to maximize the effective diffusivities.

The study is the first to present a derived equation for the diffusive hindrance factor for random porous media taking into consideration the reconstructed morphology. The equation can be applied in determining hindered diffusion in mesoporous materials whose morphology are determined by the preparation process such as the sol–gel processing, but the presented approach to diffusion hindrance expressions is general and can be applied to other materials as well. Since mesoporous silica adsorbents are used in several applications, predicting their hindrance factors will lead to a significant improvement in their performances.


Hindrance diffusion factor in random mesoporous adsorbents. Advances in Engineering

Combining physical reconstruction of the pore space morphology with pore-scale simulations of diffusion of finite-size tracers allows to derive a quantitative expression for the diffusive hindrance factor Hλ that quantifies the degree to which diffusion through a material is hindered compared with diffusion in the bulk liquid in dependence of l, the ratio of tracer size to mean pore size. Reprinted with permission from S.-J. Reich et al., Ind. Eng. Chem. Res. 2018, 57, 3041–3042. Copyright (2018) American Chemical Society.


About the author

Professor Tallarek studied chemistry at the Eberhard-Karls-Universität Tübingen (Tübingen, Germany) and obtained his Dr. rer. nat. in 1998 with an NMR imaging study of the fluid dynamics in porous media. As a Marie-Curie postdoctoral fellow from 1998 to 2000 in Wageningen (The Netherlands) he developed and applied NMR tools for the in situ characterization of transport in microfluidic devices, particularly electrokinetic microfluidics. From 2000 to 2007 he was on the faculty of the Department of Chemical and Process Engineering of the Otto-von-Guericke-Universität Magdeburg (Magdeburg, Germany), where he completed his habilitation in 2004 (mentor: Prof. Dr.-Ing. Andreas Seidel-Morgenstern) and received a venia legendi for physical chemistry. Since 2007 he is professor of analytical chemistry and a member of the board of directors of the materials science center at the Philipps-Universität Marburg (Marburg, Germany).

His research focuses on the discovery and understanding of the fundamental morphology–functionality–transport relationships in functional porous materials, ranging from solute–surface interactions to macroscale transport. Besides a variety of experimental analysis methods, this approach relies on modern simulation approaches to capture the involved widely different spatiotemporal scales (e.g., molecular dynamics at solid-liquid interfaces, large-scale simulation of flow) and the precise physical reconstruction of porous media used for storage, separation, and catalysis.


Prof. Dr. Ulrich Tallarek
Department of Chemistry, Philipps-Universität Marburg,
Hans-Meerwein-Strasse 4, 35032 Marburg, Germany
E-mail: [email protected]


Reich, S.-J., Svidrytski, A., Hlushkou, D., Stoeckel, D., Kübel, C., Höltzel, A., & Tallarek, U. (2018). Hindrance Factor Expression for Diffusion in Random Mesoporous Adsorbents Obtained from Pore-Scale Simulations in Physical Reconstructions. Industrial & Engineering Chemistry Research, 57(8), 3031–3042.

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