# Hauptseminar Porous Media SS 2021/Random walk models diffusion in porous media

- Date
- 2021-06-11
- Time
- 14:30
- Topic
- Random walks in porous media: Diffusion at the pore and at the pore-network scale
- Speaker
- Jonas Schmid
- Tutor
- Alexander Schlaich
- Handout
- [1]

## Contents

Diffusion in confined geometries - in contrast to transport - refers to the case where no driving force (chemical potential gradient, e.g. density/pressure/electric field) is applied, or where the transport due to the driving force can be neglected due to the vanishing gradients (which is typically the case in smallest pores). In this seminar, in a first step, the different diffusion mechanisms that pertain to fluid transport in nanoporous media are presented with special attention to their dependence on pore size vs. fluid molecule size and thermodynamic variables such as fluid density, pressure or temperature T. Then, in a second step, combination rules which allow one to describe the transport resulting from different diffusion mechanisms are introduced.

In detail, different diffusion mechanisms often coexist in nanopores due to the inhomogeneous density profile that results from adsorption. Of special importance are thus so-called correlation-function approaches which can be probed directly in experiments pulsed-field-gradient nuclear magnetic resonance (PFG-NMR), which allow to establish an effective diffusion constant in analogy with electrical transport in resistance networks. In a further refinement one has to take into account that the contribution of each equivalent resistor must be weighted by the local density of the fluid and that molecules can only diffuse if a diffusion path is available.

On a larger scale the molecular diffusion in a heterogeneous medium such as a fluid confined in a disordered porous solid made of different domains is considered. This problem can be treated in a rigorous fashion using theoretical frameworks such as Langevin and Fokker–Planck equations which allows predicting the dynamics in such complex media using intermediate functions which are readily obtained using experiments (typically, quasi-elastic and inelastic neutron scattering). Different random walk approaches in the confined geometries allow to assess the effective diffusivity in hierarchical porous media and shall be discussed in the second part of the talk.

## Points to be discussed

### Diffusion mechnisms

- Knudsen diffusion with slip corrections
- Molecular diffusion (Random walk and transition state theory)
- [Reed–Ehrlich model]
- Combination rules from correlation function approaches and intermittent Brownian motion
- Density-dependent diffusion and free volume theory

#### Literature

- Arya, G., Chang, H.-C. & Maginn, E. J., Molecular Simulations of Knudsen Wall-Slip: Effect of Wall Morphology. Molecular Simulation 29, 697–709 (2003).
- [Reed, D. A. & Ehrlich, G., Surface Diffusion, Atomic Jump Rates and Thermodynamics. Surface Science 102, 588–609 (1981).]
- Roosen-Runge, F., Bicout, D. J. & Barrat, J.-L., Analytical Correlation Functions for Motion through Diffusivity Landscapes. J. Chem. Phys. 144, 204109 (2016).
- Kärger, J. & Valiullin, R., Mass Transfer in Mesoporous Materials: The Benefit of Microscopic Diffusion Measurement. Chem. Soc. Rev. 42, 4172–4197 (2013).
- Levitz, P., Bonnaud, P. A., Cazade, P.-A., Pellenq, R. J.-M. & Coasne, B. Molecular, Intermittent Dynamics of Interfacial Water: Probing Adsorption and Bulk Confinement. Soft Matter 9, 8654–8663 (2013).
- Bhatia, S. K. & Nicholson, D., Hydrodynamic Origin of Diffusion in Nanopores. Phys. Rev. Lett. 90, 016105 (2003).
- Obliger, A., Pellenq, R., Ulm, F.-J. & Coasne, B. Free Volume Theory of Hydrocarbon Mixture Transport in Nanoporous Materials. J. Phys. Chem. Lett. 7, 3712–3717 (2016).

### Diffusion in porous networks

- geometrical vs. dynamical Tortuosity
- Smoluchowski equation, Brownian Dynamics and Langevin equation.
- On- and off-lattice random walks
- Hierarchical simulations with free energy barriers

#### Literature

- Noetinger, B. et al., Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale. Transp Porous Med115, 345–385 (2016).
- Tallarek, U., Hlushkou, D., Rybka, J. & Höltzel, A., Multiscale Simulation of Diffusion in Porous Media: From Interfacial Dynamics to Hierarchical Porosity. J. Phys. Chem. C 123, 15099–15112 (2019).
- Bhatia, S. K. Stochastic Theory of Transport in Inhomogeneous Media. Chemical Engineering Science 41, 1311–1324 (1986).