Diffusion in nanopores: inspecting the grounds
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Diffusion in nanopores: inspecting the grounds Jörg Kärger1 · Douglas M. Ruthven2 · Rustem Valiullin1 Received: 29 July 2020 / Revised: 6 October 2020 / Accepted: 7 October 2020 © The Author(s) 2020
Abstract This paper provides a general overview of the phenomenon of guest diffusion in nanoporous materials. It introduces the different types of diffusion measurement that can be performed under both equilibrium and non-equilibrium conditions in either single- or multicomponent systems. In the technological application of nanoporous materials for mass separation and catalytic conversion diffusion often has a significant impact on the overall rate of the process and is quite commonly rate controlling. Diffusion enhancement is therefore often a major goal in the manufacture of catalysts and adsorbents. Keywords Diffusion · Adsorption · Nanopores · Zeolites · MOFs Symbols A External surface of adsorbent host particle/crystal (m2) c Guest concentration (mol m−3) c∗ Concentration of labelled guest molecules (mol m−3) c0(∞) Initial (final) concentration (mol m−3)
c(t) Mean concentration at time t (mol m−3) cmicro Guest concentration in micropores (mol m−3) ctransport Guest concentration in transport pores (mol m−3) D Self-diffusivity (or tracer diffusivity or coefficient of self-diffusion …) (m2 s−1) DT Transport-diffusivity (or Fickian diffusivity or coefficient of …) (m2 s−1) D0 Corrected diffusivity (or Maxwell–Stefan diffusivity) (m2 s−1) Dmicro Diffusivity in micropores (m2 s−1) Dtransport Diffusivity in transport pores (m2 s−1) Dij Element of the diffusion matrix correlating the flux of component i with the concentration gradient of component j (m2 s−1) ����⃗⃗ DT Tensor of (transport) diffusion (m2 s−1)
* Jörg Kärger [email protected]‑leipzig.de 1
Faculty of Physics and Earth Sciences, University of Leipzig, Linnéstraße 5, 04103 Leipzig, Germany
Department of Chemical and Biological Engineering, University of Maine, Orono, ME, USA
2
D(T)i Principal (transport) diffusivity = ith element of the diagonalized diffusion tensor (m2 s−1) Ɖii Self-exchange diffusivity (m2 s−1) f Friction coefficient (Pa m s) fi Reaction-induced increase in the concentration of component i per time (mol m−3 s−1) j Molar flux (mol s−1 m−2) j∗ Flux of labelled molecules (mol s−1 m−2) m1 First statistical moment of molecular uptake (s) m1Barr First statistical moment of molecular uptake under limitation by surface barriers (s) m1Diff First statistical moment of molecular uptake under diffusion limitation (s) mt Fractional approach to equilibrium (–) m ∞
p Pressure (Pa) pmicro Relative amount of molecules in micropores (–) ptransport Relative amount of molecules in transport pores (–) P(x, t) Probability (density) of molecular displacement over x in x-direction during t (= propagator) (m−1) r Distance (m) R Radius of adsorbent host particle/crystal (m) Rg Universal gas constant (Pa m3 mol−1 K−1) t Time (s) T Temperature (K), overall observation time in single-molecu
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