Transport in Small Pores

(1) Knudsen Diffusion
      Knudsen diffusion is described by the Einstein relation as follows:
where Ds is the self-diffusion coefficient, which depends on the concentration c, t is time, N is the total number of particles in the system, and ri is the position vector of particle i. Knudsen diffusion occurs when the mean free path is relatively long compared to the pore size, so the molecules collide frequently with the pore wall. Knudsen diffusion is dominant for pores that range in diameter between 2 and 50 nm.

Malek, K. and M.O. Coppens, Knudsen self- and Fickian diffusion in rough nanoporous media. Journal of Chemical Physics, 2003. 119(5): p. 2801-2811.

(2) Molecular (or Fickian) Diffusion
      Molecular, or transport, diffusion occurs when the mean free path is relatively short compared to the pore size, and is described by Fick's law as follows:
where j is the mass flux, Dt is the transport diffusion coefficient (or Fickian diffusion coefficient), and Nc is the concentration gradient. The transport diffusivity relates the macroscopic flux of molecules in a system to a driving force in the concentration. This diffusion mode is applicable to Brownian motion, where the movement of each particle is random and not dependent on its previous motion.

Hosticka, B., et al., Gas flow through aerogels. Journal of Non-Crystalline Solids, 1998. 225(1): p. 293-297.

(3) Transition Diffusion
      Transition-mode diffusion has properties of both Knudsen and Fickian diffusion. A dynamic Monte Carlo simulation has been used to study both Knudson and Fickian diffusion through various pores roughened with a 3-dimensional random Koch surface and compare the results to the findings for other 2-dimensional fractal surfaces. This study predicted that the trapping zone of the pore leads to a decrease in self-diffusivity but does not affect transport diffusivity.

Roy, S., et al., Modeling gas flow through microchannels and nanopores. Journal of Applied Physics, 2003. 93(8): p. 4870-4879.

(4) Surface Diffusion
      Surface diffusion is also used to explain a type of pore diffusion in which solutes adsorb on the surface of the pore and hop from one site to another through interactions between the surface and molecules.

Jaguste, D.N. and S.K. Bhatia, Combined Surface and Viscous-Flow of Condensable Vapor in Porous-Media. Chemical Engineering Science, 1995. 50(2): p. 167-182.

(5) Hydrodynamic Flow
      Hydrodynamic flow is induced by pressure differentials along a pore channel.

Weber, M. and R. Kimmich, Maps of electric current density and hydrodynamic flow in porous media: NMR experiments and numerical simulations. Physical Review E, 2002. 66(2): p. 026306.

(6) Capillary Condensation
      Capillary condensation is known to occur when multilayer adsorption from adsorbate molecules proceeds to the point where pore spaces are filled with condensed liquid and separated from the gas phase by menisci.

Restagno, F., L. Bocquet, and T. Biben, Metastability and nucleation in capillary condensation. Physical Review Letters, 2000. 84(11): p. 2433-2436.

(7) Viscous Flow
      Viscous flow is the flow of a gas through a channel under conditions where the mean free path is small in comparison with the transverse section of the channel, so the flow characteristics are determined mainly by collisions between the gas molecules.

Andrade, J.S., et al., Inertial effects on fluid flow through disordered porous media. Physical Review Letters, 1999. 82(26): p. 5249-5252.

(8) Normal-Mode Diffusion
      One of commonly known diffusion mechanisms for nanoporous materials is normal-mode diffusion, where molecules are able to pass each other. The mathematical expressions of these diffusion modes can be induced from the Fokker-Planck equation for the probability density function. The mean squared displacement of particles following normal mode diffusion is defined as follows:
where z(t) is the displacement of a molecule at time t, A is the diffusion coefficient, and t is time. The diffusion coefficients are taken from the intercepts in plots of the log of <[z(t)-z(0)]2> versus the log of t.

Sholl, D.S. and K.A. Fichthorn, Normal, single-file, and dual-mode diffusion of binary adsorbate mixtures in AlPO4-5. Journal of Chemical Physics, 1997. 107(11): p. 4384-4389.

(9) Single-File diffusion
      Single-file mode diffusion is characterized by that molecules are unable to pass each other in pores or channels, and expressed as

where B is the diffusion mobility.

Levitt, D.G., Dynamics of a Single-File Pore - Non-Fickian Behavior. Physical Review A, 1973. 8(6): p. 3050-3054.

(10) Anomalous diffusion: Subdiffusion and Superdiffusion
      A more general form such as
can be used, where 0.5 < α < 1.0 for (normal-to-single-file) transition-mode diffusion. This relationship can also be used to describe anomalous diffusion behavior. Here the exponent α is less than unity and greater than zero for inhibited diffusion, which is also known as subdiffusion, or it can be greater than unity but less than two (ballistic motion) for accelerated diffusion, which is also called superdiffusion[1, 2] (exponents of two indicate ballistic diffusion). Superdiffusion has been observed under various conditions in many physical circumstances such as chaotic advection.[3] Subdiffusion and superdiffusion are known to occur when the molecular velocity does not vary finitely or the autocorrelation function of the Lagrangian velocities does not decay quickly enough.[4]

[1] Gitterman, M., Mean first passage time for anomalous diffusion. Physical Review E, 2000. 62(5): p. 6065-6070.
[2] Shlesinger, M.F. and J. Klafter, Accelerated Diffusion in Josephson-Junctions and Related Chaotic Systems - Comment. Physical Review Letters, 1985. 54(23): p. 2551-2551.
[3] Zaslavsky, G.M., Chaotic dynamics and the origin of statistical laws. Physics Today, 1999. 52(8): p. 39-45.
[4] Castiglione, P., et al., On strong anomalous diffusion. Physica D: Nonlinear Phenomena, 1999. 134(1): p. 75-93.