Interdisciplinary Applied Mathematics

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In the discussion so far, we have assumed that the fluid molecule size is smaller than half of the slit pore width and that the diffusion is characterized by the normal-mode diffusion; i.e., the mean-square displacement of the fluid molecules obeys the Einstein relationship. In the normal-mode diffusion, one molecule can pass another molecule within the channel. However, if the pore width decreases further and the pore is cylindrical, a molecule cannot pass another molecule because of its large size relative to the pore size, and the diffusion process is then characterized by a single-file diffusion. The mean-square displacement of a fluid molecule due to single-file

FIGURE 10.10. Diffusivity parallel to pore walls versus pore width (■ — equilibrium (a = 0), □ — Couette flow). (a) Structured pores (inset: □ — ratio of the diffusivity in Couette flow and in equilibrium simulation (a = 0), ■ — ratio of diffusivity in equilibrium for a = 0.71 and a = 0. See (Somers and Davis, 1992), for the    definition    of the wall    registry    index    a).    (b) Smooth    pores.    (Courtesy    of

H.T. Davis.)

diffusion can be expressed as

s2 = 2Bt05,

where B is the diffusion mobility. For the diffusion of methane, ethane,

FIGURE 10.11. Comparison of the mean-square displacement parallel and normal to the channel walls. (Courtesy of H.T. Davis.)

and ethylene through carbon nanotubes (see Section 13.2 for a discussion on carbon nanotubes), (Mao and Sinnott, 2000) showed that there exists a transition-mode diffusion for which the mean-square displacement of a fluid molecule due to a single-file diffusion can be expressed as

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