# Interdisciplinary Applied Mathematics

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diffusivity in the direction normal to the pore wall is zero due to the geometrical limit. To circumvent this problem, the diffusion in the z-direction is usually characterized by a mean-square displacement Az2(t) that can be calculated for a    short    time.    In this section,    we    will    discuss    the    results    for

diffusion parallel to the pore wall (characterized by Dц or Dx and Dy) and the diffusion normal to the pore wall (characterized by Az2(t)) separately.

(Magda    et    al.,    1985)    studied diffusion    in    slit    pores    with    smooth    pore

walls using equilibrium MD simulations. Figure 10.7 shows the variation of the pore-averaged diffusivity parallel to the pore (D||) with the pore width. The plot indicates that:

1. Even for the smallest pore width (h = 2), where the fluids are highly confined, the fluid atoms maintain considerable mobility.

2. When the channel width is small (h < 4), the average Dц in the pore fluctuates with the channel width, and when the channel width increases beyond h = 5, the average Dц increases smoothly toward the asymptotic bulk value.

3. For a channel width of h = 11.57, the average Dц is almost the same as the bulk diffusivity.

The second observation can be attributed to the average density variation with the change in channel width. As shown in Figure 10.7, when the channel width is small (h < 4), the average density fluctuates with the channel width,    and    when    the    channel    width    increases    beyond h =    5,    the

average density decreases smoothly toward the asymptotic bulk value. The dependence of the diffusivity on density in the pore region is much weaker compared to the quadratic dependence observed in the bulk (Levesque and Verlet, 1970). This means that the variation of diffusivity with density follows a quadratic dependence as the pore width increases and the properties of the confined fluid approach that of the bulk. However, when the pore width is low (lower than 6a), then the layering effect dominates and affects the variation of diffusivity. This leads to a weaker dependence of diffusivity on density for narrow pores. A possible explanation for this is the structured, almost solid-like form of the density profile in narrow pores.

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