# Interdisciplinary Applied Mathematics

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The diffusion coefficient must be generalized in order to describe transport in confined nanochannels. For homogeneous and equilibrium systems, the diffusion coefficient can be calculated using either the Green-Kubo equation

1 f™

D=-y (v(0 )-v(t))dt,    (10.2)

where v is the atom velocity and {) denotes the ensemble average, or by the Einstein equation

(10.3)

D = — flm ([^Qo+O — KM]2)

6 t^™    t

where r is the atom position. The Green-Kubo expression given in equation (10.2) is strictly valid only for homogeneous and equilibrium systems. However, it    is generally    accepted    that,    at    least    for    the    calculation    of the

average diffusivity in nanochannels, the Green-Kubo expression given in equation (10.2) or the Einstein relationship given in equation (10.3) is adequate. For example, Bitsanis and coworkers (Bitsanis et al., 1987) computed pore-averaged diffusitives and found that the diffusivities under flow and the equilibrium diffusivity agree within the limits of statistical uncertainly. Moreover, the diffusivities calculated from the Green-Kubo formula and the Einstein relationship agree quite well. It is important to note that in the calculation of the diffusivities under flow, the drift contribution to either equation (10.2) or equation (10.3) has to be excluded. In summary, both the Green-Kubo formula and the Einstein relationship are widely used in the calculation of diffusivity of fluids in nanochannels.

The diffusion of fluids confined in nanoscale channels has been studied extensively in slit and cylindrical pores. In a slit pore, diffusion is different in the direction parallel (the x- and у-directions) and normal (z-direction) to the    pore    wall,    especially    for    narrow    pores.    This is because,    unlike    the

diffusion parallel    to    the    pore    wall,    the    diffusion    in    the    direction    normal

to the    pore    wall    is inherently    transient;    i.e., in    the    long time    limit,    the

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