I am curious whether the size (or width) of a channel influences the rate of diffusion across the channel. Intuitively, diffusion rate would be higher for a channel with large diameter compared to the channel with small diameter, but the Fick's law (flux ~ concentration gradient) does not incorporate any geometrical factors. Which is true? Thanks.
Channel diameter affects the effective diffusion coefficient. If the bulk diffusion coefficient exceeds Knudsen diffusion coefficient than you should use latter coefficient which is influenced by channel diameter.
In addition to the above answer, there is another consideration, which involves the entrances to the channel from (presumably) a region of different size. In that case, there is an entropic contribution to the effective diffusion coefficient related to the axial derivative of the cross-sectional area, and one can derive a corrected Fick's law (see for instance the paper by Robert Zwanzig.)
If you're interested particular in diffusion modelling, take a closer look into Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation by van Genuchten and Alves. The transport equation, which considers only diffusion as a transport process, has form (1+R) dC / dt = D d^2 C / dx^2 , where R is ralated to medium porosity. So, porosity (for example width of the channels) reduces rate of the diffusion.
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