Knudsen diffusion, named after Martin Knudsen, is a means of diffusion that occurs when the scale length of a system is comparable to or smaller than the mean free path of the particles involved. An example of this is in a long pore with a narrow diameter (2–50 nm) because molecules frequently collide with the pore wall. As another example, consider the diffusion of gas molecules through very small capillary pores. If the pore diameter is smaller than the mean free path of the diffusing gas molecules, and the density of the gas is low, the gas molecules collide with the pore walls more frequently than with each other, leading to Knudsen diffusion. In fluid mechanics, the Knudsen number is a good measure of the relative importance of Knudsen diffusion. A Knudsen number much greater than one indicates Knudsen diffusion is important. In practice, Knudsen diffusion applies only to gases because the mean free path for molecules in the liquid state is very small, typically near the diameter of the molecule itself.
Mathematical description The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases:
D A A ∗ = λ u 3 = λ 3 8 R T π M A {\displaystyle {D_{AA*}}={{\lambda u} \over {3}}={{\lambda } \over {3}}{\sqrt {{8RT} \over {\pi M_{A}}}}}
For Knudsen diffusion, path length λ is replaced with pore diameter d {\displaystyle d} , as species A is now more likely to collide with the pore wall as opposed with another molecule. The Knudsen diffusivity for diffusing species A, D K A {\displaystyle D_{KA}} is thus
D K A = d u 3 = d 3 8 R T π M A , {\displaystyle {D_{KA}}={du \over {3}}={d \over {3}}{\sqrt {{8RT} \over {\pi M_{A}}}},}
where R {\displaystyle R} is the gas constant (8.3144 J/(mol·K) in SI units), molar mass M A {\displaystyle M_{A}} is expressed in units of kg/mol, and temperature T (in kelvins). Knudsen diffusivity D K A {\displaystyle D_{KA}} thus depends on the pore diameter, species molar mass and temperature. Expressed as a molecular flux, Knudsen diffusion follows the equation for Fick's first law of diffusion:
J K = − ∇ n D K A {\displaystyle J_{K}=-\nabla nD_{KA}}
Here, J K {\displaystyle J_{K}} is the molecular flux in mol/m²·s, n {\displaystyle n} is the molar concentration in m o l / m 3 {\displaystyle {\rm {mol/m^{3}}}} . The diffusive flux is driven by a concentration gradient, which in most cases is embodied as a pressure gradient (i.e. n = P / R T {\displaystyle n=P/RT} therefore ∇ n = Δ P R T l {\displaystyle \nabla n={\frac {\Delta P}{RTl}}} where Δ P {\displaystyle \Delta P} is the pressure difference between both sides of the pore and l {\displaystyle l} is the length of the pore). If we assume that Δ P {\displaystyle \Delta P} is much less than P a v e {\displaystyle P_{\rm {ave}}} , the average absolute pressure in the system (i.e. Δ P ≪ P a v e {\displaystyle \Delta P\ll P_{\rm {ave}}} ) then we can express the Knudsen flux as a volumetric flow rate as follows:
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