In physics, 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.^[1] 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:^[2]

${\displaystyle {D_{AA*))=((\lambda u} \over {3))=((\lambda } \over {3)){\sqrt ((8RT} \over {\pi M_{A))))}$

For Knudsen diffusion, path length λ is replaced with pore diameter ${\displaystyle d}$, 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, ${\displaystyle D_{KA))$ is thus

${\displaystyle {D_{KA))={du \over {3))={d \over {3)){\sqrt ((8RT} \over {\pi M_{A)))),}$

where ${\displaystyle R}$ is the gas constant (8.3144 J/(mol·K) in SI units), molar mass ${\displaystyle M_{A))$ is expressed in units of kg/mol, and temperature T (in kelvins). Knudsen diffusivity ${\displaystyle D_{KA))$ 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:

${\displaystyle J_{K}=\nabla nD_{KA))$

Here, ${\displaystyle J_{K))$ is the molecular flux in mol/m²·s, ${\displaystyle n}$ is the molar concentration in ${\displaystyle {\rm {mol/m^{3))))$. The diffusive flux is driven by a concentration gradient, which in most cases is embodied as a pressure gradient (i.e. ${\displaystyle n=P/RT}$ therefore ${\displaystyle \nabla n={\frac {\Delta P}{RTl))}$ where ${\displaystyle \Delta P}$ is the pressure difference between both sides of the pore and ${\displaystyle l}$ is the length of the pore).

If we assume that ${\displaystyle \Delta P}$ is much less than ${\displaystyle P_{\rm {ave))}$, the average absolute pressure in the system (i.e. ${\displaystyle \Delta P\ll P_{\rm {ave))}$) then we can express the Knudsen flux as a volumetric flow rate as follows:

${\displaystyle Q_{K}={\frac {\Delta Pd^{3)){6lP_{\rm {ave)))){\sqrt {\frac {2\pi RT}{M_{A))))}$,

where ${\displaystyle Q_{K))$ is the volumetric flow rate in ${\displaystyle {\rm {m^{3}/s))}$. If the pore is relatively short, entrance effects can significantly reduce to net flux through the pore. In this case, the law of effusion can be used to calculate the excess resistance due to entrance effects rather easily by substituting an effective length ${\displaystyle l_{\rm {e))=l+{\tfrac {4}{3))d}$ in for ${\displaystyle l}$. Generally, the Knudsen process is significant only at low pressure and small pore diameter. However there may be instances where both Knudsen diffusion and molecular diffusion ${\displaystyle D_{AB))$ are important. The effective diffusivity of species A in a binary mixture of A and B, ${\displaystyle D_{Ae))$ is determined by

${\displaystyle {\frac {1}((D}_{Ae))}={\frac {1-\alpha ((y}_{a))}((D}_{AB))}+{\frac {1}((D}_{KA))},}$

where ${\displaystyle \alpha =1+{\tfrac ((N}_{B))((N}_{A))))$ and ${\displaystyle {N}_{i))$ is the flux of component i. For cases where α = 0 (${\displaystyle N_{A}=-N_{B))$, i.e. countercurrent diffusion)^[3] or where ${\displaystyle y_{A))$ is close to zero, the equation reduces to

${\displaystyle {\frac {1}((D}_{Ae))}={\frac {1}((D}_{AB))}+{\frac {1}((D}_{KA))}.}$

Knudsen self diffusion

In the Knudsen diffusion regime, the molecules do not interact with one another, so that they move in straight lines between points on the pore channel surface. Self-diffusivity is a measure of the translational mobility of individual molecules. Under conditions of thermodynamic equilibrium, a molecule is tagged and its trajectory followed over a long time. If the motion is diffusive, and in a medium without long-range correlations, the squared displacement of the molecule from its original position will eventually grow linearly with time (Einstein’s equation). To reduce statistical errors in simulations, the self-diffusivity, ${\displaystyle D_{S))$, of a species is defined from ensemble averaging Einstein’s equation over a large enough number of molecules N.^[4]