# Knudsen diffusion

Schematic drawing of a molecule in a cylindrical pore in the case of Knudsen diffusion; are indicated the pore diameter (d) and the free path of the particle (l).

Knudsen diffusion 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. For example in a long pore with a narrow diameter (2–50 nm) because molecules frequently collide with the pore wall.[1]

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. This process is known as Knudsen flow or Knudsen diffusion.

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.

The diffusivity for Knudsen diffusion is obtained from the self-diffusion coefficient derived from the kinetic theory of gases.[2]

${D_{AA*}} = {{\lambda u} \over {3}} = {{\lambda}\over{3}} \sqrt{{8k_B T}\over {\pi M_{A}}}$

or for diffusivity of species j in a mixture

${D_{j}} = {8 r_{a} \over {3}} {\sqrt{{R T} \over {2} \pi M_{j}}}$

For Knudsen diffusion, path length λ is replaced with pore diameter $d_{pore}$, 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, $D_{KA}$ is thus,

${D_{KA}} = {d_{pore} \over {3}} u = {{d_{pore}\over{3}}} \sqrt{{8 R_g T}\over {\pi M_{A}}}$

Where $R_g$ is the gas constant (8.3144 J/mol K in SI units), molecular weight $M_{A}$ is expressed in units of kg/mol and temperature T has units of K. Knudsen diffusivity $D_{KA}$ thus depends on the pore diameter, species molecular weight and temperature.

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 $D_{AB}$ are important. The effective diffusivity of species A in a binary mixture of A and B, $D_{Ae}$ is determined by,

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

Where, $\alpha =1+\frac{{{N}_{B}}}{{{N}_{A}}}$

For cases where α=0, ($N_{A}$=-$N_{B}$), or where $y_{A}$ is close to zero, the equation reduces to,

$\frac{1}{{{D}_{Ae}}}=\frac{1}{{{D}_{AB}}}+\frac{1}{{{D}_{KA}}}$

## 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, $D_{S}$, of a species is defined from ensemble averaging Einstein’s equation over a large enough number of molecules,N :[3]