As described in the DiffuDict theoretical background, Knudsen diffusion models the movement of molecules in the absence of molecule-molecule collisions. This means that a molecule will move straight ahead until it encounters a solid obstacle (wall), where it is reflected in a random direction. Numerically, this is solved by a random walk method.

Each single molecule starts with a random, Maxwell-Boltzmann distributed velocity and moves through the pore space until it hits a wall. At the wall, the molecule is reflected diffusively according to Lambert’s cosine law (see Greenwood) and leaves the wall with a new random, again Maxwell-Boltzmann distributed velocity. The molecule continues its way until the desired simulation time is reached. The resulting displacement (distance between the molecules start position and end position) is compared to its travel time, which results in the diffusivity for this individual molecule. Thus, to obtain a good estimate for the diffusivity of the whole 3D structure, a high number of simulated molecules is needed.

From the displacement, the diffusivity can be determined by Einstein’s formula:

where denotes the expectation value, is the starting position of a particle and the position at time , and denotes the transposed vector.

The resulting 3x3 diffusivity matrix depends not only on the pore structure of the porous medium, but also on the mean thermal velocity of the diffusing gas. However, if one defines an intrinsic Knudsen diffusivity

with the mean thermal velocity of the fluid and the characteristic length of the porous media, the diffusivity can again be written as

Here, again is a dimensionless 3x3 matrix that is independent of the diffusing species. Therefore, the Knudsen diffusion command may determine without requiring the user to define the diffusing species.

Analogously to the approach taken for the bulk diffusivity, it is common in literature (see e.g. the papers of Tomadakis) to define a Knudsen tortuosity factor by writing

Unfortunately, this whole definition depends on the definition of the characteristic length of the porous media, which is by no means a well-defined quantity. Thus, depending on the definition of , the tortuosity factor may end up being smaller than 1.0, which disagrees with the geometric intuition.

For this reason, this definition of a Knudsen tortuosity factor has been criticized by Zalc et al, and they could indeed show, that the Knudsen tortuosity factor matches the tortuosity factor for a specific (and complicated) choice of the characteristic length.

GeoDict does not follow the approach of Zalc et al but computes the characteristic length of the structure as the mean path length between two consecutive hits of molecule and wall. Therefore, the reported Knudsen tortuosity factor and the relative diffusivity will differ from the results of the bulk diffusivity simulations, and Knudsen tortuosity factors <1 are possible for some pore geometries.