Diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). See http://en.wikipedia.org/wiki/Mass_diffusivity. It is described by Fick’s first law:

where is the diffusive flux, given in mol/m²/s and is the concentration difference over a distance . The diffusivity has the unit m²/s.

Fick’s first law also describes diffusion through a porous layer as shown above. In this case, the diffusivity depends on the properties of the diffusing species and on the properties of the porous material. A porous material may offer a different resistance to a diffusing particle depending on the direction that the particle is traveling. Therefore, a more general form of Fick’s first law

can be used to describe the diffusion in three space dimensions. In this form, is a three-dimensional vector, and the diffusivity is a 3x3 matrix which describes the full directional dependence of the diffusion. In DiffuDict, always the more general form of Fick’s first law is considered. For isotropic materials holds.

In a three-dimensional structure model, the stationary distribution of concentrations can be computed by solving

together with the appropriate boundary conditions for the local concentration at the boundaries of the domain. Here, the local diffusivity depends on the local material. When simulating e.g. the diffusion of oxygen, in the pore space is the diffusion coefficient of oxygen, in porous materials is the effective diffusivity of oxygen inside of the porous material, and in solid materials . This equation is solved in DiffuDict’s Simulate Diffusion Experiment function.

This leads to the question how the effective diffusivity of a gas inside of a porous material can be determined. In literature, the effective diffusivity is often approximated as

where is the intrinsic diffusivity of the gas, is the porosity of the porous medium, and is a tortuosity of the pores. Alternatively, the equation

can be found, where the additional factor denotes a so-called constrictivity of the pores. Various geometrical approaches are used to determine and . However, this approach is not exact because

1. the results depend vastly on the geometrical method used and
2. this approximation is only valid in the continuum and does not include Knudsen diffusion.

A mathematically consistent way to determine the effective diffusivity of a porous medium is to use an upscaling approach (see e.g. Hornung, 1997). For this, a representative 3D model of the pore structure is needed.

Under the assumption that the Knudsen number is small () and that the classical continuum mechanics approach can be used (see Knudsen number information box below), the diffusivity indeed decouples into the species-dependent part and a porous-media-dependent part :

Here, is a scalar quantity with unit m²/s and is a dimensionless 3x3 matrix. For isotropic materials, holds with a scalar diffusivity value . To determine , the Laplace equation

is solved in the pore space with Neumann boundary conditions on the pore walls, and a concentration drop in one space direction. With the computed concentration flux, it is possible to find the diffusion coefficient in this space direction.

In DiffuDict’s Bulk (Laplace) diffusion command, the Laplace equation is solved three times, each time with a concentration gradient in a different space direction, to determine the full tensor . Note, that the relative diffusivity is a dimensionless quantity and a property of the porous media alone that, as such, is independent of the diffusing species and surrounding fluid.

A comparison of the equations (189) and (192) shows that it makes sense to define a tortuosity factor through

and that this tortuosity factor is also a property of the porous media which is independent of the diffusing species and surrounding fluid. To distinguish this definition from other geometric definitions of the tortuosity, we call it tortuosity factor and denote it with . When defined like this, equation (189) holds true exactly, but a direct geometric definition of the tortuosity factor is no longer possible. Also, the constrictivity of the pores is already contained inside the so-defined tortuosity factor .

The continuum mechanics approach described above fails, when the number of molecules present in a grid cell becomes too low to define a meaningful concentration value and therefore the mass transport from one grid cell to another is no longer a diffusive process. The Knudsen number, which compares the mean free path of a molecule with the representative pore size of the medium, is a good indicator when this happens.

At ambient conditions, the mean free path of gas molecules typically lies in the range of 50 nm to 200 nm. Thus, Knudsen effects only become significant for pore sizes in the sub-micrometer range, or in case of very dilute gases. For diffusion in liquids, the Knudsen number is always small, and Knudsen diffusion is of no importance.

If the assumption () no longer holds true, the diffusing molecules experience another resistance caused by additional collisions between particles and pore walls. Due to Pollard and Present, 1948, the overall diffusivity can be approximated from the bulk diffusivity and the Knudsen diffusivity with the so-called Bosanquet approximation:

The bulk diffusivity is the effective diffusivity at Kn=0, and can be determined as described above from the result of DiffuDict’s Bulk (Laplace) diffusion command.

The Knudsen diffusivity is the effective diffusivity at , and can be determined from the result of DiffuDict’s Knudsen Diffusion command.

DiffuDict’s Bosanquet Approximation command computes from the results of the two other commands.

Therefore, to apply Bosanquet’s approximation, an algorithm is needed to compute the diffusivity at , i.e. in a situation where no particle-particle collisions happen, and the diffusion is solely caused by particle-wall interactions. This algorithm is provided in DiffuDict’s Knudsen Diffusion command.

Alternatively, the diffusivity at intermediate Knudsen numbers can be determined by a random walk method that takes particle-wall collisions as well as particle-particle collisions into account. Such an algorithm is provided in the Molecular Diffusion command. Theoretically, this algorithm provides a correct result for all Knudsen numbers, but for small Knudsen numbers the random walk algorithm becomes slow and inaccurate, and it is therefore advisable to determine the diffusivity by solving the Laplace equation in this case.