Numerous theories have been developed for diffusion in compound
systems, and they all depend, of course, on the structure of the matrix; in
other words, they are model dependent. In Section 5.3.3 a very simple model
of macroscopic regions of different diffusivity will be given. Here, a
microscopic approach is taken, where part of the material, the matrix, is
inaccessible to the diffusing species; the matrix has pores filled with solvent
or solution, through which diffusion can occur. Quite generally, the
following factors affecting the effective diffusion coefficient D* can be
distinguished:
- If thevolume fractionof the matrix isj, only a fractionð 1 jÞof
the material is available to the solute. This does not necessarily
affectD, if the concentration of solute is taken in the solvent, and
if this is also accounted for in the boundary conditions used to
solve Fick’s equations. However,jis often unknown, nor readily
determined. For one thing, the effectivejmay be larger than the
nominal value, because some of the solvent is not available to the
solute (see Section 8.3). In such cases, what will be experimentally
observed from the mass transport rate is a smallerD. - Tortuosity: the diffusing molecules have to travel around the
obstacles formed by the matrix, thereby increasing the effective
path length, hence decreasingD*. For a low value ofjthe effect is
small. At highj, say more than 0.6, the correction factor would be
of orderðp= 2 Þ^2 & 0 :4. - Constriction: if the pore radiusrp is not much larger than the
radius of the diffusing solute moleculerm, the molecule frequently
collides with the pore wall whereby its diffusion is impeded, the
more so forrm=rpcloser to unity. It is difficult to predict the
magnitude of this effect. It can be roughly estimated by the
semiempirical Renkin equation, which applies to diffusion of
reasonably spherical molecules (or particles) in a straight
cylindrical capillary:
DðlÞ
D 0
¼ð 1 lÞð 1 2 lþ 2 l^3 l^5 Þl¼
rm
rpð 5 : 26 ÞThe result is given in Figure 5.15, and it is seen that the effect is
very strong. Also more sophisticated theories, in which the pores
considered are more realistic, show that the ratioloften is the
most important factor limiting diffusion rate in porous materials.