Common sense suggests, and Eq. (5.26) predicts, that D*¼0if
rm 5 rp. This is, however, not the case. Even in a crystalline solid, diffusivity
is finite, though very small, e.g., 10^22 m^2 ?s^1. (Over a distance of 1 nmt 0 : 5
would then equal about 3 hours, and over 1 mm 300 million years.) At the
scale of molecules, pores do not have a fixed size: even immobilized
structures exhibit Brownian motion and this leads to fluctuating pore sizes,
occasionally letting even a fairly large molecule pass. This is quite obvious in
polymer gels, and something similar happens in dry materials.
All these observations imply that the prediction of diffusivity in
composite materials is far from easy and incompletely understood. Specific
interactions on a molecular scale between solute and matrix can also affect
diffusivity.
Partial Osmosis. If constriction is significantly stronger for larger
molecules—say, of a solute—than for small ones—say, of the solvent—the
diffusional fluxes are not anymore equal and opposite, as is the case in
unhindered diffusion (barring volume change upon mixing). This results in
partial osmosis. For example, if a piece of fruit is put in a concentrated
sugar solution, water diffuses faster out than sugar diffuses in, leading to a
decrease in volume. A prerequisite then is that the piece of fruit can shrink,
which means a compression of the cellular structure (see Section 17.5).
This is a way of ‘‘drying’’ the fruit, calledosmotic dehydration.Itis,
however, not pure osmosis, since the fruit is not impervious to sugar: it only
diffuses slower, say by a factor of five, than water. If the fruit is left in the
sugar solution, it will eventually obtain the same sugar content, relative to
water, as in the said solution. The fruit then becomes candied besides dried,
which is commonly applied to dates. Several other examples could be given;
for instance, when salting cheese by immersing it in brine, the loss of water
from the cheese is generally more than twice the uptake of salt.
Question
In the research department of a food company, the diffusion of salt into meat
products is studied. Large pieces are immersed in concentrated brine (salt
concentrationc 1 ), and the salt uptake per unit surface area is determined as a
function of brining timeðtÞ:As expected, it is proportional toHt. By means of Eq.
(5.19), the effective diffusion coefficientD*is estimated. For lean pork a value of
2 : 2? 10 ^10 m^2 ?s^1 results, for back fat (untrimmed bacon) only 10^11 m^2 ?s^1 .Asa
check it is determined in a separate experiment what the salt content is at a distance
of 1 cm from the outside after 5 days of brining. By using Eq. (5.21), it is expected
that in the lean porkc(expressed per kg water) will equal nearly 0: 5 c 1 , whereas it