The very low permeability of several gels is the main reason why they hold
water tenaciously.
For very narrow pores, say 10 nm, the permeability depends on
molecular size. This is comparable to, but not the same as, the hindered
diffusion discussed in the following section.
Question
A company stores margarine in 250 g packages in a cold room, temperature 5 8 C. The
packages are stacked to a height of 1 m. They can be stored for some weeks without
problems. One day, the cooling does not work properly and the temperature is 15 8 C.
Nevertheless, it is decided to store the margarine in the room. After a day it is
observed that in the packages near the bottom of the stack oiling off has occurred:
the inside of the wrappers shows free oil, about a ml per package. Can you roughly
estimate the permeability of the crystal network in the margarine? Why would no
oiling off be observed at 5 8 C? You may consider margarine to consist of a space-
filling network of aggregated fat crystals, interspersed with a continuous oil phase.
Answer
Assuming that the surface area of a package is 200 cm^2 , 1 ml of oil after one day
would correspond to a superficial oil velocity of 1/200 cm per day, or about
5? 10 ^10 m?s^1. According to Table 5.2, the viscosity of the oil would be about
0 :1Pa?s. The pressure exerted by the stack equalsrgh, which gives forh¼1m
about 10^4 Pa. The distance over which the oil has to flow would on average be about
2 cm. Inserting these values in Eq. (5.24) yields a permeability of 10^16 m^2. This is
indeed the order of magnitude experimentally determined for such fats.
A variable in Eq. (5.24) that is significantly different at 5 8 C is the oil viscosity;
it would be higher by a factor of about 2.8. Moreover, the margarine contains more
solid fat at a lower temperature, which may decrease the value ofBby a factor of 2.5.
Altogether, the flow rate may be about 7 times smaller, which would imply that also
at 5 8 C oiling off would occur after a week. However, for oil separation to occur, the
crystal network has to yield: a completely rigid network would carry all the stress
(here at most 10 kPa) applied to it, leaving no possibility for shrinkage and thereby
for oil separation. Presumably, this means that owing to the higher solid fat content
at 5 8 C, the yield stress is above 10 kPa over a time scale of weeks, whereas it is below
that value at 15 8 C over a time scale of hours. This implies that (slight) sagging must
occur in the lowest packages.