whereZgis the viscosity atTg(Tis absolute temperature).C 1 andC 2 would
be constants for each system, and often fixed values are used, viz.17.4, and
51.6 K, respectively. The curve forZain Figure 16.3b is calculated according
to Eq. (16.2) with these values for the constants andTmTg¼90 K. It is
seen that the decrease in viscosity is very strong, by a factor of about 670 for
the first 10 K increase overTg. The equation often fits results for polymer
systems well, although the two constants may vary significantly. Moreover,
the viscosity measured is an apparent one and will depend on measuring
conditions.
For other systems, the relations tend to be different and variable, and
there is currently no consensus about the theory. The most important point
is that the decrease in viscosity with increasing temperature tends to be
weaker, and in some cases much weaker, than predicted by the WLF
equation, especially near the glass transition. To be sure, even then the
temperature dependence is strong as compared to that for ‘‘simple’’ liquids,
which tend to follow an Arrhenius type relation ½Z!expðC=TÞ. For
complex mixtures, prediction of the temperature–viscosity relation from
theory is currently impossible.
Molecular Mobility. In a glass the mobility of molecules
presumably is very small. In a glass of one component, this appears
indeed to be the case. Recalling Eq. (5.16), i.e.,D¼kBT= 6 pZr, we calculate
that a molecule of radiusr¼0.5 nm, in a glass of viscosity 10^12 Pa?s, has a
diffusion coefficientD& 10 ^24 m^2 ?s^1 , which is extremely small. It would
mean that it takes a molecule 300 centuries to diffuse over 1mm distance. It
originally had been assumed that these relations were generally valid. This
would imply that the translational motion of molecules in a glassy food
cannot occur, making the food completely stable to all changes involving
diffusion, which includes nearly all chemical reactions. Moreover, it was
sometimes assumed that aboveTg, WLF viscosity could be used to calculate
Dvia Eq. (5.16). These assumptions have not been confirmed: diffusivities
tend to be far greater.
A rather trivial reason may be that the system can beinhomogeneous,
possibly containing tiny cracks, that allow much faster diffusion. Moreover,
Eq. (5.16) is poorly obeyed for highly concentrated systems, as discussed in
Section 5.3.2. A more fundamental reason seems to be that the glasses
always containmore components. Consider, for example, a sugar glass that
also contains water. At or belowTg, the sugar molecules are presumably
fully immobilized. However, smaller molecules, like water, can still move in
the spaces between the sugar molecules. An example is given in Figure 16.4a.
It is seen that well aboveTg, the calculation according to the WLF equation
is reasonably well obeyed, but that the discrepancy becomes very large close