in free energy. Flory showed that the shear modulus of an entropic gel is
simply given by
G¼nkBT ð 17 : 15 Þwherenis the number of cords (or twice the number of cross-links) per unit
volume. Further assumptions about the structure are that (a) the number of
cords per original polymer molecule is large (cf. Figure 6.2); (b) all cords
have about the same contour length; and (c) the network contains negligible
numbers of entanglements (see Figure 6.16), closed loops, and loose ends.
Moreover, the equation can only be accurate for small strains, since
considerable change in the end-to-end distance of the cords would distort
the Gaussian distribution of statistical chain elements. This happens more
readily for a smaller value ofn^0. It also implies that at increasing strain, the
chemical bonds in the primary chain become increasingly distorted.
Consequently, the increase in elastic free energy is due not merely to a
decrease in conformational entropy but also to an increase in bond
enthalpy. If the value ofn^0 is quite small, even a small strain will cause an
increase in enthalpy. (In a crystalline solid, only the increase in bond
enthalpy contributes to the elastic modulus.)
Theory has been developed that takes these aspects into account.
Figure 17.11 gives calculated relations for the elongation of a gel specimen.
For the calculation some assumptions about the network structure have to
be made, but the trends given are generally observed. It is seen that for small
n^0 , the curve readily becomes vertical, implying that the cords are fully
stretched; further stress increase would then lead to breaking of chains or
cross-links.
Although food gels are never of the rubber type discussed, the
relations given provide some qualitative insight in the factors governing the
rheological properties of polymer gels.
Junctions. Most food polymer gels have cross-links in the form of
junctions as schematically depicted in Figure 17.10b. Here no chemical
(covalent) bonds are formed between molecules, but microcrystalline
regions that involve a great number of (mostly weak) ‘‘physical bonds’’.
Van der Waals attraction, hydrophobic interaction, and hydrogen bonding
may contribute; for charged polymers, ionic bonds can be involved. The
simple type shown in the figure, i.e., a microcrystallite of stretched polymer
chains, is not very common. It appears to occur in gels of galactomannans
and possibly of xanthan. It may be that in amylose gels crystallites of
straightened single helices can provide junctions. Since nearly all of these