Physical Chemistry of Foods

cross section perpendicular tox, andDxis the deformation of the bond
between two structural elements; dFinis the change in the interaction force
between two structural elements when moved with respect to each other
over a distance dh. Further, the external stress iss ¼Fex/A, and we
postulate thatDx¼Ce;Cis a constant of dimension length, whose value
will depend on the geometry of the network. Finally, we haveFin¼
dV/dh,
where Vis the interaction free energy between two structural elements
(discussed in Section 12.1). Assuming that dFin/dh is constant, which
generally implies a very small dhand hence a very small deformation, we
arrive at

E¼
CN

d^2 V

dh^2

ð 17 : 4 Þ

Notice thatVis a (Gibbs) free energy. It can thus be due to enthalpic factors
(as in a fat crystal network) or to entropic factors (as in a rubberlike
material). Often, both enthalpy and entropy are involved.
Although Eq. (17.4) is rigorous if the prerequisites are met, even then
application is generally not easy. Analysis of micrographs may yield an
estimate of the variableN. It is more difficult to estimate the value ofC,
since precise knowledge of the structure is needed. The magnitude ofVas a
function ofhneeds to be precisely known, and this is rarely the case (see
Chapter 12). Nevertheless, in a few simple cases, reasonable predictions can
be made or, in reverse, experimentally established relations betweenEand
some variable—say, volume fraction of the network material—can be used
to derive information about the network structure.
However, most real systems do not comply with the presumptions
made in the derivation of Eq. (17.4). Generally, more than one type of
interaction force will act, and the structural elements often vary in type or
size, which implies a spectrum of interaction forces; we will see examples of
this in the following sections. The contributions to the modulus of the
various forces involved are not additive, primarily because the bonds vary in
direction. Moreover, such materials are generally not fully elastic, which
implies that the modulus will be complex [see Eq. (5.12)] and depend on
deformation rate; virtually all soft-solid foods show viscoelastic behavior of
some type. Finally, some systems are quite inhomogeneous, which further
complicates the relations.

Large Deformations. For most soft solids, the direct
proportionality between stress and strain only holds up to a very small
strain, rarely over 0.01. One may, of course, calculate an apparent modulus
Ea¼s(e)/e, which is for most foods smaller than the true modulus (cf.