Physical Chemistry of Foods

ðb=LÞvary along the chain. Nevertheless, it may be useful to have some idea
about the concentration needed for chain overlap; it can roughly be given by

jov&ðbnÞ^1 ð 6 : 13 Þ

which will roughly hold if the polymers are fairly stiff andbis not much
larger than 0.1, which is true for most polysaccharides. Clearly, chain
overlap is already reached for very small concentrations, often well below
1 %.
Thecorrelation lengthxis a parameter that quantifies the distance over
which fluctuations in a system are correlated. In a dilute polymer solution it
roughly equals the radius of gyration. For conditions of chain overlap,xcan
be defined as the diameter of the ‘‘blobs’’ illustrated in Figure 6.14. The
value of x decreases with increasing stiffness, b and j, in a manner
depending on the regime. In practice, the most important variable
determining the correlation length is thus the concentration. The degree
of polymerizationnhas no effect: if in a ‘‘semidilute’’ or a concentrated
solution some polymer molecules would be split into shorter ones, or if some
polymer molecules would be joined to form longer ones, the properties of
the solution would not or hardly change. Of course, the value ofnis an
important variable in determining whether chain overlap occurs [Eqs.
(6.12a) and (6.13)], but once in a nondilute regime, it further has little effect
on macroscopic properties.
In the‘‘semidilute’’ regime, the molecules cannot distribute themselves
at random over the volume. The polymer concentration fluctuates with a
wavelength equal to the correlation length. The system can be seen as a kind
of network with mesh size comparable tox. The ‘‘network’’ continuously
changes conformation due to Brownian motion. Over distances along the
polymer chain<x, which implies short time scales for molecular motion,
polymer sub-chains behave as in a dilute solution; interactions between two

TABLE6.4 Examples of the Critical Volume

Fractionjcrand the Critical Volume Expansion

Factorbcras a Function of the Molecular Volume

Ratio Polymer/Solventq, According to Eq. (6.12)

q jcr bcr

102 0.091  0 : 21

103 0.031  0 : 06

104 0.010  0 : 02

105 0.003  0 : 006