The sol–gel transition - Physical Gels from Biological and Synthetic Polymers - page 90

N

The zipper model: a single zipper with N links can be opened from either end. Helices may not
necessarily be double helices. Here the double helices represent the association of some ordered
structures.

Figure 3.13

Murphy, 1987 ) that if the cross-linking structure involves

more than a few k B T (the Boltzmann energy), then c 0 < c*, and, insofar as it can be
tested, this does seem to be the case. The experimental dif

- however it is constituted -

culty here is that establish-
ing c* for a system just as it undergoes a change in structure attendant with gelation is
rarely straightforward.

3.8 Zipper model

Another very useful model which is applicable to the melting of all ' junction zone ' type

gels is the so-called model of Nishinari and co-workers (Nishinari et al., 1990 ).

Figure 3.13 presents the basic model.
They treated the dissolution of such gels as the opening of molecular zippers, consist-
ing of N parallel links that can be opened from one end. First, we assume that the zipper
can be opened from only the left end. Link i from the left can be opened only when the
0th, 1st, 2nd,

' zipper '

1)th links are all open. Each link has only two energy states: the
energy of a closed link is assumed to be zero and that for the open link E. When one link is
open, we assume that it has

... ,(i −

orientations, corresponding to the rotational freedom of the
links; that is, the energy level of this link is

Ω

Ω -fold degenerate. Then, for a long enough

chain, the partition function ζ for such a single zipper which can be opened from either

end is

=

2

x N Nx N þ 1

ζ ¼ 1 ð N þ 1 Þ þ ð 1 x Þ ; ð 3 : 33 Þ

where the factor x is given by:

x ¼ Ω exp ð E = k B T Þ ð 3 : 34 Þ

and k B T is the Boltzmann energy. The partition function for the whole system containing
N

ζ p , and from this the heat capacity of the whole system is

zippers is then simply

directly found.
The advantage of this method is that the heat capacity for a gelling junction zone
system can be calculated as a function of temperature and, even allowing for the number
of parameters, a realistic fit can be obtained for a variety of systems, including PVA