Helical structures from neutral biopolymers - Physical Gels from Biological and Synthetic Polymers - page 205

mechanisms of the gels still need to be made. Assuming that the strands are much more
rigid than the links (coil segments) that hold them together, the elasticity of such a
network is then purely entropic and due to the constrained thermal agitation of the rods.
Such a case has been considered by Jones and Marques ( 1990 ) and called the freely
hinged network, where the elasticity is due to constrained thermal agitation of the rods.
Elasticity then scales as

G 0 ≈ Bk B TL 1 d 2

; ð 7 : 10 Þ

where B is a prefactor and T is the temperature. To test the concentration dependence
of elasticity, further assumptions must be made as to the geometry (topology, connectiv-
ity) of the network, and in particular as to the scaling of the strand length L with helix
concentration c helix . In the absence of such information we can make the simplest
assumption, i.e. that the length of the rods L and their typical distance d scale alike.
Since d ~ c helix − 1/2 , this leads to

G 0 ~ c 1 : 5
helix

: ð 7 : 11 Þ

The moduli are extremely sensitive to the ratio L/d. We obtain a much better agreement
with experimental values when the helices are considered longer than the mesh size d.In
fact good agreement is found using a prefactor of 1 when we assume L/d ~ 5, meaning
that the helices are much longer than the mesh size d. The intermediate domain between
percolation and a homogeneous network is treated by a simple interpolation between the
two contributions. The data for the highest helix concentrations, far from the percolation
regime, is also in agreement, giving an expected exponent close to 1.5, as shown in
Figure 7.16 .
It is possible to illustrate the structure of such a gelatin network schematically as a
collection of rigid, entangled rods (see Figure 7.17 ) whose total length per unit volume
can be measured at any moment and is the only parameter controlling the elasticity of the

10 5

Intermediate
domain

10 4

Slope = 1.48

10 3 Homogeneous network

10 2

0.01 0.1

c helix (g cm − 3 )

The power-law behaviour of the storage modulus versus helix concentration far from the
percolation regime, after the intermediate domain. In the homogeneous network of rigid rods the
exponent converges towards 1.5. Adapted from El Harfaoui et al.( 2007 ) with permission from
John Wiley & Sons.

Figure 7.16