Helical structures from neutral biopolymers - Physical Gels from Biological and Synthetic Polymers - page 203

would be more reasonable. Still needed is an estimate for the network front factor. For a
ideal rubber-like network this should lie in the range 0.5 (for a phantom network) to 1
(af

fits can be obtained with f in the
range 10 to 20 (Ross-Murphy, 1992 ). Corresponding critical concentrations are ~0.7%,
whichever value of f or front factor is applied, a value which corresponds well with light
scattering results.
More sophisticated analyses have been applied, particularly by te Nijenhuis, and are
discussed in his review (te Nijenhuis, 1997 ). For example, in the appendix to that review
he derives an equation which can be rewritten as

ne). Using these ' constraints ' we nd that very good

g
2 - 1

cRT
M

4
g

c
c 0

G eq ≈ : ð 7 : 6 Þ

-
for gelatin he assumes this will be 6 [= 3 (for a triple helix) × 2 (since he assumes the
association of two such chains)]

Here G eq is the equilibrium shear modulus, g is his so-called cross-link molecularity

and c, c 0 are concentration and critical concentration,
respectively. If we use this value, g = 6, (7.6) gives the commonly observed G ~ c 2
dependence.

-

7.2.8.2 Rod-like model

The rheological data shows that the fundamental relationship between elasticity of
gelatin gels and molecular properties is via the proportion of helical residues. The helices
have in common an identical thickness (~1 nm) and variable lengths. Consequently the
degree of perfection determined by the dispersion of melting temperatures has no
in

uence on the rigidity.
Then two main regions can be de ned: the critical region, where the modulus increases

very sharply, and at higher helical concentrations, c >> 2c critical
helix

:

(a) The critical region, where the modulus increases very sharply

Here, according to the analysis above, the shear modulus increases as a power law with
the threshold distance, c critical

helix

c critical
helix

< 0.3. The threshold c critical
helix

0.003 g cm − 3

ð c helix Þ= ≈

is very low:

t

G 0 ð c critical

helix

c helix - Þ ; t ≈ 1 : 9 : ð 7 : 7 Þ

This can be analysed in terms of a rigid-rod model, since, at the threshold, the volume
fraction occupied by the helices (assuming a protein density of 1.44 g cm − 3 ) is only
2×10 - 3 . The percolation threshold in networks made of highly anisotropic particles,
such as fibres or rods, depends on their aspect ratio (the ratio L/a between the length L and

diameter a of the rods). Balberg and co-workers (Balberg et al., 1984 ; Balberg and
Binenbaum, 1987 ) simulated homogeneous percolation in a system of randomly oriented
rods with uncorrelated contacts and established the relation

c critical L = a ≈ 0 : 7 ; L = a 1 ; ð 7 : 8 Þ