General properties of polymer networks - Physical Gels from Biological and Synthetic Polymers - page 121

where c is the number concentration of monomers, N is the number of monomers of the
chain, N ent is the number of monomers in an entanglement strand and N S is the average
number of monomers along the chain between the S stickers:

N

N S ¼

1 :

ð 4 : 24 Þ

S þ

On time scales t > , the stickers open, the stress due to the stickers relaxes and the

modulus drops to the level G 2 of the identical linear entangled chains without stickers:

τ

:

1
N ent

G 2 cRT ð 4 : 25 Þ

The second plateau G 2 persists until it reaches the terminal relaxation time T d of the
reversible network, given by

1 : 5

2S 2 N

N ent

τ

T d

p 0 2 :

ð 4 : 26 Þ

1 9 = p 0 þ 12 =

The predictions of the LRC model are summarized in Figure 4.15 . T d is enhanced
signi ed polymers T d because, for

cantly relative to that of unmodi τ > τ e , the stickers

retard the reptation process. At the same time, the G 00 (

ω ) spectrum shows two maxima,

one at high frequency, corresponding to 1/ and independent of the degree of substitution,

and the other at 1/T d for the longest relaxation time, and which depends on p 0 .
The prediction for the zero-shear viscosity from the sticky reptation model is given by

τ

1 : 5

2S 2 cRT

N ent

N
N ent

τ

¼ G 2 T d ¼

p 0 2 :

ð 4 : 27 Þ

1 9 = p 0 þ 12 =

Experimental evidence for this model is discussed by Candau et al.( 1998 ), Regalado
et al.( 1999 ) and Feldman et al.( 2009 ), according to various chain architectures. The

G 1

G 2

τ e τ T d 0 T d

log t

Figure 4.15

Time-dependent relaxation moduli of reversible networks of linear chains with stickers (solid line)
and without stickers (dashed line). Adapted from Leibler et al.( 1991 ) © 1991 American Chemical
Society.