General properties of polymer networks - Physical Gels from Biological and Synthetic Polymers - page 114

micelle

(a)

Double-bonded states, bridge One single-bonded state Free chain state

(1) (2) (3)

Loop Closed loop

(4) (5)

(b)

Superbridge

Superloop

Dangling end

Various possible states of association of telechelic polymers: (a)(1) double-bonded state, bridge;
(a)(2) one single-bonded state; (a)(3) free chain state; (a)(4) loop; (a)(5) closed loop; (b) more
complex topologies: micelles with a cross-link functionality greater than 2 are shown as filled discs.
Adapted from Annable et al. (1993) with permission of the American Institute of Physics for The
Society of Rheology.

Figure 4.8

suggests a linear increase with concentration, or G ∞ /(nk B T) to be 1 at all concentrations
(in the limit W>> k B T there are few free stickers). It was observed experimentally that
G ∞ /(nk B T) varied strongly with polymer concentration and was always smaller than 1.
Therefore, it has to be assumed that the topology of the network changes with concen-
tration. This important feature was pointed out by Annable et al.( 1993 ), who suggested
that the fundamental process underlying the concentration dependence experimentally
observed for G ∞ is the entropically driven transformation from a network composed
mainly of loops, at low concentration, to one containing mainly links, at high concen-
tration (

).
In dilute solutions, various types of associations can be generated, as shown in
Figure 4.8 . Most of the structures represented are not effective cross-links, but with an
increase in the polymer concentration the loops transform into bridges, giving rise to a
strong initial concentration dependence of the modulus, viscosity and relaxation time.
Thus the TE transient network theory represents a limiting case for HEUR solutions at
high concentrations. Annable et al.( 1993 ) proposed a statistical mechanical model,
supported by Monte Carlo simulations which show the gradual transition from micelles
built mainly from loops to micelles linked by bridging chains.
Figure 4.9 shows semi-quantitative agreement between the experimental results and
the simulation for the high-frequency modulus derived by Annable et al. from their

' ring - chain competition '