Network with Mobile Cross-Links

In the ladder model treatment of Blizard, an alternative termination of the line of springs (Fig. 10-3) was considered in which each end, rather than being fixed, is attached to three other such lines, each of these to three more, and so on indefinitely, thus reproducing the connectivity of a tetrafunctional network. This.proyision increases the equilibrium compliance by a factor of 2 (corresponding-fo the factor of (/ — 2)//mentioned in Section I above), and it modifies the frequency dependence, which is now expressed by a rather complicated combination of hyperbolic functions. This frequency dependence of J is also shown in Fig. 10-7 the maximum is slightly broader than for fixed cross-links (i.e., cross-links with affine deformation). 

Storage and loss shear compliances, normalized by the equilibrium compliance, plotted logarithmically against frequency for various network theories. (A) Mooney-Rouse theory, corresponding to Fig. 10-5 (B) Blizard model with trifurcate branching corresponding to tetrafunctional connectivity (C) tetrafunctional model of Chompff and Duiser ( ) Mooney-Rouse theory with most probable distribution of strand lengths. 

Chompff-Duiser network model. (I) Unit of tetrafunctional network (II) decoupled mathematical equivalent.  

The Chompff-Duiser relaxation spectrum for a network with mobile crosslinks is the same as equation 41 for r 2AT,r. For r 2Ar,r it is given by 

Integration over equations 41 and 42 with appropriate limits by equations 19, 23, and 24 of Chapter 3 and addition of vRT/2 in the first two cases provides the viscoelastic functions G(t), G, and G for the Chompff-Duiser theory. The corresponding curve for the loss compliance J is included in Fig. 10-7. It extends farther to the low-frequency side than the others, as would be expected from the additional slow relaxation mechanisms. 

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The glass transition is mainly governed by chain interactions and chain mobility. Cross-linking will hardly influence the interactions, but the mobility is hindered by the cross-links. Tg will, therefore, increase with increasing network density. 

Fig. 9.11. Reaciion scheme for the synthesis of network-polymeric CSPs and representative chromatograms, (a) Derivatization of A, A -diallyl-(R.R)-tartaric acid diamide (DATD) to give the bifunctional monomers used as chiral SO units, (b) Cross-linking and immobilization by hydrosilylation with multifunctional hydrosilane (alternatively, cross-linking and immobilization can be performed first with DATD followed by O-derivatization). (c) Enantioseparation of 2-(octylsulphinyl)benzoic acid. The chromatograms illustrate the column performance under non-overloadcd (left) and overloaded conditions (right). CSP network polymer from /V. -diallyl-i/il.Rl-tartaric acid diamide fc/.s-. i.S-dimethylbenzoatc bound to. ) pm 1.50 A Kromasil. Mobile phase hexane-THF (80 20 v/v) with 0.0.55 - of TFA (reprinted with permission from Ref. [194]).

On the other hand, if the cross-link density is low (the length of the chains between cross-links is large) and the mobility of the chains is high, the cross-linked material is called an elastomer. An example of a typical elastomer is cw-l,4-polyisoprene (natural rubber), which, by means of a cross-linking reaction with sulfur (vulcanization), gives rise to a network structure (see Fig. 1.4). 

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