Elastically effective chains

The crosslink density of a polymer network determines the number of elastically effective chains. Some of the chains are tied to a network and... 

The first step involves preparation of a bifunctional "living" precursor, of known molecular weight and low polydispersity. Crosslinking can be achieved either by the addition of stoichiometric amounts of a multifunctional electrophilic deactivator, or by adding small a small amount of a bifunctional monomer (such as DVB or ethylene glycol dimethacrylate), the polymerization of which will be initiated by the carbanionic sites of the precursor. In either case, the precursor chains become the elastically effective chains of the networks. The experimental conditions... 

The number of network chains active in the elastic behaviour of networks (elastically effective chains) can be obtained from the equilibrium behaviour of networks subjected to various types of stress as described in Chapters III and IV. Unfortunately, (i) the statistical... 

The discrepancy was explained by Scanlan (755) as well as by Mullins and Thomas (129). Because of the chain ends the network contains elastically effective four- and threefunctional units (in addition to ineffective two- and unifunctional units), which can be equally effective in constraints on four and three chains so that the number of elastically effective chains is given by... 

Blanchard and Wootton (13) have introduced an additional correction due to, possibly intermolecular, steric hindrances, which lead to an increase in the apparent number of elastically effective chains. They derive this effect to be approximately proportional to (vf)2. It seems... 

Fig. 13. The effect of a crosslink at P on the mechanical behaviour of a two chain network between fixed points is mathematically equivalent to a fixed point on one of the chains, thus leading to three rather than four elastically effective chains [Duiser and Staverman, Chompff and Duiser 43, 32, 33)]...

They should consist of elastically effective chains only. An elastically effective chain should connect two different crosslinks, and two such crosslinks should be tied by only one elastic chain. This means that the gel should contain no defects such as pendant chains (one end of which only is connected with a crosslink), loops (chains linked at both ends to the same crosslink), or double connections. Physical crosslinks (permanent entanglements) should be prohibited, too. 

In most real systems, energy and entropy changes can occur. The elasticity of an ideal network is entropy controlled. In this picture stresses are caused by the chain orientation. From the theory of rubberlike elasticity it can be shown that the shear modulus of an ideal network depends on the number of elastically effective cahins between the crosslinks (19) Gq = v k-T where v means the number of elastically effective chains in unit volume. 

Dependence of the Shear Modulus on the Concentration. The experimental results of Figure 3a show that the plateau values increase with the detergent concentration. Unfortunately, we were not able to reach the rubber plateau for all concentrations for lack of the frequency range. From the theory of networks it is possible to calculate the number of elastically effective chains between the crosslinks from the shear modulus Gq of the rubber plateau (12). If the network... 

As observed from these equations, both theories introduce at least one more extra parameter, which needs to be determined from experimental data the molecular weight between cross-links and the fraction of elastically effective chains/. 

Mechanical oscillation measurements could only be carried out on nonionic PAAm-BisAAm gels. Because of their high degree of swelling, the saponified swelled gels could not be cut into suitable samples without their being destroyed. Via the theory of rubber elasticity developed by Flory (32), the value of the plateau modulus (Gp ) obtained by mechanical oscillation measurements is directly related to the number of elastically effective chains per unit volume (v J. 

Figure 11 shows the dependence of the plateau modulus on the content of cross-linking agent for PAAm-BisAAm gels with 5 and 10 wt % total initial weight of monomer. The number of elastically effective chains, which is proportional to G, decreases after passing through a maximum. 

For random crosslinking rf may be assumed to be equal to r, the corresponding mean square end-to-end distance for unconnected chains of the same molecular length. Because A is inversely proportional to (Eq. (1.2)), the only molecular parameter that remains in Eq. (1.8) is the number N of elastically effective chains per unit volume. Thus, the elastic behavior of a molecular network under moderate deformations is predicted to depend only on the number of molecular chains and not on their flexibility, provided that they are long enough to obey Gaussian statistics. 

Vgff = number density of elastically effective chains (mol m ). 

Sketch showing the difference between a monosulfidic (left portion) and a polysulfidic cross link (right portion). In the latter case, the chains of sulfur atoms may act as additional, elastically effective chains in what is essentially a bimodal network. 

The gel point can also be defined using an analogy with the polymerization of multifunctional monomers [39]. For this purpose we need the number of micelles (n ) and the number of bridges, or elastically effective chains (We), in the largest cluster in the system. If this cluster is extremely large, its functionality, 4>, is defined as... 

Flory clarified the activity of subchains by using the words elastically effective chain, or active chain [ 1 ]. An elastically effective chain is a chain that connects two neighboring cross-link junctions in the network. 

Scanlan-Case criterion for elastically effective chains. Chains with i >3 and i > 3 are effective. 

The number of elastically effective chains = v(l — 2/(p) in phantom network theory is smaller than its affine value v. In an affine network, all junctions are assumed to displace under the strict constraint of the strain, while in a phantom network they are assumed to move freely around the mean positions. In real networks of rubbers, the displacement of the junctions lies somewhere between these two extremes. To examine the microscopic chain deformation and displacement of the junctions, let us consider deformation of rubbers accompanied by the sweiiing processes in the solvent (Figure4.14) [1,5,14,25]. 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

Global structure of the networks - elastically effective chains and elastic modulus... 

Fig. 8.4 (a) Scanlan-Case criterion to find the elastically effective chains. A chain with both ends... 

Converting this to the number of chains, the number of elastically effective chains can be found by... 

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