The number of elastically effective chains

The smaller solution xi of this equation can now be found as a function of Finally we can readily check that 0 l) = I holds, and its derivative is given by 

On substitution into the sol-gel transition criterion (8.25) and by the use of the relation (10.147), we find an equation 

Before studying specitic models of the junctions, we derive asymptotic forms of the network parameters in the extreme limit of complete reaction a 1. In this limit, the smaller root of (8.44) goes to zero (xi 0), and hence 

The effective chains and junctions therefore show a limiting behavior 

In contrast, for telechelic polymers, we are led to the asymptotic behavior Vend/v 0 from (8.40), again as expected. 

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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 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... 

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... 

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. 

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]. 

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

The simplest case is the pairwise association A = 2 for arbitrary functionality /. The number of elastically effective chains was calculated by Clark and Ross-Murphy [15]. Their result can be reproduced by choosing the function u x) as i<(x) = 1+x. Straightforward calculation leads to the result... 

Figure 8.9 shows the number of elastically effective chains for f = 2 with k varied from curve to curve as a function of (a) the extent of reaction, and (b) the reduced concentration [14]. The critical behavior obeys the mean-field scaling law... 

Fig. 9.3 Viscoelastic master curve of HEUR with Mw = 35K and 16 carbons in the end chain [2], The reference temperature is 5°C. The activation energy and the number of elastically effective chains can be found from the shift factors. (Reprinted with permission from Ref. [2].)...

A typical example of the master curve [2] is shown in Figure 9.3 for HEUR (polyethylene oxide) end-capped with -C16H33. The reference temperature is chosen at 5 C. From the horizontal shift factor, the activation energy is found to be 67kJmol . From the high-frequency plateau of the storage modulus, the number of elastically effective chains is found as a function of the polymer concentration, which was already studied in Section 8.2 (Figure 8.10). 

To see how these new rheological features appear, we calculate the number figff of elastically effective junctions in a unit volume from (8.31), and the number of elastically effective chains Veff from (8.32). 

On the other hand, at high frequencies o) > po, the storage modulus (7.5a) gives the equilibrium number of polymer chains in the network. This number, however, must be regarded as the number of elastically effective chains that connect two junctions in the network since only these chains can support stress. Another important feature of the network is the number and structure of elastically inactive chains that are dangling from the network. We call them dangling ends. We next study how such topological structures of the network depend on the temperature and concentration. 

Fig. 6 The number of elastically effective chains (relative to the total number of chains) as a function of the extent of association (a), and of the reduced concentration (b). The multiplicity is changed from curve to curve, while the functionality is fixed at/= 2. Each curve rises in cubic power of the concentration deviation from the transition point and approaches unity at high limit of the concentration...

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