Elastically active chains number

Cp is tire number of elasticity active chains per volume unit. The comparison between experimental data and tire prediction by (C2.1.20) shows a reasonable agreement up to large defonnation (figure C2.1.16). For large values of X, strain hardening arises because of tire limited extensibility of tire chains or because of shear-induced crystallization. 

In the random crosslinking of existing chains, the primary chains are placed in the root and nodes and each repeat unit can bear part of a crosslink (Fig. 7). In this case ve is the number of elastically active chains per primary chain. 

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

The number of elastically active chains, N, determining the equilibrium rubber elasticity, is derived from the following consideration. A chain in the gel is elastically active, if the branch points at each of its ends issue at least three paths to infinity. Such elastically active network chain (EANC) can have many long side branches but none of them may have an infinite continuation. The number of EANC s, N, is thus calculated from the number of EANC ends, i.e., branch points issuing three or more bonds with infinite continuation. The distribution of units according to the number of bonds with infinite continuation is described by a pgf T(z)... 

If the network is ideal, all hnear chains of the polystyrene precursor have been converted into a network of elastically active chains and, therefore, their number u is... 

Contrary to the extent of sweUing, the deformation of swollen model networks under uniaxial compression (or elongation) strongly depends on the quantity of danghng chains (Fig. 1.11). The elastic modules Ep of swollen samples decrease Hnearly with increasing portion of danghng chains up to about p = 0.45 and then decrease even more rapidly [129]. Indeed, the elastic modulus is proportional to the number of elastically active chains... 

X, y, z basic principal directions of strain N number of elastically active chains in unit of volume... 

The most important molecular parameter characteristic of a polymer network is the concentration of the elastic chains or that of the crosslinks connecting the macromolecules. An active junction is joined by at least three paths to the polymer network and an active chain is defined as one terminated by active junctions at both ends. There are several ways to express the extent of crosslinking (1) the concentration of the elastically active chains, r ei/Po, where v is the number of chains connecting two elastically active junctions and To is the volume of the dry network, (2) the molecular weight of the polymer chains between the junctions... 

Model silicone networks, i.e., those prepared by end-linking of functionally terminated polymer chains, have been extensively utilized to explain the influence of molecular structure on mechanical properties. An important number of studies have been focused on the contribution of elastically active chains and trapped entanglements to equilibrium properties [1-7]. In contrast, very little work has been done to explain the influence of network structure on non-equilibrium properties [8], and the contribution of some of the main structural parameters to viscoelastic properties has been poorly explored. A few qualitative studies have shown in the past that pendant chains have a strong influence on relaxation properties, but the type of contribution was not clearly understood [9]. 

At the end of the cross-hnking process, the topology of the mesh is composed of the different entities represented in Figure 6 (16,57-59). An elastically active junction is one joined by at least three paths to the gel network (60,61). An active chain is one terminated by an active jimction at both its ends. Rubber-like elasticity is due to elastically active chains and jimctions. Specifically, upon deformation the number of configurations available to a chain decreases and the resulting decrease in entropy gives rise to the retractive force. 

The value of equation (9.92) here lies in its complementary determination of the quantity n [see equation (9.4) for simplicity]. Both equations (9.4) and (9.92) determine the number of elastically active chains per unit volume (containing, implicitly, corrections for front factor changes). By measuring the equilibrium swelling behavior of an elastomer (x values are known for many polymer-solvent pairs), its modulus may be predicted. Vice versa, by measuring its modulus, the swelling behavior in any solvent may be predicted. 

The effective crosslink density, px, is defined as the concentration of elastically active chains (chains which are deformed by an applied stress) in the polymer network, and is usually reported on the basis of moles of chains per cubic centimeter of dry polymer. Network structure can also be described with a number of closely related terms [28]. For example, when linear polymers are crosslinked, it Is often desirable to express crosslinking in terms of the number average molecular weight of the polymer before crosslinking, Mp, and the number average molecular weight between crosslinks,... 

The case of Gaussian chains suggests O to be a Gaussian function as a trial, since the kernel p is itself Gaussian. After some straight forward algebra this procedure gives an explicit relation for the modulus of the network, Le. the number of elastically active chains ... 

In POLYM the output data of KINREL are used with compositional information to calculate the number and mass average molecular masses (Rn and Rm, respectively) and number and end-group average functionalities (fp and fg> respectively) in the pre-gel region in all stages. In addition, the network characteristics such as sol fraction, mj, and the number of elastically active network chains per monomer (5), Ng, are calculated in the post-gel regime of stage 3. 

The equality of the values for f=3 implies that random gel-gel reaction in fact leads to an average of two elastically active junctions or three elastic chains lost per pair of gel-gel groups reacted. For f=4, the experimentally derived value 0.03 was on the basis of one elastically active junction lost per pair of gel-gel groups reacted. Hence, the ratio of the experimentally derived and the calculated values, 0.03/0.13 = 0.23, is the average number of elastic-... 

In the following, we will briefly outline the use of the link p.g.f. (l.p.g.f.) for the calculation of the gel point in /-functional polycondensation without and with cyclization including the f.s.s.e. In Chapter II, section 2.2 we will consider an application in connection with the number of elastically active network chains in random polycondensates or in a collection of randomly crosslinked chains. 

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... 

Free chain ends (unreacted functionalities) reduce the number of active network chains in a network compared with the same network without free ends. Disregarding possible presence of loops and entanglements, C — 1 C crosslinks are necessary according to Flory (55) to connect C chains into one giant macromolecule. Additional crosslinks will be elastically effective. Their number is given by... 

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