Elastically active chains

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. 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

According to the equations derived by Miller and Macosko (24), outlined in the Appendix, the probability that a short segment selected at random in the network is part of an elastically active chain is (P /p)2 where p is the final extent of the curing reaction. If an entanglement results from pairwise interactions between chains, as proposed (13), then Te in eq 2 can be equated to (Pxi/p)4, which is the probability that two interacting chains are active. 

The statistical theory of crosslinking used in the last section also gives the theoretical concentration of elastically-active chains, N, which in turn determines the rubbery modulus E = 3NRT (R is the gas constant and T is the absolute temperature). At 70% reaction one calculates E - 2 x 10 dyn/cm1 2 3 4 5 6 7 8 9 10, in agreement with the apparent level in Figure 1. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

To a first approximation, which neglects changes in average chain structure, the loss in elastically active junction point concentration may be translated directly into loss in concentration of elastically active chains and increase in the value of M, . For a perfect network in the dry state, the concentration of elastically active chains is given by the equations... 

The structure and properties of a network polymer are determined by the relation between the inter- and intramolecular reactions of functional groups. The latter gives rise to ineffective cycles, and has been studied fairly well3-5 7°.71-8i-85.i°8 u2 Their occurrence in the system results in the gelation point being shifted towards higher conversions, higher sol fraction, fewer elastically active chains, and smaller equilibrium modulus of the network. 

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

The molecular theory of elasticity of polymeric networks which leads to the equation of state, Eq. (28), rests on the following basic postulates Undeformed polymeric chains of elastic networks adopt random configurations or spatial arrangements in the bulk amorphous state. The stress resulting from the deformation of such networks originates within the elastically active chains and not from interactions between them. It means that the stress exhibited by a strained network is assumed to be entirely intramolecular in origin and intermolecular interactions play no role in deformations (at constant volume and composition). 

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. 

Styrene-Divinylbenzene Networks. Using ionic polymerization methods, Rietsch et al. (1976) prepared polystyrene (PS) networks with a well-controlled length of elastically active chains and crosslinks of variable functionality. In a given series, the glass transition temperature obeys the classical free volume theory ... 

Polyurethane Networks. Andrady and Sefcik (1983) have applied the same relationship as Rietsch et al. (1976), to the glass transition temperature of networks based on poly(propylene oxide) diols with a controlled molar mass distribution, crosslinked by aromatic triisocyanates. They obtained a Kr value of 25 K kg mol-1, about twice that for PS networks. They showed that the length distribution of elastically active chain lengths, directly related to the molar mass distribution of the starting poly(propylene oxide), has practically no effect on Tg. 

How do we take into account its eventual internal rotations How do we distinguish between tetrafunctional (four elastically active chains) and trifunctional (three elastically active chains and one dangling chain) crosslinks ... 

Chemical aging resulting from water absorption (i.e., hydrolysis) has not been as widely studied as physical aging. It is relatively well understood at the molecular scale (chemical mechanisms). But macromolecular (kinetics of decrease of the elastically active chain concentration) and mechanical aspects (effect of chain scissions on mechanical properties) are far from being elucidated. 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

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)... 

Figure 2.32 (a) Effect of ends of molecules on network structure, (b) Network structure and defects cross-links ° terminus of a molecule (1) elastically active chain (2) inactive loop (3) inactive loose end. An arrow (—>) signifies continuation of the network strucmre. (After Flory, 1944.)... 

Figure 9.2 Main elements constituting the structure of a polymer network (1) crosslink point, (2) elastically active chain, (3) dangling chain, (4) loop or cycle, (5) multiple connection between two crosslink points, and (6) permanent chain entanglements between two adjacent crosshnks.

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