Elastically active network junction

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

Prediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions (EANJs) and chains (EANCs), respectively and [260, 261]. EANJs are the intersection of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13). 

Since at long times pendant chains do not contribute to permanent elastic properties, the elastic equilibrium behavior of networks containing these chains should not differ substantially from that of regular networks. The elastic modulus from a network with pendant chains can then be obtained from the molecular theories of rubber elasticity provided that the concentration of elastically active network chains (v) can be calculated accurately. Depending on the different approaches that can be used for the rubber elasticity theory, the calculation of some other parameters, like the concentration of junctions points (p) and trapped entanglements (Te), also may be needed. 

The ideal network structure can be envisaged as a three-dimensional array of crosslink points, each crosslink point being connected to at least three other crosslink points via linear polymer segments, which are called elastically active network chains. In practice non-ideal network elements are also present, such as loops or dangling ends (Figure 16.1). Network density, or crosslink density, is expressed as the concentration of either the crosslink joints or the elastically active network chains (those chains that are part of the infinite structure and attached to crosslink junctions at both ends) per unity of volume of the unswollen material. 

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. 

The smallest loops, as they occur in the networks at complete reaction, are Illustrated in Figure 3, together with the elastically active function points lost. For f=3, each smallest loop leads to the loss of two junction points and for f=4 only one junction point per smallest loop is lost. Notwithstanding that more complex ring structures will occur, the greater loss of junction points per smallest loop, and indeed per next smallest loop(16) for f=3 compared with f=4 networks is the basic reason why the former networks (curves 1 and 2 in Figure 1) show larger reductions in modulus per pre-gel loop than the latter networks (curves 3 to 6). 

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... 

Scanlan has suggested another criterion (282). An effective network junction point is a crosslink in which at least three of the four strands radiating from it lead independently to the network. A crosslink with only two strands anchored to the network simply continues an active strand a crosslink with only one anchored strand is part of a dangling end and can make no elastic contribution at equilibrium. An elastically effective strand is therefore one which joins two effective network junction points. Accordingly, the total number of active strands is simply one half the number of gel-anchored strands radiating from effective junction points ... 

It follows from the above discussion that a junction is elastically active if at least three paths leading away from it are independently attached to the network., A polymer chain segment, also called a strand, is elastically active if it is bound at each end by elastically active junctions. Loops in the network structure [Fig. 2.32(b)] do not contribute to the elasticity of the network as they have the two ends of a chain segment connected to the network at the same point. 

Ideal or Perfect Networks The lUPAC Commission on macromolecular nomenclature defines a perfect network as a network composed of chains all of which are connected at both of their ends to different junction points [7]. If a perfect network is in the rubbery state, then, on macroscopic deformation of the network, all of its chains are elastically active and display rubber elasticity. An ideal or perfect network can also be defined as a collection of individual Gaussian elastic chains (linear... 

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... 

Another important parameter is the crosslink functionality, /, which is the number of chains emanating from a network junction. Only junctions with functionality higher than 2 are elastically active. For perfect networks, i.e., cross-... 

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. 

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