Rubber-elastic effect

A quantitative treatment of the rubber-elastic effect has recently been proposed by Choy ei al [48], which has led to an understanding of the drastic difference in the expansivities of LDPE and HDPE. According to them, an oriented polymer may be considered as a composite made up of four phases, i.e. the crystallites, the amorphous region, bridges and tie molecules. The internal strain due to rubber-like tie molecules 

Above the amorphous transition, the modulus of the tie molecules increases with temperature ( is proportional to T) according to entropic theory of rubber elasticity. n is nearly constant so 

The tie molecules therefore make a negative contribution to a through the last term in equation (8.21). 

Though the necessary input parameters of equation (8.21) are not known accurately enough to allow detailed comparison with experimental data, equation (8.21) can be used to determine factors controlling the magnitude of the rubber-elastic contraction [32,48]. For example Choy et al. [48] consider the case of an LDPE sample with A = 4.2 for which = — 40 X 10 K at 320 K. The rubber-elastic effect is presumably fully appropriate at this temperature and equation (8.21) may be expected to apply. The first two terms of equation (8.21) are of the order of 10 and may be neglected. Using equation (8.21) for the last term and y = 0.4 yields 

The relation for the modulus of the composite in the above series-parallel model is given by  

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The effective molecular mass Mc of the network strands was determined experimentally from the moduli of the polymers at temperatures above the glass transition (Sect. 3) [11]. lVlc was derived from the theory of rubber elasticity. Mc and the calculated molecular mass MR (Eq. 2.1) of the polymers A to D are compared in Table 3.1. 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

Anseth et al. [20] have reviewed the literature dealing with the mechanical properties of hydrogels and have considered in detail the effects of gel molecular structure, e.g., cross-linking, on bulk mechanical properties using theories of rubber elasticity and viscoelasticity. 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

In our first paper, the molecular theory of rubber elasticity was briefly reviewed, especially the basic assumptions and topics still subject to discussion (21). we will now focus on the effects of the structure and the functionality f of the crosslinks and the relevant theory. 

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... 

The kinetic theory of rubber elasticity is so well known and exhaustively discussed (17, 27, 256-257, 267) that the remarks here will be confined to questions which relate only to its application in determining the concentration of elastically effective strands. In principle, both network swelling properties and elasticity measurements can provide information on network characteristics. However, swelling measurements require the evaluation of an additional parameter, the polymer-solvent interaction coefficient. They also involve examining the network in two states, one of which differs from its as-formed state. This raises some theoretical difficulties which will be discussed later. Questions on local non-uniformity in swelling (17) also complicate the interpretation. The results described here will therefore concern elasticity measurements alone. 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

Crosslink density directly affects E0 (through rubber elasticity), and has an indirect influence on E (through the antiplasticization effect). Cole-Cole plots open the way to analyzing the distribution of relaxation times (the exponents a and y or % or % are linked to the width of the distribution of relaxation times). According to the results of Table 11.3, these exponents seem to depend more on the molecular-scale structure (they vary almost... 

J.-P. Jarry and L. Monnerie, Effects of a nematic-like interaction in rubber elasticity theory, Macromolecules, 12, 316 (1979). 

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