Entropic high-elasticity theory

As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. 

Therefore, the complete methods of calculation of the characteristics of crosslinked networks was proposed, which combines the ruhher high-elasticity entropic theory, the cluster model of amorphous state structure of polymers and fractal analysis methods. The proposed method has shown that growth in statistical segment length is observed as the drawing ratio increases. This snpposes that the chain statistical flexibility depends not only on its chemical constitntion, but also on the network deformed state. The considered method can be nsed for computer simulation and prediction of the structure of crosslinked polymer networks [6]. 

The experimental data about rubbers deformation are usually interpreted within the frameworks of the high-elastieity entropic theory [1-3], elaborated on the basis of assumptions about high-elastic polymers incompressibility (Poisson s ratio V = 0.5) and polymer chains Gaussian statistics. As it is known [4], the Gaussian statistic is characteristic only for the networks, prepared by chains concentrated solution curing, in the case of their compression or weak (draw ratio A, < 1.2) tension. For such stmctures the fractal dimension d = 2 and in case of v = 0.5 the following classical expression was obtained [3] ... 

The experimental data on elastoplastics deformation are usually interpreted within the frameworks of entropic high-elasticity classical theory, which corresponds well to experiment only in a relatively small strains region (e < 0.2, where e is strain)... 

The discrepancy between high-elasticity classical theory and experimental curves o-e or o-X for elastoplastics indicated above is due to two factors firstly, by essentially non-Gaussian statistics of real polymer networks and, secondly, by the lack of coordination of two main postulates, lying in the basis of entropic high-elasticity classical theory - Gaussian statistics and elastoplastics incompressibility. The last condition is characterised by the criterion v = 0.5, where v is Poisson s ratio [46]. 

Balankin [46-48] obtained the following eqnation for the description of elastoplastics curves o-A, within the frameworks of entropic high-elasticity fractal theory ... 

In turn, the entropic high-elasticity classical theory eqnation, nsed for the same purposes, has the form [46] ... 

Hence, the results obtained above have shown that behaviour at deformation for the considered polyurethanes and nanocomposites on its basis is described within the frameworks of entropic high-elasticity fractal theory or, equivalently, within the frameworks of the classical theory approximation for long polymer chains. The considered polymer networks obey Ganssian statistics due to their preparation method. The inaccuracy of the application of the entropic high-elasticity classical theory (Equation 7.13) is defined by non-fnlfilment in the given case of a postulate about elastoplastics incompressibility [49]. 

Having sorted out the covalent bonds between the neighbors, we can now concentrate on all the other interactions. These are frequently referred to as volume interactions . As we have said, they have a typical energy E-2, and are much weaker than those in charge of the linear memory. In the crudest theory, we may completely neglect them. Then we shall end up with exactly what is called an ideal polymer chain. This is just how we handled all the calculations in the previous chapters. It worked fairly well, and we coped with quite a number of problems. We described how a chain rolls up into a loose coil, and we revealed the peculiar entropic nature of the high elasticity of polymers. 

Returning now to the left part of the curve, for a 350% extension it has been shown that, at about —70° C, it suddenly turns in the same direction as for hard materials like metals. Referring to p. 661 it may be remembered that —70° C is the temperature T, at which rubber freezes, loosing its high-elastic properties. In accordance with the above developed theories, the conditions for ideal high-elasticity are now no longer present, because the molecules are not sufficiently movable. At about —70° C the kinetic (or entropic) elasticity is therefore transformed into potential elasticity. 

Hence, the stated above results show that the classical theory of entropic high-elasticity can be used for the description of stress-draw ratio curves for rubbers with weak strain hardening, but it is incorrect in case of nanocomposites with elastomeric matrix. The correct description of deformation behavior of the latter gives the high-elasticity fractal model that is due to fractal nature of filled rubbers structure [13]. 

The application of fractal analysis for the description of the behaviour of rubbers is difficult because of the fact that these materials are (or are close to) Euclidean objects. Nevertheless, at present the theory of elasticity and entropic high-elasticity of fractals is developed, which differs principally from the classical theory. The change of molecular mobility, characterised by fractal dimension of a chain part between crosslinking nodes, is of interest for rubbers. Lastly, local order models can be used successfully for quantitative description of the nucleation process of crystalline regions and the melting temperature of rubbers. These and some other questions will be considered in detail in the present chapter. 

In this chapter, AFM palpation was introduced to verify the entropic elasticity of a single polymer chain and affine deformation hypothesis, both of which are the fundamental subject of mbber physics. The method was also applied to CB-reinforced NR which is one of the most important product from the industrial viewpoint. The current status of arts for the method is still unsophisticated. It would be rather said that we are now in the same stage as the ancients who acquired fire. However, we believe that here is the clue for the conversion of rubber science from theory-guided science into experiment-guided science. AFM is not merely high-resolution microscopy, but a doctor in the twenty-first century who can palpate materials at nanometer scale. 

At temperatures well below Tg, when entropic motions are frozen and only elastic bond deformations are possible, polymers exhibit a relatively high modulus, called the glassy modulus (Eg) which is on the order of 3 Gpa. As the temperature is increased through Tg the stiffness drops dramatically, by perhaps two orders of magnitude, to a value called rubbery modulus Er. In elastomers that have been permanently crosslinked by sulphur vulcanization or other means, the values of Er, is determined primarily by the crosslink density the kinetics theory of rubber elasticity gives the relation as... 

And at last, the third and the most fundamental factor is the ehange of nanocomposite structure at the introduction of particulate filler in high-elastieity polymeric matrix. As Balankin showed [9], classical theory of entropic high-elastieity has a number of principal deficiencies due to non-fulfilment for real rubbers of two main postulates of this theory, namely, essentially non-Gaussian statisties of real polymeric networks and lack of coordination of postulates about Gaussian statistics and incompressibility of elastic materials. Last postulate means, that Poisson s ratio v of these materials must be equal to 0.5. As it is known [10], Gaussian statistics of macromolecular coil is correct only in case of its dimension Dj=2.0, i.e., for coil in 0-solvent. Since between value Df and fractal dimension... 

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