Rubber elasticity refinements

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. 

Necklace models represent the chain as a connected sequence ctf segments, preserving in some sense the correlation between the spatial relationships among segments and their positions along the chain contour. Simplified versions laid the basis for the kinetic theory of rubber elasticity and were used to evaluate configurational entropy in concentrated polymer solutions. A refined version, the rotational isomeric model, is used to calculate the equilibrium configurational... 

The statistical theory of rubber elasticity has undergone significant and continuous refinement, resulting in a series of correction terms. These are sometimes omitted and sometimes included in scientific and engineering research, as the need for them arises. In this section we briefly consider some of these. 

The swelling of vulcanized crosslinked rubber by solvents has long been observed. This behavior was first modeled in the 1940s by Flory and Refiner [47]. They combine the Meyer-Flory-Huggins statistical thermodynamic theory of polymer solutions (Section 3.3) with the molecular theory of crosslinked rubber elasticity [48]. The molecular weight between crosslinks of the vulcanizates was predicted from the swelling to be... 

The principles were refined by Meyer in a second paper (70). In it he proposed that the micelles occurred at regular intervals. He also included an explanation of the elasticity of rubber based on the assumption that the molecular chains tended to roll together in knots in unstretched rubber, but line up when stretched. This explanation was especially elucidating since it agreed well with Katz s discovery (53) that amorphous rubber crystallizes when stretched. 

The average length (or molecular weight) of network chains in a crosslinked polymer can be experimentally determined from the equilibrium rubbery modulus. This relationship is a direct result of the statistical theory of rubber-like elasticity . In the last decade or so, modem theories of rubber-like elasticity 2127) further refined this relationship but have not altered its basic foundation. In essence, it is... 

Thus, the level of sophistication which one may consider for the application of rubber-like elasticity theory to epoxy networks may depend on the application. For highly crosslinked systems (M < 1,000), a quantitative dependence of the rubbery modulus on network chain length has recently been demonstrated , but the relevance of higher order refinements in elasticity theory is questionable. Less densely crosslinked epoxies, however, are potentially suitable for testing modern elasticity theories because they form via near quantitative stepwise reactions. Detailed investigations of such networks have been reported by Dusek and coworkers in recent studies ... 

Realizing the shortcomings of this simple model, extensive work has subsequently been performed to refine the picture of the nonlinear elastic behavior of crosslinked rubbers. Further information is available in textbooks [49] and recent reviews [50]. 

Natural rubber (NR) is elastomer (an elastic hydrocarbon polymer) that was originally derived from latex, a milky colloid produced by some plants (Hevea brasiliensis, a member of the spurge family, Euphorbiaceae). The plants are tapped by making an incision in the bark of the tree and collecting the sticky, milk-coloured latex sap, which is refined into usable rubber. The purified form of NR is the chemical polyisoprene, which can also be produced synthetically. NR is used in many applications and products, as is synthetic rubber. It is normally very stretchy and flexible and extremely waterproof. 

The theory in the Gaussian limit has been refined greatly to take into account the possible fluctuations of the junction points. In these approaches, the probability of an internal state of the system is the product of the probabilities Win) for each chain. The entropy is deduced by the Boltzmann equation, and the free energy by equation (26). The three main assumptions introduced in the treatment of elasticity of rubber-like materials are that the intermolecular interactions between chains are independent of the configurations of these chains and thus of the extent of deformation (125,126) the chains are Ganssian, freely jointed, and volumeless and the total number of configurations of an isotropic network is the product of the number of configurations of the individual chains. 

Polymer melts and semidilute and concentrated solutions of polymer are highly viscous. Even at a concentration of 1 wt %, solutions of polymer with a molecular weight greater than several million g/mol can flow only slowly. Their behaviors are even elastic like rubber at accessible time and frequency ranges. These exquisite properties had eluded researchers for decades until the tube model and the reptation theory elegantly solved the mystery. The tube model and the reptation theory were introduced by de Gennes." They were refined and applied to the viscoelasticity of semidilute solutions of polymers and polymer melts in the late 1970s by Doi and Edwards." Until then, there had been no molecular theory to explain these phenomena. We will leam the tube model and the reptation theory here. 

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