Rubber elasticity theory extending

The dominant contribution to the free energy of lengthy (rubbery) polymer chains is entropy. This is known to accoimt for rubber elasticity, which can be satisfactorily modelled by the entropy of the cross-linked pol3rmer chains alone. A simple illustrative model of copolymer self-assembly can be developed by extending rubber elasticity theory to include bending as well as stretching deformations, to calculate chain entropy as a function of interfacial curvatures in diblock aggregates. 

The firs t systematic study of the reversible networks was the transient network theory developed by Green and Tobolsky [ 11 ], in which stress relaxation in rubber-Uke polymer networks was treated by the kinetic theory of rubber elasticity suitably extended so as to allow the creation and annihilation of junctions during the network deformation. 

The terminal spectrum is furnished by cooperative motions which extend beyond slow points on chain in the equivalent system. The modulus associated with the terminal relaxations is vEkT, which is smaller by a factor of two than the value from a shifted Rouse spectrum. It is consistent with a front factor g = j given by some recent theories of rubber elasticity (Part 7). The terminal spectrum for E 1 has the Rouse spacings for all practical purposes, shifted along the time axis by an undetermined multiplying factor (essentially the slow point friction coefficient). Thus, the model does not predict the terminal spectrum narrowing which is observed experimentally. 

Kaliske, M., Heinrich, G., 1999. An extended tube-model for rubber elasticity Statistical-mechanical theory and finite element implementation. Rubber Chem. Technol. 72 (4), 602-632. Khokhlov, A.R., 1992. In Dusek, K. (Ed.), Responsive Gels Volume Transitions I. Springer, Verlag Berlin, p. 125. 

The principles of the modem physical theories of the deformation of rubber xerogels have already been dealt with in Chapter IV, p. 123. If a strip of raw rubber is rapidly extended, the molecules, which initially assumed randomly kinked forms, are stretched too. The more extended shape of the chains is a statistically less probable one corresponding to a lower entropy. en the piece of rubber is rapidly released again, it reassumes its original form in that the chains return to their, most probable configurations The entropy-character of rubber elasticity has been proven in that it exhibits a positive temperature coefficient. 

Recent experimental evidence using SANS instrumentation suggests that the ends of a network segment deform affinely, yet the chain itself barely extends in the direction of the stress and contracts in the transverse direction even less. Develop a model to explain the results, and comment on how you think the theory of rubber elasticity ought to be modified to accommodate the new finding. 

We shall now develop a statistical theory of rubber elasticity using the Gaussian formula to represent the most probable distribution of molecular end-to-end distance. The mass of rubber is assumed to consist of N chains each end of which is joined to the ends of other chains. Suppose that the mbber is extended by a factor a. 

Inspection of Figure 5 shows a very broad glass-to-rubber transition range which extends from below -100°C to above 0°C for the polyurethane adhesive. The relaxation modulus E(t) - 400 Kg/cm which occurs at the rubbery inflection temperature - 40°C - 313 K describes an effective molecular weight M as defined by kinetic theory of rubber elasticity ... 

Kaliske, M. and Heinrich, G. (1999) An extended tube-model for rubber elasticity statistical-mechanical theory and finite element implementation. Rubber Chem. 

In the 1930 s several scientists presented evidence showing that rubber elasticity is essentially an entropy phenomenon, related to the change in randomness of location of the rubber segments when the material is extended. On this basis they derived relationships between the initial elastic modulus, the average chain length between network junctions, etc. The basic idea-appealed to me. It was a natural extension of my 1922 theory of conformations in simple molecules. 

From his early youth, under his father s influence, K. H. Meyer had retained a keen interest in biological problems, as was evident from his study of the phenomena of narcosis, which he pursued during his stay in industry. As a natural consequence, he extended his thoughts to biological problems, and evolved a quantitative theory of muscular contraction (in collaboration with Picken), based on analogies with the elasticity of rubber. With J. F. Sievers, the permeability of synthetic membranes was investigated, and a mathematical treatment of the phenomenon was advanced which was later applied to living membranes. 

The assumption that the contraction process is ideally adiabatic, while perhaps not entirely permissible practically, seems indicated by modern theory of the behavior of molecular chains, which pictures these as undergoing, when freed of restraints, a sort of segmental diffusion, much like the adiabatic expansion of an ideal gas into a vacuum (155). In the case of the molecular chain, it diffuses to the most probable, randomly coiled configuration, which is much less asymmetric, hence shorter, than an initially extended chain. Because rubber most nearly presents this ideal behavior, those fibers which develop increased tension (a measure of the tendency toward assumption of the contracted form) when held isometrically under conditions of increasing temperature (favoring the diffusion ) are said to be rubber-like. Most normal elastic solids upon stress are strained from some stable structure and relax as the temperature is raised. 

Further deviations from eqn (4.13) are to be expected where segments are highly extended and Gaussian statistics no longer apply, because of neglect of the experimentally observed C2 term and by all the other imperfections of the Gaussian theory of high elasticity. In spite of this eqn (4.13) (or rather its simplifications (4.18) and (4.19)) has been of importance in the elucidation of the structure of vulcanized rubbers (see Chapter 8). 

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