Rubber elasticity, classical molecular theories

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

We refer the reader to the classic survey of rubber elasticity by Treloar (1975) and to three recent reviews that give fuller accounts of the molecular theory (Graessley, 2004 Mark and Erman, 1988,1992). The author thanks Mr. R.A. Paden for drawing several of the figures. 

The molecular theory of rubber elasticity on the basis of affine deformation assumption is the affine network theory, or the classical theory of rubber elasticity. 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

The tenet of classical rubber theory has been that the chains are in random networks and the networks comprise a Gaussian distribution of end-to-end chain lengths. However, the mechanisms and molecular bases for the elasticity of proteins are more complex than that of natural rubber. In biological systems elastomeric proteins consist of domains with blocks of repeated sequences that imply the formation of regular stmctures and domains where covalent or noncovalent cross-linking occurs. Although characterised elastomeric proteins differ considerably in their precise amino acid sequences they all contain elastomeric domains comprised of repeated sequences. It has also been suggested that several of these proteins contain p-tums as a structural motif (Tatham and Shewry 2000). 

OLo=h /h, where h (respectively, h ) the release altitude (respectively rebound height), characterizes the losses in the bulk of the viscoelastic material, without intervention of molecular attraction forces. Due to the shortness of the collision time 19) the penetration of the ball into the rubber surface occurs according to the classical theory of Hertz, even for an adhesive surface, so that maximum values of radius Omax of the contact area and the elastic penetration depth Smax are closely related to the release altitude h by and Smax-A through known prefactors depending upon mass M... 

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. 

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