Elasticity, rubber-like molecular theory

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

Priss LS (1981) Molecular origin of constants in the theory of rubber-like elasticity considering network chains steric interactions. J Pure Appl Chem 53 1581-1596 Priss LS, Gamlitski YuA (1983) Mechanism of conformation transitions in polymer chains. Polym Sci USSR 25 117-123... 

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... 

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. 

M. Shen, W. F. Hall, and R. E. De Wames, Molecular theories of rubber-like elasticity and... 

Despite of the approximations, the statistical theory is of fundamental significance for understanding of the molecular mechanisms causing rubber-like elasticity. It serves as a starting point for generalizations that agree more precisely with experiments. One generalization is the Mooney-Rivlin equation. After Equation (35), we have ... 

There are several molecular theories of rubber-like elasticity The simplest one is based on a Gaussian distribution function for the end to end separation of the network chains [23] (the dimensions of the free chains as unperturbed by excluded volume effect are represented by (r )o)... 

Since at long times pendant chains do not contribute to permanent elastic properties, the elastic equilibrium behavior of networks containing these chains should not differ substantially from that of regular networks. The elastic modulus from a network with pendant chains can then be obtained from the molecular theories of rubber elasticity provided that the concentration of elastically active network chains (v) can be calculated accurately. Depending on the different approaches that can be used for the rubber elasticity theory, the calculation of some other parameters, like the concentration of junctions points (p) and trapped entanglements (Te), also may be needed. 

Applying the kinetic theory of rubber-like elasticity (62,109) to the entanglement network one can determine the molecular weight between entanglements Me from the plateau compliance Jn or plateau modulus Gn (62) ... 

The above features of rubbery materials have long been known. The quantitative measurements of mechanical and thermodynamic properties of natural and other elastomers go back to 1805 and some of the studies were conducted by luminaries like Joule and Maxwell. The first molecular theory in polymer science dealt with the rubber elasticity (9-12). 

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. 

There are several important postulates that have been used in the development of the molecular theories of rubber-like elasticity [9]. 

The simplest molecular theories of rubber-like elasticity are based on the Gaussian distribution function... 

Experimental results on networks of natural rubber in shear deformation [134] are not well accounted for by the simple molecular theory of rubber-like elasticity. The constrained-junction theory, however, was found to give excellent agreement with experiment. Shear measurements have also been reported for some unimodal and... 

The above-mentioned general features of elastomeric materials have long been known and, in fact, the area of rubber-like elasticity has had one of the longest and most distinguished histories in all of polymer science (1,2,16). Forex-ample, quantitative measurements of the mechanical and thermodynamic properties of natural rubber and other elastomers go back to 1805, and some of the earliest studies have been carried out by such luminaries as Joule and Maxwell. Also, the earliest molecular theories for polymer properties of any kind were, in fact, addressed to the phenomenon of rubber-like elasticity. 

These elastomers also exhibited stress-strain isotherms in elongation that were closer in form to those expected from the simplest molecular theories of rubber-like elasticity. Specifically, there were large decreases in the Mooney-Rivlin 2C2 correction constant. 

In this section, the expressions for the elastic free energy for different theoretical models are reviewed. In later sections, the force deformation relations for these model are derived. Simplest molecular theories of rubber-like elasticity are based on the Gaussian network chains. These theories are referred to as the Gaussian networks. The probability W(r) that the distance between the two ends of a network chain is given by the Gaussian function (1,13)... 

A molecular theory of the relaxation properties of filled elastomers has been developed by Sato 138) on the basis of the statistical concept of rubber-like elasticity. He has derived expressions for the estimation of Young s modulus stresses and mechanical losses of filled polymers. 

The essential concept involved in the statistical theory of rubber elasticity is that a macroscopic deformation of the whole sample leads to a microscopic deformation of individual polymer chains. The microscopic model of an ideal rubber consists of a three-dimensional network with junction points of known functionality greater than 2. An ideal rubber consists of fully covalent junctions between polymer chains. At short times, high-molecular-weight polymer liquids behave like rubber, but the length of the chains needed to describe the observed elastic behavior is independent of molecular weight and is much shorter than the whole chain. The concept of intrinsic entanglements in uncrosslinked polymer liquids is now well established, but the nature of these restrictions to flow is still unresolved. The following discussion focuses on ideal covalent networks. 

In the earlier discussion of the plateau modulus, it was remarked that a liquid of long chains acts, at intermediate times or frequencies, like a network. The theory of rubber elasticity predicts a relationship between the shear modulus and the concentration of network strands (Chapter 1). This relationship is used to evaluate Me, the equivalent molecular weight of a strand in the entanglement network, which is called the entanglement molecular weight [1] ... 

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