The thermodynamics of rubber elasticity

As early as 1935, Meyer and Ferri showed experimentally that the streteh-ing force F required to maintain a rubber at constant strain is very nearly proportional to the absolute temperature T, i.e. 

The thermodynamic argument that follows shows that, if equation (6.31) holds, the internal energy of the rubber is independent of its length at constant temperature and, to this approximation, the force F thus depends only on the rate of change of entropy, S, with length. 

Applying this equality with / = A,x = I and y = T to equations (6.35) leads to 

However, equation (6.31) shows that F — T(dF/8T)i = 0, which proves that (9f7/9/)r = 0, i.e. that the internal energy is independent of length 

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These conclusions have been confirmed by Wood and Roth, who carried out measurements at both constant lengths and at constant elongations using natural rubber vulcanized with sulfur and an accelerator. Their results at constant elongation, to be considered later in connection with the thermodynamics of rubber elasticity at higher elongations, are summarized in Fig. 89. 

In this chapter, we first discuss the thermodynamics of rubber elasticity. The classical thermodynamic approach, as is well known, is only concerned with the macroscopic behavior of the material under investigation and has nothing to do with its molecular structure. The latter belongs to the realm of statistical mechanics, which is the subject of the second section, and has as its... 

In order to determine the nature of the force generated by a polymer gel, one must consider all the relevant interactions that contribute to the force or displacement. We learned from the thermodynamics of rubber elasticity that the molecular mechanism of force generation in the network chains is made up of two different contributions. In general, energetic and entropic effects must be taken into account ... 

Thermodynamics, both classical [Appendix 3.A] and statistical [Appendix 2A], have been applied to many topics in polymer science. The results have provided insights into the origin of rubber elasticity, the nature of polymer crystalline, polymeric heat capacities and the miscibility of polyblends. 

The theory of rubber elasticity is largely based on thermodynamic considerations. It will be briefly discussed as an example of how thermodynamics can be applied in polymer science. Eor more detailed information the reader is referred to the various textbooks [10-13]. It is assumed that there is a three-dimensional network of chains, that the chain units are flexible and that individual chain segments rotate freely, that no volume change occurs upon deformation, and that the process is reversible (i.e., true elastic behavior). Another usual assumption is that the internal energy U of the system does not change with deformation. Eor this system the first law of thermodynamics can be written as ... 

The theory of rubber elasticity and the theory of ideal gas dynamics show that the two equations, G = tiRT and PV = n RT, share certain common thermodynamic ideas. What are they ... 

It is suggested that the reader perform additional reading on the subjects of the theory of rubber elasticity with especial emphasis on the thermodynamic approach to the theory. 

All the powers of the French Academy of Sciences were needed chemical analysis, mechanical analysis, thermodynamics, alchemy, synthesis. But even Berthelot could not really provide much insight. The Laws of Rubber Elasticity were established by Joule, but even the wizards of British physics could not conjure up an explanation. Until the dawning of the molecular age, no real progress was likely on this subject. 

Leonov model This constitutive equation is based on irreversible thermodynamic arguments resulting from the theory of rubber elasticity [5]. Mathematically it relates the stress to the elastic strain stored in the polymer melt as ... 

These deductions from basic facts of observation interpreted according to the rigorous laws of thermodynamics do not alone offer an insight into the structural mechanism of rubber elasticity. Supplemented by cautious exercise of intuition in regard to the molecular nature of rubberlike materials, however, they provide a sound basis from which to proceed toward the elucidation of the elasticity mechanism. The gap between the cold logic of thermodynamics applied to the thermoelastic behavior of rubber and the implications of its... 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

In Geneva, he resumed with new energy his studies of macromolecules. He was able to obtain the cooperation of A. J. H. van der Wijk, who was one of his most devoted coworkers the latter s realistic criticisms were a valuable balance to Meyer s great enthusiasm. Studies on the thermodynamics of large molecules in solution, and on the structure of cellulose and chitin, were pursued with C. Boissonnas, W. Lothmar, and L. Misch. A theory of the elasticity of rubber evolved from his work with C. Ferri and his previous observations with Susich and Valk6. 

With the basic structure of polymers of macromolecules clarified, scientists now searched for a quantitative understanding of the various polymerization processes, the action of specific catalysts, and initiation and inhibitors. In addition, they strived to develop methods to study the microstructure of long-chain compounds and to establish preliminary relations between these structures and the resulting properties. In this period also falls the origin of the kinetic theory of rubber elasticity and the origin of the thermodynamics and hydrodynamics of polymer solutions. Industrially polystyrene, poly(vinyl chloride), synthetic rubber, and nylon appeared on the scene as products of immense value and utility. One particularly gratifying, unexpected event was the polymerization of ethylene at very high pressures. 

The present work is intended to survey briefly the basic thermodynamic considerations of rubber elasticity and... 

In other theories of rubber elasticity, the network structure is explicitly considered. However, the polymer on the surface is taken to be fixed (according to an affine deformation) upon deformation. - A truly statistical mechanical theory would also treat the surface statistically. More fundame ntally, however, in these theories the fixed point character of the surface i hen completely determines the behavior of the bulk material. This would appear to be nonsense in the thermodynamic limit of infinite volume, unless the fixed surface were of finite extent. In this case, the theory is no longer statistical in nature. 

The development of rubber and rubber elasticity theory can be traced through several stages. Perhaps the first scientific investigation of rubber was by Gough in 1805 (13). Working with unvulcanized rubber, Gough reached three conclusions of far-reaching thermodynamic impact ... 

The equation of state of rubber elasticity will now be calculated via statistical thermodynamics, rather than the classical thermodynamics of Section 9.5. Statistical thermodynamics makes use of the probability of finding an atom, segment, or molecule in any one place as a means of computing the entropy. Thus tremendous insight is obtained into the molecular processes of entropic phenomena, although classical thermodynamics illustrates energetic phenomena adequately. 

EXAMPLE 9.3 The thermodynamics of a rubber band. Is the retraction of a rubber band driven by a change in enthalpy or in entropy The answer to this question will help us to construct a model for the microscopic behavior of polymeric materials in Chapter 29. Suppose you apply a quasi-static stretching force that increases the length of a rubber band. The force of retraction / exerted by the rubber band is equal and opposite to the applied stretching force. To deal with elastic forces when there is no particle exchange, we have U = U S,VJ) and... 

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