Rubber elasticity free energy state

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. In this section, the elasticity of the single chain is discussed first, followed by the elementary theory of elasticity of a network. Corrections to the theory coming from intermolecular correlations, which are not accounted for in the elementary theory, are discussed separately. 

The ratios of mean-squared dimensions appearing in Equation (13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic (laboratory) state of deformation, an assumption has to be made to relate microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubber-like elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before that, however, we describe the macroscopic deformation, stress, and the modulus of a network. 

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free... 

The statistical theory of rubber elasticity discussed in the preceding section was arrived at through considerations of the underlying molecular structure. The equation of state was obtained directly from the Helmholtz free energy of deformation (or simply conformational entropy of deformation, since the energy effects were assumed to be absent), which we can recast with the aid of equations (6-45) and (6-59) as... 

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. 

For an elastic material the work done can be equated to a change in the stored elastic energy U. In the case of rubbers, it is usual to consider a reversible isothermal change of state at constant volume, so that the work done can be equated to the change in the Helmholtz free energy A, i.e. A17 = A 4. Here U is... 

A totally different approach to rubber elasticity has been developed by Stepto and co-workers [15, 16], which also accounts for the Mooney-Rivlin softening. Their approach is not phenomenological, but is based on structural considerations that give an accurate description of the moleeular eonformational states of the units in the polymer chains as the network is stretched. They have proposed a method for calculating the free energy of a stretched molecular network based on the rotational isomeric state of the network chains, with conformational energies determined from observations of conformational properties. 

As a first approach to the equation of state for rubber elasticity, we analyze the problem via classical thermodynamics. The Helmholtz free energy, F, is given by... 

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