Networks elastic free energy

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

The evaluation of Z(n) for filled rubbers has been done [15] along the line of Edwards and Deam [13,18] for unfilled networks, who used a Feynman variational principle [19]. The rigorous derivation of the elastic free energy for the filled rubber problem leads to the following expression [15] ... 

The evaluation of the elastic free energy AFei rests on the assumption that the root-mean-square distance between the ends of the chain is distorted by the same factor a representing the linear expansion of the spatial distribution. As in the treatment of the swelling of 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 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 total elastic free energy AAei of the network relative to the undeformed state is obtained by summing Equation (6) over the v chains of the network [4]... 

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. 

Comparison of the expressions for the elastic free energies for the affine and phantom network models shows that they differ only in the front factor. Expressions for the elastic free energy of more realistic models than the affine and phantom network models are given in the following section. 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

The elastic free energy AFe causes difficulty because of its sensitivity to the crystallization model assumed. To estimate AFe for lamellar morphology, consider first an important property of a network, amorphous or crystalline. Network crosslinks are considerably restricted in their fluctuations. Fluctuations of crosslinks several chains removed from a particular chain are therefore inconsequential for that chain. A chain in the interior of a path traced through several sequentially connected chains behaves as if the path ends are securely anchored at fixed positions ( 7). If Gj chain vectors make up the path, then... 

The two-network theory for a composite network of Gaussian chains was originally developed by Berry, Scanlan, and Watson (18) and then further developed by Flory ( 9). The composite network is made by introducing chemical cross-links in the isotropic and subsequently in a strained state. The Helmholtz elastic free energy of a composite network of Gaussian chains with affine motion of the junction points is given by the following expression ... 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

The elastic free energy, assuming no appreciable change in internal energy in the network, may be described as... 

In order now to obtain the difference in free energy between a swollen network and a dry network, we need to apply Eq. (111-5) for the elastic free energy twice once with respect to (r2)0s as the swollen reference state and once with respect to [Pg.39]

Since affine deformation cannot be proven for the non-Gaussian network chain defined by Eq. (IV-30), Blokland uses Eq. (IV-5) to derive the elastic free energy of the network. This yields ... 

Accordingly, the elastic free energy of a strand chain of a network with the deformed span R is given by... 

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... 

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