Network Helmholtz free energy

The Helmholtz free energy of the deformed network can thus be written as... 

It is clear that the Helmholtz free-energy change upon deformation predicted by equation (6-46) is just one-half as large as that given in equation (6-45) based on the affine deformation of crosslink points. Real networks probably exhibit behavior that is between these two extremes. Not all crosslink points move affinely however, steric interactions are strong enough to prevent the phantom approximation from being completely realistic. 

A more exact calculation involves the calculation of the energy stored in a rubber network because of network deformation. This stored energy is expressed in terms of the Helmholtz free energy (A) and is derived from entropy considerations. The force-extension relation can then be calculated by taking the derivative of A with respect to elongation, as described below and in Chapter 3.2. 

Hie total elastic free energy, or the Helmholtz free energy, of a network consists of the sum of the free energies of the individual chains and the contributions coming from intermolecular correlations. Hie elementary theories of rubber elasticity ignore contributions from intermolecular effects. Improvements in the elementary models have been made by different researchers in different ways. In this section most of the models will be covered. Since the single-chain elasticity forms the basis of rubber elasticity, it will be discussed first in some detail. 

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. 

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