Gaussian chain elastic free energy

The elastic free energy Ae of a Gaussian chain is related to the probability distribution W(r) by the thermodynamic expression [5]... 

Flere, A (T) is a function of temperature alone. Equation (6) represents the elastic free energy of a Gaussian chain with ends fixed at a separation of r. The average force required to keep the two ends at this separation is obtained from the thermodynamic expression [28]... 

If Gc chains crystallize (partially) there will remain G-Gc completely amorphous chains and G + 3Gc/2 total elastic elements (amorphous chains and subchains). This is true only for the model described with 1/2 Gc chains folding once and 1/2 Gc chains not folding at all. If each elastic element is Gaussian in its behavior, the elastic free energy Fg can be written as... 

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 ... 

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. 

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 ... 

Moreover, the equation can only be accurate for small strains, since considerable change in the end-to-end distance of the cords would distort the Gaussian distribution of statistical chain elements. This happens more readily for a smaller value of It also implies that at increasing strain, the chemical bonds in the primary chain become increasingly distorted. Consequently, the increase in elastic free energy is due not merely to a decrease in conformational entropy but also to an increase in bond enthalpy. If the value of is quite small, even a small strain will cause an increase in enthalpy. (In a crystalline solid, only the increase in bond enthalpy contributes to the elastic modulus.)... 

If the chains are crosslinked in the isotropic state and Pq is an isotropic Gaussian distribution, the additional elastic free energy of a strand of the network with deformed span R is given by (denoting Ax by A)... 

Using Fade approximation of the inverse Langevin function (4.13) the elastic free energy of a non-Gaussian chain with the chain extension h/Na expresses by the following closed formula. 

If of the G network chains crystallize, leaving G-G amorphous chains, there are G+Gc amorphous elastic elements (amorphous chains plus amorphous sub-chains) that comprise the seml-crystalllne network. It Is here assumed that a chain enters a crystallite only once or not at all. By and large, multiple entry Into crystalline regions Is probably rare except In very lightly crossllnked networks. Let each elastic element (chain or subchain) behave In a Gaussian fashion so that we may write for the elastic free energy F of a specified network... 

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)... 

The affine and the phantom network models, based on the Gaussian chain, act as the two limiting idealized models of amorphous networks in the absence of intermolecular interactions such as interactions between network chains. Expressions for the elastic free energy of more realistic models than the affine and the phantom network models are given in the following sections. 

With this modified distribution of the end-to-end vector of a network chain, the evaluation of the elastic free energy follows the same lines as given above for the affine model. When the chains are sufficiently long, the parameter p - 0, and the Gaussian affine model is obtained. 

The elastic free energy of a phantom network of Gaussian chains was obtained rigorously by Flory and is valid for networks of any functionality, irrespective of their structural imperfections. It is given in equation (101). The elastic equation of state for phantom networks may then be expressed by equation (102). Equation (84) is then recovered, as expected, because of the relationship shown in equation (103). 

The necessity to introduce a front factor into the equation for the elastic free energy of a network of Gaussian chains has been defended by different arguments. In fact, the different arguments lead to front factors with different physical meanings and although... 

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