Helmholtz elastic free energy

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 ... 

Expressed in terms of moduli, the Helmholtz elastic free energy relation is given by eq. 5. 

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. 

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. 

In the MSF theory, the function,/, in addition to simple reptation, is also related to both the elastic effects of tube diameter reduction, through the Helmholtz free energy, and to dissipative, convective molecular-constraint mechanisms. Wagner et al. arrive at two differential equations for the molecular stress function/ one for linear polymers and one for branched. Both require only two trial-and-error determined parameters. 

To understand robber elasticity we have to revisit some simple thermodynamics (the horror. the horror ). Let s start with the Helmholtz free energy of our piece of rubber, by which we mean that we are considering the free energy at constant temperature and volume (go to the review at the start of Chapter 10 if you ve also forgotten this stuff). If E is the internal energy (the sum of the potential and kinetic energies of all the particles in the system) and 5 the entropy, then (Equation 13-26) ... 

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... 

Space not permit us to review the extensive literature of these theories, including the many recent developments. Instead, we shall try to apply the free volume concept to the hard sphere system in the simplest fashion. Imagine that the diameter a of the N hard spheres in the volume V of our system is shrunk till the molecules are elastic point centers. We now have an ideal gas whose Helmholtz free energy is - (w = p )... 

The two contributions to the elastic force may be experimentally distinguished by the following means From the Helmholtz free energy analogy to equation 3.4... 

A = Helmholtz free energy (constant volume and temperature) dW = the elastic work... 

Here we begin with a sample of rabber having initial dimensions l, I2, I3. We deform it by an amount A/, A/2, A/3 and define the stretch (ratio) in each direction as A, = (/, -I- A/,)//, = ///,. The purpose of Finite Elasticity Theory has been to relate the deformations of the material to the stresses needed to obtain the deformation. This is done through the strain energy density function, which we will describe using the Valanis-Landel formalism as IT(A, A2, A3). Importantly, as we will see later, this is the mechanical contribution to the Helmholtz free energy. Vala-nis and Landel assumed [60] that the strain energy density function is a separable function of the stretches A, ... 

Finite Elasticity Theory Classical Theory. The finite elasticity theories available today are very powerful and well developed from a phenomenological perspective. Because the K-BKZ (70-72) has the form of a time-dependent finite elasticity (it was developed as a perfect elastic fluid ) it is useful to briefly outline the basics of finite elasticity theory here. In the initial sections of this article, the stress and strain tensors were discussed, and it was noted that the constitutive relationships that arise between the stress and the strain include material parameters called moduli. When a material is classified as hyperelastic then the moduli are related to derivatives of the free energy function (often the Helmholtz free... 

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... 

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... 

Since the Helmholtz free energy is the work function and the work of deformation is / dL (where L = a Li), as shown in the second equality, the elastic force may be obtained by differentiating Eq. (1.12), giving... 

Although E drops significantly as T is raised above Tg, K changes relatively little, so that K E and, from Eq. (9), v 0.5. Volume changes may hence be considered negligible compared with other types of deformation. This justifies the use of the Helmholtz free energy in the thermodynamic analysis of rubber elasticity, defined by Eq. (11). 

Most recent estimates of the crystal elastic constants for polymers follow the method of calculating the second derivatives of the Helmholtz Free Energy A with respect to deformation of a small volume element Vo [62,63]. The stiffness constants Ctmnk are defined as... 

The Helmholtz Free Energy includes both intermolecular forces and intramolecular forces and also entropic contributions. The intramolecular contributions are the same as those required for the single-chain calculation, but there is more difficulty in producing force fields that include intermolecular contributions. The intermolecular contributions are typically Lennard-Jones type interactions and to obtain plausible values that are satisfactory for a range of different chemical compositions is often debatable. It is, however, possible to obtain some confirmation of their validity in a particular instance by verifying that the calculations predict the correct crystal structure and this must be regarded as a the first step to calculating the elastic constants. 

The general and detailed constitutive relations of E.H. Lee s elastic-plastic theory at finite strain have been derived by Lubarda and Lee [5]. In this work, let the specid constitutive relations which are employed in the general purpose finite element program be listed as follows. First, the Helmholtz free energy density, E, as a function of the invariants of the elastic Cauchy-Green tensor, c/y, may be expressed as... 

Note that according to Eqs. (15) and (16) the free Helmholtz energy serves as a potential for the stresses T and for the microstructural flux S. According to the assumption of elastic material behavior, results of this type have to be expected. The additional balance equation for k [Eq. (17)] possesses the same structure as the balance of equihbrated forces obtained in Refs. [14, 20, 33] and appHed, e.g., in Ref [36]. Following the MuUer-Liu approach, Svendsen [39] also derived a generalization of Eq. (17) for a model with scalar-valued stractural parameters. 

---