Entropy Elastic Deformation

First the conformations of certain idealized chains are considered. For he chain of n equal links of length C joined entirely at random (i.e. neglecting valence angles and hindrances to internal rotation) the probability distribution p(x, y, z) to find a chain end at a point (x, y, z) — if the other end is fixed at the origin — is obtained as a Gaussian distribution  

The only parameter, b, is the inverse of the most probable chain end separation and is under the above conditions Z3/n. The extended length of the chain is, of course, equal to nC, whereas the average chain end separation (the root-mean-square length /W equals /n. 

More realistic is the model of the statistically kinked chain with valence angles and hindrances to internal rotation. Here the mean square length is derived as  

The entropy 5 of a single chain is according to Eq. (5.5) and the Boltzmann relation 

The retractive force / acting between the chain ends — and in the direction of the line joining them — is derived from Eqs. (5.2), (5.4) and (5.7). If one considers a reversible deformation and postulates that all conformational states posses the same internal energy U so that U does not depend on conformation then 

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In general, the moduli of elasticity are much lower than the lattice moduli of the same compounds. There are many reasons for this. For one thing, most chains of a sample do not normally lie in the strain direction. Consequently, deformation can occur through increase in the interchain distance, that is, through other conformational positions (entropy-elasticity). Deformations can also occur through irreversible slippage of chains past each other (viscoelasticity). 

The transition obtained under stress can be in some cases reversible, as found, for instance, for PBT. In that case, careful studies of the stress and strain dependence of the molar fractions of the two forms have been reported [83]. The observed stress-strain curves (Fig. 16) have been interpreted as due to the elastic deformation of the a form, followed by a plateau region corresponding to the a toward [t transition and then followed by the elastic deformation of the P form. On the basis of the changes with the temperature of the critical stresses (associated to the plateau region) also the enthalpy and the entropy of the transition have been evaluated [83]. 

Note at proof Recently Gaylord et al. (Polymer, 25, 1577, 1984) have shown theoretically that elastic deformation of crystalline polymers is controlled by energetic interaction rather than by entropy. 

To answer the questions stated above, a typical value for the entropy of deformation, ASdef, caused by the stress related to the velocity gradient in the gel front , must be estimated. To do so, Eqs. (17c) with AGdd = —T ASdef > 0 and (18a) are applied under the assumption of rubber-elasticity in the deformed chain. They yield the relationship... 

Surface stress — The surface area A of a solid electrode can be varied in two ways In a plastic deformation, such as cleavage, the number of surface atoms is changed, while in an elastic deformation, such as stretching, the number of surface atoms is constant. Therefore, the differential dUs of the internal surface energy, at constant entropy and composition, is given by dUs = ydAp + A m g m denm, where y is the interfacial tension, dAp is the change in area due to a plastic deformation, gnm is the surface stress, and enm the surface strain caused by an elastic deformation. Surface stress and strain are tensors, and the indices denote the directions of space. From this follows the generalized Lippmann equation for a solid electrode ... 

In our statistical treatment of an ideal elastomer, we have assumed that the elastic force is entirely attributable to the conformational entropy of deformation, energy effects being neglected. That the theory reproduces the essential features of the elasticity of real elastomers attests to the basic soundness of this assumption. On the other hand, we know that in real elastomers such energy effects cannot be entirely absent, and deviations from the ideal elastomer model may be expected to occur. Let us now examine in greater detail the extent to which the neglect of energy effects is justified. We can rewrite equation (6-28) ... 

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... 

Entropy decrease (for entropy-elasticity) of the network chains as a result of the dilation (elastic deformation term). 

Generally, the structure of the polymer-particle complex can be found from the minimization of free energy that includes the polymer-particle interaction energy, entropies of nonadsorbed monomer units and imits localized at the surface of the particle, and typically, for the system under consideration, the elastic deformation of crosslinked macromolecule. Such theoretical analysis, following the lines of [234,235], can explain the specific behavior of P(T,a) observed for the envelopes with different niunbers of crosslinks j [ 57 ]. According to [235], when the number of crosslinks is small enough, nj all jimc-... 

The moduli of elasticity determined by stress / strain measurements are generally much lower than the lattice moduli of the same polymers (Table 11-3). The difference is to be found in the effects of entropy elasticity and viscoelasticity. Since the majority of the polymer chains in such polymer samples do not lie in the stress direction, deformation can also occur by conformational changes. In addition, polymer chains may irreversibly slide past each other. Consequently, E moduli obtained from stress/strain measurements do not provide a measure of the energy elasticity. Such E moduli are no more than proportionality constants in the Hooke s law equation. The proportionality limit for polymers is about 0.l%-0.2% of the... 

Energy-elastic bodies exhibit large moduli of elasticity for small deformations of about 0.1%. In contrast, entropy-elastic bodies possess low moduli of elasticity and high reversible deformabilities of several hundred percent. 

In contrast to ideal entropy-elastic bodies, real entropy-elastic bodies have an energy-elastic component. The force Fe resulting from this component is given for a uniaxial deformation by... 

In the preceding discussion about the energy- and entropy-elastic behavior of matter, it was tacitly assumed that the body returns immediately and completely to the original state when the load is removed. In actual fact, this process always takes a certain time in macromolecular substances. In addition, not all bodies return completely to the original position in some cases they are partially irreversibly deformed. 

Macromolecular materials usually possess entropy elasticity together with viscous and energy-elastic components. Such behavior was only partly comprehensible by use of the models discussed up to now. It can be described very satisfactorily, however, by a four-parameter model in which a Hooke body, a Kelvin body, and a Newton body are combined (see the lowest figure in Figure 11-11). With this model, the deformation must again be added, i.e., with Equations (11-49), (11-52), and (11-57),... 

Under force apphcation, the deformation of purely entropy elastic bodies shows a delayed increase. The delay factor may be tiny (in the ps range). Such deformations are also completely reversible. 

Plastic flow is the persistent deformation observed in solid bodies when a certain minimum strain level (flow point) is exceeded viscous flow, on the other hand is the persistent deformation in the entropy elastic range as well as in the melt or flow range of plastics. 

The percentage of viscous flow within the die is large and the rubber elastic (entropy elastic) reverse deformation after exiting the die is small. Hardly any relaxation takes place during passage through a short die, entropy elastic predominates and the swell is large. 

The deformation characteristic of more ductile polymer materials at ambient temperatures like most thermoplastics or all elastomers is highly non-linear, e.g., either mostly viscoelastic or entropy-elastic or a combination of both. Compared to concepts of LEFM relatively rarely used for polymer materials different concepts of non-linear elastic firacture mechanics such as elastic-plastic fracture mechanics (EPFM) or post-yield fracture mechanics (PYFM) are somewhat widely applied, therefore. One of the most important concepts of EPFM is the J integral concept. Notwithstanding the J integral is primarily defined to be valid... 

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