Entropy of elasticity

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

The entropy of elasticity of a droplet is a measure of the increase in the available volume in configuration space. This increase occurs with a transition from rigid, regular structure to an ensemble of states that include many different structures. If the potential wells in the liquid state were as narrow as those in the solid state and if each of those potential wells was equally populated and corresponded to a stable amorphous structure (and vice versa), then the entropy of elasticity would be a direct measure of the increase in number of wells or a direct measure of the number of available stractures [3,5]. 

Entropy, of elasticity n. A measure of the unavailable energy in a closed thermodynamic system that is also usually considered to be a measure of the system s disorder and that is a property of the system s state and is related to it in such a manner that a reversible change in heat in the system produces a change in the measure which varies directly with the heat change and inversely with the absolute temperature at which the change takes place. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

A great many liquids have entropies of vaporization at the normal boiling point in the vicinity of this value (see benzene above), a generalization known as Trouton s rule. Our interest is clearly not in evaporation, but in the elongation of elastomers. In the next section we shall apply Eq. (3.21) to the stretching process for a statistical—and therefore molecular—picture of elasticity. 

The modulus increases with temperature. This behavior is verified by experiment. By contrast, the modulus of metals decreases with increasing T. The difference arises from the fact that entropy is the origin of elasticity in polymers but not in metals. 

We have already observed that the entropy theory of elasticity predicts a modulus of the right magnitude and possessing the proper temperature coefficient. Now let us examine the suitability of Eq. (3.39) to describe experimental results in detail. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

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]. 

The coacervation of tropoelastin plays a crucial role in the assembly into elastic fibers. This coacervation is based on the LCST behavior of tropoelastin, which causes tropoelastins structure to become ordered upon raising the temperature. The loss of entropy of the biopolymer is compensated by the release of water from its chain [2, 18, 19]. This release of water results in dehydration of the hydrophobic side chains, and this is the onset of the self-assembly leading to the alignment of tropoelastin molecules. 

The main models are described in a review by Vrhovski and Weiss [8]. For ideal elastomers in the extended mode, all the energy resides on the backbone and can therefore be recovered upon relaxation [18]. Generally, it is believed that the mechanism of elasticity is entropy-driven, thus the stretching decreases the entropy of the system and the recoil is then induced by a spontaneous return to the maximal level of entropy [8]. 

If the process is conducted reversibly, dQ = TdS where S is the entropy of the elastic body. Substitution of this expression for dQ in Eq. (5) will require dW to represent the element of reversible work. In order to comply with this requirement, the coefficients P and / in Eq. (6) must be assigned their equilibrium values. In particular, / will henceforth represent the equilibrium tension for a given state of the system, which may be specified variously by aS, F, and L, by T, F, and L, or by T, P, and L. Then... 

Until recently s the last term in the brackets in Eq. (38) was given erroneously as (viFe/Fo)t 2. This error resulted from the use of incorrect elastic entropy and free energy expressions in which the InaJ term of Eq. (35) was omitted. This term takes account of the entropy of distribution of the effective cross-linkages over the volume Foq = F. 

In Eq. (2), mi is the chemical potential of the solvent in the polymer gel and /al 0 is the chemical potential of the pure solvent. At equilibrium, the difference between the chemical potentials of the solvent outside and inside the gel must be zero. Therefore, changes of the chemical potential due to mixing and elastic forces must balance each other. The change of chemical potential due to mixing can be expressed using heat and entropy of mixing. 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

According to the theory of rubber elasticity, the elastic response of molecular networks is characterized by two mechanisms. The first one is connected with the deformation of the network, and the free energy change is determined by the conformational changes of the elastically active network chains. In the early theories, the free energy change on deformation of polymeric networks has been completely identified with the change of conformational entropy of chains. The molecular structure of the chains... 

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... 

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