Network conformational entropy

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... 

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... 

According to the current state of the theory, the deformation of polymeric networks must be accompanied not only by the intrachain conformational entropy changes but intrachain energy changes which depend on the conformational energies of macromolecules. Therefore, reliable experimental determination of these intrachain energy changes and their interpretation by means of isomeric state theory is of fundamental importance for polymer physics. 

The molecular mechanisms by which the extension of the N-terminus by the extra methionine residue destabilized recombinant a-lactalbumin remain unclear. Additional conformational entropy of the extra methionine residue in the unfolded state could account for the destabilization and unfolding-rate acceleration of the recombinant protein [22]. Ishikawa and coworkers reported the destabilization of recombinant bovine a-lactalbumin, similarly induced by the extra N-terminal methionine residue, and showed that the enthalpy change of thermal unfolding was the same for the authentic and recombinant proteins, indicating that the destabilization was caused by an entropic effect [42]. However, the destabilization by the extra methionine residue in the lysozyme homologous to a-lactalbumin was rather enthalpic and accompanied by a disruption of hydrogen-bond networks in the N-terminal region [43,44]. 

A number of different mechanisms have been proposed to account for the rubberlike properties of materials. In classical rubber theory these properties are attributed to a decrease in conformational entropy on deforming a network of kinetically free random polymer molecules. Stress orders the polymer chains and decreases their entropy by limiting their conformational freedom, thus providing the restoring force to the relaxed state. Such a theory was developed for elastin by Hoeve and Flory... 

As described above for elastin and resilin, the ability of elastomeric proteins to exhibit elasticity relies on the molecular movement, stmctural folding, and conformational freedom of individual components so that they can instantaneously respond to the applied force within a cross-linked network to distribute the stress throughout the system. Stretching initially will interrupt interactions between the loops such as hydrophobic interactions, hydrogen bonding, and electrostatic interactions, while at higher extensions a decrease in conformational entropy will be prevalent. To date, different models are proposed to explain the mechanisms of elasticity for resilin, based on the knowledge from elasticity models that have been proposed for elastin. 

Elastomers are also sometimes known as rubbers. They are also irreversibly cross-linked by covalent bonds. The elastomer raw materials (rubber base) generally have higher molar masses than the thermoset raw materials. In addition, the cross-linked network produced by cross-linking (vulcanization, hardening) is not so densely cross-linked as in the case of the thermosets. For this reason, elastomers have a high segmental mobility above the glass transition temperature. They deform readily under stress above this temperature, but because of their cross-linked structure, they rapidly return to their initial state when the stress is released because the initial state corresponds to the state of maximum conformational entropy. 

The total entropy change is a linear integration of conformational-entropy changes of all the network chains ... 

The conformational entropy of the chain between the cross-linking points works against the network swelling, which can be calculated from the classical elastic free energy of cross-linked network, as... 

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

Rubbers and gels are three-dimensional networks composed of mutually cross-linked polymers. They behave like solids, but they still have high internal degrees of freedom that are free from constraints of external force the random coils connecting the cross-links are free in thermal Brownian motion. The characteristic elasticity of polymeric materials appears from the conformational entropy of these random coils. In this chapter, we study the structures and mechanical properties of rubbers on the basis of the statistical-mechanical models of polymer networks. 

The entropy associated with placement of the junctions is reduced by the stricture of the A-B junctions at the interface, while the conformational entropy is reduced by the restriction of chain motions within the confines of the domain boundaries.This is the ordered state and is characteristic of di-blocks as well as network-forming block copolymers. This ordered state can be detected by observing the periodicity of the A-domain-interphase-B-domain stucture by the scattering of x-rays, light, or neutrons. The order-disorder transition temperature, Tc, is measured by these techniques. 

At 130°C, the temperature of the experiment, oxidative scission and cross-linking are both going on all the time. The reactions happen, however, in the condition of the network at the time. This means that the continuous stress relaxation experiment measures only the degradation step, because the new cross-links form in the stretched chains that is, the second network develops in the extended state, at equilibrium. There is no significant change in conformational entropy on formation of these cross-links, and the change in stress is near zero. 

A characteristic property of amorphous polymers is the ability to sustain large strains. For cross-linked three-dimensional networks the strain is usually recoverable and the deformation process reversible. The tendency toward crystallization is greatly enhanced by deformation since chains between points of cross-linkages are distorted from their most probable conformations. A decrease in conformational entropy consequently ensues. Hence, if the deformation is maintained, less entropy is sacrificed in the transformation to the crystalline state. The decrease in the total entropy of fusion allows crystallization, and melting, to occur at a higher temperature than would normally be observed for the same polymer in the absence of any deformation. This enhanced tendency toward crystallization is exemplified by natural rubber and polyisobutylene. These two polymers crystallize very slowly in the absence of an external stress. However, they crystallize extremely rapidly upon stretching. 

Rubber elasticity of a polymer network is one of the most distinctive features of long polymer chains. The elastic force of such a network is mainly due to, the change of conformational entropy of network strands which are connected to other strands by chemical linkages or topological constraints. The theoretical models to clarify the relationship between... 

The aim of this contribution has been to link the basic macroscopic phenomena associated with polymers with the unique features of their structure, the most obvious being the presence of long, flexible molecular chains. The important role of the conformational entropy of flexible chains, not only for rubber elasticity but for polymer dynamics in general, has been demonstrated. Moreover, the concept of an entanglement network, which underpins much of the theory of polymer dynamics in the melt, has also been shown to have important repercussions for the high strain behavior of solid polymers, namely plastic deformation, crazing, and fracture. 

For polymer networks, the conformational entropy closely depends on the cross-linking density according to the thermodynamic theory, the dimensionality of the polymer system should affect the value of Tg which is indeed observed on highly cross-linked systems, confirming the predictions of this theory. 

Elastomeric chains of sufficient regularity may crystallize under some conditions. The deformation of a macromolecule is accompanied by a decrease in its conformational entropy. Its tendency to crystallize is therefore increased. The oriented system will crystallize even above the melting point of the isotropic system (and is thus strain-induced crystallization ). The crystallinity reaches a value which depends on the degree of deformation and the crystallization temperature. The melting point = Ai/ /ASm, given by the ratio of the enthalpy of fusion to the entropy of fusion, is thus increased by the network deformation. 

The reversible recovery of a deformed elastomer to its original (undeformed) state is due to an entropic driving force. The entropy of polymer chains is minimum in the extended conformation and maximum in the random coil conformation. Cross-linking of an elastomer to form a network structure (IX) is... 

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