Entanglement models elastic free energy

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

The slip-link model incorporates the effects of entanglements along the chain contour into the elastic free energy. According to the mechanism of the slip link, sketched in Fig. 3, a link joins two different chains which may slide a distance a along the contour of the chains. The elastic free energy resulting from this model is... 

In real polymer network the effects of excluded volume and chain entanglements should be taken into account. In 1977 Hory [26] formulated the constrained junction model of real networks. According to this theory fluctuations of junctiOTs are affected by chains interpenetration, and as the result the elastic free energy is a sum of the elastic free energy of the phantom network AAph (given by Eq. (5.78)) and the free energy of constraints AA ... 

A different statistical-mechanical approach based on so called replica formalism was developed by Edwards and CO workers [29,30]. They studied the effect of topological entanglements between chains on the elastic free energy of the network and formulated the slip-link model. The elastic energy of constraints in the slip-link theory is... 

The Constrained Junction Fluctuation Model. The affine and phantom models are two limiting cases on the network properties and real network behavior is not perfectly described by them (recall Fig. 29.2). Intermolecular entanglements and other steric constraints on the fluctuations of junctions have been postulated as contributing to the elastic free energy. One widely used model proposed to explain deviations from ideal elastic behavior is that of Ronca and Allegra [34] and Hory [36]. They introduced the assumption of constrained fluctuations and of affine deformation of fluctuation domains. 

This model incorporates the effects of entanglements along the chain contour into the elastic free energy. 

Here, Nc and Ns are the number of chemical aoss-links and slip-links, respectively, and j/ = 0.2343. The first term on the right-hand side of eqn [43] is the contribution to the elastic free energy from the phantom network. The second term, similar to that in the constrained-junction model, gives the contribution from entanglements along the chains. 

One of the most interesting alternative approaches is the slip-link model, which incorporates the effects of entanglements [40,41] along the network chains directly into the elastic free energy [42]. Still other approaches are the tube model [43] and the van der Waals model [44]. 

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