Flory-Erman model

Models of rubber elasticity have been reviewed for finite deformation and compared with experimental data by Boyce and Arruda (2000). A hybrid model of the Flory-Erman model for low stretch deformation and the Arruda-Boyce model for large stretch deformation is shown to give an accurate predictive description of Treloar s classic data over the entire stretch range for all deformation states. 

The detailed calculations according to the constrained junction fluctuation model and other advanced models can only be performed numerically. The fitting of the stress-strain (or swelling) data to the Flory-Erman model, in principle, requires three parameters / ]ph, k and Here we briefly outline the steps of the fitting procedure [113,114] ... 

The corresponding equation according to the Flory-Erman constrained junction fluctuation model is... 

In the following sections typical experimental results obtained for different network systems and analyzed using several of the theoretical approaches are briefly reviewed. For a more extensive discussion, we refer the reader to a work by Han, Horkay, and McKenna [111] where a critical evaluation of many of the modem theories of molecular rubber elasticity was performed. Based on an analysis of carefully selected data sets reported in the literature, these authors concluded that, of the tested models, the Flory-Erman theory and its modified versions provided the best... 

In recent years, with the development of model networks it has been possible to prepare networks of controlled and junction functionality These are prepared by endlinking functionalized prepolymers with cross-linking agents of known functionality. Therefore, by choosing the appropriate molecular weight distribution of the prepolymers it is possible to prepare unimodal and bimodal networks. Mark and coworkers (5-11) have performed extensive studies on model networks to test the various theories of rubber elasticity. In the case of unimodal networks they find that the macroscopic properties such as stress or swelling ratios can be described reasonably well by the Flory-Erman theory (12,13). 

The theoretical equations presented above can be used to interpret stress-strain measurements in uniaxial extension and thus to fully characterize elastomeric networks. In this regard, equations (124) and (125) are of particular interest since they relate the parameter k, quantifying the entanglement constraints in the Flory and Erman model, to the polymer microstructure and conformational properties and to the network topology. An illustrative analysis of stress-strain data due to Queslel, Thirion and Monnerie is reported below. 

X) A. Recently, Erman, Flory, and Hummel carried out a detailed analysis of end-to-end vectors as a function of the chain length for />-phenylene polyamides and polyesters. The required geometrical parameters were obtained from the results for model compounds. Configurational averaging was performed on the basis of torsional potentials obtained by the same authors The value of persistence length depends on... 

B.Erman, D.C.Marvin, P.A.Irvine, and P.J.Flory, Optical anisotropies of model analogues of polycarbonates. Macromolecules 15 664 (1982). 

These two relations result from the phantom network model. Their derivations are given elsewhere (Flory, 1976 Mark and Erman, 2007). 

The extent to which entanglements contribute to network elasticity is not yet fully resolved. In the model of Langley [45], Dossin and Graessley [46-49] a contribution to the equilibrium modulus is associated with the plateau modulus of viscoelasticity. On the other hand, Flory [36] and Erman [38 0] assume that interpenetration of chains is solely reflected by suppression of the fluctuations of junctions. 

Kloczkowski, Mark, and Erman [95] compared the prediction of the diffused constraint model with the results of the Flory constrained-junction fluctuation theory [36] and the Erman-Monnerie constrained chain theory [94]. They found that the shapes of the [/ ] vs. a curves for all three theories were very similar. Rubinstein and Panyukov [101] reanalyzed the data of Pak and Flory [118] obtained for uniaxially deformed crosslinked PDMS samples. They concluded that the fit of the experimental data by the diffused... 

A molecular model for the deformation is required. The Flory-Rehner Eqs. (4) and (5) were developed for a network deforming affinely, i.e. a network without junction fluctuations. A more general treatment by Flory and Erman which includes such fluctuations is described in this Section. 

Stress-strain measurements in uniaxial extension have revealed that real networks have a behavior closest to the affine limit at small deformations and approach the phantom limit at large deformations. The recent molecular theory developed by Flory and Erman accounts for this transition. In this model, the restrictions on junction... 

Fig. 8.7 Plot of the theoretical dependence of the melting temperature on elongation ratio for K = 10 for constrained junction model and /c = oo for Flory model. (From Erman and Mark (9))...

The strain function depends on the transformations of chain dimensions with deformation in real networks. According to the model proposed by Flory and Erman the elements of the molecular deformation tensor can be expressed as the sum of contributions from the phantom network, and from the constraints, A ... 

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