Flory-Erman

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

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

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] ... 

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... 

The Flory-Erman theory has given a good account of experimental results in elongation and compression... 

Equations (33) and (34) can be used to interpret stress-strain measurements with the Flory-Erman theory The general Flory-Erman relationship for uniaxial extension of a swollen network is... 

For stress-strain measurements on swollen networks, it is advantageous to immerse the samples completely in excess solvent. Thus errors due to solvent vaporization during the stretching of a swollen network exposed to the air are thereby suppressed. However, the swelling capacity of an elastomer is not constant with deformation fortunately, a swelling equilibrium equation can also be obtained for this case making use of the Flory-Erman theory... 

It is interesting to note that junction fluctuations increase in the direction of stretching but decrease in the direction perpendicular to it. Therefore the modulus decreases in the direction of stretching, but increases in the normal direction since the state of the network probed in this direction tends to be more nearly affine. The curve of [/" ] versus 1/a is sigmoidal. The parameters k and f of poly(dimethylsiloxane) networks are determined in Figure 17 (155) the intercept of the sigmoidal curves is the phantom modulus. This Flory-Erman theory has been compared successfully with such experiments in elongation and compression (155,162,162-166). It has not yet been extended to take account of limited chain extensibility or strain-induced crystallization (167). 

Fig. 17. Determination of the parameters k and of the Flory-Erman theory for perfect trifunctional poly(dimethylsiloxane) networks (155). To convert N/mm to psi, multiply by 145. Courtesy of Springer-Verlag.

Nevertheless, the Flory-Erman theory has left a number of questions unanswered, and it has become evident that it is not possible to take account of all the physical effects of chain entanglements by considering... 

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 swelling results of these PDMS networks are consistent with the results of Mark and coworkers. This indicates that the macroscopic properties of networks such as the swelling behavior are reasonably well described by the current theories of rubber elasticity, such as the Flory-Erman theory. 

The Flory-Erman description of how these factors affect the swelling and modulus is given in equation 1, which describes polymers swollen to their maximum with a solvent. 

In the Flory-Erman treatment, polymer networks are fully characterized by their "cycle rank", The cycle rank Incorporates the crosslink density and all other structural features of the crosslinked polymer. That is convenient mathematically, but would be synthetically more useful if broken out into conventional structural parameters. For a random network such as is obtained in copolymerizations of vinyl and divinyl monomers, with crosslink functionality of four, the cycle rank density, Vq is [4]... 

The Ronca-Allegra theory (177), and Flory-Erman theory (3,178,182) are both based on the idea that effects of constraints are local and decrease with increasing strain and swelling. The basic difference between the two theories is that in the Ronca-Allegra theory the fluctuations of junctions become exactly affine as the undeformed state is approached, whereas in the Flory theory they are close to but below those of the affine state. 

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