Flory-Erman junction fluctuation

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]...

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... 

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] ... 

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. 

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). 

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. 

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. 

---