Flory-Erman theory

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 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. 

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]...

P.J. Flory and B. Erman, "Theory of Elasticity of Polymer Networks. 

The above results, which were formulated in the early thirties (TVeloar 1958), explain the main features of the elastic behaviour of polymer networks. Nevertheless, there are notable discrepancies between empirical data and the cited results. Further investigations demonstrated that the values of shear modulus and stress-strain dependence are determined substantially by topological constraints due to the proximity of the chains. The theory was improved by taking into account the discussed issue (Edwards 1967a, 1967b, 1969 Flory 1977 Erman and Flory 1978 Priss 1957, 1980, 1981). More recent developments are summarised in the work of Panyukov and Rabin (1996), where many additional relevant references could be found. 

Erman B, Flory PJ (1978) Theory of elasticity of polymer networks. II The effect of geometric constraints on junctions. J Chem Phys 68 5363—5369 Erukhimovich IYa, Irzhak VI, Rostiashvili VG (1976) On concentration dependence of swelling coefficient of weakly non-Gaussian macromolecules. Polym Sci USSR 18 1682-1689... 

A great number of papers have been published on the subject of RIS since the theory appeared in a fully developed form in 1974. The seminal work is of course by P. J. Flory, and notable contributions have been made by the research groups of A. Abe, B. Erman, J. E. Mark, W. L. Mattice, E. Saiz, and U. W. Suter, among others. These workers have contributed greatly to the further development of the theory and to the making of important applications. For an exhaustive list of references (>500), the reader is referred to a book on RIS by Mattice and Suter mentioned earlier. ... 

Applying the constrained jrmction theory on a phantom network Erman and Flory (1978) obtained ... 

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... 

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