Deformation of entanglements

Deformation of Entangled Polymer Molecules Under Shear... 

Y. Y. Wang and S.-Q. Wang. Exploring stress overshoot phenomenon upon startup deformation of entangled linear polymeric liquids. J. Rheol, 53 (2009), 1389-1401. 

Inoue, T., Uematsu, T., Yamashita, Y., Osaki, K. Significance of the longest Rouse relaxation time in the stress relaxation process at large deformation of entangled polymer solutions. Macromol (2002) 35, pp. 4718-4724... 

Such considerations appear to be very relevant to the deformation of polymethylmethacrylate (PMMA) in the glassy state. At first sight, the development of P200 with draw ratio appears to follow the pseudo-affine deformation scheme rather than the rubber network model. It is, however, not possible to reconcile this conclusion with the temperature dependence of the behaviour where the development of orientation reduces in absolute magnitude with increasing temperature of deformation. It was proposed by Raha and Bowden 25) that an alternative deformation scheme, which fits the data well, is to assume that the deformation is akin to a rubber network, where the number of cross-links systematically reduces as the draw ratio is increased. It is assumed that the reduction in the number of cross-links per unit volume N i.e. molecular entanglements is proportional to the degree of deformation. 

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 deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

It became clear in the early development of the tube model that it provided a means of calculating the response of entangled polymers to large deformations as well as small ones [2]. Some predictions, especially in steady shear flow, lead to strange anomaUes as we shall see, but others met with surprising success. In particular the same step-strain experiment used to determine G(t) directly in shear is straightforward to extend to large shear strains y. In many cases of such experiments on polymer melts both Hnear and branched, monodisperse and polydisperse,the experimental strain-dependent relaxation function G(t,Y) may be written... 

In the two classic viscometric deformations of simple shear and extension, the appropriate components of Q have very different behaviour. For small shear strains, the shear stress depends on the component Q which has the linear asymptotic form 47/15. This prefactor is the origin of tne constant v in the tube potential of Sect. 3.For large strains, however, Qxy 7 and therefore predicts strong shear-thinning. Physically this comes from the entanglement loss on re-... 

Reasons have been advanced for both an increase and a decrease of the tube diameter with strain. A justification of the former view might be the retraction process itself [38]. If it acts in a similar way to the dynamic dilution and the effective concentration of entanglement network follows the retraction then Cgjy < E.u > so that a < E.u On the other hand one might guess that at large strains the tube deforms at constant tube volume La. The tube length must increase as < E.u >,so from this effect a < E.u > . Indeed, Marrucci has recently proposed that both these effects exist and remain unnoticed in step strain because they cancel [69] Of course this is far from idle speculation because there is another situation in which such effects would have important consequences. This is in conditions of continuous deformation, to which we now turn. 

As is obvious from the above discussion, a very detailed understanding of entangling in elastomeric networks is required for interpretation of the elastic modulus, in particular its dependence on deformation and swelling. 

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