Molecular entanglement networks

Two deformation schemes are commonly used to account for molecular orientation. These are the so-called affine and pseudoaffine schemes whose description, along with their applicability to real polymers, it given in a series papers [5, 6]. However, it turns out that the behaviour of the real polymers (amorphous as well as the semicrystalline ones) differs essentially from these schemes, entailing mmierous modifications [5, 6]. The main principle of all these modified and unmodified deformation schemes is the presence of a molecular entanglement network [7, 8]. 

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 mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

Chompff and Duiser (232) analyzed the viscoelastic properties of an entanglement network somewhat similar to that envisioned by Parry et al. Theirs is the only molecular theory which predicts a spectrum for the plateau as well as the transition and terminal regions. Earlier Duiser and Staverman (233) had examined a system of four identical Rouse chains, each fixed in space at one end and joined together at the other. They showed that the relaxation times of this system are the same as if two of the chains were fixed in space at both ends and the remaining two were joined to form a single chain with fixed ends of twice the original size. 

The mass fractions of these three phases are shown in Table 14. the crystalline fraction is relatively small as 0.43 or 0.44. This low level of crystallinity may arise from relatively strong molecular entanglement due to the network structure. 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

Prior molecular orientation (either above or just below T ) should change the starting extension ratios of the entanglement network and therefore X of any crazes produced. Consider a polymer melt extended in uniaxial extension to X = J p, X2 =Xi — The new. ax in a direction parallel to the molecular orientation... 

As molecular weight is reduced below some critical value, M,., stable craze formation in PS is no longer possible and. in PMMA, both brittle strength and fracture energy fall essentially to zero. The value of M varies with composition It is usually taken as 2M and thus has a value of about 38,000 for PS and about 27,500 for PMMA . Molecules with M less than M do not contribute to strength and their presence dilutes the entanglement network and weakens the polymer . 

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