Entangled system viscosity

Unlike Williams, Chikahisa attempts to deal directly with entangling systems. The result is a viscosity expression which is not far from that observed experimentally, but the form is unfortunately more dependent on a series of intermediate assumptions about the nature of the friction forces than on the basic transport theory itself. Although not implausible, the assumptions are nevertheless arbitrary and lacking in theoretical justification. 

For the weakly entangled systems, one can expect, that the ratio E/B, that is the parameter of internal viscosity is small. It can be demonstrated in Section 4.2.3, that transition point from weakly to strongly entangled systems occurs at E B. To describe these facts, one can use any convenient approximate function for the measure of internal resistance, for example, the simple formula... 

The law for coefficient of viscosity, which was unambiguously established by Fox and Flory (1948) for polystyrene and polyisibutilene and confirmed for many polymer system investigated later (Berry and Fox 1968, Ferry 1980), determines the first critical point Mc 2Me separating entangled and non-entangled systems of linear polymers... 

While the law with index 3.4 for viscosity is valid in the whole region above Mc, the dependence of terminal relaxation time is different for weakly and strongly entangled systems (Ferry 1980) and determines the second critical point M ... 

A more quantitative interpretation of the scattering data involves molecular models. As an example, we turned to the Rouse model [20] developed for solutions and extended to melts [21]. As this model is unable to account for the molecular weight dependence of zero-shear viscosity (t o M ) above the critical molecular weight (Mc 35 000 for PS), the analysis wall be extended as a next step to other models which are more realistic for entangled systems. A basic result of the Rouse model relates the monomeric friction coefiBcient Co and the zero-shear viscosity t o ... 

Dynamic properties of a polymer system reflect the process in which the polymer chains perturbed by an external force recover their equilibrium conformations by Brownian motion. Hence it is the key for their theoretical formulation to clarify what mode of chain motions occurs in a given system. However, as mentioned in the preceding sections, there is no established consensus about this mode for entangled systems. This fact explains why the 3.4 power law for viscosity still remains a mystery. 

These features show striking similarities between ring and linear polystyrenes with respect to the molecular weight dependence of melt viscosity. Of especial interest is the virtual agreement of r/(r) and i] ) in entangled systems, i.e., in... 

While Eq. (2.33) is valid for monodispersedathermal polymers, in reality polymers are polydispersed and do interact with each other. Consequently, application of the model to a real system requires that the influence of individual molecular weight fractions on the rheological functions is taken into account. For example, in the simplest case of the zero-shear viscosity-composition dependence for entangled systems [194, 195] the prediction is ... 

In this form the expression for the stresses is valid for any dynamics of the chain. For the macromolecules in an entangled system, the elastic and internal viscosity forces according to equations (26) and (40) - (42) have the form... 

I hc volume of the tube should be the free volume of an individual chain. According to Eq. (30), the viscosity of such an entanglement system, r, can be expressed as ... 

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