Frictional factor molecular weight independence

To the extent that the segmental friction factor f is independent of M, then Eq. (2.56) predicts a first-power dependence of viscosity on the molecular weight of the polymer in agreement with experiment. A more detailed analysis of f shows that segmental motion is easier in the neighborhood of a chain end because the wagging chain end tends to open up the structure of the melt and... 

We can conclude that the ERT has accurately explained the molecular-weight dependence of the zero-shear viscosity and the steady-state compliance and their respective transition points Me and M. This success is indeed the logical consequence of the success of the theory in analyzing the G t) curves of the studied samples, a vitally important aspect of which is the molecular-weight independence of the frictional factor K. Prom the analysis of the G t) curves, it is revealed that entanglements exist between Mg and Me- This point will be further confirmed by the... 

The friction factor depends upon the same features that govern the viscosity of small-molecule liquids. At low temperatures f0 depends on T — T% (Tg < T< Tg+ 4-100° C), and at higher temperatures it depends on an activation energy for flow. The value of 3 for a solution depends on the properties of both components and their concentrations, but it is independent of the large scale structure of the polymer as long as its molecular weight is large (Mn > 104 for most linear polymers). 

It is immediately noticed that the tp s do not depend on the number of subchains chosen [cf. Ferry (96)). In the first place the friction factor f of the single bead must be inversely proportional to the number of beads chosen per unit of chain length, in order to keep the frictional resistance per unit of chain length constant. This means that ffV must be proportional to the molecular weight. In the second place, b2N is equal to the mean square end-to-end distance of the total chain in a solution at rest. Also this value must be proportional to the molecular weight and independent of the number of subchains chosen. This is in agreement with Section 2.6.3. According to eqs. (3.37) and (3.50) one obtains for the contribution of the macro-molecules to the viscosity of the solution ... 

The inherent friction factor fo is presumed constant, independent of molecular weight and temperature (see section 3.2), although it may depend on molecular structure to some extent. Knowledge of the mole-culEir weight dependence of the constants and permits the analysis of j(Z, T) at constant ljoc T— Fj) (i.e., at constant f) as is required to determine the function F Z). 

The diffusion coefficient at high cs decreases with increasing molecular weight. This is similar to neutral polymers where the diffusion coefficient is inversely dependent on the friction factor, which is proportional by the power law to polymer molecular weight, i.e., D =/ 1 = M v. The diffusion coefficient at low cs is, on the other hand, independent of polymer molecular weight. This can be also documented by a more detailed data set on molecular weight standards of NaPSS [13] (Figure 8). 

As shown in Eq. (3.62), the relaxation time ti is the product of the frictional factor K which is independent of molecular weight and a structural... 

That the obtained frictional factor If is, as expected, independent of molecular weight is a vitally important result of the analysis of the G t) curves. The very good description of the line shapes of the G t) curves by the theory would be totally meaningless if the obtained K values were not independent of molecular weight. The significance will be further demonstrated when the comparison of theory with experiment is made for the zero-shear viscosity. 

As shown above, the G t) line shapes of a series of polystyrene samples at different molecular weights above M are well described by convolut-ing Eq. (9.19) with a nearly monodisperse distribution. Furthermore, the frictional factor K obtained from the analysis of the G t) curves measmed at the same temperature is independent of molecular weight. Thus, we expect to obtain good agreement between theory and experiment for the... 

Here, it should be stressed that the theoretical basis for the scheme that enables the Jp t) results to be quantitatively analyzed is ultimately the frictional factor K in the ERT being independent of molecular weight as expected. The ERT having met this crucial criterion for its validity, theoretically there is no limit to the time range of Jp t) that can be analyzed, depending on the molecular weight of the sample under study. 

In eqns [60b] and [60c], we have neglected a power-law-type prefactor ( Alar])7 ) ° because this factor is much more weakly dependent on Alarm compared to the exponential terms shown therein. Ale(C) appearing in eqns [59] and [60] is the entanglement molecular weight specified by eqn [58], andC(C) denotes the friction coefficient of the Rouse segment that changes with C but independent of A1 (unless A1 is very small). 

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