Relaxation time terminal

It was assumed that the quantity B is a function of Mo, but, luckily, one does not need in expression for explicit dependence to obtain the final results (6.74) 

The first line is valid for the case when matrix is a weakly entangled matrix, the second line - a strongly entangled matrix. 

Watanabe (1999, p. 1354) has deducted that, according to experimental data for polystyrene/polystyrene blends, when the matrix is a weakly entangled system, terminal time of relaxation depends on the lengths of macromolecules as 

The comparing formulae (6.78) and (6.79) allows one to estimate the dependence of coefficient of enhancement on the lengths of macromolecules as 

The empirical result (6.80) does not correspond to the reliable results for monodisperse (Mo = M) system well. Indeed, taking result (6.80) into account, the terminal relaxation time (6.58) can be written as 

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Some information concerning the intramolecular relaxation of the hyperbranched polymers can be obtained from an analysis of the viscoelastic characteristics within the range between the segmental and the terminal relaxation times. In contrast to the behavior of melts with linear chains, in the case of hyperbranched polymers, the range between the distinguished local and terminal relaxations can be characterized by the values of G and G" changing nearly in parallel and by the viscosity variation having a frequency with a considerably different exponent 0. This can be considered as an indication of the extremely broad spectrum of internal relaxations in these macromolecules. To illustrate this effect, the frequency dependences of the complex viscosities for both linear... 

Tfr is transition zone relaxation time, rte is terminal relaxation time, is reptation time... 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

The mean times t and tw will be called the number-average and weight-average relaxation times of the terminal region, and tw/t can be regarded as a measure of the breadth of the terminal relaxation time distribution. It should be emphasized that these relationships are merely consequences of linear viscoelastic behavior and depend in no way on assumptions about molecular behavior. The observed relationships between properties such as rj0, J°, and G and molecular parameters provides the primary evidence for judging molecular theories of the long relaxation times in concentrated systems. 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

The expansion determines the terminal quantities the viscosity coefficient 7] and the elasticity coefficient u which, in their turn, determine the terminal relaxation time and steady-state compliance, correspondingly,... 

Now, we can try to relate the above results to the experimental data on the viscoelasticity of concentrated solutions of polymers. For the systems of long macromolecules, the estimated values of parameter are small. Having used expressions (6.40) for this case, one can evaluate the terminal relaxation time of the system... 

In the alternative case of large values of one can use the upper line of equation (6.40) to calculate the terminal relaxation time of the system, which coincides with the given relaxation time in order of magnitude... 

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 ... 

The difference in the molecular-weight dependence of the terminal relaxation time can be attributed to the change of the mechanisms (diffusive and repta-tion, correspondingly) of conformational relaxation in these systems. Further on in this section, we shall calculate dynamic modulus and discuss characteristic quantities both for weakly and strongly entangled systems. 

To provide the validity of empirical dependencies of viscosity and terminal relaxation time on the molecular length (relations (6.43) and (6.44)), the sum... 

The results of Van der Vegt suggest that for polydisperse polymer melts the terminal relaxation time might be predicted by... 

In principle, extrudate swell or die swell is dependent on the terminal relaxation time and on the time of residence in a capillary. The shorter the time of residence in the capillary or the longer the relaxation time the higher the die swell. This leads to (see, e.g. Te Nijenhuis, General References, 2007, Chap. 9.4)... 

One convenient strategy to interpret these results is to review the molecular characteristics of binary blends as extracted from polymer melt rheology [40]. The influence of short chains (M < Me) is to effectively decrease the plateau modulus and the terminal relaxation times as compared to the pure polymer. Consequently, the molecular weight between entanglements... 

The viscosity, plateau modulus, limiting compliance and maximum (terminal) relaxation time derived firom the basic D-E model are power laws of the molecular weight M ... 

In a large range of concentrations (< )S0.05) where the high component is assumed to behave as an entangled melt, the variations of the terminal relaxation time (Oni in the iso-free volume state (Fig. 29) confirms relation (6-8). As the steady-state compUance J l scales as ( )" (Fig. 30), the zero-shear viscosity tiol varies as expected and the plateau modulus Gn which reveals the entanglement network is proportional to 0 . ... 

The value of the relaxation modulus at this terminal relaxation time is of order kT per characteristic polymer. The hyperscaling ideas discussed in Section 6.5.3 require that the characteristic polymers are just at their... 

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

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