Weight-average relaxation time

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. 

Weight-average relaxation time of the terminal viscoelastic region, 7oJe°. 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

For these first experiments, a temperature relatively close to Tg, T=123°C, was chosen with the intention of minimizing the relaxation of stress and chain orientation during the quenching the weight-average relaxation time of sample SI at 123°C is calculated from that at 140°C and the thermal shift factor between 123°C and 140°C Xw(123°C) 380s. On the other hand the cooling time of the stretched specimens can be estimated to a few seconds [19], which is very small compared to the polymer relaxation time at the temperature of the experiments. 

If the Rouse time is determined from the experimental value of the zero-shear viscosity, one finds X.i 615s for sample SI at 123°C, which is not too far from the experimental value of the weight-average relaxation time (X. 380s). Clearly, the determination of and Xi from the experimental value of qo implies the assumption of monomeric friction enhancement by entanglements [20], since for sample SI M is of the order of 3xMc. 

The product r o is the characteristic relaxation time xq of the terminal region. In terms of molecular models, this time scales as the longest relaxation time. In terms of the distribution of relaxation times H(x), Xo is the "weight-average relaxation time" which is the average relaxation time related to the second order moment of the relaxation spectrum ... 

In Eq. (2) the A, values are the specific relaxation times and the G values are the corresponding weights. The knowledge of the set (G, A,) is very useful because it allows one to predict the behaviour of the material in any standard experiment. From this set of (G, A,) values the zero-shear viscosity, r o, the plateau modulus, G, the zero-shear first normal viscosity, P 2 and the weight-average relaxation time. A, can be obtained by the following equations ... 

The value of the intercept is also a measure of the width of the distribution of relaxation times, which is the most sensitive to the distribution of long times. It is convenient to derive the weight-average relaxation time... 

For diluted chains, a Cole-Cole plot of the complex viscosity (Fig. 19) exhibits a relaxation domain well-separated from the matrix allowing one to measure the same average relaxation times as above. However, the weight average time x, has to be corrected by the matrix contribution. Watanabe [20] cast it into the form ... 

Figure 29 Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptation and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]...

For M M c, if there were really a single terminal relaxation time as implied by equation 14, combination of that equation with equation 34 of Chapter 3 would make the steady-state compliance the same as the plateau compliance 7 = 7/v = /G%. with J% given by equation 2. Actually, as shown by Graessley, the ratio Je/J% is the ratio of what may be termed the weight- and number-average relaxation times in the terminal zone ... 

Janzen foimd that he could relate the value of a to the molecular weight distribution. He also found that the characteristic time of this model was equal to an average relaxation time defined in terms of the log-Gaussian (log-normal) relaxation spectrum. 

Figure 9 Treating internal dynamics during the refinement process. Due to dynamics and the weighting of the NOE, the measured distance may appear much shorter than the average distance. This can be accounted for by using ensemble refinement techniques. In contrast to standard refinement, an average distance is calculated over an ensemble of C structures (ensemble refinement) or a trajectory (time-averaged refinement). The time-averaged distance is defined with an exponential window over the trajectory. T is the total length over the trajectory, t is the time, and x is a relaxation time characterizing the width of the exponential window.

Fig. 1.24 Two examples of frequency-depen-dent relaxation times - 7"i is plotted as a function of the proton resonance frequency V = ou/2 JI, which was obtained from measurements at different magnetic fields strengths. Left polyisoprene (PI) melts and solutions of the same samples at 19wt-% concentration in cyclohexane. Numbers indicate the average molecular weight. The difference between the melt and solution increases towards lower magnetic fields strengths, the frequency dependence is more pronounced for melts.

Alternatively, the much more common situation in trNOE studies involves fast exchange on the chemical shift time scale where the observed resonance shifts are weighted averages of the corresponding shift in the free and bound state [13]. A full account of the complete relaxation matrix and conformational exchange effects for n spins has been performed by London et al. [13], and a similar treatment was later incorporated into the programme CORCEMA [14]. 

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