Relaxation strength

Plain Carbon and Low Alloy Steels. For the purposes herein plain carbon and low alloy steels include those containing up to 10% chromium and 1.5% molybdenum, plus small amounts of other alloying elements. These steels are generally cheaper and easier to fabricate than the more highly alloyed steels, and are the most widely used class of alloys within their serviceable temperature range. Figure 7 shows relaxation strengths of these steels and some nickel-base alloys at elevated temperatures (34). 

Fig. 7. Comparison of relaxation strengths at 1000 h as measured by residual stress of various steels and nickel alloys (34). To convert MPa to psi, multiply...

It is apparent that there are a considerable number of parameters to be determined. According to equation (8) and equations (2-7) there are 6N+2 parameters where N is the number of relaxations present (it is not 8N because the relaxed modulus of one process is equal to the unrelaxed modulus of the next process in a sequence). In practice, it is found that with the large number of experimental points available in a scan (typically 50-100) the determinaton usually proceeds satisfactorily. However, in coitimon with many statistical fitting situations, it can happen that parameter determination is not unique. Our experience has shown that problems can arise when the relaxation strength is small or when only part of a peak is recorded. The problem with small relaxaton strength is associated with equation (1) where it is seen that the activation energy is related to the ratio of peak area and relaxation strength E(j- Ep. When the process is quite... 

Figure 8. Relaxation strength versus crystallinity in isotactic polypropylenes of Figure 7. Unrelaxed low temperature modulus Q)> relaxed y modulus (A), relaxed 0 modulus (0), relaxed a modulus ( >). Filled symbols are for the isothermally crystallized (68%) specimen. 

The relaxation strengths are the difference between the instantaneous value and equilibrium value,... 

Fig. 14. Estimation of relaxation strength from temperature dispersion of piezoelectric constant...

Dielectric relaxation measurements for the adsorbed water have been reported by Jansen (44) the dielectric relaxation time is essentially 3r where r is the rotational jump time of the water molecule. From Figure 5 it can be seen that the dielectric and NMR mobility estimates agree rather well. All is not quite in order, however, for Jensen estimates from relaxation strength that he sees only one-third of the water molecules. 

If we define a normalized distribution of relaxation strengths H(r) in the obvious way, the relaxation function can be calculated by taking the Fourier transform of Eq.(9) to... 

If the relaxation function were a single exponential decay, then only two parameters would be necessary to characterize the results a relaxation strength determined by the fraction of the total scattered light associated with the slowly relaxing fluctuations observed with PCS obtained from the intercept, and a relaxation time t. With the... 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

The site model appears to provide an adequate description of basic features of the dynamics of molecular motions in secondary transitions. On the other hand, however, attempts to calculate and interpret the extent of energy losses, i.e. the relaxation strength, in terms of the barrier model have so far been less successful. Theories employing a model with the same depth of potential energy minima1,76,77 do not seem suitable... 

This equation implies that the relaxation strength of a molecular motion passes through a maximum at AH° = 2.4 RT and that for AH° = 0 no relaxation process can exist. An equation for the loss factor tan 6 derived from this theory was attempted81 for the... 

The y Relaxation. In common with many other polymers (8) both PPO and PS display significant loss maxima below room temperature at the frequencies under consideration. Whereas the process responsible for the a loss is at least qualitatively understood in terms of a main chain relaxation associated with the glass transition, y losses can often only tentatively be attributed to specific mechanisms. In PPO, for example, it does not seem unreasonable to propose that the y loss is associated with librations in the two pendant methyl groups this view is somewhat reinforced by the observation that in the dielectric measurements the relaxational strengths of the y and a loss processes are comparable. As the latter can be well interpreted (6) in terms of a dipolar relaxation of the main chain in which the entire dipolar contributions arise from the methyl groupings, it seems plausible to assume that the same dipoles are responsible for the y loss mechanism. In polystyrene there is a similar... 

Because the y loss maxima of the constituent polymers are so close in temperature, we would not expect to find any change in this parameter in the mixtures, and indeed none was observed. The maxima in pure PPO are somewhat stronger than in PS at 1 kHz, the c"max are 2.5 X 10 3 and 9 X 10"4, respectively. Consequently, it is not unexpected also to find a uniform change in the c"max values for the mixtures roughly proportional to the respective compositions. More information, however, can be obtained by considering the relaxational strengths themselves—i.e., of the areas under the loss peaks. The values of eR — were obtained from the following equation (10) ... 

Fig. 71 Relaxation strength of the /J transition as a function of AP concentration (from [17])...

As usual, the relaxation strength of the subglass processes increases as the frequency decreases. The relaxation strength of the y relaxation in P4THPMA is close to the corresponding to PCHMA [30] and significantly lower than that of the corresponding to PDMA [38,104],... 

The dielectric relaxation strength of the polymers follows the same trends as the mean-square dipole moments per repeating unit. Thus P26DFBM and P25DFBM which have the higher and lower values of (/x2)/x, 5.00 and 1.92 D2 at 25°C, respectively. 

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