Reduced frequency relaxation spectrum

The molecular weight dependence enters Equation 29 through tiq For this reason It Is convenient to define the reduced frequency relaxation spectrum as ... 

Figure 28. Reduced frequency relaxation spectrum for LLDPE/PP System 1 (top) and System 2.

Figure 8. Reduced Gross frequency relaxation spectrum for Serles-I and II. The arrows Indicate computed coordinates of the maximum (Hg niax>Hiiax) ...

We assume that the above-indicated drawback of the present model can be avoided (or at least reduced) if a new paradigm [mentioned below in Section X.B.4(ii)] of the molecular model will be constructed. In our opinion, this drawback of the present model is stipulated by the following. In view of Eq. (11) the libration lifetime T0r is determined by the experimental Debye relaxation time td, so variation of Tor cannot be used for other corrections of the calculated spectra. In the proposed new paradigm it is desirable to use Tor for the latter purpose, while a correct describing of the low-frequency Debye spectrum is assumed to be reached by variation of additional parameter(s). 

Thus, at low frequencies the Gross susceptibility (280) reduces to the Debye relaxation spectrum. 

Equation 30 gives the experimentally observed proportionality of [i ] to A/ /2 in a 0-solvent. Again the reduced intrinsic moduli as functions of (otj are independent of M. They are plotted in Fig. 9-8-II. As with the Rouse theory, the results are insensitive to N so long as the frequency is not too high. The slope of on the logarithmic plot at higher frequencies corresponds to a relaxation spectrum in which all Tp except the first few are proportional to instead of to as in the free-draining case. 

Figure 5.7 Storage and loss moduli versus reduced frequency for poly(vinyl acetate) with a very narrow MWD as calculated from creep data using the retardation spectrum as an intermediary (logarithmic scales). It was not possible to achieve superposition over the entire range of frequencies, and two shift factors were used to deal with data in high and low-frequency zones.The reference temperature is 60 °C. All the relaxation zones are clearly exhibited. From Plazek [31].

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

Fourier transform NMR spectroscopy, on the other hand, permits rapid scanning of the sample so that the NMR spectrum can be obtained within a few seconds. FT-NMR experiments are performed by subjecting the sample to a very intense, broad-band, Hl pulse that causes all of the examined nuclei to undergo transitions. As the excited nuclei relax to their equilibrium state, their relaxation-decay pattern is recorded. A Fourier transform is performed upon this relaxation-decay pattern to provide the NMR spectra. The relaxation-decay pattern, which is in the time domain, is transformed into the typical NMR spectrum, the frequency domain. The time required to apply the Hl pulse, allow the nuclei to return to equilibrium, and have the computer perform the Fourier transforms on the relaxation-decay pattern often is only a few seconds. Thus, compared to a CW NMR experiment, the time can be reduced by a factor of 1000-fold or more by using the FT-NMR technique. 

Actually, the spectroscopic data would more closely resemble the pattern in Figure 3.15, which is the same as the wave in Figure 3.14, except that the overall intensity of the signal decays exponentially with time. (Note that the decay does not affect the frequencies.) Such a pattern is called the modulated free induction decay (FID) signal (or time-domain spectrum). The decay is the result of spin-spin relaxation (Section 2.3.2), which reduces the net magnetization in the, y plane. The envelope (see Section 3.6.2) of the damped wave is described by an exponential decay function whose decay time is T, the effective spin-spin relaxation time. 

In this method, two pulsed lasers are used, both usually in the nanosecond regime. One (the burn laser) is operated at high power, and is scanned across the absorption spectrum. It excites molecules (or clusters) from the particular vibrational level (usually the i = 0 level) to an electronically excited state. The upper state relaxes (radiatively or otherwise) back to the ground state, but not necessarily to i = 0. Thus, depletion in the population of this species is achieved. A second, low-power laser (the probe laser) is fired after a suitable time delay (to allow complete decay of the emission induced by the pump laser). It is tuned to one of the excitation spectrum vibronic bands of the system, and the fluorescence induced by it (the signal ) is continuously monitored. Whenever the frequency of the bum laser corresponds to excitation of the species giving rise to the absorption of the probe laser, the signal is reduced. This reduction appears as a hole that is burned in the spectrum—hence the name of the method. If a different species is excited (another molecule or a different vibrational level) no change in fluorescence intensity is incurred. 

---