Relaxation spectrum correlation

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by ... 

The impurity generation route is shown in Fig. 5 to the right of the line. The interaction of the relaxed states formed from m ami unexcited impurity t results in exiplex t formation. This exiplex can decay thermally or form coupled ion-radical pairs t, which may dissociate in the electric field. For explanation of the absorption and photoconductivity spectrum correlation it is necessary to assume a very high concentration of exiplex sites. 

For higher modes, the ratio xjxt becomes sensitive to the correlations. As p increases, tp/t, decreases, as shown by Eq. (38). For illustration, this ratio is plotted semilogarithmically in Figure 2 as a function of pjN for a chain with 104 beads and for P = 0, 0.2,0.5, and 0.9. It is seen that in this one-dimensional model the relaxation spectrum is broadened as the energetic preference for extended conformations (P > 0) is increased. In particular, the longest and shortest relaxation times are related by... 

Fig. 3. Effect of bond correlations on the relaxation spectrum of a one-dimensional chain of 10,000 links. The slope d log H/d log-z of the relaxation function is plotted as a function of reduced relaxation time t/t,. Open circles indicate points at which p/N = 0.25 filled circles, pjN = 0.5.

CdO and smaller amounts of CdCU, YF3 and LaFs). The modulus relaxation spectrum has been fitted to stretched exponential function and it has been found that p has a roughly constant value in the glassy state and above Tg, it decreases rapidly. Since decoupling index also decreases rapidly above the glass transition temperature, it is suggestive of an implicit relation between and / . It is also noted that Rr Tg itself is inversely correlated to the corresponding p values and P decreases linearly with log at Tg (also see Hunt, 1994)... 

Fig. 14. Schematic of selective excitation and ID exchange spectroscopy, (a) Typical pulse sequence with a soft selective pulse centered at pulsation a>s with a frequency dispersion AcoP <3C Aoj much smaller than the typical linewidth. After an evolution time te smaller or of the order of the spin-lattice relaxation time, a reading sequence of hard pulses that covers uniformly the whole broad line is applied, (b) Effect of a selective excitation on a homogeneously broaden line, (c) Selective frequency labeling of an inhomogeneously broaden line at the irradiation pulsation cos of the first soft pulse. For a soft n pulse, the magnetizations of all the spins that can exchange energy at this pulsation are reversed. By following the difference spectra between the spectra acquired at different evolution times te and the fully relaxed spectrum AS(te) — S(t -> oo) — S(te), limits or evaluation of the correlation time tc of the motion can be achieved.

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

A complementary use of polymer viscometry is the indirect evaluation of the MWD of a polymer from dynamic viscosity measurements [28-30]. The methods used to correlate the MWD of polymers to rheological data are based on the previous determination of the polymer relaxation spectrum from linear oscillatory shear experiments [31, 32]. MWDs obtained from viscometric data analysis can help in the determination of the MWD curve from online measurements, or in cases where this curve cannot be easily determined from size exclusion chromatography (SEC) [30, 31]. 

Figure 9.15. Typical trajectories of a Gaussian stochastic process x(t) with zero mean and Gaussian (a) or exponential (i>) correlation function. Circles are crossing points of x = 0. Trajectories were generated by regular sampling in the frequency domain, (c) corresponds to the Debye relaxation spectrum with a cutoff frequency. Reorganization energy of the discarded part of the spectrum is 7% of the total. The sampling pattern was the same as in (b).

Fig. 18.18 (a) Correlation between the abscissas of relaxation spectrum maximum and the zero-shear viscosity for PE/PE Series-I (miscible LPX-30/LLDPE-10) and Series-II (immiscible LPX-30/LDPE). (b) Correlation between the ordinate of relaxation spectrum maximum and the polydispersity factor M /M for PE/PE Series-I (miscible LPX-30/LLDPE-10) and Series-II (immiscible LPX-30/LDPE)... 

These values of G and Xi constitute the relaxation spectrum of a polymer, and are useful in correlating the viscoelastic behavior of polymers with their molecular characteristics, such as long-chain branching in polyethylene (Figure 13.33) [79]. 

Initial Crack Generated Isothermally. Slow, controlled crack propagation is possible for most polymers above 160 K. Stress relaxation at the crack tips is determined by the stress-relaxation behavior at the given temperature. This can be verified when measuring Kic in a temperature range including a relaxation peak (17). The temperature dependence of Kic correlates well with that of the relaxation peak when both measurements are based on the same time scale of mechanical deformations Kic is directly related to the relaxation spectrum. 

The relaxation-time spectrum correlates with the spectrum of pore geometrical properties. On this basis, Tong et al. (2006) used induced polarization measurements on shaly sands to derive capillary pressure curves. Transformation of the normalized decay curve (comparable to NMR processing) yields a relaxation-time spectrum. The relaxation-time spectrum reflects the pore-size distribution, and also contains information about pore-body and pore-throat cmitributions. Figure 8.45 shows an example. 

In order to obtain information about molecular dynamics from a dielectric relaxation spectrum, the complex dielectric permittivity is related to the correlation function of the electric dipole moment m, of the ith species and the dipole moment Mj of a small (in comparison with the whole sample) macroscopic volume V surrounding m,. Mj is the sum of permanent dipole moments in this volume Mi N being the number of dipole moments in the volume. The... 

At present, the time correlation can be taken over a wide range of timescales. Photon correlation spectroscopy has heen established to measure the relaxation spectrum of fluctuations in complex fluids. 

In the case where x and y are the same, C (r) is called an autocorrelation function, if they are different, it is called a cross-correlation function. For an autocorrelation function, the initial value at t = to is 1, and it approaches 0 as t oo. How fast it approaches 0 is measured by the relaxation time. The Fourier transforms of such correlation functions are often related to experimentally observed spectra, the far infrared spectrum of a solvent, for example, is the Foiuier transform of the dipole autocorrelation function. ... 

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