Relaxation decay curves

When the spin-diffusion rate is comparable to the values, the relaxation decay curves become non-single exponential, and when the spin-diffusion rate is less than 10% of the values, the relaxation decay curve can be described by a single exponential with an intrinsic relaxation rate. 

Figure II. Proton spin-spin relaxation decay curve of Estane hydrolyzed at 53%RH and the correlation of aging time with molecular weight and inverse...

In all relaxation measurement experiments, spectra are recorded as a function of t, a time during which relaxation is allowed to occur. The variation of the intensity of a particular signal in the spectrum as a function of t then constitutes the relaxation decay curve, which we must analyze to retrieve motional data. As this curve will generally have an exponential form (or at least approximately), in order to define this curve properly, it is desirable to arrange to spread the t points of measurement non-linearly and to concentrate them near the t — 0 end of the curve where the signal intensity varies most rapidly. A logarithmic distribution of t points is usually optimal. It is essential that the spin system is fully relaxed before acquiring the next scan or spectrum and also that the t data measurement is continued until the complete decay curve is measured, otherwise errors will inevitably result in its analysis. 

Note that we assume fast spin diffusion within each phase in the phase-separated blend (Model D in Fig. 10.18), i.e., each phase is homogeneously mixed from the spin-diffusion point of view. In other words, one Tip or Ti value is associated with one phase. As shown in Section 10.3.1, such heterogeneity of a blend manifests itself as multiexponential relaxation decay curves. For Model D, we expect a double-exponential decay for the respective spins of polymers A and B. For spin-locking Tip experiment, the two double-exponential decay curves are given for the two component polymers as... 

Another very fast and effective method for measuring rubber processing properties is to perform a stress relaxation test. A stress relaxation decay curve can quickly quantify the viscoelastic properties of both raw rubbers and mixed stocks. The Maxwell model, shown in Fig. 35. illustrates this principle with a spring and dashpot in series [125]. A sudden... 

The RPA can also do a stress relaxation test, Figure 37 shows RPA stress relaxation decay curves from three truck tread stocks of the same formulation. Each curve indicates different qualities of mix caused by variations in mixing work history (variations in total energy at Banbury dump). As the energy at dump for these tread mixes increases, the peak torque decreases, the slope of the log-log plot of the stress relaxation curve becomes steeper, the regression line intercept decreases, the integrated area under the curve decreases, and the time to a given % drop decreases [133]. Therefore a simple stress relaxation test is a fast and effective w ay to measure variations in viscoelastic properties for both raw rubber and mixed stocks. 

Figure 37 Comparison of RPA stress relaxation decay curves for three truck tread stocks with different qualities of mix caused by indicated variations in mixing work history (energy at Banbury dump).

Troyer et al. (2005) studied differently hydrated CPG at the pore diameter of 23.7 nm using analysis of water proton spin-spin relaxation decay curves modeled as two-component exponential decays as a function of hydration. The results were consistent with a geometric model involving a... 

A rotational viscometer connected to a recorder is used. After the sample is loaded and allowed to come to mechanical and thermal equiUbtium, the viscometer is turned on and the rotational speed is increased in steps, starting from the lowest speed. The resultant shear stress is recorded with time. On each speed change the shear stress reaches a maximum value and then decreases exponentially toward an equiUbrium level. The peak shear stress, which is obtained by extrapolating the curve to zero time, and the equiUbrium shear stress are indicative of the viscosity—shear behavior of unsheared and sheared material, respectively. The stress-decay curves are indicative of the time-dependent behavior. A rate constant for the relaxation process can be deterrnined at each shear rate. In addition, zero-time and equiUbrium shear stress values can be used to constmct a hysteresis loop that is similar to that shown in Figure 5, but unlike that plot, is independent of acceleration and time of shear. 

Starting from the assumption that the geometry relaxation after excitation is of primary importance with respect to the luminescence response, we decided to employ a solid polymer matrix to suppress conformational changes of the oligomers. For the measurements, dilute blends with polysulfone as the transparent host matrix were prepared. In Figure 16-13, the PL decay curves for the two cyano compounds in both chloroform and polysulfone are presented, as are the PL spectra of Ooct-OPV5-CN in chloroform and polysulfone [69J. 

Fig. 19. Experimental spin alignment decay curves of chain deuterated PS-d3 at temperatures above and below the glass transition for various evolution times t,. Note the different timescales of t2 at the different temperatures. The straight lines indicate the decays of the plateau values on the timescale of the spin-lattice relaxation time T,. Sample characterization Mw = 141000, Mw/Mn = 1.13, atactic...

Figure 2.7. The fluorescence spectra from unrelaxed (/ = 0) and relaxed ( - oo) states and the emission decay curves at the short-wavelength edge (a), the maximum (b), and the long-wavelength edge (c) of the spectrum.

In the case of hi- or multi-exponential relaxation curves the treatment involved can be rather complex (119-123). It becomes even more problematic. Needles to say, the same is true for systems with suspected continuous distributions of relaxation rates, whose evaluation by numerical analysis of the decay curves (124-128) represents one of the most arduous mathematical problems (124-128). In general, evaluation tasks of this kind need to be treated off-line, using specific programs and algorithms. 

A single exponential curve (cf. "Equation U") has been fitted to the birefringence decay curve after removal of the electric field the agreement is good for Cq = 0.5 at 25°C (Fig.l) and moderately good for Cq = 1.6 at 60°C (Fig.2b).The fits lead to the relaxation times Tj = 1/6 Dj reported in Table I. As to the other cases, the agreement is very poor (Fig.2a) a two exponential curve is then necessary to describe the decay curves behaviour.The two-time constants are reported in Table I. 

The values of R are reported in Table I. When the decay curve of birefringence is characterized by two relaxation processes, a radius is attributed to each time constant. 

The introduction of reversible photoassociation at higher fluor (or quencher) concentrations invalidates the analysis of fluorescence decay curves of either emitting species in terms of a single decay constant since these species do not relax independently accordingly the decay curves each represent the sum of two exponential components and the corresponding decay constants Ax and A2 are related to the photoassociation rate constants kDM and kMD and to the lifetimes rp and t of both emitting species. The treatment given below follows that of Birks, Dyson, and Munro.1... 

A decay curve depicting this type of relaxation for a typical value of R is shown in Fig. 7(a), and its semilog plot is shown in Fig. 1(b). The fluorescent lifetime is defined as the time T for the population to decrease to 1 je of its initial value. This number is easily obtained from the semilog plot. 

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