Multiexponential relaxation

UV (360 nm) and blue (450 nm) light irradiations of the ultrathin azo-silane SAMs clearly induce the forth, i.e., trans—>cis, and back, i.e., cis—>trans, photoisomerization of azobenzene molecules (see Figure 4.2A). The real-time dependence of the absorbance of the sample during the thermal cis—>trans back reaction is not a monoexponential decay (see Figure 4.2B). This decay shows a complex multiexponential relaxation behavior that could be fit neither by a monoexponential decay nor by a biexponential relaxation. Nevertheless, a monoexponential decay could be fit to the data acquired over... 

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... 

In Time Resolved Fluorescence (TRF) experiments, depicted schematically in Figure 12, the emission spectrum line shape changes from a peak value of AE(0) to the final AE(oo) peak of the equilibrium emission spectrum as the solvent responds to the new solute electronic state." " " Experimental results in bulk liquids show that the nonequilibrium correlation functions initially exhibit a very fast (less than 50 fs" ) inertial component, which may account for 60-80% of the total relaxation in water. This is followed by a multiexponential relaxation on the subpicosecond to picosecond timescale," corresponding to reorientation and translation of solvent molecules, or, to particular intramolecular solvent modes" " around the solute. Slower dynamics are found in more viscous liquids." ... 

Porous rocks generally contain a variety of pore sizes and different relaxation mechanisms. The measured signal represents the superposition of aU decaying signals as a spectrum. The decay function must be formulated as a multiexponential relaxation decay function ... 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

In the second region, relatively far from the particle, the refocusing pulses are effective, which corresponds to a weak magnetization condition. The relaxation rate can then be analyzed following Majumdar and Gore 26). The result of this calculation is in remarkable agreement with the slow rates, obtained from simulated multiexponential decays 27) ... 

Relaxation to steady state of the network is multiexponential, and now we are interested in estimate of the longest relaxation time r ... 

Models for solvation in water that allow for a structured solvent do indeed predict a multiexponential response. For instance, the dynamical mean spherical approximation (MSA) for water solvation predicts that solvation of an ion in water is well represented by two characteristic times [38]. Nonetheless, the specific relaxation times differ substantially from the observed behavior [33],... 

In the case of T measurements we have mentioned that cross relaxation provides multiexponential magnetization recovery (Sections 1.7.4 and 7.2.2). A far less known analogy may occur in the linewidths, as already discussed (Section 8.8) when two protons are dipole-dipole coupled and cross correlation occurs between Curie relaxation and proton-proton dipolar relaxation. In this case, we are in the presence of two overlapping signal components with different linewidths, i.e. of biexponentiality in T2 [35], Pulse sequences are available to remove the effects of cross correlation [36]. Such effects are common in paramagnetic metalloproteins where Curie relaxation is usually relevant (in principle, such cross correlation effects can be operative also in the case of 7i, although only to the extent that Curie relaxation on T is effective). 

If several nuclei could be observed in high-resolution NMR techniques to monitor similarities or differences in both chemical shifts or integrals, other parameters can be monitored by using LF 1H NMR. In this case, relaxation parameters are usually measured as intrinsic discriminating values. As pointed out in several studies, T2 relaxation decay has a multiexponential decay in both muscles and fish tissues. This suggests the presence of different "pools" in tissues and water distribution was assumed to be present in three distinct compartments, namely (a) "bound water," (b) "entrapped water," and (c) "free water." In those three pools, water acts with different relaxation times because it can be bound to proteins, involved in the conversion of muscle to meat and entrapped by weak surface forces, showing relaxation values in the range of 1-10,10-100, and 100-400 ms, respectively. 

The dilatational rheology of the poly(vinylacetate) monolayer onto an aqueous subphase has been studied between 1°C and 25°C by Monroy et al. [59], These authors have used the combination of several techniques. By this way, the exploration of a broad frequency range was possible. The relaxation experiments have shown multiexponential decay curves, whose complexity increases with decreasing the temperature. A regularization technique has been used to obtain the relaxation spectra from the relaxation curves and the dilatational viscoelastic parameters have been calculated from the spectra. The shapes of the relaxation spectra agree with the predictions of the theoretical model proposed by Noskov [100],... 

After encoding of the we observe two regimes of relaxation. an initial, very rapid decay, occurring just after stopping of the seeding process, and the second, much slower, multiexponential decay, comparable to the decay observed in polymers poled by the corona poling method. Orientation losses vary from 10% to 20% after 15 hours in the dark at ambient temperature. 

The photoinduced susceptibility shown in Equation 11.14a is the sum of two terms one with exp(-2Dt) (relaxation of the first-order parameter A ) decay and the second with exp(-12D ) (relaxation of the third-order pammeter A3) decay. Hence, the first very rapid decay may contain the fast exp(-12D ) contribution. However, as can be seen from Figure 11.14, the relative magnitude of this initial very fast decay does not depend on the optimization of the intensity ratio between the writing beams. So, this first rapid decay may not be due to the decay of the third-order parameter A3. In addition, because the hyperpolarizability P of DRl is different in the ds and in the trans state, the first very rapid decay also contains a contribution connected with the hferime of the metastable ds form, which is due to molecules coming back to the trans form without any net orientation. A better model would have to account for a distribution of diffusion constants for molecules embedded with various free volumes, which may explain the multiexponential behavior of the decay. 

Magnetic multipoles of rank higher than one become active in spin systems with I > 5 and their contribution to relaxation depends on dynamics. The appearance of multipole terms complicates the relaxation description and supports the multiexponential behavior of relaxation. Nosel et al. presented the effects of high rank multipoles on lineshape and longitudinal relaxation of 7=3 systems. Results obtained from both numerical simulation and experimental data show that longitudinal and transverse relaxation are strongly influenced by these multipole terms, especially at lower temperatures where, due to molecular mobility, the extreme narrowing condition is not fulfilled. 

Pal et al., 2002). To understand the different solvation timescales, we have fitted the decay curves to multiexponentials. Four different solvation timescales are identified, from ultrafast to slow components. An ultrafast component with a time constant of 40-50 fs, followed by a fast component at 0.7-1.2 ps was observed. Two slower components with time constants in the range of 6-17 and 42-88 ps were also noticed. Such different solvation timescales arise from the presence of different types of water molecules within the hydration layer (Bandyopadhyay et al., 2005). The initial ultrafast relaxation arises from the high frequency librational (hindered rotation) and intermolecular vibrational (hindered translation) motions of the "free" or bulk-like water molecules. The moderately damped rotational motions of these water molecules contribute to the fast relaxation ( 1 ps). The slowest component observed (42-88 ps) arises from those water molecules which... 

The well-known continuum models and also the microscopic theories of solvation dynamics suggest a close relation between solvation dynamics and DR. This is expressed as tl = (soo/so)td where tl is the longitudinal relaxation time and td is the Debye relaxation time. However, the solvation dynamics of an ion at the protein surface is difficult to understand because of the heterogeneous environment of the protein surface. Therefore, a straightforward application of the continuum model with a multiexponential description of DR is not possible. The continuum theory suggests that at short length scales, the relaxation time is essentially given by the DR time. Therefore, we certainly expect a slow component in the solvation dynamics. 

From a phenomenological point of view, it is natural to Interpret the multiexponential curves in Equations I, II, and III on the basis of energy cascading models. Such schemes assume - parallel to a single monomer and excimer state - additional electronic dwell-stations to be involved in serial energy relaxation processes. In a quantitative treatment, one has to diagonalize, then, rate equations of the form... 

In this work the main aspect has been concerned with the problem of electronic energy relaxation in polychro-mophoric ensembles of aromatic horaopolymers in dilute, fluid solution of a "good" solvent. In this morphological situation microscopic EET and trapping along the contour of an expanded and mobile coil must be expected to induce rather complex rate processes, as they proceed in typically low-dimensional, nonuniform, and finite-size disordered matter. A macroscopic transport observable, i.e., excimer fluorescence, must be interpreted, therefore, as an ensemble and configurational average over a convolute of individual disordered dynamical systems in a series of sequential relaxation steps. As a consequence, transient fluorescence profiles should exhibit a more complicated behavior, as it can be modelled, on the other hand, on the basis of linear rate equations and multiexponential reconvolution analysis. 

For partially miscible and immiscible blends, various domain/phase structures can be invoked. Unfortunately, the resonance position of a particular spin in each domain is not appreciably affected by its respective domain structure (see Section 10.2.2.1). Therefore, we cannot expect to observe highly resolved NMR resonances for different domains. Since relaxation times are different for each domain, a relaxation curve is observed to be a featureless multiexponential one and, in most of the cases, is too monotonous to include interdomain spin diffusion. Therefore, most of the experimental results have been explained by using the simplified picture of no interdomain spin diffusion and the observed multiexponential decay is fitted to a sum of exponential functions. Practically three exponentials are enough to realize the observed decay. Each relaxation time represents one domain, thus, only a few domains can be distinguished by one resonance line. Inevitably, the heterogeneous structures deduced from NMR relaxation experiments become simple. 

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