Nonexponential relaxation kinetics

Ye, X., lonascu, D., Gruia, R, Yu, A., Benabbas, A., and Champion, P. M. 2007. Temperature-dependent heme kinetics with nonexponential binding and barrier relaxation in the absence of protein conformational substates. Proc. Nat. Acad. Sci. USA 104 14682-87. 

The stochastic description of barrierless relaxations by Bagchi, Fleming, and Oxtoby (Ref. 195 and Section IV.I) was first applied by these authors to TPM dyes to explain the observed nonexponential fluorescence decay and ground-state repopulation kinetics. The experimental evidence of an activation energy obs < Ev is also in accordance with a barrierless relaxation model. The data presented in Table IV are indicative of nonexponential decay, too. They were obtained by fitting the experiment to a biexponential model, but it can be shown50 that a fit of similar quality can be obtained with the error-function model of barrierless relaxations. Thus, r, and t2 are related to r° and t", but, at present, we can only... 

The aimealing kinetics of the light-induced defects are shown in Fig. 6.29. Several hours at 130 °C are needed to anneal the defects completely, but only a few minutes at 200 C. The relaxation is nonexponential, and in the initial measurements of the decay the results were analyzed in terms of a distribution of time constants, Eq. (6.78) (Stutzmann, Jackson and Tsai 1986). The distribution is centered close to 1 eV with a width of about 0.2 eV. Subsequently it was found that the decay fits a stretched exponential, as is shown in Fig. 6.29. The parameters of the decay-the dispersion, p, and the temperature dependence of the decay time, t - are similar to those found for the thermal relaxation data and so are consistent with the same mechanism of hydrogen diffusion. The data are included in Fig. 6.23 which describes the general relation between x and D,. The annealing is therefore the process of relaxation to the equilibrium state with a low defect density. 

Here x and x are column vectors of the components x (t) and x (t), respectively, and A(t) is now a physical reaction matrix containing time-dependent elements. Fluorescence of kinetic transients, e. g., the relaxation profiles of monomer- or excimer fluorescence are, therefore, strictly nonexponential for which closed form, analytical solutions can be found in few cases, only. A convincing manifestation of nonexponential trapping in low-temperature, solid state p-N-VCz is a recent analysis by Bassler et al. (4, ). With the use of rate function in Equation 2, the transient ps-rise profile of the low-energy excimer E2 has been satisfactorily fitted to the numerical solution of Equation 3 with a single-fit dispersion parameter a between 0.2 and 0.8 depending on the temperature of the system. 

The definition (9.1) of the ET rate constant assumes that the kinetics is exponential. In general, of course, it is not, particularly at low temperatures. Possible sources of nonexponential behavior are complex spectral density and, as a result, complex relaxation dynamics of the participating bath modes [45, 47], fluctuating tunneling matrix element [59-61], time-dependent external field modulating the energy gap [66, 67], and nonequilibrium... 

Chemical kinetics is all about bottlenecks. They determine the reaction mechanism on different timescales. We have seen that ET can be hmited by nonadiabatic transitions, solvent dynamics, intramolecular vibrational relaxation, translational diffusion, and conformational fluctuations either of the reactants themselves or of the embedding medium. The interplay between different mechanisms as well as nonequilibrium initial condition may result in rich kinetic behaviors, with strong nonexponentiality and coherence effects observed in recent experiments. 

Figure 9(a) shows the kinetics for the ak I and the methylene proton peak of (MA-St) in the coil form at af=0.87 and 35°C. Under the assumption of single exponential decay function expressed by Equation (1), ln(l - Mz/Mq) has to be proportional to x. For the phenyl protons, the proportional relationship is apparently well satisfied in a range of x less than 2 sec, and ln(l - Mz/Mq) at x= 0 is very close to the expected value (= In 2), but the kinetics for the methylene protons in styrene is not expressed by Equation (1) in the range of x. Figure 9(b) shows the kinetics for the compact form copolymer at = 0.25. For the peak I, the proportional relationship between ln(l - Mz/Mq) and x is clearly seen in a range of x less than 3 sec, and for the backbone protons the proportionality is apparently found, but not strictly. As described later, the recovery curve with a correlation time is in principle nonexponential because of coupled relaxation, and therefore we made use as before of concept of an effective relaxation time T from the initial perturbation to the time when the deviation from equilibrium falls to about 25 30% of its initial value. A dotted line in Figure 9(a) shows an example of the... 

For the individual interested in molecular motion, the important feature of spin-lattice relaxation (or other relaxation mechanisms) is the dependency on molecular motion to provide an efficient energy pathway for relaxation. Thus, molecular motions at the Larmor frequency for individual carbon atoms in a molecular framework may be mapped by T] measurements. Since the frequency of molecular motion is temperature dependent, additional thermodynamic and kinetic information may be obtained by measuring Tj values for different carbons over a range of temperatures. In the paper by Lyerla and coworkers in this volume, Tj measurements made for the first time over a range of low temperatures yielded specific information about motion in the backbone and side chains of a semi-crystalline and a glassy pol3nner. The data was taken at a Larmor frequency of 15.1 MHz for nuclei. It was also noted in this paper that the Tj values measured for the glassy polymer showed nonexponential behavior. As stated in the paper, this represented a distribution of Tj values due to the many different environments, and therefore the many different Tj mechanisms, present in the polymer. 

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