Relaxation timescale

So the criterion that the effective constraint-release must be fast enough to allow local pieces of umelaxed chain to explore any dilated tube fully confirms the assumption of dynamic dilution for nearly the whole range of relaxation timescales exhibited by star polymers. 

The initial problem arises from the fundamental Ball-McLeish [27] equation for the relaxation timescales along the star arms, which we write here from... 

For the kinetics of a reaction, it is critical to know the rough time to reach equilibrium. Often the term "mean reaction time," or "reaction timescale," or "relaxation timescale" is used. These terms all mean the same, the time it takes for the reactant concentration to change from the initial value to 1/e toward the final (equilibrium) value. For unidirectional reactions, half-life is often used to characterize the time to reach the final state, and it means the time for the reactant concentration to decrease to half of the initial value. For some reactions or processes, these times are short, meaning that the equilibrium state is easy to reach. Examples of rapid reactions include H2O + OH (timescale < 67 /is at... 

The mean reaction time or reaction timescale (also called relaxation timescale relaxation denotes the return of a system to equilibrium) is another characteristic time for a reaction. Roughly, the mean reaction time is the time it takes for the concentration to change from the initial value to 1/e toward the final (equilibrium) value. The mean reaction time is often denoted as x (or Xr where subscript "r" stands for reaction). The rigorous definition of x is through the following equation (Scherer, 1986 Zhang, 1994) ... 

The above simple formula is one of the reasons why some authors prefer the use of the mean reaction time (or relaxation timescale) instead of the half-life. The mean reaction time is longer than the half-life. 

Table 2-1 Relaxation timescale and concentration evolution for reversible reactions...

This problem explores the concept of relaxation timescale (x ) for a first-order reaction. It is simplest to use formula in Table 2-1 but you might have to do some conversion. Consider a first-order reaction H2CO3 H2O + CO2 with kf x 5 s and kb 0.043 s at 25°C. Determine x for... 

What general conclusion do you get for the relaxation timescale for first-order reactions ... 

The relaxation timescale of second-order reactions depends strongly on the initial species concentrations. 

Table 1-4 Dissolution mechanism for some substances Table 2-1 Relaxation timescale and concentration evolution for reversible reactions Table 2-2 Decay steps in decay chains...

Linear viscoelastic measurements using infrared dichroism on the compatible blend polyethylene oxide) and poly(methyl methacrylate) were reported by Zawada et al. [139]. Unlike Monnerie and coworkers [127], who reported seeing only orientation in the PMMA component, and none in the PEO, Zawada et al. observed alignment in the PEO. However, since the PEO was of lower molecular weight (as was the case for Monnerie and coworkers), its relaxation timescales were substantially faster than the PMMA. This may explain the lack of any measurable orientation by Monnerie and coworkers, who studied quenched samples, since their preparation may have allowed the PEO to relax prior to testing. 

In order to categorize the obtained Markovian model for the effective dynamics more precisely, let us denote the typical (mean) exit time from Dj to Dk by Tjk, and the typical relaxation timescale within Dj by Tj (that is, when the system enters Dj at t = 0 it has lost almost all of its memory tt = Tj). 

Using a master equation to model the VER process as a multistep reaction, the excess energy flow kinetics in FeP was examined, where the third order Fermi resonance parameters served as approximate reaction rate constants [88]. It was found that the subsequent relaxation is slow relative to relaxation of the initially excited system mode, providing an explanation for the observed difference in relaxation timescales. 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

Perhaps, it was Hynes who initiated two of the most popular so far semi classical non-Markovian approximations [84]. The first approximation was inspired by the success of the [1,0]-Pade approximant, which turns out to be exact in the Markovian limit. This approximation is sometimes referred to as the substitution approximation, because effectively one substitutes non-Markovian two-point distribution function (9.46)-(9.47) into the Markovian expressions (9.50)-(9.51) for the rate kernel. The substitution approximation was shown to work rather well for the case of biexponential relaxation with similar decay times [102]. However, as Bicout and Szabo [142] recently demonstrated, it considerably overestimates the reaction rate when the two relaxation timescales become largely different (see Fig. 9.14). They also showed that for a non-Markovian process with a multiexponential correlation function, which can be mapped onto a multidimensional Markovian process [301], the substitution approximation is equivalent to the well-known Wilemski-Fixman closure approximation [302-304]. A more serious problem arises when we try to deal with the... 

Figure 17.5. Orientational anisotropy decays of the OD stretch mode in die various samples, (a) Bulk water, Wq = 60,40, and 20, shows httle variability in the anisotropies of fliese samples. The decays are single exponential, (b) Bulk water, Wq=10, 5, and 2, anisotropies reveal the progressively longer orientational relaxation timescales in the smaller reverse micelles. The anisotropy decays of the small reverse micelles are biexponential. Figure adapted widi permission from. J. Phys. Chem. A, 110 (2006), 4985 999. Copyright (2006) American Chemical Society.

Due to the tetrahedral arrangement of water its rotational freedom is limited. Also such bound hydrogen and free hydrogen can exchange their positions by the well-known large-amplitude rotational jump mechanism with a time interval of 100 fs [13]. In bulk water the dipole, OH, and HH vector relaxation timescales are 2.05, 2.3, and 2 ps, respectively. But in confined water such relaxation times... 

The relaxation timescale for solutions of (47) is given by the reciprocals of the eigenvalues of Kf, which using the asymptotic relations 6i,l/Tc . 1 take the form... 

In Eq. 6, / and v are the length and velocity scales for the physical problem under consideration. Equation 6 can also be interpreted as the ratio of two timescales, namely, freiax = and fadvection = Uv, where treiax IS the charge relaxation timescale and fadvection is the advection time-scale. With a typical value of / 10 pm and V 200 pm s fadvection IS of the order of 0.1 s, which is several orders of magnitude higher than the typical values of freiax in an aqueous... 

The Rouse predictions are consistent with some of the observations described earlier. Thus, the longest relaxation timescales with t]oJs (tr = (15/7r )(r/o7s°)R),... 

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