Characteristic time particle

The classical bath sees the quantum particle potential as averaged over the characteristic time, which - if we recall that in conventional units it equals hjk T- vanishes in the classical limit h- Q. The quasienergy partition function for the classical bath now simply turns into an ordinary integral in configuration space. 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

A possible approach to interpretation of a low-frequency region of the G ( ) dependence of filled polymers is to compare it with a specific relaxation mechanism, which appears due to the presence of a filler in the melt. We have already spoken about two possible mechanisms — the first, associated with adsorption phenomena on a filler s surface and the second, determined by the possibility of rotational diffusion of anisodiametrical particles with characteristic time D 1. But even if these effects are not taken into account, the presence of a filler can be related with the appearance of a new characteristic time, Xf, common for any systems. It is expressed in the following way... 

The characteristic times on which catalytic events occur vary more or less in parallel with the different length scales discussed above. The activation and breaking of a chemical bond inside a molecule occurs in the picosecond regime, completion of an entire reaction cycle from complexation between catalyst and reactants through separation from the product may take anywhere between microseconds for the fastest enzymatic reactions to minutes for complicated reactions on surfaces. On the mesoscopic level, diffusion in and outside pores, and through shaped catalyst particles may take between seconds and minutes, and the residence times of molecules inside entire reactors may be from seconds to, effectively, infinity if the reactants end up in unwanted byproducts such as coke, which stay on the catalyst. 

In case when the concentration of adsorption particles increases t > -j4co ) formulas (1.108) and (1.109) become valid only for times t < tee, where tee is the characteristic time determined from equation > which is the condition of leveling-off of criti-... 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

A number of other spectroscopies provide information that is related to molecular structure, such as coordination symmetry, electronic splitting, and/or the nature and number of chemical functional groups in the species. This information can be used to develop models for the molecular structure of the system under study, and ultimately to determine the forces acting on the atoms in a molecule for any arbitrary displacement of the nuclei. According to the energy of the particles used for excitation (photons, electrons, neutrons, etc.), different parts of a molecule will interact, and different structural information will be obtained. Depending on the relaxation process, each method has a characteristic time scale over which the structural information is averaged. Especially for NMR, the relaxation rate may often be slower than the rate constant of a reaction under study. 

TABLE 5.7 Characteristic Times Associated with the Uptake of Gases into Liquid Particles and Their Reaction in the Bulk Phase"... 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

In a typical case, only two large clusters of the particles of A and B type survive in the system [88, 89]. The steady-state concentration takes a value corresponding to the time needed for the creation of a cluster of such a size. This characteristic time is given by the relation Sd/2td/2 (a M/2 where M is the full volume (number of sites) of the fractal considered. According to equation (7.3.7), at large times one gets... 

Temperatures of vitrification, Tv, and melting, Tm, for some compounds the diffusion coefficients D and the characteristic times rD at 77 K of the shift at distances of the order of the atomic size for particles with radius Ru - 3 A... 

Research on the temperature dependence of charge recombination reaction (1) has shown the P700 decay time to depend on temperature at 150 K < T < 270 K. Below 150 K only 10-30% of the total number of P1 particles accumulated under the action of light disappear after the light is switched off, the characteristic time of their decay being independent of temperature. 

From the results of the investigations discussed in this and in previous chapters it follows that tunneling recombination reactions proceed relatively slowly. Thus, at the typical values of the parameters ve = 10ir s 1 and ae = lA in the expression for the probability of electron tunneling, the characteristic tunneling times to distances of R = 10-20 A amount to 10 6-102s. These times are far longer than the characteristic time of diffusional approach of particles separated by the same distances in water solutions ... 

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