Diffusional relaxation time

Let us emphasize that as a result of scaling t with the characteristic hydrostatic time to 260 sec long as compared with that of diffusional relaxation time L2/D 10 sec, D emerges in the system (6.3.9)-(6.3.15) as a large parameter. 

On a RDE, in the absence of a surface layer, the EHD impedance is a function of a single dimensionless frequency, pSc1/3. This means that if the viscosity of the medium directly above the surface of the electrode and the diffusion coefficient of the species of interest are independent of position away from the electrode, then the EHD impedance measured at different rotation frequencies reduces to a common curve when plotted as a function of p. In other words, there is a characteristic dimensionless diffusional relaxation time for the system, pD, strictly (pSc1/3)D, which is independent of the disc rotation frequency. However, if v or D vary with position (for example, as a consequence of the formation of a viscous boundary layer or the presence of a surface film), then, except under particular circumstances described below, reduction of the measured parameters to a common curve is not possible. Under these conditions pD is dependent upon the disc rotation frequency. The variation of the EHD impedance with as a function of p is therefore the diagnostic for... 

The presence of the film decreases the amplitude of the response to the flow perturbation. If the film develops over time, then the film thickness can be calculated, knowing SD from the characteristics of the flow. Furthermore, the relaxation time for the composite system is the sum of the two diffusional relaxation times ... 

Fig. 4K is discussed later. Suffice it to note here that the relaxation time for the activation-controlled process is inversely proportional to the exchange current density while x, the diffusional relaxation time, is independent of i. Thus, a large value of x /x indicates that the electrode reaction is slow, and vice versa. 

It is convenient to define also a diffusional relaxation time X which... 

The rate of diffusion depends primarily on the product Since the diffusion coefficient in simple lutions does not usually vary by more than an order of magnitude, e reach the rather obvious conclusion that the diffusional relaxation time depends primarily on the concentrations of the reactant and the product. If the product... 

The gating period ( 1 ns) is much shorter than the diffusional relaxation time of the enzyme-substrate system [t = R /D (402)/100 = 16 ns, where... 

The experimental data following the evaporation to dryness (Days 68, 69) showed a continuation of excess power that gradually decreased, as would be expected for a process controlled by a diffusional relaxation time for deuterium inside the palladium. The cooling of this cell was also slower than expected, and there was at least one period (Day 69) during which the cell contents reheated with no apphed electrochemical or heater power [31, 33]. Illustrations of these effects are shown in Ref. [31] (Figures A22, A23, and A24). 

On the other hand, a step decrease in feed hydrogen resulted in a relatively very rapid and monotonic decline to the final steady-state ethylene concentration. It should be noted that the sum of all hydraulic and mixing lags for this system is of the order of 75 s and the diffusional relaxation time (R /Dg) is much smaller than one second. Hence, the extremely slow response observed in the step-up experiment and its asymmetry compared to the step-down result suggest that non-linear dynamics of the gas phase-catalyst surface interaction play a major role in unsteady reactor behavior. 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

The concept of a T2 cut-off that partitions the relaxation time distribution between the pores which can be displaced and those that cannot does not always apply. An exception is when there is significant diffusional coupling between the micropores that retain water at a high capillary pressure and the macropores in close proximity to the microporous system [26, 27]. A spectral BVI model or a forward model has been suggested to interpret these systems [30, 31, 53]. 

The haphazard rotational motions of molecules or one or more segments of a molecule. This diffusional process strongly influences the mutual orientation of molecules (particularly large ones) as they encounter each other and proceed to form complexes. Rotational diffusion can be characterized by one or more relaxation times, t, describing the motion of a molecule or segment of volume, V, in a medium of viscosity, 17, as shown in the following equation ... 

By assuming an Arrhenius type temperature relation for both the diffusional jumps and r, we can use the asymptotic behavior of /(to) and T, as a function of temperature to determine the activation energy of motion (an example is given in the next section). We furthermore note that the interpretation of an NMR experiment in terms of diffusional motion requires the assumption of a defined microscopic model of atomic motion (migration) in order to obtain the correct relationships between the ensemble average of the molecular motion of the nuclear magnetic dipoles and both the spectral density and the spin-lattice relaxation time Tt. There are other relaxation times, such as the spin-spin relaxation time T2, which describes the... 

The minimum in the spin-lattice relaxation time is more difficult to account for. It cannot be attributed to the onset of the diffusional motion, because the jump frequency does not match the Larmor frequency at the temperature where diffusion becomes important. For this reason it is necessary to postulate an additional kind of motion in the lithium-vanadium bronze—a side-to-side jumping from one side of the channel to the other. In the structure there are sites on both sides of the channel roughly 2 A. apart which are equivalent but only one of which is occupied to fulfill stoichiometry. This kind of motion should start at a lower temperature than the above diffusion and lead to a correlation frequency that matches the Larmor frequency at the spin-lattice time minimum. Because of modulation of quadrupolar interaction, side-to-side motion could provide an effective spin-lattice relaxation mechanism. 

Kinetic phenomena can also be used to delimit the timescales of magmatic processes and, unlike radiometric ages, do not require absolute constraints on the timing of eruption. Two important kinetic controls on crystal properties are crystal growth and diffusional relaxation of compositional heterogeneities in minerals. Rates of crystal settling are not described here but have also been used to delimit crystal storage times (e.g., Anderson et al., 2000 Resmini and Marsh, 1995). 

Another criterion for predicting if the transport in polymeric gels is controlled by diffusion (Fickian) or by relaxation, is to determine the diffusional Deborah number De), which is a ratio between the characteristic polymer relaxation time of the polymer (2) when it is subject to a swelling stress and a characteristic diffusion time (6), defined as the coefficient between the square of the sample thickness (h) and the coefficient of water diffusion in the polymeric gel... 

The application of relaxation time measurements to study segmental motion (in polymers) as well as diffusional chain motion is very well documented but is still a subject of study, particularly using the frequency dependence of relaxation times to test the detailed predictions of models (McBriety and Packer 1993). The anisotropy of reorientation can also be studied conveniently, and recent interest in motion of molecules on surfaces (e.g. water on porous silica) has been investigated with great sueeess (Gladden 1993). Since the dipolar interaction is usually both intermolecular and intramolecular, the relaxation of spin- /2 nuclei (e.g. H) in the same molecule as a quadrupolar nucleus (e.g. H) can permit a complete study of reorientation and translation at a microscopic level (Schmidt-Rohr and Spiess 1994). 

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