Relaxation time treatments

This treatment illustrates several important aspects of relaxation kinetics. One of these is that the method is applicable to equilibrium systems. Another is that we can always generate a first-order relaxation process by adopting the linearization approximation. This condition usually requires that the perturbation be small (in the sense that higher-order terms be negligible relative to the first-order term). The relaxation time is a function of rate constants and, often, concentrations. 

We have next to consider the measurement of the relaxation times. Each t is the reciprocal of an apparent first-order rate constant, so the problem is identical with problems considered in Chapters 2 and 3. If the system possesses a single relaxation time, a semilogarithmic first-order plot suffices to estimate t. The analytical response is often solution absorbance, or an electrical signal proportional to absorbance or to another physical property. As shown in Section 2.3 (Treatment of Instrument Response Data), the appropriate plotting function is In (A, - Aa=), where A, is the... 

If two complexes coexist, there will be two relaxation times, and a treatment analogous to the analyses of Schemes III and IV is required. Table 4-2 gives a few rate constants for these reactions.The mechanism of such reactions is believed to consist of at least two steps, shown in simplified form in Scheme IX. 

A more rigorous free-volume treatment is due to Cohen and Grest (CG) [34,35], according to which the material is comprised of liquid and solid-like cells. The former have free volume, but mobility requires continuity of the local empty space. The temperature dependence of the relaxation times according to the CG model is... 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

The Cr5+ ion has only one unpaired electron hence, no zero-field splitting is expected. Indeed, a well-resolved spectrum has been observed for the ion on alumina 147, 148), silica gel 149-151), silica-alumina 152-154), and magnesium oxide 155). The line may be resolved into parallel and perpendicular g values. As van Reijen and Cossee (151) have shown, the values of g range from 1.970 to 1.975 whereas, the values of g range from 1.898 to 2.002, depending on the treatment of a Cr/SiCh sample. These authors have suggested that Cr5+ in two different symmetries is present one has a long relaxation time and can be observed at room temperature, but the other has a very short relaxation time and can be observed only at very low temperatures (—253°). 

To study dipole-dipole relaxation, one must distinguish between homonuclear and heteronuclear (unlike) spin-1 pairs. The latter gives rise to the so-called 3/2 effect.29 For an isolated pair of like spin-i nuclei (/= 1) separated by an intemuclear distance r, the treatment of spin relaxation is identical to that for a spin-1 quadrupole system. The Zeeman spin-lattice relaxation time T1Z and spin-spin relaxation time T2 are given, respectively, by... 

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

The build-up of the laser pulse from the statistical fluctuation is beautifully demonstrated in 31), where a short period of statistical fluctuations was registered with a streak camera while they were traveling back and forth in the resonator. A detailed theoretical treatment of this process is given in 32> for the case of solid-state lasers and in 33> for the case of dye lasers in which the saturation of the active medium plays an important role. The halfwidth of the pulse which is finally reached in this mode-locking process is theoretically determined by the inverse spectral bandwidth of the active medium provided that the dye relaxation time rp is sufficiently short. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

This treatment will not be discussed in further detail, as a knowledge of distinct relaxation times is not required for the present purpose. Interest is focussed on the relation between reduced first normal stress difference and reduced shear stress, as expressed by eq. (3.41). The simplest way to evaluate this equation for a polydisperse polymer has been given by Peterlin (76). This procedure has extensively been used by Daum (32, 73) in his experimental investigations. 

An Evaluation of the Debye-Onsager Model. The simplest treatment for solvation dynamics is the Debye-Onsager model which we reviewed in Section II.A. It assumes that the solvent (i) is well modeled as a uniform dielectric continuum and (ii) has a single relaxation time (i.e., the solvent is a Debye solvent ) td (Eq. (18)). The model predicts that C(t) should be a single... 

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