Relaxation multiple

In 2012, Liao and coworkers [52] reported a Dy6 octahedron (24, Figure 3.17), seen as two edge-sharing [(p4-0)Dy4] tetrahedra, with two tail-to-tail calixarene ligands. The ac susceptibilities show obvious temperature-dependent both in-phase (x ) and out-of-phase (x") signals under a zero dc field, indicating multiple relaxation SMM behaviour. 

The reaction is coupled to an indicator color change which is much faster than the metal substitution step. L refers to the multidentate ligands which were employed to avoid multiple relaxations r is the relaxation time. 

These two inequalities can be proved for a more general model in which the film is assumed to consist of multiple parallel bands, each band containing an arbitrary fraction of amorphous phase which exhibits multiple relaxation processes. 

Rich et al. (79) have carried out a temperature jump relaxation study of E. coli asparaginase and reported that the finding of multiple relaxation times is qualitatively consistent with the suggested subunit structure. 

Recent ultrasonic measurements on S02 by Bass, Winter, and Evans [258] confirms the earlier observations of multiple relaxation [259,260] and extends the temperature range to 1090°K. As T increases, the relaxation time for the bending mode increases with increasing temperature, while that for the stretching modes decreases in the normal manner. Shields and Anderson [261], using an improved absorption apparatus, have studied pure S02 and its mixtures with argon. Under careful inspection, the results are shown to... 

Therefore, using multiple relaxation times. Equation 13 yields ... 

In the correlations the data reported for pure solvents were used. Solvents of low dielectric permittivity which contain salt may exhibit multiple relaxations due to ion-pair formation [199]. The above information relates to aprotic solvents which do not form hydrogen bonds and have one relaxation time. 

The simplest special case is obtained when all / are equal, giving r = / and 3/. Then (pa - pi) decays at rate/and (pa + pO at 3/ Both decays appear in the polarization if the site separation d a is not equal to daa. More elaborate cases of this sort of multiple relaxation, on rings as well as rows of sites, have been analysed by Hoffman and Axilrod, and other workers. ... 

Pure rotary diffusion of rigid dipoles in two or three dimensions, then, gives exponential decay of polarization with a single relaxation time, provided the sites are uniformly distributed and D is constant. The description of the motion in terms of D alone breaks down, as we shall see, for very short times. A three-dimensional rigid body in any case executes a more complex motion. Even an internally uniform model of rectilinear charge-carrier difiurion automatically shows multiple relaxation. More realistic models must take account of the dynamic s of molecular motion. 

Figure 2. Plot of observed frequency factors versus those calculated according to Hynes Model. (298K) Open circles indicate the calculated values taking account of the multiple relaxation of alcohols.

The plot shows a distribution closely around a slope of unity indicated by the solid line in Figure 2 except for the alcohols and nitrobenzene. Such anomaly in alcohols is also reported for other chemical processes and time-dependent fluorescence stokes shifts and is attributed to their non-Debye multiple relaxation behavior " the shorter relaxation components, which are assigned to local motions such as the OH group reorientation, contribute the friction for the barrier crossing rather than the slower main relaxation component, which corresponds to the longitudinal dielectric relaxation time, tl, when one regards the solvent as a Debye dielectric medium. If one takes account of the multiple relaxation of the alcohols, the theoretical ket (or v,i) values inaease and approach to the trend of the other solvents. (See open circles in Figure 2.)... 

However, Eq. (3-24) can be extended to allow multiple relaxation modes simply by setting mtf — t equal to a sum of exponentials (Lodge 1956.1968 Yamamoto 1956.1957. 

EXTENSIONAL FLOW. In steady extensional flows, such as uniaxial extension, the single-relaxation-time Hookean dumbbell model and the multiple-relaxation-time Rouse and Zimm models predict that the steady-state extensional viscosity becomes infinite at a finite strain rate, s. With the dumbbell model, this occurs when the frictional drag force that stretches the dumbbell exceeds the contraction-producing force of the spring—that is, when the extension rate equals the critical value Sc. ... 

It is advisable to review very briefly the preceeding work done on the viscoelastic properties of polymers subject to multiple relaxation mechanisms, as well as previous solution studies of phosphates in the presence of lanthanum on the basis of these studies La" was chosen as the ion most likely to lead to bond interchange. 

A. The Phenomenology of Multiple M dianlsm Rehixation and Separation of the Mechanisms AH of the primary experimental data were obtained by the technique of stress relaxation. Since the primary curves (before shifting) show no unusual features, they will not be reproduced here. The characteristics of multiple relaxation do become evident, however, when an attempt is made to construct a master curve (54), It is observed that while in the glassy and transition regions the behavior is completely normal, in the flow region superposition is impossible. This manifests itself in a number... 

An alternative formulation is that of Fixman, in which diffusion theory is modified to account for multiple relaxation times. This is pursued in... 

The relaxation time r is a fundamental dynamic property of all viscoelastic liquids. Polymer liquids have multiple relaxation modes, each with its own relaxation time. Any stress relaxation modulus can be described by a series combination of Maxwell elements. 

This relaxation time for the Maxwell model is an average relaxation time [see Eq. 7.136] whenever a material has multiple relaxation modes. 

It is often possible to describe the dielectric data for complex liquids in terms of multiple relaxation processes, each of which follows a Debye relaxation based on equation (4.5.2). The resulting equations for and Sout are... 

Other relationships which have been used to describe dielectric relaxation data include the Cole-Cole and Cole-Davidson equations [29]. These are preferred when a distribution of relaxation times rather than a single relaxation time is more appropriate to describe the data in a given frequency range. Nevertheless, the Debye model in its simple version or multiple relaxation versions works quite well for most of the solvents considered here. 

According to the Debye model there are three parameters associated with dielectric relaxation in a simple solvent, namely, the static permittivity s, the Debye relaxation time td, and the high-frequency permittivity Eoq. The static permittivity has already been discussed in detail in sections 4.3 and 4.4. In this section attention is especially focused on the Debye relaxation time td and the related quantity, the longitudinal relaxation time Tl. The significance of these parameters for solvents with multiple relaxation processes is considered. The high-frequency permittivity and its relationship to the optical permittivity Eop is also discussed. 

All protic solvents undergo multiple relaxation processes due to the presence of hydrogen bonding. In the case of water and formamide (F), the data can be described in terms of two Debye relaxations. For the alcohols and A-methyl-formamide (NMF), three Debye relaxations are required for the description. In all of these solvents, the low-frequency process involves the cooperative motion of hydrogen-bonded clusters. In the case of water and the alcohols the high-frequency process involves the formation and breaking of hydrogen bonds. The intermediate process in the alcohols is ascribed to rotational diffusion of monomers. Studies of dielectric relaxation in these systems have been carried out for the -alkyl alcohols up to dodecanol [8]. Values of the relaxation parameters for water and the lower alcohols are summarized in table 4.5. 

It is clear from these results that there is an approximate correlation between the value of the average relaxation time Xs and the longitudinal relaxation time Xl (see section 4.7). However, careful examination of the time-dependent Stokes shift (TDSS) data reveals that C(t) is described by more than one relaxation process. This is not difficult to understand, considering that many aprotic solvents are found to have more than one relaxation process. This can be attributed to the formation of dimers and other aggregates in these liquids. This is especially the case in aprotic solvents such as DMSO and PC, which have very high dipole moments. Alcohols also have multiple relaxation processes due to the presence... 

The Rouse model (Rouse, 1953) extends these theories to multiple beads and springs (or multiple-relaxation modes). Here the expression for the viscosity becomes... 

It is less certain that the combination of instrumental noise at such low concentrations and the existence of multiple relaxation mechanisms will allow the determination of collision frequencies accurate enough to allow a critical test of the Cooper-Mann model and its generalization to delocalized monolayers. We are exploring this possibility. 

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