Relaxation time defined

The special investigation has shown that, in particular, for xo = 0 the relaxation time defined by (5.118) is true for /X > 0.5. At the same time it would be reasonable to take the first term in (5.118)—that is, 0 = A /3D—as an upper bound for the relaxation time in the rectangular well to avoid the difficulties with the possible nonmonotonic behavior of the W( , t) regardless of the values xo and . 

The rheology of lamellar phases has attracted considerable attention. For a quenched lamellar phase it has been observed that where G = G" both scale as to112 for to < wc, where determined operationally as being approximately equal to 0.1 r1, where r is a single-chain relaxation time defined as the frequency where G and G" cross (Bates et al. 1987 Rosedale and Bates 1990). Similar dynamic moduli scaling was found with PS-PI-PS and PS-PB-PS triblocks (here... 

For a quenched lamellar phase it has been observed that G = G"scales as a>m for tv < tvQ. where tvc is defined operationally as being approximately equal to 0.1t and r is a single-chain relaxation time defined as the frequency where G and G" cross (Bates et al. 1990 Rosedale and Bates 1990). This behaviour has been accounted for theoretically by Kawasaki and Onuki (1990). For a PEP-PEE diblock that was presheared to create two distinct orientations (see Fig. 2.7(c)), Koppi et al. (1992) observed a substantial difference in G for quenched samples and parallel and perpendicular lamellae. In particular, G[ and the viscosity rjj for a perpendicular lamellar phase sheared in the plane of the lamellae were observed to exhibit near-terminal behaviour (G tv2, tj a/), which is consistent with the behaviour of an oriented lamellar phase which flows in two dimensions. These results highlight the fact that the linear viscoelastic behaviour of the lamellar phase is sensitive to the state of sample orientation. 

The structural relaxation time defines the activation enthalpy Ah for small deviation from equilibrium... 

For high temperatures ( a. c 1), the effective relaxation time defined by Eq. (4.270) reduces to... 

Movaghar et al., 1986) is recovered. Plotting the relaxation times defined by the intersection of the logarithmic branch with the asymptote ttr (inset in Fig. 4) yields the empirical relationship (Pautmeier et al., 1989)... 

This analysis is, however, a little bit partial. We would like to notice that a model like that of the system of Eq. (1.7) can also be interpreted as a model of classical activation. The relaxation time defined by Eq. (5.5) would then be interpreted as a sort of activation time, that is, the time required for a starting point condition extremely favorable for the escape from a well. An experimenter aiming at activating a chemical reaction process would be greatly disappointed if the activation time were infinite ... 

This relaxation time defines implicitly an upper bound for Nw- In fact, a free energy profile of depth F is filled by the Nw walkers in a time that is... 

Here t is the relaxation time (defined as the time needed for the stress to relax to l/e 0.37 of its initial value). In most materials the stress relaxation follows a different course, since there may be a number (a distribution) of relaxation times. Nevertheless, the relaxation time, even if it merely concerns an order of magnitude, is a useful parameter. 

There is one point important to note here, the experimental data plotted as y( - 1) must cross the ordinate at a value identical to the surface tension of the surfactant-free system, i.e. the surface tension of water for a water/air interface. This is often not the case, in particular for drop volume or maximum bubble pressure experiments where due to the peculiarities of the measurement an initial surfactant load of the interface exists. It has been demonstrated in the book by Joos [16] that even in these cases, assumed it is the initial period of the adsorption time, the slope of the plot y( /t) yields the diffusion relaxation time defined by Eq. (4.26) and hence information about the diffusion coefficient. For small deviation from equilibrium we have the relationship... 

The structural- (or a-) relaxation time ts may be somewhat arbitrarily defined by the time when the ratio between the contribution of the glassy-relaxation process (G) to the relaxation modulus G(t) and the total contributed by all the entropic processes (R), G/R reaches 3. Physically, this means that G/R has decayed nearly by factor of e (2.72) from 10, which is the G/R ratio at the relaxation time t = t ) of the highest Rouse-Mooney mode when the temperature is at the glass transition point. At the same time, the contribution from the glassy component in such a state is still significant. The structmal-relaxation time defined this way is basically equivalent physically to that defined by ... 

The structural-relaxation time as given by Eq. (14.10) is also in close agreement with the a-relaxation time defined in a usual way the time at which the relaxation modulus reaches 10 dynes/cm (see Figs. 14.17 and 14.18). The structural-relaxation time defined by Eq. (14.10), besides reflecting the effect of the glassy relaxation on the bulk mechanical properties, has the virtue of following exactly the temperature dependence of the... 

While K is a frictional factor, s having the unit Da is a structural factor. Thus the AT dependence of s (Fig. 14.14) and that of K (Fig. 14.15) are of different physical nature. With decreasing AT, the former represents the growth of a T -related (dynamic) structure while the latter represents purely the frictional slowdown of the Rouse segment. The structural relaxation (pc(t)) with relaxation time defined by Eq. (14.11) contains the effects of both the frictional slowdown and structural growth while the fiAit) or Pij(t) process is only affected by the frictional slowdown. As a result, the... 

In the physical relation (62.12) for viscoelastic material, there exists the relaxation time defined as t = rj/M, where // is termed the viscosity. The viscosity depends also on MC, as shown in Figure 62.12. 

The fluorophore was modeled by two beads that are attached as a short pendant side-chain (tag). Both the absorption and emission dipole moments of the fluorophore are defined by the direction of the tag (parallel), as indicated by the vector in Fig. 19, and the fluorescence anisotropy was calculated from its orientation autocorrelation function. For simplicity, we assumed that the reorientaional motion of the fluorophore is the only source of fluorescence depolarization. We neglected energy transfer and other processes that might occur in real systems. The fluorescence anisotropy decays were interpreted using the mean relaxation time, defined as ... 

The typical decay behavior of the dipole correlation function of the microemulsion in the percolation region is presented at Fig. 24. Figure 25 shows the temperature dependence of the effective relaxation time, defined within the fractal parameters, and corresponding to the macroscopic relaxation time Tjjj of the KWW model. In the percolation threshold T, the exhibits a maximum and reflects the well-known critical slowing down effect (131). 

Conductive-system dispersive response may be associated with a distribution of relaxation times (DRT) at the complex resistivity level, as in the work of Moynihan, Boesch, and Laberge [1973] based on the assumption of stretched-exponential response in the time domain (Eq. (118), Section 2.1.2.7), work that led to the widely used original modulus formalism (OMF) for data fitting and analysis, hi contrast, dielectric dispersive response may be characterized by a distribution of dielectric relaxation times defined at the complex dielectric constant or permittivity level (Macdonald [1995]). Its history, summarized in the monograph of Bbttcher and Bordewijk [1978], began more than a hundred years ago. Until relatively recently, however, these two types of dispersive response were not usually distinguished, and conductive-system dispersive response was often analyzed as if it were of dielectric character, even when this was not the case. In this section, material parameters will be expressed in specific form appropriate to the level concerned. 

In the Stokes number relationship x is the particle relaxation time defined as... 

The analysis of the relaxation equations for the population of different levels shows [59] that it is impossible to formulate a simple relaxation equation for (E) similar to Eq. (15.11). Moreover, the effective relaxation time defined by... 

For polyacrylamide there are two rheological effects which can be explained in terms of its random coil structure. Firstly, it was discussed above that polyacrylamide is much more sensitive than xanthan to solution salinity and hardness. This is explained by the fact that the salinity causes the molecular chain to collapse, which results in a much smaller molecule and hence in a lower viscosity solution. The second effect which can be explained in terms of the polyacrylamide random coil structure is the viscoelastic behaviour of this polymer. This is shown both in the dynamic oscillatory measurements and in the flow through the stepped capillaries (Chauveteau, 1981). When simple models of random chains are constructed, such as the Rouse model (Rouse, 1953 Bird et al, 1987), the internal structure of these bead and spring models gives rise to a spectrum of relaxation times, Analysis of this situation shows that these relaxation times define response times for the molecule, as indicated in the simple Maxwell model for a viscoelastic fluid discussed above. Thus, because of the internal structure of a flexible coil molecule, one would expect to observe some viscoelastic behaviour. This phenomenon is discussed in much more detail by Bird et al (1987b), in which a range of possible molecular models are discussed and the significance of these to the constitutive relationship between stress and deformation rate and deformation history is elaborated. 

This mean relaxation time defines an usual cooperative diffusion coefficient, which increases with the concentration. This description is only useful for times shorter than T ic q). 

The monomeric friction coefficient Cq is one of the most important properties of macromolecules, which can be calculated, using Eq. (4.94), from the measurement of Figure 4.6 gives plots of log G versus log Tjreduced variables G and G"j, respectively, are defined by G j. = G MIcRT and G"j. = G" -cor )MlcRT, where c is the concentration of polymer solution and Tj is the terminal relaxation time defined... 

---