Relaxation time, dependence

It turns out that a rather simple description of this nonlinear relaxation in terms of a single relaxation time,, depending on the final average chain length Loo, is suggested by a scaling plot of L t) for different L o, as shown in Fig. 18 for an initial exponential MWD. It is evident from Fig. 18 that the response curves, L o — L t), for different L o may be collapsed onto a single master curve, 1 - L t)lLoo = /(V Loo) measured in units of a... 

Note that h is proportional to n1/2 in 0-solvents, and thus to N112. For 0 = 0 the flow disturbance is zero, the chain is said to be free draining, and the original Rouse model is recovered. For hP, flow in the coil interior is presumed to be substantially reduced, the chain is frequently said to behave as an impenetrable coil, and the Zimm model is obtained. Equations (4.10-4.12) continue to apply for all values of h, although the distribution of relaxation times depends on h. Some results for the two limiting cases and large N are ... 

The expressions in brackets are the expansivities above and below Tg. The constant K3 is a function of bond type in chains and is really constant for every class of polymers. The physical interpretation of this equation may be consistent with the iso-free-volume concept. However, we believe that the introduction of this equality is in practise a denial of the concept. There are also other arguments against this concept. Kastner56 found, for example, that dielectric losses diminish during the isothermal volume contraction, which indicates a dependence of relaxation times on free-volume. However, if we assume that relaxation time depends exclusively on free-volume, the calculated reduction factor differs from the experimental one. 

The time evolution in Eq. (7.75) is described by the time-dependent Schrodinger equation, provided the molecule is isolated from the rest of the universe. In practice, there are always perturbations from the environment, say due to inelastic collisions. The coherent sum in Eq. (7.75) will then relax to the incoherent sum of Eq. (7.74), that is, the off-diagonal interference terms will vanish and cn 2 — pn corresponding to the Boltzmann distribution. As mentioned earlier, the relaxation time depends on the pressure. In order to take advantage of coherent dynamics it is, of course, crucial that relaxation is avoided within the duration of the relevant chemical dynamics. 

F. Ingrosso, B. Mennucci and J. Tomasi, Quantum mechanical calculations coupled with a dynamical continuum model for the description of dielectric relaxation time dependent Stokes shift of coumarin Cl53 in polar solvents, J. Mol. Liq., 108 (2003) 21 -6. 

A frequency dependence of complex dielectric permittivity of polar polymer reveals two sets or two branches of relaxation processes (Adachi and Kotaka 1993), which correspond to the two branches of conformational relaxation, described in Section 4.2.4. The available empirical data on the molecular-weight dependencies are consistent with formulae (4.41) and (4.42). It was revealed for undiluted polyisoprene and poly(d, /-lactic acid) that the terminal (slow) dielectric relaxation time depends strongly on molecular weight of polymers (Adachi and Kotaka 1993 Ren et al. 2003). Two relaxation branches were discovered for i.s-polyisoprene melts in experiments by Imanishi et al. (1988) and Fodor and Hill (1994). The fast relaxation times do not depend on the length of the macromolecule, while the slow relaxation times do. For the latter, Imanishi et al. (1988) have found... 

Dynamic up-shift of the A (l) v(CO) band of the lowest excited state is the most pronounced IR effect of excited-state relaxation. Time-dependences of the A (l) v(CO) energies measured for various complexes in different media have revealed [37, 75, 76, 86] that relaxation involves four different lifetimes (Fig. 12) ... 

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 change in the emission spectrum with time after pulsed excitation (TRES) is a method for assessing the overall response of the solvent to a change in solute geometry or polarity [22]. The precise values of the relaxation times depend upon the method of measurement. At room temperature the TRES solvent correlation times are subnanosecond and, in some cases subpicosecond. The Debye relaxation time in water is 8 ps, while the TRES correlation time is shorter [22]. Although there is not, in general, a... 

Proton relaxation times depend on the distance between the resonating nucleus and the nearest-neighbor protons. The closer the neighbors, the faster is the relaxation and the shorter is 7]. The two isomers 5-la and 5-lb [Bz = Ph(C = 0)j may be distinguished by their proton relaxation times. In 5-la, H] is axial and close to the 3 and 5 axial protons, resulting in a Ti of 2.0 s. In 5-lb, H is equatorial and has more distant nearest neighbors. 

A superstatistical equilibrium distribution is written as a superposition of Boltzmann distributions with different temperatures. We showed that the excess heat could be written as a superposition of correlation functions with different temperatures using the generalized fluctuation-dissipation theorem. When a relaxation time depends on a temperature, we can expect various behaviors for the area of the hysteresis loop from the fluctuation-dissipation theorem. 

The form of these dielectric dispersion curves is shown in Fig. 8 for a series of elongated ellipsoids of revolution of varying axial ratios. The magnitude of the contribution to the dielectric constant made by the two different portions of the curve, corresponding to the two different relaxation times, depends upon the components of the total... 

Although the timescale and the principle of mobility filters are readily outlined in this way, the validity of the representation is limited, because relaxation times depend on the spectral densities of molecular motion at more than one frequency, which is neglected in Fig. 7.1.3. In addition, the spectral densities relevant to NMR of condensed matter... 

Thus, each relaxation time depends on all four rate constants. 

The process of T relaxation can have contributions from a number of different mechanisms, e.g. dipole-dipole, scalar coupling of the first or second kind, quadrupolar, chemical shift anisotropy (CSA) and spin-rotation. The temperature dependence of the T relaxation time depends upon the relative contribution of each of these mechanisms. For example, if spin-rotation is the dominant relaxation mechanism, then the Ti value decreases linearly with increasing temperature, whereas if dipole-dipole is dominant the T value increases with increasing temperature in the extreme narrowing limit. Examples of temperature measurements using T are included in Sections 3.2.5 and 5.1. 

The values of the slow diffusion coefficient are increasing as a function of salt concentration. The amplitude of /> cff is a maximum at the B-point (see Figure 15) and strongly decreases and disappears or becomes too low to be detected. This behavior is identical to that observed in NaCl and CaCl2 [32] where the splitting in the relaxation times depends on the ratio C/Cs. 

The relaxation time depends, as seen in Table IV, on the polymer, on the solvent, on temperature, and perhaps on the concentration of solute. 

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