Relaxation spectrum, damping

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

Equations 3.4-3 and 3.4-4 form the molecular theory origins of the Lodge rubberlike liquid constitutive Eq. 3.3-15 (23). For large strains, characteristic of processing flows, the nonlinear relaxation spectrum is used in the memory function, which is the product of the linear spectrum and the damping function h(y), obtained from the stress relaxation melt behavior after a series of strains applied in stepwise fashion (53)... 

A few rheometers are available for measurement of equi-biaxial and planar extensional properties polymer melts [62,65,66]. The additional experimental challenges associated with these more complicated flows often preclude their use. In practice, these melt rheological properties are often first estimated from decomposing a shear flow curve into a relaxation spectrum and predicting the properties with a constitutive model appropriate for the extensional flow [54-57]. Predictions may be improved at higher strains with damping factors estimated from either a simple shear or uniaxial extensional flow. The limiting tensile strain or stress at the melt break point are not well predicted by this simple approach. 

Actually, the spectroscopic data would more closely resemble the pattern in Figure 3.15, which is the same as the wave in Figure 3.14, except that the overall intensity of the signal decays exponentially with time. (Note that the decay does not affect the frequencies.) Such a pattern is called the modulated free induction decay (FID) signal (or time-domain spectrum). The decay is the result of spin-spin relaxation (Section 2.3.2), which reduces the net magnetization in the, y plane. The envelope (see Section 3.6.2) of the damped wave is described by an exponential decay function whose decay time is T, the effective spin-spin relaxation time. 

Since the reduced spectrum x"( ) clearly shows the low-ftequency Raman modes, we introduced a simple model to analyze the spectral profile of x"(.v) for obtaining the quantitative information. The model is composed of two damped harmonic oscillator modes and one Debye type relaxation mode (liquid water) or one Cole-Cole type relaxation mode (aqueous solution). Cole-Cole type relaxation is usually adopted in analyzing the dielectric relaxation. The formula of Cole-Cole type relaxation is represented as ... 

Fig. 7. (A) The WEFT sequence in this sequence the tt pulse is applied to rotate all of the magnetization (i.e. both solute and solvent) to the -z-axis. A delay (I>np) of sufficient length is used to allow the water magnetization to relax to the origin ( >np = InfZ) ) whilst during the same period, by virtue of faster longitudinal relaxation, the solute resonances have reached thermal equilibrium. An excitation pulse (represented here as a tj/2 pulse) is then applied and an almost water-free spectrum is acquired. However, in the presence of radiation damping the water quicldy returns nonexponentially to the equilibrium position at a similar rate to the solute nuclei (see Fig. 2). However, if during D p a series of n very weak and evenly spaced gradient pulses are applied so as to inhibit the effects of radiation damping, the water relaxes according to its natural spin-lattice relaxation rate. This is the basis of the Water-PRESS sequence (B). An example of a spectrum obtained with Water-PRESS is shown in Fig. IB and Fig. 6.

The specific form of the light-scattering spectrum wiU depend mainly on the characteristics of the inverse susceptibilities yjj of the uncoupled modes. Taking Eq. (10) for a damped harmonic oscillator, and Eq. (11) for a relaxator, it is easy to predict the form of the scattering spectra. 

Figure 4-a illustrates how p spin in Mu and Mu evolves in a transverse magnetic field. The frequency of Mu evolution is at the limit of the time resolution and is not observed. The Mu polarization looks as if depolarized, and this is called hf relaxation , i.e. the relaxation that occurs at the rate of the hf freqiKncy. Figure 4-b is a typical asymmetry spectrum of pS Rotation measured at 18G, where the evolutions of Mu and diamagnetic muon are observed superimposed. (Note the difference of the time scale from Fig. 4a). The fast damping of the Mu asymmetry is apparent, and is caused by the two sli tly different frequencies of Mu precessions. Thus it is customary to measure the u evolution in a much lower field ( 3G) in which such a two-frequency splitting is not significant. The precession of the diamagnetic muons is usually measured around lOOG to see more p rotations.

There is also P(to) = toa(to) where a(co) is the frequency dependent conductivity (see Chapter 25). In the case of a low conductivity ionic solid, P(co) is typically of an oscillating behaviour (a peak at = coq, more or less damped, see Fig. 11.2) and there are only very rare diffusive events which contribute to P(co = 0). In a liquid, the spectrum is centred at (u = 0, since all the particles diffuse. When a liquid becomes (more) viscous a pseudo-oscillating behaviour may be observed (c), while the oscillator damping in a superionic conductor may decrease the difference between the time of flight between two sites and the time of oscillation on a site (Fig. 30.2), leading to a quasi-liquid state ". In order to simplify the model, either the diffusive or the oscillatory behaviour is assumed to be predominant. The choice may depend on the supposition of relaxation time... 

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