Complex relaxation

Transient nOe represents the rate of nOe buildup. The nOe effect (so-called equilibrium value) itself depends only on the competing balance between various complex relaxation pathways. But the initial rate at which the nOe grows (so-called transient nOe) depends only on the rate of cross-relaxation t, between the relevant dipolarly coupled nuclei, which in turn depends on their internuclear distance (r). 

To better understand the nature of the two types of dynamics, the Cole-Cole plots were plotted, and they clearly indicate the evolution from FR to SR with the changing temperature. Strikingly, the plots can be nicely fitted by the sum of two modified Debye functions (Figure 3.8b, inset), which is used to explain such a complex relaxation process. Here, the two separated relaxation processes are most likely associated with distinct anisotropic centres, that is, two Dy ions with... 

The relaxation approach has played an important role in our understanding of the mechanisms of complex formation in solution (Chap. 4) 39,i4o -pjjg qj computer programs has now eased the study of multiple equilibria. For example, four separate relaxation effects with t s ranging from 100 xs to 35 ms are observed in a temperature-jump study of the reactions of Ni with flavin adenine dinucleotide (fad) (Eqn. (8.121)). The complex relaxation... 

The filament-bound complex relaxes through the release of stored conformational energy and returns to the resting state, myosin-ADP-Pj. 

The initial steep change in the extinction is ascribed to the temperature equilibration of the sample. Following this, a multi-exponential decay in the extinction is observed, indicating the complex relaxation kinetics of the homogeneous reaction between the various cationic SE s. Finally, the long-term single exponential relaxation... 

Figure 3.3 Complex relaxation process in which internal energy transfer occurs with relaxation time t12.

Furthermore, the complex relaxation requirements of a double resonance technique mean that low temperatures are required in all cases. In general, temperatures below that of liquid nitrogen are needed. These are obtained either with a variable temperature flow cryostat using liquid helium ( 4-40 K) or with a liquid helium immersion dewar ( 4 K, or 2 K for pumped (superfluid) helium). 

This equation relates the tensile complex relaxation modulus to the bulk and shear complex relaxation moduli. In the same way, Eq. (5.95) leads to the relationship... 

Figure 6.3 Vectorial components of (a) the complex relaxation modulus G and (b) the complex compliance function J. ...

According to Eqs. (6.19), the relationships between the components of the complex compliance function and those of the complex relaxation modulus... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The shift factors are usually obtained by empirical methods that involve the horizontal translation of the isotherm representing the reduced viscoelastic functions in the time or frequency domains, in double logarithmic plots with respect to the reference isotherm. However, analysis of the components of the complex relaxation moduli in the terminal region ( 0) permits... 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

Following the same procedure, the components of the complex relaxation modulus can be expressed in terms of the relaxation spectrum. Thus, by substituting the difference Git) — Gg given in Eq. (9.5) into Eq. (6.3), we obtain the following expression for the storage relaxation modulus ... 

From Eq. (9.43), the complex relaxation modulus is given at low frequencies by... 

The Laplace transform of Eq, (10.74) gives the complex relaxation modulus of the viscoelastic material in terms of its characteristic impedance,... 

The isochrones showing the temperature dependence of the components of the complex relaxation modulus are rather insensitive to the crystallinity... 

Figure 12,27 Variation of the complex relaxation modulus of poly(ethylene ter-ephthalate) with temperature, in the vicinity of the glass-rubber relaxation, for samples of various crystallinities obtained in isothermal crystallizations ( ) 46%, (<>) 40%, ( ), (V) 26%, ( ) 2-3%, and (O) 0%. (From Ref. 33.)...

Figure 12.28 Temperature dependence of the complex relaxation modulus of poly(ethylene terephthalate), in the glassy region, for the same samples of Figure 12.27.

Figure 12.2(9. Curves showing the. variation of the complex relaxation modulus with temperature, at 1. Hz, for aliphatic polyesters of different degrees of crystallinity ( ) 60%, (A).44 /o (O) 36%, (O) 30%, and (-f) 20%. (From Ref. 37.)...

Case 1. For a standard solid in shear, the complex relaxation modulus (see Chap. 10 for details) is given by... 

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