Frequency-dependent behaviour

Between the static (time-dependent) and the dynamic (frequency-dependent) behaviour the following correlation exists ... 

The frequency dependent behaviour of e, therefore, depends on the response of the polarization to the applied field. The above relation can be recast into the form cited at the beginning. The change of polarization direction in response to the external field is characterized by a time constant of the material, known as relaxation time t. When the material has a single relaxation time, this relation can be reduced to. 

Six [Ln(Pc)2] complexes with heavy lanthanide ions (Ln = Tb, Dy, Ho, Er, Tm or Yb) were investigated by the measurements of alternating current (AC) magnetic susceptibility [18]. Out of the six compounds, [TbPc2] and [DyPc2] were found to show temperature and frequency dependence on AC magnetic susceptibility similar to that observed for the transition-metal SMMs, while the rest did not. Their SMM behaviour have been observed either in bulk, in dilute solid solutions... 

In many cases the temperature dependence of the quadrupolar coupling constant is an indicator of dynamic processes, because the symmetry around the lithium cation is affected by motions which are fast on the NMR time scale. If the rate of these processes exceeds 1/x, the effective symmetry around the lithium cation increases and a decrease in x( Li) results. In Li MAS spectra, a broadening of the satellite transitions can be observed which eventually disappear completely if the rate of the dynamic process comes in the order of the quadrupole frequency. This behaviour was observed for the THF solvated dimer of bis(trimethylsilylamido)lithium, where the Li MAS spectrum at 353 K shows only the central transition and the sidebands caused by CSA and homonuclear Li- Li dipole coupling (Figure 27) . The simulation of the high-temperature spectrum yielded —20 ppm and 1300 Hz for these quantities, respectively. The dipole coupling agrees closely with the theoretical value of 1319 Hz calculated from the Li-Li distance of 2.4 A, which was determined by an X-ray study. 

The electrical conductivity of CoOP as a function of temperature is shown in Figure 6. Above room temperature the compound exhibits metallic behaviour but coincidental with the development of the superstructure the conductivity falls rapidly with decreasing temperature. Below 250 K CoOp behaves as a semiconductor with an activation energy of meV.74 The conduction has been shown to be frequency dependent below 250 K.75 Thermopower studies have also clearly demonstrated the changeover from metallic behaviour above 300 K. to semiconductor behaviour below 250 K.72 The behaviour of ZnOP is very similar to that of CoOp, with the phase transition from the Cccm to Pccn space group occurring at 278 K. Superstructure formation is complete by about 260 K.77... 

Figure 3a-b illustrate the LCB effect on the melt rheological properties. The response of the rheological behaviour to the copolymerisation ability and vinyl end group selectivity of the siloxy-substituted metallocenes has been investigated from their dynamic modulus curves. The frequency dependency of the dynamic modulus of the polyethenes produced with catalysts 2 is demonstrated in Fig. 3a. For comparison dynamic modulus for a linear polyethene, prepared by the catalyst -BuCp2ZrCl2, is shown in Fig. 3b. 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

However, real electrochemical systems exhibit much more complex behaviours. They are not simply resistive. The electrochemical double layer adds a capacitive term. Other electrode processes, such as diffusion, are time and/or frequency dependent. Therefore, for an actual electrochemical system, impedance is used instead of resistance. The impedance of an electrochemical system (defined as Ziot)) is the AC response of the system being studied to the application of an AC signal (e.g., sinusoidal wave) imposed upon the system. The form of the current-voltage relationship of the impedance in an electrochemical system can also be expressed as... 

The slow exchange may have some unforeseen effects on the temperature and frequency dependence of lanthanide induced shifts. Such effects must be borne in mind when deviations from expected behaviour are noted. When tb C 72b the induced shift S is given by... 

The basic theory of dielectric relaxation behaviour, pioneered by Debye, begins with a macroscopic treatment of frequency dependence. This treatment rests on two essential premises exponential approach to equilibrium and the applicability of the superposition principle. In outline, the argument is as follows. 

It is also shown how x(< ) is related to the temporal behaviour of the dielectric polarization follomng the sudden application, or removal, of an electric field. Various forms of the Kramers-Kronig dispersion relations are introduced for y (o>) and x C") aod for a number of functions of The section closes with the ddOnition of the frequency-dependent complex refractive index n() = n(cu) — and a discussion of its relation... 

This polarizability involves a set of characteristic times jlkT, Id(, illoil, and (IcIIodI), between any two of which the correlation function may take a fairly simple form. However, this example indicates that for a linear system which falls to show normal mode behaviour the frequency-dependent admittance may be much more straightforward than the corresponding correlation function. 

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