High frequency behavior

The still faster ( nsec) dynamics of local motions of polymer molecules in dilute solutions have been investigated by Ediger and coworkers (Zhu and Ediger 1995, 1997). They find that the rates of these local motions of a few bonds are not proportional to the solvent viscosity, unless the solvent reorientation rate is fast compared to the polymer local motion. Thus, for local motions (such as bond reorientations) of polymer molecules, Stokes law of drag does not always hold. 

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Once the component values have been calculated, the physical construction of the transformer and the PCB layout become critical for the effectiveness of the filter stage. Magnetic coupling due to stray inductive pick-up of high-frequency noise by the traces and components can circumvent the filter all together. Added to this is the fact that the common-mode filter choke looks more and more capacitive above its self-resonance frequency. The net result is the designer needs to be concerned about the high-frequency behavior of the filter typically above 20 to 40 MHz. 

The function Am(s) should describe the low- and high-frequency behavior of AM (s ) to a desired degree. We require that Am(s) reproduces N/, high- and Ni low-frequency moments. Because Am(.v) is determined by an even number of constants an and, one needs to choose Nk I Ni 2N. We refer to the resulting... 

Figure 13.11. High-frequency behavior of TFTs built with semiconductor wires and ribbons, (a) xs-Si MOSFETs on PI substrates. (Reprinted with permission from Ref. 17. Copyright 2006 IEEE) (b) xs-GaAs MESFETs on PET substrates, (c) Dependence of fT on gate length of xs-GaAs MESFETs. The different symbols represent measurements on different devices the dashed line corresponds to calculation. (Reprinted with permission from Ref. 82. Copyright 2006 American Institute of Physics.)...

Massa,D,J., Schrag,J.L., Ferry,J.D. Dynamic viscoelastic properties of polystyrene in high-viscosity solvents. Extrapolation to infinite dilution and high-frequency behavior. Macromolecules 4,210-214 (1971). 

Osaki, K., SchragJ.L. Viscoelastic properties of polymer solutions in high-viscosity solvents and limiting high-frequency behavior. I. Polystyrene and poly(a-methyl-styrene). Polymer J. (Japan) 2,541-549 (1971). 

The imaginary part of /J exhibits the high-frequency behavior... 

At high frequencies, the viscoelastic behavior of suspensions is primarily dissipative, as the particles are forced to move through the solvent much faster than they can relax by Brownian motion. The high-frequency behavior is characterized by a constant high-frequency viscosity = lim >oo G" jay, which has been subtracted from the data plotted in Fig. 

Lionberger and Russel (1994) suggested that the stabilizing layers on the spheres of van der Werff et al. produce different lubrication forces than those between bare particles, such as those of Shikata and Pearson, and that this accounts for the differences between the high-frequency moduli of these two systems. Thus, it would appear, perhaps not surprisingly, that the high-frequency behavior of concentrated suspensions is sensitive to the details of interactions between spheres in near contact. 

The average active site dimension dact was deduced from the two characteristic frequencies corresponding respectively to the high-frequency behavior and to the low-frequency behavior, phf aitd plf respectively, i.e.. 

Phase angle The high-frequency behavior in these plots is counterintuitive due to the role of solution resistance. 

We remark that Eq. (262), unlike the form of the Rocard equation of the Levy sneaking model, Eq. (248), has an inertial term similar to the Rocard equation for normal diffusion, Eq. (249). This has an important bearing on the high-frequency behavior because return to transparency can now be achieved, as we shall demonstrate presently. The exact solution, Eq. (260), also has satisfactory high-frequency behavior. We further remark that, on neglecting inertial effects (y —> 0), Eq. (261) yields the Cole-Cole formula [Eq. (9)]—that is, the result predicted by the noninertial fractional Fokker-Planck equation. 

Dielectric loss spectra /"(m) versus logn/mq), for various values of a and y, are shown in Figs. 11 13. It is apparent that the spectral parameters (the characteristic frequency, the half-width, the shape) strongly depend on both a (which pertains to the velocity space) and y. Moreover, the high-frequency behavior of x"((o) is entirely determined by the inertia of system. It is apparent, just as in Brownian dynamics, that inertial effects produce a much more rapid falloff of x"(to) at high frequencies. Thus the Gordon sum rule for the dipole integral absorption of one-dimensional rotators [14,28], namely... 

Thus the permanent-like dipole behavior of the polyelectrolyte solutions can be explained. The high-frequency behavior of polyelectrolyte solutions is found, however, similar to the behavior of colloids with no added polymer. The mobility of bound counterions and the degree of their binding to the polyion surface can be precisely determined if a proper (complete) theory is applied. 

A dilute polymer solution is a system where polymer molecules are dispersed among solvent molecules. An assumption common to any existing theory for flow properties of polymer solutions is that the structure of solvent molecules is neglected and the solvent is assumed to be replaced by a continuous medium of a Newtonian nature. Thus, macroscopic hydrodynamics may be used to describe the motion of the solvent. Recently, some ordering or local structure of solvent molecules around a polymer chain has been postulated as an explanation of the stress-optical coefficient of swollen polymer networks (31,32) so that the assumption of a solvent continuum may not apply. The high frequency behavior shown in Chapter 4 could possibly due to such a microscopic structure of the solvent molecules. Anyway, the assumption of the continuum is employed in every current theory capable of explicit predictions of viscoelastic properties. In the theories of Kirkwood or... 

Although t] x can be derived from theories based on rigid models such as a rod or a once-broken rod (24-26), these theories can not be applied to the data shown in the preceding chapter. On the other hand, the bead-spring model which was so successful in the lower frequency range needs considerable modification before its application to high frequency behavior. Several theories or ideas have been proposed for explanation of the high frequency behavior on a molecular basis. They... 

The idea of a rigid structure was first adopted by Lamb et al. (82-84) for explanation of the high frequency behavior of solutions of flexible polymers. They assumed that a part of each polymer molecule behaves as if it were a rigid sphere at such high frequencies as exceed that corresponding to the relaxation mechanisms observed in the last chapter. 

Summary of the high-frequency behavior of the first- and second-order FRFs for different simple isothermal kinetic mechanisms... 

From Figure 11.12-Figure 11.15 and Table 11.2, it can be seen that different high-frequency behavior of the second-order FRFs is obtained for different mechanism combinations so that they can be used for model identification even for complex kinetic mechanisms. 

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