Behavior at High Frequencies

An extension to the flexible chain theory taking into account the elastic reaction of high frequencies was introduced by Marvin. Essentially, it replaces the viscous force by an elastic and a viscous force in series, so to becomes fo(l i to/tE)li + At low frequencies, the force per unit velocity is fb as before at high 

The minimum relaxation time to/tE the Marvin theory can be shown to be equal to the time at which the Rouse expression for H, as given by equation 23 of Chapter 9, has the value Gg/ r. 

Modification of ladder network by Marvin to account for limiting value of modulus at high frequencies and short times. 

Comparison of calculated curves for 7 andy for polyisobutylenc, by the ladder network model of Fig, 10-17, with experimental results reduced to 25 C. Curves without points, theory open circles, experimental J black circles, experimental J . 

A detailed comparison of theory with experiment is given for polyisobutylene in Fig. 10-18, where the components of the complex compliance are chosen for representation. The general aspects of the onset of the glassy zone are evidently semiquantitatively reproduced. However, the distinct difference in slope between theory and experiment for values of J and J less than 10 cm /dyne is apparent. A similar treatment was made by Shibayama and collaborators, who also introduced varying parameters for the springs and dashpots in the ladder model to modify the shapes of the viscoelastic functions predicted. But a more detailed picture of local molecular motions is needed to explain viscoelastic behavior near the glassy zone. 

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We found an equivalent electrical circuit that fits best the LixC6 electrode behavior at high frequency. The circuit consists of a resistor R in parallel with a constant phase element (CPE). The latter is defined with a pseudo-capacitance Q and a parameter a with 0< a <1 [6], The impedance of... 

Fig. 11-13. Effect of frequency on dynamic response of an amorphous, lighlly cross-linked polymer (a) elastic behavior at high frequency—stress and strain are in phase, (b) liquid-like behavior at low frequency—stress and strain are 90" out of phase, and (c) general case—stress and strain are out of phase.

Impedance data are presented in different formats to emphasize specific classes of behavior. The impedance format emphasizes the values at low frequency, which t5rpically are of greatest importance for electrochemical systems that are influenced by mass transfer and reaction kinetics. The admittance format, which emphasizes the capacitive behavior at high frequencies, is often employed for solid-state systems. The complex capacity format is used for dielectric systems in which the capacity is often the feature of greatest interest. 

Fluorescence in Sinusoidal Electric Fields. Fluorescein is an indicator whose fluorescence intensity changes with pH. Consequently, it is reasonable to interpret fluorescence response to applied AC voltage in terms of electrochemical reactions that alter the local pH. This AC-response of local fluorescence intensity exhibits classic relaxation behavior. At high frequencies, electrochemical reactions do not proceed long enough during each half-cycle to produce any appreciable pH change consequently, specimens exhibit time-... 

We refer to the spectrum of a function u as to the Cartesian graph of (o ) versus the frequency a. The asymptotic behavior of the spectrum of a function u, i.e. its behavior at high frequencies, is determined by the analyticity properties of u, as stated by the following lemma ... 

Another example of network formation is found in PEO (poly(ethylene oxide))-silica systems [58, 59]. At relatively small-particle concentrations, the elastic modulus increases at low frequencies, suggesting that stress relaxation of these hybrids is effectively arrested by the presence of silica nanoparticles. This is indicative of a transition from liquidlike to solidlike behavior. At high frequencies, the effect of particles is weak, indicating that the influence of particles on stress relaxation dynamics is much stronger than their influence on the plateau modulus. 

As has already been said and can be observed in Figures 12.5 and 12.7, results deviate from Maxwellian behavior at high frequencies. An upturn is observed in G", which has been ahributed in other systems to a transihon of the relaxation mode from reptation-scission (Cates model) to breathing or Rouse modes [28, 31]. In the systems presented in this chapter, if the relaxation mechanisms were just living reptation at long times -I- Rouse relaxations at short ones, results should be fitted by Equations 12.3 and 12.4, where a Rouse relaxation mode has been added to the Maxwell model, subscripts M and R referring to Maxwell and Rouse relaxations, respectively ... 

The theory of the impedance method for an electrode with diffusion restricted to a thin layer is well established [24,25,39,54,65,66,114-116,118,119,121,125,129, 130,133] however, the ideal response—separate Randles circuit behavior at high frequencies, a Warburg section at intermediate frequencies, and purely capacitive behavior due to the redox capacitance at low frequencies (see Fig. 3.7)—seldom appears in real systems. 

The Fourier spectrum (Figure 6b, presented as a log-log plot) displays an inverse power law behavior at high frequencies, i.e., the broad plateau in the region of 500 kHz to 20 MHz is followed by l//P-like behavior at still higher frequencies. A least-squares fit of the high-frequency coefficients on a log-log plot yielded (3 = 2.1 0.1. Mandelbrot has noted that fractal sets are characterized by Fourier spectra with inverse power law behavior. (74,75) He has shown that the exponent in the inverse power law, p, is related to the fractal dimension, Df, of the corresponding nominally one-dimensional data by... 

E. BEHAVIOR AT HIGH FREQUENCIES AND IN HIGH-VISCOSITY SOLVENTS... 

As neural stimulation and neural response are high frequency in nature, impedance behavior at high frequencies is of particular interest for stimulation. At... 

Note that the form for G is the same as that for a Maxwell model, eq. 3.3.31, while that for G" shows liquid rather than solidlike behavior at high frequency. Figure 10.3.9 shows good agreement between [G /j ] and [G ] and measurements on tobacco mosaic virus. DuPauw (1968) gives the virus dimensions as 2n = 300 nm and 2b = 18 nm (DuPauw, 1968). 

The Nyquist plot of this spatially restricted diffusion impedance, calculated for Ri = 0.05 cm s and D = 63 x 10 cm s , is presented in Fig. 20 for various values of the electrolyte film thickness e. The diagrams exhibit a classical Warburg behavior at high frequency (straight line with a 45° slope) followed by a capacitive behavior at lower frequencies. Electrically, the low-ffequency behavior is equivalent to a resistor-capacitor series coimection. The value of this low-frequency resistor decreases as the electrolyte film thickness decreases. An increase of the characteristic frequencies (corresponding to the transition frequency between the Warburg and the capacitive behavior, for example) is also observed as the electrolyte film thickness decreases. 

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