Relaxation as a function of frequency

If the observed property varies as a function of time in a way described by only a single constant, T, and since the phase lag is a result of the time necessary for the system to adjust, it follows that the dependence on frequency should also be described by some equation, again with only one adjustable constant. 

The following mathematical treatment is therefore completely general for any relaxation process. The mathematical technique for going from a function of a real variable, like time, to the corresponding variable, here frequency, is via the Fourier transform. However, the Fourier transform is two sided , with both positive and 

In fact, the result for a response as above is the amazingly simple equation  

So the real response, relative to R and as a function of frequency, is given by the formula + which is our familiar sigmoid curve running from 

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Observation of magnetic susceptibility relaxation after perturbation of a spin equilibrium would be the most direct way to measure the dynamics of the equilibrium. This does not appear to have been reported as measured in solution. In principle susceptibility relaxation as a function of frequency could be measured much as dielectric relaxation is examined. The requirement is for a sufficiently strong magnetic field with very sensitive detection. A nonequilibrium magnetic susceptibility has been generated by light at low temperatures in the solid state (39). 

Billisecond to aicrosecond range. Measurements of slow relaxation as a function of frequency at low temperatures which supported the thesis were made by Denney and Cole for methanol /1-propanol (87) by Denney for i-butyl bromide/i-butyl chloride and 1-propanol/i-amyl bromide <88) and by Bos (89) for brOBobenzene with benzyl alcohol and with cyclohexanol. 

Dynamic mechanical relaxation 135 10.5 Relaxation as a function of frequency... 

The dielectric permittivity as a function of frequency may show resonance behavior in the case of gas molecules as studied in microwave spectroscopy (25) or more likely relaxation phenomena in soUds associated with the dissipative processes of polarization of molecules, be they nonpolar, dipolar, etc. There are exceptional circumstances of ferromagnetic resonance, electron magnetic resonance, or nmr. In most microwave treatments, the power dissipation or absorption process is described phenomenologically by equation 5, whatever the detailed molecular processes. 

When the pulse is switched off, the excited nuclei return slowly to their original undisturbed state, giving up the energy they had acquired by excitation. This process is known as relaxation. The detector is switched on in order to record the decreasing signal in the form of the FID (free induction decay). You can observe the FID on the spectrometer s computer monitor, but although it actually contains all the information about the NMR spectrum we wish to obtain, it appears completely unintelligible as it contains this information as a function of time, whereas we need it as a function of frequency. 

Fig. 2. Calculated relaxivities as a function of the water exchange rate for various proton Larmor frequencies and rotational correlation times, tr. The simulations have been performed by using the common Solomon-Bloembergen-Morgan theory of paramagnetic relaxation.

Fig. 4. Inner sphere contribution to the proton relaxivity as a function of the proton Larmor frequency. The curves were calculated on the basis of the Solomon-Bloembergen-Morgan theory for different values of the rotational correlation time, tr, and q — 1, kex — 10 x 106 s-1, tv = 20 ps, A2 = 0.1 x 102Os-2.

The sound absorption coefficient, a, is increased when the dynamics of the chemical system are of the same order of magnitude as the frequency of the sound wave,41 and experimentally this quantity is measured as a function of frequency of the ultrasonic sound wave (Fig. 4). When the frequency of the sound wave is of the same order as the frequency for the relaxation process, effects due to relaxation of the equilibrium give rise to characteristic changes in the quantity a//2, where a is the sound absorption coefficient measured at frequency /40 The variation of a with frequency, /, has an inflection point at the relaxation frequency of the system, fr, which is related to 1/t, where r is the relaxation time (1/t = 27i/r).40,41 The expression relating the quantity... 

Material response is typically studied using either direct (constant) applied voltage (DC) or alternating applied voltage (AC). The AC response as a function of frequency is characteristic of a material. In the future, such electric spectra may be used as a product identification tool, much like IR spectroscopy. Factors such as current strength, duration of measurement, specimen shape, temperature, and applied pressure affect the electric responses of materials. The response may be delayed because of a number of factors including the interaction between polymer chains, the presence within the chain of specific molecular groupings, and effects related to interactions in the specific atoms themselves. A number of properties, such as relaxation time, power loss, dissipation factor, and power factor are measures of this lag. The movement of dipoles (related to the dipole polarization (P) within a polymer can be divided into two types an orientation polarization (P ) and a dislocation or induced polarization. 

Dielectric relaxation study is a powerful technique for obtaining molecular dipolar relaxation as a function of temperature and frequency. By studying the relaxation spectra, the intermolecular cooperative motion and hindered dipolar rotation can be deduced. Due to the presence of an electric field, the composites undergo ionic, interfacial, and dipole polarization, and this polarization mechanism largely depends on the time scales and length scales. As a result, this technique allowed us to shed light on the dynamics of the macromolecular chains of the rubber matrix. The temperature as well as the frequency window can also be varied over a wide... 

As will be shown in the theory, the electrostriction effect plays an important role in the piezoelectric effect of polymer films. Moreover, a knowledge of the complex electrostriction constant as a function of frequency reveals a new aspect of the relaxational behavior of polymers. In this review a new method for measuring complex electrostriction constant with varying frequency will be presented with some results for poly(vinylidene fluoride). 

Perturbation of a chemical equilibrium by ultrasound results in absorption of the sound. Ultrasonic methods determine the absorption coefficient, a (neper cm-1), as a function of frequency. In the absence of chemical relaxation the background absorption, B, increases with the square of the frequency f (hertz) that is, a/f2 is constant. For a single relaxation process the absorption increases with decreasing frequency, passing through an inflection point at the frequency at (radians sec-1 = 2nf) which is the inverse of the relaxation time, t (seconds), of the chemical equilibrium [Eq. (6) and Fig. 3]. 

There has been a flurry of recent activity in the study of the photodissociation dynamics of this molecule (186,187,188). Hermann and Leone used the infrared luminescence technique with a circular variable filter to determine the IR emission as a function of frequency when this molecule was photolyzed with lasers at 248 and 266 nm. From their results, they were able to show that the umbrella bending V2 mode of the CH3 radical was the only mode of CH3 that was excited in the photofragmentation of CH3I. This is in accord with the idea that the photodissociation of this molecule is unusually simple, and involves primarily the scission of the C-I bond with the simultaneous relaxation of the pyramidal structure of the CH3 part of the molecule into its final planar form. The data are used to obtain a vibrational distribution of the CH3 radical that peaks at v" = 2 and extends all the way out to the v" = 10 level. 

Fig. 5.3. Water proton longitudinal relaxivity as a function of proton Larmor frequency ( H NMRD profiles) for solutions of Fe(OH2) + at ( ) 278 K, ( ) 288 K, (A) 298 K, ( ) 308 K. High field transverse relaxivity data at 308 K >) are also shown. The lines represent the best fit curves using the Solomon-Bloembergen-Morgan equations (Eqs. (3.11), (3.12), (3.16), (3.17), (3.26) and (3.27)) [4],...

In order to visualize the effects of water exchange, rotation and electronic relaxation as well as of magnetic field on proton relaxivity, we have calculated proton relaxivities as a function of these parameters (Fig. 2). The relaxivity maximum is attained when the correlation time, tc1, equals the inverse proton Lar-mor frequency (l/rcl = l/rR + l/rm + l/Tle = a>j). The most important message of Fig. 2 is that the rotational correlation time, proton exchange and electronic relaxation rates have to be optimized simultaneously in order to attain maximum relaxivities. If one or two of them have already an optimal value, the remaining parameter starts to become more limitative. The marketed contrast agents have relaxivities around 4-5 mM1 s 1 contrary to the theoretically attainable values over 100 mM 1 s1, which is mainly due to their fast rotation and slow water exchange. 

Nuclear Magnetic Relaxation Dispersion (NMRD). The Koenig Relaxometer Relaxivity as a function of field/frequency through analysis by simulation models r, q, rR, tm, ts for inner sphere, second sphere, and outer sphere water interactions. Also equilibrium constants for BPCA-protein and BPCA-cell interactions. 

Analysis of XH T2 relaxation as a function of temperature yields information on molecular motions. Side-groups and local chain motions (secondary relaxations) cause a change in T2 below the Tg. The value of T2 increases with the amplitude and the frequency of molecular motions. The glass transition that occurs in the time scale in the NMR T2 relaxation... 

Impedance spectroscopy is discussed in depth in the monograph edited by J.Ross Macdonald [17]. It has its origins in the classical work of K.S. Cole and R.H. Cole, published more than 60 years ago, concerned with methods of plotting the response of a dielectric material to applied voltages as a function of frequency. The method assists in identifying observed relaxation effects with processes at the atomic and microstructural levels. For a system having a single well-defined... 

Materials that exhibit a single relaxation time constant can be modeled by the Debye relation which appears as a characteristic response in the permittivity as a function of frequency. The complex permittivity diagram is called Cole-Cole diagram constructed by plotting e" vs. e with frequency as independent parameter. 

Detailed examination of the relaxations requires isothermal scans of relative permittivity and dielectric loss factor as a function of frequency/ so that effective dipole movements and activation energies of relaxation times may be obtained. A typical pair of plots of d and e" values against log/is shown in Fig. 3.7. Graphs of dielectric data of this kind are sometimes called, rather... 

The main purpose of this section is to give the basis of how measurements of the dielectric constants of ionic solutions can give information on solvation, particularly primary hydration numbers. However, dielectric measurements as a function of frequency also give information on the dynamic behavior of water by allowing us to determine the relaxation time of water in ionic solutions and expressing the changes in terms of the number of water molecules bound to the ion. 

Halle et al. (1981) measured NMR relaxation for solutions of several proteins as a function of frequency and protein concentration. They estimated hydration by use of a two-state fast-exchange model with local anisotropy and with assumed values of the order parameter and several other variables. The hydration values ranged from 0.43 to 0.98 h for five proteins, corresponding approximately to a double layer of water about a protein. The correlation time for water reorientation was, averaged over the set of proteins, 20 psec, about eight times slower than that for bulk water. A slow correlation time of about 10 nsec was attributed to an ordering of water by protein at very high concentration. 

Figure 2 Dielectric loss as a function of frequency for systems permitting chemically induced dielectric relaxation d) Schematic curves...

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

Measurements of dielectric constant as a function of frequency. These dielectric dispersion measurements permit the estimation of the relaxation times or rotary diffusion constants which characterize the rotary Brownian movement of the protein molecule. 

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