Fourier transform dielectric relaxation

DC Transient-Current Method. In this method a step voltage is applied to the sample and the current response is measured by a fast-response electrometer. For the single- relaxation-time model, the current response would be given by equation (7-9). In recent years this method has been of renewed interest because with the advent of modem computing methods, it is possible to Fourier-transform the response in the time domain to obtain the frequency response. Several Fourier-transform dielectric spectrometers have been designed. We may note the one of historical significance due to Johnson et al.15, as well as modem commercial instruments.16 The method has the great... 

The elastic contribution is also called elastic incoherent structure factor (EISF). It may be interpreted as the Fourier transformed of the asymptotic distribution of the hopping atom for infinite times. In an analogous way to the relaxation functions (Eq. 4.6 and Eq. 4.7), the complete scattering function is obtained by averaging Eq. 4.22 with the barrier distribution function g E) obtained, e.g. by dielectric spectroscopy (Eq. 4.5)... 

The analysis of the dynamics and dielectric relaxation is made by means of the collective dipole time-correlation function (t) = (M(/).M(0)> /( M(0) 2), from which one can obtain the far-infrared spectrum by a Fourier-Laplace transformation and the main dielectric relaxation time by fitting < >(/) by exponential or multi-exponentials in the long-time rotational-diffusion regime. Results for (t) and the corresponding frequency-dependent absorption coefficient, A" = ilf < >(/) cos (cot)dt are shown in Figure 16-6 for several simulated states. The main spectra capture essentially the microwave region whereas the insert shows the far-infrared spectral region. 

Several comprehensive reviews on the BDS measurement technique and its application have been published recently [3,4,95,98], and the details of experimental tools, sample holders for solids, powders, thin films, and liquids were described there. Note that in the frequency range 10 6-3 x 1010 Hz the complex dielectric permittivity e (co) can be also evaluated from time-domain measurements of the dielectric relaxation function (t) which is related to ( ) by (14). In the frequency range 10-6-105 Hz the experimental approach is simple and less time-consuming than measurement in the frequency domain [3,99-102], However, the evaluation of complex dielectric permittivity in the frequency domain requires the Fourier transform. The details of this technique and different approaches including electrical modulus M oo) = 1/ ( ) measurements in the low-frequency range were presented recently in a very detailed review [3]. Here we will concentrate more on the time-domain measurements in the high-frequency range 105—3 x 1010, usually called time-domain reflectometry (TDR) methods. These will still be called TDS methods. 

Studies of the dielectric constant of solutions and the relaxation times of water in the presence of ions have been refined since the 1980 s and indeed difficulties do turn up if one looks at data from measurements over large frequency ranges. The variation of the dielectric constant with frequency has been studied particularly by Winsor and Cole, who used the Fourier transform of time domain reflectometry to obtain dielectric constants of aqueous solutions and the relaxation times in them. Their frequency ranges from over 50 MHz to 9 GHz. 

Figure 4. Dielectric loss data of I, l -di(4-methoxy-5-methyl phenyl (cyclohexane (BMMPC) at various combinations of temperature and pressure as indicated to demonstrate the invariance of the dispersion of the a-relaxation at constant a-loss peak frequency or equivalently at constant a-relaxation time The dashed line is the imaginary part of the one-sided Fourier transform of the KWW function with Pkww — (I n) — 0.55. The logarithmic ordinate scale makes evident the presence of an excess wing at higher frequencies.

Figure 22. Dielectric loss spectrum of m-FA at 279 K and 1.69 GPa ( ), 1.60 GPa ( ), 1.52 GPa data ( ), and 1.4 GPa data (A). Dielectric loss spectrum of mFA at ambient pressure and 174 K ( ), 177 K data (o), and 180 K (A). The dashed lines are fits to the data at 279 K and under GPa pressures by the one-sided Fourier transform of the KWW function. The solid fines are similar fits to the ambient pressure data. The vertical arrows indicate the calculated primitive relaxation frequencies, Vo, for all the data sets.

In the previous subsection, we have provided conceptually the rationale and experimentally some data to justify the expectation that the primitive relaxation time To of the CM should correspond to the characteristic relaxation time of the Johari-Go Id stein (JG) secondary relaxation Xjg- Furthermore, it is clear from the CM relation, Ta = ( "to)1 1- , given before by Eq. 6 that To mimics Ta in behavior or vice versa. Thus, the same is expected to hold between Xjg and Ta. This expectation is confirmed in Section V from the properties of tjg- The JG relaxation exists in many glass-formers and hence there are plenty of experimental data to test the prediction, xjG T,P) xo(T,P). Broadband dielectric relaxation data collected over many decades of frequencies are best for carrying out the test. The fit of the a-loss peak by the one-sided Fourier transform of a Kohlrausch function [Eq. (1)] determines n and Ta, and together with tc 2 ps, To is calculated from Eq. 6... 

The relaxation map of Fig. 55 shows the temperature dependence of the most probable relaxation times xa, xp, and xy of neat EPON828 obtained. The dielectric ot-loss peak of neat EPON828 was well-fitted by the one-sided Fourier transform of the KWW function with n = 0.47. It is temperature-independent near Tg and together with xa(I), the corresponding Tq(T) is calculated by Eq. (10). The calculated values of xo(7) at 7 256 and 259 K near Tg are... 

The fast Fourier transform (FFT) algorithm now readily available is economical when many real time points and derived values at many frequencies are needed. It is, however, not ideally suited to evaluations of dielectric relaxation functions, because as already mentioned one usually wants these for frequencies that are uniformly spaced on a logarithmic time scale, rather than for the constant intervals and harmonically related frequencies required by the FFT. 

The FT technique has been applied in a multitude of different areas. Starting at low frequencies, FT methods have been used for dielectric response spectroscopy of solids (sometimes called time domain reflectometry). A short picosecond voltage pulse is applied to a dielectric and the current response is measured. Fourier transformation of the current gives the dielectric response function, s v), which is typically interpreted as the Debye relaxation of dipoles. 

Dielectric relaxation measurements couple to the dynamics of the dipole moment of the sample. The dielectric permittivity is the Fourier-Laplace transform of the dipole moment autocorrelation function. 

Recently, the same behavior was demonstrated for the system water (NaCl 8%) + decane 1 1-butanol-A-octylribonamide (CsNg), by using H chemical shift and relaxation time data. At saturation, the molar ratio of bound water to OH groups is again about 1 [135]. This is somewhat surprising, as usually the water solubilization behavior revealed by spectroscopic techniques is entirely different. Thus, NMR [14] (Fig. 17), time domain dielectric spectroscopy (TDS) [136], ESR [137], and Fourier transform infrared (FTIR) [15] measurements indicate that upon the addition of even a small amount of water, an equilibrium between free and bound water is established. This apparent discrepancy is readily understood because the spectroscopic techniques sense the water molecules most near the surfactant. 

The dielectric a process is well defined in the frequency domain for a large number of polymers (McCrum et al., 1967). The complex permittivity e (w) in the frequency region of a well defined a-process may be related to the relaxation function Fa(t), say, by a Fourier transformation (WiUiams, 1972 a Williams, 1978). 

The phenomenological theory of the dielectric relaxation behaviour of linear systems is well-established [1-5]. The fundamental relationship joining the frequency-dependent complex permittivity c(cu) measured at frequency / = (ofln and the transient step-response function t) is the Fourier transform relationship... 

Hence, the relaxation strength Ae reflects the increase of the dielectric constant by the orientation polarization. If this equation is Fourier transformed, the following frequency response function is obtained ... 

Usually, the depolarization current is measured to avoid the dc conductivity contribution. The dielectric relaxation spectrum is then obtained by Fourier transform or approximate formulas, e.g., the Hamon approximation [14]. By carefully controlling the sample temperature and accurately measuring the depolarization current, precision measurements of the dielectric permittivity down to 10" Hz are possible [18]. In fast time domain spectroscopy or reflectometry, a step-like pulse propagates through a coaxial line and is reflected from the sample section placed at the end of the line. The difference between... 

Various instrumental methods applied or developed at MMI over the past ten years to study liquid state transitions are presented. T>Tg transition studies with an emf asis on polystyrene are discussed. Instrumental techniques cover the areas of dielectric relaxation, thermal methods, dynamic mechanical relaxation, and computer statistical analysis of tabulated literature data, as well as the spectroscopic methods of Fourier transform infrared and electron spin resonance. 

The fluctuation-dissipation theorem (FDT) of Callen and Welton states a general relationship between the response of a given system to an external disturbance and the internal fluctuation of the system in the absence of the dismrbance. Such a response is characterized by a response function or equivalently by an admittance or an impedance. For dielectric relaxation, the complex dielectric function, e ( u), is related to the dipole moment correlation function < >( ) via Fourier transformation ... 

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