Frequency domain, dielectric

Fig. 2.39 Dielectric permittivity (A) and loss ( ) for P3M2NBM in the frequency domain at 105°C. The discontinuous straightline represents the conductive effects. Circular black points ( ) represent the resulting loss curve after subtraction showing the dipolar a - relaxation. The continuous line correspond to the Havriliak-Negami curve fit [35,70], (From ref. [35])...

Fig. 2.48 Deconvolution of dielectric loss in the frequency domain for P3MTHFM at 168 K. Open circles represent the experimental losss factor, dashed lines the deconcolution processes, and continues lines represent the sum of both deconvolutes processes.(From ref. [64])...

This chapter concentrates on the results of DS study of the structure, dynamics, and macroscopic behavior of complex materials. First, we present an introduction to the basic concepts of dielectric polarization in static and time-dependent fields, before the dielectric spectroscopy technique itself is reviewed for both frequency and time domains. This part has three sections, namely, broadband dielectric spectroscopy, time-domain dielectric spectroscopy, and a section where different aspects of data treatment and fitting routines are discussed in detail. Then, some examples of dielectric responses observed in various disordered materials are presented. Finally, we will consider the experimental evidence of non-Debye dielectric responses in several complex disordered systems such as microemulsions, porous glasses, porous silicon, H-bonding liquids, aqueous solutions of polymers, and composite materials. 

This function is the product of the KWW and power-law dependencies. The relaxation law (25) in time domain and the HN law (21) in the frequency domain are rather generalized representations that lead to the known dielectric relaxation laws. The fact that these functions have power-law asymptotes has inspired numerous attempts to establish a relationship between their various parameters [40,41]. In this regard, the exact relationship between the parameters of (25) and the HN law (21) should be a consequence of the Laplace transform according to (14) [11,12]. However, there is currently no concrete proof that this is indeed so. Thus, the relationship between the parameters of equations (21) and (25) seems to be valid only asymptotically. 

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. 

The dielectric spectroscopy study of conductive samples is very complicated because of the need to take into account the effect of dc-conductivity. The dc-conductivity c>o contributes, in the frequency domain, to the imaginary part of the complex dielectric permittivity in the form of additional function a0/(so ). The presence of dc-conductivity makes it difficult to analyze relaxation processes especially when the contribution of the conductivity is much greater than the amplitude of the process. The correct calculation of the dc-conductivity is important in terms of the subsequent analysis of the dielectric data. Its evaluation... 

Software for dielectric data treatment and modeling in the frequency domain has been developed recently [132]. This program (MATFIT) was built around the software package MATLAB (Math Works Inc.), and it is available through an intuitive visual interface. Key features of the program include ... 

As mentioned in Section II.B, the dielectric response in the frequency domain for most complex systems cannot be described by a simple Debye expression (17) with a single dielectric relaxation time. In a most general way this dielectric behavior can be described by the phenomenological Havriliak-Negami (HN) formula (21). 

Another way of describing experimentally observed dielectric relaxation in polymers is based on an empirical function formulated by Havriliak and Negami (1967) for the frequency domain ... 

Especially at the short times, the use of time domain methods, as opposed to their polnt-by-polnt frequency domain equivalents, is advantageous in a number of ways. They can, for example, be considered as truly spectroscopic techniques because of their broad-band nature and their capacity to generate dielectric properties as a continuous function of time or, Ity appropriate transformation, frequency. In the past few years, time domain methods have received fresh Impetus from advances in two different types of method firstly, the d.c. step response technique as used by Reddish and Williams has been up-graded in sensitivity and bandwidth through... 

Experimental Methods.— The initial fleeting excursions from frequency domain into time domain (for example, ref. S) appear to have been made because, at that time, steady-state measurements at very low frequencies ( 10 Hz) were unsatisfactory. Step-up, step-down, and ramp voltages were variously applied to capacitors containing dielectric samples, and the tranaent current i(/), or charge q t), responses monitored over a wide range of times such approaches have been reviewed. Although it is now quite feasible to make steady-state measurements at very low frequencies. 

The high-frequency precision of t.d.s. methods has been tested by making measurements on water itself. The data for water at 278 K are shown on Figure 12, together with a r resentative sample of frequency domain measurements. On the basis of most of the previous dielectric S. K. Garg and C. P. Smyth, J. Phys. Chem., 1965,69, 1254. 

The newly established methods in time-domain dielectric studies are reviewed by Dr. Suggett in Chapter S. In general principle these are extensions of the step- or ramp-voltage methods which in various forms have been in occasional use over some thirty years, but the differences in the speed of operation are such as to make them qualitativdy different. For the high-frequency r on the first system was desoibed by Fellner-Feldegg in 1969, whilst a spectrometer operating from 10 Hz to 10 Hz... 

The KWW expression is convenient, but not sacrosanct other expressions can be used to approximate the spectrum. The Cole-Davidson spectrum is useful in the frequency domain, in particular for dielectric relaxation ... 

Figure 12.4 The dielectric loss in the frequency domain, at different temperatures, for poly(methyl acrylate). (From Ref. 6.)...

In the particular application to dielectric relaxation, fit) is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole-Davidson equation [Eq. (10)] [28],... 

As indicated above, exponential relaxation does not generally provide a satisfactory description of experimental data. The response of dielectric materials is instead generally found to be characterized by certain power laws, both in the time and frequency domains (95,96). As observed by Curie and von Schweidler, a long time ago (97,98), the polarization current following the application of a constant electric field typically decays as a power function of time. The same is true for the dielectric response function, since it is proportional to the current oc l/t , where a is a positive constant. This response... 

Practically, DMTA is limited to low frequencies (up to tens of hertz) and, consequently, provides information about relatively slow processes. Dielectric spectroscopy is a related approach in which an alternating electric field is applied to a sample and the complex permittivity is then obtained from phase and amplitude measurements of current and voltage again, it is possible to consider data in the frequency domain, the temperature domain, or even as frequency/temperature contour maps.2 ° 2 See Refs. 230 and 232 for a theoretical account of the underlying physics. The approach can provide information in the frequency range W -io" coupling the applied electric field... 

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