Recognition of a new permittivity function for glycerol by the use of the eigen-coordinates method
Introduction
In recent years, dielectric relaxation phenomena in glass-forming glycerol and similar hydrogen-bonded molecular materials received a lot of attention from both experimental and theoretical aspects [1], [2], [3], [4], [5], [6], [7], [8], [9]. Most of the experimental studies show that the dielectric ac response of such glass-forming materials is hardly being explained by the `classical' Debye dielectric function that gives the complex permittivity as [10], [11]. The loss-peak angular frequency ωp is generally temperature dependent and can be described by the Vogel–Fulcher–Tammann (VFT) formula [6], [7], [11], [12]. Non-Debye ac response can classically be correlated with some distribution of relaxation times quantifiable in terms of purely empirical parameters entering into the phenomenological dielectric functions being proposed to describe such behaviour [10], [11](a), [11](b). The most conventional empirical analytical dielectric expression that is often used to describe the generalized broadened asymmetric relaxation loss peak observed in many dielectric materials over a wide frequency range is the conventional Havriliak–Negami (HN) equation [10], [11](a), [13]where χHN(jω) is the HN-complex susceptibility with the real and imaginary components χ′(ω) and χ″(ω), respectively and ε0=8.854×10−12 F/m is the permittivity of free space. The parameter is a measure of the broadness of a symmetric dielectric relaxation curve and is the shape parameter of an asymmetric relaxation curve. This type of behaviour will be collectively termed as the Debye-type response [14]. However, in water mixtures with small organic compounds such as polyhydroxyl alcohol [9] the primary dielectric relaxation process exhibiting a broad and asymmetric relaxation curve has been described by the Fourier transform of the Kohlrausch–Williams–Watts (KWW) function [9], [10], [11](a) that gives the complex permittivity function in the formwithwhere the parameter βK (0<βK⩽1) is a measure of the broadness of an asymmetric loss relaxation curve. Other empirical expressions can be also used to analyze the experimental ac response of dielectrics over a wide range of frequencies, among which is the widely-used Jonscher's formula [11a] that describes the frequency dependence of χ″(ω) of the complex susceptibility below and above the loss-peak angular frequency ωp. Another empirical permittivity function, which includes all of Jonscher's `universal response', the Debye-, CC-, and CD-dielectric functions as its special cases and which also takes into account the contribution of dc conduction, which may be encountered experimentally at the low-frequency side, has been recently proposed by Raicu [14]. Though this empirical dielectric function implies that as ω→0, it has been reported [14] to cope remarkably with the experimental dielectric response of complex biological tissues over a broad range of frequencies.
In practice, dielectric relaxation processes in a material specimen can be explored by measuring complex permittivity using two experimental methods; the time-domain (TD) [11](a), [15], [16], [17] and the more popular frequency-domain (FD) techniques [11a]. Dielectric information may be presented in a number of equivalent ways [11a] and it is important to use the most appropriate form of presentation to suit particular requirements. Alternatively, the well-known plots of the imaginary component against the real component, typically on a linear presentation, but the logarithmic presentation can be also informative [11a].
A conventional non-linear curve-fitting method usually results in a best curve fit to the experimental data with a number quantifying how good the fit is and yields a set of values for the adjustable parameters involved, which are always presumed to represent the behaviour of such data. However, such fitting programs can fit, given enough adjustable variables, almost any theoretical/empirical model, but they cannot tell one which theory/model should apply. Consequently, the deduced fitting parameters might be illusive or misleading, as one often obtains different sets of values for them, corresponding to different `local' minima in the statistical function used in the minimization procedure, which give best fits to the same model chosen. Only when a `global' minimum is arrived at through the minimization procedure, the obtained set of fitting parameters can be considered to be physically well behaved and reliable for further analysis.
Recently, a new approach based on the so-called the eigen-coordinates (ECs) method, which appears to be very efficient and relatively simple to use in the data curve-fitting analysis, has been developed by one of the authors (R.R.N.) [26]. The basic ideas and the procedures of applying the ECs method for analyzing data are detailed in Refs. [26], [27]. It has been also applied efficiently to analyze the measured complex impedance of semiconducting selenium films [28] without imposing a priori any equivalent-circuit model being proposed as usually made by many researchers in the field. Also, the concept of the `universal dielectric response' proposed by Jonscher that is often expressed in terms of the CPA-function, which can be related [29], in a general way, to the properties of `fractal' structures usually exist in complex systems [30] should be exemplified further. In a recent report [31], the concept of the CPA-elements has been utilized in modeling of the complex impedance characteristics of electrical-circuit networks describing tin-oxide-based films and was related to some sort of fractality of the system studied. However, these conceptual representations can be understood, more generally and on truly physical grounds, in terms of a self-similar heterostructure [32]. Accordingly, any branching electrical circuits of the `Cauer/Foster' type with a 4-pole complex element and containing self-similar ideal RC elements can be reduced to a single recap two-pole element [32]. The abbreviation symbol recap stands for the combination of the words resistance and capacitance. A single recap element or a combination of several recap elements are considered to depict the fractal nature of a sample interior and/or interfacial/electrode phenomena [16] through a general impedance form having intermediate characteristics with a fractional power-law frequency response of the type [32]where C(ν) is a dimensional parameter and 0⩽ν⩽1, but in some cases having values that exceed unity [32]. Each recap element, besides its own exponent ν, is dominant in a certain frequency range ωmin⩽ω⩽ωmax, which cannot be sometimes achieved by the experimental set up used, over which such a recap element should reflect the existence of a fractal structure formed in some mesoscopic systems. We should remark also the papers of Schiessel et al. [40], where elements similar to the recaps have also been used for description of electric properties of self-similar systems.
The main purpose of this paper is to employ the ECs method in conjunction with a separation procedure to identify the real permittivity function that should merely describe the behaviour of the measured permittivity data of glycerol in the frequency/temperature ranges (1 mHz–1 MHz)/(188–263 K). The applicability of the HN function to the measured permitivity data will be also discussed.
Section snippets
Experimental details
The glycerol sample used in this work was purchased from Aldrich. The relative dielectric complex permittivity of glycerol has been measured in the frequency range between 1 mHz and 1 MHz at different temperatures between 188.0 and 263.0 K. In order to cover the frequency range from 1 mHz to 1 MHz, we have employed two measuring systems. One is an LCR meter (HP 4284A) that has been used for measurements between 20 Hz and 1 MHz. The other was a newly improved ac phase analysis (ACPH) method [33]
Formulation of the Problem
In most cases, the analysis of the dielectric relaxation processes is reduced to a consideration of a complex permittivity of the typewhere χHN(jω) the HN-susceptibility function given in Eq. (1) and is seven-dimensional fitting vector. At the present time, reliable theories connecting directly the involved parameters β and ν with the structure of a substance are not available, though an attempt [35] has been undertaken to relate these
Model experiments
Before using the data curve-fitting approach based on the ECs method to analyze any real experimental situation, which in many aspects is unpredictable, it is necessary to verify the basic linear relationships (which are defined as the ECs) corresponding to the original functions. For the dielectric permittivity described by Eq. (4), the required ECs relations are given in Appendix A. The ECs for linear combination of two-power functions have been analyzed in details in [28]. As it follows from
New data treatment procedure
Dielectric relaxation phenomena in glycerol over broad frequency and temperature ranges have been investigated in many research laboratories. The results obtained in the broadest frequency and temperature ranges [6], [7] can be summarized as follows: there is no one single dielectric function that enables one to describe the experimental data of the relative complex permittivity of glycerol over a wide frequency/temperature range. As already pointed out for the case of glycerol, as well as in
Discussion of the validity of new data curve-fitting approach
One can ask the question: why the problem of recognition of a theoretical hypothesis/function and its true correspondence to experimental data is not discussed in the modern scientific literature? The reason for this keeping off is now obvious, as the modern data curve-fitting techniques employing methods of applied statistics cannot give a certain answer to this question. As a proof, one can cite the words of an expert who is working in this field [20]: “To estimate the probability that a
Physical interpretation of the recognized new permitivity function
Up to now, most of the experimental permittivity data of solids and liquids, particularly those of glass-forming hydrogen-bonded materials, have been discussed and analyzed by the use of the conventional empirical dielectric functions [6], [7], [8], [9], [11](a), [13] and/or in terms of a single/superposition of the `universal' CPA-function [11a]. The concept of the CPA elements has been exemplified in modeling of electrical-circuit networks to describe ac impedance behaviour of different
Summary and basic conclusions
Several basic conclusions can be summarized as follows:
- 1.
Preliminary analysis is inevitable for diagnosing the permittivity behaviour; a procedure that often gives an indication how to choose an `acceptable' theoretical hypothesis to describe the experimental data.
- 2.
Regarding functions that originate from different processes, with one of them being suppressed by others, a direct fitting analysis of the data to the chosen theoretical formula usually leads to a good false fitting curve. Thus a
Acknowledgements
The authors wish to thank Dr Peter Lunkenheimer (Universität Augsburg, Germany) for useful comments on this paper. One of the authors (R.R.N.) specially thanks E. Axelrod (Hebrew University, Israel) for doing an independent fitting of the recognized new dielectric function given in Eq. (5) to some experimental permittivity data of glycerol by the use of the efficient `WinFit' non-linear curve-fitting program.
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