Processing and Application of Ceramics 4 [2] (2010) 87–93 Numerical differentiation methods for the logarithmic derivative technique used in dielectric spectroscopy# Henrik Haspel, Ákos Kukovecz*, Zoltán Kónya, Imre Kiricsi Department of Applied and Environmental Chemistry, University of Szeged, Rerrich 1, 6720 Szeged, Hungary Received 15 January 2010; received in revised form 28 June 2010; accepted 30 June 2010 Abstract In dielectric relaxation spectroscopy the conduction contribution often hampers the evaluation of dielectric spectra, especially in the low-frequency regime. In order to overcome this the logarithmic derivative technique could be used, where the calculation of the logarithmic derivative of the real part of the complex permittivity function is needed. Since broadband dielectric measurement provides discrete permittivity function, numerical differentiation has to be used. Applicability of the Savitzky-Golay convolution method in the derivative analy- sis is examined, and a detailed investigation of the influential parameters (frequency, spectrum resolution, peak shape) is presented on synthetic dielectric data. Keywords: dielectric spectroscopy, logarithmic derivative, Savitzky-Golay method I. Introduction ential parameters are considered, and the capabilities Dielectric relaxation spectroscopy (DRS) has be- and limitations of the approach are discussed. come a popular and powerful technique for studying II. Theoretical Background the dielectric properties of almost any kind of mate- rial. Modern measurement systems allow the acquisi- In DRS the frequency-dependent complex dielec- tion of relaxation spectra over a wide range in frequen- tric function is used to determine the electrical/dielec- cy and temperature with a high accuracy. Hence we tric properties of materials [1]: ∗ are able to obtain information about molecular relax- ()= ′ ()− ′′ ()ωεωεωε ation dynamics and conduction processes, leading to a where ε’(ω) is the real and ε”(ω) is the imaginary part deeper understanding of the materials’ nature. of the complex dielectric function. In order to extract most of the information car- If the measured sample contains mobile charge ried by the measurement data, mathematical meth- carriers, i.e. it exhibits conduction, a considerable ods may be applied. In broadband dielectric spec- increase shows up in the low frequency part of the troscopy the so-called harmonic analysis is used. imaginary permittivity spectrum (loss spectrum). In This means that the value of the permittivity func- moderately to highly conducting materials this “con- tion is only known at given measurement frequen- ductivity tail” hampers the proper analysis of slow cies. Since the spectra are discrete datasets, only nu- dipolar relaxations. Furthermore, if the charge car- merical approximations can be used. Therefore, the riers are mobile ions, an undesirable effect, the so- result of the evaluation depends strongly on the pre- called electrode polarization takes place (EP) [2]. cision of these methods. This is due to the partial blocking of the charge ex- In this paper we will present numerical derivative change at the sample/electrode interfaces which re- calculations on synthetic dielectric data. All the influ- sults in the formation of two double layers. These double layers give rise to large capacitances in series #Paper presented at 8th Students’ Meeting, SM-2009, Processing and Application of Ceramics, Novi Sad, Serbia, 2009 to the conducting bulk of the sample. This manifests * Corresponding author: tel: +380 44 424 3364, itself in a high apparent dielectric constant typically fax: +380 44 424 2131, e-mail: [email protected] in the low frequency range. There are several meth- 87 H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010) 87–93 E ­ ½ 2 ° 'H ° ª D § SD · 2D º H cc 'HEsin ) 1 2 ZW cos¨ ¸ ZW ® D E ¾ « » ¯° 1 iZW ¿° ¬ © 2 ¹ ¼ where ª § SD · º « sin¨ ¸ » © 2 ¹ ) arctan« » « D § SD ·» H. Haspel et al. / Processing and Application of Ceramics 4 [2] (2010)« ZW 87–93 cos¨ ¸» ¬ © 2 ¹¼ H. Haspelǻİ et= al.İs /– Processing İ is the relaxationand Application strength, of Ceramics İs, and 4 İ[2] is (2010) the permittivity 87–93 as the frequency Ȧĺ0, and Ȧĺ . ods which help reduce the coveringThe effect logarithmic of the con derivative- The of thislogarithmic function derivativeis: of this function is: duction and the electrode polarization. One of them which is now solely based on relaxation phenomena. It which approximately equalsαD ª απDSthe Ohmic-conduction-º is the numerical Kramers–Kronig transform, which αβDE∆' εH() ωτZW cos −() 1 +βE Φ) can be calculated by numerical techniques described in free dielectric′c loss for rather broad« peaks [5].» For non- is an elegant way to remove Ohmic conduction from ∂wεH HN ¬ 2 ¼ = − 1+βE measuredliterature loss [3,4]. spectra. It transforms the real part of distributed∂w ϖY Debye relaxations, i.e. for single-relaxation− ln 2 An alternative to the numerical Kramers–Kronig time processes,ª the derivativeαD §παSD results· in 2peaksαD º that are the complex dielectric function into the imaginary «1+ 2() ωτZW cos¨ ¸ + () ωτZW » part,transform which isis based now onsolely the logarithmic based on relaxationderivative: phe- sharper [6]. ¬ © 2 ¹ ¼ nomena. It can be calculated by numerical techniques For the double-stretched Havriliak-Negami SwhwereHc 0 < D,E 1 are the shape parameters. There described in literatureHZcc () [3,4]. wherefunction 0 < αwe,β have≤ 1 are for thethe shapeimaginary parameters. part İ": There are der w Z An alternative to the numerical2 ln Kramers–Kronig three special cases. If α,β = 1 the relaxation peak is the transform is based on the logarithmic derivative: sharpest possible and it is called a DebyeE peak caused ­ ½ by a Debye-type relaxation. α and β make2 peaks broader °π ∂ε'′H ° ª D § SD · 2D º ε′′ H()cc ω = −® ¾ 'HEsin )in a 1 symmetrical 2 ZW cos (β¨ = 1) or¸ in ZWan asymmetrically (α = 1) der ∂ ω D E « » ¯° 21 lniZW ¿° way.¬ The first case© is2 called¹ a Cole-Cole,¼ the latter a which approximately equals the Ohmic-conduction- Cole-Davidson peak. The frequency where the function freewhere dielectric loss for rather broad peaks [5]. For non- reachesfrequency its maximum where the equals function with thereaches characteristic its maximum fre- distributed Debye relaxations, i.e. for single-relaxation quencyequals for with Debye the andcharacteristic Cole-Cole frequency peak, but notfor forDebye Cole- and ª § SD · º time processes, the derivative« sinresults¨ ¸in peaks» that are DavidsonCole-Cole peak, peak, where but the not difference for Cole-Davidson between them de-peak, « © 2 ¹ » E sharper [6]. ) arctanH. 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These two techniques are special cases of 'H ''HH ª ªªD SDDD 2SDSDª D º 2 SD DDºº 2º22 HHHcccccc ®®® ° °°¾¾¾ = °'' ∆'HEHEεHEsinsinsin () β )° )Φ )° 1 1 1 + 2 2 2 ()ZW ZWωτZW cos cos cos¨¨¨ § ¸¸¸+ · ()ZW ZWωτZW§§ 1E2 ·· § logarithmic-equidistant2·22 five point spline. These two H cc ®HHcccccc DDD E®E®E ¾ 'wHElnsinY ¾¾ «« )«'' HEHE 1sinsin 2 ZW ) ) 1 1 cos 2 2 ¨ZW ZW ¸ cos cos ZW¨¨«»»» ¸¸sin ZW ¨ZW the¸ so-called» Savitzky-Golay method (SG) for differ- D E DD EEE ª « «« ©©©SD222 ¹¹¹ º 2 » techniques»» are special cases of the so-called Savitzky- ¯°¯°¯° 1 1 1° i iZWiZWZW °°¿°¿°¿° ° °° ¬¬¬ ¬ D ¬¬ § 2©· 2 ¹ 2D©© «¼22¼¼ ¹¹¼ © entiation2 ¹ ¼¼ » [7,8], which is a convolution of a uniformly- ¯ 1 iZW¯¯ 1 1 ¿ iiiZWZW ¿¿ «1 2 ZW cos¨ ¸ ) ZW arctan» Golay method (SG) for differentiation [7,8], which is a where ¬ © 2 ¹ ¼ « D spaced§ SD ·(2» m + 1) point (with a positive integer m) data wherewherewherewhere wherewhere « ZW cosconvolution¨ ¸» of a uniformly-spaced (2m + 1) point ¬ array© 2 with¹¼ a set of b coefficients derived from the least-