Numerical differentiation methods for the logarithmic derivative ...

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 analysis 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 Dielectric relaxation spectroscopy (DRS) has become a popular and powerful technique for studying the dielectric properties of almost any kind of material. Modern measurement systems allow the acquisition of relaxation spectra over a wide range in frequency and temperature with a high accuracy. Hence we are able to obtain information about molecular relaxation dynamics and conduction processes, leading to a deeper understanding of the materials’ nature. In order to extract most of the information carried by the measurement data, mathematical methods may be applied. In broadband dielectric spectroscopy the so-called harmonic analysis is used. This means that the value of the permittivity function is only known at given measurement frequencies. Since the spectra are discrete datasets, only numerical approximations can be used. Therefore, the result of the evaluation depends strongly on the precision of these methods. In this paper we will present numerical derivative calculations on synthetic dielectric data. All the influ-

ential parameters are considered, and the capabilities and limitations of the approach are discussed.

II. Theoretical Background In DRS the frequency-dependent complex dielectric function is used to determine the electrical/dielectric properties of materials [1]:

ε ∗ (ω ) = ε ′(ω ) − ε ′′(ω )

where ε’(ω) is the real and ε”(ω) is the imaginary part of the complex dielectric function. If the measured sample contains mobile charge carriers, i.e. it exhibits conduction, a considerable increase shows up in the low frequency part of the imaginary permittivity spectrum (loss spectrum). In moderately to highly conducting materials this “conductivity tail” hampers the proper analysis of slow dipolar relaxations. Furthermore, if the charge carriers are mobile ions, an undesirable effect, the socalled electrode polarization takes place (EP) [2]. This is due to the partial blocking of the charge exchange at the sample/electrode interfaces which results in the formation of two double layers. These double layers give rise to large capacitances in series to the conducting bulk of the sample. This manifests itself in a high apparent dielectric constant typically in the low frequency range. There are several meth-

Paper presented at 8th Students’ Meeting, SM-2009, Processing and Application of Ceramics, Novi Sad, Serbia, 2009 * Corresponding author: tel: +380 44 424 3364, fax: +380 44 424 2131, e-mail: [email protected] #

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İ is the relaxation strength, İs, and4 İ[2] the permittivity as the frequency Ȧĺ0, and Ȧ ǻİ et=al. İs /–Processing is H. Haspel and Application of Ceramics (2010) 87–93 logarithmic of this function is: ods which help reduce the coveringThe effect of the conlogarithmic derivativeThe of this function derivative is: duction and the electrode polarization. One of them DS is now solely based on relaxation phenomena. which approximately equals the Ohmic-conductionªαπ º is which the numerical Kramers–Kronig transform, which It ) Φ αβ ∆ − 'εH (ωτ (1 + β )» DE ZW ) αD cos « E [5]. be calculated numerical techniques described free loss for rather broad peaks ′HN c ∂wεH dielectric is can an elegant way tobyremove Ohmic conduction fromin 2 ¼ For non=− Debye relaxations,¬ i.e. for single-relaxation 1 +βE literature [3,4]. distributed measured loss spectra. It transforms the real part of − ∂w ln ϖ Y 2 ª º πα SD § · An alternative to the numerical Kramers–Kronig time processes, the derivative results in that are 2peaks α α D D the complex dielectric function into the imaginary 1+ 2(ωτ (ωτ ZW ) cos¨ ZW ) » ¸+ « transform is based on the logarithmic derivative: sharper [6]. 2 © ¹ part, which is now solely based on relaxation phe¬ ¼ For the double-stretched Havriliak-Negami nomena. It can be calculated by numerical techniques S wH c parameters. There function we have forthe the shape imaginary part İ": There are cc (Z[3,4]. H der ) where 0 < D,E 1 are the shape where 0 < α,β ≤ 1 are parameters. described in literature 2 w ln Z three special cases. If α,β = 1 the relaxation peak is the An alternative to the numerical Kramers–Kronig sharpest possible and it is called a Debye peak caused transform is based on the logarithmic derivative: E °π ∂ε'′H ½° byªa Debye-type relaxation. α and β make peaks broader 2 § SD · D 2D º ′′H (ccω ) ' ) H E ZW ZW sin 1 2 cos ε der = −® ¨ ¸ in a symmetrical (β = 1) or in an asymmetrically (α = 1) ¾ « » ω D E °¿ ¹ a Cole-Cole, °¯ 21 ∂ ln ¬ The first case© is2called ¼ iZW way. the latter a Cole-Davidson peak. The frequency where the function which approximately equals the Ohmic-conductionreaches its maximum equals with the characteristic frefree dielectric loss for rather broad peaks [5]. For nonwhere frequency where the function reaches its maximum equalsforwith the and characteristic for for Debye and quency Debye Cole-Colefrequency peak, but not Coledistributed Debye relaxations, i.e. for single-relaxation ª º SD · § Cole-Cole peak, but not for Cole-Davidson peak, Davidson peak, where the difference between them detime processes, the derivative results that are sin ¨ ¸in peaks « » © 2 ¹ where between them depends on the E on the4 [2] βdifference shape sharper [6]. ) arctan « » H. Haspel et al. / Processing and Application pends of Ceramics (2010)parameter. 87–93 § SD · » function « Havriliak-Negami D shape parameter. Fig. 1 shows the peak sharpening effect, which For the double-stretched ZW cos ¨ ¸» « 1 shows the peak sharpening effect, which ¬ part ε”: © 2 ¹ ¼ makes Fig. possible to resolve nearby peaks better. Because we have for the imaginary makes possible to resolve nearby peaks better. the derivatives are sensitive to noise, a proper numerE ǻİ = İs – İ is the relaxation strength, İs, and İ is the the should derivatives are sensitive to noise, a proper ° ½° 2 technique be chosen ∆44Hε4Ȧĺ0, ª icalBecause ' SD § · D 2D º for the differentiation pel spel peletetetal. al. al.///Processing Processing Processing and and andApplication Application Application of of of Ceramics Ceramics Ceramics [2] [2] [2] (2010) (2010) (2010) 87–93 87–93 87–93 permittivity as the frequency and Ȧĺ . −ℑ = Haspel etH. H. al.Haspel Haspel / Processing etetal. al.//and Processing Processing Application and and Application of Application Ceramicsof 4of[2] Ceramics Ceramics (2010) 4 4 87–93 [2] [2] (2010) (2010) 87–93 87–93 numerical technique should be chosen for the ' ) Hε c′c′ = H E ZW ZW sin 1 2 cos ¨ ¸ ® ¾ of the data. Wübbenhorst and van Turnhout suggested « » Eβ α this D The logarithmic°derivative of function is: 2 © ¹ ° ¬ ¼ differentiation of the data. Wübbenhorst and van ( ) + ωτ i ZW 1 ¯ ¿ to use either one based on a low pass quadratic least Turnhout suggested to use either one based on a low DS ª º filter or a quadratic logarithmic-equidistant five where 1 E ) » −EEβE E squares DE 'H ZW D cos « EEE passspline. squares filter or a cases quadratic quadratic ½ ½ ½ c 2 2 2 w H point These least two techniques are special of 2 ¬ ¼ ª ª ª º º º ' ' ' H H H SD SD SD HN ½ ··· DDD· §2§2D2SD ° ½° ª DD D D 2 SD logarithmic-equidistant 2º22 αD ªª §§§πα αSD º º º 'H °°°¾°¾°¾ =½°'' ' ' H H SD ° five point spline. These two ª § · · D D D D D 2 2 2 2 § · 1 E ' ) ) ) H H H E E E ZW ZW ZW ZW ZW ZW sin sin sin 1 1 1 2 2 2 cos cos cos ( ) ( ) ( ) ∆ Φ + + ε β ωτ ωτ ¨ ¨ ¨ ¸ ¸ ¸ the so-called Savitzky-Golay method (SG) for differ « « « » » » c c c c sin E E E ZW ¸cos ' '' HH«1sin 2EE ) ) « «11cos 22222¨ZW HH ZW ZW sin cos w Hlnsin YEEE ¾¾E ) ¨ZW ¸ »» » ¨º¨ « 2 ¸¸»ZW DDD ®® E ¾ ® techniques are special cases of the so-called Savitzky© © © ¹ ¹ ¹ ª SD D D D D 2 § · ° ° ° ¬ ¬ ¬ ¼ ¼ ¼ 2 D D ¹ ¼¼ » [7,8], which is a convolution of a uniformly°1iiZW iZW ZW 2ZW ¹arctan ¬¬ ¨ ©¸ ) ZW °¿°¿ «1 ¬ 2ZW cos ©»© «22 ¹¹¼ © entiation ¯ iZW°¯°¯ ¿1¿1¿°¿iiZW for differentiation [7,8], which is a SD ·(2m + 1) §Golay » method (SG) © 2 ¹ spaced point (with a positive integer m) data where ¬ ¼ « ZW D cos ¨convolution ¸» of a uniformly-spaced (2m + 1) point « © 2 with ¹ ¼ a set of bk coefficients derived from the least¬ array (with a positive integer m) data array with a set of bk where ª0ªª < D,E §§§SD 1 are the shape parameters. There are º º º SD SD ·ªª · πα º§§SD ºº SD·· squares-fit formulas of from n-degree polynomial: §·¸·¸SD sin sin¨¨¨relaxation ¸ ¸ »»»sin ««İ« isª« sin coefficients derived the least-squares-fit formulas sin the strength, İ , and İ is the permittivity as the frequency Ȧĺ0, and Ȧĺ . ǻİthree = İs –special sin sin ¨ ¨ ¨ ¸ ¸ s D , E = 1 the relaxation peak is the cases. If »» ¹¹¹2«« ¹ »»» »©© 222 ¹¹ «logarithmic «« « ©©© 222derivative m )) ) sharpest arctan arctan arctan © of n-degree polynomial: The of this function is: « « » » » Φ = arctan and it··is a Debye peak caused ) arctan ) )DDD arctan arctan SD SD §§§SD ·»»»−called ««« possible ′′ (ω ) = πα ε der bk ε ′( r k ω ) SD ··»» §¸¸SD §§SD ««(ωτ Dcos ZW ZW« ZW cos cos ¨relaxation. ¨¨cos ¸»» ») DαD·¸D »+cos «««ZW by a Debye-type D and E make peaks cos ZW ZW cos m ¨ ¨ ¨ ¸ ¸ ©©©«2«22©¹¹¹¼¼2¼ ¹ » ¬¬¬ «¬ k =− m 2 ¹»¼»¼DE 'H ZW D cos ªDS 1 E ) º cc ¬¬ ¼ broader in a symmetrical (©©E22= ¹1) or in an « H der (Z ) entire spectrum bk H c( r k Z ) has to slide » For differentiating the one cHN w H 2 ¬ ¼ on on n strength, strength, strength, İ İ İ , , , and and and İ İ İ is is is the the the permittivity permittivity permittivity as as as the the the frequency frequency frequency Ȧĺ0, Ȧĺ0, Ȧĺ0, and and and Ȧĺ Ȧĺ Ȧĺ . . . k m s s s D = 1) way. The first case is called a asymmetrically ( xation isisthe the relaxation strength, relaxation İsstrength, ,strength, and –İε isİİsiss,sthe , and and İrelaxation İisisthe the permittivity permittivity as theYfrequency as the the frequency Ȧĺ0, andȦĺ0, Ȧĺ0, Ȧĺ and and Ȧĺ Ȧĺpoint . 1. E thepermittivity strength, εas , and εfrequency is the this. (2m + 1) „window” through the measured data Δε = ε w ln sis: ∞ s ∞ ative tive ativeof of ofthis this this function function function is: is: Cole-Cole, the latteris: a Cole-Davidson peak. The 2 ª º SD erivative arithmic arithmic derivative of derivative this function of of this this is: function function is: § · 2 D D permittivity as the frequency ω→0, and ω→ ∞. points (except for m points at each end of the data array «1 2ZW cos¨ 2 ¸ ZW » DS DS DSªDS ª ª ª º º º © ¹ DDD ¬ ¼ DS º ªªDS ºº D D 'ZW cos )) 'HHHZW ZW 111 DDEcos DE DE DE''DE ZW E1cos E ) cos cos «¬«¬cos «¬'2'22HH«ZW )»» H ZW DE DE E«»¼»¼»¼ ) »11EE ) cHN cHN c c wwwHHHHN « c< cHND,E 1 are the wH HNwhere ¼¼ ¬ 2 shape parameters. ¬1¬11E2E2E ¼ wwH0HHN There HN 1 E 111EEE wwwln Y Y ln lnY 222 w ln Y ªªª wwln lnYYDDD §§§SD º º º SD SD ·· ·DDD 22D2D§D§SD 2 ª ZW ª¨¨ §·¸SD ºº 222 2D º·· Dª¨ cos ZW ZW ¨»¨»» SD ZW ZW cos cos ¸ZW ««1«112«212ZW ZW cos ZW ZW 22D2DD»» 2 ZW 1©cos 1© 2¨2¹¸¹ZW cos ¸ ¸ ¸ « « » 2 2 2 © ¹ ¬¬¬ ¬ ©¼©¼¼22 ¼¹¹ ¬¬ © 2 ¹ ¼¼

(

)

∑ ¦

shape parameters. parameters. There There There Dshape ,hape he ,EEshape 11parameters. are areparameters. the the shape shape parameters. parameters. There There There

Figure 1. Comparison of dielectric loss ε”(ω) (lines with symbols) and ε”der(ω) (dashed lines) for symmetric (left) and for Figure 1. Comparison of dielectric loss İ"(Ȧ) (lines with symbols) and İ"der (Ȧ) (dashed lines) for symmetric (left) and for asymmetric (right) HN functions for α,β = 0.2, 0.4, 0.6, 0.8, 1 asymmetric (right) HN functions for D,E = 0.2, 0.4, 0.6, 0.8, 1

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Figure 2. Numerical approximation of the logarithmic derivative of the Debye peak at four different spectrum resolutions Figure 2. Numerical approximation of the logarithmic (r =derivative 2, 21/2, 21/4,of 21/8the ) Debye peak at four different spectrum resolutions (r = 2, 21/2, 21/4, 21/8)

that must be treated separately, using special coefficients For differentiating the[9]). entire has to to calculate the derivative Thespectrum values of one the coeffislide this (2m + 1) point „window” through the cients depend on the degree of the fitted polynomial and measured data points (except for m points at each end the chosen width of the „window” and were published offorthe data array must be treated a broad range that of parameters in [7,9].separately, using special coefficients to calculate The We examined the effect of the the derivative polynomial[9]). degree values of the coefficients depend on the degree of the and the width of the dataset, used in one calculation fitted polynomial and the chosen width of the step, on the quality of the approximation of the deriva„window” were spectrum publishedresolutions for a broad rangebaof tive at fourand different for three parameters in [7,9]. sic dielectric peaks. A normalized Debye, a Cole-Cole examined thepeak effect of chosen the polynomial degree andWe a Cole-Davidson were for that with the and the width of the dataset, used in one calculation following parameters: Δε = 1, τ = 0.01, (β = 1, α = 0.22), step, the quality the approximation of the (α = 1,onβ = 0.2). The SGof coefficients of the used approxderivative at four different spectrum resolutions for imations are listed in Table 1. threeAtbasic peaks.the A two normalized n = 2 dielectric and (2m + 1) = 5 different Debye, set of co-a Cole-Cole and a Cole-Davidson peak were chosen for efficients are the quadratic five-point SG and the quathat with the following parameters: ǻİ = 1, W = 0.01, dratic spline, respectively [10]. r is the logarithmic = 1, D between = 0.22), (the D =measurement 1, E = 0.2). The SG chosen coefficients (E spacing points as r of= the used approximations are listed in Table 1. step 2, 21/2, 21/4, 21/8. Thus, spacing is halved in every At n =the 2 and (2m + 1) the two different (doubled resolution) on=a 5logarithmic scale. set of coefficients are the quadratic five-point SG and the quadratic spline, III. Results andrespectively discussion [10]. r is the logarithmic spacing between the measurement points chosen as r = The 1/4 results 1/8 of the numerical differentiation with , 2 , 2 . Thus, spacing is peak halved every step 2,the21/2 five techniques for the Debye areinpresented in (doubled the resolution) on a logarithmic scale.

Fig. 2. One can easily compare the exact derived peak with the numerically calculated ones visually. The inset III. Results and discussion graphs magnify the peaks at their maxima. As a general rule we state that as the resolution of the analyzed Thecanresults of the numerical differentiation with spectrum increases, all tested approximations provide in the five techniques for the Debye peak are presented good resolution, the more precise Fig. results. 2. One The can larger easily the compare the exact derived peak the technique is. At the same time we havevisually. to considwith the numerically calculated ones The erinset that graphs in broadband dielectric spectroscopy the number magnify the peaks at their maxima. As a ofgeneral measurement couldthat not as be the increased arbitrarirule wepoints can state resolution of the lyanalyzed because spectrum the measurement time depends on increases, all strongly tested approximations it,provide especially in the low frequency Rememberthe good results. The largerregime. the resolution, that at 0.001 Hzthe onetechnique total period ~17same min, time hencewe more precise is. takes At the -3 7 to measurement consider that dielectric –10 Hz range takes ahave dielectric in thein10 broadband spectroscopy the number measurement hours or even days at veryofhigh resolution. points There could are not be increased the measurement cases where this canarbitrarily be realizedbecause (with stable sample, or time strongly dependstechnique on it, especially in the low quenched measurement [11]), but with sensifrequency regime. Remember thattime at is0.001 Hz one tive or time-dependent systems [12] of primary total period takes ~17 hence a dielectric importance. Therefore, resultsmin, at lower resolutions are the 10-3–107 Hz range takes hours or ofmeasurement great practicalinsignificance. even very high resolution. There cases Thedays most at emphatic differences between the are applied where this can be realized (with stable sample, numerical techniques are at r = 2 logarithmic spacing. It or quenched measurement [11]), decreasbut with can be seen, that as the degreetechnique of the polynomial or time-dependent systemsthe [12] time is of essensitive or the width of the dataset increases, numerically primarypeakimportance. at lower derived is broadened Therefore, its maximumresults is lowered. resolutions are of great practical significance.

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Table 1. bk coefficients used for the calculation of the logarithmic derivative

(2m + 1) 5

n=2 (-2,-1,0,1,2) 10 ln r

7

n=3 (1,-6,0,6,-1) 8 ln r

(-3,-2,-1,0,1,2,3) 28 ln r

The explanation of the first observation is that a higher degree polynomial can describe rapid changes better. Since the Debye peak is the sharpest peak that theoretically exists in dielectric spectroscopy, such a function fits better onto the data. However, higher degree polynomials are not considered in the following in order to avoid overfitting which would describe random error instead of the underlying process in a real, noisy measurement. The latter statement is also easy to understand by recalling the calculation procedure applied in the convolution method. A set of data points is multiplied with a set of coefficients in order to get the smoothed derivative at the middle of the data set and then this “window” is slid forwards. Practically, this is a weighted moving average which suppresses here not only the transients (in the case of noisy data), but the whole peak. As the fitted dataset broadens, the suppression is more and more notable. The following order of decreasing accuracy of the applied methods can be deduced from Fig. 2 at r = 2: fivepoint cubic SG (5pt n = 3), five-point quadratic spline (Spline), seven-point cubic SG (7pt n = 3), five-point quadratic SG (5pt n = 2), and seven-point quadratic SG (7pt n = 2). It is also obvious that this order is just an impression from the figure and does not take the effect of the frequency into account. Furthermore, at higher resolutions the differences between the function and the numerical results decrease with each applied method, hence with increasing resolution the use of lower degree polynomials or wider datasets provides also a good result. In those measurements where huge amounts of data can be collected and these data suffer from considerable noise, the use of larger arrays should be considered. Visual comparison allows important qualitative conclusions to be drawn but it lacks the possibility of exact numerical characterization of the applied methods. In our case the absolute and the relative error is defined as:

(1,-8,0,8,-1) 12 ln r (22,-67,-58,0,58,67,-22) 252 ln r

The quantitative results for the Debye, Cole-Cole, and Cole-Davidson peaks are given in Fig. 3. Since the quadratic Savitzky-Golay approximations are the worst ones, only the three other techniques (five-point quadratic spline, five-point cubic SG, seven-point cubic SG) are the subjects of our further investigations. Fig. 3 is technically a matrix, in its rows results for the three peaks at one specific resolution, whilst in its columns calculations for one specific peak at four different resolutions are presented. As expected, the accuracy of the approximations strongly differs with changing peak shapes. A broader peak could be fitted by any of the tested methods with an acceptable error. The difference in the absolute and relative errors for the Debye and for the Cole-Cole peak is about four-five order of magnitude at any resolution. Furthermore, the accuracy depends on the measured frequency as well, so results near and far from the maximum are discussed separately. Hence the original accuracy order established on the basis of Fig. 2. must be reconsidered, or more exactly, a detailed one needs to be determined in the light of the error calculations. At r = 2 spacing, near the maximum of the Debye peak the quadratic spline provides the best result, followed by the five-point, and the seven-point SG with -0.9 %, -3.8 %, -11.7 % absolute, and -1.3 %, -5.0 % and -15.4 % relative errors. At lower and higher frequencies the five-point SG claims the first place (-0.02 % absolute, -15,1 % relative error), the spline is the second (-0.04 % absolute, -40.3 % relative error), and the seven-point SG is the third best choice (-0.1 % absolute, and -127.9 % relative error). In the case of the Cole-Cole peak at the peak maximum the five-point SG is the most accurate method (-1.1×10-4 % absolute, -1.5×10-3 % relative error) followed by the seven-point SG (-7.2×10-4 %, absolute, -8.9×10-3 % relative error) and the quadratic spline (6.7×10-3 % absolute, and 8.4×10-2 % relative error). In Fig. 3. points for spline divided by ten are plotted in order to scale all results to the same order of magnitude. Further from the maximum the same order of accuracy can be found with approximately half of absolute errors which means slightly better relative errors (5pt n = 3: 5.4×10-5 % abs., 1.2×10-3 % rel.; 7pt n = 3: 3.4×10-4 % abs., 7.2×10-3 % rel.; Spline: -1.4×10-3 abs., -3.4×10-2 % relative error) in spite of the decreasing function values.

′′ (ω ) = ε der ′′ ,num (ω ) − ε der ′′ (ω ) ∆ε der and

′′ ,num (ω ) − ε der ′′ (ω ) ′′ (ω ) ε der ∆ε der = ′′ (ω ) ′′ (ω ) ε der ε der The lack of the absolute value function in the definition of the relative error allows this quantity to take negative values that show the direction of the deviation from the approximated function.

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Figure 3. 3. Comparison Comparison ofofthe quadratic spline (Spline), the five-point cubic SG (5ptSG n =(5pt 3) and cubic SG Figure the quadratic spline (Spline), the five-point cubic n =the 3)seven-point and the seven-point (7pt n = 3) numerical derivatives for the investigated three different peaks at four resolutions cubic SG (7pt n = 3) numerical derivatives for the investigated three different peaks at four resolutions

As the Cole-Davidson function asymerror). In Fig. 3. points for spline describes divided byanten are metrically broadened Debye peak towhere the slope plotted in order to scale all results the same order of the low-frequency peak is unchanged and magnitude. Furtherwing from of thethe maximum the same order the slope of the decreases with of accuracy canhigh-frequency be found withwing approximately halfdeof creasing it could be regarded a mixture of arelative Debye absolute β,errors which means as slightly better -5 -3 % abs., in 1.2×10 % rel.; 7pt errors (5pt n = 3: peak. 5.4×10 and a Cole-Cole It behaves the low-frequen-4 -3 % abs., % high-frequency rel.; Spline: n =range 3: 3.4×10 cy like a Debye peak7.2×10 and in the -2 -3.4×10peak % relative error) from in spite the 1.4×10as-3 aabs., range Cole-Cole as observable theoferror decreasing function values. calculations. Values at the characteristic frequency are the Cole-Davidson function an best As approximated by the quadratic splinedescribes (-0.03 % abasymmetrically broadened Debye peak where the solute, -0.2 % relative error), then the five-point SG (-0.3 slope of the wing peak is % absolute, -1.7low-frequency % relative error), andofthethe seven-point SG (-1.1 % absolute, and -5.5 % relative error). Close

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to the peak value the slope five-point SGhigh-frequency is the most accurate unchanged and the of the wing (0.2 % absolute, 0.7 % relative the quadratic E, iterror) couldthen be regarded as a decreases with decreasing spline (0.3of%a absolute, 1.1a%Cole-Cole relative error) sevmixture Debye and peak.and It the behaves en-point SG (0.9 % absolute, and a3.9 % relative in the low-frequency range like Debye peak error). and in So only the accuracy order the direction of the thenot high-frequency range as but a Cole-Cole peak as difference changed. Allcalculations. tested methods behave at observablewas from the error Values at the characteristic approximated by the low frequenciesfrequency similar toare thebest Debye case: 1. five-point -3 quadratic spline (-0.03 % absolute, -0.2 % relative SG (-2.0×10 % absolute, -15.3 % relative error), 2. qua-3 error),spline then the five-point SG (-0.3-40.5 % absolute, -1.7er% dratic (-5.4×10 % absolute, % relative -2 relative error), and the seven-point SGand (-1.1 % ror), 3. seven-point SG (-1.7×10 % absolute, -130.9 absolute, -5.5and % at relative error). Close to thetopeak % relative and error), high frequencies similar the -4 value the case: five-point SG is the accurate (0.2 % Cole-Cole 1. five-point SGmost (4.6×10 % absolute, 3.2×10-3 % relative error), 2. seven-point SG (1.1×10-2 %


ror). 3. quadratic spline (0.04 % absolute, 1.1 % relative error). At r = 21/8 spacing the previous order is valid for all types of peaks used to describe dielectric relaxation processes. Even if we use the least accurate one, the quadratic five-point spline, the absolute error remains below 0.1 %, the relative error under 0.3 % for a Debye peak, and below 0.01 % absolute, and 0.25 % relative error for a Cole-Davidson peak

absolute, 7.2×10-2 % relative error), 3. quadratic spline (-2.8×10-2 % absolute, -0.2 % relative error). Doubling the resolution of the spectrum (r = 21/2) rearranges the accuracy order in the Debye and the ColeDavidson cases. At the maximum value of the Debye peak the former two best choices are interchanged: 1. five-point SG (-0.4 % absolute, -0.5 % relative error), 2. quadratic spline (0.8 % absolute, 1.0 % relative error), 3. seven-point SG (-1.8 % absolute, and -2.4 % relative error), whereas at much lower and higher frequencies the order remains the same: 1. five-point SG (-4.5×10-4 % absolute, -0.8 % relative error), 2. quadratic spline (-2.9×10-3 % absolute, -5.3 % relative error), 3. sevenpoint SG (-3.0×10-3 % absolute, -5.5 % relative error). A strong decrease in the errors could be observed, especially far from the peak maximum for the quadratic spline and the seven-point SG. In the case of the ColeCole function the order is the same as mentioned above, but the absolute and the relative errors are so small even if we use quadratic spline (0.002 % absolute, and 0.02 % relative error), that these are completely negligible in the evaluation of a real measurement. Since the errors decrease with increasing resolution, only the two other types of peaks are discussed below. The best method for the Cole-Davidson function at this resolution (r = 21/2) is the five-point SG with the highest absolute error of 0.03 % (0.35 % relative error) in the peak area, and -5.4×10-5 % (-0.8 % relative error) in the low-frequency range. The second best choice is the quadratic spline with an absolute error of 0.12 % (0.54 % relative error) near the maximum and -3.5×10-4 % (-5.3 % relative error) at low frequencies. The absolute error for the seven-point SG at the maximum is under 1 % (0.17 % absolute, 0.77 % relative error) and is close to the absolute error of the spline in the lowfrequency regime (-3.6×10-4 % abs., and -5.5 % relative error). At high frequencies both the absolute and the relative errors of all three methods are much lower, practically negligible, which is the effect of the similarity to the Cole-Cole function. At r = 21/4 logarithmic spacing for Debye peak the accuracy order is the following in the whole frequency range: 1. five-point SG with a highest absolute error of -0.04% and -0.05% relative error. 2. seven-point SG with maximum -0.2 % absolute and -0.3 % relative errors. 3. quadratic spline with maximum of 0.3 % absolute and 1.1 % relative errors. From this resolution on, all examined techniques are able to approximate the sharpest possible derived spectrum with an error (either absolute or relative) less than 1 % (even less than 0.5 %) which is an often used limit in the characterization of analytical methods. For the Cole-Davidson peak the same order holds with very similar highest error values: 1. five-point SG (-0.003 % absolute, and -0.05 % relative error). 2. seven-point SG (-0.02 % absolute, and 0.3 % relative er-

IV. Conclusions The accuracy of numerical techniques could depend on many parameters, hence we have to apply them very carefully. Here we examined the effect of the frequency, spectrum resolution and the shape of the peak on the accuracy of the so-called numerical logarithmic derivative technique, a versatile tool in the evaluation of dielectric relaxation spectroscopy measurements. A well-known numerical method, the Savitzky-Golay convolution method for differentiation was chosen for the calculations. Synthetic dielectric data were generated in the 10-3– 7 10 Hz frequency range with four different resolutions (r = 2, 21/2, 21/4, 21/8 logarithmic spacing). Three types of peaks were used to investigate the shape effect: the logarithmic derivative of the normalized Debye, a normalized Cole-Cole (α = 0.2), and a Cole-Davidson (β = 0.2) functions. These are all empirical dielectric relaxation functions used frequently. From visual comparison of the numerical results for the Debye peak it can be concluded, that with increasing polynomial degree or decreasing dataset width better approximations could be reached even at lower resolution (i.e. at r = 2 spacing). As the resolution of the spectrum increases, all of the investigated methods give good results with an acceptable error. Although applying higher-than-third degree polynomials is not recommended in order to avoid overfitting, the use of wider dataset could be beneficial due to its stronger noise suppression capability. A quantitative analysis of the investigated methods was also done for all the three types of peaks at the four different resolutions. In the case of the Cole-Cole function a definite order in the accuracy can be found. The best choice for the differentiation of such a symmetrically broadened peak is the five-point Savitzky-Golay method at all resolutions. In the case of Debye and Cole-Davidson functions, at lower resolutions there is a difference in the order of accuracy near the peak maximum and far from it. Generally the five-point quadratic spline approximates the peak values best, followed by the five-point SG, which is in turn the most accurate in the low and high frequency regime followed by the quadratic spline. As the resolution increases the five-point SG overtake the first place in every range of frequency for any of the peak shape, and the accuracy of the seven-point SG also increases. Hence if the spacing can be

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chosen short enough, wider „windows” in the convolution procedure can be used. Summarizing, numerical differentiation techniques for logarithmic derivative method must be chosen carefully, taking the shape of the relaxation peak, the resolution and possibly the frequency range where the approximation should be the most accurate into account. Although our results support the selection of a proper method, the best tactics is the generation of noisy synthetic dielectric data, which properties are almost the same as the measured spectrum. In that case we could be 100 % sure regarding the reliability of the applied numerical technique.

4. P.A.M. Steeman, J. van Turnhout, “A numerical Kramers-Kronig transform for the calculation of dielectric relaxation losses free from Ohmic conduction losses”, Colloid Polym. Sci., 275 (1997) 106–115. 5. G.G. Raju, Dielectrics in Electric Fields, Marcel Dekker, New York 2003. 6. M. Wübbenhorst, J. van Turnhout, “Analysis of complex dielectric spectra. I. One-dimensional derivative techniques and three-dimensional modelling”, J. NonCryst. Solids, 305 (2002) 40–49. 7. A. Savitzky, M.J.E. Golay, “Smoothing and Differentiation of Data by Simplified Least Squares Procedures”, Anal. Chem., 36 (1964) 1627–1639. 8. J. Steinier, Y. Termonia, J. Deltour, “Comments on Smoothing and Differentiation of Data by Simplified Least Square Procedure”, Anal. Chem., 50 (1978) 1383–1386. 9. P.A. Gorry, “General Least-Squares Smoothing and Differentiation by the Convolution (Savitzky-Golay) Method”, Anal. Chem., 62 (1990) 570–573. 10. F.B. Hildebrand, Introduction to Numerical Analysis, 2nd Ed., McGraw-Hill, New York, 1974. 11. J. Mijovic, J.-W. Sy, “Segmental and normal mode dynamics during network formation”, J. Non-Cryst. Solids, 307-310 (2002) 679–687. 12. Y. Bian, J. Mijovic, “Effect of side chain architecture on dielectric relaxation in polyhedral oligomeric silsesquioxane/polypropylene oxide nanocomposites”, Polymer, 50 (2009) 1541–1547.

Acknowledgment: The financial support of the Hungarian Scientific Research Fund (OTKA) through projects NNF-78920 and 73676 is acknowledged References 1. F. Kremer, A. Schönhals (eds.), Broadband Dielectric Spectroscopy, Springer, Berlin 2003. 2. E. Barsoukov, J.R.Macdonald (eds.), Impedance Spectroscopy: Theory, Experiment, and Applications, John Wiley & Sons, Inc., Hoboken, New Jersey, 2005. 3. A. Brather, “Numerisch einfache Bezichungen zwischen Verlust- und Speicherkomponente des dynamischen dielektrischen Faktors”, Colloid Polym. Sci., 257 (1979) 467–477.

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