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Ionic conductivity in glasses: is the window effect statistically relevant?
Journal of Non-Crystalline Solids 244 (1999) 223±231
Ionic conductivity in glasses: is the window eect statistically relevant? D.L. Sidebottom
*
University of New Mexico, Advanced Materials Laboratory 1001, University Buld. SE, Albuquerque, NM 87106, USA Received 9 June 1998; received in revised form 9 November 1998
Abstract Studies of ion motion in glasses are divided between analyses based upon the Kohlrausch description of the electric modulus and the Jonscher power law description of the ac conductivity. It was recently demonstrated that the latter approach suers from a so-called `window eect': the power law exponent obtained from optimized curve ®tting is aected by variations in the accessible frequency range of the experiment. In the present work, a statistical treatment of the curve ®tting is conducted which reveals that the changes in the exponent are accompanied by increases in the error in determining the exponent. Although deviations are evident between the data and the ®t, the observed change in the exponent generally remains within the error of determination and renders the window eect statistically irrelevant. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Frequency dependent dielectric measurements are a primary source of information regarding relaxation of charge carrying entities such as permanent dipoles and ions within a material [1]. In the case of relaxation of permanent dipoles in a liquid, for example, the imaginary part of the complex permittivity exhibits a loss peak, e00loss f , as a function of the frequency, f, whose frequency of maximum loss provides a measure of the relaxation rate for dipolar reorientation [2]. Typically, the loss peak is non-Debye and better described by low frequency and high frequency wings of the form f m and f nÿ1 , respectively, in Ref. [1].
* Tel. : +1-505 272 7613; fax: +1-505 272 7304; e-mail: [email protected]
Ionic relaxation in glasses can be investigated by similar dielectric techniques and also exhibits non-Debye behavior. The migration of a cation between anion sites is essentially equivalent to the rotation of a permanent dipole [1] and hence a dielectric loss like that encountered in the dipolar case is to be expected. However, the migration of the cation also results in dc conduction within the material which contributes e00dc ro /2pfeo to the imaginary part of the permittivity. Experimentally, there is no direct evidence of a low frequency wing e f m visible in e00 f due to the substantial nature of the dc conduction usually encountered in ionic glasses. Nevertheless, several models [3±5] predict that a loss peak does occur for ionic relaxation, albeit obscured by dc conduction. Although one could, in principle, subtract e00dc to obtain e00loss , the result suers considerably at low frequencies from the eective ampli®cation of any random error present in ro [6].
0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 0 1 7 - 4
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D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
Over the years, two major approaches for describing the non-Debye properties present in ionic relaxation have developed: the Kohlrausch±Williams±Watt (KWW) analysis of the electric modulus [7] and the Jonscher power law (JPL) analysis of the ac conductivity [1]. These approaches are empirical attempts to characterize the bulk of the dielectric response, namely the dc conduction at low frequencies and the high frequency wing e00 f nÿ1 of e00loss , caused by ionic motions. In 1972, Moynihan and co-workers [7,8] proposed a novel approach for the analysis of dielectric spectra. They suggested that a more meaningful understanding of ionic relaxation could be obtained by examining the electrical modulus, the inverse of the complex permittivity (M M0 + iM00 1/e ). In this representation the dielectric data take on forms that resemble traditional features of storage and loss moduli occurring in mechanical analogues [9]. As a function of the frequency, the complex modulus displays a broad, asymmetric shape for M 00 f and a corresponding sigmoidal shape for M 00 f . The modulus is typically too extended in frequency to be described by Debye relaxation and is better described by a Laplace transform of the KWW or stretched exponential decay: i h 1 / t exp ÿ t=sb : Later, Jonscher [1,10] cataloged a vast amount of ac conductivity data for ion conducting materials and demonstrated a power law frequency dependence that was similar for a variety of materials. This similarity became known as the `universal dynamic response'. Almond and West [11,12] developed phenomenological expressions for describing the power law conductivity which include a dc contribution. They also demonstrated relationships between the ac and dc activation energies and the power law exponent. Although there are numerous others who have developed more elaborate expressions, for the sake of this discussion the acronym, JPL, refers to ac conductivity of the form r f ro 1 f =fo n :
2
Expressed in this manner, the scaling feature of the ac conductivity, which arises for systems in which
the exponent, n, is temperature independent, is evident. In fact, Kahnt [13] demonstrated this scaling behavior by examining several dissimilar types of glasses using the JPL to show how ac conductivity data could be scaled onto a common master curve. In addition to the scaling behavior, many have proposed that the JPL is universal: not only is the exponent ®xed in a given material, but nearly the same exponent is found across diering classes of materials [1,13,14]. However, while many materials [15] show n 0.67 0.05, there are several exceptions and the issue remains unresolved. Since their inception, both approaches have received their own share of criticism, and a fair amount of controversy has arisen over which approach provides the best insight into ion dynamics [16]. Critics of the KWW point to the systematic failure of Eq. (1) to account for modulus data at high frequencies [15,17,18]. Advocates counter by suggesting that the this failure is insigni®cant since the ®t is successful at low frequencies and near the peak in M00 and accounts for better than 80% of the `strength' of the ionic relaxation [19]. Critics of the JPL point out that the Kramers±Kronig transform of Eq. (2) results in an ac permittivity which diverges at low frequencies, in contradiction to observations [20]. Still others Refs. [3,21] advocate separating the dielectric response into dc conduction and a residual dielectric loss, e00loss , as discussed earlier. At one level, the goal of many researchers in this ®eld has been not so much to describe the dielectric response with absolute precision as to describe the bulk of the response using a limited set of parameters in an eort to compare the characteristics (rates, non-Debye properties) found in one material with that found in another. At this level, empirical descriptions are often employed to generate a body of `critical experimental facts' [22± 24] which might then be used to guide the development of appropriate microscopic models and theories. These facts include the dependence of the non-Debye relaxation upon such variables as temperature, ion concentration, and glass structure, as well as the experimental probe that is used (e.g., nuclear magnetic resonance, mechanical loss). For some time, many considered the KWW
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
and JPL approaches to be essentially equivalent methods of parameterizing the degree of non-Debye behavior found in ionic relaxation. Both exponents, b and n, convey the deviation from Debye relaxation (for which b 1 and n 0). However, it has been demonstrated that these two exponents are not trivially related Refs. [14,15], and that the KWW and JPL approaches can lead to diering perspectives regarding the critical experimental facts. For example, KWW analysis indicates ionic relaxation becomes more Debye-like with decreasing ion concentration, while the JPL displays scaling (no variation in non-Debye behavior) with changing ion concentration [15,25]. The purpose of the present paper is to examine one of the more recent criticisms of the JPL, commonly referred to as the `window eect'. The window eect was illustrated by Jain and Hsieh [26] (JH) in their curve ®tting of the JPL to ac conductivity data obtained on an aluminosilicate glass. In that study, one spectrum of r f (consisting of 11 data points), was initially ®t by Eq. (2) using a standard non-linear regression program. The eect of increasing temperature, which serves to shift r f toward higher frequencies within a ®xed spectrometer window, was simulated by successive truncation of the original data set from the high frequency side. Curve ®tting of the JPL was then repeated on the remaining data. With subsequent truncation of the data set, JH observed an increase of the JPL exponent associated with the optimized ®t. Similar ®ndings were also noted by Kahnt [13] for both data truncation and for increasing temperature. Based upon this observation, JH question the validity of the JPL for describing r(x). If data were precisely of the JPL form, they argue, then the exponent should not depend upon the available frequency window and should remain unaected by data truncation. Does the increased exponent observed in this fashion actually imply an inadequacy of the JPL? What JH fail to acknowledge is that the increase observed in their value of the power law exponent is accompanied by an increased error in its determination. This increase is obvious if one takes the data truncation procedure of JH to extremes. At some point the curve ®tting algorithm used to ®t
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the data is challenged to ®t a `dc-plus-power law' to data which exhibit only a dc component. In this limiting case, the power law exponent could assume most any value between zero and in®nity without signi®cantly aecting the quality of the ®t. Hence, the error in determining this exponent must be increasing with truncation. In this paper, the window eect is examined more closely by looking at the problems inherent in ®tting the JPL to a limited range of experimental data. A formal statistical analysis of the curve ®tting is conducted. As expected, truncation of r f data at high frequencies leads to an increase in the range the exponent can have for which an acceptable curve ®t can be obtained. In ®tting data for a Li-metaphosphate glass, systematic deviations between the ®t and the data are observed and in agreement with JH and Kahnt, the exponent for the optimized ®t does initially increase with decreasing window. However, this increase is not statistically meaningful within the context of standard de®nitions [27] of the goodness of ®t. Aside from the systematic variation in the residuals, the JPL is found to be successful in describing r f to within 5% over approximately 20 square decades. 2. Experimental Procedure Curve ®tting algorithms strive to reduce the absolute deviation between the data (yD ) and the ®t (yF ). This deviation is quantitatively expressed by the sum of squared residuals or `chi square' [27]: v2
2 N X yD ÿ yF i1
Dyi2
i
;
3
where Dy is the experimental error in yD . Better still, this quantity is often normalized to the number of data points resulting in the reduced chi square: v2red
N 1X yD ÿ yF i : Dyi2 N i1 2
4
De®ned in this manner, a ®t with v2red less than one is considered an acceptable ®t [27] since on average
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the RMS deviation of the ®t from the data (yD ÿyF ) falls within the error (Dy). Speci®c algorithms vary in the method [28] by which the space formed by the ®tting parameters is searched. Most compute v2red for various parameter sets and based upon the outcome, migrate to another region of parameter space which the algorithm determines will oer an improvement in decreasing v2red . The process is repeated until some condition is triggered (usually v2red becomes less than some preset limit) and the program execution then halts and returns the last set of parameters which provided the smallest v2red . The error range of a single parameter is generally aected by the presence of other parameters in the set. However, according to Bevington [27], a standard approach to estimating the error range of a given parameter is to increase and decrease only that parameter until the v2red increases by 1. 3. Results To consider relevant features of curve ®tting with the JPL, let us begin with a simple analytical illustration. Using Eq. (2) with n 0.67, ro 1 and fo 1, a ®ctitious data set is constructed. The set is shown in Fig. 1, where a random number generator was used to add 3% noise. Fig. 1(b) shows the residuals, (rD ÿrF )/rF . Next the data set is ®t by the JPL using yD log10 rD which limits the range of yD from 0 to 1.7. Fits are performed with ro ®xed (ro 1) and n ®xed at several selected values (0:2; 0:3; . . . ; 1:5), and only the parameter fo is varied to minimize the v2red . The process is then repeated for several (A; B; . . . ; H) truncations of the original data set and the v2red s tabulated. Results are shown in Fig. 2 where the variation of the v2red -surface with n is plotted for each successive truncation. In ®ts of the entire data set (no truncation) the v2red -surface decreases abruptly in the vicinity of n 0.67. The same is true for series B through D, although the width of this v2red `valley' increases with the data truncation. As the data are further truncated (F, G, and H) the valley in the v2red -surface broadens and becomes more asymmetric, rising more rapidly with decreasing n than with increasing n. This widening indicates
Fig. 1. (a) Simulated data set discussed in the text using Eq. (2) with n 0.67, ro 1, and fo 1. Lines labeled A through H denote range of truncation. (b) residuals of best curve ®t of Eq. (2).
Fig. 2. Variation of v2red with power law exponent for various data truncations shown in Fig. 1(a). Solid line at 1.3 locates the condition where v2red v2red jmin 1.
that as less of the power law portion is made available, a larger error is incurred in the determination of the exponent.
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
Using Bevington's de®nition [27] for the parameter error, the range of n that should provide statistically suitable ®ts for any given truncation series can be estimated. This is done by locating in Fig. 2 the upper and lower ns where v2red v2red jmin 1, as shown by the solid line in the ®gure. In Fig. 3 these upper and lower limits for n are plotted as a function of the truncation as given by the maximum of f =fo of each series. Included in the ®gure are corresponding results obtained when the same analysis is performed using only 1% noise. In both instances the range of acceptable power law exponents increases as the truncation increases. For comparison, results from JH and Kahnt are included in the plot (fo was taken as the frequency where r 2ro ). Kahnt reported changes in n due to increasing temperature as well as selfimposed frequency truncation. As can be seen from the ®gure, the increases in n observed in these previous studies occur in approximately the same regime of truncation for which the error increases most rapidly. One could rightly argue that the above example, while illustrative of the inherent tendency for error in the exponent to increase with increasing truncation, is an unrealistic idealization. Since the data set was constructed from a JPL, the success of ®tting with the JPL is in some ways predestined.
Fig. 3. Range of exponent (coarse cross-hatched region) for the simulation data set (3%) determined from the v2red v2red jmin 1 condition. Finely cross-hatched region is for the same simulation run with 1% random error. Included (symbols) are variation of n at v2red jmin observed by JH [26] and Kahnt [13].
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Although many in the literature have suggested that the JPL does an adequate job of describing real conductivity data, perhaps there may exist systematic deviations between the data and the JPL that could promote the variation of the power law exponent with truncation. To consider this possibility, similar analysis on real conductivity data previously obtained [14] for a Li-metaphosphate glass is performed. A sample of the data obtained for four temperatures is shown in Fig. 4(a). The ®gure shows how changes in temperature eectively alters the window: with increasing temperature each spectrum becomes more dominated by dc conduction leaving less of the power law exposed. Fits of the JPL are also included in Fig. 4(a), and the residuals (Fig. 4(b)) indicate the ®t and data generally agree to within about 5%. Further analysis is now performed on the data set taken at 307 K. This particular spectrum is
Fig. 4. (a) Conductivity of LiPO3 glass at four temperatures. Solid curves are ®ts of the JPL. Line segments indicate successive data truncation discussed in the text. (b) residuals of ®ts for all four temperatures.
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D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
selected since it possesses a well de®ned ro 4.0 (0.1) ´ 10ÿ7 S/m and equal portions of both ro and power law frequency dependence, is located in the midrange of the resolution limits of the spectrometer (Schlumberger 1260), and is not seriously aected [29] by contributions due to `constant loss' processes [5,30]. Again curve ®ts of the JPL are performed on a series of data sets formed by truncation of the 307 K spectrum (see Fig. 4(a)). In the ®ts, ro is ®xed (4.0 ´ 10ÿ7 ) and n ®xed at selected values. Each datum was weighted evenly with Dy taken as 0.015 corresponding to a 3% error in r. The resulting v2red -surface is shown in Fig. 5. In several ways the v2red -surface for the 307K spectrum of LiPO3 is similar to that of the previous simulation study (Fig. 2). The surface again displays a minimum in the vicinity of n 0.70 for the ®rst four series (A through D) and the width of this `valley' increases with increasing truncation. Unlike the simulation results, further truncation causes the minimum to shift toward larger exponents, much in the manner of observations by JH and Kahnt. However, accompanying this shift in the minimum of v2red is an increase in the error of determining the exponent. The range of acceptable n is again estimated by the v2red v2red jmin 1 condition and is shown in Fig. 6. Also included are the ns corresponding to v2red jmin , which ®rst increase to about 1 near fmax =fo 0:3 and then decrease to less than 0.1 with increasing truncation. Note
Fig. 5. Variation of v2red with power law exponent for various data truncations of the LiPO3 spectrum at 307 K shown in Fig. 4(a).
Fig. 6. Range of exponent (cross-hatched region) for the 307 K LiPO3 spectrum determined from the v2red v2red jmin 1 condition. Included (symbols) are variation of n at v2red jmin . Line indicates truncation-independent n 0.70 which remains within the error limits.
that n at v2red jmin is approximately centered between the upper and lower limits of the error and that n (n 0.70) found for limited truncation (fmax =fo 1) would constitute a statistically suitable ®t for all cases of truncation. Hence, the observed increase in n with truncation is not statistically signi®cant. The situation may be more evident in Fig. 7 where data for series E is shown together
Fig. 7. Truncated conductivity of LiPO3 at 307 K (series E in Fig. 4(a)) together with JPL ®ts at n 0.76, corresponding to v2red jmin and at n 0.63 and 0.92, corresponding to lower and upper error limits (see Fig. 6). Inset shows enhancement with error bars (5%) included.
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
with the `best' ®t (n 0.76) and with ®ts marking the upper and lower bounds for the exponent (n 0.92 and 0.63, respectively). Although the ®ts deviate considerably at frequencies greater than 104 Hz, they are comparable over a range less than this frequency. Indeed, if an error bar corresponding to 5% (about 2% on the logarithmic scale) is superimposed on each datum, as is done in the inset to the ®gure, it is evident that all three ®ts generally remain within these bounds. Despite the statistical insigni®cance of the shift in the position of v2red jmin , it is still curious that JH, Kahnt, and the present analysis all show a similar increasing trend. The origin of this increase may be understood by comparing features of the simulation study (which exhibited no shift in the position of v2red jmin ) with the LiPO3 study. One dierence between these two cases is the trends in the residuals displayed in Fig. 1(b) and Fig. 4(b). For the simulation, the residuals, produced by a random number generator, scatter randomly. But in the LiPO3 case, a weak systematic variation can just be discerned. This variation is more visible when the temperature dependence is removed by scaling the x-axis by the frequency, fo (where r 2ro ), as is done in Fig. 8. As is more evident in this ®gure, the residuals are seen to cluster in an oscillatory manner about the ®t curve. The trend is illustrated in the ®gure by the sinusoidal curve drawn with an
229
amplitude of 3%. Most of the residuals fall within 2% of this curve. This trend clearly indicates that the best ®t is not one that `threads' evenly through the data set, but rather is one that is forced to `weave' the data slightly in order to accommodate the substantial range of conductivity data including large portions of both dc and power law behavior. This weaving indicates that the simple JPL may not be suciently precise to account for the frequency dependence of the ac conductivity. The systematic deviation between data and ®t seen in Fig. 8 does however provide some insight into the window eect. When a frequency window is available which contains approximately equal portions of dc and power law conductivity, the optimum JPL ®t weaves the data set; that is, it sacri®ces data in one region to better accommodate data in another region. As the truncation approaches fmax =fo 1, corresponding to removal of the data associated with the ®rst half cycle of Fig. 8, the JPL ®t no longer needs to sacri®ce low frequency data since the constraints at high frequency have now been removed. Presumably, this change is why the parameter n associated with the v2red jmin increases in the vicinity of fmax =fo 1 (series E and F in Fig. 5), corresponding to truncation of the ®rst half cycle in Fig. 8. This explanation is also consistent with the observed decrease that occurs for n near fmax =fo 0:1 corresponding to truncation of the second half cycle. 4. Discussion
Fig. 8. Residuals of LiPO3 spectra ®t with JPL (see Fig. 4(b)) with frequency axis scaled by fo . Solid sinusoidal curve indicates the apparent systematic variation of 3% with width (dashed curves) of 2%.
The window eect, namely the sensitivity of the JPL exponent associated with v2red jmin to the truncation of data from the high frequency edge, is a real phenomenon and indicates that the JPL is an imprecise description of the ac conductivity of ionconducting glasses. However, I consider it impressive that a simple empirical function such as the JPL performs so well, describing r f in LiPO3 (as well as other materials we have studied) to within about 5% over roughly 20 square decades (about 3 decades in r by 6.5 decades in frequency for Fig. 4(a)). The imprecision in the JPL should be contrasted with alternative descriptions such as the
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KWW of the electric modulus which typically shows substantial deviations (often more than 40%) in the high frequency wing. A more detailed comparison between the success of ®tting data with the KWW or JPL cannot be made, as analysis using the KWW typically does not involve numerical curve ®tting in the present context since the Laplace transform of Eq. (1) in non-analytic. Instead, the KWW decay is more often approximated by a sum of exponentials and then transformed to obtained tables [8] of M f for selected bs. `Fits' of the KWW often involve judicious superpositioning of these tabulated results onto the observed M 00 f . Such ®ts are subjective and most examples in the literature indicate a disposition on the part of the researcher to emphasize the peak of M 00 f more than the high frequency wing [19]. If the Laplace transform of the KWW could be expressed as an analytic function, then an objective ®t of M f could be performed. Further, if these ®ts were required to maintain similar standards as those imposed upon the JPL ®tting, wherein all the data are equally weighted with none (e.g., the peak of M 00 f ) receiving special merit, it is likely the optimized ®t would also `weave' the data set and exhibit oscillatory systematic features in the residuals similar to those seen here only of much greater magnitude than 5%. The systematic deviations observed in the present study could be due to the existence of a dielectric loss peak residing beneath the dc conductivity. The existence of such a peak is justi®ed and predicted by several models [3,4], and would introduce a weak distortion in r f in the vicinity of fo which is not accounted for by the JPL. However, the present work indicates this loss peak is of limited signi®cance in comparison to the dc conduction, contributing only about 3% above the 2% random noise level (see Fig. 8). Analysis of r f that incorporates a dielectric loss together with the dc conduction may account for the 3% deviation missed by the JPL, but will do so at the expense of a 33% increase in the number of ®tting parameters. Clearly one must exercise some caution in the analysis of data sets with fmax =fo < 10, and recognize the inherent errors present in any curve ®tting procedure. Despite a small degree of im-
precision, the JPL can still be used to accurately evaluate data sets with larger fmax =fo and obtain a meaningful exponent which can be used to test universality and scaling issues. 5. Conclusions Descriptions of the ac conductivity of ionic glasses using the Jonscher power law display a window eect in the sense that the power law exponent obtained from curve ®tting analyses generally increases as data at high frequency are successively omitted. However, this truncation of the data leads to instability in determining the exponent and a corresponding increase in the error associated with the exponent obtained for the optimized curve ®t. Consequently, the observed increase is not statistically signi®cant and acceptable ®ts can be obtained with a truncation-independent exponent. Although the window eect indicates a degree of imprecision of the JPL to account for experimental data, this simple empirical function appears to successfully describe the ac conductivity to within 5% over ranges of about 20 square decades. Acknowledgements The author thanks Dr Laura Swiler for helpful discussions regarding statistical analysis of data. These conductivity measurements were performed at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. References [1] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics, London, 1983. [2] V.V. Daniel, Dielectric Relaxation, Academic Press, New York, 1967. [3] A. Hunt, J. Phys.: Condens. Mater. 3 (1991) 7831. [4] A. Hunt, Appl. Phys. A 54 (1992) 508. [5] L.A. Dissado, R.M. Hill, J. Appl. Phys. 66 (1989) 2511.
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231 [6] A.S. Nowick, B.S. Lim, A.V. Vaysleyb, J. Non-Cryst. Solids 172±174 (1994) 1243. [7] P.B. Macedo, C.T. Moynihan, R. Bose, Phys. Chem. Glasses 13 (1972) 171. [8] C.T. Moynihan, L.P. Boesch, N.L. Laberge, Phys. Chem. Glasses 14 (1973) 122. [9] N.G. McCrum, B.E. Read, and G. Williams, Anelastic and Dielectric Eects in Polymeric Solids, Dover, New York, 1991. [10] A.K. Jonscher, Nature 267 (1977) 673. [11] D.P. Almond, A.R. West, R.J. Grant, Solid State Commun. 44 (1982) 1277. [12] D.P. Almond, A.R. West, Solid State Ionics 11 (1983) 57. [13] H. Kahnt, Ber. Bunsenges. Phys. Chem. 95 (1991) 1021. [14] D.L. Sidebottom, P.F. Green, R.K. Brow, J. Non-Cryst. Solids 183 (1995) 151. [15] D.L. Sidebottom, P.F. Green, R.K. Brow, Phys. Rev. B 56 (1997) 170. [16] Disscussion session on `Glassy Ionics', J. Non-Cryst. Solids 131±133 (1991) 1113. [17] S.R. Elliott, J. Non-Cryst. Solids 170 (1994) 97.
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[18] A.S. Nowick, B.S. Lim, J. Non-Cryst. Solids 172±174 (1994) 1389. [19] C.T. Moynihan, J. Non-Cryst. Solids 172±174 (1994) 1395. [20] J. M Hyde, M. Tomozawa, J. Non-Cryst. Solids 109 (1989) 18. [21] G.P. Johari, K. Pathmanathan, Phys. Chem. Glasses 29 (1988) 219. [22] K.L. Ngai, J. Non-Cryst. Solids 203 (1996) 232. [23] J.C. Dyre, J. Non-Cryst. Solids 135 (1991) 219. [24] S.R. Elliott, Solid State Ionics, 70&71 (1994) 27. [25] D.L. Sidebottom, J. Non-Cryst. Solids, to be published. [26] H. Jain, C.H. Hsieh, J. Non-Cryst. Solids 172±174 (1994) 1408. [27] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969. [28] T.E. Shoup, Numerical Methods for the Personal Computer, Prentice-Hall, New Jersey, 1983. [29] D.L. Sidebottom, P.F. Green, R.K. Brow, J. Non-Cryst. Solids 203 (1996) 300. [30] X. Lu, H. Jain, O. Kanert, J. Non-Cryst. Solids 172±174 (1994) 1436.
Ionic conductivity in glasses: is the window eect statistically relevant? D.L. Sidebottom
*
University of New Mexico, Advanced Materials Laboratory 1001, University Buld. SE, Albuquerque, NM 87106, USA Received 9 June 1998; received in revised form 9 November 1998
Abstract Studies of ion motion in glasses are divided between analyses based upon the Kohlrausch description of the electric modulus and the Jonscher power law description of the ac conductivity. It was recently demonstrated that the latter approach suers from a so-called `window eect': the power law exponent obtained from optimized curve ®tting is aected by variations in the accessible frequency range of the experiment. In the present work, a statistical treatment of the curve ®tting is conducted which reveals that the changes in the exponent are accompanied by increases in the error in determining the exponent. Although deviations are evident between the data and the ®t, the observed change in the exponent generally remains within the error of determination and renders the window eect statistically irrelevant. Ó 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Frequency dependent dielectric measurements are a primary source of information regarding relaxation of charge carrying entities such as permanent dipoles and ions within a material [1]. In the case of relaxation of permanent dipoles in a liquid, for example, the imaginary part of the complex permittivity exhibits a loss peak, e00loss f , as a function of the frequency, f, whose frequency of maximum loss provides a measure of the relaxation rate for dipolar reorientation [2]. Typically, the loss peak is non-Debye and better described by low frequency and high frequency wings of the form f m and f nÿ1 , respectively, in Ref. [1].
* Tel. : +1-505 272 7613; fax: +1-505 272 7304; e-mail: [email protected]
Ionic relaxation in glasses can be investigated by similar dielectric techniques and also exhibits non-Debye behavior. The migration of a cation between anion sites is essentially equivalent to the rotation of a permanent dipole [1] and hence a dielectric loss like that encountered in the dipolar case is to be expected. However, the migration of the cation also results in dc conduction within the material which contributes e00dc ro /2pfeo to the imaginary part of the permittivity. Experimentally, there is no direct evidence of a low frequency wing e f m visible in e00 f due to the substantial nature of the dc conduction usually encountered in ionic glasses. Nevertheless, several models [3±5] predict that a loss peak does occur for ionic relaxation, albeit obscured by dc conduction. Although one could, in principle, subtract e00dc to obtain e00loss , the result suers considerably at low frequencies from the eective ampli®cation of any random error present in ro [6].
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Over the years, two major approaches for describing the non-Debye properties present in ionic relaxation have developed: the Kohlrausch±Williams±Watt (KWW) analysis of the electric modulus [7] and the Jonscher power law (JPL) analysis of the ac conductivity [1]. These approaches are empirical attempts to characterize the bulk of the dielectric response, namely the dc conduction at low frequencies and the high frequency wing e00 f nÿ1 of e00loss , caused by ionic motions. In 1972, Moynihan and co-workers [7,8] proposed a novel approach for the analysis of dielectric spectra. They suggested that a more meaningful understanding of ionic relaxation could be obtained by examining the electrical modulus, the inverse of the complex permittivity (M M0 + iM00 1/e ). In this representation the dielectric data take on forms that resemble traditional features of storage and loss moduli occurring in mechanical analogues [9]. As a function of the frequency, the complex modulus displays a broad, asymmetric shape for M 00 f and a corresponding sigmoidal shape for M 00 f . The modulus is typically too extended in frequency to be described by Debye relaxation and is better described by a Laplace transform of the KWW or stretched exponential decay: i h 1 / t exp ÿ t=sb : Later, Jonscher [1,10] cataloged a vast amount of ac conductivity data for ion conducting materials and demonstrated a power law frequency dependence that was similar for a variety of materials. This similarity became known as the `universal dynamic response'. Almond and West [11,12] developed phenomenological expressions for describing the power law conductivity which include a dc contribution. They also demonstrated relationships between the ac and dc activation energies and the power law exponent. Although there are numerous others who have developed more elaborate expressions, for the sake of this discussion the acronym, JPL, refers to ac conductivity of the form r f ro 1 f =fo n :
2
Expressed in this manner, the scaling feature of the ac conductivity, which arises for systems in which
the exponent, n, is temperature independent, is evident. In fact, Kahnt [13] demonstrated this scaling behavior by examining several dissimilar types of glasses using the JPL to show how ac conductivity data could be scaled onto a common master curve. In addition to the scaling behavior, many have proposed that the JPL is universal: not only is the exponent ®xed in a given material, but nearly the same exponent is found across diering classes of materials [1,13,14]. However, while many materials [15] show n 0.67 0.05, there are several exceptions and the issue remains unresolved. Since their inception, both approaches have received their own share of criticism, and a fair amount of controversy has arisen over which approach provides the best insight into ion dynamics [16]. Critics of the KWW point to the systematic failure of Eq. (1) to account for modulus data at high frequencies [15,17,18]. Advocates counter by suggesting that the this failure is insigni®cant since the ®t is successful at low frequencies and near the peak in M00 and accounts for better than 80% of the `strength' of the ionic relaxation [19]. Critics of the JPL point out that the Kramers±Kronig transform of Eq. (2) results in an ac permittivity which diverges at low frequencies, in contradiction to observations [20]. Still others Refs. [3,21] advocate separating the dielectric response into dc conduction and a residual dielectric loss, e00loss , as discussed earlier. At one level, the goal of many researchers in this ®eld has been not so much to describe the dielectric response with absolute precision as to describe the bulk of the response using a limited set of parameters in an eort to compare the characteristics (rates, non-Debye properties) found in one material with that found in another. At this level, empirical descriptions are often employed to generate a body of `critical experimental facts' [22± 24] which might then be used to guide the development of appropriate microscopic models and theories. These facts include the dependence of the non-Debye relaxation upon such variables as temperature, ion concentration, and glass structure, as well as the experimental probe that is used (e.g., nuclear magnetic resonance, mechanical loss). For some time, many considered the KWW
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
and JPL approaches to be essentially equivalent methods of parameterizing the degree of non-Debye behavior found in ionic relaxation. Both exponents, b and n, convey the deviation from Debye relaxation (for which b 1 and n 0). However, it has been demonstrated that these two exponents are not trivially related Refs. [14,15], and that the KWW and JPL approaches can lead to diering perspectives regarding the critical experimental facts. For example, KWW analysis indicates ionic relaxation becomes more Debye-like with decreasing ion concentration, while the JPL displays scaling (no variation in non-Debye behavior) with changing ion concentration [15,25]. The purpose of the present paper is to examine one of the more recent criticisms of the JPL, commonly referred to as the `window eect'. The window eect was illustrated by Jain and Hsieh [26] (JH) in their curve ®tting of the JPL to ac conductivity data obtained on an aluminosilicate glass. In that study, one spectrum of r f (consisting of 11 data points), was initially ®t by Eq. (2) using a standard non-linear regression program. The eect of increasing temperature, which serves to shift r f toward higher frequencies within a ®xed spectrometer window, was simulated by successive truncation of the original data set from the high frequency side. Curve ®tting of the JPL was then repeated on the remaining data. With subsequent truncation of the data set, JH observed an increase of the JPL exponent associated with the optimized ®t. Similar ®ndings were also noted by Kahnt [13] for both data truncation and for increasing temperature. Based upon this observation, JH question the validity of the JPL for describing r(x). If data were precisely of the JPL form, they argue, then the exponent should not depend upon the available frequency window and should remain unaected by data truncation. Does the increased exponent observed in this fashion actually imply an inadequacy of the JPL? What JH fail to acknowledge is that the increase observed in their value of the power law exponent is accompanied by an increased error in its determination. This increase is obvious if one takes the data truncation procedure of JH to extremes. At some point the curve ®tting algorithm used to ®t
225
the data is challenged to ®t a `dc-plus-power law' to data which exhibit only a dc component. In this limiting case, the power law exponent could assume most any value between zero and in®nity without signi®cantly aecting the quality of the ®t. Hence, the error in determining this exponent must be increasing with truncation. In this paper, the window eect is examined more closely by looking at the problems inherent in ®tting the JPL to a limited range of experimental data. A formal statistical analysis of the curve ®tting is conducted. As expected, truncation of r f data at high frequencies leads to an increase in the range the exponent can have for which an acceptable curve ®t can be obtained. In ®tting data for a Li-metaphosphate glass, systematic deviations between the ®t and the data are observed and in agreement with JH and Kahnt, the exponent for the optimized ®t does initially increase with decreasing window. However, this increase is not statistically meaningful within the context of standard de®nitions [27] of the goodness of ®t. Aside from the systematic variation in the residuals, the JPL is found to be successful in describing r f to within 5% over approximately 20 square decades. 2. Experimental Procedure Curve ®tting algorithms strive to reduce the absolute deviation between the data (yD ) and the ®t (yF ). This deviation is quantitatively expressed by the sum of squared residuals or `chi square' [27]: v2
2 N X yD ÿ yF i1
Dyi2
i
;
3
where Dy is the experimental error in yD . Better still, this quantity is often normalized to the number of data points resulting in the reduced chi square: v2red
N 1X yD ÿ yF i : Dyi2 N i1 2
4
De®ned in this manner, a ®t with v2red less than one is considered an acceptable ®t [27] since on average
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the RMS deviation of the ®t from the data (yD ÿyF ) falls within the error (Dy). Speci®c algorithms vary in the method [28] by which the space formed by the ®tting parameters is searched. Most compute v2red for various parameter sets and based upon the outcome, migrate to another region of parameter space which the algorithm determines will oer an improvement in decreasing v2red . The process is repeated until some condition is triggered (usually v2red becomes less than some preset limit) and the program execution then halts and returns the last set of parameters which provided the smallest v2red . The error range of a single parameter is generally aected by the presence of other parameters in the set. However, according to Bevington [27], a standard approach to estimating the error range of a given parameter is to increase and decrease only that parameter until the v2red increases by 1. 3. Results To consider relevant features of curve ®tting with the JPL, let us begin with a simple analytical illustration. Using Eq. (2) with n 0.67, ro 1 and fo 1, a ®ctitious data set is constructed. The set is shown in Fig. 1, where a random number generator was used to add 3% noise. Fig. 1(b) shows the residuals, (rD ÿrF )/rF . Next the data set is ®t by the JPL using yD log10 rD which limits the range of yD from 0 to 1.7. Fits are performed with ro ®xed (ro 1) and n ®xed at several selected values (0:2; 0:3; . . . ; 1:5), and only the parameter fo is varied to minimize the v2red . The process is then repeated for several (A; B; . . . ; H) truncations of the original data set and the v2red s tabulated. Results are shown in Fig. 2 where the variation of the v2red -surface with n is plotted for each successive truncation. In ®ts of the entire data set (no truncation) the v2red -surface decreases abruptly in the vicinity of n 0.67. The same is true for series B through D, although the width of this v2red `valley' increases with the data truncation. As the data are further truncated (F, G, and H) the valley in the v2red -surface broadens and becomes more asymmetric, rising more rapidly with decreasing n than with increasing n. This widening indicates
Fig. 1. (a) Simulated data set discussed in the text using Eq. (2) with n 0.67, ro 1, and fo 1. Lines labeled A through H denote range of truncation. (b) residuals of best curve ®t of Eq. (2).
Fig. 2. Variation of v2red with power law exponent for various data truncations shown in Fig. 1(a). Solid line at 1.3 locates the condition where v2red v2red jmin 1.
that as less of the power law portion is made available, a larger error is incurred in the determination of the exponent.
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
Using Bevington's de®nition [27] for the parameter error, the range of n that should provide statistically suitable ®ts for any given truncation series can be estimated. This is done by locating in Fig. 2 the upper and lower ns where v2red v2red jmin 1, as shown by the solid line in the ®gure. In Fig. 3 these upper and lower limits for n are plotted as a function of the truncation as given by the maximum of f =fo of each series. Included in the ®gure are corresponding results obtained when the same analysis is performed using only 1% noise. In both instances the range of acceptable power law exponents increases as the truncation increases. For comparison, results from JH and Kahnt are included in the plot (fo was taken as the frequency where r 2ro ). Kahnt reported changes in n due to increasing temperature as well as selfimposed frequency truncation. As can be seen from the ®gure, the increases in n observed in these previous studies occur in approximately the same regime of truncation for which the error increases most rapidly. One could rightly argue that the above example, while illustrative of the inherent tendency for error in the exponent to increase with increasing truncation, is an unrealistic idealization. Since the data set was constructed from a JPL, the success of ®tting with the JPL is in some ways predestined.
Fig. 3. Range of exponent (coarse cross-hatched region) for the simulation data set (3%) determined from the v2red v2red jmin 1 condition. Finely cross-hatched region is for the same simulation run with 1% random error. Included (symbols) are variation of n at v2red jmin observed by JH [26] and Kahnt [13].
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Although many in the literature have suggested that the JPL does an adequate job of describing real conductivity data, perhaps there may exist systematic deviations between the data and the JPL that could promote the variation of the power law exponent with truncation. To consider this possibility, similar analysis on real conductivity data previously obtained [14] for a Li-metaphosphate glass is performed. A sample of the data obtained for four temperatures is shown in Fig. 4(a). The ®gure shows how changes in temperature eectively alters the window: with increasing temperature each spectrum becomes more dominated by dc conduction leaving less of the power law exposed. Fits of the JPL are also included in Fig. 4(a), and the residuals (Fig. 4(b)) indicate the ®t and data generally agree to within about 5%. Further analysis is now performed on the data set taken at 307 K. This particular spectrum is
Fig. 4. (a) Conductivity of LiPO3 glass at four temperatures. Solid curves are ®ts of the JPL. Line segments indicate successive data truncation discussed in the text. (b) residuals of ®ts for all four temperatures.
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selected since it possesses a well de®ned ro 4.0 (0.1) ´ 10ÿ7 S/m and equal portions of both ro and power law frequency dependence, is located in the midrange of the resolution limits of the spectrometer (Schlumberger 1260), and is not seriously aected [29] by contributions due to `constant loss' processes [5,30]. Again curve ®ts of the JPL are performed on a series of data sets formed by truncation of the 307 K spectrum (see Fig. 4(a)). In the ®ts, ro is ®xed (4.0 ´ 10ÿ7 ) and n ®xed at selected values. Each datum was weighted evenly with Dy taken as 0.015 corresponding to a 3% error in r. The resulting v2red -surface is shown in Fig. 5. In several ways the v2red -surface for the 307K spectrum of LiPO3 is similar to that of the previous simulation study (Fig. 2). The surface again displays a minimum in the vicinity of n 0.70 for the ®rst four series (A through D) and the width of this `valley' increases with increasing truncation. Unlike the simulation results, further truncation causes the minimum to shift toward larger exponents, much in the manner of observations by JH and Kahnt. However, accompanying this shift in the minimum of v2red is an increase in the error of determining the exponent. The range of acceptable n is again estimated by the v2red v2red jmin 1 condition and is shown in Fig. 6. Also included are the ns corresponding to v2red jmin , which ®rst increase to about 1 near fmax =fo 0:3 and then decrease to less than 0.1 with increasing truncation. Note
Fig. 5. Variation of v2red with power law exponent for various data truncations of the LiPO3 spectrum at 307 K shown in Fig. 4(a).
Fig. 6. Range of exponent (cross-hatched region) for the 307 K LiPO3 spectrum determined from the v2red v2red jmin 1 condition. Included (symbols) are variation of n at v2red jmin . Line indicates truncation-independent n 0.70 which remains within the error limits.
that n at v2red jmin is approximately centered between the upper and lower limits of the error and that n (n 0.70) found for limited truncation (fmax =fo 1) would constitute a statistically suitable ®t for all cases of truncation. Hence, the observed increase in n with truncation is not statistically signi®cant. The situation may be more evident in Fig. 7 where data for series E is shown together
Fig. 7. Truncated conductivity of LiPO3 at 307 K (series E in Fig. 4(a)) together with JPL ®ts at n 0.76, corresponding to v2red jmin and at n 0.63 and 0.92, corresponding to lower and upper error limits (see Fig. 6). Inset shows enhancement with error bars (5%) included.
D.L. Sidebottom / Journal of Non-Crystalline Solids 244 (1999) 223±231
with the `best' ®t (n 0.76) and with ®ts marking the upper and lower bounds for the exponent (n 0.92 and 0.63, respectively). Although the ®ts deviate considerably at frequencies greater than 104 Hz, they are comparable over a range less than this frequency. Indeed, if an error bar corresponding to 5% (about 2% on the logarithmic scale) is superimposed on each datum, as is done in the inset to the ®gure, it is evident that all three ®ts generally remain within these bounds. Despite the statistical insigni®cance of the shift in the position of v2red jmin , it is still curious that JH, Kahnt, and the present analysis all show a similar increasing trend. The origin of this increase may be understood by comparing features of the simulation study (which exhibited no shift in the position of v2red jmin ) with the LiPO3 study. One dierence between these two cases is the trends in the residuals displayed in Fig. 1(b) and Fig. 4(b). For the simulation, the residuals, produced by a random number generator, scatter randomly. But in the LiPO3 case, a weak systematic variation can just be discerned. This variation is more visible when the temperature dependence is removed by scaling the x-axis by the frequency, fo (where r 2ro ), as is done in Fig. 8. As is more evident in this ®gure, the residuals are seen to cluster in an oscillatory manner about the ®t curve. The trend is illustrated in the ®gure by the sinusoidal curve drawn with an
229
amplitude of 3%. Most of the residuals fall within 2% of this curve. This trend clearly indicates that the best ®t is not one that `threads' evenly through the data set, but rather is one that is forced to `weave' the data slightly in order to accommodate the substantial range of conductivity data including large portions of both dc and power law behavior. This weaving indicates that the simple JPL may not be suciently precise to account for the frequency dependence of the ac conductivity. The systematic deviation between data and ®t seen in Fig. 8 does however provide some insight into the window eect. When a frequency window is available which contains approximately equal portions of dc and power law conductivity, the optimum JPL ®t weaves the data set; that is, it sacri®ces data in one region to better accommodate data in another region. As the truncation approaches fmax =fo 1, corresponding to removal of the data associated with the ®rst half cycle of Fig. 8, the JPL ®t no longer needs to sacri®ce low frequency data since the constraints at high frequency have now been removed. Presumably, this change is why the parameter n associated with the v2red jmin increases in the vicinity of fmax =fo 1 (series E and F in Fig. 5), corresponding to truncation of the ®rst half cycle in Fig. 8. This explanation is also consistent with the observed decrease that occurs for n near fmax =fo 0:1 corresponding to truncation of the second half cycle. 4. Discussion
Fig. 8. Residuals of LiPO3 spectra ®t with JPL (see Fig. 4(b)) with frequency axis scaled by fo . Solid sinusoidal curve indicates the apparent systematic variation of 3% with width (dashed curves) of 2%.
The window eect, namely the sensitivity of the JPL exponent associated with v2red jmin to the truncation of data from the high frequency edge, is a real phenomenon and indicates that the JPL is an imprecise description of the ac conductivity of ionconducting glasses. However, I consider it impressive that a simple empirical function such as the JPL performs so well, describing r f in LiPO3 (as well as other materials we have studied) to within about 5% over roughly 20 square decades (about 3 decades in r by 6.5 decades in frequency for Fig. 4(a)). The imprecision in the JPL should be contrasted with alternative descriptions such as the
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KWW of the electric modulus which typically shows substantial deviations (often more than 40%) in the high frequency wing. A more detailed comparison between the success of ®tting data with the KWW or JPL cannot be made, as analysis using the KWW typically does not involve numerical curve ®tting in the present context since the Laplace transform of Eq. (1) in non-analytic. Instead, the KWW decay is more often approximated by a sum of exponentials and then transformed to obtained tables [8] of M f for selected bs. `Fits' of the KWW often involve judicious superpositioning of these tabulated results onto the observed M 00 f . Such ®ts are subjective and most examples in the literature indicate a disposition on the part of the researcher to emphasize the peak of M 00 f more than the high frequency wing [19]. If the Laplace transform of the KWW could be expressed as an analytic function, then an objective ®t of M f could be performed. Further, if these ®ts were required to maintain similar standards as those imposed upon the JPL ®tting, wherein all the data are equally weighted with none (e.g., the peak of M 00 f ) receiving special merit, it is likely the optimized ®t would also `weave' the data set and exhibit oscillatory systematic features in the residuals similar to those seen here only of much greater magnitude than 5%. The systematic deviations observed in the present study could be due to the existence of a dielectric loss peak residing beneath the dc conductivity. The existence of such a peak is justi®ed and predicted by several models [3,4], and would introduce a weak distortion in r f in the vicinity of fo which is not accounted for by the JPL. However, the present work indicates this loss peak is of limited signi®cance in comparison to the dc conduction, contributing only about 3% above the 2% random noise level (see Fig. 8). Analysis of r f that incorporates a dielectric loss together with the dc conduction may account for the 3% deviation missed by the JPL, but will do so at the expense of a 33% increase in the number of ®tting parameters. Clearly one must exercise some caution in the analysis of data sets with fmax =fo < 10, and recognize the inherent errors present in any curve ®tting procedure. Despite a small degree of im-
precision, the JPL can still be used to accurately evaluate data sets with larger fmax =fo and obtain a meaningful exponent which can be used to test universality and scaling issues. 5. Conclusions Descriptions of the ac conductivity of ionic glasses using the Jonscher power law display a window eect in the sense that the power law exponent obtained from curve ®tting analyses generally increases as data at high frequency are successively omitted. However, this truncation of the data leads to instability in determining the exponent and a corresponding increase in the error associated with the exponent obtained for the optimized curve ®t. Consequently, the observed increase is not statistically signi®cant and acceptable ®ts can be obtained with a truncation-independent exponent. Although the window eect indicates a degree of imprecision of the JPL to account for experimental data, this simple empirical function appears to successfully describe the ac conductivity to within 5% over ranges of about 20 square decades. Acknowledgements The author thanks Dr Laura Swiler for helpful discussions regarding statistical analysis of data. These conductivity measurements were performed at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under Contract DE-AC04-94AL85000. References [1] A.K. Jonscher, Dielectric Relaxation in Solids, Chelsea Dielectrics, London, 1983. [2] V.V. Daniel, Dielectric Relaxation, Academic Press, New York, 1967. [3] A. Hunt, J. Phys.: Condens. Mater. 3 (1991) 7831. [4] A. Hunt, Appl. Phys. A 54 (1992) 508. [5] L.A. Dissado, R.M. Hill, J. Appl. Phys. 66 (1989) 2511.
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