Dielectric loss spectra of organic glass formers and Chamberlin cluster model

]OllRb!Ab OF

llill ELSEVIER

Journal of Non-Crystalline Solids 215 (1997) 293-300

Dielectric loss spectra of organic glass formers and Chamberlin cluster model C. Hansen, R. Richert *, E.W. Fischer Max-Planck-lnstitut fiir Polymerforschung, Ackermannweg 10, 55128 Mainz, German 3, Received 28 October 1996; revised 15 January 1997

Abstract

Dielectric loss spectra d'(to) of 11 different glass-forming materials measured in the frequency range 10-3 to 106 Hz and at various temperatures were analyzed within the framework of a Gaussian distribution of independently relaxing domains as described by Chamberlin. The relaxation time distribution G(ln~-) derived from this model is shown to be paralleled by the results of an unbiased numerical transform of ¢"(to) into the most appropriate G(ln ~'). Deviations of loss spectra from a power law of the form log(d0 ct log(to) are well accounted for by the cluster-model and appear to an inherent feature of the a-process. However, the values obtained for the scaled cluster sizes x o = E/o" are too small to be compatible with the anticipated Gaussian probability density of cluster sizes, hence the applicability of the proposed dielectric function does not necessarily justify all assumptions of the model. © 1997 Elsevier Science B.V. PACS: 64.70.Pf; 77.22.Gm

I. Introduction

Dielectric relaxation spectroscopy is widely used for investigating the time scales of structural relaxations in many materials [ 1-4]. In the simplest possible case one obtains the Debye type dielectric function e *(to) = e' - i • d ' which corresponds to an exponential decay pattern in the time domain or to a 6-shaped peak in terms of the relaxation time distribution [5]. In such a situation the entire dynamics are characterized by the single value ~'. The Debye type loss spectrum e"(to) is symmetric with the asymp-

* Corresponding author. Tel.: +49-6131 379 116; fax: +496131 379 100: e-mail: [email protected].

totic slopes + 1 and - 1 for log(e") versus log(to) in the respective limits to<< T-1 and to>> 1"-~. However, practically all experimental loss curves are wider and possibly asymmetric relative to Debye behavior [1-4]. A possible method of describing such non-Debye behavior is in terms of a distribution of relaxation times which then displays a finite width. For these cases the most commonly used empirical fit function is the one proposed by Havriliak and Negami (HN) [6] e s -- 8~

e*(w)=¢=+

[1 + ( i t o ~ ' ) ' ] ' "

(1)

The HN form of e"(to) is similar, but not identical, to the Kohlrausch-Williams-Watts [7,8] (KWW) or stretched-exponential decay law exp[ - ( t / ~ ' ) t~] often employed in the time domain. In the limits

0022-3093/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 3 0 9 3 ( 9 7 ) 0 0 0 8 0 - X

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C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

to << T- 1 and to >> r - 1 the HN function also displays power laws, but in this case of the forms e" ¢t to~ and 6" ct to-~r, respectively. Although the HN type 6"(to) yields a satisfactory representation of measured results in the vicinity of the loss-peak in many cases, the high frequency wing is notorious in exhibiting systematic deviations from power law behavior [2,9-11]. Such deviations can be observed for a large class of materials and have been found previously by Dixon et al. [9] for several of the liquids analyzed here. Moreover, there exists no necessity for the high frequency portion of loss spectra to follow e" ¢x to-K. Because of the universal appearance of high frequency wings [9], a rationalization in terms of a superimposed independent loss peak does not seem appropriate. A picture leading to deviations from a e"cx to-~ pattern is the cluster model (CM) put forward by Chamberlin [10-13]. The scope of the present work is to analyze a series of different materials regarding their high frequency spectra in view of Chamberlin's cluster model. The comparison between experimental data and model function is made for both quantities, the loss spectra 6"(to) and the associated relaxation time distribution functions. Although the number of fit parameters in the CM does not exceed that of the HN function, the high frequency wings, especially the deviations from power law behavior, are well represented by the CM curves. The resulting scaled cluster sizes are often too small to comply with a Gaussian distribution of cluster sizes which is claimed to reflect the appropriate statistics of sizes for supercooled liquids, i.e., above the glass transition temperature Tg.

density centered at ~0 and subject to a standard deviation o-/v~-: p(s~)=~---1 exp[-(~-s%)2/o-2],

~>0.

(2) Without invoking a microscopic model for the molecular dynamics, the relation between cluster size ~: and relaxation frequency to or relaxation time r is assumed to be given by ~.(s¢) = to-,(s¢) = toll e x p [ - B / ~ : ]

= < l exp[C/x],

(3)

where B is a non-negative constant. The right half side of this equation is obtained after introducing the scaled sizes x = ~/tr and x 0 = ~0/o- and defining a correlation coefficient C = - B / o ' . Within the framework of the CM, the dielectric loss as a function of frequency is then given by 6"(to) = A f M 0

f0=seP(s¢)

toT(v)

1 + [toT(sO)] 2

d~, (4)

where M 0 is a normalization constant. In general, the loss spectrum e"(to) related to an arbitrary G(ln T) normalized such that fG(ln r)dln Z = 1 takes the form [5]

g'(to) = Aef+=G(lnT) -~

tot

1 + (toT)2 d l n r .

(5)

Comparing Eqs. (4) and (5) immediately constitutes the probability density G(ln r ) = r . g(T) which complies with the assumptions of the CM. Introducing the substitution s = C/x = ln(to=T) finally leads to +~

tot

2. Model predictions

6"(to) = A e f ~ 3'(s) 1 + (toT) 2ds'

The idea of the CM introduced by Chamberlin [10-13] was to formulate a physical basis which accounts for details of experimental decay pattern or susceptibility spectra. The starting point is to view the ensemble averaged response as a superposition of independently and exponentially relaxing subsystems or clusters. The inherent disorder of the addressed materials enters in terms of a distribution of cluster sizes ~. At temperatures above Tg these size statistics are assumed to follow a Gaussian probability

with T= to~l given by

exp[C/x] =

(6)

to~l exp[s] and with y(s)

=

-MoC2s-3exp[-(C/s-xo) ] O,

for s < 0 for s > 0.

(7) The value of M 0 is determined by the normalization condition fT(s)ds = 1 and has to be evaluated nu-

C, Hansen et aL / Journal of Non-C~stalline Solids 215 (1997) 293-300

8~P(,o), S~um(,o), and e~(,o). The error regarding the

100 10-1 to 10-2

295

approximation e " ( , o ) = 8~um(,o) Can b e e s t i m a t e d according to the following inequalities:

°c e x P [ - ( c / s - X o ) 2 ] /

M 0 e 2 , o e sa

-- QcS.3

10-3

2,o sg

'

(10a)

lO4 0 s;(,o)

s

merically. The probability density y(s) displays a peak at Smax positioned at "J- 6 ) 1/2 - - X 0 ] .

(8)

In the range s << Smax the shape of y ( s ) is governed by the term s-3 which emerges from the size dependence of the relaxation times r ( ~ ) . Its tendency to diverge at s = 0 is suppressed by the term e x p [ - ( C / s - x 0 ) 2] which dominates for s>> Smax such that y ( s ) becomes small near s = 0. The shape of the resulting loss spectrum s"(,o) depends only on the two parameters C and x o, whereas the absolute frequency position is determined by ,o~ via s = ln(,o~r). Fig. 1 displays an example for the function y ( s ) using the parameters C = - 4 and x0 = 1.

3. Numerical procedure The numerical difficulty in applying Eq. (6) reduces to finding appropriate upper and lower limits for the numerical integration. In order to outline the degree of approximations we define the following integral partitions ~"(`o)

As

: y ( s ) ,or ,~ y ( , ) `or f'~ ~ 1 + (`or) 2 d s + f ,7. 1 + ( `or)2ds

~o ~/(s),or ~h 1 + ( , o r ) 2 d s '

+ j.

AeMoC2

-

Fig. 1. Example for the probability density function y(s) as defined in Eq. (7) for the parameters C = - 4 and x o = 1 (M o = 0.55). The relation of y(s) to the CM prediction as regards e"(w) is stated in Eq. (6). The dashed lines outline the contributions according to the terms ors -3 and o ~ e x p [ - ( C / s - X o ) 2] as indicated.

Smax = I C [ ( x 2

<

(9)

with the respective rhs terms times A s denoted

2 ,o

[ ,o~e-'% + to - ,o~e- s, ln2(,o~/,o) sa

(lOb)

for s, > ln(,o~/,o),

Mo C2 e ; ( , o ) _< A e - ~ s ~

'

r [c exp/-L

[

- Xo

.

(lOc)

The integration limits s a and s b depend on the parameters A e, C, x 0, ,o~, and to. For the numerical integration these limits are set such that the total error remains smaller than a given maximum value, typically e"(,o) + s~(,o) _< 10 -3. The value of S'n'um(,o)= s"(,o) is calculated using Simpsons formula with a sufficient number of points on the s-scale, typically 100-200. In addition, the residue of the integration has been confirmed to be negligible relative to the data accuracy on the basis of the 4th derivative of the integrand. The fit curves for a given set of parameters obtained in this manner agree with previous numerical results by Chamberlin et al. [10,11].

4. Experiment The dielectric relaxation measurements analyzed presently was obtained for 11 different glass-forming materials using a Solartron 1260 gain phase analyzer in the frequency range 1 mHz to 1 MHz. A Chelsea Dielectric Interface has been employed for the current-to-voltage conversion. The sample capacitors were mounted in a N 2 gas stream cryostat and temperature controlled to within 0.02 K by a Novocontrol Quatro system equipped with a Pt-100 sensor. Experimental details of this technique have been stated elsewhere [3,4]. The temperature ranges for the samples have been chosen such that both peak and high frequency wing of s"(`o) are within the experimental frequency range. The list of glass-

296

C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

Table 1 Compilation of the glass-formers analyzed in terms of the cluster model together with their abbreviations and presently investigated temperature range AT AT (K)

Ref.

N-methyl-6-caprolactam Glycerol Salol Propylenecarbonate Propyleneglycol Poly(vinylacetate) m-tricresyl phosphate i-tricresyl phosphate 2-methyltetrahydrofuran Cresolphthaleindimethyl-ether

Sample NMEC GLYC SALO PRCA PRGL PVAC MTCP ITCP MTHF CPDE

175-193 195-221 223-245 161-169 173-189 309-325 211-223 217-231 93-97 321-337

this work [22] [4,22] [4,22] [22] [22] [22] [22] [22] [22]

Phenolphthaleindimethyl-ether

PPDE

301-315

[4,22]

$ v

formers analyzed according to the CM is compiled in Table 1. A Levenberg-Marquardt [14] algorithm has been used to determine the four free parameters A e, C, x 0, and to=.

5. Results A series of representative loss spectra is shown for NMEC in Fig. 2 and for PRGL in Fig. 3 together with their CM and HN fit results. Due to the logarithmic scales for e"(to) the deviations from a power

10

%

0.1

Ioglo(~/Hz) Fig. 3. Experimental results (symbols) for the dielectric loss spectra 6"(to) of propyleneglycol (PRGL) at the temperatures T = 173, 177 and 181 K in the order of increasing peak frequencies. The solid lines represent the best CM fits. The dashed lines are HN fits to the data in the vicinity of the peak.

law behavior and thus from the HN function at high frequencies are obvious. Qualitatively, this feature is found for the entire series of materials under study. The agreement between data and CM fit is similarly found for all substances. The temperature dependent fit parameters C, x 0, and to~ are depicted in Figs. 4 and 5. The ranges covered by the two shape parameters C ( - 1 2 < C < - l ) and x 0 ( 0 < x 0 < 2 . 5 ) a s a

NMEC $ 2f ,, ~ o • % . , x --60 • , ; g ° -2~.,a . . . . . a , (o~o,

0"06f 0 0.04

0 0

x° o.021

0

12[

0'001" . . . .

%

l

o 9 o o , o ,o

0.1 -2

= I

0

i

I

2

I\ "l

4

I'~ 3

6

Iogl0(m/Hz) Fig. 2. Experimental results (symbols) for the dielectric loss spectra e"(w) of N-methyl-8-caprolactam (NMEC) at the temperatures T = 175, 177 and 181 K in the order of increasing peak frequencies. The solid lines represent the best CM fits with the dc-conduetivity being disregarded. The dashed lines are t-IN fits to the data in the vicinity of the peak.

175

180

185

190

T/K

Fig. 4. Results for the variation of the CM parameters characteristic relaxation frequency to~, scaled average cluster size x 0, and correlation coefficient C with temperature for N-methyl-e-caprolactam (NMEC). The curve for log( to= / H z ) as a function of temperature parallels the equivalent data for log(tOr~ x/Hz), where tOmax is the frequency position of the loss peak with d e " / d to = 0.

C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

PRGL

'

i

,

297 [

'

I

4

. 1 K /

N 10-1

:f [] •

"-s

. . ~ o

~

-2

,

'

°

'

'

[]"1 •

,

'

%,.,

'

e--

~- 10-2

' 0)cq

i 1.0[ o o o o o x°0.5

10-3

o

0 0

0.0 . . . . . . .

(..)

o,

A

175

180

185

190

T/K Fig. 5. Results for the variation of the CM parameters characteristic relaxation frequency to~, scaled average cluster size x o, and correlation coefficient C with temperature for propyleneglycol (PRGL). The curve for log(to~/Hz) as a function of temperature parallels the equivalent data for log(tomax / H z ) , where tom,x is the frequency position of the loss peak with d e " / d t o = 0.

function of material and temperature and especially their correlation is graphically compiled in Fig. 6. For some of the dielectric results we have con2.5 %0 0 o o

2.0

o o •

•

1.5

•

o

X

• * + ~ o • v

1.0 0.5

0.0

o ,

o

CPDE ITCP MTCP PRGL PRCA NMEC GLYC PPDE SALO MTHF PVAC .

,

.

-12-10-8

~ °°

,

.

t

-6

.

i

-4

.

'

4

/

I

-2

n

I

0

2

Iog(~ / s)

A ~ ~

.S ~

i

-2

.

0

Fig. 7. Plot of the probability density G(ln~')= r . g ( r ) for finding a certain within the manifold of relaxation times for glycerol (GLYC) at T = 195 K. The symbols refer to the unbiased numerical transform e *(to) ~ G(ln r ) based directly on the experimental data. The lines represent G(ln r ) curves corresponding to the CM fit (solid line), to the HN fit (dotted line), and to the KWW fit (dashed line) to the experimental e"(to) data.

ducted a model-free numerical transform of e *(to) data into the most appropriate non-negative G(ln r) according to the inversion of Eq. (5). Because this belongs to the class of ill-posed problems, the method involves regularization techniques for solving Fredholm equations of the first kind [15]. The numerical realization of this algorithm has been initially worked out for time domain decay data [16] and later been adopted to handle frequency domain susceptibility results [17]. Roughly speaking, the scope of this algorithm is to find the smoothest allowable nonnegative G(ln~-) which still yields a satisfactory root-mean-square deviation. The self-consistency method used here can be shown to yield a more reliable estimate for the optimal regularization parameter relative to the statistical F-test used in the CONTIN program. The result of such a procedure for GLYC at T = 195 K is shown in Fig. 7, which also includes the G(lnr) curves as reconstructed from the CM and HN fits to the loss data. Fig. 7 is again representative for a large set of data analyzed along these lines.

C Fig. 6. Observed correlations between the scaled average cluster size x 0 and the correlation coefficient C for all materials compiled in Table 1 and each evaluated in the stated temperature interval. The different symbols are related to the glass formers as given in the legend. Different points for a common symbol refer to different temperatures. Note that the NMEC data lies within the range 0 _< x 0 < 0.05 (see Fig. 4).

6. Discussion

The common way of empirically representing frequency domain dielectric loss data by means of the HN function has proven to be appropriate when regarding mainly the upper most decade regarding

298

C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

e"(to) or the frequency region around the loss peak. At frequencies much above the position Wm~~ of the loss peak where e"(OJ)/e"(OJma x) can be as low as --0.005 the observed loss is often significantly higher than the HN fit. These subtle deviations become obvious only on logarithmic e" scales but appear to be a universal feature of dielectric relaxation [ 10-13]. According to our present analysis, the intent of the cluster model proposed by Chamberlin to capture such fine details of a susceptibility spectrum has been achieved to a remarkable degree of accuracy. This is underlined by the notion that the CM fits to the spectra of the nine other materials which are not shown here are as accurate as demonstrated in Figs. 2 and 3. Noting that the CM function does not involve more free parameters than does the HN type fit, such a coincidence between data and fit strongly suggests that the typical form of the high frequency wing is an inherent feature of the a-process. A drawback of comparing fits and loss data in an e"(log to) representation is the fact that a single relaxation time contribution to the entire curve already corresponds to a Debye type process which is 1.14 decades wide (FWHM) so that subtle details are obscured in the d'(log to) picture. G(ln ~-) curves are more decisive because here a single relaxation time contributes as an infinitely sharp peak. Fig. 7 indicates the almost prefect coincidence of the unbiased G(ln~-) result derived from the data with the CM equivalent probability density calculated from the CM fit parameters. This result also demonstrates the capability of the e*(to)~ G(ln~-) algorithm of resolving the details of such a typical probability density of a supercooled liquid. The HN equivalent G(lnr) included in Fig. 7 is analytically given by the expression [18] GHN(In r )

(~'/~-HN) '~'rs i n ( y ~ )

'II"[('r//"/'HN) 2a -'F 2(~'/THN) ~cos(Tra) + 1] v / z ' sin ( "rra ) with • = arctan (~./rHN) ~ + COS(WC~)

(11)

In agreement with the corresponding e"(~o) picture the G(ln~-) data in the range of small relaxation

times display components which significantly exceed the values expected on the basis of a HN fit. We conclude that the frequency dependent susceptibility including its detailed structure in the outer wing can be reduced to a few parameters for a large set of experimental data while obtaining a data representation which is superior over the HN type data reduction. To this end we have regarded the CM function only as a numerical tool for casting the information of a loss spectrum into a set of four parameter values while disregarding the physical model behind the CM fits. For the two examples, NMEC and PRGL, the CM parameters C, x 0, and oJ~ are displayed is Figs. 4 and 5, respectively. The results for log(w~) basically follow the variation of log(~Omax) with temperature, where O)max refers to the position of the e" peak on the w scale. The systematic offset between to~ and COma x arises from to~ being related to the slow relaxation time of an infinitely large cluster, i.e., for ~:-~ ~. The tendency of the difference between tog and tOmax to decrease with increasing temperature might be due to the effect of the high frequency wing being shifted to frequencies above the present experimental limit of 1 MHz. It appears attractive to derive the average ~ and standard deviation tr of the density of cluster sizes from the results of such a CM analysis. However, such conclusions are obscured by the notion that these values enter the shape parameters only in terms of the scaled quantities x 0 = ~0/tr and C = - B / t r . Therefore, absolute measures for the length scales of the clusters cannot be obtained and, secondly, they would critically depend on the assumed dependence of relaxation times on the cluster size. Especially, the qualitatively similar relaxation behaviors of NMEC and PRGL are contrasted by the very different trends of xo(T) and C(T) obtained for these materials. As shown in Fig. 6 the values of x 0 and C are highly correlated. The general trend for xo(C) is that for a certain material the variation of the two parameters with temperature can in many cases be approximated by [ C [ - 1 ct x 2. For some samples, e.g., salol, the variation of x 0 is not pronounced such that [C]ct x 0 equally holds, in accord with the previous [10] observation of Xo/ICI = 0.2 for salol. However, a general correlation of the form Xo/[C[ = constant claimed previously [10,11] can be excluded on the

C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

basis of the data now available because for the low values of x o we find C ~ - l for x 0 ~ 0 . The effect of the observed xo-C correlation is that the shape is approximately determined by a single parameter, which again emphasizes that the high frequency excess loss relative to a power law is most likely a universal signature of the a-process, in contrast to the picture of a distinct additional process at frequencies well above the e" peak position. The applicability of the CM type fit function to a wide range of materials and temperatures suggests the existence of a universal deviation from a power law behavior of the form e " ~ to -K. Such power laws are not only inherent in the empirical HN type dielectric function (and its special cases C o l e Davidson and Cole-Cole types [5]) but also claimed by the Curie-yon Schweidler-law [19-21] ~ ( t ) a t - ( t / ~ ' ) ~ and by the KWW type decay for the fast components of the relaxation. That these deviations from e " c t to -K are properly accounted for by the two CM shape parameters leads to the same conclusion regarding the universality of this feature as does a scaling argument. For instance, the scaling of dielectric loss curves proposed by Dixon and Nagel [9] leads to a material independent master plot (for systems with e" oc to for to << ~-- 1) which also collapses the power law deviations onto a single curve. Several of the systems listed in Table i have been subjected to such a scaling analysis in [9]. The values of ~0 and x 0 = ~0/o- refer to the average cluster size which lead us to limiting the fit parameter x 0 to non-negative values, i.e., x 0 >_ 0. The unexpected dependence of x 0 on temperature seen for NMEC in Fig. 4 is therefore a consequence of the x o > 0 limitation, rather than reflecting changes in the relaxation behavior. A comparison with the results for PRGL in Fig. 5 indicate that the condition of x 0 >_ 0 has a feedback effect on the behavior of C(T). The more severe problem is encountered already for positive values of the average scaled cluster size x 0 > 0. As outlined in Fig. 8, small average cluster sizes ~0 are associated with a Gaussian density for ~ which possesses a significant contribution in the unphysical range ~ < 0. Such a situation corresponds to an effective p( ~ ) which has the form of a truncated Gaussian and which is no longer considered to reflect a physically reasonable model. Since for NMEC the x o results are very

299

0.8sx.P -.-4 Q..

0.6-

,.-, 0.4 "Q"

L

0.2,/

o.0

t

Xo = 0

2

4

6

8

~/~o Fig. 8. Plot of the probability density p ( ~ ) o t e x p [ - ( ~ ~:0)2/cr'~J for finding a certain cluster size ~: according to the Gaussian type CM ((r = so0/x0). The curves are parametric in the scaled average cluster size x o for the values x 0 = 0.1, 0.2, 0.3, 0.4, 0.5, and 1 in the order from wider to narrower curves.

small within the entire experimental temperature interval, the situation of an effective cutoff at ~-- 0 is encountered also for loss spectra where the high frequency wing is well inside the present frequency range. Obviously, the quality of the CM fit is not affected by the density p( ~ ) being truncated near its maximum, emphasizing that even a perfect CM fit gives no stringent prescription for the shape of the cluster size distribution at small sizes. In summary, we have conducted an in depth analysis of the shape of dielectric loss spectra regarding the structural or a-process for a series of glassforming materials. We focus on the comparison between the empirical function established by Havriliak and Negami and the predictions of the cluster model proposed by Chamberlin. Although the number of free parameters is equivalent, the CM fit accounts in a remarkable fashion for the loss observed in the high frequency wing which appears as systematic deviation from any empirical function which conforms with a power law e" ~ to -K. Based on the qualitative universality of this high frequency behavior together with the correlation between the two CM shape parameters x 0 and C we conclude that the deviations from a power law like e"((o) are a universal and inherent feature of the a-process. To this end, however, the coincidence between loss data and CM fit alone does not provide sufficient justification for the assumptions underlying the Chamberlin model. As inferred from the optimized fit parame-

300

C. Hansen et al. / Journal of Non-Crystalline Solids 215 (1997) 293-300

ters, the probability density function p ( ~ ) of cluster sizes often deviates strongly from the initially anticipated Gaussian shape. Because also the truncated Gaussian p(~:) yields very good fit results, the assumptions regarding the cluster size distribution seem not to enter critically into the corresponding CM fit function. As a consequence, a microscopic interpretation of the obtained parameters can not be warranted at present.

7. Conclusions Employing the CM fit we find that the high frequency deviations from a power law type loss profile appear to be a universal and inherent feature of the dielectric a-process. Despite the very effective data reduction when applying the CM equations, the according physical interpretation of the results in terms of independently relaxing clusters remains ambiguous. Furthermore, any physical conclusions drawn from a CM analysis of relaxation data critically relies on the relation ~-(~:)ot e x p [ - B / ~ ] , for which further justification in terms of a microscopic model for the molecular dynamics in supercooled liquids is not yet given.

Acknowledgements The authors are grateful to R.V. Chamberlin for stimulating discussions. We also thank F. Stickel for the access to the original dielectric data. Financial support by the Deutsche Forschungsgemeinschaft (SFB 262) is gratefully acknowledged.

References [1] N.G. McCrum, B.E. Read, G. Williams, Anelastic and Dielectric Effects in Polymeric Solids (Dover, New York, 1991). [2] A. Hofmann, F. Kremer, E.W. Fischer, A. Schi~nhals, in: Disorder Effects on Relaxational Processes, ed. R. Richert and A. Blumen (Springer, Berlin, 1994). [3] F. Stickel, E.W. Fischer, R. Richert, J. Chem. Phys. 102 (1995) 6251. [4] F. Stickel, E.W. Fischer, R. Richert, J. Chem. Phys. 104 (1996) 2043. [5] C.J.F. BiSttcher, P. Bordewijk, Theory of Electric Polarization, Vol. 2 (Elsevier, Amsterdam, 1978). [6] S. Havriliak, S. Negami, J. Polym. Sci. Polym. Symp. 14 (1966) 89. [7] R. Kohlrausch, Pogg. Ann. Phys. 91 (1854) 179. [8] G. Williams, D.C. Watts, Trans. Faraday Soc. 66 (1970) 80. [9] P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams, J.P. Carini, Phys. Rev. Lett. 65 (1990) 1108. [10] R.V. Chamberlin, R. BShmer, E. Sanchez, C.A. Angell, Phys. Rev. B46 (1992) 5787. [11] R.V. Chambedin, Phys. Rev. B48 (1993) 15638. [12] R.V. Chamberlin, D.W. Kingsbury, J. Non-Cryst. Solids 172-174 (1994) 318. [13] B. Schiener, A. Loidl, R.V. Chamberlin, R. BiShmer, J. Molec. Liq. 69 (1996) 243. [14] P.R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969). [15] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977). [16] H. SchMer, U. Albrecht, R. Richert, Chem. Phys. 182 (1994) 53. [17] H. Sch':ifer, E. Sterain, R. Stannarius, M. Arndt, F. Kremer, Phys. Rev. Lett. 76 (1996) 2177. [18] S. Havriliak, S. Negami, Polymer 8 (1967) 16l. [19] E. yon Schweidler, Ann. Phys. 24 (4) (1907) 711. [20] R. Schilling, in: Effects on Relaxational Processes, ed. R. Richert and A. Blumen (Springer, Berlin, 1994). [21] W. G~tze, Vth Int. Symp. on Selected Topics in Statistical Mechanics (World Scientific, Singapore, 1990). [22] F. Stickel, thesis (Verlag Shaker, Aachen, 1995).