Comparison of shear response with other properties at the dynamic glass transition of different glassformers

Journal of Non-Crystalline Solids 307–310 (2002) 270–280 www.elsevier.com/locate/jnoncrysol

Comparison of shear response with other properties at the dynamic glass transition of different glassformers €ter *, E. Donth K. Schro Fachbereich Physik, Universit€at Halle, 06099 Halle (Saale), Germany

Abstract The dynamic shear response of six glass formers (the epoxy resin diglycidyl ether of bisphenol-A, propylene carbonate, glycerol, two bulk metallic glasses, and a soda lime glass) from different substance classes with widely differing fragility is compared: No correlation of the shear Kohlrausch exponent with fragility was observed. A comparison of shear with multidimensional NMR results on glycerol is compatible with dynamic heterogeneity in glass formers having mobile islands in a lesser mobile matrix. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 64.70.Pf; 66.20.þd; 62.20.-x

1. Introduction Two of the canonical signatures of glass transitions are [1,2]: (i) the non-Arrhenius temperature dependence of relaxation times, quantified by the fragility index m [3], and (ii) the non-exponentiality of correlation functions, quantified for instance by the b exponent in the Kohlrausch–Williams– Watts (KWW) function [4,5]. In an extensive compilation of literature data for many substances B€ ohmer et al. [3] found a certain trend: increasing non-exponentiality for increasing fragility. They used values from many different sources analyzing different physical sig-

* Corresponding author. Tel.: +49-345 552 5352; fax: +49345 552 7017. E-mail address: [email protected] (K. Schr€ oter).

nals at the dynamic glass transition (a relaxation) but did not discuss the difference between them. We will concentrate on the question of different physical signals (‘different activities’). To gain more insight into this subject we have conducted detailed dynamic shear experiments on the a relaxation in six substances from different material classes, with a wide variation in fragility. Diglycidyl ether of bisphenol-A (DGEBA) and propylene carbonate were chosen because many informations from e.g. dielectrics [6–8], light scattering [9,10], NMR [11] are available. Moreover, propylene carbonate is also the ‘outlier’ in the B€ ohmer plot. Glycerol is one of the archetypal glass formers. The relatively new bulk metallic glasses should be compared with the more traditional glass formers. The soda lime glass has a low fragility, and detailed viscosity data are available from literature. Some of the data on glycerol and the bulk metallic glasses were already published [12,13].

0022-3093/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 2 ) 0 1 4 7 6 - X

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

271

Finally, we will compare shear [12] and multidimensional NMR [14] results on glycerol to get more informations on dynamic heterogeneity.

2. Experimental DGEBA is a commercial sample (Epon 828) from Shell. Propylene carbonate was purchased from Aldrich. It is an anhydrous sample and contains less than 0.005% water. The inorganic standard glass is a sample from the Deutsche Glas Gesellschaft (DGG), distributed by the H€ uttentechnische Vereinigung der Deutschen Glasindustrie e.V. Frankfurt/Main together with a viscosity table. It is a soda lime glass with a quoted content of 71.72 wt% SiO2 , 14.95 wt% Na2 O, 6.73 wt% CaO, 4.18 wt% MgO, 1.23 wt% Al2 O3 , 0.436 wt% SO3 , 0.38 wt% K2 O, 0.191 wt% Fe2 O3 , 0.137 wt% TiO2 . All measurements were done with a rheometrics dynamical analyzer RDA II. The inorganic glass was measured in stripe geometry and the molecular liquids with an elongated sample between parallel plates. For propylene carbonate, glycerol, and the bulk metallic glasses the sample chamber was fluxed with dry nitrogen gas to avoid the influence of moisture. Prior to the measurements the samples were held at the desired temperature for at least 15 min.

3. Results Fig. 1 presents the experimental shear modulus data for DGEBA at about 2 K above Tg measured over five frequency decades. The lower part of the figure shows the modulus spectrum H calculated from the data with the help of the NLREG program of Honerkamp and Weese [15]. Note the inverse time scale with short times on the right. This spectrum and the imaginary part of the modulus shows a short time (high frequency) wing with a slope of about 0.45. For low frequencies the values of the real part fall below the lower measurement limit of the RDA II transducer. The error bars there show the correspond-

Fig. 1. Upper part: Double logarithmic representation of complex shear modulus G for DGEBA as a function of frequency x at a temperature of 256.25 K. The symbols show the measured values and the lines are recalculated from the spectrum in the lower part of the figure. Lower part: Shear relaxation spectrum H calculated from the measured data. The continuous line is the KWW spectrum calculated for bKWW ¼ 0:45.

ing uncertainty (see below). Otherwise the uncertainty is smaller than the symbol size. The spectrum terminates at frequencies below the peak frequency in G00 . Relaxation processes with longer spectral time scales are not important. This corresponds to a formulation by Buchenau: ‘‘If a . . . relaxation barrier is too high, the relaxing entity will flow away before it has a chance to jump’’ [16]. The pure viscous flow with G00 proportional to x dominates there.

272

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

Fig. 2. Double logarithmic plot of the complex shear modulus for DGEBA as a function of frequency at a temperature of 253.3 K.

The slight bend of the imaginary part at high frequencies, or, correspondingly, at the short time side of the spectrum, is real. This bend, indicated in Fig. 1, enters the frequency window of the shear measurements more clearly if we go to a slightly lower temperature (Fig. 2). All shape parameters in the following correspond to the steeper part. Fig. 3 presents the same experimental data (Fig. 1) of DGEBA at 256 K in the compliance representation. As often emphasized by Plazek [17] this representation has the advantage that the viscous contribution 1=xg can simply be subtracted from J 00 . We get also a peak in the remaining retardation part of the compliance, J 00  1=xg. The experimental problem is the evaluation of small differences which causes steeply increasing uncertainties at low frequencies, visible as scatter in the measurement points. This could be overcome by shear creep recovery experiments in an apparatus like that of Plazek [18], not available in our lab. The error margins were estimated as follows. For J 00  1=xg the scatter results from the temperature stability of the measurement and the corresponding fluctuation in viscosity (0.5% in this case) which should otherwise be constant at the lowest frequencies. For the real parts J 0 and G0 the scatter results mainly from fluctuations of the measured phase angle of the rheometer (0.3° here).

Fig. 3. Upper part: Complex shear compliance J  for DGEBA as a function of frequency x at a temperature of 256.25 K. The full symbols show the measured values for real and imaginary part of J  and the open symbols the value for the retardation contribution to J 00 , J 00  1=xg. The dotted line is the viscous contribution to J 00 . Note the linear scale for the real part J 0 . The full lines are a fit with the Havriliak Negami function. Lower part: Shear retardation spectrum L calculated from the measured data.

Note also that the real part of the compliance J 0 , shown on a linear scale, gives only a change of about a factor of four, not the many decades like the imaginary part. The lower part of the figure shows the corresponding compliance spectrum L, again with a slope of about 0.45 at the short time side. Fig. 4 shows the analogous modulus and compliance data for propylene carbonate. Fig. 5 gives the results for the soda lime glass. At low

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

273

frequencies, no plateau in the real part of the modulus [19] (as an indication of a network structure) was observed. Additionally, to the scatter of the phase angle a small temperature decrease (about 0.5 K) during the measurement at this high temperature limit of the RDA II was recognized. This is the reason for the slight bend in real and

Fig. 5. Upper part: Double logarithmic representation of the complex shear modulus G for the inorganic standard glass as a function of frequency x at a temperature of 847 K. The full symbols show the measured values and the lines are recalculated from the spectrum (not shown). Lower part: Complex shear compliance J  for the inorganic glass as a function of frequency x at a temperature of 847 K. The symbols show the measured values for real and imaginary part of J  . The dotted line is the viscous contribution to J 00 . Note the linear scale for the real part J 0 . The full lines are a fit with the Havriliak Negami function. Fig. 4. Upper part: Double logarithmic representation of the complex shear modulus G for propylene carbonate as a function of frequency x at a temperature of 159 K. The full symbols show the measured values and the lines are recalculated from the spectrum (not shown). Lower part: Complex shear compliance J  for propylene carbonate as a function of frequency x at a temperature of 159 K. The full symbols show the measured values for real and imaginary part of J  and the open symbols the value for the retardation contribution to J 00 . The dotted line is the viscous contribution to J 00 . Note the linear scale for the real part J 0 . The full lines are a fit with the Havriliak Negami function.

imaginary part of the modulus and the real part of the compliance below x  101 rad s1 . The frequency of the peaks in the imaginary parts of modulus and compliance are compared with results from other methods in the Arrhenius diagrams (Figs. 6–8). For DGEBA, dielectric measurements from literature and from our lab, with their broad

274

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

Fig. 6. Arrhenius diagram (log x versus 1=T ) for DGEBA from dielectrics ((j) and ( ) [6]), light scattering [9] (), TMDSC [20] (.), the maximum in G00 (,), the maximum in the retardation contribution to J 00 (n) and the inverse Maxwell times from viscosity (Eq. (1)) ( ).

Fig. 8. Arrhenius diagram (log x versus 1=T ) for propylene carbonate from dielectrics ((j) and ( ) [8]), TMDSC (.), the maximum in G00 (,), the maximum in the retardation contribution to J 00 (n) and the inverse Maxwell times from viscosity ( ).

Typically, all compliances for small-molecule glass formers are on a common trace in the Arrhenius diagram, with a variation of about plus minus half a decade in frequency [12,21]. The shear modulus peak is about a frequency decade above. To include the viscosity g in this diagram the Maxwell relaxation times were calculated according to sMaxwell ¼

Fig. 7. Arrhenius diagram (log x versus 1=T ) for the inorganic glass from TMDSC [22] (.), the maximum in G00 (,), the maximum in the retardation contribution to J 00 (n) and the inverse Maxwell times from viscosity ( ).

frequency window, give the guide in this diagram (Fig. 6). Their extrapolation to lower frequencies coincides with the peaks in shear compliance and calorimetry by temperature modulated DSC [20].

1 xMaxwell

¼

g G1

ð1Þ

with a temperature independent glass zone modulus of G1 ¼ 9:6  108 Pa. The value of G1 is taken from the Havriliak Negami fit (see below). This procedure gives a time scale near the G00 trace with a temperature dependence similar to the other signals. For the inorganic glass the arrangement of the traces is similar (Fig. 7). The calorimetric peaks [22] are near the peak in the shear compliance. Dielectric results were not available. For propylene carbonate there are some peculiarities (Fig. 8). First, the peak frequencies of the imaginary parts of shear modulus and compliance are different by only half a decade (compare also

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

Fig. 4). Secondly, the calorimetric point from TMDSC [23] is above the extrapolated trace of the dielectric function and shear compliance. For high fragilities the mutual arrangement is very sensitive to temperature calibration errors in the different measuring devices. The error bar for TMDSC in Fig. 8 corresponds to a temperature uncertainty of 1 K. For polymers one must pay attention to the fine structure of the main dispersion zone [24,25]. Only the contributions on the short time side of the whole dispersion zone correspond to the ‘proper glass transition’ [24] or ‘local segmental motion’ [25]. There the monomeric units get mobile. For longer times, but still within the dispersion zone, they start to flow past each other and begin to feel the chain structure: Polymeric modes develop and get stopped at the long time side of the dispersion by the chain entanglement. The huge peak in the shear compliance J 00 or the L spectrum corresponds to these polymeric modes and is at distinctly lower frequencies (typically three decades below the shear modulus peak frequency). In some cases like polystyrene, by a combination of shear measurements at widely different time scales, a separation of the ‘proper glass transition’ contribution was possible [26]. Only this contribution should be compared to the other physical signals at the glass transition. Then also polymers show a similar arrangement of the different signals as described above [27–29]. Next we analyze the shape of the dynamic glass transition peaks. It is useful to parameterize the curves with different empirical functions. One of them is the KWW function [4,5] "  bKWW # t GðtÞ ¼ G1 exp  ð2Þ sKWW used here formally for the stress relaxation data, calculated from the measured frequency dependence of the shear data. In this case G1 is the glass zone modulus. In the fit procedure this value was fixed to the result of the Havriliak Negami fit on the dynamic shear compliance (see below). Fig. 9 shows an example for DGEBA. From the parameters the corresponding shear modulus spectrum H was calculated according to

275

Fig. 9. Stress relaxation modulus GðtÞ as calculated from the dynamic data of Fig. 1. The solid line is a fit with the stretched exponential function (Eq. (2)) with an exponent of 0.46. The glass zone modulus was fixed.

Ref. [61]. It is included in the lower part of Fig. 1. The Kohlrausch spectrum matches the main features of the experimental spectrum but slight differences in shape are apparent. The KWW function with its one shape parameter can always be only an approximation. It seems not particularly useful to discuss small differences between Kohlrausch exponents from modulus or compliance representation. Another possibility is to fit the dynamic shear compliance data with the Havriliak Negami function [30]. " # 1 0 J ðxÞ ¼ J1 þ ðJs  J1 ÞRe b c ð1 þ ðixsHN Þ HN Þ HN " 1 1 þ ðJs  J1 ÞIm J ðxÞ ¼ b c xg ð1 þ ðixsHN Þ HN Þ HN

ð3Þ #

00

ð4Þ with J1 and Js the limiting values of the real part at high and low frequencies, respectively, g the zero shear rate viscosity, bHN and cHN the two shape parameters, and sHN the characteristic time scale for the dispersion. The fits are shown as continuous lines in the Figs. 3–5. The resulting parameters are listed in

276

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

Table 1 Parameters from the fits with the KWW (Eq. (2)) and Havriliak Negami (Eqs. (3) and (4)) functions Substance

bKWW

bHN

cHN

sHN (s)

m

T (K)

DGEBA Propylene carbonate Glycerol [12] Zr65 Al7:5 Cu17:5 Ni10 Pd40 Ni40 P20 Inorganic standard glass

0.45  0.02 0.42  0.04 0.435  0.01 0.45  0.02 0.42  0.04 0.45  0.04

0.65  0.05 1a 1  0.05 1a 1a –

0.7  0.1 0.36  0.04 0.44  0.05 0.37  0.15 0.36  0.05 –

34  8 12  2 35  1 45  10 1  0.3 –

153 104 53 43 (1 Hz) 41.5 (1 Hz) 35.4

256.25 159 192.5 644 591 847

a

The value has been fixed for the fit.

Table 1. The bHN parameter in some cases was fixed to 1 because of the limited frequency range on the low frequency side of the peaks. The small temperature drifts during the measurement for the inorganic glass caused greater errors of the parameters. For this reason the Havriliak Negami parameters were not included in Table 1. The fit with the KWW function is more sensitive to the short time side of the peak which was not influenced by the drift. The two Havriliak Negami exponents b and c could be converted to some ‘equivalent’ KWW exponent according to Ref. [31]. The associated problems were discussed in Ref. [32]. Of course, modulus GðtÞ and compliance J ðtÞ can not simultaneously correspond to KWW functions. This is only an approximation. But the difference between the exponents in both representations is small, especially in the region 0:4 . . . 0:5 as in our case [33]. So we use the exponents from GðtÞ.

4. Discussion For all samples we find KWW exponents from shear in the range 0.42–0.45, irrespective of their wide variation in fragility. Contrary to the expectation from Ref. [3] there is no strong correlation to fragility. Furthermore, we observe important differences between results from different physical signals (‘different activities’). In Fig. 10 the KWW exponents from shear and dielectrics, if available, are compared for our samples, together with the original results [3] for all the other samples. The shear exponents seem to ‘decouple’ from the trend for the other physical signals. Typical shear exponents near 1/3 (so-called Andrade creep) for

Fig. 10. Correlation of fragility index m and Kohlrausch exponent bKWW . The solid symbols show the results on the six substances investigated here. The symbol (j) on the left comes from the shear experiments and the symbol (d) on the right from dielectrics. The open symbols are from the original diagram in Ref. [3].

many molecular glass formers were also observed by Plazek and Bero [34,35]. For the highly fragile poly(vinyl chloride) in Ref. [36] a KWW exponent of 0.31 for short time creep and 0.33 for long time creep experiments was found. For an inorganic Y– Si–Al–O–N glass a value of bKWW ¼ 0:48 from shear near Tg was observed in Ref. [37]. From the coupling-model [38] a correlation of fragility and non-exponentiality (or coupling parameter n ¼ 1  bKWW ) is proposed. Despite some success, the problem of different bKWW parameters from different physical signals for the same substance remains [39] . Also the comparison of flow and glass transition in a series of polybutadienes of different microstructure in Ref. [40] shows

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

inconsistencies with the coupling-model. For different substituted poly(p-phenylenes) in Ref. [41] a variation of fragility with molecular structure, but a constant bKWW of 0.42–0.46 from dielectrics was found. Shear measurements on polystyrene of different molecular weight gave a variation in fragility but a constant shape of the relaxation curves at the dynamic glass transition [42]. Other theories make quantitative predictions on the value of the bKWW parameter. Some theories based on relaxation on a fractal [43] or on percolation [44] give bKWW ¼ 0:33, Brownian diffusion in a box with fixed obstacles [45,46] gives 0.43 or 0.6, statistical models [47] bKWW ¼ 0:27, or percolation models [48] result in a certain connection of bKWW and fragility. A scaling model [49] gives a short time power law for the stress relaxation with an exponent of 0.33. A cluster scaling model [50] gives similar correlation functions as the KWW function, and additionally includes also the short time wing [51]. Our constant value of the Kohlrausch exponent from shear is not compatible with the percolation model [48]. Others [52,53] suppose a linear variation of bKWW from 0 to 1 with temperature between the Vogel and the crossover temperature, respectively. In general, this quantitative correlation for the ‘ideal WLF situation’ (Ref. [2], Section 4.3) cannot be true. For instance for propylene carbonate dielectrics [7] find bKWW increasing from 0.7 to 0.9 in the temperature range 150–220 K (the crossover [54] is at 189 K). Depolarized light scattering [10] above the crossover finds a constant bKWW of 0:77  0:05 in the wide temperature range from 210 to 350 K. An important issue in glass transition research is the question of dynamic heterogeneity [2,55, 56]. Some methods like multidimensional NMR [57,58] are able to select dynamic subensembles and measure their individual relaxation. With the help of spin diffusion measurements the spatial extent of these slower regions was estimated. For poly(vinyl acetate) a ‘heterogeneity’ length scale of 3  1 nm at Tg þ 10 K was obtained [59]. But there are different possibilities for the spatial arrangement of slower and faster regions in space. If we assume one continuous matrix and one disperse phase there are two possibilities: a more mobile

277

Fig. 11. Schematic picture of a dynamically heterogeneous system composed of lesser mobile (dark) and more mobile (bright) regions. In case (a) the lesser mobile fraction is the continuous matrix, in case (b) the more mobile fraction.

matrix with lesser mobile inclusions, or, inversely, more mobile islands within a lesser mobile matrix. This is schematically depicted in Fig. 11. What would shear measurements find in the two cases? Shear on a structurally heterogeneous sample is spatially sensitive on the continuous matrix in both cases. The matrix dominates the shear response [60]. The inclusions lesser contribute to the response of the whole system. We suppose the same behavior also for shear on a dynamically heterogeneous pattern. The time scales should correspond to the characteristic a relaxation time, where the dynamic heterogeneity has not been completely averaged. The combination of the spatial selectivity of shear response and the time scale selectivity of NMR should be able to distinguish between the Fig. 11 alternatives for the spatial arrangement. NMR results [14] on glycerol give us the opportunity for a check. First the relaxation of all molecules was measured in a 2D experiment [14] and fitted with a KWW function. For instance, at a temperature of T ¼ 199 K a Kohlrausch exponent bKWW ¼ 0:62 and a time constant sKWW ¼ 460 ms were obtained. From this results we obtain the mean correlation time [61], hsiKWW ¼ sKWW

Cð1=bKWW Þ ; bKWW

ð5Þ

of 664 ms. The peak frequency in the corresponding susceptibility in the frequency domain can then

278

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

Fig. 12. Arrhenius diagram (log x versus 1=T ) for glycerol from dielectrics ((s) [8]), heat capacity spectroscopy ((þ) [63], () [64]), the maximum in G00 (,), the maximum in the retardation contribution to J 00 (n), and NMR ((m) [14]).

be calculated [2,62] as log xmax =rad s1 ¼ 0:237. The results of xmax for the three temperatures of Ref. [14] were added to the Arrhenius diagram of our shear results [12] (Fig. 12). The NMR data are on a common trace together with heat capacity spectroscopy, parallel to the dielectric trace. This enables the slight extrapolation to the temperature of the shear measurements, where the NMR peak

would practically coincide with the peak in the imaginary part of the shear compliance. Differences in the temperature scale of shear and NMR experiments are not critical because glycerol has only a moderate fragility. In the interesting temperature range an error of 1 K would correspond to a frequency shift of 0.23 decades. Then in a multidimensional NMR experiment [14], 70% of the molecules were selected as the slow subensemble. For these molecules a heterogeneity length scale of about 1 nm was estimated [14]. The distinction of ‘slow’ and ‘fast’ time scales is at the disposal of the experimenter but it also governs the proportion of the two molecular subensembles. To get an impression of the difference in time scales between ‘slow’ and ‘fast’ subensembles, we calculated a formal distribution of correlation times for the mean NMR Kohlrausch exponent bKWW ¼ 0:55 according to Ref. [61]. Then we divide this distribution in two parts according to 70% and 30% contribution, respectively. This gives a ‘cutting time’ of 0.375 sKWW . For both parts of the distribution above and below this boundary we calculated a mean correlation time. This gives sslow ¼ 2:368sKWW and sfast ¼ 0:135sKWW , i.e. a difference of about 1.25 decades. So a distinction between the two times is possible also on a logarithmic time scale. The mean correlation time of the slow subensemble sslow ¼ 2:368sKWW is similar to the

Fig. 13. Double logarithmic representation of the imaginary part J 00 and linear representation of the real part J 0 of the shear compliance for glycerol as a function of frequency x at a temperature of 192.5 K [12]. Indicated are the peak in susceptibility from NMR for the whole ensemble of molecules (large arrow) and the slow and fast selected subensembles of NMR (smaller arrows) [14].

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

total average hsiKWW ¼ 1:7sKWW for all the molecules. Now we can include these time scales to the representation of the shear response of glycerol [12] (Fig. 13). As mentioned above, the peak in the NMR susceptibility for all molecules is at the peak in the retardation contribution of the shear compliance J 00  1=xg. The location for the slow and fast subensembles were depicted according to the ratio of their mean correlation times. The slow subensemble seems to dominate the shear response because the time scale nearly coincides with the step in the real part and the beginning of the dominance of the flow contribution in the imaginary part. This supports the view of the slow subensemble as the continuous matrix or, correspondingly, the faster subensemble as the islands of mobility within (Fig. 11(a)). This corresponds to a more theoretical discussion (Ref. [2], Section 4.3) on the basis of the Levy distribution for dynamic heterogeneity. We think that dynamic heterogeneity in glass formers and the sensitivity of shear on such heterogeneous patterns is also the reason for the similar values of the Kohlrausch exponent found for shear in different substances. 5. Conclusion For six substances with widely varying fragility we find constantly a KWW exponent of 0.42–0.45 from shear experiments. The shear exponent does not seem to follow a general trend of increasing non-exponentiality with increasing fragility as for the other activities. In this respect different activities have to be strictly distinguished. In the Arrhenius diagram, however, all compliances lie usually on a common trace. A comparison of shear and multidimensional NMR experiments on glycerol supports the picture of more mobile islands within a lesser mobile matrix for the spatial arrangement of dynamic heterogeneity.

Acknowledgements The DGEBA sample has been kindly provided by the Deutsche Shell Chemie GmbH. We thank

279

the DFG and the Fonds der Chemischen Industrie for financial support.

References [1] M.D. Ediger, C.A. Angell, S.R. Nagel, J. Phys. Chem. 100 (1996) 13200. [2] E. Donth, The Glass Transition. Relaxation Dynamics in Liquids and Disordered Materials, Springer, Berlin, 2001. [3] R. B€ ohmer, K.L. Ngai, C.A. Angell, D.J. Plazek, J. Chem. Phys. 99 (1993) 4201. [4] R. Kohlrausch, Ann. Phys. 12 (1847) 393. [5] G. Williams, D.C. Watts, S.B. Dev, A.M. North, Trans. Farad. Soc. 67 (1971) 1323. [6] R. Casalini, D. Fioretto, A. Livi, M. Lucchesi, P.A. Rolla, Phys. Rev. B 56 (1997) 3016. [7] U. Schneider, P. Lunkenheimer, R. Brand, A. Loidl, Phys. Rev. E 59 (1999) 6924. [8] Stickel, PhD thesis, University of Mainz, 1995. [9] L. Comez, D. Fioretto, L. Palmieri, L. Verdini, P.A. Rolla, J. Gapinski, T. Pakula, A. Patkowski, W. Steffen, E.W. Fischer, Phys. Rev. E 60 (1999) 3086. [10] W.M. Du, G. Li, H.Z. Cummins, M. Fuchs, J. Toulouse, L.A. Knauss, Phys. Rev. E 49 (1994) 2192. [11] F. Qi, K.U. Schug, S. Dupont, A. D€ oß, R. B€ ohmer, H. Sillescu, H. Kolshorn, H. Zimmermann, J. Chem. Phys. 112 (2000) 9455. [12] K. Schr€ oter, E. Donth, J. Chem. Phys. 113 (2000) 9101. [13] K. Schr€ oter, G. Wilde, R. Willnecker, M. Weiss, K. Samwer, E. Donth, Eur. Phys. J. B 5 (1998) 1. [14] S.A. Reinsberg, X.H. Qiu, M. Wilhelm, H.W. Spiess, M.D. Ediger, J. Chem. Phys. 114 (2001) 7299. [15] J. Honerkamp, J. Weese, Rheol. Acta 32 (1993) 65. [16] U. Buchenau, Phys. Rev. B 63 (2001) 4203þ. [17] D.J. Plazek, J. Rheol. 36 (1992) 1671. [18] D.J. Plazek, J. Polym. Sci., Part A-2, 6 (1968) 621. [19] J.M. Pelletier, J. Perez, L. Duffrene, A. Sekkat, J. NonCryst. Solids 258 (1999) 119. [20] S. Corezzi et al., J. Chem. Phys., in press. [21] W.M. Du, Y.-H. Hwang, J. Korean Phys. Soc. 34 (1999) 46. [22] A. Hensel, PhD thesis, University of Rostock, 1998. [23] E. Hempel, unpublished results, 2001. [24] E. Donth, M. Beiner, S. Reissig, J. Korus, F. Garwe, S. Vieweg, S. Kahle, E. Hempel, K. Schr€ oter, Macromolecules 29 (1996) 6589. [25] K.L. Ngai, D.J. Plazek, Rubber Chem. Technol. 68 (1995) 376. [26] K.L. Ngai, D.J. Plazek, I. Echeverria, Macromolecules 29 (1996) 7937. [27] M. Beiner, J. Korus, H. Lockwenz, K. Schr€ oter, E. Donth, Macromolecules 29 (1996) 5183. [28] M. Beiner, PhD thesis, University of Halle, 1995. [29] J. Colmenero, A. Alegria, P.G. Santangelo, K.L. Ngai, C.M. Roland, Macromolecules 27 (1994) 407.

280

K. Schr€oter, E. Donth / Journal of Non-Crystalline Solids 307–310 (2002) 270–280

[30] S. Havriliak, S. Negami, J. Polym. Sci., Part C 14 (1966) 99. [31] F. Alvarez, A. Alegria, J. Colmenero, Phys. Rev. B 44 (1991) 7306. [32] C.R. Snyder, F.I. Mopsik, Phys. Rev. B 60 (1999) 984. [33] G.C. Berry, D.J. Plazek, Rheol. Acta 36 (1997) 320. [34] D.J. Plazek, C.A. Bero, I.-C. Chay, J. Non-Cryst. Solids 172 (1994) 181. [35] C.A. Bero, PhD thesis, University of Pittsburgh, 1994. [36] B.E. Read, G.D. Dean, P.E. Tomlins, J.L. LesniarekHamid, Polymer 33 (1992) 2689. [37] L. Donzel, A. Lakki, R. Schaller, Philos. Mag. A 76 (1997) 933. [38] K.L. Ngai, Commun. Solid State Phys. 9 (1979) 127. [39] P.G. Santangelo, K.L. Ngai, C.M. Roland, Macromolecules 29 (1996) 3651. [40] R. Zorn, G.B. McKenna, L. Willner, D. Richter, Macromolecules 28 (1995) 8552. [41] M. Connolly, F. Karasz, M. Trimmer, Macromolecules 28 (1995) 1872. [42] P.G. Santangelo, C.M. Roland, Macromolecules 31 (1998) 4581. [43] R.M.C. de Almeida, N. Lemke, I.A. Campbell, Eur. Phys. J. B 18 (2000) 513. [44] A. Sch€ onhals, E. Donth, Phys. Status Solidi (B) 124 (1984) 515. [45] J.C. Phillips, J. Non-Cryst. Solids 172 (1994) 98. [46] J.C. Phillips, Rep. Progr. Phys. 59 (1996) 1133.

[47] D.G. Kubat, H. Bertilsson, J. Kubat, S. Uggla, J. Phys.: Condens. Matter 4 (1992) 7041. [48] A.G. Hunt, J. Non-Cryst. Solids 274 (2000) 93. [49] R.H. Colby, Phys. Rev. E 61 (2000) 1783. [50] R.V. Chamberlin, R. B€ ohmer, E. Sanchez, C.A. Angell, Phys. Rev. B 46 (1992) 5787. [51] P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams, J.P. Carini, Phys. Rev. Lett. 65 (1990) 1108. [52] C. Schick, E. Donth, Phys. Scripta 43 (1991) 423. [53] J. Rault, J. Non-Cryst. Solids 271 (2000) 177. [54] M. Beiner, H. Huth, K. Schr€ oter, J. Non-Cryst. Solids 279 (2001) 126. [55] E. Donth, J. Non-Cryst. Solids 53 (1982) 325. [56] H. Sillescu, J. Non-Cryst. Solids 243 (1999) 81. [57] K. Schmidt-Rohr, H.W. Spiess, Phys. Rev. Lett. 66 (1991) 3020. [58] R. B€ ohmer, G. Hinze, G. Diezemann, B. Geil, H. Sillescu, Europhys. Lett. 36 (1996) 55. [59] U. Tracht, M. Wilhelm, A. Heuer, H. Feng, K. SchmidtRohr, H.W. Spiess, Phys. Rev. Lett. 81 (1998) 2727. [60] M. Takayanagi, Memoirs Fac. Eng. 23 (1963) 41. [61] C.P. Lindsey, G.D. Patterson, J. Chem. Phys. 73 (1980) 3348. [62] D. Lellinger, E. Donth, unpublished results, 1994, 2000. [63] N.O. Birge, S.R. Nagel, Phys. Rev. Lett. 54 (1985) 2674. [64] J. Korus, PhD thesis, University of Halle, 1997.