Relaxation spectroscopies of viscous liquids

32

Journal of Non-Crystalline Solids 131-133 (1991) 32-36 North-Holland

Relaxation spectroscopies of viscous liquids Lei Wu, Paul K. Dixon and Sidney R. Nagel The James Franck Institute and the Department of Physics, The University of Chicago, Chicago, IL 60637, USA

Bruce D. Williams and John P. Carini Department of Physics, University of Indiana, Bloomington, IN 47405, USA

Data on supercooled liquids obtained by specific heat spectroscopy, dielectric susceptibility, ultrasonic attenuation and frequency-dependent shear modulus are reviewed. Where these different measurements have been made on the same sample, they measure the same average relaxation times. In addition, dielectric data are presented covering 13 decades in frequency. These data can all be scaled such that, independent of temperature, they all fall on a single curve. Data for the primary relaxation of different samples can be overlaid to give a universal curve determined by only two parameters, peak frequency and width. This sealing form is different from any as yet proposed in the literature. Interesting scaling is also found when the secondary relaxation is included.

1. Introduction If the glass t r a n s i t i o n were a true thermodyn a m i c phase transition, we would expect universal properties to a p p e a r as the t r a n s i t i o n t e m p e r a t u r e is approached. I n this paper, we review some results o n the spectroscopy of supercooled liquids a n d try to emphasize which aspects of these measurements do seem to have universal properties. There appears [1] to be n o change i n the structure of a liquid as it is cooled through the glass transition; if there is a true phase transition, displaying universal critical p h e n o m e n a , it is characterized b y an order p a r a m e t e r which is subtle a n d does n o t show u p in the density. W e have studied liquids a p p r o a c h i n g the glass transition using a n u m b e r of different probes: ultrasonic a t t e n u a t i o n [2], freq u e n c y - d e p e n d e n t shear m o d u l u s [3], specific heat spectroscopy [4] a n d dielectric susceptibility [5]. A l t h o u g h each of these m e a s u r e m e n t s is sensitive to a different property of the system, they all show the same qualitative behavior: the spectrum is d o m i n a t e d b y a relaxation peak that is char-

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33

Lei Wu et al. / Relaxation spectroscopies of viscous flquids i

acterized by a timescale that increases rapidly as the temperature is lowered. Figure 1 shows such a characteristic relaxation spectrum using the dielectric susceptibility for different temperatures. Indeed, most of the properties of the glass transition are simply a manifestation of the timescales for relaxation becoming larger than the duration of the experiment [6]. The question that naturally occurs is whether there is only a qualitative similarity between the different relaxation functions examined by different measurements or whether there is universal behavior that appears in all the susceptibilities. Thus, we are led to ask whether the different probes are governed by a single relaxation time, ~', or whether they are each dominated by a different time. Likewise, we are interested in whether the relaxation functions themselves are related to each other in so far as their width and shape are concerned. Finally, we are interested in whether different samples of glass-forming liquids all have the same characteristic behavior.

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We first present data showing how the average relaxation times measured by the different spectroscopies compare with one another. In fig. 2 we show the data for pp, the frequency of the peak of the relaxation, as a function of temperature, T, obtained from ~, the dielectric susceptibility [5], and specific heat spectroscopy [4,7]. The results are plotted for four different samples: glycerol, o-terphenyl, propylene glycol, and salol. Since the dielectric susceptibility is adiabatic and the specific heat data is isothermal, we have multiplied pp for the dielectric data by the factor (ce(v = oo)/Cp(V = 0)) in order to make a valid comparison [2]. For all four samples it is quite clear from fig. 2 that the two different spectroscopies give very similar values for vp over the frequency range where both measurements can be made. A similar comparison can be made for the data derived from the shear modulus and ultrasonic data. For glycerol, it was found that the equation that was used to fit the values of Vp determined from the shear modulus data [3] was identical to that used to fit the

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specific heat spectroscopy data [4]. Likewise, the ultrasonic data for the relaxation, after correction by the isothermal/adiabatic factor, and the specific heat data, although they did not overlap in the frequency range covered, could be fit by a single scaling or Vogel-Fulcher equation (fig. 4 of ref. [2]). Thus, for glycerol it was found [8] that the peaks in the relaxation for the dielectric susceptibility, specific heat spectroscopy and ultrasonic attenuation could all be placed on top of each other on a common plot (fig. 6 of ref. [8]). We have emphasized above that, as far as one can tell, the relaxation time measured by each spectroscopy is the same as that measured by others. However, it is evident that, although the average time is the same, the width of the different relaxation peaks measured by different probes is significantly different as reported elsewhere [5]. We now turn our attention to the shape of the dielectric susceptibility relaxation data. As shown in fig. 1, we have obtained dielectric data over 13 decades of frequency. As the temperature is lowered, the width of c ' , the imaginary part of e, increases. This is characteristic of data using any of the spectroscopies.

Lei Wu et aL / Relaxation spectroscopies of viscous liquids

34

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3. Discussion

Our aim is to try to scale all of our dielectric data in such a manner that it can be fit onto a single curve. Clearly, the curve for each temperature must be shifted on the loglo 1, axis in order to bring the peak positions to the same point. Likewise, since the width varies as a function of temperature, each curve should be plotted so that the half-widths are matched. This corresponds to making the abscissa ( l / w ) lOglo(V/Vp). H e r e w is the full width at half maximum W normalized to the Debye width, W D = 1.14 decades (i.e., w IV/WD). In order for the peaks to have the same height, we have chosen * the ordinate to be w("/A(, where A( - {((v = 0) - ((v = oo)}. In fig. 3(a) we plot the data over 13 decades of frequency for glycerol on such a plot. This scaling works quite well for this liquid. However, when we plot the data for salol (fig. 3(b)), also over 13 decades,

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we see that the curve is considerably less good: at frequencies near and below the peak the datapoints for different temperatures do not lie on top of each other very well. The difference between glycerol and salol is that the width, w, does not vary very much in the case of glycerol, whereas it does vary significantly for salol. In order to obtain a single curve that superimposes all the data even in the cases where w has a significant temperature dependence, we have to try another scaling form. The scaling form that we have found to work the best [5] is to plot ( l / w ) lOglo(C"Vp/VAc) versus ( l / w ) ( 1 + 1/w) logl0(v/ vp). Again there are only two parameters that need to be chosen for each temperature: the peak position, vp, and the width, w. In fig. 4 we show the plot for all the salol data when plotted in this manner. This plot leads to an excellent collapse of all the data. A similar quality curve was found for the glycerol data [5]. We have data on seven samples: glycerol, propylene glycol, salol, dibutylphthalate, a-phenyl-o-cresol mixed with 13% oterphenyl and o-terphenyl mixed with 9% and 33% o-phenylphenol. (For propylene glycol and oterphenyl, the data were only taken over 10 decades. For o-terphenyl, ot-phenyl-o-cresol and dibutyl-phthalate a secondary relaxation peak occurs at high frequencies. This limited our range where we could get good data on the primary

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relaxation peak alone.) What is remarkable is that the dielectric data on the primary relaxation peak for all seven samples can all be superimposed onto a single curve [5]. This is shown in fig. 5. If we do not limit our data to that covering only the primary relaxation peak but include the data on the secondary relaxation peak as well, we find that the secondary relaxation peak gives rise to additional weight in the high-frequency tail of the susceptibility curves. In fig. 6 we show the

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data for a-phenyl-o-cresol. At high frequencies the secondary relaxation peak appears and there is a break away from the universal curve shown in fig. 5. For each temperature, the break occurs at a different position. It is remarkable that the envelope of this secondary relaxation peak is a smooth curve when scaled with the same parameters that were used to fit the primary relaxation data. We can now ask whether different probes of the relaxation can be scaled in the same manner. In fig. 7 we show the specific heat spectroscopy data [7] for salol scaled as ( l / w ) 1 O g l 0 ( ( C p r ) " P p / vA(cpK)) versus (1/w)(1 + 1/w) logl0(v/~,p). The data is consistent with the same scaling fit used to fit the dielectric spectroscopy results. Unfortunately, in the high-frequency tail the data are too noisy to determine if it overlays the dielectric results or if it is more similar to a stretchedexponential form (i.e., the Fourier transform of the time derivative of a response function of the form ~ ( t ) = ~0 exp[ - (t/-r) #]). The stretched-exponential fit with fl = 0.65 is shown for comparison. We are presently obtaining better quality specific heat data in this regime in order to ascertain whether or not this scaling is universal, independent of the type of probe used.

36

Lei Wu et al. / Relaxation spectroscopies of viscous liquids

4. Conclusions

Data are presented that indicate several universal features of relaxation in supercooled liquids. First, the average relaxation time and its temperature dependence is independent of the type of probe used in the measurement. Second, the shape of the primary relaxation time for the dielectric response on different liquids can all be superimposed onto a single scaling curve. Only two parameters, the peak position and the peak width, are involved in parametrization of the data. These results were obtained with liquids with quite different chemical bonding and structures, yet the scaling appears to be universal between the different samples. Finally the shape of the relaxation curve measured by specific heat spectroscopy is similar to that found with dielectric susceptibility. The authors thank Raymond Goldstein, Leo Kadanoff, John Marko and Tom Witten for many

helpful discussions. This work was supported by NSF Grant D M R 88-02284.

References [1] L.E. Busse and S.R. Nagel, Phys. Rev. Lett. 47 (1981) 1848; L.E. Busse, Phys. Rev. B29 (1984) 3639. [2] Y.H. Jeong, S.R. Nagel and S. Bhattacharya, Phys. Rev. A34 (1986) 602. In our samples the c~rrection, (cp(p = ~ ) / c e ( v = 0)), has a value of - 2. [3] Y.H. Jeong, Phys. Rev. A36 (1987) 766. [4] N.O. Birge and S.R. Nagel, Phys. Rev. Lett. 54 (1985) 2674; N.O. Birge, Phys. Rev. B34 (1986) 1631; P.K. Dixon and S.R. Nagel, Phys. Rev. Lett. 61 (1988) 341. [5] P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams and J.P. Carini, Phys. Rev. Lett. 65 (1990) 1108. [6] C.A. Angell and W. Sichina, Ann. NY Acad. Sci. 279 (1976) 53. [7] P.K. Dixon, Phys. Rev. ]342 (1990) 8179. [8] N.O. Birge, Y.H. Jeong and S.R. Nagel, Ann. NY Acad. Sci. 484 (1986) 101.