Scattering experiments for the microscopic understanding of the glass transition

] O U R N A L OF

ELSEVIER

Journal of Non-Crystalline Solids 192 & 193 (1995) 384-392

Section 10. Relaxation and glass transition

Scattering experiments for the microscopic understanding of the glass transition E. Bartsch * Institut fiir Physikalische Chemie, Universitiit Mainz, Jakob-Welder-Weg 15, D-55099 Mainz, Germany

Abstract Scattering experiments studying the dynamics of glass-forming systems have become of special importance for understanding glass transition phenomena. They monitor the density autocorrelation function which has become a key observable since it is calculated within a new concept of the glass transition, the mode coupling theory (MCT). After a short reference to the main predictions and examples of scattering experiments where these phenomena have been observed, results for two model systems will be analyzed with MCT: a recently reported analysis of neutron scattering data for orthoterphenyl in the 13-regime is improved and photon correlation spectroscopy data on colloidal spheres are described by simultaneously analyzing the 13- and a-relaxation dynamics using the MCT scaling laws.

I. Introduction When a liquid is supercooled, it freezes into an amorphous solid state of matter, termed a glass. This glass transition is accompanied by several characteristic 'macroscopic' phenomena. The most prominent are, in addition to the temperature discontinuity in the specific heat (and other second-order derivatives of the free energy) at the so-called caloric glass transition temperature, Tg: (i) the dramatic slowing of the dynamics in a narrow temperature range above Tg, which can be seen in the increase of the shear viscosity over 14 orders of magnitude, and (ii) the appearance of the stretching phenomenon, i.e., the relaxation functions of certain observables no longer follow the Usual Debye or exponential behaviour but

* Corresponding author. Tel: +49-6131 39 2490. Telefax: +49-6131 39 4196. E-mail: [email protected].

instead show a Kohlrausch or stretched exponential behaviour. The glass transition is not connected to any structural anomaly. It is not possible to decide between liquid and glass just by measuring the static structure factor, since it is qualitatively the same in both cases. Thus it must be a dynamic transition and the question arises as to whether it can be connected with a characteristic temperature in the same way as for ordinary phase transitions. As Tg depends on the cooling rate [1], it cannot qualify as a characteristic physical transition temperature. It merely signals that point where the system drops out of equilibrium because the timescale of structural rearrangements becomes too slow to follow the temperature changes [2]. Another candidate that has been proposed to characterize a glass-former is the Vogel-Fulcher temperature, TO, which is connected with the shear viscosity, 7. The latter has been used to classify glass-

0022-3093/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI 0 0 2 2 - 3 0 9 3 ( 9 5 ) 0 0 3 7 9 - 7

E. Bartsch/Journal

of Non-Crystalline

slowing down on cooling towards T, produced by the movement of the inflection point to longer times as T -+ T,’ , a cusp in t, at T, results. Such cusps have been reported in only two systems so far, for a hard-sphere colloidal system [9] (Fig. l(b)) and for an ionic glass [lo]. The results, however, are still controversial [ 111. (iv) The glass transition dynamics are the result of an interplay between the cage effect and thermally activated hopping processes of particles in a ‘frozen’ potential energy landscape. The former tends to arrest structural relaxation at T,. It is counteracted by the latter which restores ergodicity at T,. The idealized MCT neglects hopping processes and, thus, predicts a complete freezing of structural relaxation with a concomitant divergence of the shear viscosity, q, at T, > Tg, in contradiction to experimental evidence. The hopping processes have been incorporated in an advanced version of the theory, called extended MCT [12], thus replacing the ideal glass transition by a change of the mechanism of motion from liquid-like diffusional motion to solid-like thermally activated processes. However, from the few studies of experimental data with extended MCT available [13,14], it can be inferred that the (more easily applicable) idealized MCT can still serve as a good first approximation for the description of the relaxation scenario in glass-formers as long as the data analysis is restricted analysis to temperatures not too close to T, or - near T, - to the p-relaxation regime. A more complete account of experimental tests of mode coupling predictions can be found in some recent reviews [5,15] and conference proceedings [6,71. 3. Mode coupling analysis of time domain data: two examples

Within the idealized MCI the dynamics in the region of the plateau in Q,(t) is described by the scaling law [2] @e(t) =fe” ++,g,(t/r,), (3) which results from an asymptotic solution of the mode coupling equation in the vicinity of T, and is exact in the limit T + T, and t, << t -=z T, (to is the microscopic timescale governing the short-time de-

Solids192&193 (1995)384-392

387

cay of the density fluctuations; ru is the structural relaxation time). The time dependence is determined by a wave vector and temperature invariant scaling function, g +(t>, which is expressed in terms of two power expansions with t-* and - Btb as leading terms [16]. The critical exponents, a and b, as well as the expansion coefficients are specified by a single number, the exponent parameter, A, which can be calculated from the static structure factor [2]. The Q dependence is contained in the non-ergodicity parameter, fi, which represents that part of the density fluctuations which is frozen, and the amplitude factor, ho. The temperature dependence enters via two scales, the amplitude scale, c,, and the timescale, t, , of the @process, which depend critically via the power laws c, = o’/2

(4a)

and rV= t0(+-‘/2a

(4b) on the distance from the transition point, as measured by the separation parameter, u, o= co ](T-

T,)/T,

I.

(5) cc, denotes a material-dependent constant, which describes the proportionality between the control parameters used by theory and experiment, when a system is driven through the transition. Eq. (31 can be used for a first check as to how far the idealized MCT is applicable to glass-formers. The ususal procedure is to use Eq. (31 as a fourparameter fit function and to test whether a description of a set of experimental data for various temperatures or densities is possible with the same values for A and $. In the next step it is checked whether the power laws of Eq. (4) are obeyed and yield the same T,. In the following we wish to demonstrate how this procedure can be improved to achieve more stringent tests of the theory, using as an example two systems where the so-called P-analysis has been applied with some success: the molecular glass former orthoterphenyl and a particular type of colloidal system. 3.1. /3-analysis of orthoterphenyl The organic liquid orthoterphenyl (T, = 329 K; Tg = 243 K> has been considered to be a good test

E. Bartsch /Journal of Non-Crystalline Solids 192& 193 (1995) 384-392

386

leading to a cusp at T~. This cusp has been found for a variety of different glass-formers by neutron scattering [5-7] and is the most direct verification of the existence of a characteristic temperature. One example is shown in Fig. l(a). (iii) The ~-relaxation has a crossover time, t~, which is either that time when the density autocorre-

0

100

200

lation function has decayed to the plateau (T ~< Tc) or the inflection point separating I~- from a-relaxation (T >t To). Since the decay to the plateau is essentially temperature independent [2], the increase of fQ(T) below Tc necessarily implies a decrease of t,~, leading to a counterintuitive critical slowing down of [~-dynamics on heating. Together with the critical

300

Temperature (K) 10

b

EXPT ~', EXPT ~'p •

MCT ., MCT

T,=t01ol -v r e = t d o l "~

I

t'-

t..,.J 0

o

6

-0.2

-0.1

--0.0

0.1

O"

Fig. 1. Examples of glass transition phenomena predicted by the idealized MCT and observed in fragile glass formers by scattering techniques: (a) incoherent Debye-Waller factors, fQ(T), of orthoterphenyl (OTP) for a set of Q-values obtained by neutron scattering (from Ref. [8]); (b) relaxation times, ~'~, ( [] ), and crossover times, 1-13- t~ ( • ), versus the separation parameter, rr, of Eq. (5), as obtained from MCT fits. The dashed and solid lines represent MCT predictions for ~'~ (Eq. 6(b)) and to (Eq. (4b)), respectively (from Ref. [9]).

E. Bartsch /Journal of Non-Crystalline Solids 192& 193 (1995) 384-392

slowing down on cooling towards Tc produced by the movement of the inflection point to longer times as T ~ T~+ , a cusp in t~ at Tc results. Such cusps have been reported in only two systems so far, for a hard-sphere colloidal system [9] (Fig. l(b)) and for an ionic glass [10]. The results, however, are still controversial [11]. (iv) The glass transition dynamics are the result of an interplay between the cage effect and thermally activated hopping processes of particles in a 'frozen' potential energy landscape. The former tends to arrest structural relaxation at To. It is counteracted by the latter which restores ergodicity at Tc. The idealized MCT neglects hopping processes and, thus, predicts a complete freezing of structural relaxation with a concomitant divergence of the shear viscosity, r/, at Tc > Tg, in contradiction to experimental evidence. The hopping processes have been incorporated in an advanced version of the theory, called extended MCT [12], thus replacing the ideal glass transition by a change of the mechanism of motion from liquid-like diffusional motion to solid-like thermally activated processes. However, from the few studies of experimental data with extended MCT available [13,14], it can be inferred that the (more easily applicable) idealized MCT can still serve as a good first approximation for the description of the relaxation scenario in glass-formers as long as the data analysis is restricted analysis to temperatures not too close to Tc or - near T~- to the 13-relaxation regime. A more complete account of experimental tests of mode coupling predictions can be found in some recent reviews [5,15] and conference proceedings [6,7].

3. Mode coupling analysis of time domain data: two examples Within the idealized MCT the dynamics in the region of the plateau in ~ o ( t ) is described by the scaling law [2]

ClgQ(t) =f~ + hQC~ g± ( t / t ~ ) ,

(3)

which results from an asymptotic solution of the mode coupling equation in the vicinity of Tc and is exact in the limit T --->Tc and t o << t << r~, (t o is the microscopic timescale governing the short-time de-

387

cay of the density fluctuations; ~-~ is the structural relaxation time). The time dependence is determined by a wave vector and temperature invariant scaling function, g ±(t), which is expressed in terms of two power expansions with t -~ and - B t b as leading terms [16]. The critical exponents, a and b, as well as the expansion coefficients are specified by a single number, the exponent parameter, A, which can be calculated from the static structure factor [2]. The Q dependence is contained in the non-ergodicity parameter, f~, which represents that part of the density fluctuations which is frozen, and the amplitude factor, hQ. The temperature dependence enters via two scales, the amplitude scale, c,,, and the timescale, G, of the [3-process, which depend critically via the power laws (4a)

Co- = 0 " I / 2

and

t~ = tour-1/2~

(4b)

on the distance from the transition point, as measured by the separation parameter, ~r, ° ' = c0 I ( r -

L)/L

I.

(5)

c o denotes a material-dependent constant, which describes the proportionality between the control parameters used by theory and experiment, when a system is driven through the transition. Eq. (3) can be used for a first check as to how far the idealized MCT is applicable to glass-formers. The ususal procedure is to use Eq. (3) as a fourparameter fit function and to test whether a description of a set of experimental data for various temperatures or densities is possible with the same values for A and f~. In the next step it is checked whether the power laws of Eq. (4) are obeyed and yield the same Tc. In the following we wish to demonstrate how this procedure can be improved to achieve more stringent tests of the theory, using as an example two systems where the so-called B-analysis has been applied with some success: the molecular glass former orthoterphenyl and a particular type of colloidal system.

3.1. fl-analysis of orthoterphenyl The organic liquid orthoterphenyl (Tm = 329 K; Tg = 243 K) has been considered to be a good test

388

E. Bartsch / Journal of Non-Crystalline Solids 192& 193 (1995) 384-392

system since it consists of well-defined molecular units and belongs to the class of 'fragile' glassformers, for which M C T can be expected to apply best. In a series of neutron scattering experiments, it could be shown that most of the features predicted by mode coupling theory appear in the dynamics of orthoterphenyl (OTP), the most prominent being the cusp in the temperature dependence of the plateau value, fo(T), yielding Tc = 290 K (see Fig. l(a)) (for a review of the neutron scattering experiments on OTP see Ref. [17], where references to the original literature and to related work on OTP can be found). Since neutron scattering spectrometers are highly specialized, data from several instruments had to be combined in order to achieve a dynamic range large enough to study glass transition phenomena [18]. Incoherent spectra (monitoring single-particle dynamics) measured on three instruments were converted into the time domain in order to take care of the different resolution functions and then combined to yield the incoherent intermediate scattering function, S(Q, t)ct cI)~(t) (the index, s, indicates that incoherent scattering monitors self-motion; since the equations relevant to this paper are formally analogous for cI)o(t) and q ~ ( t ) , the index, s, is dropped and it is understood that by ~Q(t) self-motion is meant throughout the remainder of this section), spanning 2.5 decades in time (an example is given in Fig. 2(a)). In a first attempt to analyze the data with Eq. (3), the exponent parameter was fixed at h = 0.77 (a = 0.295; b = 0.526) as determined from an analysis of the short-time behaviour of the a-relaxation [8] and f~, G and hQC~ were allowed to vary freely. The fits, restricted to 1 ps at short times to account for the effect of the boson peak (which is shown by dynamic light scattering to superimpose on the B-relaxation [19]), describe the neutron data quite well. f~ was found to be weakly temperature-dependent which was attributed to the O(T) term in Eq. (2). The temperature variation of hQc~ and t,~ was compatible with the power laws Eq. (4) and a Tc of about 290 K. However, the fit parameters showed a considerable scatter and the power-law behaviour was found to extend only up to 313 K. Thus a direct extraction of Tc by fitting (hQC,r) 2 and t~ 2a to a straight line was not attempted. Here we discuss a different approach. Given the

0.8 0.6

d ~0.4

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1

10

100

t /ps 80

cq

,

t

,

i

,

i

,

i

,

i

,

0.8

70

0,7

60

0.6

50

O.S

~40

0.4

U C~30 ~z 20

~0 (,q i

0.3

b

0.2

10

0.1

0

0 i

290

i , i

300

310

i

i

320

330

340

T/K Figl 2 " ~ I a~aly sis O~ the i~cohe ~ent intermedlat e scatte ~ g

~unc I

tions S(Q, t) of OTP: (a) S(Q, t) for (top to bottom) 293, 298, 305, 312, 320 and 327 K as obtained by combining data from three different neutron spectrometers. The solid lines are MCT fits using Eq. (3). (b) Timescale, t,~ (A), and amplitude scale, hQC~ (D), plotted as t~_2" and (hQCa)2. The solid straight lines are linear fits to the data points corresponding to the linearized power laws (Eq. (4)) of MCT (error bars are shown only where larger than the symbol size).

restricted time range over which S(Q, t) is available and described by Eq. (3), a fit with three of four parameters varying freely allows too much flexibility. This is underlined by the fact that f~ and t,~ are related via tPQ(t = G ) = f 6 [16] and should not really be treated as independent quantities. Alternatively an attempt can be made to fix as many parameters as possible using data obtained independently [2O]. In Ref. [8] f~ was determined assuming (i) that the D e b y e - W a l l e r factor, f~, can be decomposed into a vibrational part and a relaxational part accord-

E. Bartsch/Journal of Non-Crystalline Solids 192& 193 (1995) 384-392

ing to f 0 ( T ) = fdvib(T).f~rel(T) and (ii) that the temperature dependence of the vibrational part, In fovib(T) o t - T , prevailing at low temperatures where fffl(T) = 1 can be extrapolated to higher temperatures. Since Eq. (2) is expected to be valid for the relaxational part only, f~ was extracted by dividing by f~ib(T) and fitting the remainder with Eq. (2). The values obtained for f~, plotted versus Q, followed the theoretical values for a hard-sphere system when the Q-axis was rescaled such that the first maxima in S(Q) of OTP and the hard-sphere system coincided (cf. Fig. 6(a) of Ref. [8]). Motivated by this result, the weak temperature dependence of f~ above Tc found in Ref. [18] can be tentatively interpreted as being completely due to the vibrational part and f~ in Eq. (3) can be replaced by fQ(T)=f~2 jQrVib(T). Using g o ( t = t,~)=fo(T), t,~ can then be directly read off the graph. With three parameters fixed, only the amplitude scale, hoCo., remains to be adjusted. As is demonstrated in Fig. 2(a), varying this single parameter is sufficient to fit the time decay curves for all temperatures in the range 1-100 ps (solid lines in Fig. 2(a)). Small deviations observed at times larger than 100 ps are most probably due to resolution effects or to numerical errors introduced by the Fourier transformation necessary to convert the spectra into the time domain. The significant deviation for T = 327 K signals the onset of the c~-relaxation which cannot be described by Eq. (3). Plotting (hoc,,) 2 versus temperature should by virtue of Eq. (4) lead to a straight line intersecting the abscissa at Tc, which is indeed the case (Fig. 2(b)). The timescales, t,~, taken from the time decay curves are included in Fig. 2(b) as t~ 2a and now follow the power law (Eq. (4)) over the entire temperature range. Extracting Tc via linear regression yields Tc = 290 K in both cases, in agreement with previous work [17]. Thus the neutron data in the time range 1-100 ps are consistent with an interpretation in terms of the 13-relaxation of the idealized MCT. Some critical comments are, however, in order. The above analysis is based on the assumptions that the vibrational and the relaxational contributions are independent and that the harmonic behaviour of vibrations extends to higher temperatures. This is only an approximation. It has been argued [21] that, for a correct theoretical description of the dynamics in the

389

range intermediate between structural relaxation and microscopic excitations, a coupling of relaxational and vibrational modes (the boson peak) has to be taken into account. In Ref. [21] the concept of fragility is connected with the relative contributions of vibrational and relaxational modes, a glass-former being the more fragile the weaker the vibrational part. In a fragile system like orthoterphenyl, the boson peak is less pronounced than in strong glassformers. Thus it can be expected that any additional temperature dependence of f~ib(T), caused by the coupling to relaxational modes, is negligible compared with the strong decrease of the relaxational part of fo(T) above Tg. Nevertheless, it will lead to some errors in the value of f~ in addition to experimental noise, with a concomitant effect on the timescales t,~. A deviation of f~ from the derived value of more than 3-4%, however, leads to timescales that do not allow any consistent description of the above data by Eq. (3). Another limitation is seen in the cut-off values, which are somewhat arbitrary. To overcome this one would like to have a theoretical description that covers both the short and long times. This is not possible for the short-time dynamics of molecular systems, requiring an incorporation of the boson peak into the equations of MCT which has not been done so far. An extension to longer times is, however, possible. As this requires good experimental data in the e~-relaxation region as well, it will be discussed for the example of colloidal suspensions where, owing to the large dynamic range of photon correlation spectroscopy, data covering ten decades of time are available.

3.2. a, fl-analysis of colloidal polymer micronetwork spheres Colloidal suspensions are ideal test systems for the idealized MCT. They can be thought of as macroatoms in a structureless background medium, where even hard-sphere interactions can be realized [9], thus allowing one to model the glass transition of hard sphere atoms with the temperature replaced by the volume fraction as control parameter. Because of their Brownian nature, thermally activated processes and the boson peak are expected to be absent and the

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E. Bartsch/Journal of Non-Crystalline Solids 192&193 (1995) 384-392

large shift of timescales by some nine orders of magnitude (atoms: "rc~(Tg)(I 102S -q' colloids: z~(~bg) ct 10 al) accompanying the size increase from ~ 0.1 nm to ~ 100 nm should move any ergodicity-restoring processes out of the accessible time window. Accordingly, full quantitative agreement was found between the glass transition dynamics of hard-sphere PMMA colloids and predictions of the idealized MCT for hard spheres [9]. In a different type of colloidal system, consisting of polystyrene micronetwork spheres, a glass transition was observed showing quite similar features to the colloidal PMMA spheres [22]. Below thg, the density autocorrelation functions, qbo(t), as obtained by photon correlation spectroscopy, could be interpreted over seven decades in time, using Eq. (3), in terms of the 13-relaxation of idealized MCT. The long-time part of the decay curves obeys the analogue of the time-temperature principle, the obtained master curve being well described by a Kohlrausch function in accordance with MCT [23]. The time and amplitude scales followed the power laws of Eq. (4), yielding qbc = 0.64 = ~g. However, by contrast with the hard-sphere colloids, no interpretation of the decay curves above ~bg was possible within MCT. This was attributed to either a significant size distribution of the particles or deviations from the hard-sphere character [23] which is also indicated by the exponent parameter, A, being larger than the hard-sphere value (0.88_ 0.02 [22] versus 0.76 [2]). The question arises as to whether the discrepancies between the theoretical prediction and experiment arise suddenly at ~b> ~bg or whether there are subtle effects already on the 'liquid' side which are masked by the remaining flexibility of the [3-analysis. For instance, the power laws of Eq. (4), which are actually connected by the separation parameter, tr, in Eq. (5), were checked independently allowing different ~b¢ values. Here, the aim is to extend our former analysis by socalled a-13 fits, a procedure first applied to the hard-sphere colloids [24] and later used successfully to extend the MCT analysis of ~Q(t) for the ionic glass Ca0.4K0.6(NO3)l.4, obtained by neutron spin-echo techniques, to higher temperatures where the 13-relaxation and the a-relaxation are no longer well separated [25]. The main idea is to match the scaling law, ~Q(t)

=f~FQ(t/T~), of the a-relaxation, which is valid for t > t,~ and ~b< ~bc, to that of the 13-relaxation (Eq. (3)) via dPo(t) =f~FQ( t/z~) + hQO'l/2[g_(t/t.) + B(t/t.)b], (6a) Ta=(--O" ) (1/2b)t~r-~-tolO'l-(l/2a+l/2b)

(6b)

The von Schweidler law (ct -Bt b) is shared by both the a- and 13-processes, linking the timescales through Eq. (6b). Thus, the additional term in Eq. (6a) is introduced to cancel out the corresponding term in Eq. (3), since it is already contained in the master function of the a-process, FQ(t/r~). For the latter no analytical formula can be given. It has to be calculated numerically, a task so far solved only for the hard-sphere system [26]. For other systems it can generally be well represented by Kohlrausch functions, FQ(t) ~ exp(t/rp~O) with temperature- or volume-fraction-independent coefficients, ~-Q and ~Q < 1. Thus, application of Eq. (6a) requires f~, ho, r0 and ~Q tO be volume-fraction-independent, while the only volume fraction dependence is set by the scale, o-, connecting ~-~ and t,, via Eq. (6b). A and f~ were taken from the previous 13-analysis (0.88 + 0.02 and 0.855 + 0.015, respectively) [22], while hQ = 0.159 + 0.013 was obtained by a 13-analysis with connected scales t~ and c~ producing the same fit curves as in Ref. [22]. ~-Q and flQ were adjusted by fitting the data for 4' = 0.629 and the resulting values, 4.39 _ 0.18 and 0.42 + 0.03, respectively, were kept constant for other volume fractions. The remaining parameters, o- and t 0, were then varied producing the a,13-fits in Fig. 3(a) which describe the data better than the 13-fits in Ref. [22]. t o showed weak decrease from 5 × 10 -5 s to 3 × 10 -5 s, which is insignificant with respect to experimental error. The separation parameter, tr, follows the expected linear variation with th, leading to ~bc = 0.643 via Eq. (5). In the inset of Fig. 3(b), it (solid squares) is compared with the results obtained by a pure 13-fit with connected scales (open squares). While both fits yield the same qbc, the slopes, i.e., c o in Eq. (5), are different. The reason for this can be seen in Fig. 3(b), where the fit curves of the 13-fits are compared with the 13 and KWW parts of the a,13-fits. It can be seen that, because of lack of knowledge of the upper

E. Bartsch /Journal of Non-Crystalline Solids 192& 193 (1995) 384-392

1,0 0,8

~

0,6 0,4

I° °

A_

~, ,=0.584 *=0.003

a) . ~ ~_ a-p-m

.

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0,00.7 10-6 10-5 10-4 10 4

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t/s

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10 3 10 4

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~¢

O'.

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' ~ . dp=0.603.

•. . . . . ~ . . . . . . . . . . . . 10-710-610-510-410-310-210-l 10° 10 t 102 l0 s 104

t/s

Fig. 3. e~,13-analysis of the density autocorrelation functions,

cPo(t), for colloidal polystyrene micronetworkspheres of 200 nm diameter in an organic liquid at q~< d~g obtained from photon correlation spectroscopy. (a) for five different values of 4~ indicated; (b) data for ~ = 0.603 compared with 13-fit ( . . . . . . ), 13 part of et,13-fit ( ) and KWW of a,13-fit ( . . . . . ). Data from Ref. [22]. cut-off of the 13-scaling regions, 13-fits are usually extended too far, thereby leading to wrong separation parameters or to errors in t,~ and c,~, in case of fits with independent determination of these parameters, as noted in Refs. [24,25]. Since this effect becomes more important with distance from the critical point, o-= 0, this might also be a reason for the failing of the power laws seen in the earlier [3-analysis of orthoterphenyl [18] at higher temperatures.

4. C o n c l u s i o n

Neutron and light scattering experiments on molecular and colloidal glass-formers monitor the

391

dynamics close to the glass transition at a microscopic level, i.e., at nearest neighbor distances, and allow quantitative comparisons with the predictions of mode coupling theory. Fixing three out of four parameters in the analysis of the 13-relaxation of orthoterphenyl to values determined independently and restricting the fits to Eq. (3) to the 13-scaling region, the results of a previous analysis [18] could be improved at higher temperatures. For T~, the value of 290 K was confirmed and the power laws of Eq. (4) were found to apply up to 327 K. The dynamics of colloidal polystyrene spheres in the 13and a-regimes below q9c can be simultaneously described over a time range of seven decades by combining the or- and 13-scaling laws of the idealized MCT. This contrasts with the situation above ~bc where no MCT description was possible [23]. Scattering experiments on different glass-formers have been compared with MCT with varying success. The common features are scaling laws and the critical slowing down of the dynamics on the ' liquid' side towards a specific temperature or volume fraction where the dynamics changes its character, irrespective of whether this happens in the form of a complete structural arrest, as stated for hard-sphere colloids [9], or by the appearance of thermally activated processes or other ergodicity-restoring processes. The best agreement has been stated for calcium potassium nitrate [10] and hard-sphere colloids [9]. For less 'fragile' systems, other phenomena such as the boson peak become more important. To assess their relevance and to understand their interplay with the phenomena treated by MCT will be a task for future experimental and theoretical work.

The author is grateful to his colleagues F. Fujara, M. Kiebel, W. Petry, H. Sillescu and J. Wuttke in connection with the neutron scattering part and M. Antonietti, V. Frenz, W. Schupp and H. Sillescu with respect to the light scattering part of the work. Stimulating discussions with W. G6tze, M. Fuchs, W. van Megen and J. Baschnagel are acknowledged. Financial support of the neutron scattering work by the Bundesministerium fiir Forschung und Technologie (Contract No. 03-SI3MAI) and of the light scattering work by the Deutsche Forschungsgemeinschaft (SFB262) is gratefully appreciated.

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E. Bartsch /Journal of Non-Crystalline Solids 192&193 (1995) 384-392

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