The dynamics of strong and fragile glass formers: vibrational and relaxation contributions

) O U R N A L OF

ELSEVIER

Journal of Non-CrystallineSolids 172-174 (1994) 138-153

The dynamics of strong and fragile glass formers: vibrational and relaxation contributions A.P. Sokolov a' 1.,, A. Kisliuk a, D. Quitmann a, A. Kudlik b, E.

R6ssler b

a Institutfftr Experimentalphysik, FU Berlin, Arnimallee 14, D-14195, Berlin, Germany b Lehrstuhlj~ftr Experimentalphysik II, Universit~t Bayreuth, Universit~tsstrasse 30, D-95454 Bayreuth, Germany

Abstract There are two contributions to the low-frequency excitation spectra (Raman and neutron scattering, specific heat, etc.) of glass formers: relaxations and vibrations. It is shown from the analysis of low-frequency Raman spectra that the relative weight of vibrational over relaxational excitations is larger for less fragile (in Angell's classification) glass formers. The spectra are compared with predictions of mode coupling theory (MCT) for relaxation processes. Qualitatively the predicted behaviour is observed in all analyzed systems. However, some quantitative disagreement due to significant vibrational contribution to the spectra is found for intermediate glass formers. The spectra are also analyzed using a model of vibrations coupled with relaxations. It is found that the temperature at which overdamping of the low-frequency vibrations happens is essentially the critical temperature of MCT. New details of a scenario for the liquid-glass transition are suggested.

1. Introduction The nature of the glass transition phenomena has been discussed for a long time. This discussion focuses mainly on an analysis of relaxation processes and on their temperature variation during the liquid-glass transition. During the past decade, important developments of the theory were achieved by G6tze, Sjrgren and

Permanent address: Institute of Automation and Electrometry, Russian Academy of Sciences, Novosibirsk, Russian Federation * Correspondingauthor. Tel: +49-6131 379 218. Telefax: +496131 371 100.

others, using the mode coupling approach (see Ref. [1] and references cited therein). The so-called mode coupling theory (MCT) explains the occurrence of two relaxation processes for a single variable: slow, so-called a-relaxation with strong temperature variation and fast so-called, 13-relaxation, which varies only slightly with temperature. MCT predicts a divergence of the time scale of the arelaxation process at some critical temperature, To. Up to now the experimental tests of the M C T predictions were carried out for fragile (in Angelrs [2] classification) glass-forming liquids only. These are ionic and van-der-Waals liquids, and they are closest to hard sphere systems for which M C T has been developed. Analysis of different experimental data in these liquids supported qualitatively and

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A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

quantitatively several of the main predictions of MCT and found that Tc of MCT is significantly higher than the conventional, thermodynamic glass transition temperature, T~ [1]. In the extended MCT, which includes an additional activated process [1], the divergence of the m-process timescale at Tc is suppressed, and additional details are predicted for the shape of the relaxation function [3]. Also at temperatures which are higher than T~ by similar amounts as for Tc, other peculiarities of relaxation processes were found to occur: decoupling of rotational and translation diffusion [4]; branching off of the so-called slow t-relaxation (Johari [3 process) [4]; change of the Kohlrausch stretching exponent [5]. These experimental results point again to the existence of a distinguished temperature higher than T~ where a non-monotonous change in the relaxation processes occurs. Here we want to direct attention to the fact that just in the frequency range of the fast relaxation process (fast B-relaxation of MCT) a significant part of the dynamic structure factor S(q, to) of supercooled liquids is contributed by low-energy vibrational excitations. These excitations are a general feature of the vibrational spectra of glasses, observed as the so-called boson peak in Raman spectra or excess density of states in inelastic neutron scattering spectra. Thus any MCT analysis, addressing the relaxation spectrum, ought to consider also this vibrational contribution. In many cases, it is simply subtracted from S(q, to) [6], although it is not clear whether the two contributions are sufficiently independent of each other to justify this procedure. In the present paper, the low-frequency Raman spectra of glass-forming liquids having different degree of fragility are analyzed. The difference in dynamic properties of strong and fragile glassforming liquids is shown from the spectra analysis. The next point is an analysis of the relaxation part of the Raman spectra in the framework of MCT. We want to demonstrate that not only for fragile, but also for intermediate glass formers the qualitative predictions of MCT are in agreement with the experimental results. However, quantitative disagreement appears as one is going from fragile towards strong glass formers. It is shown that this disagreement occurs due to the increase of the

139

vibrational contribution (the boson peak). The next point is an analysis of the vibrational contribution and its variation during the glass transition. It is shown that at some temperature, T*, above Tg a change of dynamic behaviour sets in: the lowfrequency vibrational motion transforms to overdamped state (relaxation type of motion). The relation of this transition to Tc of MCT, as well as a new version of the glass transition scenario, is discussed.

2. The difference in dynamics of strong and fragile glass-forming liquids A general feature of the low-frequency Raman spectra of glasses is the so-called boson peak. It was observed in the spectra of all investigated glasses [7]. It was shown that the peak in Raman spectra appears due to excess density of vibrational states [8-10], which was observed also as an excess in low temperature specific heat, Cp, around T ~ 5-10 K, and in inelastic neutron scattering. It is generally ascribed now to quasilocalized collective atomic vibrations involving ~ 30-100 atoms. The frequency of the boson peak maximum, tomax, is usually related to a so-called dynamic correlation radius Re -~ V/tomax~ 10--20 /~ (V is a sound velocity) [11-13]. This idea was recently supported by comparison of the Raman spectra with the X-ray diffraction data [14, 15]. Fig. 1 shows typical low-frequency Raman spectra of a glass-forming system at different temperatures. The spectra are normalized by the temperature factor to[n(to) + 1] ~ k T for the Stokes side, with n(to) = [exp(hto/kBT)-l)]-i,

I, = I/[to(n(to) + 1)3.

(1)

The low-frequency spectra have, as usual in amorphous systems, essentially two contributions: a vibrational one (boson peak), which dominates at frequencies beyond roughly 0.5 THz, and a relaxational one (quasielastic scattering) which dominates at lower frequencies. It has been shown recently [16] that the latter corresponds to the fast 13-process analyzed within MCT. At low temperatures ( T < 50-100 K), the relaxation processes are strongly suppressed and the spectrum is solely

140

A.P. Sokolov et al. / Journal o f Non-Crystalline Solids 172-174 (1994) 138-153

tional contribution C(o)) oc o) [8,18]. However, for the relaxation contribution (quasielastic light scattering) it was shown [16] that

1.4

,..-:,.1.2

I(6o)/(n(co) + 1) o¢ Xq(C~)

~_~0.8

~

where Xq(O)) is the density-density susceptibility function obtained from the neutron scattering spectra. In terms of Eq. (2), Eq. (3) corresponds to C(o))---const. This result has been obtained also from the direct comparison of light and neutron scattering spectra of glasses [10]. The mechanism of the low-frequency Raman scattering is still not known [16,18,19] and we will not go here into the discussion, because it is not important for the results presented in this paper. The relaxational contribution (or quasielastic light scattering), which is negligible at low T, increases with temperature faster than (n(o))+ 1) (Fig. 1). Thus, with temperature increase the relaxational contribution to l,(o)) spectrum rises, first at very low frequencies, so that a minimum appears in l,(to) at a frequency, o9, lower than the boson peak (Fig. 1). In order to characterize the ratio of the relaxational to the vibrational contributions, one may use the ratio of the intensity in the minimum to the intensity of the boson peak maximum [20]:

0,G

~0.4 0.2

8Kj 0.2

0.4

0.6

0.8

1.2

1

1.4

1.G

v (THz) Fig. 1. The low-frequency Raman spectra of B203 at different temperatures. The spectrum at T = 8 K is essentially the vibrational contribution (the boson peak) only.

determined by the spectrum of vibrational excitations, i.e., by the boson peak (for example, spectrum at T = 8 K, Fig. 1) [7,8]. At temperatures T < Tg, the vibrational contribution follows well the Bose statistic I(to) oc (n(o)) + 1) (Fig. 1). This corresponds to the first order light scattering process. In this case, the intensity of the Raman spectra will be proportional to the density of vibrational states, #(o9), multiplied by the light-to-vibration coupling coefficient, C(og) [17]: I(o9) = g(oo)C(oo)[n(o)) + 1]/o9

(3)

(4)

R 1 = (ln)min/(ln)max.

(2)

Analysis of a number of glass formers has shown [20] that an essential difference of the Raman spectra is this ratio Rt (see Fig. 2). For B203, the

Comparison of light scattering data with neutron and heat capacity data shows that for the vibra-

10 O0

(t) a#, ~., 100

:t.,

435~ (e) m.TCP

3s2~(b) glycerol

5 (d)

27

622 59O 525

29253~x, 2

1

9

236 21

~

8 I

0.01

0.1

1

10

. . . . ,...'"~r.[ ...... , . . . ~ 0.1 1

.............. 10 0.1

i. . . . . . . . X ..... t 10

0.1

1

10

v f rz~) Fig. 2. The low-frequency Raman spectra, l.(o)), of different glass formers in double logarithmic plot. The numbers are temperature in K, The arrows show the boson peak position.

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153 1.8 ¸

-

1.4

0.2 0

/4' 20

40

(comparing with the quantity F = E,(T~)/T s used in Ref. [20], F ,~ 2.3Rm, R is the gas constant). It is seen in Fig. 3 that systematically more fragile glass formers have higher values of R1. As has been shown [20], this tendency is not specific for the Raman spectra only, but is also found in inelastic neutron scattering spectra (Fig. 3, Table 1). Moreover, this systematic difference in the dynamic properties of glass-forming liquids shows up also in the low temperature ( < 20 K) heat capacity, Cp, of the glasses. In the framework of the Debye model (gDeb(09)OC (.O2), it is expected that Cp(T) = CD=b OC T 3. For most of the crystals, the temperature variation of heat capacity follows to the Debye model: CffCD~b ~ 1, while for all glasses Cp/CD=b > 1 and an additional peak, related to the excess density of vibrational states, is found at T ,~ 5-10 K [7] (Fig. 4). At still lower T ( T ~< 1 K), Cp/CDcb increases again (Fig. 4) due to the linear term in heat capacity. The latter is usually ascribed to so-called two-level or tunneling systems (TLS) [7]. Thus similarly to the Raman case, a minimum develops roughly between 1 and l0 K and one may introduce the quantity

-1

7•8•/10•

11

0.8 -0.6 -0.4 "0.2

60

80

100

120

140

160

(5)

m = - {d[log(z)]/d[T~/T]}r=r,

1.2

~ 0.8 0.4

1.6

-1.4

13A

E3

1.2

0.6

transition:

-1.8

1.6

141

0 180

IT1

Fig. 3. Ratios of relaxational and vibrational contributions for different glass formers: RI(Tg) (taken from the Raman spectra ([-7) and neutron data (&), see Eq. (4)) and R 2 (from the lowtemperature heat capacity ( I ) , see Eq. (6)) versus degree of fragility, m. The numbers correspond to various glasses as listed in Table 1.

boson peak dominates the spectra even at T higher than the melting point, Tin; in the spectrum of glycerol at Tm it can be seen only as a shoulder, it disappears in the spectra of m-tricresyl phosphate (m-TCP) already at T slightly above Tg, while for Ca2Ka(NO3)7 (CKN) it cannot be seen even at Tg. The corresponding correlation of RI(T~) with fragility is demonstrated in Fig. 3 and Table 1. As a measure of the degree of fragility we use, following B6hmer et al [21], the temperature behaviour of the average relaxation time, z, close to the glass

(6)

R2 = (Cp/Co,b)mi./(Cn/CD,b) . . . .

Table 1 Degree of fragility, m, R1 at T~ estimated from the Raman and neutron data and R 2 from Cp for different glass formers (for explicit definition see text, all data from Ref. [20], excluding PVC from Ref. 122]). Also TJTg (from Ref. [28]) and T*/Tg are presented N

Substance

Ts (K)

m

1 2 3 4 5 6 7 8 9 10 11 12 13

SiO2 GeO2 B20 3 4SiO2-Na20 3SiO2-Na20 glycerol PS (M = 37 000) salol PB m-TCP o-terphenyl CKN PVC

1446 820 526 760 735 186 370 218 181 206 243 333 358

23 33 39 44 44 58 59 63 64 76 81 94 157

R1 Raman

R1 neutron

R2

0.5

0.28 0.43 0.44 0.56 0.63 0.63 0.75

0.41

0.39

0.6

0.85 0.7 0.7 1.3

0.95

0.65

1 1.7 1.4

1

To~Ts

T*/T s

1.68

1.9

1.17 1.19 1.24 1.19 1.14

1.3 1.2

142

A.P. Sokolov et al. / Journal o f Non-Crystalline Solids 172-174 (1994) 138-153

(single particle diffusion and weakly perturbed phonons). t

/ I

3

/

3. The relaxation contribution and MCT analysis r.,.)

~a¢~ ~ . ~ . 7 S ~ - ~ - ~ - . ~ _

0

4

8

r (K)

12

glycerol

16

20

Fig. 4. Low-temperature heat capacity of different glasses normalized to the Debye values (data from Ref. [20]).

Now it is known that more fragile systems have lower values of the maximum of Cv/Coeb, but on the other hand the TLS contribution was found to increase with fragility [20-1, again in parallel with the light and neutron scattering (RI). When we now plot this value, R2, against m, it also correlates quantitatively with the degree of fragility of the systems (see Fig. 3 and Table 1). It is thus found [20] that a peculiarity of the dynamic structure factor of glass-forming liquids, i.e., the ratio of the vibrational to the relaxational contributions, is related to differences of the low-temperature anomalies of glasses. In particular, the relative intensity of the quasielastic scattering (fast 13-relaxation process of MCT) in the liquid correlates with the density of tunneling centers in the glassy state. The results presented above show that the main difference in the dynamics between strong and fragile glass formers is the type of the collective atomic motion: for fragile systems it is relaxation (fast 13-relaxation in terms of MCT), while for strong systems it is vibrations (boson peak). This difference determines the scenario of the glass transition and we can speculate that for fragile liquids the decisive dynamic process in the glass transition is local relaxational motion, while for strong liquids it is vibrations. Moreover, this difference of the liquids has its correspondence even at very low temperature, seen in the low-temperature properties of glasses. However, both the relaxational and the vibrational process are most probably different from those occurring in a crystal before melting

The difference discussed above between strong and fragile liquids cannot be explained in the framework of the mode-coupling theory [1]. MCT focuses on the relaxation part of S(q, co) and does not include explicitly vibrational contribution of comparable frequency. Indeed the relaxation contribution dominates S(q, to) of fragile systems. Thus, these glass formers are ideal for testing MCT, and its predictions were supported by many neutron [6,23-25] and light [5,16,26,1 scattering experiments on fragile liquids. The theory is formulated in terms of the normalized density~lensity correlation function, Oq(t): • q(t) = (~p*(0) ~pq(t)), where 8pq is the q-th component of the microscopic density fluctuation and S(q) is the static structure factor. At high temperature, the idealized version of the theory predicts a two-step decay of O~(t): a fast 13-process, with z~ not depending on temperature, and a slow or-process, with strong temperature dependence of ~. From the analysis of the correlation function, several predictions for the relaxation behaviour in the time and in the frequency domains have been derived. [1]. In the latter case, the dynamic susceptibility Z~(to) is usually analyzed. MCT predicts a stretched spectral form of Z~(to) for the at-relaxation, where the extrapolation of the high frequency tail shows the power-law dependence for the imaginary part: Z~(to) oc to-b.

(7a)

For the 13-relaxation, MCT predicts a spectral form with a power-law for the low-frequency tail: Zg(to) oc toa.

(7b)

The exponents a < 0.4 and b ~< 1 depend on the coupling between relaxation modes, Oq(t), and they are related to each other via the transcendental equation involving the gamma function, F,

r2(I - a)/ro - 2a) = F2(1 + b)/F(1 + 2b).

(8)

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

Thus, at high temperatures (above the MCT critical temperature, T¢) the theory predicts a minimum in the spectra of X~(og)which opens up between at- and ~-relaxations. The minimum is predicted to exhibit an universal spectral form (master curve): ~"(o9) OC )~min[b(og/ogmin)a .-]- a(og/ogmin)-b]/(a +

for T > To,

b), (9)

where ogmi, and Xm~. are the frequency and the amplitude of the minimum, respectively. According to MCT, these two parameters will show a critical temperature dependence:

Zmi~ oc ( T - T¢)1/2 ]'T ogmin OC(T-- Tc)l/2aJ > To.

(10)

Further, the frequency of the maximum of the ~relaxation o9~ = 1/r~ is also predicted to exhibit a critical temperature variation above T¢: o9, oc (T - T~)~, with exponent y = 1/2a + 1/2b.

(11)

In fact, Eq. (9) follows from Eqs. (7a), (7b) and (10) together with one very important prediction of MCT: At T > T~, the fast dynamics (15-relaxation) is fixed in amplitude and frequency (it does not vary with temperature), while the slow a-relaxation is fixed in amplitude, but varies in frequency according to Eq. (10). This prediction was also formulated in terms of the Debye-Waller factor for the amplitude of the fast dynamics, which must be fixed at T > T¢, and then decreases oz (T~ - 7) 1/2 [1]. We do not discuss this point in the present paper and stress only that this prediction was supported by many neutron scattering [23], Brillouin [26,27] and Raman [28] measurements data. Thus MCT describes the glass transition as a purely kinetic or dynamic phenomenon. The theory develops not only a qualitative scenario, but leads also to quantitative predictions for the spectra X"(og). The first tests of MCT predictions were carried out in neutron scattering experiments [23-25], which give without serious additional assumptions ~q(t) (from neutron spin-echo spectra) and X~(og) (from time-of-flight spectra). For both spectra, good agreement was found with the MCT

143

predictions in the cases C K N [23,25], O T P [6,23] and in some other fragile liquids. The two relaxation processes - fast, with slight temperature variation of the characteristic time, za, and slow, with strong temperature dependence of z~ - have been observed. More specifically, the minimum of Z"(o9) with the predicted spectral form (Eq. (9)), as well as the critical temperature variations of the parameters /(rain and o9min (Eq. (10)) have been found. The estimate of the critical temperatures lead to values significantly higher than Tg: Tc ~ 370 K (Tc/Tg "~ 1.12) for C K N [25] and Tc ~, 290 K (Tc/Tg ~ 1.2) for OTP [6]. It was shown later [5,16] that the same analysis (Eqs. (7)-(11)) may be applied to the light scattering spectra using the assumption that X~(o9) oc I/(n(o9)+ 1) (Eq. (3)). Figs. 5(a) and 6(a) show the typical spectra of the minimum of 1/(n(o9) + 1). These are the spectra presented in the Fig. 2 multiplied by an additional factor co: X"(o9) = I,(o9)o9. We would like to stress the point that the analyzed minimum appears here between two relaxation contributions, while for spectral density, 1,(o9), (Fig. 2) it appears between vibrational (boson peak) and fast relaxation (MCT I]process) contributions. The analysis of the minimum of 1/(n(og) + 1) has been carried out first by Cummins and co-workers for C K N [16] and salol [5], and later by others also for m-TCP and glycerol [28] and O T P [29]. The scaling behaviour of the susceptibility minimum (Eq. (9)) has been found in all these substances. In the case of very fragile CKN, the master curve obtained is well approximated by a sum of two power laws (Eq. (9)) with exponents a = 0.27 and b = 0.46 fixed by the transcendental Eq. (8) [16]. The critical behavior of X~i, and ogmi, Eq. (10) has been found, and the critical temperature was estimated to be Tc ~ 378 K in full agreement with the value of T¢ estimated from the a-relaxation time Eq. (I 1) [16]. In the case of fragile OTP, the authors of Ref. [29] found a = 0.33 and b = 0.65. The critical temperature was estimated as Tc ~ 290 K in full agreement with the previous neutron measurements [6]. In the case of the less fragile m-TCP, the master curve of the susceptibility minimum may also be approximated by Eq. (9) assuming that a and b are related through Eq. (8); this leads to exponents

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

144

X"(v)/X'~. 2 2 X"(v) X~in 1

m-TCP 0.5 (a)

_

............... I0H

1

~

V/Vmin

10

tl 1

0.5

. . . . . . . . . . . .

(a)

10-~

4o?y.,,,~bl

1

10 v/Vmin

(Vmin/GHz)Za/103 (Vmin/GHz)za 50

X~n/10 s

m-TCP

/

120

40 a=0.346

1000

*-

• 80

Tc=255,7 K J , ~ ,

"e/

0 (b)

240

260

280

.

,""

40 .

.

.

.

300 320 T/K

I~

,,"

a.u

6O

--* •

40

•

500

a:1.1

.

.

340

0 360

20

f

o oo .

/./

/'x

--

10 "

. ,x"/ /

x"

30 20

glycerol

a.o

200

24-0 280

(b)

320

360

400

4-40

T/ K

O1T-1/pa sK-1)-l/Y

(Xrot/S)-1/Y

(tiT -1/PasK'l) -1/¥

glycerol 160

m-TCP

120

y=2.14

rl/

/

60 200

80

/ / /

/ Xrot

40

100

4-0 J

20

0 --r----r----~----,'----~

200 (c)

240

280 320 T/K

.... T .... r"

360

400

0 (c)

320

360 400 T/K

440

Fig. 5. M C T analysis of the susceptibility Z"(co) ~ I/(n(co) + 1) spectra of m-TCP (from Ref. 1-28]). (a) Master plot of X"(co)/Zmi, versus V/Vm~,,double logarithmic plot. - . . . . , interpolation by MCT master curve (Eqs. (8) and (9)) with a = 0.35 and b = 0.72. (b) Temperature dependence of (Xmin)2 a n d (Vrnin)2a. (C) Temperature dependence of the viscosity timescale, ~//T, and the timescale Z ot (from NMR) with ? = 1/2a + 1/2b.

Fig. 6. M C T analysis of the susceptibility ;("(co) ~ l/(n(co) + 1) spectra of glycerol (from Ref. 1-281). (a) Master plot of g"(co)/Xmi, versus V/Vmi., double logarithmic plot. - . . . . , interpolation by M C T master curve (Eqs. (8) and (9)) with a = 0.34 and b = 0.71. . . . . . . best fit by a sum of two power laws (Eq. (9)) with a = 1.16 and b = 0.37. (b) Temperature dependence of(Av~in)2 a n d ( v i , ) 2° with a = 0.34 ( x ) and a = 1.16 (0). (c) Temperature dependence of the viscosity timescale, ~//T, with ? = 2.17 fixed by Eq. (11) (©) and best fit, 7 = 2.5 (0).

a = 0 . 3 5 a n d b = 0 . 7 2 (see F i g . 5(a)). H o w e v e r , some deviation appears on the high frequency side o f t h e m i n i m u m ( F i g . 5(a)). A n a l y s i s o f t h e t e m p e r -

ature variation of the parameters describing the minimum s h o w s t h e p r e d i c t e d ( E q . (10)) c r i t i c a l b e h a v i o u r w i t h Tc ,~, 255 + 7 K ( F i g . 5(b)) [ 2 8 ] .

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

This value agrees reasonably well with the estimate of T¢ from viscosity and N M R data extrapolated according to Eq. (11) (Fig. 5(c)) [28]. In the case of the intermediate glass former glycerol, the proposed procedure for constructing a master curve, namely scaling of Z~i,(T) and ~omi,(T), leads also to a master curve around the minimum of l/(n(~o) + 1) (Fig. 6(a)). However, the curve deviates significantly from the shape expected in the framework of MCT (Eqs. (8) and (9)). Accepting the relation between exponents a and b (Eq. (8)), the best fit in this case gives a master curve with a = 0.34 and b = 0.71; here b was first fixed by a fit to the low frequency side, However, this curve (dashed curve in Fig. 6(a)) deviates strongly from the experimental spectra on the high frequency side of the minimum. Alternatively, one can still approximate well the spectrum by a sum of two power laws (Eq. (9)), which then get exponents a = 1.16 and b = 0.37, values which do not satisfy MCT Eq. (8). Fig. 6(b) shows the temperature variation of the intensity and of the frequency of the minimum. Clearly, T¢ taken from extrapolating (Zmi,)2 is significantly higher than that taken from (~Om~,)2" with a = 0.34. Nevertheless, using a = 1.16 (as given by the interpolation after dropping the MCT restriction Eq. (8)), a good coincidence is found from both extrapolations, and T¢ ~ 300 K may be estimated. This result agrees reasonably well with the value Tc ~ 320 K (Fig. 6(c)) estimated from the power law behaviour of the viscosity timescale ~/T using = 2.17, the value ~ being calculated from the MCT values a = 0.34 and b = 0.71 (Eq. (11)). Even better agreement is found by performing a best fit of the viscosity data to ~/Toc ( T - T¢) -~ with free y; this gives y ,~ 2.5 and T¢ ~ 310 K (Fig. 6(c)). The results presented above show that the spectra of less fragile systems have some deviations from the master curve predicted by MCT: the stronger the liquid, the larger the deviation at the high frequency side of the minimum (Figs. 5(a) and 6(a)). It is obvious that this deviation appears due to the non-negligible vibrational contribution (the boson peak) to the Raman spectra, which increases for stronger glass formers (see the previous section). This contribution disturbs not only the spectral form, but also the temperature dependence of the minimum parameters (Eq. (10)). In all cases, Tc

145

estimated from ((Drain) 2a is smaller than that estimated from tZmi,~, . . . . 2 and the difference increases with increase of the vibrational contribution: a small difference may be seen for CKN [16], salol [5], OTP [29], m-TCP in Fig. 5(a), but it is already large for glycerol (Fig. 6(a)). The reason is that the estimate of Tc from (t~mi,)2" depends strongly on the exponent a. If, as an alternative, the value of the exponent a is taken from the best fit to the spectra (but without restriction of MCT, Eq. (8)), reasonable agreement is found with the estimate from the x2, as well as with temperature dependence of t'" ,~min) the estimate from the temperature dependence of the 0t-relaxation time (Eq. (11), Figs. 5 and 6). The problem of the exponent a had already been found in the cross test of Eq. (10), i.e., from the plot of :t~i, vs. t~mi, for the cases CKN [16], salol [5] and OTP [29]. The value of a estimated in this way was once again higher than that estimated directly from the shape of the master curve. Thus, the above analysis shows that two features, the master curve for the minimum of the spectra and the critical behaviour of its parameters, may be found for all investigated liquids (Figs. 5 and 6). These regularities are in fact consequences of the fact that, according to MCT for high temperatures (T > To), the fast dynamics (~-relaxation) are fixed and have no temperature variation, and only the timescale of slow Qt-relaxation varies critically with temperature. In this respect, one may conclude that, even in intermediate glass formers, like glycerol, the high temperature dynamics (T>~ T~) follows qualitatively to the MCT predictions. However, the vibrational contribution disturbs the high frequency part of the dynamic spectra and some quantitative disagreement of MCT with the experimental results appears. This disagreement is negligible for extremely fragile systems, like CKN, and increases for stronger glass formers due to the increase of the vibrational contribution (see the previous section). One may expect that simple subtraction of the vibrational spectra can restore the MCT spectral form. Unfortunately, this is not the case: The analysis made for glycerol in Ref. [28] and also for OTP in Ref. [6] showed that the subtraction improves the situation, but the quantitative disagreement remains.

146

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

4. The vibrational contribution and its transformation during the glass transition The problem is that the boson peak and fast relaxation not only occur in the same frequency range, but are apparently strongly interrelated. This point of view is first suggested by the smoothness of the transformation from (almost or indeed) purely relaxation spectral shape at high temperatures, well above To, to (almost) purely vibrational behavior at low temperatures (T<
-37.0°C

--?

_~-~i~.................................... 24.2°C

0

I-i f f I ~ J , , I , , , , I , , j i J i , , i 10 20 30 4 0 to/cm-~

Fig. 7. T h e low-frequency Raman spectra of GeSBr2 (T B = - 30°C) and their approximation by a sum of the boson p e a k a n d L o r e n t z i a n c u r v e s ( f r o m Ref. [32]).

this analysis that Ogmaxvaries with temperature approximately in accord with the variation of sound velocity, v, i.e., COmax/V ~ const. The conclusion was made that the dynamic correlation radius, Ro, which at low temperature will be a characteristic of the vitreous structure Rc ~ v/o9 .... is formed already in the liquid state at T > T~ and remains essentially constant upon decrease of temperature ([31,32]; see also Ref. [141). Another interesting result obtained from this analysis is the disappearance of the boson peak contribution to the spectra at some temperature significantly above Tg (e.g., at 1.3Tg in the GeSBr2 system; see Fig. 7) [32,33] and that the increase of the relaxational contribution upon increase of T happens, at least in part, at the expense of the vibrational contribution. However, as was mentioned above, the simple decomposition of the spectra into two contributions is most probably not correct, because they are strongly interrelated. In Ref. [34] another approach to the description of the low-frequency Raman spectra has been proposed. The authors of Ref. [34] assumed that the low-frequency vibrations change drastically with temperature due to coupling with some relaxation channel. The susceptibility function for a single vibration with

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

frequency, t2, may than be expressed as D(o9) = It22 - o92 _

82/(1

_

io9z)]- 1

(12)

Here the third term describes the coupling of the vibration, 12, to some relaxation channel for the vibrational energy, 82 is the temperature-dependent strength of the coupling and z is the relaxation time. The light scattering spectrum becomes then I(og)/(n(o9) + 1) = C(f2) Im {D(o9)} = C(f2)G/[(K22 _ o92 _ G/Ogz)2+ G2], G = 82O9"~/(032"L"2 "+"

1).

(13)

Here we have again introduced explicitly the lightto-vibration coupling coefficient, C(o9), which was discussed in Section 2. It determines the depolarization ratio (the corresponding indices have been omitted for simplicity). The well known shape Eq. (13) consists for small ~2<
(14)

According to the soft-mode formalism [35] (which should not be confused with soft-mode approach of Ref. [36]), o90 has a power law critical temperature dependence at temperature close to T*, T < T*: o9o ~ oc (T* - T).

(15)

It corresponds to a transition of the vibration to an overdamped state at T = T*, i.e., to the transition of the soft-mode from vibrational to relaxational type of motion. Here we would like to stress

147

a qualitative difference with the usual hydrodynamic approach, where overdamping occurs due to variations of z at o9z ~ 1. In our case (Eqs. (12)-(15)), overdamping occurs due to variation of the relaxation strength, 82, rather than due to the variation of z. The authors of Ref. [34] assumed that the boson peak in the Raman spectra of glasses is formed by distribution of vibrational modes, each of them coupled to the relaxation channel, in the way described by Eq. (12). This distribution (or inhomogeneous broadening) is what we call here for short boson peak, typical of each glass. The low frequency spectrum including the relaxation (i.e., with homogeneous broadening, Eq. (13)), is then I(o9)/(n(og) + 1) = f d f 2 C(f2)a(f2)G/{(f~ 2 - o92 _ G/o9z)2 + G2}. tt

(16) Here 9(t2) is the density of the vibrational states. In order to make this a usable expression, one has to know the spectrum C(I2)O(I2 ). It may be obtained from the low-temperature (T<< T+) spectra, where 8 2 ~ 0 because all relaxations will be strongly suppressed (Fig. 1). In this case Eq. (16) gives I.o(o9) = Io(o9)/o~(n(o9) + 1) =

C(o9)9(o9)/o92,

(17)

which is essentially the well known Shuker and Gammon [17] formula. Thus the final expression for the low-frequency Raman spectra may be expressed through the known spectrum I.o(o9): l(~o)/(n(o9) + 1) =

fda

l.o(f2)f22G/{(f22 - o92 _ 6/o9T) 2 + 62}.

(18) We have used this expression (Eq. (18)) for an analysis of the low-frequency Raman spectra of glycerol, dibutyl phthalate, alcohols [34] and it has also been used for GeSBr2 [33] and As-S-P [37]. Using 8(7) 2, T and o9m,~(T) as fit parameters it was found that Eq. (18) describes well the spectra of all analyzed liquids in a broad temperature range. As a new result, Fig. 8 shows the comparison of experimental and calculated (from Eq. (18)) spectra for

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

148

(c)

glycerol, m-TCP and OTP. The shapes I,o have been obtained by an extrapolation of the low-frequency tail of the boson peak in spectra of the substances at T < T~ (Fig. 8). It was found that tom.x, the position of the maximum of I,o, varies in accord with the variation of sound velocity, i.e.,

I

I

I

~lyeerol

+~~322 K

[]

1

(a)

I

I

I

OTP OK

.....

245 K 270 K

<> +

/ I "

I

I

I

500

I000

1500

.

1

Gnz)

(Fig. 8. continued)

/ / /

"/

0

I

I

I

250

500

750

"l OH,.) (b)

I

I

~,

-

m-TCP .

205 235 262 287

t~ ~n ~

K K K K

.

.

.

.

o + x

+

0

I

I

I

250

500

750

.

GH,,I

Fig. 8. The low-frequency Raman spectra of different glass formers: experimental data (points, only every third point is plotted); . . . . . , I,o(to); , fit with the model 134] (Eq. (18)).

once again Rc ~ v/tom,x ~ const, was obtained. Fig. 8 shows that the model 1-34] describes well the low-frequency Raman spectra, i.e., the boson peak and the high frequency (to > 100 GHz) part of the quasielastic scattering, of different glass-forming systems in a large temperature range, from T<< Tg up to T/> Tin. The model explains also the temperature and spectral behaviour of the depolarization ratio: This ratio is determined by the depolarization ratio of I,o(0), i.e., the spectra of the boson peak at low temperature. Thus it is the same for the vibrational and relaxation part, and does not depend on temperature. The most interesting result, from our point of view, is the temperature variation of the renormalized frequency of the boson peak maximum 0)2 = (/)max2 __ ~2 (Fig. 9), where tom.~ may be considered characteristic for the entire boson peak. It shows a critical behavior (Eq. (15)) in all analyzed liquids and gives an estimate of a temperature, T*. At this temperature according to the model [34], the low-frequency vibrations giving the boson peak transform into overdamped state. It means the transition from essentially vibrational to essentially relaxational type of collective molecular motion, i.e., transition from solid-like (vibrational)

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

(a)

150

,

,

,

I

I

I

1

1.1

1.2

100 CO2

50

0 ....

~a~

T/T, i

(b) 200

i

150

W2

100 50 0 I

I

1

1.2

1.4

T/Tg (c)

I

i

I

1500

1000

149

range scale ( ~ 10-30 ~). Thus, the onset of a different type of dynamics occurs on this intermediate range scale. This first sign of the glass transition (appearance of characteristic solid state motions on short time and small spatial scales) appears thus already in the liquid state. The temperatures, T*, are well above Tg, (see Fig. 9, Table 1) and do not coincide for all systems with the melting temperatures, Tin. The value of T*/Tg differs for different liquids and correlates with the degree of fragility of the liquids (Table 1): in fragile liquids, where already at T,~ T8 the relaxational motion dominates, transition of the vibrations to overdamped state occurs closer to Tg, while in strong and intermediate glass formers, where the relaxational motion is strongly suppressed, the transition may happen even at temperature above the melting point. We mention here, that the estimate of T* is somewhat model dependent. The problem is that the relaxation part in Eq. (12) is taken in the simplest Debye form with a single z, but it is known that the relaxations in glass-forming systems are more complicated. Thus, probably, the value of T* will vary slightly with changes of the relaxation term in Eq. (12). However, this will probably not change the conclusions significantly. The nature of the coupling, the mechanism of its temperature dependence and any possible interrelation between 8 and, for example, g(og) remain open at present.

CO 2

5. A scenario of the glass transition

500

I

I

I

1

1.4

1.8

T/T, Fig. 9. Temperature dependence of the renormalized frequency of the boson peak m a x i m u m , co2o, (Eq. (14)) for different glass formers.

to liquid-like (relaxational) behaviour. This transition concerns primarily the motions seen in the boson peak, which - as we have argued in Section 2 - are motions localized on a medium

Let us now try to conceive a unified picture by combining the results of the two previous sections, namely, the analysis of the relaxation processes within the framework of MCT, and the analysis of the vibrational part within the framework of the model [34]. The most exciting point is the fact that T¢ of MCT and T*, estimated from boson peak analysis, nearly coincide for different glass formers (Table 1). It means that we have two independent supports for the idea of a non-monotonous change of the dynamics in glass forming liquids at temperature above Tg and that the term dynamic phase transition may be appropriate: (i) The relaxational

150

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

motion, which is typical for the liquid state, has a critical temperature variation (slowing down) when one approaches Tc from above; (ii) The vibrational excitations, which are a sign of solid state, also have a critical temperature variation (becoming overdamped), but when we are coming from below T*. Using these facts we can speculate about the scenario of the glass transition. First, at high temperature, only relaxational motions are significant. They are well characterized by two relaxation processes: fast process, the so called fast 13-relaxation, which is virtually temperature independent, and slow ct-relaxation with strong temperature variation. Upon decrease of T, a critical slowing down of s-relaxation develops. All over this temperature range, MCT [1] predicts well the features of the relaxation motion. But the contribution of the overdamped low-frequency vibrations has to be included. At (nearly) the same temperature as the critical temperature, T~, ( T * ~ T¢) of MCT, the boson peak starts to develop upon cooling. However, the boson peak is caused by a dynamic correlation over ,-, 10-20/~. This leads us to the following hypothesis: it is in dynamically coupled regions of intermediate size that relaxational motions (overdamped) turn into vibrational motions (underdamped on the timeseale of the boson peak frequency). In this way, at some temperature a type of dynamic phase transition from relaxational to vibrational type of collective atomic motion occurs in the liquid. Also, this is a sign of the transition to solid state, which happens at first on a spatial scale of ~ 10-20/~ [13,14], and on a timescale of one vibration ~ 10-12 s. It is then tempting to speculate that this first transition on an intermediate scale induces other strong variations of the dynamics of the liquids which are known to occur near T¢: the translational and rotational diffusion deeouple; some additional so-called slow 13-relaxation (Johari process) appears (in these solid-like mieroregions); the width of the relaxation time distribution increases (decrease of the stretching exponent). The mechanisms for these induced processes are not clear yet. Between Tc and T~, the solid fraction (in a dynamic sense) increases steadily when T is lowered

further. This leads to a dramatic increase in viscosity, as is discussed below. Finally at T = Tg a solid phase appears on a macroscopic spatial and time scale. The increase in viscosities, which is the defining characteristic of glass-forming liquids, shows a clear differentiation between so-called [2] fragile and strong glass formers: Upon cooling from T>> (T~, To, T*), the temperature dependence of r/(T) is simple Arrhenius-like down to Tg in strong systems, whereas in fragile systems the increase is first much slower, and changes to a much steeper increase (compared with strong systems) around To. The differentiation may be expressed by the fragility parameter, m (Eq. (5)). To our knowledge, no proven clear model exists for this differentiation. One explanation of the difference between strong and fragile glass formers was recently suggested by Vilgis [38]. Here the fragility parameter was related to fluctuations, Az, of the coordination number z: Az ~ 0 for strong glass-forming liquids, but in fragile systems one has an increase of Az. This increase leads to an increase of fragility because higher barriers (than average) occur which lead to a high apparent activation energy, in fragile systems just above Tg. This idea is carried through quantitatively in Ref. [38]. In a way, this model overlaps with the picture presented here: the dominance of vibrational motion in strong glass formers corresponds to the motion around fixed positions and does not change the arrangement of atoms, i.e., Az/At = 0. On the other side, there is mostly relaxation motion in fragile systems, corresponding to permanent structural changes; as a consequence, Az/At is large, and it increases with increase of the relaxation contribution. Thus the static picture proposed in Ref. [38] correlates with the difference in dynamics of strong and fragile glass formers found in Ref. [20]. We turn now to an interesting idea [39] for calculating the viscosity from S(q, 09) in supercooled liquids, which will be modified to include the varying contributions of vibrational and relaxational motion. In Ref. [39], Buchenau and Zorn supposed that r/may be related to the mean-square displacement of atoms (u 2) by q = qo exp(u2/(u2),o¢),

(19)

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153 T (K) 400 500

300

-15

b

G"

measured

- - Vogel - Fulcher lit Se

O

3,0

k

F

1000

calculated from neutron data

5 o

5

,

Tm IO00/T (K -1)

Fig. 10. Temperature variation of viscosity in Se: Comparison of the experimental data ( - - - - ) with calculated (Eq. (19)) values ( . . . . . , from Ref. 1-39]).

151

analysis of neutron scattering data [6,39,40] supports this idea: In the extremely fragile OTP, (u 2) increases much faster and has a value larger than in the intermediate systems of glycerol and Se. As a result, viscosity of strong glass formers, where the vibrational contribution dominates, has a slow and smooth temperature variation (from Eq. (21)), while in more fragile systems the increase of the relaxational contribution over and above Bose-Einstein population (see the beginning of Section 2) leads to a much stronger temperature dependence of t/ in that temperature range where the relaxational contribution (u2), displays its strong rise. This rise sets in at Tg, (see Fig. 2). Using (Eq. (20) we can express the degree of fragility, m: m = - Tg{d[log(t//T)]/dT}r=r,

where (u2)lo, was defined as a difference between the full (U2)liq of the supercooled liquid and ( u 2 ) , of the crystal, at the same T, and was estimated from the neutron scattering spectra in the frequency range 50-500 GHz. Using % and Uo as fit parameters, the authors reached good agreement of viscosity as calculated from Eq. (19) with experimental data in a broad temperature range (Fig. 10). Reasonable values were found for the fit parameters, namely Uo = 1.59.~ and % = 3.1 x 10 -+ Pa s for Se [39]. This approach recalls old ideas of free volume theory. Since, however, the physical reasoning for a linear relation between In(t/) and 1 / ( U 2 ) l o c is not unambiguous, one may assume this relation to be of the more general form, ln(t//t/o) oc F(u~/(u2)~o¢),

(20)

where F is a not yet specified monotonous function. Now we do the important step presenting (u2),q as a sum of vibrational (harmonic) and relaxational (strongly anharmonic) terms: (u2)v and (u2), respectively, so that (u 2) = (u2)v + (u2),. The first term varies nearly proportional to temperature: (u2)v oc k T .

(21)

The second term, (u2),, increases with T much faster. Since the second term is large for fragile systems, one has to expect that for them (u 2) will increase with T much faster and will have a value higher than for strong glass-forming liquids. An

oc { d F / d ( u 2 ) ( d ( u 2 ) v / d T + d ( u 2 ) , / d T ) } r = r,.

(22)

Generally, the relaxation term increases with temperature much faster than the vibration term. As a result, a larger relaxation contribution to (u 2) will lead to a larger degree of fragility, m, of the system. By contrast, for strong systems, where the first term in Eq. (22) dominates, m will be small and the activation energy of the viscous flow will be only slightly temperature-dependent. Moreover, the dependence of m on the temperature variation of (u 2) (but not on its value) through Eq. (22) may explain the scattering of the points in Fig. 3, in particular, for glycerol, which has a comparatively small weight of relaxation contribution to the Raman spectra at T,, but then it increases strongly with temperature (Fig. 2).

6. Conclusions

The above analysis has shown that not only relaxational motions determine the scenario of the glass transition, but that collective vibrational excitations also influence the dynamics of this process. It was demonstrated that the main difference in dynamics between strong and fragile glass-forming liquids is the ratio between the contributions to the dynamic structure factor which come from fast

152

A.P. Sokolov et al. / Journal of Non-Crystalline Solids 172-174 (1994) 138-153

13-relaxation and those from low frequency vibrations. The analysis of the temperature variations of both contributions strongly suggests that a type of dynamic phase transition on a microscopic scale occurs in glass-forming liquids at Tc significantly above Ts. This is a first sign of the glass transition in liquid state, being a qualitative change of the dynamic properties. The ratio of Tc/Tg depends on the degree of fragility of the liquid due to the mentioned difference in dynamics between strong and fragile liquids. We propose the hypothesis that this change of dynamics occurs at first on an intermediate range scale ( ~ 10-30/~). After observing that strong and fragile glass formers differ in the relative strength of vibrational over relaxation motions, the connection to the shape of the r/(T) curve (Arrhenius vs. Vogel-Fulcher, say) is made. To that end, we extend the considerations in Ref. [39] by considering the vibrational and relaxational contributions to the relevant mean square amplitude. The high temperature behaviour of the relaxation part of S(q, o9) is well described in the framework of MCT [1] (T~> T¢). However, at high frequencies some quantitative disagreement with the experimental results appears due to the contribution of the vibrational excitations (the boson peak). The disagreement is higher for stronger systems, which have a higher vibrational contribution to S(q, o9). Mode coupling theory has significant problems at T < T¢ and has been extended by including an activated process [1,3]. The observation which was emphasized in the present paper, namely, the rise of the vibrational contribution below T¢, suggests another important detail of the dynamics in this temperature range, which ought to be included in MCT analysis. The model [34] which was used here to describe the relaxational and the vibrational motions together does not include the driving force for the relaxation in a satisfactory way. Perhaps, a coupling of both approaches can describe better the full dynamics of the glass transition. One of the authors (A.P.S.) is grateful to the Alexander von Humboldt Foundation for a Re-

search Fellowship. This work was supported by Deutsche Forschungsgemeinschaft trough SFb 337 and partly by Russian Fund of Fundamental Research (grant N93022171).

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