Dynamical phase transition in simple supercooled liquids and polymers - an NMR approach

Physica A 201 (1993) 237-256 North-Holland SDI:

037%4371(93)E0255-D

Dynamical phase transition in simple supercooled liquids and polymers an NMR approach E. RGssler, A.P. Sokolov, P. Eiermann lnstitut fiir Experimentalphysik, Germany

and U. Warschewske

Freie Universitiit Berlin, Arnimallee

14, 1419.5 Berlin,

We present NMR investigations on m-tricresyl phosphate (m-TCP; 31P NMR) and on polybutadiene (PB; ‘H NMR). Relaxation studies have been combined with the analysis of the stimulated echo, and a reorientational correlation function is probed over the entire supercooled regime providing correlation times in the range 10-l’ s-10 s. Furthermore, we have performed Raman scattering (RS) experiments on m-TCP and glycerol. Testing predictions of mode coupling theory (MCT), the scaling behaviour observed by RS on m-TCP is well described by MCT for T > T, with T, -260K. The idea of a change of dynamics at T - T, is further supported by phenomena which - although not yet emerging from MCT are revealed by combining results from NMR, RS and neutron scattering. (i) The time scale of reorientational and translational motion separates below T, (m-TCP). (ii) The stretching parameter of the a-relaxation changes significantly near T, (m-TCP). (iii) NMR and dielectric relaxation probe a third process which exists only below T,. This process has all properties of the secondary relaxation discussed by Johari, however, it is well distinguished from the fast p-process analyzed within MCT and observed by RS. Thus, above T, the dynamics are described by a two-step correlation function with all the features predicted by MCT, whereas below 7’. a three-step function holds. As a consequence, a second order parameter appears, and the third process is described in terms of an activated process with a distribution of local environments.

1. Introduction Mode coupling theory (MCT) has introduced a new concept for the understanding of molecular dynamics in supercooled liquids [l-3]. Quantitative predictions have been made in terms of the density correlation function. In particular, a two-step function is predicted. Between the so-called microscopic time scale and the time scale of the structural relaxation (cu-process) a secondary relaxation process (P-process) is predicted. Within the idealized version of the theory, a- and p-process exhibit certain scaling relations with diverging time scales at a critical temperature T,. Thus, the transition at T, is of kinetical nature only, and the term dynamical phase transition has been 0378-4371/93/$06.00 fQ 1993 - Elsevier Science Publishers B.V. All rights reserved

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E. Riissler et al. I Dynamical phase transition in supercooled

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chosen. To account for the fact that in real systems a divergence of the transport coefficients is not observed, an extended version of the theory [4-61 has incorporated so-called hopping or activated processes. The additional relaxation processes restore ergodicity below T,. It is believed that the main features of the idealized version of the theory are retained above T,. However, below T, new features show up. All in all, a change of transport mechanism is expected in the supercooled liquid at T,. Indeed, recent neutron and light scattering experiments support the basic ideas of the theory [7-111. Features of the dynamical phase transition show up at a temperature well above the calorimetrically measured glass transition temperature Tg. Furthermore, it is well established that the temperature dependence of the viscosity n of supercooled liquids cannot be described within a single approach. At least two formulas have to be applied for interpolating (VFT) viscosity data [12,13]. For example, the Vogel-Fulcher-Tammann descriptions holds near Tg whereas a power law as predicted by MCT may interpolate the high temperature behaviour of n [14]. This is demonstrated in fig. 1, where the viscosity data of several van der Waals liquids are plotted on a reduced temperature scale T/T,. T, is the minorly corrected Tg. More precisely, T, has been taken in such a way to yield the best coincidence in the range lo*
n-butyl

P

set-butyl

a

cis-tram

.

di-n-butyl

v

3-methyl

benzene benzene decaline phthalate pentone

132

I

135 140 182 66 215 219

0

n-propyi

x

i-propyl

benzene benzene

129 130 255 313 244 206 206 345

T/T, Fig. 1. Viscosity of van der Waals glass formers plotted as function of the reduced temperature TIT,. T, is the minorly corrected T, and listed in the inset. Solid line: interpolation for o-terphenyl by a power law (PL) (cf. eq. (5)); dashed line: VFT description.

E. Riissler et al. I Dynamical phase transition in supercooled

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near Tg, whereas individual curves interpolated by power laws are observed at high temperatures. Independently of theoretical predictions these observations suggest a change of transport mechanism at T - l.l5T,. Correspondingly, in the following contribution we shall address the question: are there experimental facts in addition to features predicted by MCT which support the idea of a change of dynamics in the supercooled liquid above Tg? Due to the strong slowing down of molecular dynamics upon supercooling a broad range of excitation frequencies u has to be covered by the different relaxation methods applied to probe molecular fluctuations. We have performed NMR experiments on the van der Waals glass former m-tricresyl phosphate (31P NMR) and on the linear polymer polybutadiene (*H NMR). The NMR results - providing information on reorientational processes-are compared with those of Raman (RS) and neutron scattering (NS) experiments and with those of dielectric relaxation and viscosity studies. Whereas RS and NS probe the fast dynamics in the liquid, and predictions of MCT can be checked, NMR and dielectric relaxation provide information also on slower processes. Only by combining the results from different methods on the chosen glass formers a complete picture of the dynamics in the liquid may emerge. In some aspects this paper has a character of a review, thus, for details the reader is referred to the literature cited.

2. Results 2.1. 31NMR on m-tricresyl phosphate Fig. 2 shows typical 31P NMR spectra observed while supercooling a simple liquid, here m-tricresyl phosphate (m-TCP, cf. fig. 8a for the chemical formula) [16]. Starting with a narrow Lorentzian line at the highest temperature (244 K) the spectra continuously broaden, and finally at the lowest temperature (212 K), a characteristic solid state spectrum shows up. In the case of m-TCP the dominant interaction of the 31P nucleus with its surroundings is the chemical shift anisotropy (CS). The CS interaction originates from the specific electron density in the molecule leading to an anisotropic shielding of the external magnetic field. The spectra in fig. 2 also indicate the temperature ranges chosen for applying the different NMR methods. While the spectra are described by a Lorentzian line (T > 230 K) the spin-lattice (T,) and the spin-spin relaxation time (T2) are measured. At lower temperatures (T < 226 K) the analysis of the so-called stimulated echo is carried out in addition to studying T,. The stimulated method allows to probe slow processes close to Tg (T, z 210 K for m-TCP [16]).

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E. Riissler et al. / Dynamical phase transition in supercooled

400

250

200

T in K

-20

0

20

300

liquids and polymers

v/W Fig. 2. Fig. 2. “P NMR

spectra

of m-tricresyl

phosphate

while supercooling;

0,/2~r

= 121.5 MHz.

Fig. 3. “P spin-lattice (T,) and spin-spin relaxation times (T,) of m-tricresyl phosphate at two Larmor frequencies o, as a function of reciprocal temperature. (0) and (0): T, ; (+) and (0): T, at w,/2~r = 47.3 MHz and 121.5 MHz, respectively.

For m-TCP we have measured T, and T, relaxation times at two frequencies (w0/2n = 121.5 MHz and 47.3 MHz); their temperature dependences are shown in fig. 3. T, exhibits the typical minimum observed in many viscous liquids, and T2 continuously decreases. Below T - 230 K the temperature and frequency dependence of T, changes, which is an indication that a second process takes over the NMR relaxation close to Tg. As given by NMR theory [17] the relaxation rates are related to the spectral density @i(w) at the Larmor frequency w,, l/TFS = KCS[@;(w,)] , l/T;’

=+KCS[4@;(0)

+ 3@;(w,)] .

(1)

The coupling constant K ” is connected with the width of the NMR solid state

E. RBssler et al. I Dynamical phase transition in supercooled liquids and polymers

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spectrum. We note that in the case of 2H NMR similar relations as quoted in eq. (1) hold, and the corresponding coupling constant KEFG specifies the interaction of the electrical nuclear quadrupole moment with the electric field gradient (EFG) (cf. 2H NMR on polybutadiene, below). @i(w) is given by the cos-transform of the normalized reorientational correlation function F2(t) corresponding to the second Legendre polynomial P2(f) = +[3 cos29(t) - 11. The angle 6 describes the orientation of a molecular fixed axis - defined by the respective NMR interaction -with respect to the external magnetic field. Thus, molecular reorientations are probed, or in the case of polymers, segmental motions. Due to the different dependence of l/T, and 1 /T2 on the spectral density @i(w) (cf. eq. (1)) the non-exponential character of the correlation function F2(t) is usually reflected by different magnitudes of the temperature dependence of T, and T2, respectively (cf. fig. 3, T < 280 K). This experimental feature helps to determine the degree of non-exponentiality of F,(t), provided a proper model function is assumed for @z(w). A spectral density of the Cole-Davidson (CD) type has been found to describe both T, and T2 in many supercooled liquids [16,18-221. We write [23] @I(o) = 0-l sin&,

arctan(oTc,)][l

7*a = 72 = PCDTCD= @i(O) )

PCD is the so-called distribution

+ (07cD)2]-BCD’2 ,

O<&,Sl.

(2)

parameter, and T,,~ the reorientational correlation time. Thus, measuring T, and T2 allows to fix-besides the coupling constant Kc’ - T,,~ and pcD at each temperature. Kc’ can be compared for consistency with the one obtained from the solid state spectrum. However, save information gathered on &n is restricted to the conditions ti,,rrot B 1 (in the case m-TCP, T < 270 K), otherwise NMR in simple liquids probes only the time constant T,,~. As is demonstrated in fig. 4, a consistent description of the relaxation data for both Larmor frequencies is possible by applying the CD spectral density with a temperature independent pcD = 0.375 (T > 230 K) [16]. The correlation times extracted from the four sets of T, and T2 data all fall on a single curve. Below 230 K the CD function fails to describe the relaxation behaviour because it provides the unphysical result of correlation times depending on the Larmor frequency. This implies certain pecularities at the high frequency wing of @z(o) which will be discussed below. For T > 230 K the correlation times show the typical non-Arrhenius temperature behaviour, and thus are related to the a-process in the supercooled liquid. Below T - 230 K the long time behaviour of F,(t) can be studied by analyzing the decay of the stimulated echo amplitude. By applying a three-

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cl-process ';; \ -4-

liquids and polymers

TinK

ox* : 0.

m-TCP f

5 I-*

H-6-

06

-8-lo-/

I

2.5

I I I , I I I

3.5 4.5 l/T in 103Km' Fig. 4.

I , 1.

5.5

Fig. 5.

Fig. 4. Reorientational correlation times of m-tricresyl phosphate as given by different NMR methods plotted versus the reciprocal temperature. (0) and (0): T,; (0) and (X): T, at o,/2n = 47.3 MHz and 121.5 MHz, respectively. (m) analysis of stimulated echo. Below 230 K the correlation times provided by T, assuming the CD spectral density become o,,-dependent, thus indicating a failure of the CD function to describe the relaxation near T,, i.e. the &-process. Fig. 5. Normalized reorientational correlation function F,(t) as given by the analysis of the stimulated echo. Solid lines: interpolation by a Kohlrausch-Williams-Watts function with p,,, = 0.60.

pulse sequence a single particle correlation function is monitored directly in the time domain, and for a special experimental set-up, again, the correlation function F,(t) is measured [16,24-261. The results from this analysis are shown in fig. 5. As demonstrated (solid lines) F*(t) is well interpolated by assuming a Kohlrausch-Williams-Watts function, explicitly F,(t) = exp[ - (tIrkWW)PKWW], with &ww = 0.60. The KWW decay defines a correlation time T,,~ = rkwwT(l/ &ww)I&ww [27], with r standing for the gamma function. These correlation times are also plotted in fig. 4. Combining relaxation studies and the analysis of the stimulated echo, more than eleven decades of correlation times are covered. The correlation times define the time scale of the main reorientational process in the liquid (a-process) and demonstrate the slowing down of molecular dynamics within a narrow temperature range. Furthermore, it is now obvious that T, below 230 K is dominated by an additional process (&-process, cf. below). In order to check the relation of the reorientational dynamics studied by

E. Rijssler et al. I Dynamical phase transition in supercooled

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l/(qD) Pa-’

2

lo-

26

32

36 1000/T

40

44

40

521°”

in K-’

Fig. 6. Check of the Stokes-Einstein-Debye relations for m-tricresyl phosphate. (0) ~,,~/q (left scale) and (0) 11Dq (right scale) plotted as a function of the reciprocal temperature. Separation of times scales is observed below 270 K. (Diffusion data by the courtesy of F. Fujara.)

NMR and the dynamics governing the shear flow [28] we plotted as proposed [29] the ratio T,,~/Q in fig. 6. Provided the Stokes-Einstein-Debye (SED) relation holds, this ratio should increase linearly with l/T. However, a continuous small decrease is observed, at least in the supercooled regime. On the other hand, comparing the translational diffusion coefficient D (courtesy of F. Fujara [30]) and viscosity by plotting l/Dq, clearly a separation of the time scale of translational and reorientational motion is indicated below 270 K. These observations suggest a change of transport mechanism around 270 K. At higher temperatures all time constants of the different fluctuations are virtually proportional to each other, and the SED relations hold; whereas at lower temperatures some solid state like transport mechanism sets in keeping translational diffusion faster than viscous flow and molecular reorientations. The small separation of reorientational and viscosity time scales might be due to experimental artefacts of comparing different transport coefficients with very high temperature dependences. 2.2. Fast dynamics probed by Raman scattering (m-TCP) In the foregoing section we have focussed on the reorientational part of the cu-relaxation, i.e. on the long time behaviour of a correlation function. However, important predictions of MCT concern the fast dynamics in the liquid, and this has lead neutron scattering experiments to provide first confidence in the validity of the basic ideas of MCT. Only recently, Cummins and coworkers [6,10,11] have reported Raman (RS) and Brillouin scattering experiments which clearly demonstrate the potential of light scattering to check quantitatively the predictions of the theory over a broad range of frequencies (0.3 GHz-500 GHz). Although the mechanism of inelastic and quasielastic light scattering in this frequency range is still not clear (for details cf. the contribution by Sokolov et al. [58]), it was shown in [lO,ll] that correcting the RS intensity spectrum IRS(w) by the Bose factor n(w) + 1, a susceptibility

244

E. Rlissler et al. I Dynamical phase transition in supercooled

O2 00

Tg 1

liquids and polymers

m-TCP

1

240

/

280

320

360

T/K v/GHz

Fig. 7. (a) Susceptibility X”(V) of m-TCP as given by Raman scattering (solid lines) and interpolation from NMR by applying eq. (3); dashed lines: interpolation with /3cn = 0.72; dash and dots: interpolation with continuously decreasing &n while the temperature is lowered; for comparison: T, probes x”(u) at v - 1OOMHz (arrow). (b) Stretching parameter &n of the a-relaxation given by the different methods as a function of temperature: (W) stimulated echo, (- -) T,IT, and (0) matching Raman scattering and NMR results.

spectrum may be obtained, explicitly ,&(w) = Z,,(w)l[n(o) + 11, which has all the features of x”(w) predicted by MCT. In fig. 7a we display xks(v) of m-TCP in a double logarithmical plot [31] (V = w/2rr). At high frequencies the data show a broad maximum corresponding to the microscopic excitation band. In particular, a shoulder is observed at Y- 500 GHz which is an indication of the so-called Boson peak, i.e. some vibrational pecularities typical for disordered systems [32]. For lower frequencies a minimum is observed with its position and depth changing with temperature. Guided by MCT [3], these features are interpreted in the following way. On the low-frequency side of the minimum the wing of the a-process is observed, and on the high-frequency side the &-process sets in which then crosses over to the vibrational part of the spectrum. We have introduced a subscript f in the name of the fast P-process for reasons which will be clarified below. Also in fig. 7a, we show the results from 31P NMR. Assuming that again a CD susceptibility x’&,(~,,,) holds for the a-relaxation, we have applied this function with a time constant (r,,J interpolated from the NMR results in fig. 4; the &process is interpolated as proposed by MCT [3,10,11]. Thus, we write

J&, describes the height and vmin the position of the susceptibility minimum. In this representation the parameters a and & are the characteristic exponents of the two relaxation processes (PC-. = b as defined by MCT). The temperature independent constant A is chosen in such a way to match the NMR results with the one of RS. As is demonstrated in fig. 7a (dashed lines),

E. Riissler et al. I Dynamical phase transition in supercooled

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in the temperature range 312-355 K an interpolation is possible with & = 0.72 and II = 0.35 corresponding to an exponent parameter of A = 0.65. Below 312 K (dash and dots) pcD has to be continuously decreased while the temperature is lowered to match NMR and RS results (we have kept a = 0.35 for all temperatures). The thus obtained temperature dependence of the stretching parameter &, is plotted in fig. 7b. Whereas at high and again at low temperatures - as indicated by the NMR results - /3cD is approximately constant, a significant change is observed near T - 260 K. We have used the similarity of the CD and the KWW susceptibility to relate &, and PKWW [22,27]. As a further test of the predictions of MCT we show in fig. 8 a resealed plot of xlS(~)/xmin versus yIvmi, and an interpolation guided by MCT [3,10,11]. Clearly, a master curve is observed around the minimum; only at higher frequencies some deviations from the MCT interpolation occur. This is due to the not negligible vibrational contributions to x;~(v). For T > T, MCT implies the following predictions on the parameters used for the resealed plot in fig. 8a:

(XLi,)”0~T - Tc

v,$,,aT-

7

In addition, as mentioned, power law (T > T,)

T,.

(4)

the time scale of the a-process

is described by a

with y = 112~ + l/26.

(5)

The first two relations are checked in fig. 8b. Within the accuracy of the experiment all predictions are reproduced and the critical temperature can be

(“mm /GHzJ2=

T/K

Fig. 8. (a) Resealed plot of the susceptibility ,&(v) as provided by Raman scattering; dashed line: interpolation by MCI master curve. (b) The parameters vmin and XL,. used in (a) plotted in a linearized way as a function of temperature to check the predictions of MCT (cf. eq. (4)). Solid lines: fit curves.

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estimated to T, - 255 + 7 K. Furthermore, the analysis of viscosity data yields T, = 266 2 3 K with y = 2.14. Thus, the results presented in this section demonstrate that in the case of m-TCP MCT well describes the fast dynamics at high temperatures, i.e. for T > T,. Furthermore, we find that upon cooling the separation of the time scale for reorientational and translational motion starts close to T, and also the decrease of the stretching parameter &n. The last two findings are not explicitly part of the MCT predictions, however, they support the picture of a change of diffusion mechanism. Deviations from the SED relations have also been observed by other groups [26,29,33-361 and significant changes of the stretching parameter near T, by Dixon et al. [37] and by Fischer et al. [59]. 2.3. A third relaxation process Having determined for m-TCP the parameters corresponding to the aprocess and the &-process, we are not yet able to understand the NMR results which demonstrate a change of spectral density near Tg (cf. figs. 3 and 4). According to eq. (1) the spin-lattice relaxation time probes the dynamics at w = w,, with O,/~IT of the order of 100 MHz. As indicated in fig. 7a (arrow) this time window is well separated from the frequency range at which the fast &-process shows up, and thus it is not likely that the latter dominates the NMR relaxation near Tg. To further substantiate this point we present in fig. 9 the dielectric relaxation data [38] for glycerol together with our RS results [31]. We

IgX”(v1

lg k&i’(v)

I

I

I

-1

-15.

0

0 o*

2

4

6

8

10

12 -’

-20

;

p

Ig(v/Hz)

Fig. 9.

;

0 polymethylacrylaie

0”

.O4

1

6

’

8

’

10

’

12

Ig(v/Hz)

Fig. 10.

Fig. 9. Double logarithmical plot of the susceptibility x”(v) as provided by dielectric relaxation (open circles) and by Raman scattering (small dots) as a function of frequency Y for glycerol; numbers indicate the temperature in K. Solid lines: interpolation of the a-relaxation with the Cole-Davidson function (dielectric relaxation data by the courtesy of F. Kremer). Fig. 10. Double logarithmical plot of the dielectric susceptibility function of the frequency Y, taken from [39]. Numbers indicate

X”(Y) of polymethylacrylate the temperature in K.

as a

E. Riissler et al. I Dynamical phase transition in supercooled

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have adjusted the intensity of RS data to match RS and dielectric data. Whereas at high temperatures the results from RS and dielectric relaxation fit well together (T- 322 K), at lower temperatures a gap appears between the a-relaxation - interpolated by the CD function - and the RS results (cf. 253 K and 193 K). The gap is filled by a shoulder at the high-frequency wing of the a-relaxation. Above all, because no explicit relaxation peak is observed it is not evident whether these deviations from the CD fits stem from an additional relaxation process or whether they are due to some fine structure of the a-relaxation. However, as the temperature is lowered, the shoulder at the high-frequency end of the dielectric data separates more and more from the a-relaxation peak. This suggests that two distinguished time scales are involved. Hence, we might conclude that in the case of glycerol a third process with a very small amplitude comes in between the (Y- and the &-process (T<300K)? This interpretation is supported by recent dielectric studies on e.g. polymethylacrylate (PMA) [39]. There, clearly a second relaxation peak is found at frequencies similar to the ones at which the deviations from the CD fits show up for glycerol, cf. fig. 10. Moreover, the data suggests a convergence of the time scales of the two processes at high temperatures and a change of the relaxation strengths upon cooling. These features of dielectric susceptibilities have been stressed since long by Johari and Goldstein [40], and the name P-process has been given to this comparatively slow secondary relaxation. In this respect, the only difference between glycerol and PMA is the relaxation strength of the additional process as compared to the one of the n-process. Furthermore, it is now obvious that this p-process is not detected by RS; there, a very fast secondary process (&-process) is observed and described within MCT. Similar conclusions have been made by Wu and Nagel [41]. Thus, a hierarchy of secondary processes exists in the supercooled liquid, and we have introduced the name &-process for the slow P-process to distinguish it from the fast &-process [42]. As has been emphasized recently [29,42], in the case of o-terphenyl the first appearance of the &-process upon cooling is very close to T, as determined by NS experiments [43], in other words, LY-and &-process bifurcate at T - T,. It is tempting to assume that this is a general feature of the dynamical phase transition in supercooled liquids, however, it does not yet emerge from MCT. This will be checked for a polymer in the next section. ‘I Concerning the high frequency shoulder of the dielectric x”(w) found in fig. 9 for glycerol an analysis of the data yields the result that the deviations from the CD susceptibility cannot he fully explained by the appearance of a &-process with a small relaxation strength; some pecularities in the susceptibility function of the a-process itself, which are not described by the CD function, have to be assumed. They presumably appear only below T, (cf. also [3,7,38]).

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2.4. Polybutadiene (2H NMR) As suggested by the dielectric data of PMA the relaxation strength of the &-process may be significantly bigger in polymers as compared to simple liquids. For this reason, we have studied deuterated cis-trans polybutadiene (PB, (-C’H,-C’H = C2H-C2H2-),) by 2H NMR [44]. The fast dynamics of PB has been characterized by NS [9,45], and T, - 216 K and T, = 181 K have been determined. In fig. 11 we compare the intermediate scattering function S(q, t)lS(q, 0) from neutron spin-echo experiments with the results from the stimulated echo experiment of 2H NMR. The reorientational correlation function F,(t) is multiplied by a factor of 0.67 for reasons which will be clear soon. NS experiments allow for measuring the absolute correlation loss, and as seen in fig. 11, the NS decay curves start well below 1. Thus, some correlation loss takes place at shorter times (r < 1 ps) beyond the time window of the spin-echo method. Qualitatively, as has been checked by time-of-flight measurements [46], this fast process exhibits all features of the &process predicted by MCT and discussed for m-TCP above (cf. section 2.2). The decay curves above T - 220 K are due to the (~-process, its time constant exhibits the same temperature dependence as the viscosity. However, as pointed out by Richter et al. [45], below T - 220 K NS probes a faster process. On the other hand, 2H

S(q,t)/S(q,O)

F,(t)

0 67

10

10

08

08

06

06

04

04

02

c

Q-process

10-e

1o-6

Ig(
lg(r/s)

rx-process

.*

02

-

00

00 10-m

t

lo-’ I”

set

Fig. 11.

10-z

1

0 002

0 003

0 004 l/T

0 005

0 006

I” K-’

Fig. 12.

Fig. 11. Polybutadiene: intermediate scattering function S(q, t)lS(q, 0) provided by neutron spin-echo measurements at q = 1.88 A-’ [45] (full points) and reorientational correlation function F,(t) from *H NMR (open symbols). Dashed lines: guide for the eyes; solid lines: interpolation with stretched exponential taking PKWW= 0.37. The amplitude of F*(t) has been reduced to make a matching of neutron scattering and NMR data possible. Fig. 12. Correlation times (left scale) for polybutadiene as provided by different relaxation methods as a function of reciprocal temperature. (0) T,, (0) stimulated echo, (e) neutron scattering; solid line: viscosity time scale l/T (right scale); dotted line: Arrhenius law interpolating &-process; dashed line: interpolation of &-process.

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NMR monitors a much slower process at similar temperatures (cf. curves for T = 190 K and T = 180 K). Again, we draw the conclusion that a third process appears below T - 220 K connecting the fast &-process and the slow a-process. In all cases the long-time behaviour of the correlation function (cu-process) can be interpolated by the KWW function with &,w = 0.37 (solid lines in fig. 11). Fig. 12 shows the corresponding time constants. We also have added the correlation times from the analysis of the ‘H spin-lattice relaxation [44]. Above T - 220 K NS and NMR show similar correlation times which follow the same temperature dependence as the viscosity time scale 7T which is given by the temperature dependence of the monomeric friction coefficient l(T), i.e. T,,= l(T)IT [47]. Below 220 K NS probes correlation times governed by an Arrhenius law, and the corresponding process can also be found below Tg. Again, this is a typical behaviour of the &-process [40]. The correlation times provided by the stimulated echo method follow the trend of the friction coefficient, thus, the cr-process is probed. As in the case of o-terphenyl [29,42] the bifurcation of a- and &-process happens close to T - 216 K. The time scale of the fast &-process is of the order of 0.3 ps [45]. We note that the correlation times from T, (T > T,) have been obtained by implying that reorientational correlation times and viscosity (i.e. l(T)/T) have the same temperature dependence. Under these preconditions, the CD spectral density is not able to describe T, of PB. We have chosen the Havriliak-Negami (HN) function [23] to extract correlation times which are proportional to l(T)/T. The HN function has the feature that for W,,T4 1 a frequency dependence is still observed for @z(w). As a consequence, the long time behaviour of the a-relaxation shows some pecularities in the case of PB which are not observed for simple liquids. Similar features have been reported for other polymers [48]. 2.5. NMR relaxation below T, In fig. 12 the results from T, studies on PB only for T > T, are displayed. According to the discussion conducted above, we think that the &-process is responsible for the pecularities observed in the spin-lattice relaxation of m-TCP and PB for T < T,. As pointed out recently [22], the NMR relaxation in all organic glass formers studied so far exhibits such features. This again is demonstrated in fig. 13 where we present the full T, data of o_terphenyl[21,49] and of PB [44] as a function of the reduced temperature scale T,/T. For T,/T > 0.85, T, deviates from the behaviour expected from results of the stimulated echo analysis. The relaxation times are too short. Furthermore, at T - Tg a discontinuous change of the temperature dependence is observed, i.e. in the glass (T < T,) the temperature dependence of T, is smaller as compared

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E. Rlissler et al. / Dynamical phase transition in supercooled

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Fig. 13. Spin-lattice relaxation time T, for o-terphenyl (OTP) and for polybutadiene (PB) plotted versus the reduced reciprocal temperature T,IT, T, = 243 K for OTP and T, = 181 K for PB. Dashed lines: relaxation times as expected if a-process dominates T,. Notice the discontinuous change of the temperature dependence of T, at T,.

to the one above Tg. The only significant difference in the relaxation behaviour between o-terphenyl and PB is that the relaxation of PB in the glass is much faster. We think that because dielectric and NMR relaxation probe similar reorientational processes, any NMR analysis should reflect the basic results reported for the dielectric susceptibility x:(m). For example, we know from dielectric studies [39,40,50] (cf. also fig. 10) that the relaxation strength of the &-process gets smaller as the temperature is lowered (as is the case for the &-process below T,), and it is smaller than the one of the a-process. In terms of reorientational motions this means that only small angular displacements are responsible for the &-process. Furthermore, NMR indicates that below T, the spin-lattice relaxation becomes inhomogeneous [49] (cf. also contribution by Fujara et al. [60]). Since T, is dominated by a faster process the condition 7, > T, is reached close to Tg, and a non-exponential 2H spin-lattice relaxation is observed because the different sites corresponding to the &-process in the topologically disordered liquid are not any more averaged by the now too slow cY-process. Concluding from these findings, we think that for T < T,, T, has to be described by an inhomogeneous distribution of correlation times G(ln TV) and by a relaxation strength which decreases with l/T. Hence, the description of the &-process involves several parameters, and the present NMR experiments are not able to provide all necessary information. Nevertheless, using also results from other techniques we want to propose a simple model as a first approach. In some aspects similar ideas have been presented by Wu and Nagel [41,50] and Bohmer and Loidl [51]. Before getting explicit we have to address the question whether the &process is also responsible for the NMR relaxation below T,. From fig. 13 it is evident that a significant change of relaxation behaviour occurs at T - T,. In

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the case of o-terphenyl and glycerol it has been proposed that the fast &-process takes over the NMR relaxation below T, [49,52]). This conclusion is suggested by the change of the T, dispersion observed for o-terphenyl if one passes Tg, which implies that 0~7 4 1 holds below Tg. However, we want to list some facts which do not support this idea. (i) There are no traces of the &process in the dielectric data surprisingly even at highest frequencies where the time window overlaps with the one of RS and Brillouin scattering (as stressed by Cummins, oral discussion). Instead, in the case of o-terphenyl an explicit &-relaxation maximum is observed for x L(w) below Tg [40,41], and generally a change of the temperature dependence of the relaxation strength is reported near Tg [50,53]. (ii) *H NMR line shape analysis on PB glass reveals a process on the time scale of about 100 l.~s[44] in accordance with the NS data. (iii) ‘H NMR studies on e.g. propanol show a second relaxation maximum implying that in certain cases the &-process is strong and fast enough for yielding a NMR relaxation maximum in the glass [22,54]. (iv) Because the &-process is a very fast process it should be observed at higher temperatures (T- T,) by a so-called preaveraging effect, i.e. the NMR coupling constant K is supposed to be reduced by the fast motion. However, K extracted from the T, IT2 and from the spectra analysis coincide within less than 2% [16,49]. Concluding, we think - although some experimental findings are not completely understood and further investigations of the NMR relaxation in the glass are needed - that many experimental facts favour the idea of a &-process dominating T, above and below T,, and we tentatively assume that the @,-process is not probed by NMR at all. According to a molecular dynamics study by Wahnstrom and Lewis [61], the &-process is mainly attributed to the center-ofmass motion. This might explain its absence in the NMR (and dielectric) relaxation which reflects only reorientational motions. As seen from fig. 12, the correlation times of the &-process do not exhibit any pecularities at Tg; they regularly follow an Arrhenius law [42,50]. Hence, we propose that the discontinuous change of the temperature dependence of T, always observed very close to Tg stems from a freezing of the &-relaxation strength, i.e. the amplitude of the angular displacements does not change any more in the glass due to the absence of structural relaxation below T,. As mentioned this is supported by a discontinuous temperature dependence of the &-relaxation peak [50,53]. Following the ideas sketched we describe the NMR relaxation below T, by a two-step correlation function representing the &- and the a-process. The loss of correlation is shared by both processes, and we introduce a second generalized order parameter f,(T) [55] ( in addition to the non-ergodicity parameter introduced by MCT) as a formal description of the two-step correlation function [56]

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f,(t) describes the relaxation strength of the a-process and 1 -f,(T) the one of the &-process. Both processes are believed to be statistically independent and we can assume that 1 /T, = 1 IT,, + 1/Z’,@ (T < r,) holds. The relaxation contribution of the &-process can be written as l/T,,

= KEFG[l -f,(Y’)][@;(w,)

+ 4@;(2w,)]

.

(7)

Dielectric relaxation studies suggest a symmetric susceptibility function for the &-process [40,41,56]. To arrive at this point we have assumed a Gaussian distribution of Debye processes, and the distribution of correlation times G(ln ra) is believed to originate from a distribution of activation energies g(E) representing the different local environments surrounding the &-relaxation centres in the structurally disordered liquid. Thus, simple activated jumps over an energy barrier E are assumed for the entities undergoing the &-relaxation, and the spectral density may be expressed by

an approach also taken in refs. [41,50,51]. To limit the number of fit parameters for the NMR analysis we have taken the correlation times reported by dielectric relaxation experiments (re,) on o-terphenyl [40] and on PB by NS [45] as the mean correlation times (T,) of the Gaussian distribution of correlation times, i.e. r,__= 7, = r(E,), with E, being the mean activation energy of the distribution g(E). Then, we are left with only two free quantities for describing the NMR relaxation, namely f,(T) and the temperature independent width of the distribution of activation energies. The following temperature dependence of the order parameter f,(T) is proposed to provide an interpolation of the NMR relaxation:

X0-J = const. ‘(T)

=

f,(T,)

LG”A f,U’,) andA

,

-t A(T, - T)‘.5 ,

T
(9)

are properly chosen constants. As can be judged from fig. 14 this phenomenological approach is able to provide an interpolation of the NMR results on PB down to SOK. In particular, the power-law behaviour observed below ;rg naturally emerges from the concept of a distribution of activation energies. In the case of o-terphenyl the fit is less satisfactory: below

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I Dynamical phase transition in supercooled liquids and polymers

10-2

253

a0004

0006

0012

0016

Fig. 14. Measured spin-lattice relaxation times T, (open symbols) and relaxation times corrected for the a-process (full symbols) of o-terphneyl (OTP) and polybutadiene (PB) below T,. Solid lines: interpolation by assuming a distribution of activation energies for the &-process in addition to a relaxation strength described by eq. (9). For PB the solid line also represents the fit by a power law as indicated.

150 K T, shows a further decrease of its temperature dependence. Here, relaxation by the two-level system might take over [49]. We note that a distribution of correlation times G(ln r) originating from a distribution of activation energies g(E) may be written G(ln r) = RTg(RT In r) provided a simple Arrhenius law, i.e. In r m E/T, holds [57]. As a consequence the width of G(ln r) and thus the width of the apparent dielectric susceptibility should increase linearly with l/T. Indeed, this behaviour has been reported for the &-process by Wu and Nagel [41,50] and also by Bohmer and Loidl [51].

3. Conclusions Combining results from different relaxation methods we have demonstrated that several phenomena - although not yet emerging from MCT - support the idea of a change of transport mechanism at a critical temperature T, well above T,. (i) m-TCP is the second experimental example reported for which the time scales of reorientational and translational motions separate below T - T, (in addition to OTP, cf. the contribution by Fujara et al. [60]). (ii) The stretching parameter of the a-relaxation changes upon passing through T, (simple liquids). (iii) We have demonstrated that a third process (&-process) appears near T, upon cooling. This process has been called p-process since the early days of dielectric and mechanical relaxation studies, however, has to be distinguished from what has been called P-process by MCT analyses. The latter process (&-process) is a much faster process and is probed by NS and RS experiments as has been shown here for m-TCP. Thus, above T, the dynamics in the supercooled liquid are described by a two-step correlation function with all the features predicted by MCT, whereas below T, a three-step function

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F,(t) 10

pf-process

08 06

04 00

1 -12

1 ‘-10

1 -8

-6

1

-4

1

-2

0

Ig(t/s) I

Fig. 15. Shape of the time correlation function for an observable A above and below the critical temperature T,, schematically. The two order parameters f,(T) and f,(T) describe the respective plateaus of the correlation function.

holds (cf. fig. 15). As a consequence, a second order parameter appears, and the dynamics of the &-process have been described in terms of a thermally activated process with a distribution of local environments, i.e. the &-process is believed to be a typical solid state relaxation process. The results presented here imply certain points. It has been demonstrated in many studies and here again that, for T > T,, the predictions of MCT are well reproduced by experiments performed on liquids containing no directed intermolecular bonds, e.g. Ca,,,K,,,(NO,),~,, salol, o-terphenyl, m-TCP and also the polymer PB. As has been pointed out by Sokolov et al. [58] the predictions work out the better, the lower the vibrational contribution to x”(w) is. In the extended version of MCT hopping processes are included to allow a description of the dynamics also below T,. Concluding from our analysis, we think, this extended theory has to scope with two experimental features below T,: (i) the &-process and (ii) the not diverging time scale of the a-process. Thus, the features of the hopping processes have to be clarified. Does the latter process exhibit properties of a structural relaxation process (cY-process) or of a further secondary process (&-process)? Furthermore, since MCT is basically a liquid state theory, the fast &-process discussed by the theory is a process which directly follows the microscopic time scale. Thus, in our opinion, it is prohibitive to test the scaling predictions of MCT on the slow &-process in order to determine a critical temperature.

Acknowledgements

E.R. appreciates helpful discussions by Prof. D. Quitmann and Prof. H.-M. Vieth and providing the possibility to join their research group; thanks also to Prof. D. Richter and B. Frick for supplying deuterated polybutadiene, and to

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D. Plazek, F. Fujara and F. Kremer for providing unpublished data on viscosity, diffusion and on dielectric relaxation, respectively. The authors are grateful to A. Kisliuk, A. Kudlik, and S. Loheider for discussions and computeral help. Financial support of the Deutsche Forschungsgemeinschaft is acknowledged by E.R., and A.P.S. is grateful to the Alexander von Humboldt foundation for a fellowship.

References E. Leutheusser, Phys. Rev. A 29 (1984) 2765. U. Bengtzelius, W. Giitze and A. Sjolander, J. Phys. C 17 (1984) 5915. W. Gotze and L. Sjogren, Rep. Prog. Phys. 55 (1992) 241. W. G&e and L. SjBgren, J. Phys. Cond. Matter 1 (1989) 4203. M. Fuchs, W. Gotze, S. Hildebrand and A. Latz, J. Phys. Cond. Matter 4 (1992) 7709. H.Z. Cummins, W.M. Du, M. Fuchs, W. Giitze, S. Hildebrand, A. Latz, G. Li and N.J. Tao, Phys. Rev. A 47 (1993) 4223. I71 F. Fujara and W. Petry, Europhys. Lett. 4 (1987) 921. F. Mezei, W. Knaak and B. Farago, Phys. Rev. Lett. 58 (1987) 571. ;i; D. Richter, B. Frick and B. Farago, Phys. Rev. Lett. 61 (1988) 2465. PO1 G. Li, W.M. Du, X.K. Chen, H.Z. Cummins and N.J. Tao, Phys. Rev. A 45 (1992) 3867. [Ill G. Li, W.M. Du, A. Sakai and H.Z. Cummins, Phys. Rev. A 46 (1992) 3343. WI D.J. Plazek and J.H. Magill, J. Chem. Phys. 45 (1966) 3038. 1131 W.T. Laughlin and D.R. Uhlmann, J. Phys. Chem. 76 (1972) 2317. [I41 E. Rossler, J. Chem. Phys. 92 (1990) 3725. P51 E. Rossler, J. Non-Cryst. Solids 131-133 (1991) 242. [161 E. Rossler and P. Eiermann, J. Chem. Phys. (1993), accepted. u71 A. Abragam, The Principles of Nuclear Magnetic Resonance (Oxford Univ. Press, Glasgow, 1961). I181 J.P. Kintzinger and M.D. Zeidler, Ber. Bunsenges. Phys. Chem. 77 (1969) 98. 1191 M. Wolfe and J. Jonas, J. Chem. Phys. 71 (1979) 3252. PO1 E. Rossler and H. Sillescu, Chem. Phys. Lett. 112 (1984) 94. WI Th. Dries, F. Fujara, M. Kiebel, E. Rossler and H. Sillescu, J. Chem. Phys. 2139 (1988) 88. P21 E. Rossler, K. Biirner, J. Tauchert, M. Taupitz and M. Poschl, Ber. Bunsenges. Phys. Chem. 95 (1991) 1078. v31 C.J.F. Bottcher and P. Bordewijk, Theory of Electric Polarization, vol. II (Elsevier, Amsterdam, 1978). 1241 F. Fujara, S. Wefing and H.W. Spiess, J. Chem. Phys. 84 (1986) 4579. PI E. Rossler, Chem. Phys. Len. 128 (1986) 330. P61 F. Fujara, B. Geil, H. Sillescu and G. Fleischer, Z. Phys. B 88 (1992) 195. [271 C.P. Lindsey and G.D. Patterson, J. Chem. Phys. 73 (1980) 3348. WI A.J. Barlow and A. Erginsav, Proc. Roy. Sot. London A 237 (1972) 175; D.J. Plazek, unpublished data, private communication; E. Riande, H. Markovitz, D.J. Plazek and N. Raghupathi, J. Polym. Sci. Symp. 50 (1975) 405. 1291 E. Rossler, Phys. Rev. Lett. 65 (1990) 1595. [301 F. Fujara, unpublished diffusion data on m-TCP, private communication. 1311 A.P. Sokolov, E. Rossler, A. Kisliuk and D. Quitmann, Phys. Rev. B (1993), submitted. i321 A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Quitmann, Phys. Rev. Lett. 69 (1992) 1540. [331 S.S.N. Murphy, J. Molec. Liquids 44 (1990) 211.

256

E. Rijssler

et al. I Dynamical

phase transition

in supercooled

liquids and polymers

[34] D. Ehlich, H. Sillescu, Macromolecules 23 (1990) 1600. [35] G. Fytas, A. Rizos, G. Floudas and T.P. Lodge, J. Chem. Phys. 93 (1990) 5096. [36] W. Steffen, A. Patkowski, G. Meier and E.W. Fischer, J. Chem. Phys. 96 (1992) 4171. [37] P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams and J.P. Carini, Phys. Rev. Lett. 65 (1990) 1108. [38] A. Hofmann, F. Kremer and E.W. Fischer, Dielectric data on glycerol, private communication; N. Menon, K.P. O’Brien, P.K. Dixon, L. Wu, S.R. Nagel, B.D. Williams and J.P. Carini, J. Non-Cryst. Solids 141 (1992) 61. [39] F. Kramer, A. Hofmann, E.W. Fischer, Polymer Preprints 33 (1992) 96. [40] G.P. Johari and M. Goldstein, J. Chem. Phys. 53 (1970) 2372. [41] L. Wu, S.R. Nagel, Phys. Rev. B 46 (1992) 11198. [42] E. Rossler, Phys. Rev. Lett. 69 (1992) 1620. [43] W. Petry, E. Bartsch, F. Fujara, M. Kiebel, H. Sillescu and B. Farago, Z. Phys. B 83 (1991 175. [44] E. Rossler and U. Warschewske, data on Polybutadiene, to be published. [45] D. Richter, R. Zorn, B. Farago, B. Frick and L.J. Fetters, Phys. Rev. Lett. 68 (1992) 71 [46] B. Frick, D. Richter, W. Petry and U. Buchenau, Z. Phys. B 70 (1987) 73. [47] G.C. Berry, T.G. Fox, Adv. Polym. Sci. 5 (1968) 261. (481 A. Schonhals, F. Kremer and E. Schlosser, Phys. Rev. Lett. 67 (1991) 999. [49] W. Schnauss, F. Fujara, H. Sillescu, J. Chem. Phys. 97 (1992) 1378. [50] L. Wu, Phys. Rev. B 43 (1991) 9906. [51] R. Bohmer and A. Loidl, in: Basic Features of the Glassy State, J. Colmenero and A. Alegria, eds. (World Scientific, Singapore, 1990) p. 215. [52] F. Fujara, W. Petry, R.M. Diehl, W. Schnauss and H. Sillescu, Europhys. Lett. 14 (1991) 563. [53] S.S.N. Murphy, J. Chem. Sot. Faraday Trans. II 85 (1989) 581. [54] M. Poschl, G. Althoff, S. Killie, E. Wenning and H.G. Hertz, Ber. Bunsenges. Phys. Chem. 95 (1991) 1084. [55] G. Lipari and A. Szabo, J. Am. Chem. Sot. 104 (1982) 4546. [56] E. Rossler and W. Schnauss, Chem. Phys. Lett. 170 (1990) 315. [57] E. Rossler, M. Taupitz, K. Borner, M. Schulz and H.-M. Vieth, J. Chem. Phys. 92 (1990) 5847. [58] A.P. Sokolov, A. Kisliuk, M. Soltwisch and D. Quitmann, Physica A 201 (1993) 295, these proceedings. [59] E.W. Fischer, Physica A 201 (1993) 183, these proceedings. [60] E. Bartsch, F. Fujara, B. Geil, M. Kiebel, W. Petry, W. Schnauss, H. Sillescu and J. Wuttke, Physica A 201 (1993) 223, these proceedings. [61] G. Wahnstrom and L.J. Lewis, Physica A 201 (1993) 150, these proceedings.