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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 22 8 JUNE 2000
Light scattering study of the liquid–glass transition of meta-toluidineA. Aouadi, C. Dreyfus, M. Massot, and R. M. PickL.M.D.H., B.P. 86, Universite P. et Marie Curie, 4 Place Jussieu, F-75005 Paris, France
T. Berger and W. SteffenMax-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany
A. PatkowskiMax-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany, and Institute ofPhysics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
C. Alba-SimionescoL.C.P., Universite´ Paris-Sud, Orsay 91400 France
~Received 30 August 1999; accepted 3 March 2000!
An experimental study of the glass transition of meta-toluidine combining several light scatteringtechniques was performed. The structural relaxation time is measured in depolarized geometry fromthe glass transition temperature up to well above the melting point and found to vary over 13 timedecades. An analysis by means of the idealized Mode Coupling Theory shows that, as found in otheraromatic liquids, experimental results obtained in depolarized light scattering can be described bythis theory aboveTc in a two-decade frequency range. The polarized Brillouin doublet, measured inthe backscattering geometry between 176 K and 300 K, is also analyzed. None of the sets ofparameters we obtained in fitting those spectra could fulfil all the requirements of this ModeCoupling Theory. ©2000 American Institute of Physics.@S0021-9606~00!52220-X#
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I. INTRODUCTION
The existence of supercooled liquids which remain luid below their melting temperature during macroscotimes without crystallization, and which, under slow coolirates~typically 10 deg/min! can be transformed into glasse~the liquid–glass transition! is the oldest unsolved problemof condensed phase physics. In this regime, the mostmatic effect which characterizes the approach of the tration is the change in the shear viscosity:hs varies, typically,over more than 15 orders of magnitude from a normal liqphasehs;1 cP to a value ofhs of the order of 1013P at thecalorimetric glass transition temperature,Tg . This thermaldependence has thus been the subject of many theorestudies.1,2 It does not display a unique behavior and,though this aspect has been known for a very long timewas only in the mid-eighties that Angell3 rationalized all thedata with an Arrhenius plot of loghs(T) vs Tg /T, thus allow-ing for a classification of the various types of glass-formiliquids. This Angell plot shows in particular that many liquids have a very similar behavior in which loghs(T) containstwo approximately linear parts separated by a regionrather strong curvature in the vicinity of some crossover teperatureTcross; such liquids are the so-called fragile glasforming liquids. Many nonalcoholic organic liquids belonto this class.
Although the shear viscosity may appear to be a stquantity, a dimensional argument allows us to write it asproduct of a shear modulus and a relaxation time. The Mwell theory of viscoelasticity makes clear the rationale bhind this dimensional analysis by proposing that the relatiship between the local shear strain and the corresponshear rate would be nonlocal in time in a supercooled liqu
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it involves a memory function the ingredients of which aprecisely the shear modulus and the~relaxation! time justmentioned. This theory thus bridges an experiment pformed in a steady state regime with a dynamical approand proposes that it is essentially the corresponding reation time, attributed to the ‘‘a relaxation process,’’ whichvaries under cooling. Thisa relaxation process is, indeeddetected by many spectroscopic techniques: such techniare always mostly sensitive to the dynamics of some spevariable~s!, and it has been recognized for more thanyears~see, e.g., Ref. 4 for a review! that, although thisarelaxation process is visible in any experiment, the corsponding averaged relaxation time can differ from one tenique to another, even at temperatures aboveTcross.
When lowering the temperature, fitting these spectscopic data frequently becomes impossible without introding, at least, another relaxation channel. In the classanalysis where one identifies each channel with a relaxatime, the latter appears to be much faster than thea processextrapolated at that temperature and is governed byArrhenius law unrelated to thea process.
Such a channel has been named by Johari and Golds5
a b relaxation. Yet, the analyses which reveal thisb processdo not rely on any theoretical basis, while the experimenevidence tends to show a dispersion of the correspondtimes with the experimental technique larger than for theaprocess6 and also a dispersion in the corresponding Arrheius energies,7 two facts which may cast some doubts on tgeneral validity of their existence.
During the mid-eighties the first microscopic theori
0 © 2000 American Institute of Physics
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9861J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
appeared proposing an explanation of the liquid–gltransition,8–10which incorporates the dynamical aspect of tliquid. For the past ten years, these theories, called MCoupling Theories~in short, MCT! have received a lot oattention. In their simplest version,4 they are written for afluid made of shapeless particles, for which the only dynacal variables are the positions of the centers of mass.earliest version, the idealized MCT, predicts at a temperaTc a dynamical transition between two regimes: in onethem (T.Tc) the time correlation function of any densitfunction will eventually decay to zero while in the other o(T,Tc), all these functions will remain finite at infinitetime, this limit being called their nonergodicity parametThis theory makes specific predictions for the dynamicsthese correlation functions in the vicinity ofTc . In particu-lar, it predicts, forT.Tc , the existence of ana relaxationprocess with a wave vector dependent relaxation time, wthe numerical analysis of the corresponding solutions shthat this dynamics can be represented by a stretched enential with a stretching coefficient independent of tempeture but wave vector dependent. A second prediction oftheory is that, close toTc , for intermediate times~i.e., fortimes much longer than those characteristic of the micscopic motions, but much shorter than the correspondinarelaxation times!, a wave vector independent dynamishould take place. This intermediate dynamics has bcalled11 the ‘‘b fast process,’’ the letterb recalling the factthat this process has a characteristic time much shorterthea-relaxation process, but this ‘‘b-fast process’’ has nothing in common with theb Johari–Goldstein process5 dis-cussed above.
A first and standard criticism of this theory is that, whits predictions are compared with the experimental resuapproximate agreements can be obtained only ifTc has alarger value than the glass transition temperatureTg , thisvalue turns out to be close to the temperatureTcrossat whichloghs(T) has its maximum curvature in the Angell plot. Thmay be attributed to the neglect of the role of the particurrents as additional relevant variables. As soon as tcontribution to the time correlation functions are introduceeven in a schematic form, in the MCT equations,Tc becomesa crossover temperature between a high temperature rewhere the predictions of the ideal MCT remain appromately valid and a lower temperature regime wherestructural arrest is not complete and thea relaxation timestill goes on diverging.
A second criticism is that no simple liquid ever becomsupercooled, in the sense described at the beginning ofsection. The simplest liquids which undergo a liquid–glatransition and could correspond to this fragile glass-formscheme are formed of more or less complex molecuMoreover, most of the experimental techniques either msure, or are largely sensitive to, the orientational motionsthe molecules. Extensions of the theory adding specificthose other degrees of freedom appeared as necessarprovements. They have recently started to be develop12
and they basically confirm12b the predictions of the simpleratomic like, MCT. Molecular Dynamics computations bason asymmetric dumbbells have also been rece
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performed12c and the time correlation functions of variablerelated to different values of the indexl of the sphericalharmonics characterizing their orientation have been coputed. These calculations largely confirm the MCT predtions extended to rotational dynamics, though there aretain reservations: for instance, the time temperatsuperposition principle, which supposes that the stretchcoefficient of thea relaxation process does not dependtemperature, appears not to hold properly forl 51.
Yet, the corresponding fluids~such as CO or NO, forinstance, or larger asymmetric dumbbell molecules, suchOCS! do not undergo a liquid–glass transition, so that thonumerical studies cannot represent an experimental situaConversely, the real fragile glass-forming liquids always ehibit a more complex dynamics, originating from internlow frequency modes, and/or from weak intermolecubonding, ingredients not yet incorporated in those MD coputations.
Performing precise measurements of the overall dynaics of a given glass-forming liquid, over a very large frquency range, using polarized and depolarized dynamic lscattering which gives access to different observables,comparing those results with the MCT predictions may hto answer the following question: when dealing with a reglass-forming liquid, do those additional dynamical featutotally mask ~or modify?! the signatures predicted by thMCT?13 If this is not the case, to what extent does the msured dynamics agree with the predictions of MCT?
The present paper aims at discussing this problem incase of a very simple fragile glass-forming liquid, mettoluidine (CH3–C6H4–NH2!. This has been achievethrough a light scattering study of this system with the hof different instruments. Our paper is organized in the flowing way. Section II will summarize the results of thMCT we wish to test here. Section III is devoted to thdescription of the experiments and of the results. In Sec.the depolarized light scattering results are analyzed inspirit of the MCT, making use of a phenomenological dscription of thea relaxation. Section V is devoted to thanalysis of the longitudinal Brillouin line along the sampattern. A brief conclusion ends the paper.
II. THEORETICAL BACKGROUND
Comparison between experimental results and the idMCT predictions have been frequently performed on a vaety of substances. Extended reviews can be found in Reand references therein. Without entering into a detailed psentation of the theory, let us briefly recall, in this sectiothe theoretical results of the MCT we shall use in this papWe shall state them for the simplest version of the theothe idealized MCT, in the canonical case where the orelevant dynamical variables are the different normalizdensity–density correlation functions of a monomolecufluid:
Fq~ t !5S~q,t !
S~q,0!, ~2.1a!
where
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9862 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
S~q,t !5K (i , j
exp~ iq•~r i~ t !2r j~0!!!L . ~2.1b!
The fundamental set of equations of the theory reads:
Fq~ t !1Vq2Fq~ t !1E
0
t
(q1
mq,q1~ t2t8!Fq~ t8!dt850,
~2.2!
whereVq is called the microscopic frequency of the mowith wave vector q and where the leading order termq,q1
(t) is given by
mq,q1~ t !5Vq,q1
Fq1~ t !Fq2q1
~ t !. ~2.3!
AboveTc , and for (T2Tc)/Tc small compared to unity, oneexpects thatFq(t) will decay, for t→`, as:
Fq~ t !→ f q0 expS 2S tYtq
a D bqD , ~2.4!
where
tqa5tq
a0~T2Tc!2@~1/2a!1~1/2b!# ~2.5a!
and where the nonergodicity parameterf q0, the relaxation
time factortqa0 and the stretching coefficientbq depend onq
but not on temperature. For the same temperature regionfor Vq
21!t!tqa , the time dependence ofFq(t) is predicted
not to depend onq; this is theb fast dynamics, which isgoverned by a single parameter,l; the latter fixes the valueof the two positive numbers,a andb, which appeared in Eq~2.5a!. In particular, the asymptotic behaviors ofFq(t) arepredicted to be given by:
Fq~ t !5 f q01hqAT2Tc Al S tYtb D 2a
t!tb,
~2.6a!
Fq~ t !5 f q02hqAT2Tc Bl S tYtb D b
t@tb, ~2.6b!
with:
tb5tb0 ~T2Tc!21/2a. ~2.7a!
The experiments we are going to analyze were pformed in the frequency domain. This makes the compariwith the formulas given above quite difficult. The possibiliof obtaining a conclusive information from this comparisin the frequency space is based on two hypotheses:
• the time domains where the microscopic motionst'Vq
21), the b fast relaxation regime (Vq21!t!tq
a)and thea regime (tq
a<t) take place are well separate
• the relative amplitudes of those three motions are ofsame order of magnitude.
If those two conditions are fulfilled, similarly to the timdomain case, there exist three well separated frequencygions in each of which one of the Laplace transforms~inshort LT! of the three preceding regimes dominates. Moprecisely, ifxq8(v) is the real part of the LT of the susceptibility function related toFq(t):
xq8~v!5v Fq9~v! ~2.8a!
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9863J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
imply that, for q21 of the order of an optical wavelengthmq(t) must be proportional toq2 while its dynamics shouldbe independent ofq, at the lowest order inq. Finally, for thisvalue ofq, Vq should be replaced byvq , which is the fre-quency of a sound wave~i.e., of a longitudinal propagatingmode! and is expressed as:vq5c0q, c0 being the~relaxed!sound velocity. The LT of Eq.~2.2! then reads:
Fq~v!5v2mq~v!
v22vq22vmq~v!
. ~2.9a!
With the definition Eq.~2.8b! of the Laplace Transform, thespectrum measured in a light scattering experiment is pportional toFq9(v); this last quantity is easily found to be
Fq9~v!5vq
2mq9~v!
~v22vq22vmq8~v!!21~vmq9~v!!2
[vq
2
vIm
1
v22vq22vmq~v!
. ~2.9b!
The line shape of the spectrum corresponding to Eq.~2.9b!,i.e., the Brillouin line shape, is then entirely determinedthe frequency and thermal behavior ofvmq(v) wheremq(v) is the LT of mq(t). At long times, the latter can bcharacterized by a relaxation timetL. Consequently:
• At high temperature, wherevqtL!1, vmq(v) is a
small imaginary quantity which simply contributesome linewidth to the Brillouin peaks located at6vq56c0q.On the contrary, at low temperature wherevqt
L@1, asthe v→` limit of vmq(v) must be written asD2q2,the spectrum is made up of two lines located at:
6vq56Ac021D2q[6c`q. ~2.10!
The idealized MCT adds two aspects to those classresults:
• First,16 (D/c`)2 is the nonergodicity parameterf q re-lated to the longitudinal [email protected]., the equivalent, aany temperature, of thef q
0 term of Eq.~2.6a!#. Whence:
fq512S c0
c`D 2
. ~2.11!
Meanwhile, the idealized MCT predicts that:
f q5 f q05cte T.Tc , ~2.12a!
f q5 f q01kAT2Tc
TcT,Tc . ~2.12b!
Analysis of the Brillouin line shape, i.e., the determinationc0 , c` , andD is thus, in principle, a fourth way of measuing Tc and is, at the same time, a method for having soinformation on the validity of one specific MCT predictiobelow Tc . Unfortunately this analysis requires the introdution of a numerically tractable expression formq(v) in orderto obtain D and c` . This cannot be done without makinassumptions on the analytic form of this memory functioSome of them will be presented in Sec. V.
o-
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-
.
III. EXPERIMENTAL ASPECTS
A. Former experimental studies
Meta-toluidine (CH3–C6H4–NH2) belongs to the familyof the meta bisubstituted benzenes, X–C6H4–Y, where X~resp. Y! may be CH3,NH2,OH..., which remain very easilysupercooled17 down to their thermodynamic glass transitiotemperature,Tg ~except for the case X5Y5CH3). In thepresent case,Tg5187 K ~measured in DSC,17 with a tem-perature decrease of 10 deg/min! while the melting tempera-ture is Tm5243.5 K. Under normal conditions, crystallization never happens in the supercooled phase: one neebring the glass at least 50 deg belowTg to see microcrystalsgrowing in the supercooled phase under re-heating.18 Thisimplies that in our experiments, we will not need to worabout the possible existence of microcrystals in the supcooled phase.
The thermal variation of the index of refraction andthe density are given by19:
n~T!51.692~5!24.5 1024T~°K!, ~3.1!
[email protected]~0!28.1 1024T~°K!#g/cm3. ~3.2!
Among other quantities, the thermal variation of the shviscosity has been measured by different authors.18,20,21Di-electric measurements between 198 K and 233 K have bperformed in the frequency range 1022 Hz–107 Hz byLegrand17 and repeated by us, and elastic neutron scatteexperiments have given the static structure factor,S(Q), as afunction of temperature and pressure.22 Raman spectra wererecorded by Alba-Simionesco and Krauzman15a who care-fully deduced from their measurements the light scatterdepolarisation ratio,23 as a function of frequency, betwee150 GHz and 5 THz, for temperatures ranging from 175~i.e., belowTg) to 290 K. They found that this ratio did nodepend on the frequency and was equal to 0.7360.01.
B. Experiments
Meta-toluidine, purchased at Merck, was distilled twibefore the light scattering experiments were performed. Tlatter were made with the help of a coherent Ar1 laser emit-ting at 514.5 nm, with vertical polarization. Backscatteriexperiments were performed in Paris, in the VV geome~polarized scattering! between 1 GHz and 30 GHz, and in thVH ~depolarized! geometry between 1 GHz and 600 GHz, oan 8 pass, Sandercock-type,24 Tandem Fabry–Pe´rot interfer-ometer already described.25 The sample was placed in a cryostat which ensured a temperature regulation better thanK. The polarization of the scattered light was analyzthrough a combination of a quartz half-wave plate followby a Glan prism which allowed to keep a constant directof the electric field of the scattered light inside the spectroeter. This procedure eliminated any possible difference intransmission of the interferometer for the two different plarizations. At each temperature, a large band depolarspectrum was obtained by joining together scans with a laoverlap and made with different spectral ranges of the Tdem interferometer. Five different spectral ranges were nessary for the Fabry–Pe´rot experiments at every temperatu
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9864 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
which were chosen to be within 1 K of those used in theRaman experiments.15a Those Raman spectra were finaljoined with the Tandem ones, using the same overlap tenique as above~see Fig. 1!. The final composite spectrcover a three-decades frequency domain~1 GHz–1 THz! andthe series of spectra recorded at different temperatureshown on Fig. 2 in the form of susceptibility spectra:
x8~v!5I VH~v! ~11n~v!!21, ~3.3!
wheren(v) is the Bose–Einstein factor.As it can be seen in that figure, the maximum of t
susceptibility spectra, which represents the most clear sigture of thea relaxation process, is only visible down to 26K. In order to follow the evolution of this relaxation proceat lower temperatures, two more series of experiments wperformed in Mainz. One of them consisted of 90 degconfocal, depolarized Fabry–Pe´rot measurements, performewith a free spectral range of 0.75 GHz. Because no attewas made to match those spectra with those obtainedthe Tandem Fabry–Pe´rot instrument, there was no constraion the choice of the temperatures; those actually usednoted with a subscriptc in Table I.
At still lower temperatures, a third technique had toused in order to record thisa relaxation process. This consisted in a series of experiments made with a PCS instrum
FIG. 1. Composite susceptibility curve for depolarized light scattering
FIG. 2. Thermal variation of the susceptibility spectra:~1! T5300 K, ~2!T5290 K, ~3! T5280 K, ~4! T5266 K, ~5! T5256 K, ~6! T5239 K, ~7!T5233 K, ~8! T5223 K, ~9! T5213 K, ~10! T5193 K.
h-
is
a-
ree
ptith
re
nt
in Mainz. They were also performed in a 90° scattering gometry, with a vertically polarized incident beam and a dpolarized scattered beam. The spectra were recorded,then analyzed using correlation times ranging from 1024 s to102 s, at temperatures lower than those used in the TanFabry–Pe´rot experiments. Those temperatures are notedTp
in Table I.The polarized spectra were recorded in another serie
experiments, on the Paris instrument~Fig. 3, Table II!. Inthat case, we used a single spacing for the interferomecorresponding either to a 18.75 GHz, or to a 25 GHz fspectral range, depending on temperature, in order to aan overlap between the instrument ghosts and the mostevant part of the Brillouin spectrum. A series of corresponing depolarized spectra was recorded on the same instrumat the same time, simply turning the half-wave plate placin the scattered beam. Analysis of the Brillouin line shaalso requires the knowledge of the relaxed sound velocThis quantity was measured in the Paris Laboratory
TABLE I. Thermal variations of the relaxation timetK and of the stretchingparameterbK .
T ~K! bK tK ~ns!
300 0.66 0.01290 0.63 0.02280 0.61 0.04266 0.61 0.06245c - 0.36
240.5c - 0.5236c - 1.0232c - 1.6227c - 2.8198p 0.45 4.0E6195p 0.42 0.9E7193p 0.46 8.0E7191p 0.43 2.7E8189p 0.48 1.36E9187p 0.38 8.14E9185p 0.37 1.07E11183p 0.31 4.05E11
FIG. 3. VV spectra at different temperature and fit~logarithmic scale!. Inset:Polarized and depolarized spectra of metatoluidine at 290 K.
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9865J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
Bonello,26 by an ultrasonic technique, between 258 K a296 K. His measurements can be accurately representethe linear formula:
c0~T!5~2723.523.85T! m/s, ~3.4!
whereT is measured in Kelvin.
IV. ANALYSIS OF THE DEPOLARIZED SPECTRA
A. The a relaxation process
There is a general agreement on the fact that thefrequency part of the depolarized spectra of liquids formof anisotropic molecules are mostly sensitive to the orientional motion of these molecules. We have exemplified tstatement in Fig. 4 which compares the susceptibility speof meta-toluidine and CCl4, a spherical top molecule, at thsame temperature, 293 K. As the rotation of sphericalmolecules does not contribute to the scattered intensity,corresponding intensity and susceptibility spectrum of C4can only originate from pure center-of-mass interactionduced effects. This pure center-of-mass integrated intenscales with (ra2)2, wherer is the number density anda themolecular polarizability. Figure 4 shows that the metoluidine spectrum, in which the molecular reorientatio
TABLE II. Characteristics of the longitudinal phonons and the parameof their fit by thea only model~see text!. The three last columns represerespectively, in this model, the nonergodicity parameter,f q , the linewidthdue to thea relaxation term@Eq. ~5.5!# and the reliability factor of the fit.
T ~K!nB
~GHz!nq
~GHz!Dq
~GHz!n`
~GHz!tCD
L
~ns! f q g1 xv2
290 10.09 9.74 7.27 12.16 0,015 0.357 3,8 2.1280 10.38 10.01 7.49 12.50 0,021 0.358 3,4 2.1266 11.53 10.38 7.72 12.93 0,029 0.356 2,5 1.3256 11.90 10.64 8.02 13.32 0,046 0.362 2,05 1.246 12.68 10.91 8.31 13.71 0,081 0.367 1,5 1.6236 13.58 11.18 8.87 14.27 0,17 0.386 1,1 1.6226 14.24 11.45 9.26 14.72 0,42 0.395 0,69 1.6216 14.78 11.72 9.77 15.26 0,88 0.409 0,50 1.9206 15.66 11.99 10.48 15.93 1,80 0.433 0,37 3.196 16.16 12.26 10.92 16.42 3,3 0.442 0,28 2.4186 16.54 12.54 11.12 16.76 8,5 0.440 0,18 2.0176 16.62 - - 0,05 -
FIG. 4. Relative susceptibilities of pure metatoluidine and pure CCl4 mea-sured at room temperature.
by
wd-
sra
phel-ity
-s
contribute to the light scattering mechanism, has a mlarger intensity, particularly at low frequency, than the CC4
one. As the ratio @(ra2)2#met/@(ra2)2#CCl4for meta-
toluidine and CCl4, respectively, is approximately 2.5,means that the pure center-of-mass integrated intensitytribution is much smaller than the contributions originatifrom orientational fluctuations~including induced ones! aswe just stated. This rotational dynamics is mostly visiblevery low frequency, as a maximum, clearly identifiedsome of the spectra of Fig. 2. This is thea relaxation part ofthe susceptibility spectra that we analyze in Sec. IV A, aone of its characteristics is its relaxation time that we shcall tR. This analysis was performed through the three seof experiments described in Sec. III:
• We first analyzed the low frequency part of the TandeFabry–Pe´rot experiments for the four highest tempertures (266 K<T<300 K). They are the only onewhere the corresponding susceptibility spectra@see Eq.~3.3!# exhibit a visible maximum at a frequencyvmax,and this maximum is one of the most precise signatuof this relaxation process. Those low frequency pawere fitted to a Cole Davidson function:
x8~v!'ImF12S 1
11ivtCDR D bCDG . ~4.1!
This expression provides a good fit to the LT ofstretched exponential up to frequencies of the order10 times the maximum of the r.h.s. of Eq.~4.1!, pro-vided that the corresponding relaxation timetCD
R andstretching coefficientsbCD are related to the relaxatiotime tK
R and stretching coefficientbK of the correspond-ing stretched exponential through linear relations givby Lindsay and Patterson.27
Excellent fits to those low frequency parts ofx8(v)were obtained for these four temperatures~see Fig. 8!with nearly the same value forbCD, the agreement being very good up to frequencies of the order of 10vmax.
• As a second step, the correlation functionsg(2)(t) re-corded in the PCS experiments were fitted with tstandard formula:
g~2!~t!511ASG expS2S tYtKRDbKD1IbgD 2
, ~4.2!
where I bg is a background intensity supposed notdepend on time,A andG being scaling factors. CoefficientstK
R(T) and bK(T) were deduced from those experiments and then transformed27 into the correspond-ing tCD
R (T) andbCD(T).
• Finally we analyzed the confocal FPI data. The frspectral range of this instrument turned out to benarrow with respect to the relaxation timestCD
R to allowfor an independent measurement of both parametCD
R and bCD. Because of the small, and presumabsmooth variation of the latter, we decided to interpolaits value at each temperature with the help of the resobtained in the first two experiments. Using again E~4.1!, tCD
R was then determined for each spectrumcorded with the confocal Fabry–Pe´rot instrument, tak-
s
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9866 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
ing into account both the existence of a flat backgrouand of the overlap between the different orders ofspectrum inherent to this instrument.
The values resulting from these three series of expments are given asbK andtK
R in Table I. The different valuesof tK
R are also represented in Fig. 5 together with a VogFulcher fit to these relaxation times:t5t0 exp(TA /(T2T0))5t0 exp(DT0 /(T2T0)), whereD is the strength index measuring the fragility. This fit correctly describes the resultsan activation energy represented byTA5843 K and aVogel–Fulcher temperatureT05163.5 K, giving D'5, avalue characteristic of a very fragile glass-forming liquFinally, we have also represented in Fig. 5 the dielecrelaxation times measured by Legrand18 and by us: thesetimes turn out to be close to the light scattering ones.
FIG. 5. Activation plot of meta-toluidine obtained by light scattering tecniques and dielectric measurements:~j! tK depolarized geometry,~s!tKPCS, ~h! tK dielectric measurements~this work!, ~.! tKVV dielectricmeasurements~Ref. 17!, ~- - -! Vogel–Fulcher fit;~d! tK polarized geom-etry.
de
i-
–
r
.c
B. The b fast relaxation process and the determinationof Tc through the depolarized spectra
1. Data analysis
Figure 2 shows that, with decreasing temperatureminimum tends to develop at a frequencyvmin(T) locatedbetweenvmax(T) and a high frequency secondary maximuthe frequency of which is essentially temperature indepdent, approximately located at 3 THz and can be consideas the equivalent of the microscopic frequenciesVq of Eq.~2.2!. The value ofx8(vmin) as well as the position ofvmin
decrease with decreasing temperature, which are somedictions of MCT for theb relaxation process.
We have thus to find out whether this spectral regmay be adequately fitted by the formulas given by ttheory, Eqs.~2.6c! and ~2.7b!, in which the amplitude ofx8(vmin) is proportional toAT2Tc. Following proceduressimilar to others already used in this case,28,29 we first deter-mined by direct inspection approximate values ofvmin andx8(vmin) for all the spectra where this was possible, i.e.,233 K,T,290 K, and we fitted the region of the minimumwith Eq. ~2.6c! ~see Fig. 6!, letting the coefficientsa andb beindependent free parameters. This technique produced aries of those coefficients. The correspondinga andb coeffi-cients are never too far from the theoretical curve whrelates them@Fig. 6~b!#, the variation ofb being, as expectedlarger than that ofa. Nevertheless, for most spectra, the cefficient a is not well defined, the values ofb being moreaccurate, therefore, we fixedb as the mean value of its independent determinations and deduced the corresponding vof a from the relationship between those two coefficienThis gaveb50.58,a50.31, andl50.73. The whole set ofx8(v) curves was then fitted again with Eq.~2.6c! as a func-tion of v/vmin using the values ofa and b as fixed inputparameters. This procedure determined new values ofvmin
and x8(vmin) at each temperature. Figure 6~c! representsx8(v)/x8(vmin) vs v/vmin at different temperatures as weas the fitting curve, Eq.~2.6c!. The range over which the dif
FIG. 6. Comparison of the susceptibility spectrum with MCT predictions:~a! fit of the T5266 K susceptibility spectrum by the schematic model:~ !experiment,~1! numerical fit~Ref. 15!, ~ ! asymptotic approximation to theb fast correlator;~b! dispersion of thea andb coefficients in the individualfits of theb fast part of the susceptibility spectra, for the temperatures at which the fit was possible,~ ! theoretical curve;~c! x8(v)/xmin8 vs v/vmin forthe same spectra, with the theoretical asymptotic susceptibility spectrum.
desicer
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9867J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
ferent curves superpose is slightly larger than two decaAs already pointed out29 for a similar monomolecular glasforming liquid, salol, the agreement between the theoretcurve and the experimental one is quite good at all temptures forv,vmin . However, forv.vmin the frequency do-main for which Eq.~2.6c! represents the data decreases wincreasing temperatures and the agreement with the theical curve is much poorer.
2. Determinations of the critical temperature T c
The results onvmin andx8(vmin) just obtained, as welas the values oftK obtained in Sec. IV A, yield three different determinations ofTc :
Eq. ~2.6c!: x8~vmin!'AT2Tc, ~4.3a!
Eq. ~2.7b!: vmin'~T2Tc!1/2a , ~4.3b!
Eq. ~2.5a!: tK'~T2Tc!2~1/2a11/2b!. ~4.3c!
Figures 7~a! and 7~b! represent, respectively,vmin2a together
with x8(vmin)2 and tK
22ab/a1b vs T. Those three curvesshould display a linear temperature variation and they shohave the same intercept with the abscissa axis atT5Tc . This
FIG. 7. ~a! Power laws forxmin andvmin ; ~b! power law fortmax.
s.
ala-
het-
ld
is approximately the case for the linear variation but thecurves do not yield a singleTc value; we obtain, respectivelyTc5233 K (x8(vmin)), 228 K(vmin) and 220 K(tK), the or-dering of those temperatures being independent of the evalue of b used for the fits. It may be worthwhile pointinout that most of the data points used in Fig. 7~a! correspondto temperatures aboveTm . As the MCT predictions we testhrough Eqs.~4.3! are in principle asymptotic predictions,is not clear that they should hold for such largeT/Tc values.
C. Discussion
1. Validity of the previous analysis
Two points need to be discussed. The first concernsspecificity of the former analysis. Indeed, in the past, MCanalyses of depolarized light scattering spectra have geally been carried out assuming that, at each temperaturetotal light scattering spectrum was only related to centersmass induced scattering, which allowed to connect thspectra in a rather direct manner to the density fluctuasusceptibility spectra deduced from the MCT equations. Twas done by means of expressions of the correlator valithe asymptotic T2Tc→0 limit ~using the whole bcorrelator4 of the idealized or of the extended MCtheory30!, or by using more sophisticated MCT formulationsuch as some schematic MCT models,15 introducing in thelatter case parameters which describe some specific feaof the system under study. For instance, Fig. 2 shows thathe 500 GHz region, there exists a small spectral featcalled the boson peak, which is more and more visible whthe temperature is lowered. This feature was specifictaken into account as a second microscopic frequencyAlba-Simionescoet al.15a in their fit of the total Raman spectrum of meta-toluidine between 150 GHz and 4 THz byschematic two-correlators model. The agreement betwthese purely center-of-mass formulations and the expments was found to be good in a more or less importfrequency range for several glass-forming liquids.28–31,15bInthe case of the meta-toluidine broadband spectra, oneextend the agreement obtained in Ref. 15a and coverwhole frequency range by using the same technique aRef. 15b, i.e., by restricting the set of Eqs.~2.2! to a systemof two nonlinear integrodifferential equations. An exampof such a fit is given in Fig. 6~a! for T5266 K. Such fits canbe performed at each temperature, and they yieldTc
5233 K andl50.67, a somewhat smaller value than thel50.73 value obtained through the asymptotic formuanalysis shown in Figs. 6~b! and 6~c!. The problem of suchan approach is that its validity is very questionable whenlight scattering mechanism is mostly related to orientatiofluctuations~see the first paragraph of Sec. IV A!. The intrin-sic complexity12b of the MCT equations including both thcenters-of-mass and the orientational motions is too imptant to ascertain if schematic models similar to the one ufor the construction of Fig. 6~a! are still meaningful; it ispossible that such models are flexible enough to providerect numerical fits to the spectra even if the parameters udo not properly represent the physical problem. This
heio-ouo
as
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io
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9868 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
why we shall consider only the fits of our spectra to tasymptotic formulas and shall rather focus the discuss~Sec. IV C 2! on the possibility of finding a systematic behavior in the low frequency part of the depolarized spectraseveral molecular glass-forming liquids. Let us just point ohere that, in the case of meta-toluidine, neglecting the secmicroscopic mode does not prevent analyzing the regionthe minimum of the susceptibility spectra with the formulgiven in Sec. II. This was already the case for salol29 andsuch a situation must be contrasted, for instance, withcase of a long chain polymer7 ~1-4 cis-transpolybutadiene!,where the same low frequency region (v,50 GHz) wascompletely modified by the existence of some~internal?!molecular motions.
A second problem is the relationship between the regof thea relaxation and the region of the minimum ofx8(v).The low frequency part of this minimum is usually referrto as the von Schweidler regime;4 as pointed out in this ref-erence, the existence of that regime has little to do with tharelaxation process: the combination of Eqs.~2.5a!, ~2.6b! and~2.7! shows that one can write:
Fq~ t !5 f qc2hqBlS tq
a0
tb0D bS t
tqaD b
~4.4!
so that the r.h.s. of Eq.~4.4! does scale withtqa but with
powerb and not with powerbK , contrary to the first orderexpansion of the Kohlrausch function. In the susceptibispectra, the von Schweidler regime, which describes a slodecay in frequency than thea relaxation process, is thucharacterized by anv2b law with b,bK .
In the present case, the relation betweenb andbK is onlymarginally satisfied for the temperatures at which a fit ofregion of the minimum ofx8(v) is meaningful: we obtainb50.58 andbK>0.61. In fact, as it is apparent in Figs. 6~a!and 8, some part of the spectra have been used simneously to obtain a fit of thea relaxation process with Eq~4.1! and of the region of the minimum ofx8(v) through Eq.~2.6c!. In the present case where the ratiox8(vmax)/x8(vmin)
FIG. 8. Scaled susceptibility curves~a peak region!: ~ ! experiment,~* !Debye fit,~m! CD fit.
n
ftndof
e
n
er
e
ta-
has typical values of 2 or 3, the determination ofbK may besomewhat biased by that aspect of our fitting technique;bias could be more important than in the case of two otfragile glass-forming liquids, salol and OTP~ortho-terphenyl!, also based on phenyl ring substances, which hbeen recently analyzed within the same MCT scheme29–31bythe same depolarized light scattering techniques. In thtwo cases wherexmax8 /xmin8 are substantially larger, similavalues ofb were obtained, while the values ofbK were muchlarger (bK50.8 in salol,29 0.8,bK,0.95,41 andbK50.7930,31 in OTP!. More work has to be done to elucdate the importance of this point.
2. Dispersion in the T c values
The approximate value ofTc'226 K places meta-toluidine in the same category as the two other glass-formliquids just mentioned. Previous measurements have indgiven:
Salol:29 Tm5315 K;Tg5218 K;Tc5258 K;Tc /Tg51.18;~4.5a!
OTP:30,31 Tm5329 K;Tg5244 K;Tc5290 K;
Tc /Tg51.19. ~4.5b!
Such aTc /Tg value ;1.20 is in agreement with an earlestimate of this ratio for fragile glass-forming liquids bason a corresponding states analysis of viscosity data,32 andalso with a recent depolarized light scattering study of thebfast relaxation of toluene,33 another member of that family.
We obtain here a similar ratio,Tc /Tg51.21. Note thatwhen those three liquids are studied using the same depized light scattering technique, and when their spectraanalyzed by the same method, the distribution of the diffent values ofTc is notably larger~13 K! for meta-toluidinethan for the other two liquids ~6 K and 5 K,respectively!28–30 but the order of the three different temperatures remains the same:Tc(x8(vmin)).Tc(vmin).Tc(tK). It may be worth studying if this order exists alsoother similar glass-forming liquids.
V. ANALYSIS OF THE ISOTROPIC SPECTRA
A. General considerations
As mentioned at the end of Sec. III, polarized spectI VV(v), were recorded in the back scattering geometry ev10 K between 178 K and 298 K using one single free sptral range. The corresponding depolarized spectra,I VH(v),were measured in exactly the same geometry, at the stemperature~see Fig. 3!. Those two spectra are not indepedent. Indeed, in a liquid composed of optically anisotropmolecules, it is generally found that the spectrum is depoized in a frequency range going from 50 GHz to a few TH~the upper limit corresponding to frequencies up to whichmolecules can be considered as rigid!. This means that:
I VV~v!54/3I VH~v! ~5.1!
lauc
aiqizeteen
oe
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ino
thuthey
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;
-bili-
tude,
ined
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9869J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
and we mentioned in Sec. III that such a constant depoization ratio was indeed approximately measured for sfrequencies in the Raman study of meta-toluidine.15 This re-sults from the light scattering mechanism in such liquidswell as from the macroscopic isotropic properties of the luid. The situation changes at lower frequencies: the polarspectrum contains a second contribution which originafrom the propagation of density fluctuations. The latters gerate, through the Clausius–Mossotti relation, a variationthe isotropic part of the dielectric tensor. The LT of its timcorrelation function,I iso(v), appears only in the polarizespectrum and is proportional to the density–density corrtion function at the scattering wave vector, a quantity whwas discussed in Eq.~2.9!. One may obtain the corresponding Fq9(v) from:
AFq9~v
o
ic
in
es
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e
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or
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the
s
of
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alheir
rn-
9870 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
enough forvmq1(v) to reduce to its first term and to give n
contribution to the Brillouin linewidth. This yieldedg0
50.175 GHz.
2. Results for the ‘‘ a only’’ relaxation model
Good fits were obtained at each temperature and typexamples of the quality of the fit, corresponding toT5290 K, 256 K, 236 K, 216 K, and 192 K, are givenFig. 3. The corresponding values ofDq andtCD
L , as well asthose ofnB , nq5vq/2p, andn`5c`
q /2p are given in TableII, while Fig. 10 showsc0 , c` , andcB52pnB /q as a func-tion of temperature.cB(T) varies from values close toc`(T)nearTg , to values close toc0(T) at the highest temperatur(T5290 K), as it has been found in many other systemFinally, the variation oftCD
L is represented in its Kohlrauscform, tK
L , in Fig. 5. This figure shows a quite typical Arrhenius behavior with an activation energyEa5kBTa with Ta
'650 K. Such a thermal behavior, which has also befound in salol34 for instance, calls for two comments:
~a! The procedure used to determineg0 implied that thecontribution of tCD
L to the linewidth would be negli-gible at 178 K while this is clearly not the case. Tevaluate how inaccurate that assumption was, wereport, in the last but one column of Table II, the valof the imaginary part ofmq
1(v) for v5vB , supposingtCD
L to be large enough for approximating this imagnary part byg1 , with g1 given by:
g15D2q2
2p sinpbCD
2
vB~vBtCDL !bCD
~5.5!
an approximation valid forT,266 K. This yields avalue ofg150.053 GHz at 176 K, i.e., nearly 1/3 of thvalue ofg0 . The former determination ofg0 was thuspartly inconsistent.
~b! A more important point is that this Arrhenius behaviimplies thatvmq
1(v) does not represent ana relaxationprocess but rather ab process, similar in its thermabehavior to a Johari–Goldstein process which wobe already detected in the supercooled liquid. Asdetermination of the nonergodicity parameter,f q ,
FIG. 10. Thermal variation of the different sound velocities:~m! infinitefrequencyc` , ~d! apparentcB , ~ ! zero frequencyc0 .
al
.
n
so
de
through Eq. ~2.11! implies, in its derivation, thatvmq
1(v) is ana relaxation process, the value off q(T)obtained through this equation is very dubious andfit of this curve by Eq.~2.12b! has no reason to yield areliable value ofTc . Neither the fact that such a fit wapossible~see Fig. 11!, nor the proximity of the corre-sponding value ofTc523762 K with the different val-ues ofTc determined in Sec. IV can be taken as teststhe validity of such an approach.
C. Search for the possible contribution of a ‘‘ b-fast’’relaxation process
1. The model
In order to circumvent the obvious inconsistencies otained in Sec. V B, we have tried to exploit more systemacally the possibilities of MCT. In the case of CKN,35 themaximum of the Brillouin linewidth was obtained at the temperature at whichnB coincided with the maximum of theapeak of the susceptibility spectrum detected in the depoized geometry. This coincidence led Liet al.35 to proposethat this susceptibility spectrum would be the imaginary pof vmq(v). Performing a Kramers–Kronig transformatioof that imaginary part to obtain its real part, they thus detmined the completevmq(v), which they injected in Eq.~5.3!. Quite a good fit of the whole Brillouin spectrum waobtained in that case. Unfortunately, the present situationot the same but is similar to the case of propylecarbonate;30 we observe the same type of difference betwethe temperature variations of the width of the Brillouin peand of the maximum of the depolarized susceptibility sptrum. In such a situation, Duet al.30 proposed a differentstrategy in order to mimic a relaxation function in partiagreement with the MCT predictions. We have repeated tmethod in the present Sec. V C.
We admitted that Eq.~5.3! represents the fulla relax-ation part ofvmq(v); in order to insure a thermal behavioof tCD
L compatible with that hypothesis, we imposed this lo
FIG. 11. Nonergodicity parameter: second model:b fast relaxation in-cluded,~d! r 5tR/tL51, ~m! r 52, ~.! r 53, ~ ! fits with Eq. ~2.12b!.Inset: first model;~j! CD function only,~ ! fit with Eq. ~2.12b!.
a
in
V
.,
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c-186lar-gu-rizederre-h
pri-
9871J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Liquid–glass transition of meta-toluidine
gitudinal relaxation time to be proportional to the rotationrelaxation time written under its Cole Davidson form,tCD
R ,and we deducedtCD
R from the measurements performedSec. IV interpolated through a Vogel–Fulcher formula.
We then assumedvmq(v) to be written as:
vmq~v!5vmq1~v!1ga~T!S tgS p
2~12a! D1 i Dva,
~5.6!
wherea is the MCT critical exponent determined in Sec. I(a50.31): the second term of Eq.~5.6! represents the highfrequency part of theb fast relaxation process of MCT, i.ethe Fourier transform of the critical decay of theb-fast re-laxation process taken under its asymptotic form@Eq.~II.6a!#. ~Other formulations for the high frequency part habeen proposed by Loheideret al.38 and used for severasystems.39! We have thus assumed in fact:
• that the von Schweidler part of the sameb fast relax-ation process was properly taken into account byhigh frequency part ofvmq
1(v), in agreement with theremark made in Sec. IV C;
• that the main role of the second term in Eq.~5.6! is togive an additional contribution to the linewidth of thBrillouin peak through its imaginary part; yet the rolethe corresponding real part plays a role and slighshift the position of the Brillouin peak.42
Fits were thus performed usingD(T) and ga(T) as fit-ting parameters, fixing the ratior 5tCD
R /tCDL to some given
values; a series of fits were thus performed withr 51, 2, 3,and 5, respectively.
2. Discussion of the approximate ‘‘ b fast’’ model
Equally good fits of the whole series of spectra~from290 K to 186 K! were obtained for all the values ofr listedabove. Furthermore, the values ofD r(T) deduced from thosefits led to corresponding nonergodicity parameters whichproduced rather acceptable thermal behavior, as is showFig. 11. In particular, ther 52 curve is of the same quality athe curve related to the ‘‘a only’’ model, while the r 53curve is reminiscent of the result obtained for propylene cbonate by Duet al.30 Nevertheless, Fig. 11 also shows ththe position of the square root singularity which indicatesvalue ofTc increases with the value ofr, all of them beinghigher than the value obtained in the ‘‘a only’’ model. Fix-ing any reasonable value forr thus increases the dispersioof Tc with respect to the values obtained in the study ofdepolarized spectra.
Another aspect of the problem is revealed by Fig.which represents the thermal variation ofga for the variousvalues ofr. One sees that:
• ga(T) is practically independent ofr, i.e., of the relax-ation time, below 240 K; in this thermal range, thebfast process is responsible of the entire linewidth ofmode;
• Above 240 K, bothga(T) and tCDL @throughmq
1(v)],contribute to this linewidth. But, whatever is the valu
l
e
y
llin
r-te
e
2
e
of r, the maximum of the Brillouin linewidth around270 K cannot be accounted for by a monotonous vation of ga(T): under the form used in Eq.~5.6!, the bfast contribution has to decrease, at least abovetemperature, in order to explain this behavior. Thwould force us to link the origin of the linewidth maximum to a nonmonotonous thermal behavior of thebfast contribution. In the frame of MCT, this thermabehavior is unexpected because the memory funcshould itself be connected to correlation functiowhich should obey Eq.~2.6c! at least in a limited ther-mal range aroundTc . When one identifies Eqs.~2.6c!,~2.7b! with Eq. ~5.6!, one sees easily thatga(T) shouldbe constant, at least aroundTc , which is obviously notthe result shown in Fig. 12. Even if the memory funtion which couples to the longitudinal modes does nfollow the MCT predictions up to as high temperaturas the correlation function detected in the depolarizspectrum, such a change from a predicted constanthavior to a curve with a positive and then negative fiderivative and an always negative second derivativevery unlikely.
In view of the results described in the two previous pagraphs, we must conclude that Eq.~5.6! is still a poor repre-sentation of the real memory function in the case of the lgitudinal propagative modes in meta-toluidine.
D. General discussion
The preceding two attempts of fitting the Brillouin spetra recorded over a 1–27 GHz range, between 298 K andK, and obtained through a careful subtraction of the depoized contribution to the polarized spectra, lead to an ambious answer. Although these spectra cannot be characteonly by their linewidth, at least above 240 K, their entire linshape can be described by many sets of two strongly colated parameters~see Fig. 12! as it has been shown througthe two models tested in the present section.
While the first model~Sec. V B! is certainly inconsistent~it does not include properly theb-relaxation function!, thesecond one remains largely unsatisfactory. In fact the
FIG. 12. Thermal variation ofga(T) for different values ofr 5tR/tL: ~j!r 51, ~d! r 52, ~m! r 53.
to
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.
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.
.
9872 J. Chem. Phys., Vol. 112, No. 22, 8 June 2000 Aouadi et al.
mary purpose of the study performed in this section wasfind out whether these spectra could be used as tests of saspects of MCT. Our results have been negative in the sthat we found no ‘‘single channel’’ memory function whicfitted them and was compatible with that theory. Fromtheoretical point of view, this is not really surprising. Somof us40 have recently shown, with the help of a phenomenlogical theory which is expected to mimic some importaaspects of MCT, that the memory functionvmq(v) to beused in the present section should be a linear combinatioa bulk deformation memory function and of a shear defmation memory function. Equations of the same type as~5.4! are thus not expected to hold in the case whererelaxation times of these two functions are unequal, whmay very well be the case. Yet, even if one could expecobtain the value of the shear relaxation times from the ansis of the transverse phonon spectra,40 it is unclear that thiswill be sufficient to break down the correlation between tdescriptions of the bulka-relaxation process and of the corespondingb fast process found in Sec. V C. It is very likethat one will have to use the information which can be drived from stimulated Brillouin scattering~which measuresdecays in the time regime, and is mostly sensitive for lorelaxation times, i.e., for the low temperature situation!, toobtain a better understanding of the role of thisb process.
VI. CONCLUSION
We have recorded the low frequency depolarized aisotropic light scattering spectra of meta-toluidine in a bascattering geometry from 298 K to 183 K and comparthem with the predictions of the idealized MCT. The deplarized spectra mostly probe the orientational dynamicsthe molecules through its coupling to the total light scattermechanism, i.e., including interaction induced effects. Twork of Schilling et al.12 indicates that the idealized MCTpredictions should remain essentially valid for this orientional dynamics. We found that a reasonable agreemcould be obtained in the form of unique set ofa and b pa-rameters describing approximately the correspondingnamics at various temperatures. Yet, the critical temperaTc which can be deduced from three distinct features of thspectra do not really coincide: they exhibit a dispersionDT567 K with respect to a mean valueTc5226 K. We havealso noted that the ordering of the corresponding three vais the same as already found in two other simple molecglass-forming liquids also containing benzene rings asimportant ingredient.
The isotropic spectrum~Brillouin spectrum! deducedfrom the subtraction of the depolarised spectra from thelarized ones could not be fitted using simple ingrediecompatible with MCT. It must be emphasized that suchexercise is much more difficult than the preceding onecause it consists in guessing which memory function copatible with MCT should be used to fit the spectra while tlatter contain little information. We have pointed out that tmemory function we used in our analyses was too crudeapproximation to the realistic one. Important efforts combing, at least, other light scattering techniques~analyses oftransverse propagative waves,40 stimulated Brillouin scatter-
omese
a
-t
of-q.ehoy-
-
g
dkd-f
ge
-nt
y-ree
esrn
-sn--
n-
ing! and presumably realistic MD simulations may appearthe only way of obtaining more information from these Brlouin spectra.
ACKNOWLEDGMENTS
This work has benefited from the Procope Project N94034 and of a MPI-CNRS agreement which allowed a fruful collaboration between the Mainz and the Paris grouWe thank B. Bonello, from LMDH, who performed the utrasonic measurements quoted in Sec. III. We also thankF. Lautier for the preparation of the samples, C. Caray anP. Franc¸ois for their experimental help. We finally thank VKrakoviak for allowing us to present the results of his fitthe T5266 K spectrum with a schematic model.
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