Jun 8, 2000 - This theory thus bridges an experiment per- formed in a steady state ...... expansion of the Kohlrausch function. In the susceptibility spectra, the ...
JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 22
8 JUNE 2000
Light scattering study of the liquid–glass transition of meta-toluidine A. Aouadi, C. Dreyfus, M. Massot, and R. M. Pick L.M.D.H., B.P. 86, Universite P. et Marie Curie, 4 Place Jussieu, F-75005 Paris, France
T. Berger and W. Steffen Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany
A. Patkowski Max-Planck-Institut fu¨r Polymerforschung, Postfach 3148, D-55021 Mainz, Germany, and Institute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland
C. Alba-Simionesco L.C.P., Universite´ Paris-Sud, Orsay 91400 France
共Received 30 August 1999; accepted 3 March 2000兲 An experimental study of the glass transition of meta-toluidine combining several light scattering techniques was performed. The structural relaxation time is measured in depolarized geometry from the glass transition temperature up to well above the melting point and found to vary over 13 time decades. An analysis by means of the idealized Mode Coupling Theory shows that, as found in other aromatic liquids, experimental results obtained in depolarized light scattering can be described by this theory above T c in a two-decade frequency range. The polarized Brillouin doublet, measured in the backscattering geometry between 176 K and 300 K, is also analyzed. None of the sets of parameters we obtained in fitting those spectra could fulfil all the requirements of this Mode Coupling Theory. © 2000 American Institute of Physics. 关S0021-9606共00兲52220-X兴
I. INTRODUCTION
it involves a memory function the ingredients of which are precisely the shear modulus and the 共relaxation兲 time just mentioned. This theory thus bridges an experiment performed in a steady state regime with a dynamical approach and proposes that it is essentially the corresponding relaxation time, attributed to the ‘‘␣ relaxation process,’’ which varies under cooling. This ␣ relaxation process is, indeed, detected by many spectroscopic techniques: such techniques are always mostly sensitive to the dynamics of some specific variable共s兲, and it has been recognized for more than 30 years 共see, e.g., Ref. 4 for a review兲 that, although this ␣ relaxation process is visible in any experiment, the corresponding averaged relaxation time can differ from one technique to another, even at temperatures above T cross . When lowering the temperature, fitting these spectroscopic data frequently becomes impossible without introducing, at least, another relaxation channel. In the classical analysis where one identifies each channel with a relaxation time, the latter appears to be much faster than the ␣ process extrapolated at that temperature and is governed by an Arrhenius law unrelated to the ␣ process. Such a channel has been named by Johari and Goldstein5 a  relaxation. Yet, the analyses which reveal this  process do not rely on any theoretical basis, while the experimental evidence tends to show a dispersion of the corresponding times with the experimental technique larger than for the ␣ process6 and also a dispersion in the corresponding Arrhenius energies,7 two facts which may cast some doubts on the general validity of their existence. During the mid-eighties the first microscopic theories
The existence of supercooled liquids which remain liquid below their melting temperature during macroscopic times without crystallization, and which, under slow cooling rates 共typically 10 deg/min兲 can be transformed into glasses 共the liquid–glass transition兲 is the oldest unsolved problem of condensed phase physics. In this regime, the most dramatic effect which characterizes the approach of the transition is the change in the shear viscosity: s varies, typically, over more than 15 orders of magnitude from a normal liquid phase s ⬃1 cP to a value of s of the order of 1013 P at the calorimetric glass transition temperature, T g . This thermal dependence has thus been the subject of many theoretical studies.1,2 It does not display a unique behavior and, although this aspect has been known for a very long time, it was only in the mid-eighties that Angell3 rationalized all the data with an Arrhenius plot of log s(T) vs T g /T, thus allowing for a classification of the various types of glass-forming liquids. This Angell plot shows in particular that many liquids have a very similar behavior in which log s(T) contains two approximately linear parts separated by a region of rather strong curvature in the vicinity of some crossover temperature T cross ; such liquids are the so-called fragile glassforming liquids. Many nonalcoholic organic liquids belong to this class. Although the shear viscosity may appear to be a static quantity, a dimensional argument allows us to write it as the product of a shear modulus and a relaxation time. The Maxwell theory of viscoelasticity makes clear the rationale behind this dimensional analysis by proposing that the relationship between the local shear strain and the corresponding shear rate would be nonlocal in time in a supercooled liquid; 0021-9606/2000/112(22)/9860/14/$17.00
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appeared proposing an explanation of the liquid–glass transition,8–10 which incorporates the dynamical aspect of the liquid. For the past ten years, these theories, called Mode Coupling Theories 共in short, MCT兲 have received a lot of attention. In their simplest version,4 they are written for a fluid made of shapeless particles, for which the only dynamical variables are the positions of the centers of mass. The earliest version, the idealized MCT, predicts at a temperature T c a dynamical transition between two regimes: in one of them (T⬎T c ) the time correlation function of any density function will eventually decay to zero while in the other one (T⬍T c ), all these functions will remain finite at infinite time, this limit being called their nonergodicity parameter. This theory makes specific predictions for the dynamics of these correlation functions in the vicinity of T c . In particular, it predicts, for T⬎T c , the existence of an ␣ relaxation process with a wave vector dependent relaxation time, while the numerical analysis of the corresponding solutions shows that this dynamics can be represented by a stretched exponential with a stretching coefficient independent of temperature but wave vector dependent. A second prediction of the theory is that, close to T c , for intermediate times 共i.e., for times much longer than those characteristic of the microscopic motions, but much shorter than the corresponding ␣ relaxation times兲, a wave vector independent dynamics should take place. This intermediate dynamics has been called11 the ‘‘ fast process,’’ the letter  recalling the fact that this process has a characteristic time much shorter than the ␣-relaxation process, but this ‘‘-fast process’’ has nothing in common with the  Johari–Goldstein process5 discussed above. A first and standard criticism of this theory is that, when its predictions are compared with the experimental results, approximate agreements can be obtained only if T c has a larger value than the glass transition temperature T g , this value turns out to be close to the temperature T cross at which log s(T) has its maximum curvature in the Angell plot. This may be attributed to the neglect of the role of the particle currents as additional relevant variables. As soon as their contribution to the time correlation functions are introduced, even in a schematic form, in the MCT equations, T c becomes a crossover temperature between a high temperature regime where the predictions of the ideal MCT remain approximately valid and a lower temperature regime where the structural arrest is not complete and the ␣ relaxation time still goes on diverging. A second criticism is that no simple liquid ever becomes supercooled, in the sense described at the beginning of this section. The simplest liquids which undergo a liquid–glass transition and could correspond to this fragile glass-forming scheme are formed of more or less complex molecules. Moreover, most of the experimental techniques either measure, or are largely sensitive to, the orientational motions of the molecules. Extensions of the theory adding specifically those other degrees of freedom appeared as necessary improvements. They have recently started to be developed12 and they basically confirm12b the predictions of the simpler, atomic like, MCT. Molecular Dynamics computations based on asymmetric dumbbells have also been recently
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performed12c and the time correlation functions of variables related to different values of the index l of the spherical harmonics characterizing their orientation have been computed. These calculations largely confirm the MCT predictions extended to rotational dynamics, though there are certain reservations: for instance, the time temperature superposition principle, which supposes that the stretching coefficient of the ␣ relaxation process does not depend on temperature, appears not to hold properly for l⫽1. Yet, the corresponding fluids 共such as CO or NO, for instance, or larger asymmetric dumbbell molecules, such as OCS兲 do not undergo a liquid–glass transition, so that those numerical studies cannot represent an experimental situation. Conversely, the real fragile glass-forming liquids always exhibit a more complex dynamics, originating from internal low frequency modes, and/or from weak intermolecular bonding, ingredients not yet incorporated in those MD computations. Performing precise measurements of the overall dynamics of a given glass-forming liquid, over a very large frequency range, using polarized and depolarized dynamic light scattering which gives access to different observables, and comparing those results with the MCT predictions may help to answer the following question: when dealing with a real glass-forming liquid, do those additional dynamical features totally mask 共or modify?兲 the signatures predicted by the MCT?13 If this is not the case, to what extent does the measured dynamics agree with the predictions of MCT? The present paper aims at discussing this problem in the case of a very simple fragile glass-forming liquid, metatoluidine (CH3 –C6H4 –NH2兲. This has been achieved through a light scattering study of this system with the help of different instruments. Our paper is organized in the following way. Section II will summarize the results of the MCT we wish to test here. Section III is devoted to the description of the experiments and of the results. In Sec. IV, the depolarized light scattering results are analyzed in the spirit of the MCT, making use of a phenomenological description of the ␣ relaxation. Section V is devoted to the analysis of the longitudinal Brillouin line along the same pattern. A brief conclusion ends the paper. II. THEORETICAL BACKGROUND
Comparison between experimental results and the ideal MCT predictions have been frequently performed on a variety of substances. Extended reviews can be found in Ref. 4 and references therein. Without entering into a detailed presentation of the theory, let us briefly recall, in this section, the theoretical results of the MCT we shall use in this paper. We shall state them for the simplest version of the theory, the idealized MCT, in the canonical case where the only relevant dynamical variables are the different normalized density–density correlation functions of a monomolecular fluid: ⌽ q共 t 兲 ⫽ where
S 共 q,t 兲 , S 共 q,0兲
共2.1a兲
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S 共 q,t 兲 ⫽
冓兺 i, j
冔
exp共 iq• 共 ri 共 t 兲 ⫺r j 共 0 兲兲兲 .
共2.1b兲
with: ⌽ q共 兲 ⬅⌽ q⬘ 共 兲 ⫹⌽ q⬙ 共 兲 ⫽i
The fundamental set of equations of the theory reads: ¨ q共 t 兲 ⫹⍀ q2 ⌽ q共 t 兲 ⫹ ⌽
冕兺 t
0 q1
˙ q共 t ⬘ 兲 dt ⬘ ⫽0, m q,q1 共 t⫺t ⬘ 兲 ⌽ 共2.2兲
where ⍀ q is called the microscopic frequency of the mode with wave vector q and where the leading order term m q,q1 (t) is given by m q,q1 共 t 兲 ⫽V q,q1 ⌽ q1 共 t 兲 ⌽ q⫺q1 共 t 兲 .
共2.3兲
Above T c , and for (T⫺T c )/T c small compared to unity, one expects that ⌽ q(t) will decay, for t→⬁, as:
冉冉冒冊冊
⌽ q共 t 兲 → f q0 exp ⫺
t
q
q␣
共2.4兲
,
where ␣
q␣ ⫽ q 0 共 T⫺T c 兲 ⫺ 关共 1/2a 兲 ⫹ 共 1/2b 兲兴
共2.5a兲
and where the nonergodicity parameter f q0 , the relaxation ␣ time factor q 0 and the stretching coefficient  q depend on q but not on temperature. For the same temperature region but for ⍀ q⫺1 ⰆtⰆ q␣ , the time dependence of ⌽ q(t) is predicted not to depend on q; this is the  fast dynamics, which is governed by a single parameter, ; the latter fixes the value of the two positive numbers, a and b, which appeared in Eq. 共2.5a兲. In particular, the asymptotic behaviors of ⌽ q(t) are predicted to be given by: ⌽ q共 t 兲 ⫽ f q0 ⫹h q冑T⫺T c A
冉冒冊
⌽ q共 t 兲 ⫽ f q0 ⫺h q冑T⫺T c B
冉冒冊
t
⫺a

t

tⰆ  , 共2.6a兲
b
tⰇ  , 共2.6b兲
with:
 ⫽  0 共 T⫺T c 兲 ⫺1/2a .
共2.7a兲
The experiments we are going to analyze were performed in the frequency domain. This makes the comparison with the formulas given above quite difficult. The possibility of obtaining a conclusive information from this comparison in the frequency space is based on two hypotheses: • the time domains where the microscopic motions (t ⬇⍀ q⫺1 ), the  fast relaxation regime (⍀ q⫺1 ⰆtⰆ q␣ ) and the ␣ regime ( q␣ ⭐t) take place are well separated; • the relative amplitudes of those three motions are of the same order of magnitude. If those two conditions are fulfilled, similarly to the time domain case, there exist three well separated frequency regions in each of which one of the Laplace transforms 共in short LT兲 of the three preceding regimes dominates. More precisely, if q⬘ ( ) is the real part of the LT of the susceptibility function related to ⌽ q(t):
q⬘ 共 兲 ⫽ ⌽ q⬙ 共 兲
共2.8a兲
冕
⬁
0
e ⫺i t ⌽ q共 t 兲 dt, 共2.8b兲
• the low frequency part of q⬘ ( ) is related to the LT of Eq. 共2.4兲 and, in particular, its maximum should take place at a position proportional to: 共2.5b兲 共q␣ 兲 ⫺1 ⬀ 共 T⫺T c 兲 共 1/2a 兲 ⫹ 共 1/2b 兲 . 关Note that, due to the form of the LT used in Eq. 共2.8b兲, q⬘ ( ) is a quantity related to the light scattering spectra, see Eq. 共3.3兲兴. • An intermediate frequency region should correspond to the  fast relaxation process and an approximate form of q⬘ ( ) based on the asymptotic expressions Eqs. 共2.6a, b兲 is: ⫺b a a ⫹b T⫺T c min min q⬘ 共 兲 ⫽ q0 , Tc a⫹b 共2.6c兲 with 0 min⬀min 共2.7b兲 共T⫺Tc兲1/2a , q⬘ ( ) having its minimum at min with an amplitude proportional to 冑T⫺T c . • A high frequency region dominated by the microscopic motion at frequency ⍀ q .
冉冑
冉冊 冉
冊 冉 冊冊
The measurement of q⬘ ( ) should thus lead, through the study of the value of its minimum and through the set of Eqs. 共2.5b兲, 共2.6c兲, 共2.7b兲 to three independent measurements of T c ; the extension of the simple MCT to the case of rigid molecules12b suggests that those conclusions should remain valid for the correlation functions related to orientational variables. Equation 共2.2兲 can also be solved numerically, in the case of monoatomic liquids, evaluating the coefficients V q,q1 with the help of the static structure factor of the liquid in the vicinity of T c . Their resolution provide correlation functions ⌽ q(t) in good agreement with the direct MD calculations at short time, and, at long time, they yield numerical results which agree with the theoretical predictions, Eqs. 共2.4a兲– 共2.7a兲. This has been tested for some very simple potentials.14 We also mention, for further use, that the same general form for ⌽ q(t), and, in particular, for its asymptotic behavior (T→T c , t→⬁) can be obtained from much more schematic models containing only one or two correlation functions, provided that the r.h.s. of Eq. 共2.3兲 also contains linear terms in those correlators 共see Ref. 15a and references contained in Ref. 4兲. No calculation corresponding either to the full MCT or to these schematic models exists, presently, for molecular liquids. One last aspect of Eq. 共2.2兲 will be used in the present paper. The sum on q1 appearing in its last term may be performed at once, thus introducing a memory function ˜ q(t⫺t ⬘ ). Under that form, Eq. 共2.2兲 is a quite general exm pression, which does not depend on a mode coupling analy˜ q(t) corresponds to the sis. Yet, from general arguments, m coupling of a mode with wave vector q with the other degrees of freedom. Also, the basic concepts of hydrodynamics
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imply that, for q⫺1 of the order of an optical wavelength, m ˜ q(t) must be proportional to q 2 while its dynamics should be independent of q, at the lowest order in q. Finally, for this value of q, ⍀ q should be replaced by q , which is the frequency of a sound wave 共i.e., of a longitudinal propagating mode兲 and is expressed as: q⫽c 0 q, c 0 being the 共relaxed兲 sound velocity. The LT of Eq. 共2.2兲 then reads: ⌽ q共 兲 ⫽
⫺m ˜ q共 兲 . 2 ⫺ q⫺ m ˜ q共 兲 2
共2.9a兲
With the definition Eq. 共2.8b兲 of the Laplace Transform, the spectrum measured in a light scattering experiment is proportional to ⌽ q⬙ ( ); this last quantity is easily found to be ⌽ q⬙ 共 兲 ⫽
q2 m ˜ q⬙ 共 兲 2 2 ˜ q⬘ 共 兲兲 2 ⫹ 共 m ˜ q⬙ 共 兲兲 2 共 ⫺ q⫺ m
q2 1 ⬅ Im 2 . ⫺ q2 ⫺ m ˜ q共 兲
共2.9b兲
The line shape of the spectrum corresponding to Eq. 共2.9b兲, i.e., the Brillouin line shape, is then entirely determined by the frequency and thermal behavior of m ˜ q( ) where m ˜ q( ) is the LT of m ˜ q(t). At long times, the latter can be characterized by a relaxation time L . Consequently: ˜ q( ) is a • At high temperature, where q L Ⰶ1, m small imaginary quantity which simply contributes some linewidth to the Brillouin peaks located at ⫾ q ⫽⫾c 0 q. On the contrary, at low temperature where q L Ⰷ1, as the →⬁ limit of m ˜ q( ) must be written as ⌬ 2 q 2 , the spectrum is made up of two lines located at: 共2.10兲 ⫾q⫽⫾ 冑c 20 ⫹⌬ 2 q⬅⫾c ⬁ q. The idealized MCT adds two aspects to those classical results: • First,16 (⌬/c ⬁ ) 2 is the nonergodicity parameter f q related to the longitudinal phonon 关i.e., the equivalent, at any temperature, of the f q0 term of Eq. 共2.6a兲兴. Whence: c0 2 . 共2.11兲 f q⫽1⫺ c⬁
冉 冊
Meanwhile, the idealized MCT predicts that: f q⫽ f q0 ⫽cte f q⫽ f q0 ⫹k
冑
共2.12a兲
T⬎T c , T⫺T c Tc
T⬍T c .
共2.12b兲
Analysis of the Brillouin line shape, i.e., the determination of c 0 , c ⬁ , and ⌬ is thus, in principle, a fourth way of measuring T c and is, at the same time, a method for having some information on the validity of one specific MCT prediction below T c . Unfortunately this analysis requires the introduction of a numerically tractable expression for m ˜ q( ) in order to obtain ⌬ and c ⬁ . This cannot be done without making assumptions on the analytic form of this memory function. Some of them will be presented in Sec. V.
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III. EXPERIMENTAL ASPECTS A. Former experimental studies
Meta-toluidine (CH3 –C6H4 –NH2) belongs to the family of the meta bisubstituted benzenes, X–C6H4 –Y, where X 共resp. Y兲 may be CH3 ,NH2 ,OH..., which remain very easily supercooled17 down to their thermodynamic glass transition temperature, T g 共except for the case X⫽Y⫽CH3). In the present case, T g ⫽187 K 共measured in DSC,17 with a temperature decrease of 10 deg/min兲 while the melting temperature is T m ⫽243.5 K. Under normal conditions, crystallization never happens in the supercooled phase: one needs to bring the glass at least 50 deg below T g to see microcrystals growing in the supercooled phase under re-heating.18 This implies that in our experiments, we will not need to worry about the possible existence of microcrystals in the supercooled phase. The thermal variation of the index of refraction and of the density are given by19: n 共 T 兲 ⫽1.692共 5 兲 ⫺4.5 10⫺4 T 共 °K兲 ,
共3.1兲
共 T 兲 ⫽ 关 1.225共 0 兲 ⫺8.1 10⫺4 T 共 °K兲兴 g/cm3 .
共3.2兲
Among other quantities, the thermal variation of the shear viscosity has been measured by different authors.18,20,21 Dielectric measurements between 198 K and 233 K have been performed in the frequency range 10⫺2 Hz–107 Hz by Legrand17 and repeated by us, and elastic neutron scattering experiments have given the static structure factor, S(Q), as a function of temperature and pressure.22 Raman spectra were recorded by Alba-Simionesco and Krauzman15a who carefully deduced from their measurements the light scattering depolarisation ratio,23 as a function of frequency, between 150 GHz and 5 THz, for temperatures ranging from 175 K 共i.e., below T g ) to 290 K. They found that this ratio did not depend on the frequency and was equal to 0.73⫾0.01. B. Experiments
Meta-toluidine, purchased at Merck, was distilled twice before the light scattering experiments were performed. The latter were made with the help of a coherent Ar⫹ laser emitting at 514.5 nm, with vertical polarization. Backscattering experiments were performed in Paris, in the VV geometry 共polarized scattering兲 between 1 GHz and 30 GHz, and in the VH 共depolarized兲 geometry between 1 GHz and 600 GHz, on an 8 pass, Sandercock-type,24 Tandem Fabry–Pe´rot interferometer already described.25 The sample was placed in a cryostat which ensured a temperature regulation better than 0.25 K. The polarization of the scattered light was analyzed through a combination of a quartz half-wave plate followed by a Glan prism which allowed to keep a constant direction of the electric field of the scattered light inside the spectrometer. This procedure eliminated any possible difference in the transmission of the interferometer for the two different polarizations. At each temperature, a large band depolarized spectrum was obtained by joining together scans with a large overlap and made with different spectral ranges of the Tandem interferometer. Five different spectral ranges were necessary for the Fabry–Pe´rot experiments at every temperature
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TABLE I. Thermal variations of the relaxation time K and of the stretching parameter  K .
FIG. 1. Composite susceptibility curve for depolarized light scattering.
which were chosen to be within 1 K of those used in the Raman experiments.15a Those Raman spectra were finally joined with the Tandem ones, using the same overlap technique as above 共see Fig. 1兲. The final composite spectra cover a three-decades frequency domain 共1 GHz–1 THz兲 and the series of spectra recorded at different temperatures is shown on Fig. 2 in the form of susceptibility spectra:
⬘ 共 兲 ⫽I VH共 兲 共 1⫹n 共 兲兲 ⫺1 ,
共3.3兲
where n( ) is the Bose–Einstein factor. As it can be seen in that figure, the maximum of the susceptibility spectra, which represents the most clear signature of the ␣ relaxation process, is only visible down to 260 K. In order to follow the evolution of this relaxation process at lower temperatures, two more series of experiments were performed in Mainz. One of them consisted of 90 degree confocal, depolarized Fabry–Pe´rot measurements, performed with a free spectral range of 0.75 GHz. Because no attempt was made to match those spectra with those obtained with the Tandem Fabry–Pe´rot instrument, there was no constraint on the choice of the temperatures; those actually used are noted with a subscript c in Table I. At still lower temperatures, a third technique had to be used in order to record this ␣ relaxation process. This consisted in a series of experiments made with a PCS instrument
FIG. 2. Thermal variation of the susceptibility spectra: 共1兲 T⫽300 K, 共2兲 T⫽290 K, 共3兲 T⫽280 K, 共4兲 T⫽266 K, 共5兲 T⫽256 K, 共6兲 T⫽239 K, 共7兲 T⫽233 K, 共8兲 T⫽223 K, 共9兲 T⫽213 K, 共10兲 T⫽193 K.
T 共K兲
K
K 共ns兲
300 290 280 266 245c 240.5c 236c 232c 227c 198p 195p 193p 191p 189p 187p 185p 183p
0.66 0.63 0.61 0.61 0.45 0.42 0.46 0.43 0.48 0.38 0.37 0.31
0.01 0.02 0.04 0.06 0.36 0.5 1.0 1.6 2.8 4.0E6 0.9E7 8.0E7 2.7E8 1.36E9 8.14E9 1.07E11 4.05E11
in Mainz. They were also performed in a 90° scattering geometry, with a vertically polarized incident beam and a depolarized scattered beam. The spectra were recorded, and then analyzed using correlation times ranging from 10⫺4 s to 102 s, at temperatures lower than those used in the Tandem Fabry–Pe´rot experiments. Those temperatures are noted T p in Table I. The polarized spectra were recorded in another series of experiments, on the Paris instrument 共Fig. 3, Table II兲. In that case, we used a single spacing for the interferometer, corresponding either to a 18.75 GHz, or to a 25 GHz free spectral range, depending on temperature, in order to avoid an overlap between the instrument ghosts and the most relevant part of the Brillouin spectrum. A series of corresponding depolarized spectra was recorded on the same instrument at the same time, simply turning the half-wave plate placed in the scattered beam. Analysis of the Brillouin line shape also requires the knowledge of the relaxed sound velocity. This quantity was measured in the Paris Laboratory by
FIG. 3. VV spectra at different temperature and fit 共logarithmic scale兲. Inset: Polarized and depolarized spectra of metatoluidine at 290 K.
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TABLE II. Characteristics of the longitudinal phonons and the parameters of their fit by the ␣ only model 共see text兲. The three last columns represent respectively, in this model, the nonergodicity parameter, f q , the linewidth due to the ␣ relaxation term 关Eq. 共5.5兲兴 and the reliability factor of the fit. T 共K兲
B 共GHz兲
q 共GHz兲
⌬q 共GHz兲
⬁ 共GHz兲
L CD 共ns兲
290 280 266 256 246 236 226 216 206 196 186 176
10.09 10.38 11.53 11.90 12.68 13.58 14.24 14.78 15.66 16.16 16.54 16.62
9.74 10.01 10.38 10.64 10.91 11.18 11.45 11.72 11.99 12.26 12.54 -
7.27 7.49 7.72 8.02 8.31 8.87 9.26 9.77 10.48 10.92 11.12
12.16 12.50 12.93 13.32 13.71 14.27 14.72 15.26 15.93 16.42 16.76
0,015 0,021 0,029 0,046 0,081 0,17 0,42 0,88 1,80 3,3 8,5 -
fq
␥1
2v
0.357 0.358 0.356 0.362 0.367 0.386 0.395 0.409 0.433 0.442 0.440
3,8 3,4 2,5 2,05 1,5 1,1 0,69 0,50 0,37 0,28 0,18 0,05
2.16 2.16 1.39 1.96 1.65 1.62 1.67 1.96 3.10 2.40 2.04 -
Bonello,26 by an ultrasonic technique, between 258 K and 296 K. His measurements can be accurately represented by the linear formula: c 0 共 T 兲 ⫽ 共 2723.5⫺3.85T 兲 m/s,
共3.4兲
where T is measured in Kelvin. IV. ANALYSIS OF THE DEPOLARIZED SPECTRA A. The ␣ relaxation process
There is a general agreement on the fact that the low frequency part of the depolarized spectra of liquids formed of anisotropic molecules are mostly sensitive to the orientational motion of these molecules. We have exemplified this statement in Fig. 4 which compares the susceptibility spectra of meta-toluidine and CCl4, a spherical top molecule, at the same temperature, 293 K. As the rotation of spherical top molecules does not contribute to the scattered intensity, the corresponding intensity and susceptibility spectrum of CCl4 can only originate from pure center-of-mass interaction induced effects. This pure center-of-mass integrated intensity scales with ( ␣ 2 ) 2 , where is the number density and ␣ the molecular polarizability. Figure 4 shows that the metatoluidine spectrum, in which the molecular reorientations
FIG. 4. Relative susceptibilities of pure metatoluidine and pure CCl4 measured at room temperature.
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contribute to the light scattering mechanism, has a much larger intensity, particularly at low frequency, than the CCl4 one. As the ratio 关 ( ␣ 2 ) 2 兴 met / 关 ( ␣ 2 ) 2 兴 CCl4 for metatoluidine and CCl4, respectively, is approximately 2.5, it means that the pure center-of-mass integrated intensity contribution is much smaller than the contributions originating from orientational fluctuations 共including induced ones兲 as we just stated. This rotational dynamics is mostly visible at very low frequency, as a maximum, clearly identified on some of the spectra of Fig. 2. This is the ␣ relaxation part of the susceptibility spectra that we analyze in Sec. IV A, and one of its characteristics is its relaxation time that we shall call R . This analysis was performed through the three series of experiments described in Sec. III: • We first analyzed the low frequency part of the Tandem Fabry–Pe´rot experiments for the four highest temperatures (266 K⭐T⭐300 K). They are the only ones where the corresponding susceptibility spectra 关see Eq. 共3.3兲兴 exhibit a visible maximum at a frequency max , and this maximum is one of the most precise signatures of this relaxation process. Those low frequency parts were fitted to a Cole Davidson function:  CD 1 ⬘共兲⬇Im 1⫺ . 共4.1兲 R 1⫹iCD This expression provides a good fit to the LT of a stretched exponential up to frequencies of the order of 10 times the maximum of the r.h.s. of Eq. 共4.1兲, proR and vided that the corresponding relaxation time CD stretching coefficients  CD are related to the relaxation time KR and stretching coefficient  K of the corresponding stretched exponential through linear relations given by Lindsay and Patterson.27 Excellent fits to those low frequency parts of ⬘ ( ) were obtained for these four temperatures 共see Fig. 8兲 with nearly the same value for  CD , the agreement being very good up to frequencies of the order of 10 max .
冋 冉
冊 册
• As a second step, the correlation functions g (2) (t) recorded in the PCS experiments were fitted with the standard formula: K 2 t ⫹Ibg , 共4.2兲 g共2兲共t兲⫽1⫹A ⌫ exp ⫺ R K where I bg is a background intensity supposed not to depend on time, A and ⌫ being scaling factors. Coefficients KR (T) and  K (T) were deduced from those experiments and then transformed27 into the correspondR (T) and  CD(T). ing CD
冉 冉 冉冒 冊 冊 冊
• Finally we analyzed the confocal FPI data. The free spectral range of this instrument turned out to be too R to allow narrow with respect to the relaxation times CD for an independent measurement of both parameters R CD and  CD . Because of the small, and presumably smooth variation of the latter, we decided to interpolate its value at each temperature with the help of the results obtained in the first two experiments. Using again Eq. R was then determined for each spectrum re共4.1兲, CD corded with the confocal Fabry–Pe´rot instrument, tak-
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B. The  fast relaxation process and the determination of T c through the depolarized spectra
1. Data analysis
FIG. 5. Activation plot of meta-toluidine obtained by light scattering techniques and dielectric measurements: 共䊏兲 K depolarized geometry, 共䊊兲 K PCS, 共䊐兲 K dielectric measurements 共this work兲, 共䉲兲 KVV dielectric measurements 共Ref. 17兲, 共- - -兲 Vogel–Fulcher fit; 共䊉兲 K polarized geometry.
ing into account both the existence of a flat background and of the overlap between the different orders of the spectrum inherent to this instrument. The values resulting from these three series of experiments are given as  K and KR in Table I. The different values of KR are also represented in Fig. 5 together with a Vogel– Fulcher fit to these relaxation times: ⫽ 0 exp(TA /(T⫺T0)) ⫽0 exp(DT0 /(T⫺T0)), where D is the strength index measuring the fragility. This fit correctly describes the results for an activation energy represented by T A ⫽843 K and a Vogel–Fulcher temperature T 0 ⫽163.5 K, giving D⬇5, a value characteristic of a very fragile glass-forming liquid. Finally, we have also represented in Fig. 5 the dielectric relaxation times measured by Legrand18 and by us: these times turn out to be close to the light scattering ones.
Figure 2 shows that, with decreasing temperature, a minimum tends to develop at a frequency min(T) located between max(T) and a high frequency secondary maximum the frequency of which is essentially temperature independent, approximately located at 3 THz and can be considered as the equivalent of the microscopic frequencies ⍀ q of Eq. 共2.2兲. The value of ⬘ ( min) as well as the position of min decrease with decreasing temperature, which are some predictions of MCT for the  relaxation process. We have thus to find out whether this spectral region may be adequately fitted by the formulas given by this theory, Eqs. 共2.6c兲 and 共2.7b兲, in which the amplitude of ⬘ ( min) is proportional to 冑T⫺T c . Following procedures similar to others already used in this case,28,29 we first determined by direct inspection approximate values of min and ⬘ ( min) for all the spectra where this was possible, i.e., for 233 K⬍T⬍290 K, and we fitted the region of the minimum with Eq. 共2.6c兲 共see Fig. 6兲, letting the coefficients a and b be independent free parameters. This technique produced a series of those coefficients. The corresponding a and b coefficients are never too far from the theoretical curve which relates them 关Fig. 6共b兲兴, the variation of b being, as expected, larger than that of a. Nevertheless, for most spectra, the coefficient a is not well defined, the values of b being more accurate, therefore, we fixed b as the mean value of its independent determinations and deduced the corresponding value of a from the relationship between those two coefficients. This gave b⫽0.58, a⫽0.31, and ⫽0.73. The whole set of ⬘ ( ) curves was then fitted again with Eq. 共2.6c兲 as a function of / min using the values of a and b as fixed input parameters. This procedure determined new values of min and ⬘ ( min) at each temperature. Figure 6共c兲 represents ⬘ ( )/ ⬘ ( min) vs / min at different temperatures as well as the fitting curve, Eq. 共2.6c兲. The range over which the dif-
FIG. 6. Comparison of the susceptibility spectrum with MCT predictions: 共a兲 fit of the T⫽266 K susceptibility spectrum by the schematic model: 共 兲 experiment, 共⫹兲 numerical fit 共Ref. 15兲, 共 兲 asymptotic approximation to the  fast correlator; 共b兲 dispersion of the a and b coefficients in the individual ⬘ vs / min for fits of the  fast part of the susceptibility spectra, for the temperatures at which the fit was possible, 共 兲 theoretical curve; 共c兲 ⬘ ( )/ min the same spectra, with the theoretical asymptotic susceptibility spectrum.
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is approximately the case for the linear variation but these curves do not yield a single T c value; we obtain, respectively T c ⫽233 K ( ⬘ ( min)), 228 K( min) and 220 K( K ), the ordering of those temperatures being independent of the exact value of b used for the fits. It may be worthwhile pointing out that most of the data points used in Fig. 7共a兲 correspond to temperatures above T m . As the MCT predictions we test through Eqs. 共4.3兲 are in principle asymptotic predictions, it is not clear that they should hold for such large T/T c values. C. Discussion
1. Validity of the previous analysis
FIG. 7. 共a兲 Power laws for min and min ; 共b兲 power law for max .
ferent curves superpose is slightly larger than two decades. As already pointed out29 for a similar monomolecular glass forming liquid, salol, the agreement between the theoretical curve and the experimental one is quite good at all temperatures for ⬍ min . However, for ⬎ min the frequency domain for which Eq. 共2.6c兲 represents the data decreases with increasing temperatures and the agreement with the theoretical curve is much poorer.
2. Determinations of the critical temperature T c
The results on min and ⬘ ( min) just obtained, as well as the values of K obtained in Sec. IV A, yield three different determinations of T c : Eq. 共2.6c兲:
⬘ 共 min兲 ⬇ 冑T⫺T c ,
共4.3a兲
Eq. 共2.7b兲:
min⬇ 共 T⫺T c 兲 1/2a ,
共4.3b兲
Eq. 共2.5a兲:
K ⬇ 共 T⫺T c 兲 ⫺ 共 1/2a⫹1/2b 兲 .
共4.3c兲 2a min
Figures 7共a兲 and 7共b兲 represent, respectively, together with ⬘ ( min)2 and K⫺2ab/a⫹b vs T. Those three curves should display a linear temperature variation and they should have the same intercept with the abscissa axis at T⫽T c . This
Two points need to be discussed. The first concerns the specificity of the former analysis. Indeed, in the past, MCT analyses of depolarized light scattering spectra have generally been carried out assuming that, at each temperature, the total light scattering spectrum was only related to centers-ofmass induced scattering, which allowed to connect these spectra in a rather direct manner to the density fluctuation susceptibility spectra deduced from the MCT equations. This was done by means of expressions of the correlator valid in the asymptotic T⫺T c →0 limit 共using the whole  correlator4 of the idealized or of the extended MCT theory30兲, or by using more sophisticated MCT formulations such as some schematic MCT models,15 introducing in the latter case parameters which describe some specific features of the system under study. For instance, Fig. 2 shows that in the 500 GHz region, there exists a small spectral feature, called the boson peak, which is more and more visible when the temperature is lowered. This feature was specifically taken into account as a second microscopic frequency by Alba-Simionesco et al.15a in their fit of the total Raman spectrum of meta-toluidine between 150 GHz and 4 THz by a schematic two-correlators model. The agreement between these purely center-of-mass formulations and the experiments was found to be good in a more or less important frequency range for several glass-forming liquids.28–31,15b In the case of the meta-toluidine broadband spectra, one can extend the agreement obtained in Ref. 15a and cover the whole frequency range by using the same technique as in Ref. 15b, i.e., by restricting the set of Eqs. 共2.2兲 to a system of two nonlinear integrodifferential equations. An example of such a fit is given in Fig. 6共a兲 for T⫽266 K. Such fits can be performed at each temperature, and they yield T c ⫽233 K and ⫽0.67, a somewhat smaller value than the ⫽0.73 value obtained through the asymptotic formulas analysis shown in Figs. 6共b兲 and 6共c兲. The problem of such an approach is that its validity is very questionable when the light scattering mechanism is mostly related to orientational fluctuations 共see the first paragraph of Sec. IV A兲. The intrinsic complexity12b of the MCT equations including both the centers-of-mass and the orientational motions is too important to ascertain if schematic models similar to the one used for the construction of Fig. 6共a兲 are still meaningful; it is possible that such models are flexible enough to provide correct numerical fits to the spectra even if the parameters used do not properly represent the physical problem. This is
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has typical values of 2 or 3, the determination of  K may be somewhat biased by that aspect of our fitting technique; this bias could be more important than in the case of two other fragile glass-forming liquids, salol and OTP 共orthoterphenyl兲, also based on phenyl ring substances, which have been recently analyzed within the same MCT scheme29–31 by the same depolarized light scattering techniques. In those ⬘ /min ⬘ are substantially larger, similar two cases where max values of b were obtained, while the values of  K were much larger (  K ⫽0.8 in salol,29 0.8⬍  K ⬍0.95,41 and  K ⫽0.79 30,31 in OTP兲. More work has to be done to elucidate the importance of this point.
2. Dispersion in the T c values FIG. 8. Scaled susceptibility curves 共␣ peak region兲: 共 Debye fit, 共䉱兲 CD fit.
兲 experiment, 共*兲
why we shall consider only the fits of our spectra to the asymptotic formulas and shall rather focus the discussion 共Sec. IV C 2兲 on the possibility of finding a systematic behavior in the low frequency part of the depolarized spectra of several molecular glass-forming liquids. Let us just point out here that, in the case of meta-toluidine, neglecting the second microscopic mode does not prevent analyzing the region of the minimum of the susceptibility spectra with the formulas given in Sec. II. This was already the case for salol29 and such a situation must be contrasted, for instance, with the case of a long chain polymer7 共1-4 cis-trans polybutadiene兲, where the same low frequency region ( ⬍50 GHz) was completely modified by the existence of some 共internal?兲 molecular motions. A second problem is the relationship between the region of the ␣ relaxation and the region of the minimum of ⬘ ( ). The low frequency part of this minimum is usually referred to as the von Schweidler regime;4 as pointed out in this reference, the existence of that regime has little to do with the ␣ relaxation process: the combination of Eqs. 共2.5a兲, 共2.6b兲 and 共2.7兲 shows that one can write:
冉 冊冉 冊 ␣
⌽ q 共 t 兲 ⫽ f cq ⫺h q B
q 0
0
b
t
The approximate value of T c ⬇226 K places metatoluidine in the same category as the two other glass-forming liquids just mentioned. Previous measurements have indeed given: Salol: 29 OTP: 30,31
T m ⫽315 K;T g ⫽218 K;T c ⫽258 K;T c /T g ⫽1.18; 共4.5a兲 T m ⫽329 K;T g ⫽244 K;T c ⫽290 K; T c /T g ⫽1.19.
共4.5b兲
Such a T c /T g value ⬃1.20 is in agreement with an early estimate of this ratio for fragile glass-forming liquids based on a corresponding states analysis of viscosity data,32 and also with a recent depolarized light scattering study of the  fast relaxation of toluene,33 another member of that family. We obtain here a similar ratio, T c /T g ⫽1.21. Note that when those three liquids are studied using the same depolarized light scattering technique, and when their spectra are analyzed by the same method, the distribution of the different values of T c is notably larger 共13 K兲 for meta-toluidine than for the other two liquids 共6 K and 5 K, respectively兲28–30 but the order of the three different temperatures remains the same: T c ( ⬘ ( min))⬎Tc(min) ⬎Tc(K). It may be worth studying if this order exists also in other similar glass-forming liquids.
b
共4.4兲
␣q
q␣
but with so that the r.h.s. of Eq. 共4.4兲 does scale with power b and not with power  K , contrary to the first order expansion of the Kohlrausch function. In the susceptibility spectra, the von Schweidler regime, which describes a slower decay in frequency than the ␣ relaxation process, is thus characterized by an ⫺b law with b⬍  K . In the present case, the relation between b and  K is only marginally satisfied for the temperatures at which a fit of the region of the minimum of ⬘ ( ) is meaningful: we obtain b⫽0.58 and  K ⭓0.61. In fact, as it is apparent in Figs. 6共a兲 and 8, some part of the spectra have been used simultaneously to obtain a fit of the ␣ relaxation process with Eq. 共4.1兲 and of the region of the minimum of ⬘ ( ) through Eq. 共2.6c兲. In the present case where the ratio ⬘ ( max)/⬘(min)
V. ANALYSIS OF THE ISOTROPIC SPECTRA A. General considerations
As mentioned at the end of Sec. III, polarized spectra, I VV( ), were recorded in the back scattering geometry every 10 K between 178 K and 298 K using one single free spectral range. The corresponding depolarized spectra, I VH( ), were measured in exactly the same geometry, at the same temperature 共see Fig. 3兲. Those two spectra are not independent. Indeed, in a liquid composed of optically anisotropic molecules, it is generally found that the spectrum is depolarized in a frequency range going from 50 GHz to a few THz 共the upper limit corresponding to frequencies up to which the molecules can be considered as rigid兲. This means that: I VV共 兲 ⫽4/3I VH共 兲
共5.1兲
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and we mentioned in Sec. III that such a constant depolarization ratio was indeed approximately measured for such frequencies in the Raman study of meta-toluidine.15 This results from the light scattering mechanism in such liquids as well as from the macroscopic isotropic properties of the liquid. The situation changes at lower frequencies: the polarized spectrum contains a second contribution which originates from the propagation of density fluctuations. The latters generate, through the Clausius–Mossotti relation, a variation of the isotropic part of the dielectric tensor. The LT of its time correlation function, I iso( ), appears only in the polarized spectrum and is proportional to the density–density correlation function at the scattering wave vector, a quantity which was discussed in Eq. 共2.9兲. One may obtain the corresponding ⌽ q⬙ ( ) from: A⌽ q⬙ 共 兲 ⬅I iso共 兲 ⫽I VV共 兲 ⫺4/3I VH共 兲 .
共5.2兲
In this section, we analyze ⌽ q⬙ ( ) deduced from Eq. 共5.2兲 and expressed in Sec. II through Eq. 共2.9b兲. Inset of Fig. 3 shows that at 290 K, the depolarized contribution to the polarized spectrum cannot be neglected with respect to the first one, and this is true for all temperatures above 210 K. To be sure that the depolarized spectrum was properly taken into account, we carried out two analyses of the data: the first one was performed on the total I VV spectrum, introducing in the fitting procedure the corresponding I VH spectrum, fitted by a Lorentzian and weighted by a fit parameter, instead of the 4/3 coefficient of Eq. 共5.2兲; the second one was carried out on the I ISO spectrum obtained from a direct subtraction of the I VH spectrum according to Eq. 共5.2兲. Both analyses gave very similar results. As for any other supercooled liquid, the line shape of the Brillouin spectrum, I ISO( ), is temperature dependent. The frequency of the maximum, B , increases continuously with decreasing temperature while the full width at halfmaximum, ⌬ B , passes through a maximum at 275 K, and decreases steeply at lower temperature. The values of B and ⌬ B are shown in Fig. 9 as a function of temperature as well as the frequency max of the maximum of the ␣ peak when it is visible in the Tandem Fabry–Pe´rot range. The metatoluidine case is very similar to those of salol34 and OTP:31 the maximum width of the Brillouin line occurs at a temperature at which its frequency is much larger than the frequency of the ␣ peak seen in the depolarised spectrum. This contrasts with the case of the fragile ionic glass former CKN35 where these two frequencies are almost equal, the rotational dynamics of the NO⫺ 3 ions giving possibly little contribution to the depolarized backscattering spectrum. In order to take into account the possible existence of an unresolved background, I iso( ) was fitted with:
冉
I iso共 兲 ⫽F 共 兲 丢 I bg⫹
冊
I0 1 Im 2 , 共5.3兲 ⫺ q2 ⫺ m ˜ q共 兲
where F( ) is the resolution function of the instrument and 丢 represents a convolution product. The relaxed sound velocity, c 0 , entering Eq. 共5.3兲 through q ⫽c 0 q, was obtained from Eq. 共3.4兲 at each temperature by extrapolating down to the glass transition temperature T g the linear thermal variation measured by ultrasonics between 258 K and 296 K;26
FIG. 9. Thermal variations of the frequency and width of the Brillouin peak, and of the depolarized ␣ peak frequency, when visible. 共䊏兲 B , Brillouin peak frequency, 共䊉兲 ⌬ B , Brillouin width, 共䉱兲 max , DLS ␣-peak frequency.
the background level, I bg , and the Brillouin spectrum intensity, I 0 , were used as adjustable parameters. Two possibili˜ q ( ). They both rely on a splitting ties were explored for m of this function into a sum of two terms. The first is represented by a Cole Davidson function:
冋
m q1 共 兲 ⫽⌬ 2 q 2 1⫺
1 L  CD 兲 共 1⫹i CD
册
.
共5.4兲
This expression depends on three parameters, an amplitude, L , and a stretching coefficient,  CD . ⌬, a relaxation time, CD The first was taken as a fit parameter, while  CD was assumed to be equal, at each temperature, to the value obtained in Sec. IV for the stretching coefficient of the rotational ␣ relaxation process. Two different hypotheses were used for L CD and we may already point out that, depending whether L CD will be of the order of, or much shorter than 102 s at T g , Eq. 共5.4兲 will represent either an ␣ relaxation process or another 共 relaxation兲 process which would persist in the supercooled liquid phase. The two different forms of the sec˜ q ( ) will now be discussed. ond contribution to m B. Search for an ‘‘␣ only’’ relaxation process
1. Description of the model
This model has been frequently used in order to analyze spectra related to the propagation of acoustic waves in supercooled liquids and to compare the results with the predictions of MCT, either in the frequency34,35 or in the time36,37 space. It consists in supposing that Eq. 共5.4兲 describes an ␣ relax˜ q ( ) corresponds ation process while the second part of m to so fast a relaxation process that it can be represented by a delta function in time, or by an ␥ 0 (T) contribution to m ˜ q ( ). The information contained in our spectra is not rich enough to allow for a determination of the thermal variation of ␥ 0 . We thus assumed ␥ 0 to be temperature independent and obtained its value from a fit of the Brillouin spectrum measured at 178 K, i.e., at a temperature below T g . L is supposed to be long Indeed, at that temperature, CD
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FIG. 10. Thermal variation of the different sound velocities: 共䉱兲 infinite frequency c ⬁ , 共䊉兲 apparent c B , 共 兲 zero frequency c 0 .
enough for m ˜ q1 ( ) to reduce to its first term and to give no contribution to the Brillouin linewidth. This yielded ␥ 0 ⫽0.175 GHz. 2. Results for the ‘‘␣ only’’ relaxation model
Good fits were obtained at each temperature and typical examples of the quality of the fit, corresponding to T ⫽290 K, 256 K, 236 K, 216 K, and 192 K, are given in L , as well as Fig. 3. The corresponding values of ⌬q and CD q those of B , q ⫽ q /2 , and ⬁ ⫽c ⬁ /2 are given in Table II, while Fig. 10 shows c 0 , c ⬁ , and c B ⫽2 B /q as a function of temperature. c B (T) varies from values close to c ⬁ (T) near T g , to values close to c 0 (T) at the highest temperature (T⫽290 K), as it has been found in many other systems. L is represented in its Kohlrausch Finally, the variation of CD L form, K , in Fig. 5. This figure shows a quite typical Arrhenius behavior with an activation energy E a ⫽k B T a with T a ⬇650 K. Such a thermal behavior, which has also been found in salol34 for instance, calls for two comments: 共a兲
共b兲
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The procedure used to determine ␥ 0 implied that the L to the linewidth would be neglicontribution of CD gible at 178 K while this is clearly not the case. To evaluate how inaccurate that assumption was, we also report, in the last but one column of Table II, the value ˜ q1 ( ) for ⫽ B , supposing of the imaginary part of m L CD to be large enough for approximating this imaginary part by ␥ 1 , with ␥ 1 given by: CD 2 sin 2 共5.5兲 ␥1⫽⌬2q2 L  CD B 共 B CD兲 an approximation valid for T⬍266 K. This yields a value of ␥ 1 ⫽0.053 GHz at 176 K, i.e., nearly 1/3 of the value of ␥ 0 . The former determination of ␥ 0 was thus partly inconsistent. A more important point is that this Arrhenius behavior ˜ q1 ( ) does not represent an ␣ relaxation implies that m process but rather a  process, similar in its thermal behavior to a Johari–Goldstein process which would be already detected in the supercooled liquid. As the determination of the nonergodicity parameter, f q ,
FIG. 11. Nonergodicity parameter: second model:  fast relaxation included, 共䊉兲 r⫽ R / L ⫽1, 共䉱兲 r⫽2, 共䉲兲 r⫽3, 共 兲 fits with Eq. 共2.12b兲. Inset: first model; 共䊏兲 CD function only, 共 兲 fit with Eq. 共2.12b兲.
through Eq. 共2.11兲 implies, in its derivation, that m ˜ q1 ( ) is an ␣ relaxation process, the value of f q (T) obtained through this equation is very dubious and the fit of this curve by Eq. 共2.12b兲 has no reason to yield a reliable value of T c . Neither the fact that such a fit was possible 共see Fig. 11兲, nor the proximity of the corresponding value of T c ⫽237⫾2 K with the different values of T c determined in Sec. IV can be taken as tests of the validity of such an approach.
C. Search for the possible contribution of a ‘‘-fast’’ relaxation process
1. The model
In order to circumvent the obvious inconsistencies obtained in Sec. V B, we have tried to exploit more systematically the possibilities of MCT. In the case of CKN,35 the maximum of the Brillouin linewidth was obtained at the temperature at which B coincided with the maximum of the ␣ peak of the susceptibility spectrum detected in the depolarized geometry. This coincidence led Li et al.35 to propose that this susceptibility spectrum would be the imaginary part ˜ q ( ). Performing a Kramers–Kronig transformation of m of that imaginary part to obtain its real part, they thus deter˜ q ( ), which they injected in Eq. mined the complete m 共5.3兲. Quite a good fit of the whole Brillouin spectrum was obtained in that case. Unfortunately, the present situation is not the same but is similar to the case of propylene carbonate;30 we observe the same type of difference between the temperature variations of the width of the Brillouin peak and of the maximum of the depolarized susceptibility spectrum. In such a situation, Du et al.30 proposed a different strategy in order to mimic a relaxation function in partial agreement with the MCT predictions. We have repeated their method in the present Sec. V C. We admitted that Eq. 共5.3兲 represents the full ␣ relaxation part of m ˜ q ( ); in order to insure a thermal behavior L compatible with that hypothesis, we imposed this lonof CD
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gitudinal relaxation time to be proportional to the rotational R , relaxation time written under its Cole Davidson form, CD R and we deduced CD from the measurements performed in Sec. IV interpolated through a Vogel–Fulcher formula. ˜ q ( ) to be written as: We then assumed m
冉冉
m ˜ q共 兲 ⫽ m ˜ 1q 共 兲 ⫹ ␥ a 共 T 兲 tg
冊 冊
共 1⫺a 兲 ⫹i a , 2 共5.6兲
where a is the MCT critical exponent determined in Sec. IV (a⫽0.31): the second term of Eq. 共5.6兲 represents the high frequency part of the  fast relaxation process of MCT, i.e., the Fourier transform of the critical decay of the -fast relaxation process taken under its asymptotic form 关Eq. 共II.6a兲兴. 共Other formulations for the high frequency part have been proposed by Loheider et al.38 and used for several systems.39兲 We have thus assumed in fact: • that the von Schweidler part of the same  fast relaxation process was properly taken into account by the ˜ q1 ( ), in agreement with the high frequency part of m remark made in Sec. IV C; • that the main role of the second term in Eq. 共5.6兲 is to give an additional contribution to the linewidth of the Brillouin peak through its imaginary part; yet the role of the corresponding real part plays a role and slightly shift the position of the Brillouin peak.42 Fits were thus performed using ⌬(T) and ␥ a (T) as fitR L / CD to some given ting parameters, fixing the ratio r⫽ CD values; a series of fits were thus performed with r⫽1, 2, 3, and 5, respectively.
2. Discussion of the approximate ‘‘ fast’’ model
Equally good fits of the whole series of spectra 共from 290 K to 186 K兲 were obtained for all the values of r listed above. Furthermore, the values of ⌬ r (T) deduced from those fits led to corresponding nonergodicity parameters which all produced rather acceptable thermal behavior, as is shown in Fig. 11. In particular, the r⫽2 curve is of the same quality as the curve related to the ‘‘␣ only’’ model, while the r⫽3 curve is reminiscent of the result obtained for propylene carbonate by Du et al.30 Nevertheless, Fig. 11 also shows that the position of the square root singularity which indicates the value of T c increases with the value of r, all of them being higher than the value obtained in the ‘‘␣ only’’ model. Fixing any reasonable value for r thus increases the dispersion of T c with respect to the values obtained in the study of the depolarized spectra. Another aspect of the problem is revealed by Fig. 12 which represents the thermal variation of ␥ a for the various values of r. One sees that: • ␥ a (T) is practically independent of r, i.e., of the relaxation time, below 240 K; in this thermal range, the  fast process is responsible of the entire linewidth of the mode; L 关through m 1q ( )], • Above 240 K, both ␥ a (T) and CD contribute to this linewidth. But, whatever is the value
FIG. 12. Thermal variation of ␥ a (T) for different values of r⫽ R / L : 共䊏兲 r⫽1, 共䊉兲 r⫽2, 共䉱兲 r⫽3.
of r, the maximum of the Brillouin linewidth around 270 K cannot be accounted for by a monotonous variation of ␥ a (T): under the form used in Eq. 共5.6兲, the  fast contribution has to decrease, at least above that temperature, in order to explain this behavior. This would force us to link the origin of the linewidth maximum to a nonmonotonous thermal behavior of the  fast contribution. In the frame of MCT, this thermal behavior is unexpected because the memory function should itself be connected to correlation functions which should obey Eq. 共2.6c兲 at least in a limited thermal range around T c . When one identifies Eqs. 共2.6c兲, 共2.7b兲 with Eq. 共5.6兲, one sees easily that ␥ a (T) should be constant, at least around T c , which is obviously not the result shown in Fig. 12. Even if the memory function which couples to the longitudinal modes does not follow the MCT predictions up to as high temperatures as the correlation function detected in the depolarized spectrum, such a change from a predicted constant behavior to a curve with a positive and then negative first derivative and an always negative second derivative is very unlikely. In view of the results described in the two previous paragraphs, we must conclude that Eq. 共5.6兲 is still a poor representation of the real memory function in the case of the longitudinal propagative modes in meta-toluidine. D. General discussion
The preceding two attempts of fitting the Brillouin spectra recorded over a 1–27 GHz range, between 298 K and 186 K, and obtained through a careful subtraction of the depolarized contribution to the polarized spectra, lead to an ambiguous answer. Although these spectra cannot be characterized only by their linewidth, at least above 240 K, their entire line shape can be described by many sets of two strongly correlated parameters 共see Fig. 12兲 as it has been shown through the two models tested in the present section. While the first model 共Sec. V B兲 is certainly inconsistent 共it does not include properly the -relaxation function兲, the second one remains largely unsatisfactory. In fact the pri-
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mary purpose of the study performed in this section was to find out whether these spectra could be used as tests of some aspects of MCT. Our results have been negative in the sense that we found no ‘‘single channel’’ memory function which fitted them and was compatible with that theory. From a theoretical point of view, this is not really surprising. Some of us40 have recently shown, with the help of a phenomenological theory which is expected to mimic some important ˜ q ( ) to be aspects of MCT, that the memory function m used in the present section should be a linear combination of a bulk deformation memory function and of a shear deformation memory function. Equations of the same type as Eq. 共5.4兲 are thus not expected to hold in the case where the relaxation times of these two functions are unequal, which may very well be the case. Yet, even if one could expect to obtain the value of the shear relaxation times from the analysis of the transverse phonon spectra,40 it is unclear that this will be sufficient to break down the correlation between the descriptions of the bulk ␣-relaxation process and of the corresponding  fast process found in Sec. V C. It is very likely that one will have to use the information which can be derived from stimulated Brillouin scattering 共which measures decays in the time regime, and is mostly sensitive for long relaxation times, i.e., for the low temperature situation兲, to obtain a better understanding of the role of this  process. VI. CONCLUSION
We have recorded the low frequency depolarized and isotropic light scattering spectra of meta-toluidine in a back scattering geometry from 298 K to 183 K and compared them with the predictions of the idealized MCT. The depolarized spectra mostly probe the orientational dynamics of the molecules through its coupling to the total light scattering mechanism, i.e., including interaction induced effects. The work of Schilling et al.12 indicates that the idealized MCT predictions should remain essentially valid for this orientational dynamics. We found that a reasonable agreement could be obtained in the form of unique set of a and b parameters describing approximately the corresponding dynamics at various temperatures. Yet, the critical temperature T c which can be deduced from three distinct features of these spectra do not really coincide: they exhibit a dispersion ⌬T ⫽⫾7 K with respect to a mean value T c ⫽226 K. We have also noted that the ordering of the corresponding three values is the same as already found in two other simple molecular glass-forming liquids also containing benzene rings as an important ingredient. The isotropic spectrum 共Brillouin spectrum兲 deduced from the subtraction of the depolarised spectra from the polarized ones could not be fitted using simple ingredients compatible with MCT. It must be emphasized that such an exercise is much more difficult than the preceding one because it consists in guessing which memory function compatible with MCT should be used to fit the spectra while the latter contain little information. We have pointed out that the memory function we used in our analyses was too crude an approximation to the realistic one. Important efforts combining, at least, other light scattering techniques 共analyses of transverse propagative waves,40 stimulated Brillouin scatter-
ing兲 and presumably realistic MD simulations may appear as the only way of obtaining more information from these Brillouin spectra.
ACKNOWLEDGMENTS
This work has benefited from the Procope Project No. 94034 and of a MPI-CNRS agreement which allowed a fruitful collaboration between the Mainz and the Paris groups. We thank B. Bonello, from LMDH, who performed the ultrasonic measurements quoted in Sec. III. We also thank M. F. Lautier for the preparation of the samples, C. Caray and J. P. Franc¸ois for their experimental help. We finally thank V. Krakoviak for allowing us to present the results of his fit of the T⫽266 K spectrum with a schematic model.
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Liquid–glass transition of meta-toluidine 38
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