Light scattering and calorimetric study of the thermal diffusivity of glycerol, liquid and glass

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Light scattering and calorimetric study of the thermal diffusivity of glycerol, liquid and glass

C. Allain, P. Lallemand

To cite this version:

C. Allain, P. Lallemand. Light scattering and calorimetric study of the thermal diffusivity of glycerol, liquid and glass. Journal de Physique, 1979, 40 (7), pp.693-700. �10.1051/jphys:01979004007069300�.

�jpa-00209153�

Light scattering ^and calorimetric study of the thermal

diffusivity ^of glycerol, liquid ^and glass

C. Allain and P. Lallemand

Laboratoire de Spectroscopie Hertzienne de l’E.N.S., 24, ^rue Lhomond, 75231 Paris Cedex 05, France

(Reçu ^{le 28 décembre} 1978, accepté ^{le 12 mars} 1979)

Résumé.

On présente les résultats d’expériences ^{de diffusion} de la lumière en vue de déterminer la diffusivité

thermique ^{de la} glycérine ^dans diverses situations. Si le temps de relaxation interne, qui dépend ^de T, ^est long ^ou

court devant le temps ^{de diffusion de la} chaleur, qui dépend ^de q-2, on ^dira qu’on a un verre ou un liquide. ^On

décrit d’abord des mesures de diffusion spontanée de la lumière. On trouve une très faible dispersion ^du temps ^de relaxation de la raie Mountain, et une très faible intensité pour la composante Rayleigh ^{dans le verre. On détermine}

ensuite la dispersion ^du temps de relaxation Rayleigh à l’aide de la diffusion Rayleigh forcée. On présente ^les

résultats d’une expérience ^de calorimétrie. On compare ^ces divers résultats à des prédictions théoriques. ^{On trouve}

que le modèle viscoélastique généralisé permet ^{d’obtenir une} représentation cohérente des mesures acoustiques, thermiques ^et celles obtenues par diffusion de la lumière de part ^et d’autre de la transition liquide-verre ^{dans la} glycérine.

Abstract.

Light scattering experiments have been used to determine the thermal diffusivity ^of glycerol ^{in a} variety ^of situations, changing ^the temperature ^{T and the} wave-vector q. Depending ^on whether the internal relaxation time (depending ^on T) is very short ^or very long compared ^{to the heat diffusion time} (depending ^on q-2)

we say that ^{we are} studying ^{either a} liquid ^{or a} glass. Spontaneous light scattering experiments ^{are described first.}

They give essentially information concerning ^the dispersion of the Mountain relaxation time : the ratio of its values in the liquid ^{and the} glass. They also show that the intensity ^{of the} Rayleigh component ^is negligible ⁱⁿ

the glass. ^Forced Rayleigh scattering experiments ^are then related. They ^{allow a} determination of the dispersion

of the Rayleigh relaxation time. Using ^a calorimetric technique, ^the dispersion of the thermal diffusivity ^{is then}

found to be the inverse of that of the Rayleigh relaxation time. These results concerning relaxation times and

amplitudes ^{of the} Rayleigh ^{and Mountain} components ^{are then} compared ^to theoretical models. It is shown that the generalized viscoelastic model gives ^a consistent description ^of acoustic, thermal and light scattering ^measu-

rements in glycerol.

Classification Physics ^Abstracts 62.60

44.50

In this _paper, we shall report experimental ^data

obtained by light scattering techniques concerning

the thermal properties ^of glycerol ^on either side of the glass transition, ^defined by the condition that the heat diffusion time is of the order of the structural relaxation time. The data, together ^{with the results}

of a calorimetric experiment, ^{will then be} analysed

in terms of phenomenological ^{models that have been}

discussed in detail in ^a _paper referred to as 1 [1].

It was shown there, ^that much information can be obtained from the values of the decay ^{rates and of the}

relative amplitudes ^{of the} Rayleigh ^{and Mountain} components ^{observed on} either side of the glass

transition in spontaneous ^{or forced} Rayleigh light scattering experiments.

This transition, ^{that we} shall often call the Rayleigh-

Mountain coupling ^condition (by analogy ^{with the}

Brillouin-Mountain coupling ^which corresponds ^to

ultrasonic dispersion) ^takes place ^{when the} experi-

mental situation is such that the heat diffusion time TH is of the order of the internal relaxation time _zo.

As Tu

I/Kq2, where ^K is the thermal diffusivity ^and

q the wave-vector defined by ^the geometry ^{of the} experiment ^{and as} Io varies with the temperature proportionally ^{to the} viscosity [2], ^the coupling ^can

be observed by varying q at constant T or by varying ^T

at constant q. To indicate typical conditions for

glycerol, ^{we mention that for a} spatial wavelength (1

² n/q) of the order of _{0.5 pm} we have _{ru - ’to} at 0 °C, ^{and at - 60 °C for a} wavelength ^of 100 pm.

The variety ^of conditions in which the coupling ^can

take place ^{is a clear} indication of the relaxational character of the liquid glass transition as revealed by changes in the thermal properties.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004007069300

694

We shall describe successively the various types ^of experiments ^{which we have} performed [3] : spon- taneous light scattering, ^forced Rayleigh scattering

and calorimetry. ^{We shall} give results obtained outside the coupling region, ^{that is for} ’rH > 10,ro ^and

’tH 10-1 To- Results obtained in the coupling region using the forced Rayleigh scattering (which yields ^the

best experimental data) ^{will be} presented ^elsewhere.

1. Light scattering experiments.

^A variety ^of spontaneous light scattering spectra ^{have been} obtained using ^different experimental methods. Some of this work has already ^been published, ^{so in some}

cases we shall just recall the main features of the results to give ^as complete ^a picture ^of glycerol ^as possible,

to be able to make a choice _{among the} models _pre- sented in I.

A first type ^of experiment [4] ^was performed using

a 90° scattering angle geometry ^and spectral analysis

of the scattered light ^{with a} Fabry-Perot. ^{We recall}

the ess,ential result that was obtained in this _way.

We found that when lowering ^the temperature, ^the spectrum below - 10°C (down ^{to -} 80°C) ^exhibits only ^one very ^narrow spectral component (apart

from the Brillouin lines). ^{We thus} concluded that the

intensity ^{of the} Rayleigh line, ^{in the} glass state, ^is very

small, ^{so that} (IR/Im). - ^{0. Here, we use the same}

notation as in I to indicate in which condition a

quantity ^{has been} determined. If the time scale of the

experiment ^is very long compared toro, ^{as in a} liquid,

we use the index 0, whereas if it is very short compared

to To ^as in a glass, ^{we use the index oo.}

In addition to this result, using ^the depolarized light scattering (of ^{time constant} ’tdep) ^{as a way to determine} accurately ^the internal relaxation time To, it ^was possible ^{to determine} [5] ^{the time constant of the}

Mountain line on either side of the coupling. Working

at fixed scattering angle (90°), ^and comparing ^the polarized ^and depolarized spectra for various values of the temperature, ^{we determine} L&/Ldep ^{for T} > 30 °C and ’rm- /.rdep ^{for T O°C. Assuming} that the reorien- tational motions of glycerol ^{molecules are not} coupled

to the long wavelength hydrodynamic excitations of the fluid (which ^is plausible ^as this fluid does not exhibit _any structure in the depolarized spectrum ^such

as that attributed to the coupling between molecular

alignment ^and vorticity) ^{we concluded from the} experimental data that iM/iM

^1.0 ± ^0.2.

As this important ^{result was obtained} quite ^indi- rectly ^and by changing ^the temperature ^{of the} fluid,

we have performed ^{a more direct} measurement at fixed temperature, ^but changing ^the wave-vector.

The set-up is shown in figure ^{1. It involves} usual light scattering techniques, ^{so that we shall} give ^details only concerning ^the peculiarities ^{of the} set-up. ^As

we want to make measurements at fixed temperature and different wave-vectors, ^{it was} found convenient to make recordings simultaneously ^{for a} large q (900 scattering) ^{and a} small g (small scattering angle 0).

Fig. ^1.

^Schematic diagram ^{of the} experimental set-up ^{used to} record the correlation function of the spontaneously ^scattered light simultaneously ^{for near} forward and 90° scattering.

This allowed us to exclude _any difficulty ^{due to} possible temperature drift and to work on a single sample, ^so avoiding any change ^{in the} properties ^of glycerol ^from sample ^to sample ^{due to} variation of the water content.

The temperature ^{was chosen} (T ~- 40°C) ^such that,ro ^{is not too small due to} limitations in the speed

of the correlators that were available and ^to signal/

noise considerations that were quite critical, ^as glycerol ^does not scatter much light. ^The heat diffusion time _TH for 900 scattering ^{was short} enough (- 60 ns)

to satisfy ^{the condition} r, » TH. Now, ^to satisfy

the other condition Io « TH for small angle scattering,

we just ^{needed to take a small} angle 0, ^as

(n being the index of refraction of glycerol ^and kL

the wave-vector of the input ^laser beam). ^In practice ⁰

could not be taken too small due to parasitic light

scattered by ^the optics (essentially by the windows of the cryostat). ^{As we did not use an elaborate} set-up such that described by ^Lastovska [6], ^{and as our laser} intensity ^{was not well} stabilized, ^{we had to use 0 - 10.}

This is because we had to keep ^{the amount} of elastically

scattered light ^{to a small} enough ^{level so that the} component ^{of the} photoelectric ^{current due to the} beating between scattered and laser light ^was larger

than that due to the intensity noise of the laser. In

addition, ^{it tums} out that ^{it is} quite ^{difficult to} prepare

a dust and bubble free glycerol sample, ^{so that it}

would have been out of reach to prepare ^a very large sample compatible ^{with a wide beam} experiment ^such

as that of Lastovska.

Very significant improvements ^{were obtained} through ^{the use of} very thick optical windows for the cryostat, schematically ^{shown in} figure ^{2. The} glass cylinders ^{were chosen to have an} index of refraction close enough ^{to that of} glycerol ^{so that there was}

almost no stray scattering at ^the glass-glycerol ^inter-

faces. We verified by making measurements of the

Mountain line scattered at 900 at various places ^{in the}

cell, that the heat leak due to the windows did not

Fig. ^2.

Details of the cryostat ^{used for near} forward spontaneous light scattering experiments.

produce ^severe temperature gradients ^{inside the cell.}

(We ^{recall that} around - 40 °C a 5 °C change ^leads

to a variation of the intemal relaxation time by ^a

factor 10.) With suitable temperature ^control devices,

we were thus able to perform ^{near forward} scattering experiments ^at temperatures ^{of the order of - 40} OC, fluctuating by less than ± 10-2 °C over several hours.

For 90° scattering, ^{we used the same} equipment ^as previously [2]. ^{Our first} problem ^{was to find out}

whether these experiments ^{were disturbed} by stray elastically ^scattered light. ^{We thus} performed ^{a series}

of experiments ^on the Mountain line at lower tempe-

ratures so that both for near forward (10) ^{and 90°}

scattering, ^{the heat} diffusion time _Tu was much shorter than _To. By ^careful analysis ^{of the} data, ^we established with great confidence that the spectra ^{recorded at 90°}

were of the homodyne type, ^{as we verified that the} corresponding correlation function

was proportional ^{to the} square of {G1o(t) - G1o(oo)}

which was of the heterodyne type.

We also looked for the best conditions to obtain

a heterodyne detection in the forward direction.

In particular ^{we made a} theoretical analysis ^{of the}

influence of ^a finite amount of homodyne signal ⁱⁿ

addition to the heterodyne signal. ^This problem ^is

well treated in the case of a simple exponentially decaying correlation function [7]. However, here,

our correlation functions are by ^{no means} exponential,

due to the existence of a very broad distribution of relaxation times. We thus considered the following

correlation function :

where gl(t) ^and g2(t) ^{are the} normalized first and second order correlation functions of the scattered

light, ^and ILO and Is > ^{the mean} intensities of the local oscillator and scattered light respectively. Taking

for gl(t) ^and g2(t) expressions calculated from a Cole- Davidson distribution of relaxation times, ^with 03B2

0.5, ^we found that we have to take ILO/ Is > ²⁰

to get ^a determination of _TcD with an accuracy better

than 1 %, Tcd being ^the longest time involved in the Cole-Davidson distribution. We also looked for the influence of the small amount of depolarized light

which is always present ^{and that was not considered} when analysing previous experiments [2]. ^{In fact it}

seems that due to the small depolarization ^{ratio of} Rayleigh scattering ⁱⁿ glycerol (~ 0.1), ^the presence of depolarized light ^could only ^{lead to a} very small

error ( ¹ %) ^{in the} determination of the Cole- Davidson parameter fi.

We then recorded a series of correlation functions with the following conditions T

40°C, 0

^{1 °.}

As glycerol ^{exhibits a} very broad distribution of relaxation times, ^{it was} necessary ^to determine the

experimental correlation function over a very wide time domain. This was done by combining ^several recordings, ^{as our} correlators give information for

only ¹⁰⁰ equally spaced ^{values of the time.}

We show in figure ^{3 a} correlation function recorded at 90° with a clipped digital correlator. It was analysed

as by Allain-Demoulin et al. [2]. Setting p

^{0.4 we}

obtained TS)

12.7 us, where ^we put ^{the index o0}

Fig. ^3.

Homodyne correlation function of the light ^scattered by glycerol ^{at - 40} oC, ⁰

900 and fit to a Mountain line using ^a

Cole-Davidson function.

as Th~ 60 ns LM. Note that in the case of a Cole- Davidson distribution of relaxation times, ^{the mean} relaxation time, ^{that we can call} Tm, is equal ^to 03B2TCD,

if _TCD is the longest relaxation time of the distribution.

We show in figure ^{4 a} correlation function recorded at 0

1.09° for the same sample ^{and the same tem-} perature ^{as for} figure ^{3. To} analyse ^this recording,

we tried to fit it to

where f(T) ^{is a} Cole-Davidson distribution with

03B2

^{0.4 and} 1:R the Rayleigh relaxation time. To increase the _accuracy in the determination of _1:CD, we

set the value of _TR from known values of the thermal

diffusivity ^{and of the} wave-vector. The thermal

diffusivity ^{was derived} by extrapolating ^{the results}

696

Fig. ^4.

Heterodyne correlation function of the light ^scattered by glycerol ^{at - 40 °C ; 0}

1.09° and fit to a sum of a Rayleigh

line and a Mountain line using ^a Cole-Davidson function.

of a series of measurements performed ^{at various} temperatures (20 OC, ⁰ OC, - ²⁵ oC, - ³⁵ OC) using

the same set-up. ^{For these} temperatures ’t’cD is so short that the signal observed involves only ^the Rayleigh

line. For the conditions in which figure ^{4 was} recorded,

ÏR

133 us. We could thus adjust TCD 0 using ^{a least}

square fit procedure. ^{We found} ’t’go

11.3 us and

(IR/IM)o

^{0.26 for the ratio of the} intensities. As in these experiments Tp/ïM ^is significantly larger ^than 10,

we estimated that the data had been collected beyond

the coupling region, ^{so that we could use the index 0 to}

indicate the conditions in which these results were

obtained.

A series of such experiments ^were performed, ^to

determine TCÎ/TCOD ^{for the same} sample, ^{at the same} temperature (so ^{that we do not need to know the}

exact value of the temperature). ^The average value of this ratio, ^{which we set to be} equal ^{to the ratio of the}

Mountain times (as fi ^{is the same for the two measu-} rements) ^{leads to}

The rather large ^{error bar comes from the} dispersion

of the measurements and takes into account possible

uncertainties due to non perfect homodyne ^{or hete-} rodyne ^detection.

As discussed in I, ^some models lead to a value close to 2 for the ratio ’tM /7:&, ^{so that we} also show in figure ⁴

the best fit obtained by setting TO - zcDI2

^5.7 ils.

The corresponding ^curve represents poorly ^the experi-

mental correlation function. It is thus obvious that such a value of ’tM/’tMo. ^is unacceptable. This work thus confirms the results indicated above conceming ^the

small difference of the Mountain relaxation times on

either side of the Rayleigh-Mountain coupling. ^The implications of this result will be discussed later.

We also made several recordings ^{of the} experimental

correlation function at small scattering angles, ^for

lower temperatures. ^{All the} recordings could be fitted very well using just ^a Mountain line. This means

that the intensity ^{of the} Rayleigh component ^becomes

very small in the glass ^state. This confirms the result that we had obtained using ^a Fabry-Perot ^interfero-

meter :

Apart from the small value of this intensity ratio, these measurements showed that spontaneous light scattering ^{does not seem to be} applicable ^{to determine}

the thermal diffusivity ^of glycerol ^{in the} glass ^state.

To overcome this difficulty ^{we have} performed ^forced Rayleigh scattering experiments which ^{allowed us to}

obtain complementary ^results.

2. Forced Rayleigh scattering.

^The experimental set-up is similar to that used in the early ^{work on forced} Rayleigh scattering [8]. ^{It involves two lasers} operated

at different wavelengths Âp ^and Âs, ^{such that one beam}

can be absorbed to heat-up ^{the medium, whereas}

the other _one, used to monitor index changes, ^{is such}

that the medium is transparent ^at Âs. We ^{used an} argon ion laser, ^whose output ^{beam is} mechanically chopped

to give pulses ^{as short as 20} ps, with a duty cycle ^of 1/2 ^{000. It is then} split ^{into two beams that interfere}

inside the sample. ^{A real} image of the interference pattern ^{can be made on a screen to} determine the

fringe spacing ¹

² n/q. ^{In order to} get ^sufficient heating ^{of the} glycerol, ^{we added a small} quantity

of iodine such that the absorption coefficient at the argon ion wavelength ^was of the order of 1 cm-1.

We shall show later that this does not affect signifi- cantly the thermal properties ^of glycerol. ^The resulting phase grating is detected by using ^{a He-Ne} laser whose output is diffracted by ^the grating. ^The amplitude ^of

the diffracted light ^{was small} enough ^{so that the}

detection is of the heterodyne type. ^The local oscillator is provided by stray light elastically ^scattered by ^the

windows of the cryostat, ^or by ^{some dust} particles

inside the sample. ^{It is worth} mentioning ^{that the} temperature fluctuations induced by ^the heating ^beam,

of the order of 10 - 3 to 10 - 2 °C, ^are very large compar- ed to those due to thermal agitation. ^{As a result the} signals ^are quite large ^so that there is no need to be as

careful as for spontaneous light scattering experiments

to get rid of bubbles or dust particles. ^{In addition,} glycerol being very viscous at low temperature, ^there is hardly any motion of the defects of the sample ^so

that there is no fluctuation in the relative phase ^{of the}

diffracted light ^{and local} oscillator, provided ^{the device}

used to generate the interference pattern ^{at the blue} frequency is stable. One can then sum in a multi- channel analyser ^the signal ^over many successive

cycles, usually ^a few hundred to a few thousand.

The MCA that we used includes 1 024 channels, ^with

a smallest time increment of 10 gs. In ^{some cases} it was operated ^{with a variable} clock, ^{so that we could}

record the early parts ^{of the} heat-decay signal ^with

more detail than its later parts. ^The output S(t) ^{of the}

MCA was recorded on paper tape ^{and with a} strip-

chart recorder either as S(t) ^{or as} Log (S(t) - A ),

where A was adjusted manually.

For each temperature ^{we made a series of measu-}

rements for various values of the interfringe ^{1. We}

show in figure ⁵ the results of such measurements far from the coupling region, plotting Ti/2 ^{vs. l. When} using ^the largest ^{values of} 1, ^{the number of} fringes ⁱⁿ

the heating ^{beam was} quite ^{small. To} decide whether this could lead to difficulties [9], ^{we made a} theoretical

analysis ^{of the} possible ^errors from this. Taking ^a sinusoidally modulated Gaussian beam, ^{we found}

that provided ^{there were more than 5} fringes ^within

the waist of the beam, ^the distortions could be

neglected ^{in view of the usual} signal ^to noise ratio accessible in ^our experiments.

We see in figure ^{5 that we can} determine the thermal

diffusivity ^with very good accuracy. The value of 1.03 ^x 10-3 cm2/s ^{is in} good agreement ^{with that} which we determined by spontaneous light scattering, using ^{either a} Fabry-Perot interferometer ^or light beating techniques.

Fig. ^5.

Square ^{root of the} Rayleigh relaxation time, ^vs. interfringe spacing ^obtained by ^forced Rayleigh scattering : ^{+ at 20} °C ; 0 at - 74 oC ; D derived from Fabry-Pérot analysis ^{at 20°C and A}

derived from near forward spontaneous scattering ^{at 20 OC.}

We show in figure ^{6 the} determination of the relaxation time of the Rayleigh component for various temperatures. ^The important point ^{to note is that}

away from the coupling region ^to be discussed later, the limiting ^{values of} high ^{and low} temperatures ^are quite different. An analysis ^{of the} data leads to

We obtain a variation of the Rayleigh relaxation time that is much larger ^{than that} of Mountain relaxation time.

In the coupling region, ^{it is no} longer possible ^to separate S(t) ^{into a Mountain and a} Rayleigh ^line.

However, ^it turns out that the long time behaviour of the signal ^{can be fitted rather well to an} exponential,

so that it is still possible ^{to define a time constant for}

all values of the temperature. ^The corresponding

Fig. ^6.

Relaxation time of the Rayleigh line measured by ^forced Rayleigh scattering. ^{0 : with a} 70 J.1m interfringe ; ^{+ : with a 120 gm} interfringe ; A : value derived from the calorimetric experiment.

values are plotted ⁱⁿ figure 6, they correspond ^{to a} sharp increase in the coupling region. ^To get ^more information out of the data, ^{it is} necessary ^to compare the detailed shape ^{of the} relaxation function with that calculated using ^{the models} developed ^{in I.}

This will be presented ^elsewhere.

In addition to the results conceming ^{the time cons-}

tant of the Rayleigh component, ^{we have used the} signal S(t) ^to determine the ratio of the amplitudes ^of

the Mountain and Rayleigh components ^at high temperature such that _{Tm -} 0.1 _TR. Again, ^{due to}

limitations in the time scales of the MCA, ^{we had to} take into account the finite duration of the heating part of the experimental cycle. ^{As was} indicated in our

previous ^{work on the} subject [10], ^{a uniform} heating during time à leads to the following change ^{in the} signal. If the undistorted signal ^is

the measured quantity ^{will be :}

This leads to significant distortions if the Oi ^{are not} large compared ^{to ô. This occurs in our} experiments

as the distribution function of relaxation times of

glycerol ^includes very short relaxation times. ^It will therefore be somewhat difficult ^to derive precise

values for the ratio of the amplitudes ^{of the Mountain}

and Rayleigh contributions. Using ^the generalized

viscoelastic model with 5 relaxation times, ^{as discussed} in I, ^we find that if the mean relaxation time is 1 ms

(near - ⁶⁰ OC), taking ^a heating ^{time ô}

20 ps, the signal just ^at the end of the heating pulse (for

t

ô) ^is only ⁵ % smaller than if the ^{same amount} of heat had been dissipated ^{in the fluid in a very}

short time. The experimental situation is probably

less favorable ^{as we} should take into ^{account the}

698

jitter ^{of the} electronics, ^{so that the first} measured

point may very well be taken 30 ps after the beginning

of the heating pulse, ^{due to the} fact that our MCA records one point only every 10 us. In that ^case the correction will be as much as 10 %. ^{We can therefore} only get ^an upper limit for the ratio of the amplitudes

of the Mountain and Rayleigh contributions, ^and

estimate that (AmIAR)’ ^{- 0.3. We thus} get ^a

negative ratio, but it is not accurate enough ^{to allow}

us to determine the ratio Co2 /C2, ^{as we showed in 1}

that (AM/AR)°

¹

C20/C2 both for the purely

structural and the generalized viscoelasticity ^models.

We may remark that in our experiments, ^{it is in} principle straightforward ^to take into account the finite duration of the heating pulse, ^{due to the fact}

that we consider only linear differential equations ^for

the evolution of the fluid. This is a trivial example

of the influence of the past history ^{of a} glass upon its

properties, ^{which is} very important ^{in usual} experi-

ments on glasses ^{when the} departures ^from equilibrium

are large ^{so that it is} necessary ^to consider nonlinear differential equations.

We now summarize the results obtained on either side of the coupling :

- in spontaneous scattering :

for transition temperatures ^of

+ 200 (polarized/depolarized scattering)

-

45 OC (spontaneous scattering)

-

in forced Rayleigh scattering

for a transition temperature of - 60 OC.

We can thus see that the dispersion ^{of the time}

constant of the Mountain mode is not equal ^{to that}

of the Rayleigh mode. As this result plays ^a key ^role

in the choice of a model, ^{we have made a third} expe- riment to get another value of the dispersion ^{of the}

thermal diffusivity.

3. Calorimetric measurements.

Although ^inde- pendent measurements of the thermal conductivity Â

and of the specific ^heat Cp ^{have been reported for} glycerol ^{in the} glass ^and liquid ^states [11], ^{we built}

a simple system ^{to determine} ÂlpCp. ^{A copper cylinder}

of height ^{5 cm, and} internal and external diameters 5 and 10 mm is filled with glycerol. ^Two thermocouples

allow us to monitor the temperature difference bet-

ween the central part ^{of the} glycerol ^and the copper

wall, following ^a short heat input produced by ^a

wire resistor in contact with the outside diameter of the cell. The diameter of the thermocouples (0.5 mm)

allows a very short _response time compared ^{to the}

times we want to measure. A typical recording ^is

shown in figure 7, ^{where we also show the} temperature of the _copper cell, ^{which is} weakly coupled ^{to the}

cold finger ^{of the} cryostat. ^{Far from the} glass-liquid transition, ^the signal ^{can be} analysed using ^the

standard solution of the heat diffusion equation ⁱⁿ cylindrical geometry [12], ^{if one} takes into account the slow heat leak ^to the cryostat. ^{However we have}

not made a detailed analysis ^{of the} experimental recordings, ^{but we} just determined the time 7c ^after which the signal has decreased by ^{a factor 4 with} respect ^to its value at the end of the heating pulse.

This time, of the order of 30 _s, was found to be inver-

sely proportional ^to the thermal diffusivity ^{of a series}

of liquids ^{that were studied at room} temperature.

The reason why ^we considered only this time T,,,

is that we have not been able to solve the heat diffusion

equation ^{in the} coupling region, where the thermal

diffusivity ^is frequency dependent. ^The analysis ^that

was used for the simple geometry ^{of the forced} Ray- leigh experiment ^{becomes very} complicated ⁱⁿ cylin-

drical geometry. This leads to significant distortions in the decay signal, ^{so that the} experimental ^time Tc is not simply ^{related to the thermal} diffusivity ^{in the} coupling region.

Fig. ^7.

Typical recording in the calorimetric experiment using ^a

40 J.1V/oC thermocouple. ^The noisy ^curve represents ^the tempe- rature of the _copper block with a temperature scale indicated by ^the

arrow.

We shall therefore just ^use the results of such experi-

ments on either sides of the coupling. ^{We show in} figure 8 the time Tc ^{measured as a} function of tempe-

rature for two samples of glycerol : ^one is pure glycerol,

the other contains iodine with the same concentration

as in the forced Rayleigh scattering experiments

discussed in section 2 above. Two important ^results

can be deduced from this figure concerning ^{the effect}

of iodine and the dispersion of the thermal conduc-

tivity.

The data shown in figure ^{8 were obtained with two}

cells of slightly different diameters. When this is taken into account, ^we find that the thermal diffusivities of pure glycerol and of the iodine glycerol ^mixture

are equal throughout the whole temperature range

Fig. ^8.

Temperature dependence ^{of the} decay ^time 7c ^observed in the calorimetric experiment : ^{0 : in} pure glycerol ; + : in the

iodine-glycerol mixture. The data is not corrected for 5 % change ⁱⁿ

the diameter of the cell between the two measurements.

studied. This means that the small amount of iodine added to glycerol ^{does not} affect either the thermal

or the relaxational properties ^of glycerol.

We find that there is a large change ^{in the thermal} di.ffusivity between the low temperature ^value (À/Cp)oo

and the value that can be extrapolated ^{from the} high température ^{value. We conclude that}

From this value we can calculate the corresponding dispersion ^{of the} Rayleigh relaxation time : TR R using ^one of the models discussed in I. The corres-

ponding numerical values will be shown in table I, but we can note here that the generalized viscoelastic

model, that will be preferred, yields

which is _very close to the value determined from the forced Rayleigh scattering experiment.

Table 1.

Summary of ^the experimental ^results obtained for ^various quantities ⁱⁿ glycerol : glass (00)

or liquid (0). Comparison ^{with the} corresponding quantities calculated using the model due to Mountain structural (a), ^thermal (b), ^mixed (c) ^{or the} generalized

viscoelastic model (d).

Note that as we purposely ^{chose a} set-up ^{in which} the heat capacity of the copper cell ^was quite large compared ^{to that} of glycerol (about ^{15 times} larger),

we did not attempt ^{to determine} Cpo ^and Cpoo.

4. Comparison with theoretical results.

We have

developed ^{in I several} hydrodynamic ^{models to}

describe the acoustical and thermal properties ^of glycerol. ^{There we} calculated for each model the relaxation time and the amplitude ^{of the various}

relaxation modes either in the liquid ^{or in the} glass.

We shall now compare ^our experiments ^{with the}

theoretical results obtained for the simpler ^{case of}

a single relaxation process. This simplification ^is justified ^{as we showed in 1 that} away from the coupling region, ^the simpler ^models yield approximately ^the

same predictions ^{as the} complete ^models involving

a Cole-Davidson distribution of relaxation times.

Before testing ^the predictions ^{of the models, we}

should make some comments about the experimental

data. As we have not been able to perform ^{all the}

necessary experiments ^{at the same} temperature (and ideally ^{on the same} sample), ^{and as} ultrasonic data

are only available for rather high temperatures [13],

we have to explain why it is reasonable to assume that the relevant quantities : essentially ^the dispersion ^of

the velocity ^{of sound} Cj§ /C) and that of the thermal

conductivity (À/Cp)oo/(À/Cp)o ^{do not change very much}

with the temperature ^{of the} sample, ^{at least} probably

not enough ^to invalidate the conclusions that will be reached below. In addition we assume that our

experimental results obtained at some temperatures should hold at other temperatures. ^For example ^we

have found that TM /tMoO ^{is close to 1 both at + 20 °C}

and at - 45 °C, ^{so that we set} Iù/IQ = ^{1 at all}

temperatures. ^We have also found that (À/Cp)oo/(À/Cp)o

has roughly ^{the same value at -} 90 °C and - 60 °C.

The ratio (IR/IM)OO measured in spontaneous light scattering experiment ^{has been found to be} very small from 0 °C to about - 80 °C. Finally ^the Cole-David-

son parameter p ^does vary from about 0.6 at room

temperature ^{to 0.4 at - 80} OC, but this variation should not affect in _any significant manner our

conclusions. Some information can be obtained for the variation of yo ^from measurements of the Landau- Placzeck ratio or from the ratio (IR/Im)o ^{of the} ampli-

tudes of the Rayleigh and Mountain lines _{in spon-} taneous light scattering (Litovitz et ^al. [14] give

yo

1.15 at 20 °C ; ^{we have} determined (IR/IM)o

^0.3

at 20 °C and (IR/Im)o

^{0.26 at -} 40 °C). ^{It is thus}

reasonable to believe that it does not increase markedly

from 1.15 when the temperature is lowered. Acoustic measurements performed ^{at various} temperatures indicate that C2’/C2o’ ^{increases when the} temperature is lowered, ^{so that} using ^{a value of 2} for this ratio is conservative.

We now summarize in table I the experimental

results and the theoretical predictions obtained either

with the Mountain model including ^a relaxation of

700

mixed character, ^{or the} generalized viscoelastic model which is a generalized hydrodynamic ^model involving

both a frequency dependent viscosity ^{and a} frequency dependent ^thermal conductivity. ^{In both cases the} parameters ^are fixed in order to recover the right dispersion ^{of the} velocity of sound and the right dispersion ^{of the thermal} diffusivity. ^We then compare the predictions concerning TM0’/,tMO ^{and the ratio of the}

contributions of the Rayleigh ^{and Mountain} compo- nents : (IRlIm)- ⁱⁿ spontaneous light scattering ^and (AM/AR)° in ^forced Rayleigh scattering.

We can observe in table I that the generalized

viscoelastic model is satisfactory to account for these three experimental results. Whereas if we look at the

predictions of the model due to Mountain with a

mixed relaxation, ^{we find that} they ^are incorrect for the ratio t&/’tM. ^In addition the amplitudes ^{of the} Rayleigh ^{line in} spontaneous scattering is incorrect

as is the amplitude of the Mountain line in forced

Rayleigh scattering. Concerning these last results,

we should mention that correct predictions ^{can be}

obtained in the mixed model provided the condition

is satisfied. This would be the case if C2/C2 ^was

smaller than 1.25, ^{which cannot be the case as we} expect that it is at least larger ^{than 2. We can thus}

conclude that the model due to Mountain [15] ^is totally inadequate to account for the results of light scattering experiments ^{on either} side of the liquid- glass transition. On the contrary ^{we have} good ^results using ^the generalized viscoelastic model. It will be tested in more detail in the coupling region ^{and the}

results will be presented elsewhere, together ^{with a}

discussion of the possibility ^of experimental ^deter-

mination of the time _{’t ¡} which was included in the

frequency dependence of the thermal conductivity.

These results show that light scattering techniques

are very powerful ^for study ^{of the} dynamics ^{of collec-}

tive motions of fluids, provided ^{one chooses} systems that exhihit long enough time scales. They ^are espe-

cially useful for fluids in which the dispersion ^{of the} velocity of sound is large and that become _very viscous when the internal relaxation time increases. This

requirement ^{for a} large dispersion ^{of the} velocity ^of

sound that is connected to the amplitude ^{of the}

Mountain line, ^will probably preclude ^{the extension}

of this work to dilute solutions of large polymers,

which exhibit large relaxation times whilst having fairly small viscosities. In addition the scales of time and length accessible in our experiments ^are probably

out of reach of the molecular dynamics methods,

so that it is doubtful that one could make computer calculations [16] ^of ’1(ro), Â(co) ^and S(k, ro) ⁱⁿ glycerol

in the region ^{of the} liquid/glass transition at low temperature. However, ^it might ^be very instructive to make a computer simulation of the liquid-glass

transition of some hypothetical ^{fluid that would}

have a short relaxation time. This would have to be

very accurate to lead to a significant ^{result for the}

value of the dispersion ^of À/Cp.

5. Conclusion.

In this paper ^we have given ^a

summary of experimental measurements performed

in glycerol either in the liquid ^{state or the} glass state, with a transition defined by ^{the condition that the}

heat diffusion time _TH is of the order of the internal relaxation time _Tm. We first described the work done

using spontaneous light scattering experiments. ^It

leads to the following ^{results :}

We then performed ^forced Rayleigh scattering expe- riments. We found that :

We finally ^{used a} calorimetric set-up which allowed

us to determine (ÂICP).I(ÂIC,,)o - ^{1.7 in agreement}

with TR’/TR

These results are then compared ^{to the} predictions

of two models : the Mountain model for a mixed

relaxation, ^{and the} generalized viscoelastic model.

It is found that only ^the generalized viscoelastic model

can give satisfactory predictions of all the acoustic,

thermal and light scattering experiments performed

to date in glycerol.

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