Phenomenological models compatible with acoustical
and thermal properties of viscous liquids
C. Allain, P. Lallemand
To cite this version:
C. Allain, P. Lallemand. Phenomenological models compatible with acoustical and thermal properties of viscous liquids.
Journal de Physique, 1979, 40 (7), pp.679-692.
<10.1051/jphys:01979004007067900>. <jpa-00209152>
HAL Id: jpa-00209152
https://hal.archives-ouvertes.fr/jpa-00209152
Submitted on 1 Jan 1979
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LE JOURNAL DE
PHYSIQUE
TOME
40, JUILLET 1979, :
679
Classification
Physics Abstracts
62.60
44.50
-
Phenomenological models compatible with acoustical
and thermal properties of viscous liquids
C. Allain and P. Lallemand
Laboratoire de
Spectroscopie Hertzienne de l’E.N.S., 24,
rue
Lhomond, 75231 Paris Cedex 05,
France
(Reçu le 26 décembre 1978, accepté le 12 mars 1979)
Résumé.
Après un très bref rappel sur la dispersion des propriétés acoustiques et thermiques des liquides visqueux à la transition liquide-verre, on donne quelques idées sur les modèles usuels : théorie viscoélastique et
équations de Mountain déduites de la thermodynamique des phénomènes irréversibles.
La partie 1 est consacrée à la théorie des expériences de diffusion Rayleigh forcée en utilisant le modèle de Mountain à un temps de relaxation. La partie 2 étend ce travail au cas d’un fluide à plusieurs temps de relaxation. On
calcule explicitement des fonctions de corrélation en diffusion spontanée ou Rayleigh forcée à partir des résultats
de mesures acoustiques ou thermiques. On indique alors que ces prédictions ne sont pas compatibles avec les
résultats expérimentaux. On présente dans la partie 3 un modèle viscoélastique généralisé qui inclut à la fois une
viscosité et une diffusivité thermique dépendant de la fréquence. On l’utilise pour calculer des fonctions de corrélation pour des fluides à un ou plusieurs temps de relaxation internes. On donne des exemples pratiques dans le cas
de la glycérine en tenant compte d’une distribution de temps de relaxation du type Cole-Dayidson.
2014
After a very brief review on the dispersion of acoustical and thermal properties of viscous liquids
liquid-glass transition, some information is given about the standard models : viscoelastic theory and irreversible thermodynamics as used by Mountain.
Part 1 of the paper is devoted to the theory of forced Rayleigh scattering experiments using the model due to
Mountain with one relaxing variable. Part 2 extends this work to several relaxing variables both in spontaneous
and forced Rayleigh scattering. Here time correlation functions are calculated explicitly given the results of acoustical and thermal experiments. It is then indicated that the predictions of these models are not consistent with
experimental data. Part 3 is devoted to a generalized hydrodynamics model in which both the viscosity and the
thermal diffusivity can be frequency dependent. This model, called generalized viscoelasticity, is then applied to a
variety of experimental situations first for a single relaxation process and then for a Cole-Davidson distribution
of relaxation times. It is shown that in glycerol the acoustical properties depend mainly on the viscosity while the
thermal properties depend essentially upon the thermal diffusivity.
Abstract.
2014
at the
Viscous liquids have been the object of many detailed work on this material that will be presented
experimental and theoretical studies. Special interest elsewhere, but we shall often refer to our results and
has been devoted to liquids that can be easily super- to those of other workers to justify the choice of
cooled as the interpretation of their properties leads various phenomenological models that are discussed’
to some understanding of the corresponding glasses.
in this paper.
Viscous liquids have been studied using several
Although high temperature glass forming liquids are
more interesting as far as applications are concemed,
macroscopic type experiments. Among them, we
we shall deal here with organic materials that are
shall consider the ultrasonic properties studied by
more easily amenable to detailed experimentation.
acoustic techniques [1], or by Brillouin scattering [2]
these
has
a
and
the thermal properties derived from calorimetric
liquids, glycerol
Among
prominent
as
it
exhibits
in
sound
very large dispersions
place
techniques [3] or by Rayleigh-Mountain scattevelocity and in specific heat. We have performed ring [4], [5]. Until recently the most accurate data
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004007067900
680
obtained for ultrasonic properties and several
theories have been developed for their interpretation.
Among them, the most successful is viscoelasticity [6].
With the availability of modem light scattering
techniques, it has been possible to obtain accurate
data conceming the thermal behaviour of viscous
liquids in a large time domain. This has led us to
look for phenomenological models that could account
not only for acoustical properties but also for the
thermal properties of such liquids. Following the
early attempts of Litovitz et al. [7], we have already
attacked this problem in a paper that will be referred
to as 1 [8] at a time when we had only limited experimental results conceming the thermal properties.
The data described in paper 2 [4] has led us to test
in a much more detailed manner the models used
in 1 and to propose a new model that we call generalized viscoelasticity [5]. Following these developments, part 1 of this paper will analyse the model
due to Mountain [9], for a viscous liquid with one
relaxation time, part 2 will extend it to the case of
several relaxation times and part 3 will present the
generalized viscoelastic model. It is important to
note that although many of the equations used here
have already been considered by other authors, the
originality of this work lies in the fact that in all
cases we have set the adjustable parameters of the
models starting from the known acoustic and heat
capacity measurements and that we have tested the
predictions of the models by comparison with the
data obtained in light scattering experiments.
were
l.
Hydrodynamic model for viscous liquids with a
single relaxation time. We shall first recall the
equations due to Mountain [9] that are commonly
used to describe the hydrodynamic motions of viscous
liquids and then briefly describe the results of our
previous analysis [8] in which we had considered
the acoustical properties and light scattering spectra
together with the results of calorimetric measurements that show a large dispersion of the specific
heat. Here we shall consider the forced Rayleigh
experiment [10] in order to test the models by comparison with the data presented in 2. Serious discrepancies between experiment and theory occur and
lead us to the developments of the later parts of this
define the local state of the fluid, we shall then use
scalar variables. If the molecules were very anisotropic, we would need to consider tensorial quantities,
usually second-rank tensors, but we shall neglect
them as our aim is to account for experiments performed in glycerol. In this liquid, one can detect a
small amount of depolarized light [11], [12], [13],
but to date there is no observation of the so-called
shear wave structure produced by coupling between
a tensorial variable and vorticity [14].
Following Mountain, we shall use as local variables
the following quantities X
{ p, 03C8, T, 03BE }, where p
is the density, ik the divergence of the velocity, T the
temperature (these are the three hydrodynamic
variables) and 03BE a local variable that is introduced
to account for the relaxation properties of the fluid.
(We shall use the same symbol for the average quantity and its departure from the average, unless the
expressions are too confusing. In addition, we shall
usually work with spatial Fourier components of
wave vector q.) If the relaxing fluid is assumed to
behave like a mixture of liquids in which may take
place a chemical reaction [15], the degree of advancement of which is ç, we shall consider the free
energy of the system :
=
where A is the chemical affinity. Its value is zero at
equilibrium. We assume that the equilibrium value
03BE of 03BE may depend both on density and temperature.
It relaxes at constant volume and temperature
according to
:
-
where To is the intrinsic relaxation time of the system.
In the usual discussions [16], To is noted tV,T (relaxation time at constant volume and temperature). It
can be varied widely by changing the mean value of
the temperature : for instance from 10-8 s at 0 °C
to 102 s at - 90 0C in glycerol. It will be very useful
to define the quantities
paper.
1.1 EQUATIONS OF MOTION.
When one deals
with a fluid composed of molecules that are not
very anisotropic, it is usually sufficient to consider
the longitudinal excitations of the fluid in the direction of the wave vector q. q is defined either by the
direction of sound propagation in the case of ultrasonic measurements or from q
k;nc - kscat in the
case of the scattering of an input light wave of wave
vector kinc into a light wave of wave vector kscat.
We take z as coordinate in the direction of q. To
-
=
and following other authors, we shall say that the
relaxation is structural if ÇT
0, thermal if çp 0
or mixed in other cases.
Starting from equations (1) and (2), and applying
the general principles of the thermodynamics of irreversible processes [15], one gets, neglecting mass
diffusion :
=
=
681
where :
with
Typical
ultrasonic
Here 11L
=
4 ns
+ Ilb;
C oo
velocity of
is the
sound, Yoo the ratio of the specific heats and oc. the
thermal
expansion
cY,A
q
quency,
A03BE -
coefficient taken for infinite fre-
aç p,T
2A
.
To obtain these
équations,
q
values of the quantities derived from
calorimetric studies on glycerol are :
or
We shall use these to calculate the relevant quantities
to be defined later.
We shall now briefly recall the main results of 1.
Using the thermodynamic relationships given in
the appendix of 1, one determines
T
have made use of the free energy, and applied
Maxwell relationships to relate thermodynamic derivatives. To simplify the notations, we use the indices
0 and oo to indicate whether partial derivatives are
taken at A or at ç constant. This can be interpreted
in the following manner : A constant means that j
has its equilibrium value and this can take place
when To is much shorter than the relevant time
scale : the fluid then behaves as a liquide ç constant
corresponds to the opposite situation, and the fluid
now behaves like a glass.
The equation of motion (3) involves partial derivatives with respect to time, and it will be very convenient to take a Laplace transform. This will introduce
im.
the variables related to frequency by s
Numerous consequences can be obtained from
eq. (3). Starting from the following data,
Coo/Co derived from ultrasonic measurements
Cpo/Cpoo derived from calorimetric measurements, we shall try and account for the results of
spontaneous light scattering and of forced Rayleigh
we
=
-
-
experiments.
from the dispersion of C2 and Cp. Then we find
that there is no difficulty in accounting for the ultrasonic or light scattering data at high frequency and
that little information is obtained on the nature
(thermal, structural or mixed) of ç. The situation is
quite different for the central part of the light scattering spectrum which depends very much on the
ratio çp/ ç T. It is thus very fruitful to study in detail
the Rayleigh-Mountain coupling region.
1. 2 LIGHT SCATTERING SPECTRA.
We find that
the central part of the spectrum is made of two
lines, one of width r R that varies roughly as q2 (the
Rayleigh line) and one of width TM that varies roughly
as ’tÕ 1 (the Mountain line). We give, in table I, the
ratios of the widths and intensities of these lines on
either side of the coupling for various kinds of
relaxation. Note that here the relevant time is the
heat diffusion time TH so that the index 0 applies for
TO « TH. One important feature of these results is
that :
-
These two kinds of experiments yield useful information in the vicinity of the Rayleigh-Mountain
coupling which occurs for values of the temperature
and of the wave vector such that
irrespective of
It will be
where :
particularly significant to look for the
decay times of the Rayleigh and Mountain lines and
for their amplitudes on either side of the coupling
situation.
the nature of the relaxation. This
from the fact that, when reduced to long
times, the dispersion equation is given by :
comes
682
Table I.
lines in a
Limiting values of the roots of the dispersion equation and of the intensities of the Rayleigh and Mountain
singly relaxingfluidfor spontaneous light scattering.
-
It is obvious from eq. (6) that the product of the two
line widths is constant. We show in 2 that this prediction is not consistent with the experimental results.
1.3 ANALYSIS OF FORCED RAYLEIGH EXPERIMENTS.
In such experiments, one measures the intensity (or
amplitude, when using a heterodyne detection scheme)
of the light diffracted by a phase grating initially
induced by a short heat pulse with a periodic spatial
distribution. We then have to calculate the variation
of the density p(t) induced by the excitation T(0).
As the time scales involved are usually very long
compared to the period of sound waves (of wave
vector q), we can simplify the analysis. Taking a
-
time
Laplace transform,
we
get :
We show in figure 1 the variation of the amplitudes
of the lines as a function of TH/TO for several values
of Cpo/Cpoo, corresponding to the structural, thermal
and two mixed cases. One can see that the ratio of
the amplitudes of the two lines depends very much
on the nature of the relaxation when Io « 1:H’ which
corresponds to the high temperature situation. We
may note that this ratio changes sign for :
in our numerical example). We recover the
critical value of Cpo/Cpoo as in the light scattering spectra where we observed a qualitative
change in the low temperature behaviour of the ratio
of Rayleigh to Mountain intensities.
(1.3
same
where D(s) is given by eq. (6).
The roots of the dispersion equation are obviously
the same as that involved in spontaneous light scat-
tering experiments. However, forced Rayleigh scattering yields very interesting information conceming
the amplitudes of the two lines. Furthermore, the
very good signal to noise ratio which is easily accessible leads to accurate values of the decay times.
We obtain as amplitudes of the Rayleigh and
Mountain lines :
N for
a
for
1.
Variation of the amplitudes of the Rayleigh (R) and
Mountain (M) contributions in a forced Rayleigh experiment vs.
tH/t0. S represents the pure structural case, T the pure thermal case,
1.2 or 1.8.
Ml and M 2 are mixed cases with Cpo/Cpoo
Fig.
-
=
683
One can also see from these results that useful
information could be obtained if the ratio AR/ARO
could be determined (AR being the amplitude of the
Rayleigh line). However this involves the comparison
of signals observed in different conditions (change
of q or change of T) and this is extremely difficult
to achieve in a reliable manner.
We can indicate now that the experimental results
conceming the intensities presented in 2 are in
favour of a model involving a small value of Cpo/Cpoo
(preferably the structural case). This is not consistent
with the large value found for FR/FRO.
This difficulty, together with the fact that the
relationship TM/TM TR /TR is not verified experimentally, has led us to see whether the defects of
the model are not due to the very crude way in which
we describe the relaxation process using a single
relaxation time.
rent point of view. We shall first generalize the
Mountain model assuming that the set of slow
variables of the problem (still of longitudinal character) is :
Each variable j; is such that its equilibrium value
depends only upon p and T. In addition, we assume
that the free energy
can
be written in the form :
=
2. Extension of Mountain’s model to several relaxaIt is well known that one cannot
tion processes.
account for ultrasonic measurements or dielectric
relaxation measurements in viscous fluids using
models that involve a single relaxation process of
exponential character. Historically one has tried to
get a better description of the dielectric measurements by replacing the Debye formula for the dielectric constant by an empirical formula of the type :
-
At fixed values of p and T, each
to
equilibrium according
variable 03BEi
relaxes
to :
This is a straightforward generalization of the model
discussed in part 1 of this paper, and we shall look
for the changes this brings about in the theoretical
light scattering or forced Rayleigh spectra. Starting
from the preceding hypothesis, one can determine
the equations of motion of the set Y, and the statistical properties of the fluctuations of the system,
which will be required when we determine the spontaneous light scattering spectra. The general equations have already been written down by other
authors [20], and we have studied in detail, in 1,
the case when all the variables j; are of structural
nature (ÇTi
0 for all i).
As in section 1,
=
with 0 03B2 1.
This expression due to Cole and Davidson [17]
leads to very satisfactory results, but it is by no
unique.
Similarly the simple expression :
means
longitudinal compliance
replaced [18] by :
for the
of
a
fluid had to be
In the rest of this paper, we shall fix the parameters
of our models in order to recover this expression
which allows an accurate summary of the results of
the ultrasonic measurements that we consider to be
very precise.
In paper 1,
we started from this expression of J(s)
derive 1(s) and then calculated the light scattering
spectra for To &#x3E; rH, in order to interpret the data
obtained by Allain-Demoulin et al. [19] using photon
correlation techniques. Here, we shall adopt a diffe-
to
Due to the very large number of parameters that
appear in the general equation of motion, it is necessary to use some simplifying assumption. Here we
shall extend the analysis of 1 to the case where all
the relaxation processes are of the same nature. By
this, we mean that Çpi/ÇTi is independent of i. We
take :
where x is a parameter to be adjusted from the experimental data.
This choice is not as arbitrary as it seems, as we
could interpret it as representing a system with an
intemal process of a given type, that does not relaxe
exponentially. Its temporal evolution, at p and T
fixed, could be decomposed into a sum of exponentials.
With these assumptions, the equation of motion is :
684
with :
o
o
As in the case of a single relaxation, we may deduce
many thermodynamic relationships from the free
energy. Again, we shall consider quantities at 0 or oo
frequency depending upon whether all the Çi are in
equilibrium with p and T or are constant. Among
these relationships, we select a few that are quite
useful. One has :
From these equalities, we
For instance, we obtain :
which is identical to the
value of y 00 as before.
x is given by :
can
determine Yoo and
x.
relationship that we had used for a singly relaxing fluid. Therefore, we obtain the same
Once cxoo’ Yoo and x are obtained, we must determine the contributions of the various relaxation processes.
For that purpose, we apply eq. (13) to study the acoustic properties. This is most easily done if we first contract
eq. (13) to the hydrodynamic variables, in order to get :
685
These
expressions
viscosity :
involve
a
frequency dependent
Y’t’.,(1 /n)i - 1, where
n is usually equal to 6, and y
We
then determine the A 03BEi
scaling parameter.
for a given y by a least square fitting technique and
finally minimize the differences between the two
members of eq. (21), with respect to y. This numerical
procedure leads for each set of values of fi, CO2/C.2,
Cpo/Cpoo and yo to a time scale y and to weights for
the relaxations. These quantities will then be used to
calculate the spontaneous light scattering spectrum
and the forced Rayleigh signal. This procedure works
only because we have assumed that all the relaxations
are of the same nature.
As an example, we give some numerical results for
C2/IC02 2 ; yo 1.15; fi 0.5 ; Cpo/Cpoo 2. We
1.224 4 with weights given by :
get y
=
1;
03BE2pi
a
It will be convenient to define
One then obtains the
a
related
quantity :
dispersion equation :
=
=
=
=
=
To obtain the ultrasonic
situation s =
qco »
we
consider the
means
that it will
properties,
JcQ2 .
.
pcp
This
be sufficient to consider the first three terms in
eq. (20). Furthermore, if the acoustic attenuation is
small, we obtain the velocity of sound from an
equation similar to that used in acoustics. Therefore,
we can write :
With these numbers, eq. (21) is verified to better
than 10-3 which is good enough considering that the
accuracy of the acoustic measurements is not perfect.
Starting from these numbers, we have determined
numerically the roots of the dispersion equation.
They are shown in figure 2 as a function of TH/TOWe have also determined the contributions of the
We have the index u to indicate that the corresponding
values are directly derived from acoustic measurements.
principle, eq. (21) allows us to determine U(s)
exactly. However, J and U being known in the frequency domain, we can easily determine only the
spectral behaviour of the spontaneous or forced
Rayleigh signals. Now, it tums out that all our experiments have been performed using correlators or
multichannel analysers, leading to signals in the time
In
Fig. 2. Roots of the dispersion equation for a viscous fluid with
5 relaxation processes of mixed character as a function of TH/TO-
domain.
We could obviously take a Fourier transform of the
theoretical spectra to allow comparison of theory
and experiments. We have preferred to use a different
method which allows an easier discussion of the theoretical results. We have decomposed U(s) into a sum of
homographic functions (as in eq. (18)) such that the
equality in eq. (21) is only approximate, but we then
obtain a polynomial for the dispersion equation,
so that we can calculate the time constants of the
various modes of the problem.
In practice, we have used the following procedure.
We have taken 5 processes with relaxation times
various modes. We show the amplitudes of the various
modes respectively for spontaneous light scattering
and for forced Rayleigh scattering in figures 3 and 4.
One finds a series of couplings analogous to those
discussed in part 1. We shall now discuss these results
to see whether they allow any improvement compared
to the single relaxation model.
2 .1 AMPLITUDES OF THE LINES. - We see in figure 1
4 that the amplitude of the Rayleigh and Mountain
or
686
Furthermore we find that the time constants 0, of
the Mountain modes are little affected by the coupling,
except the first one. However the relative weights pi
of the modes significantly change near the couplings.
If we define a mean relaxation time by :
the changes in pi leads to
found that :
Fig. 3.
-
Variation with
TH/To of
to the
the intensities of the contributions
a model
spontaneous light scattering correlation function in
supposed to represent glycerol.
a
change in iM.
We have
We thus obtain approximately the same result (eq. (5))
as for the single relaxation case. We have reached
the same conclusion when studying a hypothetical
fluid in which the 5 relaxations would have the same
weights. This result together with those obtained in
other cases, show that the conclusions we reach are
not specific for the Cole-Davidson formula used for
J(s).
We may conclude this discussion of the multirelaxation model by saying that it leads to little change
concerning the amplitudes or relaxation rates and
that a different model must be sought.
3. Generalized viscoelasticity.
Before discussing
a model which will be shown in 2 to give very satisfactory predictions, we shall make some remarks
concerning the generalized hydrodynamics.
-
3. 1 GENERALIZED HYDRODYNAMICS.
When we
supposed, in part 1 of this paper, that one can describe
the local state of the system with the set X
{ p, 03C8,
T, 03BE }, we meant that this ensemble contained all the
slow variables of the problem, so that it is legitimate
to write the following Langevin equation for the
fluctuations of the system :
-
=
Fig. 4.
-
Variation with rH/TO of the amplitudes of the contributions
to the forced
Rayleigh scattering signal.
on either side of the coupling region are
the
qualitatively same as seen in part 1. A more detailed
analysis has shown us that in order to get a weak
Rayleigh line in spontaneous scattering at low temperature, one cannot choose a value of Cpo/Cpoo larger
than about 1.3. A similar conclusion is reached in
analysing the forced Rayleigh signals at high temperature. We thus see that the N relaxation model
does not lead to any improvement as far as the amplitudes are concerned.
components
2.2 RELAXATION TIMES.
shown us that we get :
-
A detailed
analysis
has
where the matrix elements of the hydrodynamic
matrix G contain no time operator, and where F(t)
is a random driving term. The correlation time of F
is very short compared to the hydrodynamic time
scale.
Starting from such a set of equations, it is often
convenient to look for the corresponding generalized
hydrodynamic equations that relate p, 03C8, T and their
time derivatives. The hydrodynamic matrix J1 can
now include frequency dependent terms. We have
where :
687
Note that
some care is required when calculating
it contains terms proportional to ç(O) which
is correlated with p(O) or T(O). This has to be taken into
account when calculating the spontaneous scattering
ç(0)
as
spectra.
If we inspect A, we see that it includes frequency
deperident terms both in transport coefficients and
in non diagonal terms. We are going to give a discussion to suggest that it is preferable not to include
such non diagonal terms.
Let us consider quantities like the compressibility
the
specific heat
the thermal
pressure
or
of the system and will introduce a frequency dependence only in transport coefficients. These quantities
are directly related to time correlation functions,
so that they may very well depend upon the frequency.
To illustrate these remarks, we shall consider in
more detail the forced Rayleigh experiment in a
relaxing fluid.
The principle of the experiment [21 ]] is the following :
A sudden input of heat leads to a periodic temperature variation ôT(t) exp(iqz). After a time equal to
the period of sound of wave vector q, the pressure is
equalized in the fluid, leading to a density variation
ôp(t) exp(igz) and thus to an index of refraction
grating that may diffract a light beam. Following
a very simple argument, one is tempted to use the
relationship :
expansion coefficient
at constant
to write
For
Each of these can be expressed in terms of ensemble
average, or one time average in the Boltzmann
approach.
It is therefore not
expected that they might be
frequency dependent. However, one can consider
the limiting values at zero or infinite frequencies,
if these notions are interpreted in the following way.
At zero frequency one takes ensemble averages over
the entire phase space, at infinite frequency one limits
the phase space to values of ç corresponding to A
0.
Even though such obvious remarks are often made,
one finds in the literature expressions like M(w),
J(cv), Cp(ro), etc... It seems to us that, even though
such expressions can be useful for some types of
calculations, it is important to remark that they do not
satisfy automatically the same relationships as the
corresponding thermodynamic derivatives, and that
they can be used only in particular cases. We shall
therefore exclude them from the equation of motion
a
simple fluid,
one
has
and one can explain the experimental results using
eq. (28).
We consider next a relaxing fluid in a situation such
that TR » ro. Experiment [10] then leads to a signal
shown in figure 5. It is then tempting to interpret
the rapid initial rise in bp by using a frequency dependent a in eq. (28) :
=
keeping the same temporal dependence for ô T. One
will say that a relaxes from its glass-like value oc.
to its liquid-like value ao.
This way of interpreting the data is incorrect. It is
more satisfactory to take a constant a and to write
that ô T no longer varies exponentially. To find its
time evolution, one will have to solve the complete
dynamical equations (eq. (25)) for p(t).
In a similar manner, it is much preferable to interpret
acoustic relaxation experiments using a longitudinal
viscosity 1(s) that is frequency dependent, rather than
a compliance J(s) or a modulus M(s).
These qualitative arguments show that one should
be quite suspicious when using generalized hydrodynamic equations that involve frequency dependent
terms elsewhere than in transport coefficients.
We shall now see what are the consequences of the
equations used in part 1.
If ç T
3. 1. 1 Pure structural relaxation.
0,
J1 contains only a frequency dependent viscosity, ,
which is acceptable. This is the original model due to
-
5.
Typical forced Rayleigh scattering signal in
fluid in conditions such that ’tH &#x3E; ’to.
Fig.
-
a
relaxing
Mountain [22].
=
688
3.1.2 Pure thermal relaxation.
If çp 0, M
contains only one frequency dependent term in 1,XTT.
We have :
-
=
3 .1. 3 Mixed relaxation. - In that case tÂt contains
4 frequency dependent terms, among which Mp03C8,
and M03C8p. These two terms appear as corrections to
expressions involving the thermal expansion coefficient. In view of the previous discussion, we are going
to discard these terms, and thus use generalized
hydrodynamic equations that involve only a viscosity
and a thermal diffusivity that are frequency dependent.
3 .2.1 SinWe have chosen the following
équations to describe the time evolution of small
excitations of the fluid [23] ::
3.2 GENERALIZED
gly relaxing fluids.
with
[5] :
Adjustable parameters
appear in :
VISCOELASTICITY. 2013
-
As we shall see later, it is very difficult to determine
both in and T 03BB, so that we shall make the simplifying
assumption that I, m’t ¡, where m is a constant of
the order of unity, independent of temperature.
Following the same procedure as in part 1, we first
determine the dispersion equation :
=
where
with
which is what we expect as the fluid then behaves like
an ordinary fluid. In this particular case, the amplitudes of the Mountain lines, corresponding to r, and r03BB
are negligible, so that one cannot determine T. and r..
a
roots
We then determine the roots of the
equation in various limiting cases :
: we
get :
dispersion
(qco)- 1 «
as
mine Ilr :
± i
in ;t 03BB;rH :
Cô g2
we
thus get the Brillouin
il2
+
rtn) ,
so
that
we can
deter-
We recover the value that was used by Mountain [22].
If we now look for the central part of the spectrum,
we may deal with the reduced dispersion equation :
689
The
a
for Tn., -r  TH
yield the following results
(high temperature case)
a
for r,, ’t). &#x3E; rH
(low temperature case)
limiting
cases
For most viscous
liquids, y is close to unity,
C2
the
quantity y -
C2Coo0 (y - 1) is also
:
so
close to
that
one.
In our usual numerical example, it is equal to 1.075.
We therefore deduce from these results that :
and :
if tMÂ and ’tM., represent the time constants associated
with relaxation of the thermal diffusivity and the
viscosity respectively. We see that we can now have
a large change in the Rayleigh width and a small
change in the width of the viscosity Mountain line.
Figure 6 shows the roots of the dispersion equation
as a function of tH/’tn". Let us now consider the amplitudes of these lines, first for limiting cases and then
as a function of TH/Tq*
N For light scattering experiments.
For this purpose, we use the common
where
Npp(s) is the pp cofactor of the 3C
This leads to the
expression :
+ sI matrix.
following intensities :
Fig. 6.
-
for the
Variation with T./T. of the roots of the dispersion equation
viscoelastic model, with a single relaxation
generalized
process.
(This is due to the fact that the fluctuation p(O) is
uncorrelated with T(O) and gi(0).)
We have first calculated the Landau-Placzek ratio.
We get the following results :
the result obtained with the model of
to give a good fit with experimental data.
Let us now consider the central part of the spectrum
for Tn » (Co q) -1. In that case, it is sufficient to
consider a reduced expression of Np p(s),
We
recover
part 1, which is known
690
where IR, IMn, lM), are the intensities of the modes
associated with ’tR, ’tM" and ’tM). respectively.
It is important to note that the intensity of the new
mode, due to the relaxation of the thermal diffusivity
has a negligible contribution away from the coupling
region. We show in figure 7 the intensities in the
intermediate case. It is interesting to note that now
we get a small Rayleigh intensity at low temperature.
These conclusions hold for any value of m.
Fig. 8. Variation with iH/io of the amplitudes of the contributions to the forced Rayleigh scattering signal for the generalized
viscoelastic model, with a single relaxation process.
-
If
compare the matrix X, with the expression
in eq. (17) corresponding to the contraction of
the Mountain model with several relaxations, we might
be tempted to say that generalized viscoelasticity is
just the same as taking one purely structural variable ç
and one purely thermal variable Ç, because this
combination would not give rise to any off diagonal
element in eq. (17). We have tested this idea and
found that it was not correct. The reason is that if we
keep the Mountain model, the existence of a free
energy leads to thermodynamic relationships that
give rise to inacceptable predictions concerning the
amplitudes of the Rayleigh and Mountain modes for
both spontaneous and forced light scattering.
we
given
Fig. 7.
to the
-
Variation with
TH/TO of
the intensities of the contributions
spontaneous light scattering correlation function for the
generalized viscoelastic model, with
a
single
relaxation process.
a For forced Rayleigh experiments.
We now have to calculate the quantity
To account for the usual
sufficient to consider :
experimental situation,
it is
3. 3 EXTENSION
TO A REAL FLUID. -
In order to
account for the acoustical
data, we need to take more
elaborate expressions for il(s) and K(s) in the matrix je,
This leads to the
following
results :
given in eq. (30). As we have now no direct way to
determine K(s), we shall keep the simple form given
by eq. (32). Concerning n(s), we shall follow the
procedure of part 2. That is, we set :
with Nusually equal to 5 and as before r, ylo(1 n)i - 1,
with n
6.
We then consider the part of the dispersion equation
that leads to the acoustic modes and try and fit the
acoustical data represented by J(s) given by eq. (9).
Using a least square procedure (as in Part 2), we can
determine y, { qi }. Once these numbers are known,
we calculate the roots of the dispersion equation
and the amplitudes of the various lines in spontaneous
or forced Rayleigh scattering. As an example, we
show in figures 9,10,11 how all these quantities vary
with TH/to, for the same values of the parameters as
in part 2. A detailed comparison of figures 2, 3, 4
and figures 9, 10, 11 will show that we get significant
=
=
The results in the intermediate case are shown in
8. We see that again the new relaxation mode
has a negligible contribution away from the coupling
region. It is non zero near the coupling, but one would
require extremely accurate data to be able to extract
a significant value of m from experiments in such a
regime. It is interesting to note that this model gives
We thus
satisfactory predictions for the ratio
have a model which is quite promising and which
will be tested in detail in 2.
figure
A’mn,/ARo.
691
We can see that as in part 2, the use of several
relaxation processes does not lead to results qualitatively different from those obtained in the case of a
single relaxation.
Such results will be exploited in 2 to make comparisons with experimental data.
It is useful to note here that if we try and analyse
the Brillouin scattering experiments of Litovitz [9],
the present model yields the same predictions as the
usual viscoelasticity model. In the same way, if we
analyse the experiments of Demoulin et al. [19] which
led to the shape of the Mountain line at low temperature, the present model yields :
Fig. 9.
for the
-
Variation with TH/to of the roots of the dispersion equation
viscoelastic model, with 5 relaxation processes.
generalized
which is exactly the same expression as used in 1.
This shows that including the frequency dependent
thermal diffusivity is only required for study of the
thermal properties of the fluid, especially in the
Rayleigh-Mountain coupling region.
4. Conclusion.
In this work, we have presented
the predictions of various models to describe the
linearized hydrodynamic motions in a viscous liquid.
We first studied the model due to Mountain, which
can be derived by applying the thermodynamics of
irreversible processes to chemical reactions. Although
the general equations used here have already been
written by several authors, this paper brings new
results as we have tried to obtain definite predictions
-
Fig. 10. Variation with TH/t0 of the intensities of the contributions to the spontaneous light scattering correlation function for the
generalized viscoelastic model, with 5 relaxation processes.
-
Fig. 11. Variation with TH/TO of the amplitudes of the contributions to the forced Rayleigh scattering signal for the generalized
viscoelastic model, with 5 relaxation processes.
-
changes conceming
main results
the
in forced
-
are
the
the
amplitudes
following :
and roots. The
amplitude of the Mountain contribution
Rayleigh experiments has the right sign for
’To « TH ;
-
the roots of the dispersion equation are such that
for spontaneous or forced light scattering, starting
from the experimental data deduced from ultrasonic
or calorimetric measurements. We have indicated
that no model of this type seems acceptable in view
of our experimental data to be presented elsewhere [4],
even if we consider a distribution of relaxation times.
We have then discussed a possible reason for the
failure of these models. The fact that they lead to a
frequency dependence in thermodynamic derivatives,
when they are contracted into generalized hydrodynamics equations, make them very suspicious.
Consequently we have developed a model, that we call
generalized viscoelasticity, in which both the longitudinal viscosity and the thermal diffusivity depend
upon the frequency. The model is then developed to
calculate the relevant spontaneous pnd forced Rayleigh
light scattering spectra. This is done, first for a singly
relaxing fluid, and then for a real fluid. We shall show
in 2 that the study of the Rayleigh-Mountain coupling
region leads to new insight concerning the liquidglass transition, as the usual temperature recovery
experiments [24] in glasses are performed in the limit
to &#x3E; TH. However, these experiments give information
about the non linear behaviour of glasses, which we
did not need to consider here as we deal only with
very small departures from equilibrium.
692
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