Analysis of a heterodyne-detected transient-grating experiment on a molecular supercooled
liquid. I. Basic formulation of the problem
A. Azzimani, C. Dreyfus, and R. M. Pick
IMPMC, Université P. et M. Curie et CNRS-UMR 7590, F-75015 Paris, France P. Bartolini, A. Taschin, and R. Torre
LENS and Dipartimento di Fisica, Università di Firenze, 1-50019 Sesto Fiorentino, Italy and INFM-CRS-Soft Matter (CNR), c\o Università la Sapienza, Italy
共Received 19 July 2006; revised manuscript received 16 February 2007; published 24 July 2007兲
We present the basic equations necessary to interpret heterodyne-detected transient-grating experiments performed on a supercooled liquid composed of anisotropic molecules. The final expressions are given under a form suitable for their direct application to a test case.
DOI:10.1103/PhysRevE.76.011509 PACS number共s兲: 64.70.Pf, 78.47.⫹p, 61.25.Em
I. INTRODUCTION
During the last decade, time resolved light-scattering spectroscopy has been recognized as a very important tool for the study of the dynamics of supercooled molecular liq-uids. One such technique is the optical Kerr effect共OKE兲, in which a very intense laser beam with short duration 共⬃10−4ns兲 共the pump兲, whose frequency is not absorbed, travels through the liquid. Its electric field slightly orients the molecules, creating an optical anisotropy. The anisotropy is probed at a sequence of later times by the change it produces on the polarization of a second beam 共the probe兲, which propagates in the previously illuminated region along the same direction as the pump. This technique gives very precise information on the pure rotational relaxation dynam-ics of molecules with an anisotropic polarizability tensor, over a time scale extending typically from 10−4ns up to 10 ns. OKE has been used to study, inter alia, salol 关1兴,
orthoterphenyl 共o-TP兲 关2兴, m-toluidine 关3兴, benzophenone
and biphenyl methanol关4兴, and, more recently, supercooled
water关5兴. The OKE time window is well-suited to compare
the recorded relaxation dynamics with the predictions of mode coupling theory共MCT兲 above its critical temperature, Tc, because Tctypically corresponds, in real supercooled
liq-uids, to relaxation times of the order of 10– 102ns. This comparison has been quite fruitful and has demonstrated the existence, in the relaxation dynamics, not only of the primary ␣ relaxation process, but also of its precursors, the von-Schweidler and the so-called-fast processes关6兴 and, more
recently 关7兴, at yet shorter times, of pseudologarithmic
dy-namics evidenced in关4兴 in the 10−3– 10−2ns regime, that had escaped previous theoretical analyses.
The other important time-resolved optical spectroscopy method for the study of supercooled liquids is the transient-grating共TG兲 technique, which explores a much longer time interval and is better adapted to the lower temperature do-main. In this method, two pulses originating from a single laser propagate in nearly parallel directions, crossing at a small angle and interfering in the liquid. The duration of the pulses is typically 103 longer than in an OKE experiment. The technique turns out to be most powerful when the pump frequency is such that the liquid slightly absorbs the pump
beams. In the classical description of this experiment关8兴, the
interference between the electric field of the two pump beams creates a density grating, both directly, through an electrostrictive effect, and indirectly, through a temperature grating created by the energy absorption. The time evolution of the amplitude of a continuous probe beam diffracted by the grating monitors its decay. The duration of the recorded signal corresponds either to the lifetime, B, 共typically
10 to 103ns兲, of the longitudinal phonons with wave vector qជ launched by the pumps or, when energy absorption takes place, to the lifetime,⬘h,共104– 105ns兲 of the thermal grat-ing, which is limited by the heat diffusion process. One thus records signals in a time domain quite different from the OKE domain and very different information can be deduced from such experiments. This was recognized as early as 1995 by Nelson et al. 关8兴 who proposed, on the basis of an
el-ementary analysis of the origin of the signal, that an␣ relax-ation time,⬘L, could be directly accessed when the
condi-tion B⬍⬘L⬍⬘h is fulfilled. The other pieces of
information to be deduced would be the apparent phonon frequency,B, and its lifetime,B= 1 /⌫B, as well as the value
of ⬘h. The latter turned out to exhibit an apparent strong
increase for⬘L⬇⬘h, followed by a more important decrease
below its high temperature value when ⬘L⬎⬘h; this last effect had, in fact, been already noticed, more than 20 years before, by Allain et al.关9兴, through an early version of a TG
experiment. Salol 关10兴, CKN 关11兴, glycerol 关12兴, and o-TP
关13兴 have been analyzed within the Nelson et al. framework,
all revealing similar features. The theory had to be revisited 关14兴 when the accuracy introduced by the use of the
heterodyne-detection method共HD-TG兲 allowed one to show that, in some supercooled liquids composed of sufficiently anisotropic molecules, the transient grating was also opti-cally anisotropic; different polarizations of the probe beam led to signals with different shapes 关14,15兴. An elementary
theory of the signals recorded in m-toluidine, at different temperatures and with different polarizations of the probe, was presented in关14兴.
The complete expression of the HD-TG signals recorded when studying a supercooled molecular liquid formed of axi-ally symmetric molecules was given in 关16兴. The theory
made use of a set of equations, deduced from
logical considerations. Its major improvement over the pre-vious formulation 关8兴 was the introduction, following a
method already used by some of us in关17兴,
共1兲 on the one hand, of a coupling between the longitu-dinal phonons and the mean local orientation of the molecules and
共2兲 on the other hand, of the influence of these two quan-tities on the modulation of the local dielectric tensor. Relaxation mechanisms were also introduced in all the cou-plings, and the whole set of equations was later derived关18兴
with the help of a Zwanzig Mori technique. The theoretical expressions showed that different relaxation mechanisms, re-lated, in particular, to the propagation of the longitudinal phonons and to their coupling with the mean orientation could, in principle, be deduced from the signals measured with different polarizations.
In the current paper, we present the results of the first analysis of a HD-TG experiment, performed at LENS on m-toluidine and conducted along the lines proposed in关16兴
in two successive papers. This first paper, Paper I, describes the way the theory has to be approximated and adapted to the experimental situation in order to extract some specific infor-mation. Section II recalls the basic ingredients of the theory and the physical meaning of the results obtained in 关16兴.
Section III gives the formulas which will be compared with the experimental signals and the principal approximations necessary for fitting the data in a meaningful way. In the second paper, paper II关19兴, the results of paper I are tested
on m-toluidine, and are found to provide many different results.
II. THE THEORETICAL FORMULATION A. The HD-TG response function
In this section, we briefly present the theoretical formula-tion leading to the expressions needed to interpret HD-TG data. The details of the method have been previously pre-sented in 关16兴 and we mostly emphasize here the physical
aspects necessary for the understanding of the fitting proce-dure. References关16,17兴 assumed that the local fluctuations
of the dielectric tensor, ␦ញ共rជ, t兲, are linearly related to both the local mass density fluctuations,␦共rជ, t兲, and to the local orientational fluctuations, Qញ共rជ, t兲, of the liquid where Qញ共rជ, t兲 is a traceless, symmetrical, tensor which characterizes the mean orientation of molecules with axial symmetry at point rជ and time t
␦ញ共rជ,t兲 = a␦共rជ,t兲Iញ+ bQញ共rជ,t兲. 共2.1兲 In a HD-TG experiment, one directly monitors the time evo-lution of␦ញ共rជ, t兲 after an initial perturbation at time t=0, so that a and b characterize the two detection channels of the signal. Let Eជ1and Eជ2be the electric fields associated with the two coherent pump beams with wave vectors, respectively, k
ជ1= qជ0+2qជ and kជ2= qជ0−
q
ជ
2, with qជ0= q0zˆ, qជ= qxˆ, and qⰆq0. The two vectors kជ1 and kជ2define the scattering plane. In the TG experiment we discuss here, Eជ1 and Eជ2 are both either
per-pendicular to that plane 共VV polarization兲, or in that plane 共HH polarization兲 thus always 共nearly兲 parallel to each other. Interference between the two coherent electric fields of the pumps creates in the liquid a spatially modulated electric field, Eជint共rជ, t兲, which is the origin of a spatially modulated energy density, U共rជ, t兲. This density reads, in the impulsive limit we consider in this section,
U共rជ,t兲 = mEជ1· Eជ2cos共qx兲␦共t兲, 共2.2兲 where ␦共t兲 is a Dirac function and m the mean dielectric
constant of the liquid. A small fraction, H, of U共rជ, t兲 is ab-sorbed by the liquid and produces a thermal grating which instantaneously共on the time scale of the experiment兲 gener-ates a density grating.
The dielectric grating has two other origins, both linked to Eq.共2.1兲. Because the dielectric tensor changes with the
den-sity关the first term of the right-hand side 共rhs兲 of this equa-tion兴, the liquid tends to further minimize its modulated en-ergy density by creating a mass density modulation with the same wave vector qជ. The amplitude of this modulation is proportional to a =m/. Similarly, another minimization
of the energy density can be obtained by orienting the mol-ecules with respect to Eជint共rជ, t兲. The latter creates a torque on the molecules that eventually generates a nonzero, modu-lated, Qញ共rជ, t兲 with the same wave vector as above1 and with an amplitude proportional to b. This spatially modulated mo-lecular orientation is the third source of the dielectric grating, the corresponding contribution to the dielectric tensor being locally anisotropic, see Eq.共2.1兲 This anisotropy depends on
the common direction of Eជ1 and Eជ2: an index ex= 1 共ex = −1兲 indicates a VV 共HH兲 polarization of the pumps, thus the two possible independent torques. Similarly, the continu-ous, polarized, probe beam diffracted by this transient dielec-tric grating has amplitude and a change in polarization that decay as these density and orientational gratings decay. When Eជ1and Eជ2are parallel, with H or V polarization, there are关16兴, for each polarization, two, and only two,
indepen-dent signals that can be obtained by varying the polarization of the probe and of the diffracted beams: the first corre-sponds to the two beams having the same V polarization, the second to their having H polarization; the indexdtakes the
value 1 for VV polarization, and −1 for HH polarization. Let us now turn to the expression of the equations of motion of all the variables of the problem. Reference 关16兴
showed that, in the absence of sources, the Navier-Stokes equations which govern the dynamics of a molecular super-cooled liquid formed of axially symmetric molecules should be written as:2 ញ=共− ci2␦+b丢divvជ−m丢␦T兲Iញ+s丢ញ−丢Qញ ˙ , 共2.3a兲 1
This can be described as a spatially modulated OKE. 2
A somewhat more complex expression of Eq.共2.3c兲 was derived
in关18兴 but that paper also showed that the extra term does not play
Qញ¨ = –R2Qញ–⌫丢Qញ
˙
+⌳⬘丢ញ, 共2.3b兲 CV丢T˙ − Tm丢˙ −ⵜ2T = 0, 共2.3c兲
whereញ共rជ, t兲 is the stress tensor, related to the mass density current, or momentum density, Jជ共rជ, t兲, by the momentum conservation equation:
Jជ˙= divញ, 共2.4兲
while Jជ共rជ, t兲 is related to the mass density by the mass con-servation equation
˙ + div Jជ= 0. 共2.5兲 Here,␦T共rជ, t兲 is a local temperature fluctuation,ញ共rជ, t兲 is the strain rate tensor, defined through the mean velocity field v
ជ共rជ, t兲, and 丢 indicates a time convolution product while b共t兲,s共t兲,共t兲, and ⌫共t兲 are, respectively, the bulk
viscos-ity, the shear viscosviscos-ity, the rotation-translation coupling, and the pure orientational relaxation functions. ci is the
isother-mal sound velocity,Ra共very high兲 libration frequency, is
the heat diffusion coefficient,⌳⬘ a rotation-translation cou-pling constant,mthe mean mass density, and Tmthe mean
temperature of the liquid. Finally, CV共t兲 and 共t兲 are the
time-dependent specific heat at constant volume and tension 共or thermal pressure兲 functions. References 关16,18兴 showed
that those two functions contain not only a time dependent part but also an instantaneous part and should be written as Cv共t兲 = CV⬁␦共t兲 −␦C˙V共t兲, 共2.6a兲
共t兲 =⬁␦共t兲 −␦˙ 共t兲, 共2.6b兲 where CV⬁and⬁are these instantaneous parts, while␦CV共t兲,
and␦共t兲 are normal relaxation functions. Consequently, the specific heat at constant volume, CVth, and the pressure
coef-ficient, th, which are measured in a usual thermodynamic measurement, are given by
CVth=
冕
0 ⬁ CV共t兲dt = CV⬁+␦CV共t = 0兲 ⬅ CV⬁+␦Cv0, 共2.7a兲 th=冕
0 ⬁ 共t兲dt =⬁+␦共t = 0兲 ⬅⬁+␦0. 共2.7b兲 A long calculation关16,18兴 based on the Navier-Stokesequa-tions and involving the three sources described above gives the Laplace transform3共LT兲 of the response of the liquid to the pumps. In the impulsive limit of Eq.共2.2兲, and for the
detection mechanism described by Eq.共2.1兲, this LT reads,
up to a factor proportional to E1E2, as the sum of three terms.4 R dex共qជ,兲 = Rdex 1 共兲 + R dex 2 共qជ,兲 + R dex 3 共qជ,兲, 共2.8兲 where R dex 1 共兲 = i1 + 3dex 3 关b兴D −1共兲
冋
⌳⬘ m b册
, 共2.9a兲 R dex 2 共qជ,兲 =冋
a +⌳⬘
m b3d− 1 3 r共兲册
PL共qជ,兲 ⫻冋
m共兲 H 1 + ih共qជ,兲册
, 共2.9b兲 R dex 3 共qជ,兲 = − iq2冋
a +⌳⬘ m b3d− 1 3 r共兲册
PL共qជ,兲 ⫻冋
a +⌳⬘ m b3ex− 1 3 r共兲册
, 共2.9c兲whereh共qជ,兲 is the heat diffusion time for wave vector q
and frequency h共qជ,兲 ⬅h 0共q兲 CV共兲 CV共= 0兲 = − iCV共兲 q2 , 共2.10兲 this equation defining the heat diffusion timeh0共q兲.
Equations共2.9兲 are written in such a way that, in each of
them, the rhs is the product of a detection channel共s兲 共first square bracket兲, the source共s兲 共second square bracket兲, and, between the two brackets, a dynamical response function. The first term of Eq.共2.8兲, R
dex
1 共兲, is the LT of an OKE signal; it describes the pure relaxation dynamics of the mo-lecular orientation after their initial orientation by Eជint共rជ, t兲. This dynamics is fast enough not to be studied in the HD-TG experiments, so that D共兲 may be approximated by
D共兲 =R
2
. 共2.11兲
The second term, R
dex
2 共qជ,兲, and the third term, R
dex
3 共qជ,兲, of Eq. 共2.8兲, have been called, respectively, in 关16兴 the generalized ISTS 共impulsive stimulated thermal
scattering兲 signal5
and the generalized ISBS 共impulsive stimulated brillouin scattering兲 signal. Both contain three frequency-dependent factors. As the left-hand side factor
3
We define the Laplace transform of a response function f共t兲 关f共t兲 real, equal to zero for t⬍0 and tending to zero for t→⬁ by 关f共兲兴⬅LT关f共t兲兴共兲= i兰0⬁f共t兲exp共−it兲dt, which implies that its in-verse can be written as f共t兲=2兰0⬁lm关f共兲兴cos共t兲dt.
4
Equations共2.8兲 and 共2.9兲 formally differ from those appearing in
关16兴, Eqs. 共3.16兲 and 共3.17兲, because, on the one hand, we have
already separated here, under the form of R
dex
2 共qជ,兲 and
R dex
3 共qជ,兲, the two signals that will be called below the general-ized ISTS and ISBS signals, and, on the other hand, we have made use of the proportionality of the electrostrictive source with a, and of the orientational source with b.
5In the language of关16兴, this is rather the phonon part of the ISBS signal, the other part being the OKE signal, but, as the two signals do not appear in the same time domain, we shall here stick to this more physically meaningful denomination.
共first square bracket兲 involves a, b, and d, the shape of those
two signals depends on the polarization of the probe and of the detection beams, and the two channels contribute to the detection mechanism. The second factor, PL共qជ,兲, is the
propagator of a longitudinal phonon with wave vector qជand the right-hand side factor共second square bracket兲 depends on the sources. R
dex
2 共qជ
,兲 is generated by the heat absorption and thus does not depend on ex. Conversely, R
dex
3 共qជ ,兲 originates, as R
dex
1 共兲, from the coupling of the pumps to ␦ញ共rជ, t兲 and depends on exbecause the pumps tend to orient the molecules. This orientational aspect appears through the quantity br共兲, present in Eqs. 共2.9b兲 and 共2.9c兲, where
r共兲 =共兲 R2
共2.12兲 and thus involves the rotation-translation coupling function, 共兲. This coupling generates, in Eq. 共2.9c兲, a modulated
stress that launches longitudinal phonons; vice versa, the lat-ter orient the molecules and induce a contribution to R
dex
2 共qជ,兲 and R dex
3 共qជ,兲 that depends on the polarization of the probe and diffracted beams. In fact, as shown in关16兴,
the function br共兲 mostly influences the shape of the R
dex
2 共qជ
,兲. While its a contribution is always negative and does not change much with temperature, its b contribution exhibits a strong thermal variation, both in amplitude and in shape: in amplitude because br共兲 has a negligible value as long as the rotation-translation lifetime,, of共兲, is such thatBⰆ1 while it cannot be neglected at lower
tempera-tures whereBⰇ1; in shape because the duration of r共兲
is limited by the relaxation time . Conversely, as also shown in关16兴, the br共兲 term affects much less the shape of
R
dex
3 共qជ,兲: all its components oscillate at the phonon fre-quency around a zero mean value and decay as the phonon lifetime,B.
Finally, PL共qជ,兲, the second factor of Eqs. 共2.9b兲 and
共2.9c兲, is the longitudinal phonon propagator, the inverse of
which can be expressed关16兴 as
PL −1共qជ,兲 =2− q2
冉
c a 2 +m −1˜ L共兲 + imTm 2共兲 CV共兲 1 1 + ih共qជ,兲冊
. 共2.13兲 Here, cais the adiabatic sound velocity while˜L共兲 is calledthe “longitudinal” viscosity and contains contributions from all the relaxation functions entering Eqs.共2.3兲. The last term
of Eq.共2.13兲, called 共see below兲 the isothermal contribution
to PL共qជ,兲, is the counterpart of the change from the
isother-mal velocity, ci, to the adiabatic velocity in the propagator;
this aspect is recalled, as well as the exact form of˜L共兲, in
the Appendix. At high temperatures, this isothermal contri-bution can be neglected and the corresponding PL共qជ,兲 is
called the adiabatic phonon propagator关16,18兴. Conversely,
this contribution will be necessary to explain the final part of the ISTS signal at low temperatures, i.e., when the relaxation times involved in˜L共兲 are of the same order of magnitude
as h共qជ,兲. This more complete phonon propagator takes
into account the heat diffusion process that accompanies a quasistatic isothermal strain, and is thus called the isothermal phonon propagator, which explains the name given to the last term of Eq.共2.13兲.
III. THE EXPERIMENTAL SIGNALS AND THEIR ANALYTICAL EXPRESSION
Equations 共2.9b兲, 共2.9c兲, and 共2.13兲 represent exact
ex-pressions for the different signals, in the ideal case of perfect instrumentation and when the three sources contribute to the formation of the grating. They need, on the one hand, to be adapted to a realistic experimental situation and, on the other hand, to be transformed into expressions that will allow one to extract numerical information from the experimental sig-nals.
A. Theoretical expressions for the experimental signals
1. Experimental aspects
The experimental signals that are represented in, e.g., 关13兴, or in Paper II where we will present an analysis of
m-toluidine 关19兴, are not the inverse Laplace transforms of
Eqs.共2.9b兲 and 共2.9c兲 for three different reasons.
First, there is an experimental uncertainty on the time the pumps start to act. In关19兴, this time will be negative and we
thus write it as −t0.
Second, the response function of the experimental setup, including the duration of the pulses of the pumps, is not a ␦共t兲 function, but a function F共t兲, which extends over a time long enough not to be neglected.
Finally, the long-time part of the experimental signals is dominated by the aH共thermal grating detected by the density modulation兲 term of R
d,ex
2 共qជ,兲 which, as shown in 关16兴, is negative. Conversely the experimental signals, see, e.g.,关13兴,
are always represented with a positive long time part. A fac-tor 共−1兲 has to be added to the preceding formulas before comparing them to the experimental signals.
2. The isotropic and anisotropic signals
Equations 共2.9b兲 and 共2.9c兲 are too general because, as
already indicated in 关14兴, when one limits the study of the
HD-TG signals of supercooled liquids to times longer than, say 5 ns for q = 0.63m−1, either the signals exhibit no po-larization effects共e.g., for glycerol, or o-TP兲, or these effects are limited to the role of the probe and detection beams共e.g., for salol, or m-toluidine兲. No case is known where explays a role in this time regime, which implies that the molecular orientation source关the b3ex−1
3 r共兲 term of Eq. 共2.9c兲兴 may be neglected and the indexexdropped from any formula. The roles of␦共rជ, t兲 and Qញ共rជ, t兲 can then be disentangled if one defines the two following linear combinations of the experi-mental signals SVVexpt共qជ,兲 共d= 1兲 and SHHexpt共qជ,兲 共d= −1兲:
Sisoexpt共qជ,t兲 =
冋
2SVVexpt共qជ,t兲 + S HH expt共qជ,t兲
Sanisoexpt共qជ,t兲 =SVV
expt共qជ,t兲 − S HH expt共qជ,t兲
2 . 共3.1b兲
In view of the three points made in Sec. III A 1 and Footnote 3, those two signals have to be compared with
S␣共qជ,t兲 =2
冕
0⬁
Im关S␣共qជ,兲兴cos共t + t0兲d, 共3.2兲 where␣ stands either for “iso” or for “aniso” with
Siso共qជ,兲 = iaI0PL共qជ,兲
冋
m共兲 H 1 + ih共qជ,兲 − iaq2册
F共兲, 共3.3a兲 Saniso共qជ,兲 =⌳⬘ m b ar共兲Siso共qជ,兲, 共3.3b兲 I0representing the intensity on the detector.B. Fit formula for Siso„qᠬ,… and Saniso„qᠬ,…
In order to express Siso共qជ,兲 and Saniso共qជ,兲 under forms adapted to a fit of the experimental signals, it is convenient to write, e.g.,共兲 in the form, see Eqs. 共2.6b兲 and 共2.7b兲,
共兲 = ith关1 −g
共兲兴, 共3.4兲 with
g共兲 ⬅␦共兲
th , 共3.5兲
where, by definition, whatever the index i, a function gi共i兲
is the LT of a relaxation function gi共t兲 which has a value
equal to 1 for t = 0 and whose final relaxation dynamics is governed by a relaxation timei. Following Eq.共3.5兲,i is
then the →⬁ limit of the corresponding rhs. In practical applications, gi共t兲 will be, for instance, a Kohlrausch
func-tion, orgi共i兲 will be a Cole-Davidson function.
Using similar notations for CV共兲 and共兲, Eqs. 共3.3a兲
and共3.3b兲 may then be expressed as
Siso共qជ,兲 = − IisoPL共qជ,兲
冋
1 −g共兲 1 + ih共qជ,兲
− ia0
册
F共兲, 共3.6兲 Saniso共qជ,兲 = Kanisog共兲Siso共qជ,兲 共3.7兲 with h共qជ,兲 =h 0共q兲关1 − CVgCV共CV兲兴, 共3.8兲 a0= q2 mth a H, 共3.9a兲 Kaniso=⌳⬘ m R 2 b a 共3.9b兲 and Iiso= I0mth Ha; 共3.10兲for a given wave vector qជ, a0thus characterizes the relative efficiencies of a and H as sources of the signal while Kaniso characterizes the relative efficiencies of the molecular orien-tation and mass density channels in the detection mechanism.
PL
−1共qជ
,兲, Eq. 共2.13兲, also needs to be parametrized. We
shall write its˜L共兲 term under the approximate form
˜L共兲 = ⌬L2gL共L兲 + i␥¯ , 共3.11兲
where gL共L兲 is the LT of a “longitudinal” relaxation
func-tion, governed at long times by a unique relaxation time,L,
while␥¯ represents a weakly temperature dependent instanta-neous normal damping process. Also, as the isothermal con-tribution to PL共qជ,兲 represents a correction to the adiabatic
phonon propagator, we neglect the frequency dependence of CV共兲 and of共兲 in the first factor of this correction. One
thus obtains for this isothermal contribution
imTm 2共兲 CV共兲 1 1 + ih共qជ,兲 ⬵ −mTm 共th兲2 CVth 1 1 + ih共qជ,兲 = ca2
冉
1 ␥− 1冊
1 1 + ih共qជ,兲 . 共3.12兲 Here,␥ is the usual ratio between the specific heat at con-stant pressure and at concon-stant volume, in their thermody-namic limit, and the relation between the two last terms of Eq. 共3.12兲 is discussed in the Appendix. PL−1共qជ,兲 is thus approximated by PL−1共qជ,兲 =2− q2冋
ca2+⌬L2gL共L兲 + i␥¯ + ca2冉
1 ␥− 1冊
⫻ 1 1 + ih共qជ,兲册
, 共3.13兲where the expression ofh共qជ,兲 is given by Eq. 共3.8兲: the
adiabatic approximation of PL共qជ,兲 then consists simply in
making␥= 1 in Eq.共3.13兲 while␥takes a larger value in the isothermal case. This remark is of practical importance be-cause, as shown in关16兴, the long time part of the IST signal,
thus of Siso共qជ, t兲, relaxes on the time scale
¯h0共q兲 =␥h0共q兲. 共3.14兲 Equations共3.2兲, 共3.6兲–共3.8兲, and 共3.13兲 represent the
formu-las that have to be used to interpret a HD-TG series of ex-periments performed on a supercooled liquid formed of an-isotropic molecules. However, those expressions still contain too many temperature dependent parameters to allow for their reliable determination; some of them have to be deter-mined from independent experiments. We shall illustrate this point in Paper II 关19兴 where we shall make use of these
equations to interpret m-toluidine data recorded between 330 and 190 K at LENS.
APPENDIX
It was shown in关16兴 that the inverse of the phonon
propa-gator, PL −1共qជ,兲, was given by PL−1共qជ,兲 = 关2− q2„ci2+m−1L共兲 + g共q,兲…兴, 共A1兲 where L共兲 =b共兲 + 4 3T共兲 共A2兲 and g共q,兲 = − imTm 2共兲 CV共兲 ih共qជ,兲 1 + ih共qជ,兲 . 共A3兲
In Eq.共A2兲,b共兲 is the LT of the bulk viscosity relaxation
function while T共兲, called the “transverse” viscosity, see
关20兴, is given by
T共兲 =s共兲 −
⌳⬘
D共兲r2共兲. 共A4兲 The second term of the rhs of Eq.共A4兲 shows that this
trans-verse viscosity results from the renormalization ofs共兲, the
LT of the shear viscosity, by the coupling of the transverse phonon to the orientational density. Furthermore, g共q,兲 can be expressed as g共q,兲 = − imTm 2共= 0兲 CV共= 0兲 − imTm
冉
2共兲 CV共兲 −  2共= 0兲 CV共= 0兲冊
+ imTm 2共兲 CV共兲 1 1 + ih共qជ,兲 . 共A5兲The first term of the rhs of Eq.共A5兲 can be added to ci2 to
give ci2− imTm 2共= 0兲 CV共= 0兲 = ci2+mTm 共th兲2 CV th = ci2+ ci2共␥− 1兲 ⬅ ca 2 , 共A6兲
where we have made used of the classical thermodynamic relation mTm 共th兲2 CV th = ci 2共␥− 1兲 共A7兲 with ␥=CP th CV th. 共A8兲
This introduces the adiabatic phonon velocity. Furthermore, adding the second term of the same rhs tom−1L共兲 in Eq.
共A1兲 yields ˜L共兲 =b共兲 + 4 3T共兲 − im2Tm
冉
2共兲 CV共兲 −  2共= 0兲 CV共= 0兲冊
, 共A9兲 which is the “longitudinal” viscosity appearing in Eq.共2.13兲,the third term of the same rhs being the isothermal contribution.
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