Mirage technique in anisotropic solids
ABSTRACT Theoretical and experimental analysis of heat diffusion in an anisotropic medium are presented. The solution of the 3D thermal conduction equation in an orthorhombic medium is calculated by the mean of a Fourier transforms method. Experiments were performed on an orthorhombic polydiacetylene single crystal sample. The temperature field at the sample surface was determined using the photothermal probe beam deflection technique. Then the 3 coefficients of the thermal conductivity tensor have been measured.
JOURNAL DE PHYSIQUE IV
Colloque C7, suppltment au Journal de Physique 1 1 1 , Volume 4, juillet 1994
Mirage technique in anisotropic solids
X. Quelin, B. Perrin and G. Louis
Universitk Pierre et Marie Curie Paris VI, Dkpartement ak Recherches Physiques, URA 71 du CNRS, T22,
Boite 136,4 place Jussieu, 75252 Paris cedex 05, France
Abstract: Theoretical and experimental analysis of heat diffusion in an anisotropic medium are
presented. The solution of the 3D thermal conduction equation in an orthorhombic medium is
calculated by the mean of a Fourier transforms method. Experiments were performed on an
orthorhombic polydiacetylene single crystal sample. The temperature field at the sample surface
was determined using the photothermal probe beam deflection technique. Then the 3 coefficients of
the thermal conductivity tensor have been measured.
Many theoretical models for thermal diffusion of heat have been reported [1,2], but only a few
experimental investigations concerning anisotropic media are available. We can point out the flash method
 and the "steady-state potentiometric" technique  which have been used to analyse thermal properties
of orthotropic polymer samples.
For our study, the experimental device used is the photothermal probe beam deflection technique [5,
61. This method requires no mechanical contact with the sample and thus experiments on very small
samples can be performed.
We present a numerical simulation of the probe beam deflection for pure thermal waves progressing
in an orthorhombic sample, using the Fourier transform of the generalized 3D thermal conduction equation.
The sample under study is a polydiacetylene single crystal of ply-4BCMU (BCMU for
butoxycarbonylmethylurethane). This orthorhombic crystal of conjugated unbranched polymer chains
extending in the crystallographic c direction , is in a (b,c) planar configuration.
We have determined the 3D thermal conductivity tensor of this sample from the analysis of the
tangential component of the deflection angle, versus the offset between the pump and the probe beams.
2.1 Basic analysis
In the case of an orthorhombic geometry, the second order conductivity tensor can be written as
where K ~ ,
direction, and K, perpendicular to the (b,c) plane. Then, according to the energy conservation law, the
stationary 3D thermal conduction equation in such a medium, in the reference frame relative to the
crystallographic axis (Fig.1) and for a modulation frequency f = t0/2n is :
K,, denote the thermal conductivity coefficients respectively parallel and perpendicular to the chain
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jp4:1994765
JOURNAL DE PHYSIQUE IV
where I,, and rg are respectively the power and the radius of the pump beam. The density, the specific heat
and the optical-absorption-rate constant of the sample are denoted by p, Cp and P. The right hand term
describes the heat source which keeps the cylindrical symmetry of the gaussian pump beam profile; this
term is equal to zero in the thermal conduction equation of the surrounding media (respectively tg and xb for
the front and backing air).
"front medium g" 4
"backing medium b"
FIG. 1. Schematic description o f the geometrical configuration. The experimental device is attached
to the (X, Y, 2) axis, whereas the (U, V, 2) ffame refers to the crystallographic (c, b, a) axes (where c
denotes the chain direction). The angle (c, X) is denoted by 6.
In order to obtain the temperature distribution within the sample, we used a 2D Fourier transform in
the (u,v) plane, taking account of the boundary conditions at the interfaces (continuity of temperature and
heat flow) between the sample and the air. We can consider the sample as infinite in the (u,v) plane,
because of its dimensions larger than the caracteristic thermal wavelength associated to our frequency
modulation range. The temperature distribution tg in the probed atmosphere is deduced from the integral in
Fourier space :
where Tg(k) is the temperature in the Fourier space  and a : = kz +k: +2ili2. In this formula,
1 , = (2 K , l p, C , CO)
denotes the thermal diffusivity length in the probed air.
The probe beam deflection 4 is given by the line integral of the temperature gradient along the whole
optical path L :
Then, with the appropriate expression of tg in the (X,Y ,Z) reference frame, the tangential component +t is :
L d y , which reduces to : $, aiIdkksin(h) ~,(k) exp[-og(k)z]
where a:(k) = k2 +2i1i2. Equation (1) is numerically calculated to compare our theoretical model with
2.2 Asymptotic derivation
Physical considerations lead to simplifications in the theory presented above
in our case, the high value of the optical absorption constant (P = 2x105 cm-1 compared to a sample
thickness d = 70 pm), and the existence of a mismatch between the densities of the sample (p,) and the
probed medium (pg), (ps((p,),
lead to the following assumptions : exp(-bd)=0,
ogKg / o,K, S~6~b
/ o,Ka -0;
we introduce an effective thermal conductivity coefficient in the (b,c) plane, relative to the scanning
direction x :
K~=K,COS B+~,,sin 8
and a rescaled sample thickness 2 = d ,/X,
equation ( l) becomes
where - k2 +2 i1i2 ,and 1, = 4
Under such conditions, equation (2) has the same form as in an isotropic medium f9].
Furthermore, in the asymptotic limits of a thermally thin (d<ds), and of thermally thick (&>l,) samples,
the integrand is proportional to cri2and a , ' , and independant of K,. So, for low frequency experiments,
equation (3) reduces to :
denotes the thermal diffusion length in the direction 0.
Under these conditions, it can be schown [g] that the value of K~ can be derived from the dependence of
the amplitude A, of in terms of the transverse offset X (for large values of X).
Experiments were performed on a conventional mirage detection setup, which is described in detail
in ref. 181. Its summarized main features are the following: a 8 mW-power argon laser beam, tuned on 488
nm wavelength, provides the polarized gaussian profile pump beam. The scanning of the distance X.
between the pump and the probe beams is monitored by a 0.1 pm precision step motor. The low-frequency
modulation (0.1-20 Hz) is accomplished by an electromechanical shutter (nm Laser Products LS 200).
Monocrystals of ply-4BCMU [l01 are obtained by irradiating monomer single crystals with y rays.
These monocrystals are then cleaved along the (b,c) plane and typically 50 to 100 pm thick, 10 mm long
and 6 mm wide samples were available. These samples were hold by their edges on a Pyrex substrate, with
an air layer between the sample and Pyrex holder. The chain direction was found from the strong optical
dichroism of the 4BCMU. The maximum absorption of light occurs when the polarization of incident light
is parallel to c.
Experiments have been performed on samples of differents thickness. As the results were the same
within experimental error, we present data for only one sample of 70 pm thickness. In Fig. 2 we have
plotted the experimental data for 0 = 90 " (which means that X . is progressing in the perpendicular chain's
direction). The curves exhibit a maximum which is closely related to the pump beam radius rg but does not
influence the plot for large values of Xo. For the low frequency experiments (0.5 and 1 Hz), the slopes of
the lines give K~ = 0.42 W m-'K-1. The different behavior of the high-frequency experiments (4 and 10
Hz) allows us to pinpoint the crossover between the thermally thin and thick regimes. A numerical analysis
of these higher-frequency measurements, according to Eq.(l), leads to an estimation of Ka = 0.025 W m-'
K-'. Computed lines show quite good agreement with experiments.
In the same way, a value of K~ = 2.2 W m-1 K-1 can be deduced from low frequency experiments
with 0 = 0. For intermediate directions, d~fferent values of KO derived from the slope of low-frequency
JOURNAL DE PHYSIQUE IV
experiments have been found in good agreement with the theoretical dependence of K~ with 8, according to
Eq.(2) with K, and ~b previously determined.
FIG. I. Logarithm of the amplitude (At) of & (for 9 = 90°), is plotted for different modulation
frequencies (symbols represent experimental data). According to Eq.(l), solid lines were computed for
each frequency with the values rg = 360 pm, d = 70/cm, z = 150 pm, p, = 11 70 kg m-3, C, = I480 J kg-l
K-1, K~ = 0.42 W m-1 K-/ and K ~ =
0.025 W m-/ K-/.
We have shown that photothermal-probe-deflection is a suitable technique to study anisotropic
thermal conductivity behavior of materials, even when only small samples are available, due to the
contactless character of this method. The 3D thermal diffusion equation of heat in an orthorhombic medium
was solved using a 2D Fourier transform. So the thermal field in the probed front air can be numerically
The thermal conductivity tensor has been measured by the analysis of the transverse component of
the probe beam deflection. We have showed that for optically thick and thermally thin samples, the
complete expression reduces to the same one as in an isotropic medium, provided that an effective thermal
conductivity relative to the chain direction and a rescaled sample thickness are defined. Moreover, physical
considerations lead to simplifications in the theory so that direct measurements of the thermal conductivity
Various effects may be invoked to explain the large anisotropic behavior of 4BCMU (presence of
hydrogens bonds along the chains axis, lamellar structure), but further experimental data (sound velocities,
temperature dependence measurements) would be necessary for a complete heat diffusion analysis.
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[l01 samples provided by J. Berrehar, C. Lapersonne-Meyer and M. Schott, GPS, Univers. Paris 7 and 6.