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SIMULTANEOUS MEASUREMENTS OF THE THERMAL DIFFUSION COEFFICIENT AND OF
THE THERMAL CONDUCTIVITY OF TRANSPARENT MEDIA, BY MEANS OF A
THERMAL LENS EFFECT
P. Calmettes, C. Laj
To cite this version:
P. Calmettes, C. Laj. SIMULTANEOUS MEASUREMENTS OF THE THERMAL DIFFUSION COEFFICIENT AND OF THE THERMAL CONDUCTIVITY OF TRANSPARENT MEDIA, BY MEANS OF A THERMAL LENS EFFECT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1 125C1129. �10.1051/jphyscol:1972122�. �jpa00214912�
JOURNAL DE PHYSIQUE Colloque Cl, supplément au no 23, Tome 33, FévrierMars 1972, page Cl125
SIMULTANEOUS MEASUREMENTS OF THE THERMAL DIFFUSION COEF FICIENT AND OP THE THERMAL CONDUCTIVITY OF TRANSPARENT
MEDIA, BY MEANS OF A THERMAL LENS EFFECT
P. CALMETTES and C. LAJ
Service de Physique du Solide et de Résonance Magnétique Centre d'Etudes Nucléaires de Saclay, 91, GifsurYvette
Résumé.  Nous exposons le principe d'une nouvelle technique optique permettant de mesurer simultanément, de façon rapide et précise, le coefficient de diffusion thermique et la conductibilité thermique de tout corps transparent isotrope. Cette méthode, qui utilise l'échauffement produit dans i'échantillon par l'absorption non radiative de la lumière servant à la mesure, ne perturbe pas sensiblement le milieu.
Nous donnons aussi des résultats préliminaires obtenus sur un mélange et des liquides purs normaux, sur des solides amorphes ou cristallins et sur un mélange binaire critique.
Abstract.  We expose the principle of a new optical technique which allows the simultaneous measurements, in a precise and fast way, of the thermal diffusion coefficient and the thermal conductivity of any transparent isotropic medium. This method, which makes use of the heat produced by the nonradiative absorption of the light used for the measurement, perturbs the medium to a negligible amount.
We report here as well some preliminary results obtained on pure and multicomponent normal liquids, on amorphous and cristalline solids and on a critical binary liquid mixture.
1. Introduction.  High resolution spectroscopy, both by the photon beating or by interferometric techniques, can be used to perform the spectral ana lysis of the light scattered by a transparent fluid, and in particular to measure the Rayleigh linewidth.
It is thus possible, both for pure normal or critical fluids and for normal mixture, to obtain with an accuracy of about 10 % the value of the heat diffusion coefficient [l] :
where A is the thermal conductivity, C p the specific heat at constant pressure and p the density of the fluid.
On the other hand this experimental technique doesn't apply neither to solids nor to critical binary mixtures. In these cases indeed the entropy fluctuation line is either extremely difficult to observe because of signal to noise ratio problems or completely hidden by the Rayleigh component arising from concentration fluctuations, respectively.
A solution to the problem could be to measure independently the specific heat and the thermal conductivity by the usual methods. However these kinds of measurements, while they give good results, are quite long and delicate to perform and do not allow to sufficiently approach to the critical tempe rature, especially as far as the second technique is concerned.
Being especially interested in the measurement of the thermal diffusion coefficient of a critical binary mixture, we have thus deviced a new experimental.
technique which doesn't have the drawbacks of the ones just described. I t is an optical method which makes use of the heat produced by the absorption of the light which is used for the measurement.
Our preliminary results, obtained for several pure normal liquids, a multicomponent liuid, some isotro.
pic solids and a binary critical mixture show t h e advantages of this method.
II. Principle of the method.  Light and matter can interact in a non linear way, thus modifying the refractive index proportionally to the light intensity..
Different mechanisms are responsible for this : a) the Kerr effect, related to molecular reorienta tion in the applied electric field and thus essentially~
anisotropic,
b) the electrostriction, due to the hydrostatic pressure induced by the electric field inside the mate rial,
c) the electrocaloric effect, which is due to the ther mal dissipation of a fraction of the stored energy in the dielectrics,
d) the nonradiative absorption which creates a heating al1 along the path of the light beam.
The relative importance of these mechanisms depends on both the nature of the dielectric and
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972122
Cl126 P. CALMETTES AND C. LAJ of the time scale considered [2]. For times longer than
the caracteristic times of molecular reorientational processes, the Kerr effect can be neglected. Only the three other effects need be considered because they involve density modifications, which appear more slowly. But for an illumination of 10"1 second with an incoming power density of 1 W/cm2, the non radiative absorption, whose value is typically between
10~^{2} and 1 0^{ 4} c m^{ 1}, causes modification of the refractive index of about 10~5 to 1 0 1 0, i. e. 109 to
104 times greater than the electrostriction and the electrocaloric effect.
Thus about 10 ~6 second after the laser beam passes through the medium, the nonradiative absorption dominates inducing a transverse gradient of refractive index in it which acts then, usually, as a time depen dent divergent lens.
This effect has been noted and explained by Gordon
•et al. [3] who placed the samples directly in the laser cavity and observed long transient effect in the output power. Putting our samples outside the laser cavity we have tried to use this thermal lens effect to study the dynamics of propagation of heat in some trans
parent media and especially to measure both their thermal diffusion coefficient and thermal conductivity.
III. Theoretical background. — A pure normal fluid or an isotropic solid of absorption coefficient
is illuminated, starting at time t = 0, with a light beam along the zaxis and of total power P0. If the radial intensity distribution of the beam is gaussian,
•of radius ca0 at e~2, the sample will absorb per unit length the energy :
P = a P^{0}A e x p (  ^ ) e — H{i) where H(t) is the step function. This absorbed energy will produce a slight local heating 5T(r, z, t) whose dynamics is described by the heat diffusion equation :
where Dd is the mutual diffusion coefficient and t'^{c} = G>O/8 E>d^{ 1& a} second chacacteristic time, about
10^{2} times longer than t^{c}; k^{T} is the thermal diffusion ratio [4], P, T and c are the pressure, temperature and concentration respectively.
For most of binary liquids the second term of this expression is negligible, even in the vicinity of the critical point, and thus it reduces to the expression for a pure liquid.
jt6T(r,z,t) = DTV25T(r,z,t) +
aP0 2 I 2r2\ _a3
+ ZT — 2^{e x}P T^{ e} H(t).
pCP nco0 \ co0 I
In the limit of very small absorbancies the tempe rature distribution is given in terms of exponential integrals by :
8T(r, z, t) =
.^o{^{B I}(_^)_^{B}(__^L_)}^{e}.
4nA\ \ ml) \ <o% + 8 DT tn This temperature distribution gives a corresponding distribution of refractive index :
5n(r, z, 0 = — ST(r, z, f) = An
4nAdT\ \ mil \ 8D^{T}(t + t^{c})l ) where t^{c} = coo/8 D^{T} is a characteristic time.
As stressed in [1], this model has the weakness that ST > oo when t * co. This is because the heat dissi pation on the cell boundaries has not been taken into account. However we may expect that the heat dissi pation will not significantly change the phenomena in the illuminated volume as long as the cell diameter is great compared to the beam diameter 2 co^{0} and its length very large compared to 2 \JDt^{obs} where f^{obs } is the duration of an experiment.
Also the heat diffusion equation does not account for the phonons induced by the light beam and thus the results will only be valid as a good approximation for times longer than that necessary for sound to travel radially across the illuminated volume.
In similar conditions for a binary mixture the indu ced change in refractive index is :
For pure normal fluids and binary mixtures, a rigou rous calculation taking into account all the equations of hydrodynamics, shows that the description given above is sufficient for the interpretation of the expe rimental results.
IV. Experimental. — Our experimental set up, shown schematically in figure 1, has been designed to meet the above conditions. A laser beam of small
SIMULTANEOUS MEASUREMENTS O F THE THERMAL DIFFUSION COEFFICIENT Cl127
Multichanr~l Memory
S w i p l e cell
FIG. 1.  Schematic diagram of the experimental set UP. FIG. 2.  Path of the light beam passing through the sample cell.
divergence 2 O, with a gaussian radial intensity dis tribution [ 5 ] , is deflected vertically by a plane mirror M.
The sample cell, an optical glass cylinder 2 cm in diameter and 10 cm long, containing the liquid to be studied, is thus placed vertically, a necessary condition in order to avoid a breakdown of the cylindrical symmetry of the ce11 and light beam together owing to the small convection currents which arise when the fluid is locally heated. A second plane mirror M' deflects the beam towards the photomultiplier through a pinhole aperture of radius p, ^{= }5 x IO' cm, which is very carefully centered on the light beam.
A shutter S, placed between the laser and the first mirror allows to switch the beam on and off.
In a typical experiment the liquid, carefully filtered to avoid dust particles, was let unperturbed for at least 30 minutes, then the shutter was opened and the time dependent photocurrent stored in the memo ries of a multichannel apparatus, whose scanning begun a few milliseconds before the shutter was opened. In this way the time t = O was directly recor ded.
As soon as the shutter is opened, the sample heats up progressively and a light ray entering the ce11 parallel to the z axis will follow a circular path whose radius of curvature R (in the limit R B ce11 length) is given by [6] :
Substituting for R, one obtains :
with
and
Only the rays passing through the pinhole will contri bute to the photocurrent, i. e. the rays entering the ce11 at a maximum distance from the axis :
A ray entering at a distance r from the axis of the For usual fluids dn/dT < 0, and the fluid will act beam with an angle (O,/w, r) will, in the plane of the as a time dependent divergent lens for light rays near pinhole aperture, be at a distance p from this axis. the Correspondingly the photocurrent will As shown in figure 2 this distance p is given by : decrease from its initial value to an asymptotic value
when the fluid reaches thermal equilibrium.
The above calculations are however valable in the limit of the conditions mentioned earlier, which in the
 , % . case of our experimental set up means
< ^{t,,, } ^{6 }^{30 s for } ^{o, }= 0.5 mm.
where al1 the symbols are explained in the figure 2 V. Experimental results.  Figure 3 shows a typi and no is the refractive index of air. cal recording of the time dependent photocurrent
Cl128 P. CALMETTES AND C. LAJ obtained for methanol. This recording was given by
an XY recorder associated to the multichannel memory.
 l\ ^{Methanol }
ooo theoretical points
t seconds
1 2
FIG. 3.  Typical recording of the time dependent photocurrent.
Open circles represent the theoretical points.
In order to use such curves we first verify that the radial intensity distribution is gaussian. This was done by measuring the relative intensity passing through a set of calibrated pinholes of different diameters both very close to the ce11 and 1.50 meters farther away.
In this way, we also find the values of the beam radius w0 and its full divergence 2 6,.
and 2 6, = 0.81 f 0.01 mrad .
The results, such as the one shown in figure 3, were then computer analyzed by groups of 4 experiments simultaneously in order to increase the accuracy, using a nonlinear statistical refining program deve loped by Tournarie [7]. Besides the results this program also gives a quality factor [7] of 0.42 to 0.77 for the simultaneous fit of 4 independent experiments. This is quite satisfying and shows that the theory given above describes the phenomena correctly [SI. We must
point out however that for the: experiments concerning benzene this quality factor was significantly lower (0.2 to 0.5). No reliable explanation was found for this.
Some of the experimental results obtained for several pure fluids at room temperature are reported in Table 1.
In this table the first and second columns show the values of the absorption coefficient a and thermal conductivity A as found in references [9] and [IO].
The following columns, relative to the present work, give the values of the power Po entering the ce11 during the measurement, of the thermal diffusion coefficient D, and of the thermal conductivity A obtained using our experimental values of D,. and tabulated values of C, [IO]. The agreement with the values of the second column is quite satisfactory.
It was not possible to obtain directly the values of the thermal conductivity A, because of the difficulty of measuring absorbancies smaller than cm' by usual spectrophotometric methods. A simple way to get rid of this problem is to add a colored die, for instance a few p. p. m. of iodine, to the liquid to study and them measure its absorption. The addition of a few p. p. m. of any impurity should indeed not change the thermal conductivity to an appreciable amount. Once this value is known a new experiment on the pure sample allows to measure its real absorp tion coefficient.
The last column of Table 1 gives the values of the absorption calculated using the values of A of the preceding column. There is a slight discrepancy between these values and those reported by Leite et al. [9]. We believe that this is due to the fact that Leite et al. used small cells placed horizontally in the laser cavity and measured at equilibrium the diver gence of the laser beam. In these conditions the results are very much affected by the heat dissipation at the boundaries as well as by convection currents.
The last line of Table 1 is relative to a multicompo nent fiuid of unknown composition : the Chablis wine.
In this case the phenomena are also well described by the preceding theory.
a,,, x 104 A,,, x IO" P, x IO" orne,, ^{>( } lo3 A,,, x IO" ameas x IO4
cml erg/cm s OC erg s' cm2 s1 erg/c:m s OC cm'

Benzene 4.3
4.3 4.3
Toluene 4.7
CS2 5.9
Acetone 
Methanol 
Chablis (*) 
(*) For comparison purposes the experiment on the Chablis 1962 white burguntly wine was conducted at room temperature. However for better results the temperature of 8 OC is highly recornmended.
SIMULTANEOUS MEASUREMENTS OF THE THERMAL DIFFUSION COEFFICIENT Cl129
We have also successfully applied this technique to of the thermal diffusion coefficient DT is of the isotropic solids. A glass of unknown composition and order of 35 %, giving a weak divergence for C p . industrial plexiglas have given respective values for
D, of 9 . 2 x and 6.7 x cm2 sl. In a similar way we obtained the value 3 x 103 cm2 sl for a plastic crystal of succinonitrile.
But in this last case, the accuracy is poor because metallic mirrors were used. By local heating they also give a small beam divergence which has not been taken into account.
Preliminary experiments have also been performed on a critical binary mixture of cyclohexane and aniline in the temperature range of 0.2 to 30OK above the critical temperature. These measurements of D , which is proportional to the entropy fluctuations linewidth, motivated the establishment of this tech nique. It is probably impossible to perform directly such measurements by any other means. The sample cell, still placed vertically, was inside as small furnace temperature stabilized t o + ^{IOe2 }^{OC. } ^{The same }
computer analysis described above, shows that again the preceding theory correctly describes the obser ved time dependent photocurrent.
Unfortunately, the existence of small thermal gradients probably due to a bad thermal contact between the furnace and the sample ce11 has some what perturbed Our measurements. It was thus impos sible to obtain a set of measurements at different temperatures sufficiently precise to obtain a value of the critical exponent cc, characteristic of the divergence of the specific heat at constant pressure C, in the vicinity of the critical point [ I l ] :
where C: is a constant and C i the regular part of the specific heat. However we have been able to show that in the explored temperature range the variation
VI.  Accuracy of the measurements.  When the computer analysis of the time dependent photocurrent is performed, the characteristic time t , and the quan tity yPo are treated as unknown values. The accuracy on these two values is always better than 10 % in a single typical experiment. The accuracy can be greatly improved when a group of experiments js analyzed simultaneously.
For the absolute values of D T and A to the preced ing errors one must add those arising from the mea surements of the different experimental parameters 1, L, a,, O,, no, Po and also those concerning the values of a, dn/dT and a. Al1 these quantities can be measured with an accuracy of about 1 %, with the exception of the absolute power of the beam, where one may have a few percent incertitude, and cc whose measure ment is very difficult. But with colored samples an accuracy of a few percent is expected. Thus the lower limit of the incertitude is about 1 % on D, and 5 %
to 10 % for A in the best cases.
VII. Conclusions.  In spite of the present limita tions, to be ascribed essentially to the use of a ther mostat not adapted to this kind of experiment, this method seems extremely interesting. The thermal coefficient, the thermal conductivity and consequently the specific heat at constant pressure can be measured simultaneously in a precise and very fast way in a variety of isotropic transparent media : pure liquids, mixtures, amorphous and cristalline solids.
In addition this technique perturbs the sample to a negligible amount : in al1 Our experiments the local heating created by the laser beam has never exceeded 3 x OC. Thus the technique can be applied even in the immediate vicinity of a transition point, such as the critical point of a pure fluid, of a binary mixture of a ferroelectric crystal and so on ...
[l] LASTOVSKA (J. B.) and BENEDEK (G. B.), Phys. Rev.
Letters, 1966,17, 1039.
BERGÉ (P.), CALMETTES (P.), DUBOIS (M.) and LAJ (C.), Phys. Rev. Letters, 1970,24, 89.
[2] GIRES (F.), Thèse, Paris, 1968.
[3] GORDON (J. P.), LEITE (R. C.), MOORE (R. S.), PORTO (S. P.) and WHINNERY (J. R.), J. Appl. Physics, 1965,36,3.
141 LANDAU (L.) and LIFCHITZ (E.), Fluids Mechanics, Addison Wesley Publ. Co., 1959, 224 & 225.
[5] YARIV (A.) and GORDON (J. P.), Proc. I. E. E. E., 963,51,14.
[6] BORN (M.) and WOLF (E.), Principles of Optics Pergamon Press, 123.
[7] TOURNARIE (M.), J. Physique, 1969, 30, 737.
[8] For a simultaneous analysis of n independent expe riments the resulting global quality factor Q is at most equal to the product of the single experiment quality factors Qc(l < ^{i }< n) and unity
Qt(1 < i < n) corresponds to an undistorted fit.
[9] LEITE (R. C.), MOORE (R. S.), WHINNERY (J. R.), Appl. Phys. Letters, 1964,5,141.
[IO] LASTOVSKA (J. B.), Thesis M. 1. T., 1969.
[Il] SWIFT (J.), Phys. Rev., 1968, 173, 257.