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PA EFFECT AT PHASE TRANSITIONS :
INTERFERENCE OF THERMOACOUSTIC AND THERMOELASTIC CONTRIBUTIONS
K. Junge, B. Bein, J. Pelzl
To cite this version:
K. Junge, B. Bein, J. Pelzl. PA EFFECT AT PHASE TRANSITIONS : INTERFERENCE OF THER- MOACOUSTIC AND THERMOELASTIC CONTRIBUTIONS. Journal de Physique Colloques, 1983, 44 (C6), pp.C6-55-C6-60. �10.1051/jphyscol:1983608�. �jpa-00223167�
JOURNAL DE PHYSIQUE
Colloque C6, suppl6ment au nO1O, Tome 44, octobre 1983 page C6- 55
PA EFFECT A T PHASE TRANSITIONS : INTERFERENCE OF THERMOACOUSTIC AND THERMOELASTIC CONTRIBUTIONS
K. Junge, B. Bein and J . P e l z l
I n s t i t u t fiir ExperimentaZphysik VI, Ruhr-Universitdt, 0-4630 Bochm, F . R. G.
R&sum& - Le comportement de fins Gchantillons de Gd au voisina- ge de l'ordre magngtique est bien expliquC par la comp6tition des effets thermoglastiques et thermoacoustiques.
Abstract: Distinct perturbations of the PA signal due to the competition of the thermoelastic and thermoacoustic effects are observed in thin Gd-samples near their magnetic ordering tran- sition. The experimental results are well reproduced by a linear theoretical approach.
1. INTRODUCTION
Thermoelastically forced motions of the solid surface can yield an important contribution to the photoacoustic (PA) signal of thin solids provided a time dependent temperature gradient is built up inside the sample. Experimental observations demonstrating the existence of the thermoelastic influence have been reported by several groups /1,2,3/.
In extending these investigations some authors inspected the potentia- lity of a purely thermoelastic measurement to determine thermal cha- racteristics / 4 / . The emphasis of this work is put on the interplay between the thermoacoustic and thermoelastic effects. Near phase tran- sitions the relative contributions of both mechanisms change markedly, thereby providing a sensitive tool for the study of thermal properties in the vicinity of a transition point. Here, we report on measurements performed on a thin Gd-disk near the magnetic ordering transition of this rare earth metal at about 292X. The experimental results present- ed in the last section are analyzed on the basis of a theoretical model described in the following.
2. THEORETICAL CONSIDERATIONS
An appropriate arrangement for the study of the interplay between thermoacoustic and thermoelastic effects is shown in Fig. 1 . The PA cell is formed by a cylindrical tube of gas length kg, the ends of the tube being closed by an optical window and the sample, respectively.
The sample has the shape of a thin disk with the radius a and the thickness L. An intensity modulated light beam hits the sample and generates the PA signal which is detected in the body o£ the gas by a microphone. Depending on the direction the light impinges on the disk, the pressure change in the cell cavity results from a "thermal re- flexion" process ( R ) or a "thermal transmission" process (T) of the heat waves in the sample. In both cases R and T, the acoustic response of the gas is composed of thermoacoustic and thermoelastic contribu- tions: The thermoacoustic part results from the heat flow at the SO- lid-gas interface; The thermoelastic contributions are due to elastic deformations of the disk contrabalancing the stress fields related to the nonuniform temperature distribution in the sample. The theoretical description of the complete problem may proceed from the basic rela- tion of the PA effect given by the integro-differential equation / 5 , 6 /
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983608
JOURNAL DE PHYSIQUE
Here, p(x,t) describes the pressure distribution in the gas volume, + y is the adiabatic coefficient, and vs (xs,t) and Fs (xs ,t) are the dis- placement velocity of the walls and the heat flux through them, res- pectively. For the solid sample, Fs is obtained from the solution of the heat diffusion equation
a
6 +P c - at = - div F + ~ ( 2 , t ) (2) where the effect of deformation on the temperature field has been neg- lected / 5 / . vs is obtained by solving the momentum balance of the iso- tropic elastic solid,
~.r V V ~ + (I.~+A) grad div
t
= grad [ ( 3 ~ + 2 ~ ) a $ ] (3) in the quasistatic limit / 5 / . Here, the Lam6 parameters p and A are assumed to be constant, the linear thermal expansion coefficient a is temperature depsndent, 6 is the driving oscillating temperature dis- tribution, and a is the displacement vector which serves to determine vs'l a .
w i n d o w
Fig. 1 Schematic representation of the PA cell (left) and equivalent mechanical model (right)
Both problems, equ. (2) and equ. (3), have been frequently treated in the literature, both in textbooks /7,8,9/ and special publications /lo/. Consequently, we here only present final results of our analy- sis. Evaluating equation (I), the combined effect of the thermoacou- stic and thermoelastic contributions to the PA effect can be repre- sented by an equivalent mechanical model sketched in Fig. Ib. The de- flection of the disk due to the thermoelastic force For corresponding to the linear displacement Axo, is contrabalanced by the elastic re- storing force of the plate (force constant D2) and the pressure of the gas body (force constant Dl). Axa represents the barothermal expansion of the gas piston in the scope of the RG model. The PA signal is given by
The suffices R,T stand for the thermal reflexion and thermal trans- mission configuration, respectively. In the simplest approximation the heat diffusion problem is treated in one dimension and the disk is assumed to be simply supported. The acoustic pressure is then given by the relation
where
Dl = yp0A/Rg r
and where a , o and K are the thermal expansion coefficient, Poisson's ratio and the bulk modulus of the solid, respectively.
a0
anda L
describe the magnitude of the oscillating temperature at the front and at the rear of the illuminated plate. E is a factor which takes care that the actual experimental configuration may deviate from the assumption of a simply supported plate. With regard to the expression of the thermoelastic force F,, it may be emphasized that the thermal expansion coefficient a is treated as a spatially dependent quantity.
3. EXPERIMENTAL AND RESULTS
The PA signal was measured as a function of temperature in the range from 27oK to 330K. Gd-disks with a thickness varying between 80p and 2 0 0 ~ where prepared from polycrystalline material which had been puri- fied by arc melting. The Gd-samples were attached to the PA cell shown in Fig. 2. The cell was mounted into a cryostat where the temperature of the whole assembly was slowly and continuously changed. The tem- perature was controlled by thermocouples attached to the sample and the cell body. The Fig. 3 and Fig. 4 show recordings of the amplitude and of the phase angle of the PA signal measured at different chopping frequencies. The theoretical curves at the right hand side of the res- pective figures are obtained with relation (4). For the calculations the temperature dependence of the specific heat capacity and of the thermal expansion coefficient of Gd are taken from literature /11,12/.
The parameter E which has been adjusted to the experimental values, was found to vary only slightly for the different disks.
At low frequencies the thermal reflexion as well as the thermal trans- mission signal essentially reproduce the temperature variation of the specific heat capacity. With increasing frequencyxthe thermoelastic motion of the disk starts to dominate the change of the transmission signal (T) at the phase transition point.
sample support M?Ple
!
/Fig. 2 PA cell design used in the present experiments
JOURNAL DE PHYSIQUE
, I , I ' L I
Gd : I = O.?l mm , thermal transmission
temperature T I K
Fig. 3 Temperature dependence of the PA amplitude (top) and P A phase angle (bottom) recorded from a Gd-foil. With the thickness of 1 1 0 ~ in thermal transmission at different modulation frequen- cies: In experiment (left) and theory (right).
I I I I I I I ~ " I I
Gd : I = 0.11 m m , thermal reflexion
temperature TI K
Fig.4 Temperature dependence of the PA phase angle recorded from a Gd-foil with a thickness of 11op in thermal reflexion at dif- ferent modulation frequencies: In experiment (left) and theo- ry (right)
.
A constant value of +8K has to be added to theright temperature scale.
4. CONCLUSIONS
The good agreement between the recorded curves and the theoretical calculations give proof that in an inverse procedure these measure- ments can be used to determine cp(T) and a ( T ) in the vicinity of the phase transition. Choosing a modulation frequency where in thermal transmission configuration the thermoelastic contribution competes with the thermoacoustic one, these resulting dramatic changes of the phase angle may be used to investigate the critical behaviour of ther- mal quantities near transition points.
Acknowledgements: The authors would like to thank D. Kruger for his help in the experiments.
C6-60 JOURNAL DE PHYSIQUE
References
/I/ CAHEN D., Proc. Workshop on Photoacoustics, Bad Honnef, FRG, 1981 /2/ PELZL J., Proc. Workshop on Photoacoustics, Bad Honnef, FRG, 1981 /3/ ROUSSET G. and LEPOUTRE F., Rev. Phys. Appl. =(1982)201.
/4/ CHARPENTIER P., LEPOUTRE F. and BERTRAND L., J. Appl. Phys.
53
(1982)608.
/ 5 / PELZL J. and BEIN B.K., Zeitschr. fiir Phys. Chemie Neue Folge,
1983, in print.
/6/ BEIN B.K. and PELZL J., in this conference.
/7/ CARSLAW H.S. and JAEGER J.C., Conduction of Heat in Solids, Cla- rendon Press, Oxford, 1973.
/8/ NOWACKI W., Dynamic Problems of Thermoelasticity, Noordhoff Int.
Publ., Leyden, 1976.
/9/ PARKUS H., Thermoelasticity, Springer Wien-New York, 1976.
/lo/ JACKSON W. and AMER N.M., J. Appl. Phys. 51(1980)3343.
/I 1 / TOULOUKIAN Y .S. et al. (Ed.)
,
~hermo~hysical Properties of Matter, IFI/Plenum, New York-Washington, 1970./12/ DARNELL F.J., Phys. Rev. E(1963)1825.
