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ACOUSTIC PROPAGATION IN AN EPOXY RESIN AT VERY LOW TEMPERATURES
P. Doussineau, W. Schön
To cite this version:
P. Doussineau, W. Schön. ACOUSTIC PROPAGATION IN AN EPOXY RESIN AT VERY LOW TEMPERATURES. Journal de Physique Colloques, 1982, 43 (C9), pp.C9-509-C9-511.
�10.1051/jphyscol:19829100�. �jpa-00222404�
JOURNAL DE PHYSIQUE
CoZZoque C9, suppZhnent au n012, Tcme 43, dicembre 1982 page C 9-509
ACOUSTIC PROPAGATION I N AN EPOXY R E S I N A T VERY LOW TEMPERATURES
P. Doussineau and W. Sch6n
Laboratoire d '~ltrasons * , Universite' Pierre et Marie Curie, Tour 13, 4 place Jussieu, 75230 Paris Cedex 05, France
Rdsumd. - Les variations d'attenuation et de vitesse d'ondes acoustiques longitudinales de frequences entre 150 et 2140 MHz ont Qte mesurdes dans une resine epoxy pour des temperatures entre 0,l et 10 K. Les comportements habi- tuellement observes dans les amorphes isolants ont PtB aussi observes dans ce matdriau : en particulier la variation en T3 de l'attenuation et la croissance logarithmique de la vitesse aux plus basses temperatures. Les resultats expe- rimentaux sont bien ddcrits dans le cadre du modele des systemes Si d e w niveaux.
Abstract. - The attenuation and velocity changes of longitudinal acoustic waves in the 150 - 2140 MHz range have been measured in an epoxy resin in the temperature range 0.1 K to 10 K. The behaviour usually seen in amorphous insulators was observed : in particular the T3 variation of the attenuation at the lowest temperatures, logarithmic increase of the velocity. All the features are well explained in the framework of the TLS model.
Introduction. - Amorphous materials have low temperature properties quite diffe- rent from those observed in crystals [ I ] . Moreover the behaviour is the same what- ever is the amorphous material : insulator, metal, semi-conductor or polymer. These low temperature properties are well explained in the framework of the two-level system (TLS) theory which assumes the existence of a particle moving in a double well potential by quantum-mechanical tunnelling [I]. The acoustic behaviour is governed by two processes : first the resonant absorption of phonons by the TLS and secondly, the relaxational absorption due to the perturbation of the TLS population out of equilibrium by the acoustic wave.
The relaxational absorption leads to a T~ acoustic attenuation at low tempera- tures when the relaxation is via the thermal phonons (direct process), and a temperature independent plateau varying linearly with the frequency at higher temperatures. These behaviours have been observed in many amorphous or disordered insulators [1,2]. The resonant absorption has been observed in amorphous polymethy- methacrylate (PMMA) by Brillouin scattering [3]. However the acoustic absorption in the 100 MHz range near 1 K in amorphous selenium [41 and polystyrene (PS) 151 is well described by a wT law. Moreover Brillouin scattering experiments in PMMA and polycarbonate (PC) have revealed a linear temperature behaviour of the attenuation of 18 GHz acoustic waves in the 2 to 10 K range [31. Therefore the question arises of the existence of a new and specific relaxation mechanism for the TLS in polymers.
In that view we have performed ultrasonic experiments in an amorphous epoxy resir down to lower temperatures (0.1 K) and up to higher frequencies (2.14 GHZ) than in previous works [4,5,6].
*~ssociated with the Centre National de la Recherche Scientifique.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19829100
C9-510 JOURNAL DE PHYSIQUE
Experiments. - Our measurements were made on a sample of commercial epoxy resin with longitudinal ultra- sonic waves of frequencies between 150 MHz and 2.14 GHz. In Figure 1 the relative change of the phase velocity of the ultrasonic wave was plotted as a function of the logarithm of the temperature. The usual Rn T increase of the velocity canbe seenasprevious- ly observed in PS or epoxy resin [5,6].
The results of the attenuation measurements of acoustic longitudinal waves are shown in Figure 2 in a log- log plot for five different frequen- cies. It is clear from this figure that at the lowest temperatures
(T < 0.8 K ) the attenuation -i- does not vary linearly with the temperatu- re, -ii- is only sligthly frequency dependent. Moreover the acoustic absorption at low temperatures and high frequencies is well described by a W'T~ law.
In this temperature range we have searched for a dependence of the absorption as a function of the acous- tic intensity by varying the acoustic intensity by more than 20 dB. No change of the attenuation was detected.
Consequently and owing to the tempera- ture dependence of the attenuation, we conclude that we have measured the high intensity (or saturated) part.
EPOXY RESIN
EPOXY RESIN .xxXx
LONGITUDINAL Xx jxx
x 530 MHz
x ~ X
+ 270 .. 1
1 + I I I I
.I -2 .5 1 2
TEMPERATURE (K)
Figure 1. - Semi-log plot of the relative phase velocity of longitudinal acoustic waves propagating in an amor- phous epoxy resin as a function of the temperature.
Figure 2.
.2 .5 1 2 5 10
TEMPERATURE (K)
Temperature dependent part of the attenua- tion of
longitudinal acoustic waves propagating in an amorphous epoxy resin as a function of the temperature on a log-log scale.
Solid lines are calculated curves.
I n t e r p r e t a t i o n . - Our experimental findings are explained in the framework of the TLS theory. We recall here only the results useful to what follows. The resonant interaction between TLS and ultrasonic wave leads to a velocity change [I] given by Av/vo = C RnT/_To where To is an arbitrary reference temperature, vo the sound velocity, C = P y 2 / pv; with p the density, y a deformation potential measuring the strength of the phonon-TLS coupling and P the density of states of the TLS. In addition to the resonant interaction, the acoustic wave undergoes a relaxational absorption (and dispersion) [I]. In terms of the complex change of the elastic constant c, it is given by
where B = l/kT , E is the energy splitting of the TLS, r = (no /E)' with A, the tunneling matrix element, w / 2 ~ the frequency of the ultrasonic wave, and TI is the longitudinal relaxation time of the TLS. TI is a characteristic time for the return to equilibrium of the TLS population perturbated by the ultrasonic wave. In an insulator the relaxation arises via the absorption or emission of one thermal phonon In that case TI is given by [ 7 1 :
- I BE 4 k3 YT
TI = r R3 T3 [F] coth - 2 with K 3 = n p h 1 r vT
(T = L or T stands for the polarization). In the general case the attenuation ( a = 1 m F ) and the velocity change ( = ~ e - ) are given by an analytical Ac
C n
calculatioz of the integral over r and a numerical integration of the integral over E. But some limiting behaviours can be obtained exactly. In the low temperature
K
regime the attenuation is given by a = d- C 2 T3 which is the behaviour observed 96 V
in the epoxy resin at the lowest temperatures (see Fig. 2).
The solid curves in Fig. 2 were obtained by the procedure mentioned just above with the following parameters : C = 5 x ; K g = 2.4 x lo9 K - ~ s-' ; and using vL = 3.25 x lo5 cm s-l , p = 1.2 g ~ r n - ~ . The agreement can be considered as very good up to 5 K. At higher temperatures there is a systematic discrepancy between the calculated curves and the experimental points. It can be ascribed to a supplementary process (Arrhenius peak) often observed in amorphous materials [1,41.
The velocity change has been calculated using the same set of parameters. Adding the resonant logarithmic contribution which cannot be saturated, it was not possible to obtain a reasonable fit. Moreover it seems necessary to include values of the parameter C different for the resonant and the relaxational part. Such a problem has already been encountered in the interpretation of the amorphous like properties of Na B-A1,03 [ 3 1 . Presently the velocity measurements are restricted to the low tempe- rature range (T 6 1.5 K). Measurements at higher temperatures are in progress and are expected to ~rovide supplementary tests of the theory.
Thus the attenuation of acoustic wave in an amorphous epoxy resin shows the behaviour usually observed in amorphous insulators. As a consequence the TLS theory applies well and it is not necessary to introduce a new relaxation process in amorphous polymers.
R e f e r e n c e s
[I] Amorphous S o l i d s , edited b y W. A. Phillips (Springer-Verlag, 1981).
[21 See for example, DOUSSINEAU P., F&NOIS C., LEISURE R. G., LEVELUT A., and PRIEUR J.-Y., J . Physique 41 (1980) 1193 , and references therein.
131 VACHER R., PELOUS J., SUSSNER H., SCHMIDT M., and HUNKLINGER S., in PY'oceedings of t h e EPS Conference, Anvers (1980)
[ 4 ] DUQUESNE J. Y., and BELLESSA G., J . Phys. C 13 (1980) L-215.
151 DUQUESNE J. Y., and BELLESSA G., J . Physique L e t t . 40 (1979) L - 193.
[61 MATSUMOTO D. S., REYNOLDS Jr C. L., ANDERSON A. C., Phys. Rev. B 19 (1979) 4277.
[ 7 ] JACKLE, J., 2. Phys. 257 (1972) 212.
