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Submitted on 1 Jan 1978
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PHONON SCATTERING AND THE LINEAR
SPECIFIC HEAT TERM IN EPOXY-RESINS AT
LOW TEMPERATURES
S. Kelham, H. Rosenberg
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-982
PHONON SCATTERING AND THE LINEAR SPECIFIC HEAT TERM IN EPOXY-RESINS AT LOW TEMPERATURES
S. Kelham and H.M. RosenbergThe Clarendon Laboratory, Oxford, 0X1 3PU3 U.K.
Résumé.- La chaleur spécifique et la conductibilité thermique d'une rësine-epoxy ont été mesurées de 0,1 - 80 K pour des échantillons bien recuits et aussi pour ceux avec des recuits différents. Les résultats ne sont pas en complet accord avec les théories actuelles.
Abstract.- The specific heat and the thermal conductivity of an epoxy—resin has been measured from 0.1 - 80 K for specimens with a normal cure and for those with other cures. The results are not in complete agreement with current theories.
INTRODUCTION.- This paper describes the results of experiments on the thermal conductivity and speci-fic heat of an epoxy-resin in the range 0.1 to 80 K in which the characteristics of the resin were chan-ged by altering the curing cycle. This appears to affect the specific heat and the thermal conducti-vity in different ways.
SAMPLES AND EXPERIMENTS.- The epoxy-resin used was Shell Epikote 828 with Epikure NMA hardener and BDMA accelerator in the proportions 100 : 90 : 0.5 by weight respectively. The normal curing cycle of this resin is a precure at 100°C for 2 hours fol-lowed by a cure at 200°C for 4 hours. During the precure polymerization occurs and during the final cure cross-linking between neighbouring polymer chains is the main effect, although of course both mechanisms will occur to a certain extent during each of the curing stages.
Two other methods of preparation were used. In one (A-cure) the resin was only cured at 67°C for 36 hours and in the other (B-cure) it was cured at 200°C for 1 hour.
In the range 0.1 to 2 K the thermal conduc-tivity and the specific heat were each measured di-rectly but from 2 to 80 K the thermal conductivity and the diffusivity were measured by applying a constant and an alternating heat input respectively to one end of a conventional Searle's bar specimen.
SPECIFIC HEAT RESULTS AND DISCUSSION.- The results for the specific heat, c, which were obtained for the three types of sample are plotted as c/T3
against T in figure 1. It should be noted that (1) the specific heats of all the samples are the same above 1 K and (2) below 1 K both the A and B-cure
Fig. 1 : A plot of c/T3 against T (log scales) for
the epoxy-resins with different cures. The curve is calculated as discussed in the text.
specimens have almost the same specific heat and this is higher than the normally-cured material.
At the lowest temperatures the curves have a slope of approximately - 2 , which would suggest that c is of the form XT + YT3, where YT3 is the
ordinary Debye term. Since at higher temperatures, where the T3 term would become dominant, c is the
same for all specimens, we can assume that each does have the same Debye term. This is confirmed by the fact that the sound velocity of these samples' is the same for each of them /l/. The best fit to the results below 1 K yields the following values for X and Y
A and B cure 7.6
Normal cure 4.2
The Y values are very close to the Debye value of 28.9 which has been calculated using mea-
sured sound velocities /I/.
The X term is generally considered to be due to some type of local excitation of a set of two- level systems which have a frequency-independent distribution /2,3/. It would therefore appear that in the normal specimens the number of these systems is reduced because of the stronger cross-linking.
At higher temperatures where one would expect the Debye specific heat to start flattening off,the c/T3 plot againhas a slope of
-
2. It has been sug- gested / 4 / that the vibrational modes of linear chains might modify the Debye density of states so that it is constant beyond a certain frequency. The curve in figure 1 was actually calculated using a Debye spectrum up to Hulk = 60 K plus a constant density of two-level systems (1.4 x l o 6 s ~ r n - ~ ) up to $w/k = 1 1 K plus a constant density of harmonic states (2.3 x 10' s ~ m - ~ ) from Hw/k = 1 1 to 60 K. For higher frequencies beyond the Debye cut-off of $w/k = 60 K, a constant density of harmonic states (1.3 x lo9 s ~ m - ~ ) was used.THERMAL CONDUCTIVITY RESULTS AND DISCUSSION.- Figu- re 2 shows the results of the thermal conductivity measurements. From 0.1 to 15 K the A and B samples have a conductivity about 25 % greater than the normal cure material. Below 1 K the conductivity varies as TIs8
.
This approximately T' behaviour has been explained /2,3/ as being caused by the scattering of the Debye phonons (T3) by the locali- zed states discussed in the previous section. These should give a mean free path varying as T-', there- by yielding a T~ conductivity. However, this mecha- nism is not in accord with our specific heat results at the lowest temperatures since the higher density f localized states for the A and B samples should ive them alower
thermal conductivity than the ully-cured sample-
the opposite of what is obser- ed. Since we have shown that the Debye spectrum is he same for all the samples it would appear that ven if the localized states are responsible forthebehaviour at the lowest temperatures, the scat- ering is stronger in the normal-cure material.
The plateau in the conductivity has been ex-
x n o v r n d cure
. A c u r e
Fig. 2 : A plot of the thermal conductivity against T (log scales) for the epoxy-resin with normal cure and with A-cure. The results for B-cure are almost the same as for A-cure and for clarity they have been omitted. The curve is calculated as discussed in the text.
plained /5/ as being due to a cut-off in the propa- gation of the higher frequency phonons (s 10" Hz). The curve shows an attempt to fit the fully-cured sample results by assuming a scattering of Debye phonons by the two-level systems and by Rayleigh scattering 151. For frequencies $u/k > I 1 K we used the same density of states as in the heat capacity analysis, multiplied by a constant factor which was equivalent, using accoustic velocities, to a mean free path of about 0.1 nm. The curve is a good fit at the plateau region and above, but it is not real-
ly satisfactory below I K.
It would therefore appear that our results are not entirely consistent with current ideas on heat transport in glassy materials.
References
/I/ Yap, B.C., private communication
/2/ Anderson, P.W., Halperin, B.I., and Varma, C.M., Phil. Mag.
25
(1972) 1/3/ Phillips, W.A., J. Low Temp. Phys. z(1972) 351
141 Tarasov, V.V., Phys. Stat. Sol.
