PHONON SCATTERING AND THE LINEAR
SPECIFIC HEAT TERM IN EPOXY-RESINS AT
LOW TEMPERATURES
S. Kelham, H. Rosenberg
To cite this version:
S. Kelham, H. Rosenberg. PHONON SCATTERING AND THE LINEAR SPECIFIC HEAT
TERM IN EPOXY-RESINS AT LOW TEMPERATURES. Journal de Physique Colloques,
1978, 39 (C6), pp.C6-982-C6-983. <10.1051/jphyscol:19786435>. <jpa-00217912>
HAL Id: jpa-00217912
https://hal.archives-ouvertes.fr/jpa-00217912
Submitted on 1 Jan 1978
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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. Rosenberg
The 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 specific heat of an epoxy-resin in the range 0.1 to 80 K
in which the characteristics of the resin were changed by altering the curing cycle. This appears to
affect the specific heat and the thermal conductivity 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 followed 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
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.
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 conductivity and the specific heat were each measured directly 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.
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 + YT 3 , where YT 3 is the
ordinary Debye term. Since at higher temperatures,
where the T 3 term would become dominant, c is the
same for all specimens, we can assume that each
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
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
above 1 K and (2) below 1 K both the A and B-cure
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786435
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 measured sound velocities /I/.
The X term is generally considered to be due
to some type of local excitation of a set of twolevel systems which have a frequency-independent
x n o v r n d cure
.A
cure
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 suggested / 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
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 propagation 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
$w/k = 60 K, a constant density of harmonic states
(1.3 x lo9 s ~ m - ~was
) used.
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
THERMAL CONDUCTIVITY RESULTS AND DISCUSSION.- Figu-
analysis, multiplied by a constant factor which was
equivalent, using accoustic velocities, to a mean
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
This approximately T
' behaviour
varies as TIs8
.
has been explained /2,3/ as being caused by the
scattering of the Debye phonons (T3) by the locali-
free path of about 0.1 nm. The curve is a good fit
at the plateau region and above, but it is not really satisfactory below I K.
It would therefore appear that our results
are not entirely consistent with current ideas on
heat transport in glassy materials.
zed states discussed in the previous section. These
should give a mean free path varying as T-', thereby yielding a T~ conductivity. However, this mechanism is not in accord with our specific heat results
/I/ Yap, B.C., private communication
at the lowest temperatures since the higher density
f localized states for the A and B samples should
ive them a lower thermal conductivity than the
/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
ully-cured sample - the opposite of what is obsered. Since we have shown that the Debye spectrum is
141 Tarasov, V.V., Phys. Stat. Sol. 3 (1967) 37
151 Zaitlin, M.P., and Anderson, A.C., Phys. Rev. B
he same for all the samples it would appear that
ven if the localized states are responsible forthe
behaviour at the lowest temperatures, the scatering is stronger in the normal-cure material.
The plateau in the conductivity has been ex-
References
12 (1975) 4475