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Cryogenics 35 (1995) 713-7160 1995 Elsevier Science Limited
Printed in Great Britain. All rights
reservedINIl-2275/95/$10.00
Thermal properties of polymer/particlecomp osites at low temp
eraturesM. JiickelDresden University of Technology, Institute of
Low Temperature Physics, D-01062Dresden, Germany
With different models, the thermal conductivity of composite
materials is calculatedon the basis of the thermal conductivity of
the matrix and filler, of the volume fractionand shape of the
fillers and with consideration of the thermal boundary resistance
inthe temperature range below 20 K. Measurements of the thermal
conductivity andspecific heat of epoxy resins with different
fillers (needle-shape Ag, HTSC powders)in the temperature range
2-80 K are presented. Comparison of the measured and cal-culated
thermal conductivity of these composites shows that above 20 K, the
thermalconductivity is determined to a high degree, by the shape of
the fillers.Keywords: thermal conductivity; particle composites;
specific heat; epoxy resins
A significant advantage of polymers in low-temperaturetechnology
is the possibility of modifying the polymerswith fibres or powders,
and thus, of modifying the mechan-ical or thermal properties. In
this paper, the influence offillers in polymers on the thermal
properties, especially thethermal conductivity in the
low-temperature range, is dis-cussed. A large number of theoretical
works have dealt withdifferent models of calculating the
properties.
Specific heatFor the calculation of the specific heat of
polymeric com-posites (cc) with fibre or powder reinforcement (cF)
in amatrix (c,), the simple linear equation generally agreeswell
with experimental data (denoted by subscripts C, Fand M,
respectively):cc = ( 1 -fic&+ +fc, (1)where f is the volume
fraction of the reinforcement. Forthe glass-fibre reinforcement in
epoxy resins, the specificheat, which is calculated from the
specific heats of glassfibre and epoxy resin, is only 10 to 20%
higher than themeasured specific heat of the compound (Figure I
).The reason is that the specific heat is a bulk effect of
thesolids. The deviation between the measured specific heatof the
composites and the specific heat calculated by Equ-ation (1) shows
that filler particles or fibres interact withthe surrounding
polymer layer and that they are able tomodify the properties of the
bulk material. This interactionmodifies the intrinsic energy of the
matrix, and, therefore,changes the specific heat. This has been
shown in an earlier
Figure 1 Specific heat of glass-fibre composite as a functionof
temperature: (----) pure epoxy resin; (*) glass fibre;
(0)glass-fibre composite; (-_) calculated specific heat by
Equ-ation (1)
paper, n which the specific heat of
epoxy-resin/particlecomposites decreases with increasing filler
content at tem-peratures above 40 K.Thermal conductivity of
particle-compositematerialsFor the calculation of the effective
conductivity of a com-posite (h,) it is necessary to know the
conductivities of the
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Thermal properties of polymer/particle composites: M.
JBckelfiller and matrix (h,, A,,,,), the volume fraction of the
filler,J the size and shape of the inclusion, and the arrangementof
fillers in the matrix. Most real composites have
irregulararrangements and nonhomogeneous sizes. Nevertheless,simple
models often give useful equations for the calcu-lation of the
thermal conductivity of composite materials2,.For many models,
thermal-conductivity calculations arebased on the assumptions that
the properties of the matrixare not altered by any interaction
between filler and matrixand that the heat transfer between filler
and matrix is ideal.The supposition of an ideal transfer of heat is
fulfilled whenthere is perfect adhesion between the different
phases, andthe wavelength of the propagated phonons is very
smallcompared with the microscopic irregularities of the
fillersurfaces or the filler diameter. This is realized for
tempera-tures above 20 K. For temperatures T < lo-20 K, the
ther-mal boundary resistance at the interface between matrix
andfiller becomes increasingly important owing to the
differentacoustic impedances of the two materials. The
acousticimpedance (p.u) is the product of the ultrasonic velocity
vand density p of the material. This boundary resistance (orKaptiza
resistance) increases strongly with decreasing tem-perature (R, -
T).A commonly used model for the calculation of the ther-mal
conductivity of composite materials containing dis-persed spheric
inclusions is that of Meredith and Tobias.They considered the case
of a cubic array of uniform-sizespheres and analysed the effect in
the neighbourhood of asphere by 248 of its close neighbours of the
thermal stream.The final equation for the effective thermal
conductivity ofthis arrangement ish = h A - 2f + 0.409.B.f - 2.1
33C.f3c MA +f+ 0.409.Blf3 - 0.906C~f03 (2)
2+k 6 + 3k 3 -with A = __-k ;B=--- 3k4 + 3k ;c= -+; k=$ M
This model has been found to be in a reasonable agree-ment with
experiments at temperatures above 20 K4,. Chenet al. developed the
following mode16, which incorporatesboundary resistance for
spherical inclusions at high concen-trations by a simple extension
of the work of Meredithand Tobias2.
h = h A*-2f-0.41 B*f 13-2. 13C*f c M A*+f+0.41B*j=0.91
C*f3with
2+k+2kr 6-3k+24krA*=_. l-k+kr B = 4-k+ I6krc* = 3-3k+9kr
4+3k+l2kr k=F; r-h,%M
(3)
where d is the diameter of the filler and Rg, the
boundaryresistance. For a compound of spherical Cu fillers
(diameter5.5 or 50 pm) in epoxy resin, Equation (3) is in
reasonableagreement with experiments between 2 and 300 K6. If
therelative thermal conductivity &J.Jh, 2 102, then the
thermalconductivity of the compound depends only on the concen-
tration f of the filler. As an example, with Ag or steelspheric
fillers, one obtains nearly the same thermal conduc-tivity for the
two compounds. Below 20 K the boundaryresistance increases with
decreasing diameter of the fillerparticles, and with that, the
thermal conductivity of thecompound decreases. With embedment of
very small par-ticles in the matrix in this temperature range, one
obtainssolids with very low thermal conductivity7,.For a compound
system consisting of a continuous phasewith a discontinuous phase
as particles of various shapesin either regular or irregular
assays, Hamilton and Crossergive for the thermal conductivity:
&=A, A,+(n- l)A,-(n- 1~(A,$4-A~)A,+(n- l)hM+flA,-hF) (4)
In this equation, n depends on the shape of the particlesand
upon the ratio of the conductivities of the two phases.Equation (4)
can be used with n as an empirical constant(shape factor) that must
be determined experimentally formixtures containing particles of
arbitrary shapes. The shapefactor n corresponds to the sphericity
rC, defined as the ratioof the surface area of the inclusion to
that of a sphere ofequal volume):
1 1-?V
With Equation (4), it was possible to describe success-fully the
thermal conductivity of a compound of high-tem-perature
superconductor (HTSC) particles (YBa,Cu,O,,,)in epoxy resin for
temperatures above 20 K with n = 8(Figure 2). Since the particles
do not diverge much fromthe spherical shape, this shape factor is
possible only if theporosity of the HTSC particles is taken into
consideration,which clearly can be seen in REM pictures of the
particles.If in Equation (4) the boundary resistance through a
ser-ies arrangement of the heat resistances of filler and bound-ary
area is taken into consideration, then1 2R, 1
h;;=7+A,
With this A;, one obtains from Equation (4):t
(6)
t ,, I10 T/K looFigure 2 Thermal conductivity of pure Epilox T
20-20 (0) withHTSC-powder-filled Epilox T 20-20 and pure HTSC (0)
as afunction of temperature. Filler content: (A) 31%; (x) 20%; (0)
9%
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Thermal properties of polymer/particle composites: M. JBckel
(7)
Equation (7) is also valid for temperatures below 20 K.With
Equation (7), it was possible to describe successfullythe phonon
part of the thermal conductivity of a particlecompound of Ag
needles in epoxy resin (Epotek 3 1D) fortemperatures between 0.1
and 80 K with 12= 55 (Figure 3).The composite investigated here is
a single-compoundepoxy that is filled with an estimated 17 ~01% Ag
needles.The REM picture shows that the Ag needles are 10 to30 pm
long and 1 to 3 pm thick. The shape factor wasdetermined from
measurements of the thermal conductivityabout 30 K. In this
temperature range, the electronic contri-bution to the thermal
conductivity amounts to only 20 to25%. The electronic contribution
is estimated from theWiedemann-Franz law. Clearly, the thermal
conductivityby electrons dominates in the composite below -0.5 K.
Forcomparison, the thermal conductivity of epoxy resin with56.9% Ag
spheres (48 pm) by de Araujo and Rosenberg5is shown in Figure 3.
Though the composite with needle-shape fillers includes only 17%
Ag, its thermal conductivitybetween 20 and 80 K is about three to
four times higherthan that of the composite with Ag spheres. Here
is exper-imental evidence of the large influence of filler shape
onthe thermal conductivity of composites.If the ratio of thermal
conductivities A,,,/A, 5 lo*, thenEquation (4) can be reduced toA
=A l+(n-1)fc M 1 -.f (8)
Here, the thermal conductivity of the composite is depen-dent
only on the shape factor 12and the filler concentrationJ The large
influence of the shape of the fillers is alsoshown in Figures 4 and
5, where the ratio AJAM in Equa-tions (4) and (8) is represented
with n as a parameter. Alarge AJA,,, ratio is obtained only with
large shape factorsn; this means the shape of the filler must
strongly divergefrom the spherical shape (e.g. needle shape). From
Figure
10'1:E2 IO0E.-z
1 0 - I
1 0 ;
10:Figure3 Thermal conductivity of a Ag-filled epoxy as a
func-tion of temperature: (0) Ag needleApotek 31 (17%); (0) Ag
pow-der (25%) by Reynolds and AndersonlO; f-.-.-_)
electroniccontribution of Ag; (....) pure epoxy resin; (-1
calculated curveby equation (7) with electronic contribution; * Ag
sphericalpowders (56.9%) by de Araujo and Rosenbergs
152 n
I I10 IO IO2 103
hr 1 h,Figure 4 Thermal conductivity (relative) of the compound
as afunction of the relative thermal conductivity of fillers for f=
0.2with shape factor n as parameter by Equation (4)
Figure 5 .Thermal conductivity (relative) of the compound as
afunction of filler factor f and for Ad&., > IO2 with n as
parameterby Equation (8)
4 it can be seen that for nearly spherical fillers (n <
lo),the thermal conductivity of the filler does not influence
thethermal conductivity of the composite when Aflh,,, > lo*.
Conclusions
For many technical applications, the specific heat ofpolymeric
particle composites can be calculated withsimple linear equations,
which are in sufficient agree-ment with experimental data.The
equation by Hamilton and Crosser3 and experimen-tal measurements
(Figure 3) have shown that the shapeof the particles can influence
the thermal conductivityof a composite quite strongly if the
conductivity of thefiller material is much greater (say, >lOO
times) thanthat of the matrix for temperatures above 20 K.For
spherical particles (n < 10) with AdA,,,, 2 lo*, onecan increase
the thermal conductivity of the compositeto not more than fivefold
the thermal conductivity ofthe matrix (Figure 5). In contrast, one
can increase thethermal conductivity much higher for composites
con-taining nonspherical particles (e.g. needles withn > 20).If
the boundary resistance between filler and matrix isknown, one can
also calculate the thermal conductivityfor a composite at
temperatures below 20 K with themodified Equation (7).
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Thermal properties of polymer/particle composites: M.
JBckelReferences
Jiickel, M. and Scheibner, W. Boundary layer induced
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J Appl Phys ( 1960) 31 I270- 1273Hamilton, R.L. and Crosser, O.K.
Thermal conductivity of hetero- 8geneous two-component systems Ind
Eng Chem Fundam (1962) 1187-191Garrett, K.W. and Rosenberg, H.M.
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