International Journal of Modern Nonlinear Theory and Application, 2012, 1, 40-46
doi:10.4236/ijmnta.2012.12005 Published Online June 2012 (http://www.SciRP.org/journal/ijmnta)
Effect of Geometry of Filler Particles on the Effective
Thermal Conductivity of Two-Phase Systems
Deepti Chauhan*, Nilima Singhvi, Ramvir Singh
Department of Physics, Heat Transfer Laboratory, University of Rajasthan, Jaipur, India
Received March 24, 2012; revised April 22, 2012; accepted April 30, 2012
ABSTRACT
The present paper deals with the effect of geometry of filler particles on the effective thermal conductivity for polymer
composites. In the earlier models, less emphasis has been given on the shape of filler particles. In this paper, expres-
sions for effective thermal conductivity has been derived using the law of minimal thermal resistance and equal law of
the specific equivalent thermal conductivity for three different shapes i.e. spherical, elliptical and hexagonal of filler
particles respectively. Calculated values of effective thermal conductivity for various samples using the derived expres-
sions then compared with experimental data available and other models developed in the literature. The results calcu-
lated are in good agreement with the earlier experimental data and the deviation, is least in our expressions showing the
success of the model.
Keywords: Effective Thermal Conductivity; Polymer Composites; Minimal Thermal Resistance; Shape of Filler
1. Introduction
Particle filled polymer composites have become impor-
tant because of their wide applications in science and
engineering for technological developments. Incorporat-
ing inorganic fillers into a matrix enhances various physi-
cal properties of the materials such as mechanical strength,
elastic modulus and heat transfer coefficient [1,2]. In gen-
eral, the mechanical properties of particulate filled poly-
mer composites depend strongly on size, shape and dis-
tribution of filler particles in the polymer matrix [3].
Composite materials have primarily been used for stru-
ctural applications. Polymer composite materials have
been found extremely useful for heat dissipation applica-
tions like in electronic packaging, in computer chips [1,
2]. Polymer composites filled with metal particles are of
interest for many fields of engineering. The interest in
these composite materials arises from the fact that the
thermal characterization of such composites are close to
the properties of metals, whereas the mechanical proper-
ties and the processing methods are typical for plastics [3,
4]. Adding fillers to plastics, changes the behaviour of
polymers and a significant increase in the effective ther-
mal conductivity of the system has been observed [5,6].
Therefore, it is very important to understand the heat
transfer mechanism in polymer composites. Maxwell [7]
calculated the effective thermal conductivity of a random
distribution of spheres in a continuous medium, which
worked well for low filler concentrations. Bruggeman [8]
derived another model for the effective thermal conduc-
tivity, under different assumptions for permeability and
field strength. Hamilton and Crosser [9] extended Max-
well’s model to include the empirical factor n to account
for the shape of the particles (n = 3 for spheres and n = 6
for the cylinders). Liang and Liu [10] gave a theoretical
model for evaluating the effective thermal conductivity
of inorganic particulate polymer composites. Liang and
Lia [11] measured the effective thermal conductivity
(eff
K
) of hollow glass-bead (HGB) filled polypropylene
composites by means of a thermal conductivity instru-
ment to identify the effect of the content and size of the
HGBs on the effective thermal conductivity (eff
K
). Tekce
et al. [12] studied the thermal conductivity of copper
filled polyamide composites using the Hot-Disk method
in the range of filler content 0% - 30% by volume for
short fibers and 50% - 60% by volume for particle shape
of plates and spheres. Nielson [13], Cheng and Vachon
[14] and Agari et al. [15] proposed various theoretical
models for describing the heat transfer mechanism in
polymer composites. Liang and Lia [16] studied the heat
transfer mechanism in inorganic hollow micro-spheres
filled polymer composites and proposed a heat transfer
model.
In the present paper, expressions for effective thermal
conductivity (eff
K
) of various inorganic particles filled
polymer composites has been developed for different
shapes of filler particles as spherical, elliptical and hex-
*Corresponding author.

D. CHAUHAN ET AL. 41
agonal using the law of minimal thermal resistance and
equal law of the specific equivalent thermal conductivity.
The calculations were done for samples and compared
with the experimental results available in the literature.
2. Theory
There are several mathematical models for predicting the
effective thermal conductivity of particle filled polymer
composites like Maxwell model, Russell model, Hamil-
ton and Crosser model and others [7-17]. Some of them
are as under:
Maxwell model


22
2
fm fm
eff m
fm fm
K
KKK
KK
KK KK


 
 (1)
Russell model
22
33
22
33
1
1
m
ff
f
eff m
m
ff f
f
K
K
KK K
K

 



 






 





(2)
Hamilton and Crosser model
  
 
11
1
fm fm
eff m
fmfm
KnKn KK
K
K
KnKKK


 
  (3)
Here m
K
and
f
K
are the conductivity coefficients
of the polymer matrix and the filler particles, and n is an
empirical factor to account for the shape of the filler par-
ticles, (n = 3) for spheres and (n = 6) for cylinders. These
models are basically developed for dilute dispersion.
Liang and Liu [10] established a theoretical model of
inorganic particulate-filled polymer composites. This mo-
del is based on the specific equivalent thermal resistance
of the element of composites, when only heat conduction
is considered. Therefore, the calculation of the equivalent
thermal conductivity for composites can be attributed to
the determination of the equivalent thermal conductivity
of the unit cell with the same specific equivalent thermal
resistance.
In this paper, we start with the assumption that an over-
all composite consists of a number of small squared unit
cells having sides H and each cell contains only one filler
particle which can be of different shapes as spherical,
elliptical and hexagonal, kept at the center of the unit cell.
We also assumed that the heat flow into the element is
from the top of the square. A series model of heat con-
duction through the unit cell in an inorganic particulate
filled polymer composite is shown in Figure 1.
To study the effect of various shapes of filler particles
on the effective thermal conductivity (eff
K
) of the sys-
(a)
(b)
(c)
Figure 1. Heat transfer series model for different shapes of
filler particles.
tem, we assumed that the filler particle is located at the
center of the unit cell. The cell is then divided into three
parts as shown in Figure 1. The mean thermal conduc-
tivity of the matrix and the filler are
p
K
and
f
K
re-
spectively. The quantity of heat flowing through a body
depends upon the heat transfer route in the materials. The
equivalent thermal resistance of these three parts for dif-
ferent shapes is given as:
For spherical shape
13
p
h
RR
K
A
 (4)
and

2
2
4
p
pff
r
R
K
VKV
 (5)
Here
2
21.33π
p
VHr r3
r
(6)
and (7)
3
1.33π
f
V
For elliptical shape

D. CHAUHAN ET AL.
42
Here
1
13
p
h
RR
K
A
 (8) 2
22.433
p
VHr r 3 (14)
a nd

2
2
p
pff
r
R
K
VKV
 (9) a
nd 3
2.43 3
f
Vr (15)
Here
A
is the total cross-sectional area of the unit
cell,
p
V and
f
V are the volume of polymer matrix and
filler particles, and r is a variable parameter defined in
Figures 1(a)-(c) respectively.
Here
2
2.828 0.166π
p
VHr
3
r
r
(10)
The total equivalent thermal resistance is then given
by
a nd (11)
3
0.166π
f
V
For hexagonal shape
12
RRR R
3

 (16)
2
13
p
h
RR
K
A
 (12) Solving Equations (4)-(15) and keeping the volume of
the unit cell for different shapes same as that of the
spherical shape, the expressions for effective thermal con-
ductivity are obtained for spherical, elliptical and hex-
agonal shapes as under:
a nd

2
2
4
p
pff
r
R
K
VKV
 (13)
For spherical shape of filler particles, the equation is

1
3
11
33
1
6
11 2
π2
4ππ
39π
eff
f
pp f
p
fp
f
K
KK
K
KK







 







(17)
For elliptical shape of filler particles, the equation is

1
3
11
33
1
6
11 2
π2
ππ
69π
2
eff
f
pp pf
f
p
f
K
KK K
K
K







 







(18)
and for hexagonal shape of filler particles, the equation is

 
1
3
1
3
1
3
1
11.23 2
1.62 1.29
eff
fp
pp
f
fp
f
K
K
KK
K
K






(19)
Here
f

is the volume fraction of filler in the resin
matrix.
3. Results and Discussion
In this paper, we have obtained semi empirical relations
for effective thermal conductivities given by Equations
(17)-(19). These equations have been derived taking into
account the different shapes of filler particles as spherical,
elliptical and hexagonal. And then the calculations were
done considering these shapes. The calculated values are
then compared with the experimental data available in
the literature for phenol-aldehyde/graphite, phenol-alde-
hyde/aluminum oxide, polypropylene/aluminum and poly-
propylene/copper composites with volume fraction rang-
ing from 0 - 0.5. Figures 2-5 show the comparison be-
tween the predictions by Equations (1)-(3) and the ex-
perimental data of the effective thermal conductivity of
the phenol-aldehyde/graphite, phenol-aldehyde/aluminum
oxide, polypropylene/aluminum and polypropylene/cop-
per composites. The values of the thermal conductivities
of the samples used are given in Table 1.
The graphs represented in Figures 2-5 show that value
of eff
K
calculated by the Equations (17)-(19) were
closer to the experimental data then calculated with other
models given by Equations (1)-(3). Most of the models
fail to predict the effective thermal conductivity over the
ntire range of filler concentration. It is seen that Maxwell e

D. CHAUHAN ET AL. 43
Figure 2. Variation of effective thermal conductivity and filler volume fraction of phenol-aldehyde/graphite composites for
different shapes.
Figure 3. Variation of effective thermal conductivity and filler volume fraction of phenolaldehyde/aluminum oxide compos-
ites for different shapes.

D. CHAUHAN ET AL.
44
Figure 4. Variation of effective thermal conductivity and filler volume fraction of polypropylene/aluminum composites for
different shapes.
Figure 5. Variation of effective thermal conductivity and filler volume fraction of polypropylene/copper composites for dif-
ferent shapes.

D. CHAUHAN ET AL.
45
Table 1. Thermal conductivity of various samples used in our computations.
Polymer Matrix Thermal Conductivity (W/m-K) Filler Thermal Conductivity (W/m-K)
Phenol Aldehyde 0.111 Graphite 120
Phenol Aldehyde 0.111 Aluminium Oxide 204
Polypropylene 0.25 Aluminium 237
Polypropylene 0.25 Copper 387
and Russell equations under estimate the experimental
data over the entire range of filler concentrations, while
the Hamilton and Crosser model (with n = 3, spherical
shape of fillers) over estimates the effective thermal con-
ductivity of the materials for phenol-aldehyde/graphite
and phenol aldehyde/aluminum oxide composites. It can
also be seen that, the calculations done by our model for
different shapes are closer to the experimental data then
done by any other theoretical model predicted earlier
Equations (1)-(3). It is seen from the Figures 2-5 that
with the increase in the filler concentration, the effective
thermal conductivity of the composite increases. The
effective thermal conductivity values of the samples used
for our computations are mentioned in Table 1. Table 1
show that the thermal conductivity of the fillers is sig-
nificantly higher than that of the matrix. There is a sig-
nificant change in the eff
K
of the composite systems as
we change the shape of the filler particles. It is seen from
the figures that the eff
K
increases nonlinearly with an
increase in the volume fraction of filler particles.
Sphericity is a measure of the roundness of the particle,
denoted by

. The value of

is one for spherical
particles and less than one for other shapes of the parti-
cles. The change in the geometrical configuration of the
particles changes the value of sphericity. An increase in
the value of

means that there is decrease in the an-
gularity, which brings the particles closer and there is an
increase in the eff
K
of the system. This change in the
sphericity can be seen in the form of small deviations
among the theoretical curves drawn for different shapes
as spherical, elliptical and hexagonal Figures 2-3.
Hence, it is observed that the theoretical values for
eff
K
calculated by our model proposed for different
shapes, with other existing mathematical models Equa-
tions (1)-(3) are in good agreement with each other for
low volume fractions i.e.

< 0.3. This is because of
the low dispersion of particles in the matrix, due to which
the particles are not able to interact with each other.
However, when the volume fraction increases more than
0.3, there is a rapid increase in the eff
K
of the system as
the particles come closer as such the interaction between
filler particles become stronger.
4. Conclusions
Theoretical expressions for effective thermal conductiv-
ity for the polymer composites which were established,
was based on the law of minimal thermal resistance and
the equal law of the specific equivalent thermal conduc-
tivity. The different shapes of filler particles as spherical,
elliptical and hexagonal were considered for the calcula-
tions. Equations (17)-(19) gives the relationship between
the effective thermal conductivity (eff
K
) of filled poly-
mer composites in terms of inclusion volume fraction as
well as other physical parameters of polymer matrix and
fillers when filler volume fraction is less than 60%.
The values of the effective thermal conductivity (eff
K
)
of phenol-aldehyde/graphite, phenol aldehyde/aluminum
oxide, polypropylene/aluminum and polypropylene/cop-
per composites were then calculated. Good agreement
was found between the theoretical estimations and ex-
perimental data available in the literature. We further
plan to verify these formulae for some more samples.
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