Effect of Geometry of Filler Particles on the Effective Thermal Conductivity of Two-Phase Systems

doi:10.4236/ijmnta.2012.12005

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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

Email: *[email protected]

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

) 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

). 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

) 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



fm fm

eff m

fm fm

KKK

KK KK



 

 (1)

Russell model

eff m

ff f

KK K



 





















 









(2)

Hamilton and Crosser model

 



fm fm

eff m

fmfm

KnKn KK

KnKKK



 

  (3)

Here m

and

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

) 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

and

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

 (4)

and



pff

VKV

 (5)

Here

21.33π

VHr r3

(6)

and (7)

1.33π

V

For elliptical shape

D. CHAUHAN ET AL.

Here

 (8) 2

22.433

VHr r 3

(14)



pff

VKV

 (9) a

nd 3

2.43 3

Vr (15)

Here

is the total cross-sectional area of the unit

cell,

V and

V are the volume of polymer matrix and

filler particles, and r is a variable parameter defined in

Figures 1(a)-(c) respectively.

Here

2.828 0.166π

VHr

(10)

The total equivalent thermal resistance is then given

nd (11)

0.166π

V

For hexagonal shape

RRR R



 (16)

 (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:



pff

VKV

 (13)

For spherical shape of filler particles, the equation is



11 2

π2

4ππ

39π

eff

pp f











 









(17)

For elliptical shape of filler particles, the equation is



11 2

π2

ππ

69π

eff

pp pf

KK K











 









(18)

and for hexagonal shape of filler particles, the equation is



 

11.23 2

1.62 1.29

eff









(19)

Here



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

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.

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.

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

of the composite systems as

we change the shape of the filler particles. It is seen from

the figures that the eff

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

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

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

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

) 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

)

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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