This document summarizes research on modeling the thermal conductivity of 2D silicon-germanium nanocomposites. The researchers developed a computational approach using finite element analysis and the heat conduction equation to calculate effective thermal conductivity. They accounted for interfacial thermal resistance and size-dependent phonon mean free paths. The model was used to study the effects of inclusion size, shape, and random distribution on thermal conductivity. Predictions agreed well with results from the Boltzmann transport equation and effective medium approximation models. The proposed approach can efficiently handle complex inclusion geometries and distributions.

1. Proceedings of IMECE 2009
ASME 2009 International Mechanical Engineering Congress and Exposition
Nov 13-19, 2009, Lake Buena Vista, Florida, USA
IMECE2009-10946
EFFICIENT CALCULATION OF PHONON THERMAL CONDUCTIVITY FOR 2-D
NANOCOMPOSITES WITH RANDOMLY DISTRIBUTED INCLUSIONS
Vidya Sagar Bachina
Department of Mechanical Engineering
Clemson University
Clemson, SC, USA
Email: vbachin@clemson.edu
Gang Li
Department of Mechanical Engineering
Clemson University
Clemson, SC, USA
Email: gli@clemson.edu
ABSTRACT
In this work we propose an approach built upon the classical
heat conduction theory and taking into account the interfacial
thermal boundary resistance (Kapitza resistance) and the effect
of the interface scattering on the ballistic phonon transport in
individual phases, to predict the effective thermal conductivity
of two-dimensional nanocomposites in the cross sectional
direction. We perform the finite element analysis of the
nanocomposites to calculate the effective thermal conductivity.
We study the size, shape, and distribution effects of the
inclusions on the effective thermal conductivity of the 2-D
nanocomposites. Predictions from this approach are in good
agreement with the results obtained from the Boltzmann
transport equation (BTE) and a modified analytical EMA
model proposed recently. The proposed approach is very
efficient compared to the BTE approach, and can handle
inclusions with complex geometries and random distributions.
INTRODUCTION
Manipulation and control of thermal conductivity in
nanostructured materials such as nanocomposites have
impacted a variety of applications [1]. Accurate and efficient
prediction of the thermal conductivity is one of the central tasks
in the design and optimization of nanocomposite materials for
various applications. In nanostructured materials, when the
characteristic length is less than several hundred nanometers,
ballistic phonon transport dominates. Interface and boundary
scattering play major roles in the thermal resistance of the
material. For this reason, phonon thermal conductivity of
nanocomposites can be altered by a variety of parameters,
including the thermal conductivity of individual phases, size,
shape, distribution, volume fraction and topology of the
inclusions. To investigate these effects, proper physical models
along with efficient computational algorithms are required.
Depending on the characteristic length of the system, a variety
of models are available for computational analysis of thermal
transport in solid state materials. Typically, when the
characteristic length decreases, the models become more
sophisticated by taking into account more phonon scattering
mechanisms. However, the model complexity and/or the
required computation intensity increase significantly. Due to
the ballistic phonon transport in nanocomposites, the diffusive
Fourier conduction equation may not be directly applicable.
When the characteristic length of the system is comparable or
smaller than the phonon mean free path (MFP), phonon
Boltzmann transport equation (BTE) based on a particle view
of phonon transport can be employed. However, the
computational cost of the BTE approach can be very high for
systems containing many phases and randomly distributed
inclusions. Lower level atomistic methods such as molecular
dynamics (MD), Monte Carlo (MC) and ab initio approaches
are impractical for the computational analysis of complex
nanocomposites. Recently, an analytical approach based on a
modified effective medium approximation (EMA) model was
proposed for the calculation of the thermal conductivity of
nanocomposites [2]. While this model effectively incorporates
the ballistic transport characteristics and the interface
scattering, it is difficult to represent the geometrical parameters
of such as inclusion shape and distribution in the model.
In this work, based on kinetic theory of thermal transport,we
propose a simple computational approach taking into account
of the size and thermal boundary resistance effects and predict
the thermal conductivity of Si-Ge nanocomposites within the
continuum framework. The advantages of this approach
include: (1) it is very efficient compared to other numerical
analysis methods since only the linear heat conduction equation
is solved over the computational domain; (3) it is general
compared to the EMA based analytical models since complex
geometries, multiple phases and random distributions of the
inclusions can be straightforwardly modeled in the
computational domain.
1 Copyright © 2009 by ASME

2. 2. Theory
2.1 General model
In the proposed approach, the continuum heat conduction
equation is used as the solution framework. However, since the
thermal conductivity reduction in nanocomposites is mainly
due to the reduced phonon mean free path, we calculate an
effective phonon mean free path for each component material
of the composite by using the approach described in [2]. From
the effective phonon mean free path, we calculate the effective
thermal conductivity for each component material. The
modified effective thermal conductivity of Si and Ge are then
used in the heat conduction equation. In addition, a boundary
thermal resistance model is employed at the interface of the Si
inclusion and Ge host. Figure 1 shows the schematic of the
model.
Figure 1: Uniform distribution of a nanocomposite with
boundary conditions applied.
The squares are cross sections of the Si nanowires embedded in
the Ge host. The diffusion equation is to be satisfied in both Si
and Ge regions,
−∇ keff
i
∇ T=0 (1)
where is the modified thermal conductivity of the i-th
phase in the composite material. On the boundary of the
computational domain, the temperature is applied on the left
edge. A heat flux is applied to the right edge. Periodic boundary
conditions are imposed in the y-direction by setting a zero heat
flux on the top and bottom edges of the domain. The thermal
boundary resistance conditions for the Si-Ge interface will be
described in Section 4.
Equation (1), along with the boundary and interface conditions,
is solved by using the finite element package COMSOL to
obtain the temperature and heat flux over the domain. Once the
temperature and heat flux distributions are obtained, the
effective thermal conductivity is determined by dividing the
average heat flux by the average temperature gradient
determined between the boundaries of the domain.
2.2 Modified bulk thermal conductivities
The modified bulk thermal conductivity can be computed by
using the kinetic theory of thermal transport, i.e.,
k=
1
3
Cv  (2)
where C is the specific heat of the material and v is the phonon
group velocity, and Λ is the phonon mean free path. As the
phonon mean free path is reduced in nanocomposites due to the
increased interface scattering, it is necessary to account for the
MFP reduction due to the interface scattering in the Fourier
model. With C and v remaining constant, the reduction of the
MFP leads to a modification of the bulk thermal conductivity.
In this work, we consider several models for the modification
of the bulk thermal conductivity of the inclusion phase.
Model 1. In this model the MFP of the inclusions is taken as a
function of the bulk mean free path and the characteristic length
of the inclusion phase. The modified effective mean free path
is determined by applying Matthiesen’s rule as
1
eff
=
1
b

1
d
(3)
where is the modified MFP, is the original bulk MFP, d
is a parameter given by 4Ac/P where Ac is the cross sectional
area of the inclusion, and P is the perimeter of the inclusion.
Once the modified MFP is obtained, the bulk thermal
conductivity is computed by using Eq. (2).
Model 2. In this model, the mixture rule for a serial model of
the interface-barrier/inclusion/interface-barrier [3] is applied to
calculate the equivalent transverse thermal conductivity. For a
nanowire of diameter d, surrounded by interface-barrier with
Kapitza resistance Rk, the modified thermal conductivity is
given by
keff =
kb
1
2Rk
d
(4)
where kb is the original bulk thermal conductivity of the Si
nanowire and Rk is the Kapitza resistance.
Model 3. The modified effective thermal conductivity in Model
3 is a function of Knudsen number kn [1] given by
keff
kb
=1
kn
1−4kn
−1

−1
kn5 (5)
keff
kb
=1
kn
m

−1
kn1 (6)
where m is a parameter depending on the shape of the cross
section of the nanowire. Note that, effk is determined by
interpolation for 1kn5 .
2 Copyright © 2009 by ASME

3. The thermal conductivity of Ge host phase is assumed to be a
function of bulk MFP and a characteristic length that depends
on nanowire density. The appropriate length scale of the host
phase is calculated based on the interface density, defined as
surface area of the inclusions per unit volumes of the
composite. More details on the modification of the bulk thermal
conductivity of the host phase can be found in [2].
4. Thermal Boundary Resistance
Thermal boundary resistance or Kapitza resistance is applied
through the parameter β in the following boundary condition
which is applied at the matrix filler interface [4]:
keff
1 ∂T1
∂n1
=keff
2 ∂T2
∂n2
=T1−T2
(7)
where keff
1
and keff
2
are the effective bulk thermal
conductivities of the matrix and inclusion phase respectively, n1
and n2 are the outward normal vectors of the matrix and
inclusion materials at the interface Γ, and β is the inverse of the
interfacial resistance. The interface condition given in Eq. (7)
states that, for a low conducting interface, the normal
component of the heat flux is continuous across the interfaces
while the temperature T undergoes a discontinuity which is
proportional to the normal component of the heat flux. Here
both the phases (matrix and inclusion materials) obey Fourier’s
heat conduction law.
5. Results
In this section we study the size effect, shape effect, and
distribution effect on the effective thermal conductivity of the
Si-Ge nanocomposites.
5.1 Size effects
The thermal conductivity is calculated for various Si inclusion
dimensions at a constant volume fraction of 20%. In the
calculations we use a square cross-section and a uniform
distribution of the silicon nanowires. The three models
presented in section 2.2 are used in the calculations of modified
bulk thermal conductivity of Si. The width of the silicon
nanowire varies from 10 nm to 300 nm. The calculated
effective thermal conductivities for different inclusion sizes are
listed in Table 1. The results obtained by using the BTE [5] and
the modified EMA analytical model proposed in [2] are also
listed for comparison. It is shown that there is considerable
effect of variation in characteristic lengths on the effective
thermal conductivity. When the width of the silicon on the wire
reduced from 300 nm to 10 nm, the thermal conductivity
reduced more than 80%. All the numerical and analytical
models produce fairly good results compared to the BTE
solution. Among the three effective bulk thermal conductivity
models, Model 1 and Model 3 give almost identical results.
Although the three models for calculating the modified bulk
thermal conductivity of inclusions are all applicable for
uniformly distributed inclusions with regular cross sections,
Model 1 is more appropriate for inclusions with complex
geometries and random distributions. For this reason, Model 1
is adopted here for the rest of the calculations.
Table1: Thermal conductivity values (W/mK) for different
characteristic lengths (nm) of Si nanowire inclusion with square
cross sections.
Size Model 1 Model 2 Model 3 Modified
EMA [2]
BTE
10 8.40 8.48 8.40 8.732 7.88
20 13.93 14.09 13.92 14.32 13.90
50 23.42 23.75 23.39 23.42 23.12
100 31.17 31.65 31.13 30.14 31.75
150 35.66 36.20 35.62 33.68 35.65
200 38.79 39.35 38.76 36.05 41.76
250 41.17 41.73 41.16 37.83 44.83
300 43.07 43.62 43.08 39.27 47.62
5.2 Shape Effects
Next we investigate the effect of Si nanowire’s shape on the
effective thermal conductivity of the Si-Ge nanocomposites. In
this calculation, a uniform distribution of circular Si nanowire
inclusions with a constant volume fraction of 20% is
considered. The diameter of the silicon wire inclusion is
increased from 10nm to 300 nm. The effective thermal
conductivity of the nanocomposites as a function of the
diameter of the Si nanowire is listed in Table 2. Similar results
are obtained from the numerical solution and the EMA model.
While still in reasonable agreement with the BTE results, the
numerical and EMA models underestimate the thermal
conductivity. Comparing these results with those listed in Table
1, except for the smallest characteristic size of the Si
nanowires, only minor effect of the shape is observed.
Table2: Thermal conductivity values (W/mK) for different
characteristic lengths (nm) of Si nanowire inclusion with
circular cross sections.
Size Numerical Modified
EMA []
BTE
10 7.068 7.2477 7.98
20 11.783 12.2825 14.10
50 21.099 21.1952 23.91
100 28.940 28.3101 32.38
150 33.539 32.1994 35.96
200 36.749 34.8108 42.17
250 39.192 36.7693 45.27
300 41.150 38.3394 47.98
5.3 Combined size, shape and distribution effects
Finally, we study the combined size, shape and distribution
effect of the inclusions on the effective thermal conductivity of
the nanocomposites. We investigate three cases: (1) random
distribution of square Si nanowires with the same size of 10 nm
x 10 nm, (2) random distribution of square Si nanowires with
different sizes, and (3) random distribution of irregular shaped
Si nanowires with different sizes. The volume fraction of Si in
all the three cases is fixed at 20%. To obtain a statistical view of
the results, the inclusions are randomly distributed multiple
times. The thermal conductivity is recalculated for each
distribution. The average and median values of the calculated
3 Copyright © 2009 by ASME

4. results along with the error range are then obtained. One of the
configurations and the corresponding heat flux profile of the
three cases are shown in Fig. 2. Figure 3 shows the variation of
the thermal conductivity for the three cases. It is shown that,
even in the random distribution cases, the size effect plays a
major role. The thermal conductivity is clearly lower for
smaller inclusions. However, the thermal conductivity variation
is significantly larger for larger inclusions with irregular shapes.
This result implies that the thermal conductivity can be altered
significantly with different distributions/arrangements of the
inclusions of larger sizes.
Figure 2: Configuration and heat flux in Si-Ge nanocomposites
with random inclusion distributions.
8
9
10
11
12
13
14
A B C
Random distributions
ThermalConductivityW/mk
Figure 3: Random distribution effects
6. CONCLUSION
In this work, we propose a computational approach to calculate
the effective thermal conductivity of 2D Si nanowire-Ge
composites. It is shown that thermal boundary resistance
profoundly influences the effective thermal conductivity of the
composites. By using the classical heat transfer theory and
finite element analysis, at constant volume fraction we
investigated the nanowire size, shape and distribution effects on
the effective thermal conductivity of the composites.
Predictions from our model are in good agreement with the
BTE results. We also show that the proposed model can
effectively handle randomly distributed inclusions with
complex geometries.
REFERENCES
1. Zhoumin Zhang, Nano/ Miroscale Heat Transfer, Mc Graw­
Hill, (2007).
2. A. Minnich and G. Chen, Appl. Phys. Lett. 91,073105 (2007).
3. Q. Z. Xue, Nanotechnology 17, 1655 (2006).
4. T. Y. Chen, G. J. Weng, and W. C. Liu, Appl. Phys. Lett. 97,
104312 (2005).
5. Y. Xu and G. Li, submitted for publication, (2009).
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