ABSTRACT Although the thermal conductivity of nanoporous materials has been investigated in the past, previous models have overestimated the small pore limit. Various authors had proposed a cylindrical boundary geometry to mimic the pore&apos;s environment. This permits to solve the phonon Boltzmann equation analytically [1]or numerically [2], but for fixed porosity it leads to a saturation of the thermal conductivity at small pore diameters. We show that such saturation is a spurious effect of the cylindrical boundary approximation. By implementing a Monte Carlo calculation with correct boundary conditions, we obtain considerably different thermal conductivities than predicted by the cylindrical boundary geometry. The approach is illustrated in the case of Si and SiGe nanoporous materials

Chandan Bera

Natalio Mingo

S Volz

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Proceedings of MNHMT2009
ASME 2009 2nd Micro/Nanoscale Heat & Mass Transfer International Conference
December 18-22, 2009, Shanghai, China
MNHMT2009-18249
REVISITING THE THERMAL CONDUCTIVITY OF NANOPOROUS MATERIALS
Chandan Bera1,2, Natalio Mingo1, and S. Volz2
1CEA, LITEN, 17 rue des Martyrs, BP166, 38042 Grenoble, France
2Laboratoire dEnergetique Moleculaire et Macroscopique, CNRS UPR 288, Ecole Centrale Paris, France
ABSTRACT
Although the thermal conductivity of nanoporous materials
has been investigated in the past, previous models have overes-
timated the small pore limit. Various authors had proposed a
cylindrical boundary geometry to mimic the pore’s environment.
This permits to solve the phonon Boltzmann equation analyti-
cally [1]or numerically [2], but for fixed porosity it leads to a sat-
uration of the thermal conductivity at small pore diameters. We
show that such saturation is a spurious effect of the cylindrical
boundary approximation. By implementing a Monte Carlo calcu-
lation with correct boundary conditions, we obtain considerably
different thermal conductivities than predicted by the cylindrical
boundary geometry. The approach is illustrated in the case of Si
and SiGe nanoporous materials.
1 INTRODUCTION
In many recent publications Ref. [3–7], we have seen that
nano structuring of bulk material can reduce the thermal con-
ductivity that can increase the figure of merit, ZT of the ma-
terial. The figure of merit is ZT = S2σT/κ, where S Seebeck
coefficient,σ electrical conductivity and κ is the thermal conduc-
tivity.There has been enormous research in this direction to de-
velop new thermo electric material with a low value of thermal
conductivity. Nano and Micro structure base material such as
superlattice and quantum dot superlattice have shown significant
reduction in thermal conductivity values compared to their bulk
values. Nanocomposite porous material can also reduce the ther-
mal conductivity and enhance the nano scale effects observed in
supperlattice [1,2,8–10]. In the Ref. [1,2],We have seen theoreti-
cal study on periodically aligned nanoporous composite material.
In fact these model have over simplified some of the dependen-
cies due to their boundary condition on their model. A Monte
Carlo Simulation for periodically aligned cylindrical nano pores
inside a host matrix shows that thermal conductivity is strongly
decreased with the decrement of the pore radius. Here We have
also noticed that thermal conductivity of nano porous alloy ma-
terial behaves different way than the non alloy porous materi-
als. Recently we have seen some experimental effort and ad-
vance in the synthesis of nanostructured semiconductor. Chen et
al [11, 12] demonstrated that n-type and p-type SiGe bulk alloy
can enhance the figure of merit due to their decrement in the ther-
mal conductivity. Porous material will play a bigger role due to
their low thermal conductivity. All those applications where very
low thermal conductivity is essential, porous composite material
can be very useful [13–15].
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Proceedings of the ASME 2009 2nd Micro/Nanoscale Heat & Mass Transfer International Conference 
MNHMT2009 
December 18-21, 2009, Shanghai, China 
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In nanoporous composite material the characteristic dimen-
sion d is comparable to the mean free path(Λ) of the heat carrier;
therefore the heat transport becomes affected by the pore size
and pore arrangement. In such a case, heat conduction model
known as Fourier law becomes invalid. In literature, the ap-
proaches used for the nanoporous composite material are based
on the solution of the Boltzmann transport equation for phonons.
BTE has been used to model the thermal conductivity of porous
silicon constituted of cylindrical pores. The thermal conductiv-
ity in the cylinder longitudinal direction was predicted by Yang
and Chen [2] using the DOM, while perpendicular to the cylinder
surface was simulated by Prasher [16] using the Loyalka’s vari-
ational method [17]. In these models, the two dimensional(2D)
heat transport is considered. Here we used Monte Carlo simu-
lation method for the phonon transport through a three dimen-
sional(3D) array of parallel pores.
2 MONTE CARLO SIMULATION
Monte Carlo simulation technique has been widely used to
solve the transport equation. In this technique we can easily
follow different scattering processes, such as phonon phonon
scattering, phonon particle scattering, and boundary scattering
[18–20]. For our Monte Carlo simulation of the cylindrical par-
allel pores in side a host matrix, we consider a unit cubic cell
surrounding the cylindrical pore Fig.1(e). This simulation is fre-
quency independent. we launch phonons from the one square
wall and calculate the flux in the opposite wall. Before launch-
ing the phonon we define some initial conditions.
2.1 Position:
we arbitrarily choose the position of each phonon inside the
square wall and out side the inner circle, Fig.1.(c).
2.2 Velocity:
Each phonon we gave a random velocity in three dimen-
sion(3D). For the direction of the phonon we consider Cartesian
coordinate, so we defined three random number. As we were
launching phonon from the wall so two random number are be-
tween -1 to 1 and one random number which define the direction
along the pores length is between 0 and 1.
2.3 Free Path:
Before launching the phonon from the wall, we define the
free path(free flight) length of each phonon from the probabilistic
equation s = −lnRs×Λ0 [21]. Where Λ0 is the mean free path
of the bulk and Rs is the random number between 0 and 1.
Figure 1. (a) Previous model considering cylindrical outer surface sur-
rounding the pores. (b) Top look of the periodic cylindrical pores inside a
host matrix. Phonon transport is directed towards the length of the cylin-
der. (c) Phonon are launched from the shaded area of the wall. (d) Pe-
riodic arrangement of the cylinder. We took Periodic boundary condition
on outer surface and diffusive boundary condition on inner pore surface.
(e) Pores embedded in a host medium, described geometry we used for
numerical simulation.
2.4 Boundary condition:
For the Monte Carlo simulation boundary condition is very
important. We put a periodic boundary condition on the four side
of cube surrounding the cylindrical pore. For the boundary of the
cube we use Cartesian coordinate and for the cylindrical pore we
used cylindrical coordinate.
2.5 Scattering:
We consider diffusive boundary condition on the cylindrical
pores surface. So if phonon collide with the cylinder, its scat-
tered diffusively in all direction outside the cylinder with a new
free path(s). If the phonon cross any of the four surface of the
cube surrounding the pore, we again inject a phonon from the op-
posite side with same velocity and with the remaining free path.
If the free path is less than the length(L), then after the traveled
distance we generate a new phonon with new velocity and new
free path(s).
2.6 Length:
Length(L), Fig.1(b), of the system should be long enough
such that we can achieve the diffusive type dependence of the
mean effective free path(Λ) of the system. The number of
phonon(N), which are launched from the one side after evolv-
ing the whole length when come out through the other side, we
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Figure 2. a)Comparisons of the non dimensioned mean free path be-
tween the numerical model with periodic boundary conditions using
MC simulation(gray line with solid symbol), cylindrical geometry model
based on analytical (black line) and MC (symbol) predictions for v f =
0.2,0.4and0.6.
can define the mean effective free path, Λ as
Nthrough/N '
(
1+L/Λ
)−1
. (1)
Where Nthrough is the number of transmitted phonon.
In our simulation we used 106 number of phonon and the
L was made to vary upto 100.7 × Λ0, until we had a good
accuracy(≥ 99%). To verify our model, we first simulate the
phonon effective mean free path for the previous cylindrical
model as shown in Fig.1.(a) and we got the same result as the
previously predicted Fig.2.
Here we have computed the Mean free path reduction due to
the presence of the pore, Λ/Λ0. Only three independent param-
eters are involved: the bulk mean free pathΛ, the pore radius r,
and the porosity v f . The inter pore distance, δ is related to the
pore radius and porosity as δ = 2r((0.5/
√
v f
π
)−1). Fig.2 shows
that mean free path decrease with the decreasing of pore radius,
contrary to the previously calculated cylindrical model [1], no
saturation occurs at small pore size.
The mean free path keep decreasing as the the pores be-
comes smaller because for a fixed porosity when we decreased
the pores radius, we also increase the density of pores inside the
medium and decreased the inter pore distance. So there is more
diffusive scattering by the pores and the phonon effective mean
free path decreased depending on the inter pore distance.
3 EQUATION FOR THE THERMAL CONDUCTIVITY
In the relaxation time approximation, the thermal conductiv-
ity can be expressed as [22]
κ =
k4b
2π2vh̄3
T 3
∫ xc
0
τ
x4ex
(ex−1)2
dx (2)
Where τ is the relaxation time, x stands for h̄ω/kBT , v is the
averaged speed of sound. Now if the relaxation time only depend
on the frequency, we can define
τ(ω) =
Λ(ω)
v
(3)
Now we can use the result of Monte Carlo simulation to
compute the Λ(ω). In the Monte Carlo simulation we have
Λ/Λ0 frequency independent, but it depend on the zero poros-
ity mean free path of bulk Λ0, pore size r, and porosity v f , so,
Λ
Λ0
= F(Λ0,r,v f ). The zero porosity mean free path, i.e., the bulk
mean free path of the matrix can be expressed as a frequency de-
pendent form [23]
Λ0(ω) = v(BT ω2e−C/T )−1 (4)
Where B and C are two parameters. In this Anharmonic scat-
tering procedure both umklapp and normal scattering processes
are included. For the higher frequency, umklapp scattering is
proportional to ω2, and it thus dominate over normal processes
which depend linearly on frequency [24]. This justifies the form
of Λ0(ω) in Eqn.4. we choose the same value of constant B and C
as Ref. [23]. We employed this approach for the thermal conduc-
tivity calculation of pure Si. Therefore the effective frequency
dependent mean free path of phonon for porous Si became
Λ(ω) = Λ0(ω)F(Λ0,r,v f ) (5)
For the SiGe alloy we use the effective medium base approached
same as Ref. [25]. Λ0(ω) in this expression is combination of the
average anharmonic scattering rate of bulk Si and bulk Ge with
the alloy scattering term of SiGe, first introduced in Ref. [26]. So
for the SiGe alloy the effective mean free path became
Λ(ω) =
1
1
Λ0(ω)F(Λ0,r,v f )
+α4ω4
(6)
where α is due to the atomic disorder in alloy and it proportional
to the volume fraction of Ge in the SiGe alloy.
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Figure 3. Thermal conductivity of porous Si at 20% and 40% porosity
for different pores radius. Black line is for 40% and gray line is for 20%
porosity.
4 RESULTS
Using the Eqs.2, 5 and 6 we can study the dependence of
thermal conductivity on the temperature for different porosity
and pore radius. As shown, in Fig.3, from Eqs.2, 5 we see that
the thermal conductivity of porous Si is lower than the thermal
conductivity of bulk material and as we increase the porosity the
thermal conductivity decreased. On the other hand, in the Fig.4,
from Eqs. 2, 6 we see that the thermal conductivity of porous
SiGe alloy is not affected by the pore at all temperature range. It
remains almost temperature indeependent.
Here we report the thermal conductivity on the term of tem-
perature. Since in the alloy smaller wave length phonon is scat-
tered by the atomistic disorder, the longer wave length phonon
is responsible for heat transport. So very large size of pores are
also able to block a significant amount of phonon. But in the
non alloy material, larger frequency range carry the heat and al-
though long wave length phonon is blocked by the pores size, a
significant heat come from the short wave length phonon. The
effect of pores size on the thermal conductivity of alloy and non
alloy material was discussed in detail in another paper.
5 CONCLUSION:
We report the three dimensional (3D) heat transport for
porous alloy and non alloy material. We found that porous al-
loy material is more effective to reduce the thermal conductiv-
Figure 4. Thermal conductivity of porous Si0.8Ge0.2 at 20% and 40%
porosity for different pores radius. Black line is for 40% and gray line for
20% porosity.
ity than non alloy materials. Using Monte Carlo simulation we
also improved the effective mean free path of the material for
lower pores size. In Si0.8Ge0.2 alloy, at 300K we see that ther-
mal conductivity is nearly 5 times and 3 times smaller than the
bulk thermal conductivities value for 20% porositi at 100nm and
500 nm pores radius respectively. Though Ge have limited re-
sources and it is not cheap material, still the SiGe alloy with a
very low percentage of Ge can be used to reduce thermal con-
ductivity very effectively. So the application where the low ther-
mal conductivity is important, porous alloy material can be very
advantageous [28, 29]. The difficulty of fabrication of nano size
pores may be an obstacle for this kind of materials. Porous alloy
will allow to take advantage of the large pore size which will be
easier to produce.
ACKNOWLEDGMENT
We acknowledge the CNRS, CEA and ANR ThermaEscape
for their financial support.
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MNHMT2009-18249 REVISITING THE THERMAL CONDUCTIVITY OF NANOPOROUS MATERIALS

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MNHMT2009-18249 REVISITING THE THERMAL CONDUCTIVITY OF NANOPOROUS MATERIALS

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