This document presents a numerical effective medium approximation approach to efficiently calculate the phonon thermal conductivity of 2-D nanocomposites with randomly distributed inclusions. The approach modifies the bulk thermal conductivity of both the inclusions and host material. It also accounts for the thermal boundary resistance between materials. Results show the approach can capture the effects of inclusion size, shape, and distribution on thermal conductivity, with reasonable agreement to Boltzmann transport equation solutions. The numerical effective medium approximation provides a more general method compared to analytical models.

1. Efficient Calculation of Phonon Thermal
Conductivity for 2-D Nanocomposites with
Randomly Distributed Inclusions
Vidya Bachina, Gang Li
Department of Mechanical Engineering
Clemson University
November 19, 2009
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 1 / 12

2. Outline g
Introduction
Numerical Effective Medium Approximation Approach
General model
Modified Bulk Thermal Conductivities
Thermal Boundary Resistance
Numerical Results
Conclusion
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 2 / 12

3. Introduction g
Thermal transport in nanocomposites
Hot Cold
Thermal transport
Modeling Challenges
Multiscale
Unique physics
Modeling Approaches
Accuracy: Molecular Dynamics (MD) >
Boltzmann Transport Equation (BTE) >
Effective Medium Approximation (EMA)
Computational cost: MD>BTE>EMA
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 3 / 12

4. Introduction g
Effective Medium Approximation (EMA) for Nanocomposites (Minnich and
Chen, APL, 2007)
Modification of a classical EMA
Good accuracy compared to the BTE/Monte Carlo solutions
Analytical model
Difficult when there are multiple inclusion materials with non-uniform
sizes and shapes.
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 4 / 12

5. Numerical Effective Medium Approximation Approach g
General approach
modified effective k (inclusions)
modified effective k (host)
thermal boundary resistance
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 5 / 12

6. Numerical Effective Medium Approximation Approach g
Modifying the bulk thermal conductivity of inclusion materials
Model 1 (Minnich and Chen, 2007)
k =
1
3
CvΛ
1
Λeff
=
1
Λb
+
1
d
for general shapes d = 4Ac/P, Ac: cross sectional area of the
inclusion; P: perimeter of the inclusion.
Model 2 (Xue, 2006)
keff =
kb
1 + 2Rk/d
Rk: Kapitza resistance
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 6 / 12

7. Numerical Effective Medium Approximation Approach g
Modifying the bulk thermal conductivity of inclusion materials
Model 3 (Zhang, 2007)
keff
kb
= 1 +
kn
(1 − 4kn)−1
−1
kn > 5
keff
kb
= 1 +
kn
m
−1
kn < 1
kn: Knudsen number, m: cross-sectional shape parameter
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 7 / 12

8. Numerical Effective Medium Approximation Approach g
Modifying the bulk thermal conductivity of the host material (Minnich and
Chen, 2007)
1
Λeff
=
1
Λb
+
1
Λcoll (Φ)
Φ: interface density ⇐ obtained from the mesh geometry
Thermal boundary resistance
k1
eff
∂T1
∂n1
= k2
eff
∂T2
∂n2
= β (T1 − T2)γ
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 8 / 12

9. Results g
Size effects
0 50 100 150 200 250 300
5
10
15
20
25
30
35
40
45
Inclusion Size (nm)
ThermalConductivity(W/mK)
Model 1
Model 2
Model 3
BTE
Si20Ge80 nanocomposite Thermal conductivity
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 9 / 12

10. Results g
Shape effects
0 50 100 150 200 250 300
5
10
15
20
25
30
35
40
45
Inclusion Size (nm)
ThermalConductivity(W/mK)
Model 1
Model 2
Model 3
BTE
0 50 100 150 200 250 300
5
10
15
20
25
30
35
40
45
Inclusion Size (nm)
ThermalConductivity(W/mK)
Model 1
BTE
Square cross section Circular cross section
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 10 / 12

11. Results g
Distribution effects
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 11 / 12

12. Conclusion g
A numerical EMA method is developed for thermal
conductivity analysis of nanocomposites
Various bulk thermal conductivity modification models are
tested and compared
The numerical approach is more general and can account for
the shape, size and distribution effects of the inclusions.
G. LI gli@clemson.edu ASME IMECE 2009 Nov 2009 12 / 12