Boltzmann Transport Equation - COMSOL.com

Presented at the COMSOL Conference 2009 Boston
Nanoscale Heat Transfer using
Phonon Boltzmann Transport Equation
Sangwook Sihn
Air Force Research Laboratory, Materials and Manufacturing
Directorate, AFRL/RX, Wright-Patterson AFB, OH, 45433
University of Dayton Research Institute, Dayton, OH
email : sangwook.sihn@udri.udayton.edu
Ajit K. Roy
Air Force Research Laboratory, Wright-Patterson AFB, OH
2009 COMSOL Conference, Boston, MA
October 8-10, 2009

Outline
• Background information.
• Description of phonon Boltzmann transport equation (BTE).
• Modeling and solution procedure of BTE using COMSOL.
• Results
– Steady-state and transient problems.
– Issues of refinement in spatial and angular domains.
• Summary and conclusions.
2

Fourier Equation (FE)
• For last two centuries, heat conduction has been modeled by Fourier Eq (FE).
∂T
– Conservation of energy: ρc = −∇ ⋅q
∂t
Diffusive
– Fourier’s linear approximation of heat flux: q = −k∇T L >> Λ T
‣
T 2
1 ∂
= ∇
2 T
α ∂t
x
• Parabolic equation —> Diffusive nature of heat transport. T 1
• Heat is effectively transferred between localized regions through sufficient
scattering events of phonons within medium.
• Does not hold when number of scattering is negligible.
– e.g., mean free path ~ device size (chip-package level).
– Boundary scattering at interfaces causing thermal resistance.
• Admits infinite speed of heat transport —> Contradict with theory of relativity.
L
‣ Fourier Equation cannot be used for small time and spatial scales.
3

Hyperbolic Heat Conduction Equation (HHCE)
• Resolve the issue of the Fourier equation with the infinite speed of heat carrier.
1
∂
2
‣
2
C ∂t
T
2
1 ∂T
+
α ∂t
= ∇
– Definition of heat flux:
2
T
∂q
τ
o
+ q = −k∇T
, ( τ o
: relaxation time)
∂t
• Hyperbolic equation —> Wave nature of heat transport.
• Called as Cattaneo equation or Telegraph equation.
• Finite speed of heat carriers.
• Ad hoc approximation of heat flux definition.
• Violates 2 nd law of thermodynamics.
– If heat source varies faster than speed of sound, heat would appear to be moving from
cold to hot.
( C
2
= α
τ o
)
‣ HHCE: could be used for short time scale, but not for short spatial scale.
4

Small Scale Heat Transport (Time & Space)
• Fourier Equation cannot be used for small time and spatial scales.
• HHCE: could be used for short time scale, but not for short spatial scale.
• Needs equations and methods for small scale simulation in terms of both
time and space.
– Molecular dynamics simulation.
• Accurate method.
• Computationally expensive.
• Suitable for systems having a few atomic layers or several thousands of atoms.
• Not suitable for device-level thermal analysis.
– Boltzmann Transport Equation (BTE).
– Ballistic-Diffusive Equation (BDE).
• Similar to Cattaneo Eq. (HHCE) with a source term.
• Derived from BTE.
• Good approximation of BTE without internal heat source, disturbance, etc.
5

Boltzmann Transport Equation (BTE)
• BTE: also called as equation of phonon radiative transfer (EPRT).
• Equation for phonon distribution function:
∂f
∂t
+ v ⋅∇f
• Can predict ballistic nature of heat transfer.
• Neglects wave-like behaviors of phonon.
– Valid for structures larger than wavelength of phonons (~ 1 nm @ RT).
• Solution methods:
⎛ ∂f
⎞
= ⎜ ⎟
⎝ ∂t
⎠
scat
– Deterministic: discrete ordinates method, spherical harmonics method.
– Statistical: Monte Carlo.
• Much more efficient than MD.
f o
− f
τ
• Agrees well with experimental data.
≈
o
L

Details of Boltzmann Transport Equation (BTE)
• Phonon intensity:
Iω ( t,
v,
r)
= v ωf
( t,
v,
r)
D(
ω) / 4π
• BTE becomes EPRT:
∂I
∂t
ω
+ v ⋅∇I
ω
=
I
ω o
τ
− I
o
ω
,
Iω
o
=
1
∫
I
ω
4π
Ω= 4π
dΩ
*Equilibrium phonon intensity
determined by Bose-Einstein statistics
• For 1-D,
∂I
∂t
ω
∂I
+ v cosθ
∂x
ω
=
I
ω o
τ
− I
o
ω
x
θ
Ι ω
L
• For each angle (θ ), solve non-linear equation with iterations for
– Solving for Ι ω .
– Updating Ι ωο .
• Heat flux:
q
=
Ω=
∫∫
ω
D
Iω
cosθ dω
dΩ
4π
0
• Internal energy:
u
1 ω D
( ≈ cT ) = ∫ ω
f D(
ω)
dω
dΩ = ∫ ∫
4
π
0
Ω= 4π
I
v
ω
dω
dΩ
7

Modeling & Solution Procedure using COMSOL
∂I
∂t
∂I
+ v cosθ
∂x
=
I o
− I
τ
o
• Use a built-in feature of COMSOL, “Coefficient Form PDEs”.
• The spatial domain is discretized using FE mesh.
• The angular (momentum) domain is discretized using Gaussian quadrature
points.
• For each angle (θ ), set up the BTE with corresponding coefficients (µ i =cosθ i )
and BCs (Neumann vs. Dirichlet).
• Calculate equilibrium phonon intensity (I o ) by numerical integration of (I i ) using
Gaussian quadratures.
• Solve.
– Direct solver (UMFPACK).
– Max. BDF order = 1.
• Postprocess and visualize the results.
8

Details of Solution Procedure
• Original 1-D BTE:
∂I
∂t
∂I
I −
+ µ = o
I
v
, ( µ = cosθ
)
∂x
τ
o
• Nondimensionalize with
t
*
t x
= , η = ,
τ L
o
Kn =
Λ
L
∂I
∂t
*
∂I
+ Kn µ + I
∂η
=
I o
• Split into (+) and (-) directions:
• Discretize angular space at
Gaussian quadrature points:
+
+
⎧∂Ii
∂Ii
⎪ + Kn µ +
*
i
I
⎪
∂t
∂η
⎪
−
−
∂Ii
∂Ii
⎨ + Kn µ + I
*
i
⎪ ∂t
∂η
⎪
+
⎪Dirichlet BCs : Ii
⎩
+
i
−
i
η = 0
= I
= I
o
o
,
,
σ
= T
π
4
( µ > 0)
( µ < 0)
η = 0
i
i
,
I
−
i
η = 1
σ
= T
π
4
η = 1
• After FE run, postprocess:
I
o
( t,
η)
=
1
2
⎡
⎢
⎣
⎡
q(
t,
η)
= 2π
⎢
⎣
n
gp
∑
i=
1
n
2
gp
∑
i=
1
w I
2
i
i
+
i
+
i
+
w µ I
n
+
i
gp
∑
i=
1
+
2
w I
n
i
gp
∑
i=
1
−
i
2
⎤
⎥
⎦
w µ I
i
−
i
−
i
⎤
⎥
⎦
9

Coefficient Form for BTE
(60 Finite elements, 16 Gaussian Points)
10

Steady-State Problem: Analytic vs. Numerical Solutions
Emissive power ~ Temperature
Gradient
e
*
0
( η)
=
e
( η)
− J
o
+
J
q1
− J
−
q2
−
q2
* *
*
∆e0 = e0
( η = 0) − e0
( η = 1)
Nondim. emissive power, eo*
1
0.8
0.6
0.4
0.2
Kn =
Λ
L
Small temperature gradient
at smaller scale
Kn=0.1
Kn=1
Kn=10
Kn=100
Gradient of nondim. emissive power,
∆eo*
1
0.8
0.6
0.4
0.2
Analytic
Numerical (ngp=16)
Numerical (ngp=128)
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
0
0.1 1 10 100
Knudsen number, Kn=Λ/L
11

Steady-State Problem: Analytic vs. Numerical Solutions
Heat flux
Thermal conductivity
*
* q
*
q =
k =
+ −
*
J q 1
− J q
∆e 2
o
q
L
Nondimensional heat flux, q*
1
0.8
0.6
0.4
0.2
0
Analytic
Numerical (ngp=16)
Numerical (ngp=128)
0.1 1 10 100
Knudsen number, Kn=Λ/L
k* = q*L/∆eo*
60
50
40
30
20
10
0
Analytic
Numerical (ngp=16)
Numerical (ngp=128)
0.1 1 10 100
Knudsen number, Kn=Λ/L
12

Transient Problem: Effect of Spatial Refinement
(More Finite Elements)
• Refine spatial (x-) direction with n finite elements (n=15, 60, 120).
• Divide angular direction with 16 Gaussian points (ngp=16).
Nondimensional temperature, θ
0.6
0.5
0.4
0.3
0.2
0.1
0
Temperature @ t*=0.1
15 elements
60 elements
120 elements
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
Nondimensional temperature, θ
0.55
0.5
0.45
0.4
0.35
15 elements
60 elements
120 elements
Ray effect.
0 0.1 0.2 0.3
Nondimensional coordinate, η
• Spatial refinement leads to a smoother
solution.
• However, it does not solve ray effect.
13

Transient Problem: Effect of Angular Refinement
(More Gaussian Points)
• Refine spatial (x-) direction with 240 FE elements.
• Divide angular direction with ngp Gaussian points (ngp=4,8,16,32,64,128).
Nondimensional temperature, θ
0.6
0.5
0.4
0.3
0.2
0.1
Temperature @ t*=0.1
ngp=4
ngp=8
ngp=16
ngp=32
ngp=64
ngp=128
Nondimensional temperature, θ
0.55
0.5
0.45
0.4
0.35
ngp=4
ngp=8
ngp=16
ngp=32
ngp=64
ngp=128
0 0.1 0.2 0.3
Nondimensional coordinate, η
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
• Angular refinement resolve ray effect.
• Spatial and angular refinements are
independent.
• Highly refined spatial mesh with coarse
angular mesh alleviates solution.
14

Results of BTE
(Temperature & Heat Flux Distributions with Time Increase)
Nondimensional temperature, θ
Nondimensional heat flux, q*
1
0.8
0.6
0.4
0.2
0
0.4
0.32
0.24
0.16
0.08
t*=0.01
t*=0.1
t*=1
t*=10
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
t*=0.01
t*=0.1
t*=1
t*=10
Nondimensional temperature, θ
Nondimensional heat flux, q*
1
0.8
0.6
0.4
0.2
0
0.4
0.32
0.24
0.16
0.08
t*=0.01
t*=0.1
t*=1
t*=10
Kn=10 Kn=1 Kn=0.1
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
t*=0.01
t*=0.1
t*=1
t*=10
Nondimensional temperature, θ
Nondimensional heat flux, q*
1
0.8
0.6
0.4
0.2
0
0.4
0.32
0.24
0.16
0.08
t*=0.1
t*=1
t*=10
t*=100
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
t*=0.1
t*=1
t*=10
t*=100
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
15

Comparisons of FE, HHTC, BDE vs. BTE
(Temperature & Heat Flux Distributions for 3 Kn)
Nondimensional temperature, θ
1
0.8
0.6
0.4
0.2
BTE
BDE
Fourier
Cattaneo
Kn=10
t*=1
Nondimensional temperature, θ
1
0.8
0.6
0.4
0.2
BTE
BDE
Fourier
Cattaneo
Kn=1
t*=1
Nondimensional temperature, θ
1
0.8
0.6
0.4
0.2
BTE
BDE
Fourier
Cattaneo
Kn=0.1
t*=10
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
0
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
Nondimensional heat flux, q*
5
4
3
2
1
0
BTE
BDE
Fourier
Cattaneo
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
Nondimensional heat flux, q*
0.4
0.32
0.24
0.16
0.08
0
BTE
BDE
Fourier
Cattaneo
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
Nondimensional heat flux, q*
0.12
0.09
0.06
0.03
0
BTE
BDE
Fourier
Cattaneo
0 0.2 0.4 0.6 0.8 1
Nondimensional coordinate, η
16

Summary and Conclusions
• Nanoscale simulation is conducted for phonon heat transfer using
Boltzmann transport equation.
– Phonons for dielectric, thermoelectric, semiconductor materials.
– Electrons for metals.
– Gas molecules for rarefied gas states.
• Numerical solution of BTE has been obtained for 1-D problem (both
steady-state and transient problems).
• Temperature and heat flux distributions from the nanoscale simulation
yield completely different results from the solutions from Fourier and
Cattaneo equations.
– Thermal conductivity will be different, too.
17