Efficient multiscale methods for micro/nanoscale
solid state heat transfer
by
Jean-Philippe Peraud
co
Dipl6m6 de l'Ecole Polytechnique, Palaiseau, France (2010)
S.M., Massachusetts Institute of Technology (2011)
C
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
I
Doctor of Philosophy in Mechanical Engineering and Computation
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2015
@ Jean-Philippe Peraud, MMXV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document
in whole or in part in any medium now known or hereafter created.
Signature redacted
Author............................................
Department
.. ............
echanical Engineering
T ,
Q 9fnl5
Signature redacted
Certified by...........................
Ni
1
-iconstantinou
Professor
ThesisAunervisor
Signature redacted
Accepted by ............................
ASignature
redacted
.................
''
*
David Hardt
Chairman, Department Committee on Graduate Theses
Accepted by ........................... tz... *.....
w
=
Youssef Marzouk
Co-Director, Computational Science and Engineering
Efficient multiscale methods for micro/nanoscale solid state
heat transfer
by
Jean-Philippe Peraud
Submitted to the Department of Mechanical Engineering
on July 8, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering and Computation
Abstract
In this thesis, we develop methods for solving the linearized Boltzmann transport
equation (BTE) in the relaxation-time approximation for describing small-scale solidstate heat transfer. We first discuss a Monte Carlo (MC) solution method that builds
upon the deviational energy-based Monte Carlo method presented in [J.-P. Praud
and N.G. Hadjiconstantinou, Physical Review B, 84(20), p. 205331, 20111. By linearizing the deviational Boltzmann equation we formulate a kinetic-type algorithm in
which each computational particle is treated independently; this feature is shown to be
consequence of the energy-based formulation and the linearity of the governing equation and results in an "event-driven" algorithm that requires no time discretization.
In addition to a much simpler and more accurate algorithm (no time discretization
error), this formulation leads to considerable speedup and memory savings, as well as
the ability to efficiently treat materials with wide ranges of phonon relaxation times,
such as silicon.
A second, complementary, simulation method developed in this thesis is based on
the adjoint formulation of the linearized BTE, also derived here. The adjoint formulation describes the dynamics of phonons travelling backward in time, that is, being
emitted from the "detectors" and detected by the "sources" of the original problem.
By switching the detector with the source in cases where the former is small, that is
when high accuracy is needed in small regions of phase-space, the adjoint formulation provides significant computational savings and in some cases makes previously
intractable problems possible.
We also develop an asymptotic theory for solving the BTE at small Knudsen numbers, namely at scales where Monte Carlo methods or other existing computational
methods become inefficient. The asymptotic approach, which is based on a Hilbert
expansion of the distribution function, shows that the macroscopic equation governing
heat transport for non-zero but small Knudsen numbers is the heat equation, albeit
supplemented with jump-type boundary conditions. Specifically, we show that the
traditional no-jump boundary condition is only applicable in the macroscopic limit
where the Knudsen number approaches zero. Kinetic effects, always present at the
3
boundaries, become increasingly important as the Knudsen number increases, and
manifest themselves in the form of temperature jumps that enter as boundary conditions to the heat equation, as well as local corrections in the form of kinetic boundary
layers that need to be superposed to the heat equation solution. We present techniques for efficiently calculating the associated jump coefficients and boundary layers
for different material models when analytical results are not available. All results are
validated using deviational Monte Carlo methods primarily developed in this thesis.
We finally demonstrate that the asymptotic solution method developed here can be
used for calculating the Kapitza conductance (and temperature jump) associated with
the interface between materials.
Thesis Supervisor: Nicolas G. Hadjiconstantinou
Title: Professor
4
Acknowledgments
A PhD is a long and demanding process, and the fact that I am writing this part now
means that I am seeing the end of it. Over the last four years, I have often thought
about the moment when I would write this last section. This moment has come.
I am particularly grateful to my thesis advisor, Nicolas Hadjiconstantinou, for
being a great mentor during all these years at MIT. Nicolas, thank you for your
incredible help over all these years. Five (almost six!) years ago, it took me several
months to choose a research group and a research project, initially for my Master
degree, and I am glad that I made the right choice. Your understanding, your infinite
patience and the care you show for your students make it a pleasure to work with you.
You helped me a lot to develop my ideas, and your editing skills never cease to amaze
me. You constantly oriented me towards fruitful and exciting research directions, you
always helped me get the big picture of what I was doing, and I am grateful that you
encouraged me to stay in this field for my PhD thesis.
I next would like to thank the two other members of my thesis committee. Professor Chen, I first got to know you by taking your class, back in 2010. In some
ways, this project started in your classroom. As a committee member, you provided
many meaningful insights from the field of nanoscale heat transfer and helped me
highlight the practical implications of my work. Professor Akylas, your expertise
in asymptotic methodologies have been truly valuable for developing and organizing
the asymptotic theory presented in this thesis and we had numerous interesting and
fruitful discussions on this topic. I unfortunately did not have a chance to attend one
of your lectures, but my peers never cease to tell me how great they are.
I also want to thank Professor Austin Minnich. Austin, I first knew you as a
great TA in Professor Chen's class. You then became a great collaborator whose help
has been invaluable for highlighting the relevance of my work and giving it the right
practical orientations, and you also became a great friend.
The research group that I had the privilege to be part of has significantly contributed to making my experience at MIT more comfortable and enjoyable. Thank
you very much Husain, Gregg, Ghassan, Toby, Jessica, Harold, Michael, Matt, Jerry
and Nisha. Thank you Colin for being a great collaborator and friend. Mojtaba,
thank you very much for your careful proofreading of my thesis. I am sure you will
enjoy working on this topic.
Being a teaching assistant has occupied a significant portion of my time as a
graduate student, and I want to thank Professor Samuel Allen for making this an
even more memorable time. Professor Allen, it was an immense honor and pleasure
to work with you. You cared a lot about your mission as a teacher and it showed.
Being your teaching assistant for three terms had a great impact on my personal
development and made me build memories that I will forever cherish. Professor
Thompson, I have been your graduate instructor for a term and the trust you placed
in me when you appointed me as an instructor and let me teach 30% of the lectures
meant a lot to me.
This whole PhD thesis would not have existed without the intervention of Dr.
5
Eric Dumonteil and Dr. Cheikh Diop back in 2011. Eric and Cheikh, I am infinitely
grateful for the help you provided when I needed to convince the administration of the
Corps des Ponts, des Eaux et des Forets to let me take a leave of absence and pursue
my PhD project. I clearly owe you this. I also want to thank the administration of the
Corps for their understanding in this matter. My various funding sources, namely the
MIT Energy Initiative and Total (through the MIT-Total fellowship) and the Solid
State Solar Thermal Energy Conversion Center (U.S. Department of Energy) have
also made this project possible. I should finally mention that the administrative staff
of the Department of Mechanical Engineering has been incredibly helpful in numerous
occasions.
Going through these years would have been quite harder without the support of
my friends. In her poem "Footprints In Your Heart", Eleanor Roosevelt wrote: "Many
people will walk in and out of your life, But only true friends will leave footprints in
your heart"; at MIT, I have met a lot of incredible people, and a significant number
of them have become great friends. I cannot list them all here because space is
limited, but I want them to know that their friendship and the memories that we built
together are among the best gifts MIT has given me. Uwe, Liz, Satoru and Ahmed, I
am particularly grateful for your friendship all along these years, and although such
opportunities became rarer as years passed, I was always looking forward to hanging
out with you. Rafa and Amparo, I am forever grateful that you encouraged me to
stay at MIT and go on with a PhD.
Soyoung, thank you very much for your constant support, for our discussions,
laughs and all the great moments.
Last, I want to thank my family. Maman and Papa, thank you for your unconditional support and care, in the bad moments as much as in the good ones. I wish I had
been able to visit you more often than I did. Sorry for not calling home more often
either. Olivier, thank you for your unwavering sense of humor. Nicolas, thank you
for constantly keeping in touch with emails full of pictures of beautiful landscapes,
airplanes and for the childhood references. Anne-Lyse, when I started my time at
MIT you were still in high school. What a long and impressive way you have come
since then! Congratulations for all your accomplishments and thanks for always being
supportive.
6
A mes parents, Marie-Josse et Jean-Michel.
A la m'moire de mon grand-pire, Christian Albert,
qui nous manque terriblement.
7
8
Contents
23
1.2
O utline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
.
.
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Boltzmann transport equation and current simulation methods 27
Importance of multiscale methods . . . . . . . . . . . . . . . . . . .
27
2.2
The phonon Boltzmann transport equation . . . . . . . . . . . . . .
28
2.3
Timestep-based Monte Carlo methods for phonon transport
. . . .
32
2.4
Deviational energy-based Monte Carlo simulation
. . . . . . . . . .
36
2.5
Variance of non-deviational vs deviational algorithm . . . . . . . . .
39
2.6
Limitations of the deviational method . . . . . . . . . . . . . . . . .
40
.
.
.
.
.
.
2.1
Linearization of the Boltzmann transport equation and kinetic-type
43
Monte Carlo
3.1
Independent phonon trajectories under linearized conditions.....
3.2
Kinetic-Monte-Carlo algorithm
3.3
.
. . . . . . . . . . . . . . . . . . . .
46
Example of problem setup . . . . . . . . . . . . . .
48
3.2.2
Sampling
. . . . . . . . . . . . . . . . . . . . . . .
49
3.2.3
Steady state sampling
. . . . . . . . . . . . . . . .
50
. . . . . . . . . . . . . . . . . .
51
Spatially variable controls
.
.
.
3.2.1
3.3.1
Simulation of periodic nanostructures using spatially variable
controls
3.4
44
.
3
1.1
. . . . . . . . . . . . . . . . . . . . . . . .
53
. . . . . . . . . . . . .
56
.
2
23
Introduction
Termination of particle trajectories
9
.
1
4
5
3.5
V alidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
3.6
Consistency and comparison with the solution to the heat equation
60
3.6.1
Problem statement and solution of the heat equation
. . . . .
61
3.6.2
Monte Carlo results . . . . . . . . . . . . . . . . . . . . . . . .
62
3.7
Application to transient thermoreflectance
3.8
Range of validity
. . . . . . . . . . . . . . .
63
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Collision-induced sources and the kinetic-type method
69
4.1
Termination of particle trajectories
. . . . . . . . . . . . . . . . . . .
69
4.2
An alternative justification for independent particle trajectories
. . .
75
Adjoint formulation and "backward" simulation method
77
5.1
N otation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
5.2
The adjoint Boltzmann equation
. . . . . . . . . . . . . . . . . . . .
78
5.2.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.2.2
The fundamental relation
. . . . . . . . . . . . . . . . . . . .
80
5.2.3
Adjoint particle dynamics and simulation . . . . . . . . . . . .
82
5.2.4
Reflecting boundaries . . . . . . . . . . . . . . . . . . . . . . .
83
5.2.5
Proof of the fundamental relation (5.12) for the prescribedtemperature boundary conditions
5.3
5.4
6
. . . . . . . . . . . . . . . .
84
A pplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
5.3.1
Surface temperature in a transient thermoreflectance experiment 86
5.3.2
Highly resolved calculations of mode-specific thermal conductivity calculations . . . . . . . . . . . . . . . . . . . . . . . . .
90
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
An asymptotic solution of the Boltzmann transport equation in the
limit of small Knudsen number
99
6.1
Asymptotic analysis for the bulk . . . . . . . . . . . . . . . . . . . . .
100
6.1.1
Bulk solution
. . . . . . . . . . . . . . . . . . . . . . . . . . .
101
6.1.2
Governing equation for the temperature field . . . . . . . . . .
103
10
6.2
6.3
6.4
6.5
Order 1 boundary layer analysis . . . . . . . . . . . . . . . . . . . . .
105
6.2.1
Boundary conditions for prescribed temperature boundaries
106
6.2.2
Boundary condition for a diffuse adiabatic boundary
. . . . .
113
Order 2 boundary layer analysis . . . . . . . . . . . . . . . . . . . . .
114
6.3.1
Order 2 analysis for prescribed temperature boundaries . . . .
114
6.3.2
Order 2 analysis of a diffusely reflective boundary . . . . . . .
118
Summary and discussion of results
6.4.1
Prescribed temperature boundaries
6.4.2
Diffuse reflection
6.4.3
A one-dimensional example
6.4.4
"Implicit" boundary conditions
6.4.5
A two-dimensional example
7
125
. . . . . . . . . . . . . . . . . . . . . . . . .
125
. . . . . . . . . . . . . . . . . . .
126
. . . . . . . . . . . . . . . . .
129
. . . . . . . . . . . . . . . . . . .
131
. . . . . . . . . . . . . . . . .
Application to a transient problem
133
. . . . . . . . . . . . . . .
136
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
Two applications of the asymptotic theory
141
7.1
Application to interfaces . . . . . . . . . . . . . . . . . . . . . . . . .
141
7.1.1
146
7.2
Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementing asymptotically-derived controls through the adjoint approach . . . .. . . . . . . ..
. . . . . . . . . . . . . . . . . . . . . . .
147
7.2.1
Spatially variable control temperature in the adjoint framework 148
7.2.2
Asymptotic control for multiscale problems . . . . . . . . . . .
151
7.2.3
Validation and accuracy
152
7.2.4
Using a "hybrid" control for models with widely variable free
paths.. . . . .
8
124
. . . . . . . . . . . . . . .
Extension to time-dependent problems
6.5.1
6.6
. . . . . . . . . . . . . . . . . . .
. ..
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
155
Final remarks and future directions
163
8.1
Sum mary
163
8.2
Future directions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
165
A Derivation of the governing equation for the order 1 and order 2
bulk temperature fields
167
B Numerical solution of the kinetic boundary layer problem
169
C Well-posedness of the discretized boundary layer problem
173
C.1 Eigenvalue determination . . . . . . . . . . . . . . . . . . . . . . . . .
174
C.2
175
W ell-posedness
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D Proof of relation (6.62)
179
E On the convergence rate of the ACA method: mathematical justification and discussion
181
F Material models
189
12
List of Figures
2-1
Adequate description models for heat conduction depend on the scale
of the system of interest. The diffusive assumption (Fourier's Law) can
only be applied if the Knudsen number (ratio between the phonon mean
free path and the smallest characteristic lengthscale of the system) is
small.
More details on the transition from the kinetic (Boltzmann-
based) description to the diffusive regime can be found in Chapter 6.
2-2
Ratio oau/q" as a function of AT/Tq at fixed (and equal) number of
computational particles. Image reproduced from [1]. . . . . . . . . . .
3-1
28
39
Transient and steady-state temperature profiles between two boundaries at different temperatures. Here, AT = T(x, t) - Teq as defined
in section 3.2.1.
The dots represent the solution calculated with the
timestep-based method while the continuous line was obtained using
the kinetic-type method presented in this chapter. Reproduced from [2]. 52
3-2
Nanoporous structure with cylindrical pores organized in a triangular
lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
55
x and y components of the heat flux in the elementary cell of a triangular network of cylindrical pores subjected to a temperature gradient of
106 K
M-
1
in the y-direction. The pore surfaces are assumed diffusely
reflective. Reproduced from [3].
. . . . . . . . . . . . . . . . . . . . .
13
56
3-4
(a) Average contribution of the j-th trajectory segment (defined as the
portion of trajectory before two successive randomizing events, namely
scattering event or diffuse collision with a boundary) to the final heat
flux estimate in the y-direction for the problem sketched in Figure 3-2.
The applied temperature gradient is 106 K.m- 1 . (b) Logarithm of the
absolute value and uncertainty of the average contribution of the j-th
trajectory segment, all expressed in W m 2 . . . . . . . . . . . . . . . . 58
3-5
Average heat flux contributions between scattering
j,
j
- 1 and scattering
for the calculation of the thermal conductivity along a thin film.
After the first collision, the expected heat flux is zero because of the
symmetry of the problem.
3-6
. . . . . . . . . . . . . . . . . . . . . . . .
Solid line: theoretical heat flux in a 100nm wide thin film. Dots: heat
flux computed with the proposed method . . . . . . . . . . . . . . . .
3-7
60
61
(a) Solution of the Boltzmann equation (dots) and the heat equation
(solid lines) to the problem with cubic initial heating. (b) Ratio between the two solutions. Arrows indicate solutions for different characteristic sizes (L). The solutions are calculated in the single relaxation
tim e approxim ation.
3-8
. . . . . . . . . . . . . . . . . . . . . . . . . . .
63
(a) Solution of the Boltzmann equation (dots) and the heat equation
(solid lines) to the problem with cubic initial heating. (b) Ratio between the two solutions. Arrows indicate solutions for different characteristic sizes (L). The solutions are calculated using the frequencydependent relaxation time model.
3-9
. . . . . . . . . . . . . . . . . . . .
Schematic of a transient thermoreflectance experiment.
63
Point 0 de-
notes the center of the heating pulse, also taken to be the origin of the
cartesian (x, y, z) set of axes. The system is assumed semi-infinite in
z > 0 direction and in the x - y plane.
14
. . . . . . . . . . . . . . . . .
65
3-10 Comparison of the surface temperature (average temperature in a cylinder of height 5nm and radius 2pim right beneath the surface), calculated
by the method explained in [4] and by the proposed algorithm. The
two methods give essentially the same results.
. . . . . . . . . . . . .
66
3-11 Surface temperature (here, average temperature in a cylinder of height
5nm and radius 10[m right beneath the surface) as a function of time
for late times (only possible with the proposed algorithm).
. . . . . .
67
3-12 Normalized error between the linearized and exact collision operator, in
the case where T = 300 K. For AT/Tq = 0.1, i.e. for a temperature
difference AT = 30 K, the relative error, 6, is about 2.2%. Reproduced
from [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
68
Fictitious material with diffusely reflective interfaces normal to the
direction of the applied temperature gradient. Dashed lines denote the
cell-grid used to calculate 'Vj(x), defined in section 4.1.
4-2
. . . . . . . .
70
a. 01 and L4 . b. Heat flux contributions resulting from q, L(ed)/T and
L(ed)/T. These results were obtained using an "imposed temperature
gradient" of 106 K .m . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
Using the termination technique based on spatial discretization, the
number of particles in the system decreases exponentially . . . . . . .
4-4
73
73
Error in the effective thermal conductivity of the structure in figure
4-1 as a function of the cell-grid size Ax. The convergence order is
quadratic.
5-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
Temperature as a function of time in a transient thermoreflectance
experiment. Results shown for the pulse center, 6pim away from the
pulse center, and 12pam away from the pulse center.
5-2
. . . . . . . . . .
89
Ratio between the particle standard deviation of the temperature, and
the temperature, for 3 cylindrical detectors with height 10nm, 5nni and
nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
91
5-3
Sketch of the nanoporous structure studied in section 5.3.2. The adjoint method is used to accurately calculate the heat flux contributions
from each "frequency bin".
Teq(x),
A spatially variable control temperature
with uniform gradient, is used, as in Ref. [5].
The dashed
square represents the boundary of the computational domain, along
which periodic boundary conditions are applied (see Ref. [5,6]).
5-4
. .
.
92
Frequency-resolved differential contribution to the thermal conductivity (measured heat flux per unit temperature gradient) from the longitudinal acoustic (LA) modes in the problem defined in figure 5-3.
Result is normalized by the corresponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using
the adjoint method. Results shown for square pores of side 25nm and
50nm; the spacing between the pores is 2 microns in both cases. These
calculations used 28000 particles per frequency cell, for a total of 1399
frequency cells (thus a total of approximately 40 million particles).
5-5
95
.
Frequency-resolved differential contribution to the thermal conductivity (measured heat flux per unit temperature gradient) from the longitudinal acoustic (LA) modes in the problem defined in figure 5-3.
Result is normalized by the corresponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using
the forward method. Results shown for square pores of side 25nm and
50nm; the spacing between the pores is 2 microns in both cases. These
calculations used a total of 40 million particles.
6-1
Temperature profile associated with
K1,1
=
. . . . . . . . . . . .
TK1(OTGO/&1 r=O) for
a single-free-path material. . . . . . . . . . . . . . . . . . . . . . . . .
6-2
109
Temperature boundary layer function TK11 for a material with silicon
dispersion relation and a single relaxation time.
6-3
96
. . . . . . . . . . . .111
Temperature boundary layer function TK11 for the modified Born-VonKarman model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111
16
6-4
Difference between the order 1 asymptotic solution and a Monte Carlo
solution for the heat flux between two boundaries at different temperatures. ........
6-5
...................................
112
Contour plot of the solution of Laplace's equation in a thin film with
X2
sinusoidal Dirichlet boundary conditions (top boundary
Neumann boundary conditions (bottom boundary x 2
6-6
Absolute value of the difference between the order
=
0).
= 1) and
. . . . . .
121
1 asymptotic tem-
perature solution and the Monte Carlo results, at point A of the problem depicted in Figure 6-5. The difference is clearly an order 2 quantity.122
6-7
Schematic of order two energy balance at a boundary along which a
temperature gradient exists.
6-8
. . . . . . . . . . . . . . . . . . . . . . .
Order 1 solution (plain line) compared to the solution computed by
highly resolved Monte Carlo simulation at (Kn) = 0.1.
6-9
124
. . . . . . . .
128
Order 1 solution (dashed line) and "infinite order" solution (solid line)
compared to the solution computed by a finely resolved Monte Carlo
simulation for
KKn) = 0.5. At this Knudsen number the boundary
layer contribution is clearly visible (the solution is no longer a straight
lin e ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
6-10 Contour plot of the solution (7.28) of Laplace's equation in a thin film
with sinusoidal Dirichlet boundary conditions.
. . . . . . . . . . . .
6-11 Convergence of asymptotic temperature solutions at (x 1
=
0, x 2
=
133
1)
in the two-dimensional example considered in section 6.4.5. . . . . . .
134
6-12 (a) The problem is a square particle with side L = 10(A). The temperature is calculated at the centre point (0,0) (b) Temperature at the
center of the square after initial heating.
7-1
. . . . . . . . . . . . . . . .
138
Temperature profile in a ID system with an Al//Si interface . . . . . .
148
17
7-2
Deviational temperature profile along the segment AB in Figure 610. Comparison between the solution to the BTE calculated using the
adjoint method of section 5.2.1, and the ACA method of section 7.2.2.
The analytical solution of Laplace's equation is also shown. . . . . . .
7-3
153
Standard deviation au of the particle contributions to the estimate
of the heat flux at point A of Figure 6-10 in the X2 direction, in the
single mean free path model. The standard deviation is proportional
to (Kn) with the ACA method.
method with fixed control for
7-4
The latter outperforms the adjoint
KKn) < 0.2 . . . . . . . . . . . . . . . . 156
(a) Standard deviation ciqu of the particle contributions to the estimate
of the heat flux at point B of Figure 6-10 in the X 2 direction, in the
variable mean free path model.
(b) Standard deviation aq1
of the
particle contributions to the estimate of the heat flux at point A of
Figure 6-10 in the
7-5
x 2 direction, in the variable mean free path model.
Standard deviation aoq
160
of the particle contributions to the estimate
of the heat flux at point A of Figure 6-10 in the X2 direction, with
artificial material parameters made such that the maximum free path
corresponds to different orders of magnitudes. While the onset of the
order 1 behavior is clear in the problem with largest mean free path
690nm, the onset is progressively delayed as the largest mean free path
is increased. ........
E-1
................................
161
Evolution of the contributions of particles to the final estimate, when
the adjoint method is used with the control (E.1) along with a temperature field which is the solution of Laplace's equation for (Kn) = 0.01
(Left) and Kn) = 0.001 (Right). This calculation was performed using
the single-mean-free-path model.
. . . . . . . . . . . . . . . . . . . .
18
187
E-2
Evolution of the contributions of particles to the final estimate, when
the adjoint method is used with the control (E.1) along with a temperature field which is not a solution of Laplace's equation, for (Kn) = 0.01
(Left) and (Kn) = 0.001 (Right). This calculation was performed using
the single-mean-free-path model.
. . . . . . . . . . . . . . . . . . . .
19
188
20
List of Tables
6.1
Boundary conditions and boundary layers for prescribed temperature
boundaries, up to order 2. Symbol ej refers to the unit vector corresponding to direction xi in a right-handed set where x, represents the
coordinate normal to the boundary and pointing into the material. . .
6.2
Boundary conditions and boundary layers for diffusely reflective walls,
up to order 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. 1
126
126
Source terms appearing in the second order boundary layer problem,
and the associated solutions. . . . . . . . . . . . . . . . . . . . . . . .
21
180
22
Chapter 1
Introduction
1.1
Motivation
Heat transport plays a central role in the design, manufacturing and control of most
engineered systems as well as the dynamics of a variety of physical processes of interest. From internal combustion engines to computer CPUs, accurately predicting and
designing for the heating/cooling load is essential for reliable operation and avoiding
catastrophic consequences. Fortunately, for the last two centuries, heat transport in
devices of interest could be reliably predicted using Fourier's law, which along with
energy conservation leads to the heat conduction equation.
Fourier's law is based on the empirical observation that the heat flux is proportional to the negative of the temperature gradient. It relies on the assumption that
the main heat carriers, phonons and electrons, exhibit diffusive behavior [7]. However, at the micron or sub-micron scale, the mean free paths of these carriers are no
longer negligible. As a result, the recent drive towards systems and devices of ever
decreasing lengthscales has exposed limitations in Fourier's law, namely its inability
to describe heat transport when carrier mean free paths are on the order of or larger
than characteristic lengthscales. The kinetic effects that manifest themselves at these
scales need to be captured by more "fundamental" approaches, either molecular or
mesoscopic (81.
Solving heat transfer problems at the sub-micron scale is essential in a number
23
of applications of practical interest. The semiconductor industry may be the most
revealing example. On the one hand, the size of electronic components is ever decreasing. On the other hand, as heat dissipation becomes more localized efficient heat
dissipation mechanisms become critical.
A better understanding of the behavior of the heat carriers is also needed in the
field of thermoelectrics. Thermoelectric materials convert a heat flux into an electric
current and, as such, are seen as means of harvesting part of the energy lost as wasted
heat (from Joule effect, exothermic reactions, etc). They can also be used for passive
cooling by exploiting the Peltier effect. The efficiency of thermoelectric materials is
quantified by the Figure of Merit, which is proportional to the electrical conductivity
and inversely proportional to the thermal conductivity. Decreasing the thermal conductivity without modifying the electrical conductivity is one of the primary avenues
currently explored for the purposes of improving the Figure of Merit of thermoelectric
materials [9,10].
In this thesis we investigate and develop a number of computational and analytical techniques for describing small scale heat transport.
We concentrate on a
kinetic-theory-based description, which has the Boltzmann equation as the governing
equation. Although not as fundamental as ab-initio quantum mechanics, this description has the advantage that it is sufficiently coarse-grained to be able to extend up to
and beyond the micrometer scale and is thus useful in describing actual devices and
systems of practical interest. A considerable part of this thesis is devoted to developing methodologies for bridging this description with Fourier-based descriptions, thus
providing seamless multiscale tools for describing heat transport over a wide range of
scales.
1.2
Outline
In Chapter 2, we present the Boltzmann transport equation for phonon transport and
review the existing literature on solution methods for this equation. We emphasize
in particular Monte Carlo methods and discuss in detail the "energy-based devia24
tional Monte Carlo" method presented in [4,11]. In Chapter 3, we discuss conditions
under which the BTE can be linearized, namely when deviations from equilibrium
are small. The developments presented in the remainder of the thesis deal with the
linearized BTE. In this same chapter, we show that the linearized formulation enables the use of the "kinetic-type" Monte Carlo (KMC) algorithm where timestep
and space discretization are no longer needed. In Chapter 4, we present some more
theoretical considerations related to the KMC-type method of Chapter 3. Chapter 5
presents an adjoint formulation of the linearized Boltzmann equation. Based on this
formulation, we develop and present an adjoint algorithm which simulates particles
traveling "backwards", and which results in significant computational benefits in situations where enhanced (e.g. spectral) resolution is required in small regions of phase
space. Chapter 6 presents an asymptotic solution method for solving the Boltzmann
equation at scales where both stochastic and deterministic methods become computationally expensive. Two particular applications of this method with clear multiscale
implications are presented in Chapter 7. The first is the study of interfaces between
materials, while the second explores the use of an asymptotic solution as a control
within the deviational framework. We conclude with some final remarks and possible
future work directions in Chapter 8.
25
26
Chapter 2
The Boltzmann transport equation
and current simulation methods
2.1
Importance of multiscale methods
In this section, we briefly review the various lengthscales of interest and the corresponding ad hoc methods for solving heat conduction problems at these scales. As
explained in the Introduction, the heat equation is the model of choice whenever heat
transport can be considered diffusive, namely when the mean free path of the heat
carriers is small or negligible with respect to the characteristic lengths of the system
of interest. We define the Knudsen number (Kn) as the ratio between the mean free
path A and the length scale L. Fourier's law can be used when the Knudsen number
is very small - this is discussed in more detail in Chapter 6.
At the other end of the spectrum of Kn values (see Figure 2-1), when characteristic
lengthscales are close to the atomic size, or a few nanometers large, one can rely on
molecular dynamics simulations [12,13], in which individual atoms and their motions
are simulated in real time using the laws of classical mechanics.
In some cases,
molecular dynamics simulations can incorporate elements of quantum mechanics. In
fact, when systems only include a few atoms (molecular level), the whole system may
be solved by ab-initio methods based on quantum mechanics (e.g. Density Functional
Theory 114]). Molecular dynamics, ab-initio or otherwise, is in general very costly
27
and, without resorting to massively parallel computing, three-dimensional systems
with characteristic lengthscales larger than O(100)nm are intractable.
The idea of modeling microscale and nanoscale heat conduction processes through
the lens of kinetic theory dates back to Peierls 115, 16]. The key idea is that, when
the characteristic length scales of the system are larger than the wavelength of the
phonons, the latter can be represented by particles and the effects that are inherently associated with their wave-like nature can be neglected. Based on this premise,
phonons are modeled as pseudo-particles whose ballistic or diffusive behavior naturally arises from the interplay between advection and collision processes as their
relative importance is depending on the system length- and timescales. This interplay is captured by the Boltzmann transport equation (BTE) used to describe phonon
transport in this regime.
Boltzmann transport equation for
phonons
Molecular
Dynamics
I
109
Diffusive
regime,
Fourier's law
I
Length scale (m)
o
Figure 2-1: Adequate description models for heat conduction depend on the scale of
the system of interest. The diffusive assumption (Fourier's Law) can only be applied
if the Knudsen number (ratio between the phonon mean free path and the smallest
characteristic lengthscale of the system) is small. More details on the transition from
the kinetic (Boltzmann-based) description to the diffusive regime can be found in
Chapter 6.
2.2
The phonon Boltzmann transport equation
The BTE describes the behavior of a population of discrete particles undergoing free
flights punctuated by scattering processes. It was initially introduced by Boltzmann
to describe rarefied gases 1171, and was later expanded to other types of particles and
applications. The Boltzmann description of neutron transport has been a critical step
28
in the design of modern nuclear reactor technologies [18,19]. Similarly, a Boltzmann
description is often used for electron transport or radiation (photon transport) [7]. In
its most general formulation but neglecting acceleration due to body forces, the BTE
may be written as follows:
Of
-- + c -
atscatt
The quantity
f describes
Of
f = Of(2.1)
the distribution of the particle population in the phase space
of particle position and (broadly defined) momentum. The physical meaning of
f (x, p, t)
f
=
3
3
can be conveyed by defining an elemental volume in the phase space A xA p
around a point (x, p), where p here refers to the generalized momentum coordinate.
At time t, the number of particles in the elemental volume is f(x, p, t)A 3 xA 3 p. The
BTE is simply a conservation equation which balances particle fluxes to and from the
elemental volume through advection or scattering.
Here we note that Boltzmann equations (balance laws of the form (2.1)) may
be written for quantities other than particle distribution functions.
For example,
f
- g can be devised, where g is an arbitrarily defined
function of the phase space.
This aspect is of particular importance, not only for
a Boltzmann equation for
deviational descriptions in general [20-22], but also for phonons in particular since
we will show in Chapter 3 how these considerations enable the introduction of a
"deviational energy-based" BTE.
The right hand side of 2.1 is a generic expression referring to the contribution
of scattering processes to the flux of particles in (or out of) an elemental volume.
Clearly, this term depends on the type of particle transport and on the type of scattering model. For instance, if the scattering term is absent, motion is ballistic and
is described by the advection equation. For rarefied gases, the right hand side of the
BTE is the nonlinear expression [23]:
I1
Of-at
scatt
2mn Jff
all
ac,
all i.
(f'f' + f'f' - ff. - f f)B( aV /V, V)dQ(a)d .
~
(2.2)
This term, referred to as the collision term in the rarefied gas dynamics literature, de-
29
scribes the elastic collision of pairs of particles (three particle collisions are neglected).
It conserves the number, the momentum and the energy of the colliding particles.
In the case of phonons, the phase space is usually the cartesian product of the
position space and the wavevector space. The wavevectors take their values in the
first Brillouin zone of the reciprocal lattice [81 and are related to the phonon vibration
frequencies through the dispersion relation w = w(k). The group velocity, defined by
Vg = Vkw, gives the traveling (group) velocity of the phonon. In this thesis, we
will only consider models where the dispersion relation is assumed isotropic, and as
a consequence the group velocity is assumed to only depend on the frequency and
polarization, namely, Vg = V(w, p)f.
As commonly done in the field of phonon
transport [7], we will use the frequency L, polarization p and traveling direction Q as
base parameters of the momentum space, instead of the wavevector. The corresponding isotropic density of states [7] is denoted by D(w, p) . The distribution of phonon
frequencies at thermal equilibrium is the Bose-Einstein distribution
1
ff (L) =
(2.3)
exp ()-
where T is the (equilibrium) temperature, hw is the energy of a phonon with frequency L, kB is Boltzmann's constant and h is the reduced Planck constant. In the
rest of this thesis, the frequency dependence of f '(w) will be implicit (the equilibrium
distribution will be denoted f
,).
Similarly, we will frequently omit the parameter de-
pendencies (x, L, p, Q) of the distributions in phase space in order to keep expressions
concise.
Phonons may scatter due to the presence of material impurities ("impurity scattering") or grain boundaries ("boundary scattering").
These events both involve a
single phonon interacting with the underlying medium and changing its properties
(for example, traveling direction) as a result of this interaction.
Other scattering
modes involve several phonons interacting due to the anharmonic part of the interaction Hamiltonian.
Such events either involve two phonons merging into one or,
inversely, one phonon spontaneously splitting in two. More details on three-phonon
30
processes may be found for instance in [8, 24]. Higher-order processes are typically
negligible [25].
In this thesis, we will exclusively rely on a semi-empirical approximation that has
been known for giving reasonable agreement with experimental results. This approximation, formally similar to the Bhatnagar-Gross-Krook (BGK) model sometimes
used for modeling collision in rarefied gases [26], amounts to assuming that a given
phonon mode relaxes towards thermal equilibrium at a characteristic relaxation rate,
namely,
Of
Ot
scatt
where w(w, p, T) is the relaxation-time.
floc - f
T)
p, ,T)(2.4)
T(W
T (L,=
We will refer to it as the relaxation-time
approximation. In this model, the relaxation time depends on the phonon frequency
and polarization, and on the temperature, T; the latter is implicitly defined by the
relation
jz D(w, p) hwfd 2Qd
D (,p) h
p
p
exp(
dw.
(2.5)
)-1
The local distribution fl" is a Bose-Einstein distribution with temperature parameter
equal to the pseudo-temperature, Ti1 c; the latter is defined by the relation
SD(w, p)hw
J~LJ
D(L p)h(
2
4wr(o.c, p, T)
T (w,
p, T) (exp
dBw
(26
which requires that energy is conserved when averaged over all scattering processes.
The pseudo-temperature is different from the temperature because of the frequency
dependence of the relaxation time r(W, p, T). Single relaxation time models do not
require the introduction of the pseudo-temperature because in such a case, equation
(2.5) implies (2.6) with T10c = T.
The main difficulty associated with solving the BTE stems from the large dimensionality of phase space. A deterministic simulation method would require the discretization of the three dimensions of wavevector space in addition to physical space.
In the time-dependent case, time discretization adds another dimension.
Due to
these limitations, recent deterministic methods have introduced approximations [27]
31
to obtain efficient solution methods.
The popularity of Monte Carlo methods is a result of their ability to avoid the
full discretization of the computational domain for problems featuring a large dimensionality. As a result, Monte Carlo simulations in kinetic transport have already been
extensively used in the 1960's for neutron transport calculations. In comparison it
is only more recently that their potential for solving phonon transport problems has
been discovered.
2.3
Timestep-based Monte Carlo methods for phonon
transport
A Monte Carlo method for simulating phonon transport was first introduced by Klitsner et al. [28], who developed a method for studying the behavior of crystal free
surfaces in a heated silicon rod. His results were compared to experimental results
for the purpose of calculating the coefficient of specular reflection of phonons on
crystal surfaces. Since the sample was studied at low temperature (1K and below),
phonon-phonon scattering was not included in this study and the simulation technique was essentially a direct adaptation of Monte Carlo methods used for radiation
transfer simulations. Later, Peterson included phonon specific features such as the
density of states and relaxation time [29]. The resulting method was reminiscent of
the Direct Simulation Monte Carlo (DSMC) used in the field of rarefied gas dynamics [30]. Between 2000 and 2010, the method was further improved and was more
widely adopted.
Mazumder and Majumdar included nonlinear dispersion relations
including accurate dispersion relations accounting for phonon polarizations [31]. The
correct treatment of frequency dependent relaxation times was emphasized in several
publications, most notably in Refs. [6, 32]. By the early 2010s, Monte Carlo methods for simulating phonon transport in the relaxation time approximation were well
established.
In this "mature" form, the method simulates a number of computational parti-
32
cles, each representing an effective number of phonons Neff. It is essentially a "split"
algorithm where, at each timestep At, an advection and a scattering routine are sequentially applied. Representing the unknown distribution in terms of computational
particles amounts to introducing the approximation:
Df
N
NeffZ6 3 (x - xi(t))6(w - wL(t))6p,,g()6 2 (f - Qi(t))
(2.7)
i=1
The algorithm may be broken down into 4 distinct major components: initialization,
advection, sampling and scattering. We briefly describe each below:
-
Initialization:At time t = 0, the system is characterized by an initial population
of particles
fi.
The algorithm therefore begins by initializing N particles whose
properties (position, frequencies, polarization, traveling directions) are drawn
from the distribution
D (L,D p) A (X,
47
A Q)
(2.8)
Drawing particle properties from this distribution can be done in various ways,
such as acceptance-rejection, analytical inversion of the probability density function [3] or numerical inversion of the probability density function by discretization [311. Most frequently,
fi is
a Bose-Einstein distribution (2.3) at temperature
Ti(x).
The number of computational particles, N, is usually chosen as a trade-off
between the amount of computational resources available and the desired statistical accuracy. From N, the "effective number of particles" is deduced by first
calculating the total number of phonons
N
jotj
fid2Vdlwd3x
(2.9)
in a system, and then by requiring Neff = Ntot/N.
-
Advection:
During this step, particles move ballistically according to their
respective velocity vectors for a timestep At.
33
In other words, if a particle
position at the start of this step is xi(t), then the position is updated to
xi(t + At) = xi(t) + AtVg,,. This corresponds to solving
Of + Vg *Vxf =
0
(2.10)
between t and t + At. The timestep At must be chosen such that it is smaller
than the smallest relaxation time for the time discretization error to be small.
-
Sampling: Since the relaxation times depend on the temperature, it is necessary
to evaluate this quantity everywhere in the system before proceeding to the
scattering step. The temperature is linked to the energy density as expressed
by relationship (2.5). The energy density is estimated in a volume-average sense
in a grid of spatial computational cells. In other words, for a given cell C, the
energy estimate is:
Ec= Neff
hwi
(2.11)
iGC
The temperature of the cell is retrieved by inverting formula (2.5). For computational efficiency purposes, it is often useful to store a table matching energy
densities and temperature.
In practice, the pseudo-temperature is also required for performing the scattering step. This quantity may be calculated at this stage by summing
EC
= Neff E
T(WipiTC)
(2.12)
iEC
and by using (2.6). The pseudo-temperature may also be evaluated in the next
step.
-
Scattering: This step simulates the scattering process. In the relaxation time
approximation, this corresponds to solving
Of
floc- f
T (, p, T)
&t
34
If floc is assumed independent of time between t and t + At, then we have the
solution
f(t + At) = f (t) + (floc - f(t))
exp (.
-
(2.14)
Equation (2.14) states that f(t + At) is obtained by removing
f (M)
t(2.15)
- exp
rF(a, p, T)
(.5
from f(t) and by adding
floC
1
-
exp
()
( -(L,,
(2.16)
p, T)
The former can be done by defining, for each computational particle i, the
scattering probability
P
1 =-exp
'A
t
(2.17)
,
(
(L4i,
pi, TC))
where Tc is the temperature of the local cell, and by randomly selecting the
particle to be scattered according to this probability. This is typically done by
choosing a random number R uniformly between 0 and 1 and by comparing
it to Pi. Note that, although Pi was obtained here from the expression of the
right hand side of the Boltzmann equation, it can also be interpreted as the
probability of one or more scattering event happening between t and t + At
(Poisson process).
The selected particles are removed from the cell and replaced by particles whose
properties are drawn from the distribution
floc
which can be approximated as
-
exp
whc (ct, P, Th
.
(2.18)
f 'At/r without changing the order of the ap35
proximation (order 1 in time) that is inherently associated with the split algorithm.
It is very important that energy is conserved in each cell.
This cannot be
achieved by simply controlling the number of new particles to be generated
because the expression for the energy includes the frequency, which for each
particle is chosen randomly. The strategy usually employed for ensuring energy
conservation in a cell consists of adding or deleting randomly selected particles
until the post-scattering energy approximately equals the pre-scattering energy
[6,32].
Finally, we note that the energy of the scattered particles provides an alternative estimate of the pseudo-energy mentioned in the previous point.
Since
this estimate involves less samples than the one obtained by using (2.12), the
variance of the former is likely to be larger; for this reason the latter is usually
preferred.
2.4
Deviational energy-based Monte Carlo simulation
In Ref.
[4], Peraud and Hadjiconstantinou presented a variance reduction scheme
that accelerated Monte Carlo methods for phonon transport by several orders of
magnitude.
This improvement is a result of using deviational particles to simulate
the difference between the unknown solution and a chosen equilibrium distribution
(which is typically a Bose-Einstein distribution at a temperature chosen such that it
is close to the system reference temperature).
A second improvement follows from
simulating computational particles representing a fixed amount of energy rather than
a fixed number of particles. In the relaxation time approximation, the quantity that
is conserved is the energy, while the number of phonons can fluctuate in time. This
strategy leads to strict energy conservation provided the net number of computational
particles is conserved.
36
In the relaxation time approximation, the deviational energy-based Boltzmann
equation is found by defining the deviational distribution ed = hw(f -
fjq), and
writing the BTE in the form
aed
e -eq -ed
Vg
e
= .e
1"
(2.19)
where Tq is assumed to be independent of time and space.
The case of spatially
at
+ Vg,e
-
T(,p, T)
dependent Tq will be discussed in section 3.3.
In deviational simulations, we proceed with the approximation
D(L, p)ed =
effSi6(W
-
wi)62 (
__
R)6 3 (x -- xi)6pp
(2.20)
i=1
where the sign si can take the value
+I or -1 and arises because the deviational
quantity ed can be positive or negative. The quantity Eef now refers to the "effective
energy" that each deviational particle represents. Below, we outline the algorithm that
ensues from this formulation, with an emphasis on the major differences compared to
the algorithm mentioned in section 2.3. More details can be found in [4].
- Initialization:The effective energy is calculated similarly to the effective number
of particles in section 2.3, with the exception that f is replaced by ed
.
Initial
properties of the N initial particles are drawn from the distribution hw(f- f-q).
- Advection: Because the left hand side of the Boltzmann equation is unchanged in
the deviational formulation, computational particles are simply advected without scattering.
- Treatment of boundaries and source terms: Rigorous treatments of the boundary
conditions usually require one to write the associated formulation in a deviational sense. For instance, a boundary at a prescribed temperature Tb emits
the following distribution of deviational energy:
1
ed =
1
I
exp
.
-
(b,
-1
37
exp
kBT
q
(2.21)
For a diffuse reflective wall with inward pointing normal n, deviational particles
are treated like traditional (non-deviational) particles because for all frequency
and polarizations, the following relationship holds:
edQ' - nd 2
ed -7T
for Q - n > 0
',
(2.22)
2'-n<o
Sampling: The energy estimate in a computational cell is given by the sum of
a deterministic and a stochastic term:
E
Eq + EFg Y si
(2.23)
iecell
-
This equation directly results from the algebraic decomposition e = e' Teq + (e
e 2Teq ).
The deterministic information contained in the equilibrium distribution
is the key ingredient leading to variance reduction.
Scattering: The scattering step differs from the traditional method of section 2.3
in two ways. While the method to select scattered particles is strictly identical
to the previous method, the post-scattering properties are now drawn from the
distribution
kB T1.c
1
_
exp
(k~q
k BTeq
(.4
)L
-
1
D(w, p)hw
DL ~
p (2.24)
T(a;,T($J~,T
p, T) kexp (k h)-i
_
In addition, using energy-based particles ensures that energy is rigorously conserved throughout the scattering step by simply conserving the particles (only
their frequency, polarization and traveling directions are reset). For further efficiency gains, one may cancel particles of opposite signs that are scattered at the
same time and in the same computational cell. While this process is not always
strictly necessary, it prevents the number of particles from growing indefinitely
in the case where a source term continuously emits new particles and where
particles cannot exit the system through an absorbing boundary.
A typical
example of this situation arises in the study of periodic nanostructures 14,5].
38
2.5
Variance of non-deviational vs deviational algorithm
In the non-deviational case, for a fixed number of computational particles and small
deviations from equilibrium, the uncertainty associated with property estimates is
fairly independent of the temperature deviations
[3]. In contrast, as shown in Figure
2-2, the statistical uncertainty in the variance-reduced case is generally smaller and
proportional to the amplitude of the deviation from equilibrium.
This results in
considerable speed-ups ranging from 4 to 8 orders of magnitude (depending on the
problem) and characteristic temperature deviations on the order of a few degrees K
[4,11]. Moreover, the fact that the uncertainty in deviational methods is proportional
to the deviation from equilibrium means that the method can solve problems with
arbitrarily low deviations from equilibrium at no additional cost. Problems that were
previously intractable with the more traditional algorithm can now be solved.
0
-
-
Variance-reduced MC
Non variance-reduced MC
103
a.-
orX
0.-
10
10-1
4
1
AT
1 n-
2
In
-
1n-
Figure 2-2: Ratio oa//q" as a function of AT/Tq at fixed (and equal) number of
computational particles. Image reproduced from [1].
39
2.6
Limitations of the deviational method
Although already quite efficient, the energy-based deviational Monte Carlo method
still suffers from a few limitations. The first of these limitations is common with deterministic methods: since the system needs to be fully discretized in physical space,
very large systems cannot be simulated. The transient thermoreflectance experiment
(TTR), detailed for instance in Ref. [331 and briefly reviewed in section 3.7, is particularly illustrative of this issue: although the system is (theoretically) semi-infinite,
the computational domain must be artificially terminated. The resulting approximation can lead to neglecting important multiscale effects.
For instance, introducing
an artificial boundary can lead to incorrect interpretations of effective thermal conductivity measurements. In addition, introducing a space discretization is contrary
to the "philosophy" of Monte Carlo methods in general, which usually aim at replacing deterministic discretization by simple functional evaluation in a given probability
space, hence providing estimates that converge towards the exact answer in the limit
where the number of samples is very large.
The second major limitation lays in the use of a timestep.
Since the latter is
inherently associated with an approximation, care must be taken that the chosen
timestep be small enough for the approximation to be accurate. In the rarefied gas
dynamics literature, it is generally acknowledged that the timestep chosen for DSMC
simulations must not be larger than around a third of the relaxation time 134]. Since
the relaxation time of phonons in common materials, e.g. silicon, can span many
orders of magnitude (for instance, in silicon at 300K, the smallest relaxation time
is of the order of a few picoseconds while the largest relaxation time is around a
microsecond), this requirement leads to multiscale limitations. More specifically, the
timestep is common to all phonon modes and must therefore be smaller than the
smallest relaxation time. This means that a very large number of steps are required
to effectively simulate the dynamics of such systems. As a result, although variance
reduction has enabled the simulation of systems such as the TTR, simulating these
systems for long physical times is still a challenge. Chapter 3 of the present thesis
40
will show how, by taking full advantage of a linearized formulation of the BTE in the
relaxation time approximation, the two limitations mentioned above can be overcome.
The method that we present is of "kinetic", event-driven type and gets rid of the
necessity of a timestep and spatial discretization. Chapter 4 will present additional
theoreticaly considerations related to the method presented in Chapter 3.
41
42
Chapter 3
Linearization of the Boltzmann
transport equation and kinetic-type
Monte Carlo
In the present chapter, we describe a simulation method that is considerably simpler
to code and more computationally efficient that the one described in [4,111 and in the
previous chapter. Moreover, due to the absence of discretization in space and time, it
is expected to be, by some measures, also more accurate. The algorithm is based on
the observation that under the linearized Boltzmann dynamics, deviational particles
may be treated independently and thus sequentially (one at a time). The theoretical
developments and some of the applications /examples discussed in this chapter can
also be found in Refs. [2,3,5].
43
3.1
Independent phonon trajectories under linearized
conditions
We start from the energy-based deviational BTE (2.19). More specifically, we focus
on the non-linear scattering operator which may be written in the form [3]:
e 10Ceeq1
~ed
e_-
T(,
-
Teq)2
-- Te)
+ 0__Tc - Te)
(3.1)
T ( ,p, Te) + O(AT)
(Tioc
-
p, T)
ebOCe q- ed
-F(L,,
(TZa
e_(Tc
Teq
-
T T
p, T)
_T (L', p, Teq)
ed
TT+ O(AT 2 )
(3.2)
because ed scales with AT. In the expansions above, the frequency and polarization
dependence are implied; it is also assumed that AT and TIc - Teq are of the same
order of magnitude.
For small temperature differences, the BTE can therefore be
linearized to yield
+Vd
Oed+
V9 *Ved
Ve d - Ld)
'C(d --ed
at
r (,p, Te)
3.3
(3.3)
where
C(ed) = (Toc - Teq)
As discussed for example in
(3.4)
Teq
OT
[6] and in section 2.2, T 0c is the pseudo-temperature,
required in the relaxation time approximation because of the frequency-dependence
of the relaxation time
T(,
p,
T). Under linearized conditions ((T - Teq)/Tq
r<
0.1)
the energy conservation statement
D(L IP)
D(I
___4)
OT
(L, P,Teq))
p)ed
47r(wp,
4i
Tq)
(3.5)
can be solved explicitly, yielding
Toc -
Teq =
[LD(,
P)e
d2Q)
P47r(c,, p, Teg)
DL,
P
44
(L, ,
T
,Tg) OT
]'d;
I
(3.6)
Moreover, the term Oe
q
/&T can be calculated analytically from the Bose-Einstein
2kBTe
T
sinh 2
)
k Tq
)
distribution:
eqkBsin
We recall that, in the deviational algorithm outlined in the previous chapter and
detailed in [41, the properties of the post-collision particles are drawn from the distribution
-- e
D (L , p)A t e'oc
T)
where At is the timestep.
eq
(3.8)
T(, p, T)
47r
In the linearized approximation introduced above, this
distribution becomes
(Tioc -
D(c, p)At
')T
4
Teq)
47
T(L
, P, T,)'
.pC
(3.9)
Algorithmically, this means that, at each timestep, for each particle that is scattered,
the frequency and polarization are reset by drawing from the normalized distribution
ae"q
D(w,p)
47r
QT
(3.10)
9
r(w,p,Teq)
4T(P,Teq)
eg9
S
p
QT
D(W-p)
4r
T(WpTeq)
d,
We note that the term (T]0 c - Teq) cancels as a result of the normalization.
This
temperature term is the primary reason why a timestep-based algorithm has been used
for simulating phonon transport in the relaxation time approximation: temperature
information needs to be calculated everywhere and updated at every time step so
that the scattering term be evaluated. In the linearized approximation, however, the
energy-based deviational particles have the following properties:
- The relaxation time is independent of temperature since it is given by r(w, p, Teq).
- The normalized post-collision distribution is independent of temperature.
In other words, in the linearized approximation, for each particle the trajectory may be
calculated without any temperature information. Particle trajectories can be treated
45
as independent of each other and can therefore be simulated separately (in contrast
to the algorithm presented in Chapter 2 where the whole particle population needed
to be simulated at the same time). This also removes the need for a timestep-based
algorithm, as well as the need for spatial cells and for meshing the system. Instead,
we may use a kinetic-type or event-driven algorithm that we outline below, where
the time between each scattering event for a particle can be calculated directly by
inverting the exponential probability distribution.
3.2
Kinetic-Monte-Carlo algorithm
As explained above, the linearized formulation allows particles to be simulated one by
one and independently. Below, we describe the algorithm for calculating an individual
particle trajectory from t = 0 to t = tfin.
In the interest of simplicity we have
suppressed the particle index.
I Draw the initial properties (sign s, position xo, frequency wo, polarization po,
direction Qo, and the resulting group velocity vector Vg,o) of the particle. For
time-dependent calculations, also set up the initial time to of the particle; this
is discussed in more detail below.
II Calculate the traveling time until the first scattering event: uniformly draw a
random number R between 0 and 1, and calculate
At = -IT(Lo,
Po' Teq) In (R)
(3.11)
We note that some random number generators have a finite probability of drawing the number 0, which as a result can make a code unstable, while the number
1 is never drawn. One should therefore carefully check the specifications of the
random number generator that is used. In the case mentioned here, it is important, to replace ln(R) with ln(1 - R), especially when a large number of particles
is simulated.
46
III Calculate
knew
=
xo
+ Vg,oAt. Search for encounters with system boundaries
in the time interval At.
IVa If an encounter with a system boundary occurs, say at
update the internal time te, = to +
(Xb -
Xo)J/Vg,o
xb,
set
Xnew
= Xb and
. Depending on the
nature of the reflection (specular or diffuse), set the new traveling direction
appropriately (as explained for example in [311).
IVb If no encounter with system boundaries occurs, the particle undergoes scattering at position
xne,
=
Xnew.
The internal time is updated to te, = to + At.
New frequency Onew and polarization pne,
are then drawn from (3.10). A new
traveling direction is also chosen: in this work, we consider isotropic scattering,
but this can easily be generalized to non-isotropic scattering. From these parameters, a new velocity vector
Vg,new
can be defined. The particle sign remains
unchanged by this scattering.
V The contribution of segment [XOXnew] to macroscopic properties is sampled.
Sampling is discussed in more detail below.
VI If t >
t
flnal,
set {.}=
proceed to step I to begin simulation of the next particle; otherwise,
{.}new, where {.} denotes the set of all properties of particle i, and
return to step II.
This algorithm significantly contributes to reducing the number of operations between
scattering events and, depending on the problem of interest, the resulting algorithm
is several orders of magnitude more computationally efficient. Memory use is also
considerably reduced since a particle is always discarded when its trajectory is terminated.
The total number of particles processed, N, is determined by balancing simulation cost and desired statistical uncertainty. From this choice, the effective energy
carried by each computational particle,
Leff,
can be calculated as the total amount of
deviational energy involved in the phenomenon of interest divided by N. The total
deviational energy includes the magnitude of the deviational energy present in the
47
initial condition Eic = Y, f(47)-'D ed(t
0) d3 xdLd 2 Q (the integral range is the
=
whole phase space), as well as the magnitude of the deviational energy associated with
boundary conditions or other source terms over the course of the simulation; the former will be denoted here as Ebc = J,
f~t'a fs"o'f.n>0(47
IDV,
n edd2xdwd 2 f1dt
where S denotes the surface area and n the surface normal of the boundary, while the
latter will be denoted by E, = Y
fig
f(47) -
eDd3xdwd 2
dt; concrete examples
of these quantities will be given below. The number of particles associated with E
where i E {ic, be, s} is given by EjN/(Eic + Ebc + Es). The initial time for particles
due to the initial condition is set to to = 0, while
t
o for the remaining particles is
drawn randomly in the interval [0, tfilal] and from a distribution that reflects the rate
of deviational energy input/removal from the simulation (e.g. for a source term, this
can be achieved by inverting the absolute value of the cumulative deviational energy
magnitude input/removal function
3.2.1
Z, ft,_
f (47)-'D e d 3xdwd2 Qdt').
Example of problem setup
As in Ref. [21, let us illustrate these concepts with a ID example, namely transient
heat transfer between two boundaries at fixed and different temperatures
=
L, and their temperatures are T and T2
,
boundaries are located at x = 0 and x
[32]; the
respectively. In order to illustrate the implementation of initial conditions, we take
the initial system temperature to be
To (x)
For T1 - T2 1
T1 + T2 T1 - T2 .
Si
2
+
2
Wx
-
(3.12)
< (T + T2 )/2, deviations from a suitably chosen equilibrium temper-
ature will be small.
For example, an obvious choice, is Teq = (T + T2)/2; other
equally valid choices are Teq = T1 or Teq = T2 . In our calculations, the first possibility
Teq = (T + T2 )/2 was used as it makes the initial condition easier to treat.
The two source terms introduced above are associated with the following energy
48
terms, which are expressed in J m-2:
Ee
=
To(x)
=Ti - T2
-
jL
Teq\
ZD
dx
(3.13)
D(,P) Tdo
j
tfinal
( Ti - Teq + JT2
Ebc =n
-
Teq )
p d
)
Dwp) Vg
D
(.4
;
(3.14)
In this example, if a given particle is determined as being emitted from the initial
condition, then its initial internal time is t = 0, and its initial position is randomly
drawn between 0 and L according to the spatial distribution of the initial condition,
namely
P(x) = To(x) - TeqjZ7D(w,P)
(3.15)
Td
which, once normalized and given the expression we chose for Teq, can be written
as p(x) =
(2L)- 1 sin(7x/L). This can, for example, be implemented by drawing
a random number R uniformly between 0 and 1 and by setting xO such that R =
fXj.
p(x)dx, or xO = L cos-'(1 -2R)/7r.
If the particle is emitted from the boundary
condition, its initial time is uniformly (randomly) drawn between 0 and
tfinal,
while
its initial position is either 0 or L depending from which boundary it originates. The
sign of the emitted particle is determined by the sign of Ti - Te and
T2
- Teq at each
boundary, respectively.
3.2.2
Sampling
We now discuss the sampling process in more detail. Let
h (t') = E
(47)-'Dhed (t')d3 xdd2 q
(3.16)
be the macroscopic property of interest (at time t') in terms of a general microscopic
property h = h(x, L, p, Q). Recalling that the deviational simulation approximates
the distribution ed in the phase space using deviational (computational) particles [4],
49
inserting expression (2.20) in (3.16) yields
Zh (t') = Eeff
s ih [X (t'), L4;i(t'), ~it'),
(3.17)
Qi(t')]
For example, if the quantity of interest is the x-component of the heat flux vector in
some region of space R with volume p(R) and defined by the indicator function
then h = Vg,o - 21R/1p(R) and thus particle i only contributes to Ih(t') if
position at L'-calculated by linear interpolation between (xo,Lo) and
in R. The contribution of such a particle to
3.2.3
Ih(t')
is si[Vg,o]
1
R,
xi(t') [its
(Xnewtnew)] is
- dxeff /P(R).
Steady state sampling
As in standard Monte Carlo methods, steady state problems can be sampled by
replacing ensemble averaging with time averaging, provided sufficient time has passed
for steady conditions to prevail. The proposed method is particularly suited to this
type of problem because it can be used to directly solve for the steady state without
requiring relaxation of the system from some initial condition, if the latter is of no
interest. This can be achieved by including only the steady particle sources (Ebc and
E,) and considering each simulated particle to represent a fixed amount of energy per
unit time
4
ff; particles are emitted from the time-independent sources and propagate
according to the rules explained above.
Since the steady state is constituted of particles at all stages of their history,
one can sample the values of interest by computing curvilinear integrals along the
complete particle trajectories, from their emission to their termination (for example,
in a problem with prescibed temperature boundaries, termination would occur when
the particle is absorbed by a boundary). In this case the estimate
Z1h
= Se ff
s if
h[x (t), wi (t), pi(t),Q i(t) dt
h
is:
(3.18)
Jx, (t)C
The rigorous mathematical proof leading to expressions of this type can be found
in linear transport theory literature (see for example
50
[35]) and allows to directly
compute the steady state response to the sources, contrarily to methods based on
timestep. Although formulation (3.18) makes use of an internal time for each particle,
simplifying the integral usually leads to an expression that does not include time
(due to the steady-state nature of the problem). In the case of the heat flux in the
x-direction (h = Vx/p(R)) averaged over the domain R discussed above, equation
(3.18) reduces to
s il,
(R
(3.19)
where 1xj is the total algebraic length traveled in the x-direction by particle i while
in R. In the case of the average temperature over R, this becomes
Seff
(3.20)
[1,(R)
Vg~
where 1i is the total length traveled by particle i while in R.
Figure 3-1 shows both transient and steady state solutions of the one-dimensional
problem described in section 3.2.1. The steady state solution is obtained as described
above, rather than integrating in time from an initial condition. In these simulations,
+
Ti = 301K, T 2 = 299K, Teq = 300K leading to an initial condition To(x) = 300
sin(x7r/L), while L = 400nm; materials parameters (dispersion relations, scattering
rate) are the same as in [4j and are recalled in Appendix F; optical phonons were
taken into account (with zero group velocity, as described in [4] and in Appendix F).
3.3
Spatially variable controls
So far, we have only discussed the case where the control is not a function of space, and
where the system is linearized with respect to the uniform equilibrium temperature.
Previous work in the rarefied gas domain [21,36] has used spatially variable controls
as an acceleration method
136] or for imposing external driving forces. Below we
show that spatially variable controls can be used for the same purposes in the case of
phonon transport calculations.
Let us consider Teq = Teq(x). In this case, the deviational form of the energy-based
51
Initial condition
0.5
steady state
-0.5
-1
4
3
2
1
0
x 10'
x (m)
Figure 3-1: Transient and steady-state temperature profiles between two boundaries
at different temperatures. Here, AT = T(x, t) - Teg as defined in section 3.2.1.
The dots represent the solution calculated with the timestep-based method while the
continuous line was obtained using the kinetic-type method presented in this chapter.
Reproduced from [2].
BTE becomes
eloc - e
Oed
Vd
e
T
-q e d
q(Led - Vg *VxeTq
(3.21)
r~~T, p, T
)
V
at+ 81
In other words, introducing a spatially variable control introduces a volumetric source
term in the deviational Boltzman equation. This source term may be further simplified by writing
-V9
Vxe
= -V9 - VxTeq
OT
(3.22)
The above equation is valid for arbitrary Teq (x). In the context of the linearized
Boltzmann equation, two choices for Teq(x) are particularly appealing:
- We may directly linearize with respect to the spatially variable control temperature Tq(x). In this case, the relaxation time and the post-collision distribution
will be spatially dependent, which removes most of the convenient aspects of
the linearized algorithm.
52
- We may linearize with respect to a uniform temperature Teqo chosen within the
range where Teq(x) takes its values. In such a case, we are essentially using the
control function
econtrol
eq____
eTqO + (Teq (x) - Teq,o)
OT'
(3.23)
and the linearized deviational equation is now given by
d)
&ed
t
+
d
*,V"e - r(L, , Teq,o)-
Vge
V9 - 17-.Teq (X)
-
OT
'3.24)
The latter formulation conserves the benefits of a spatially uniform control (notably,
the spatially independent relaxation times) without decreasing the order of the approximation (order 2 in both cases). We will frequently use this formulation in the
remainder of this thesis.
3.3.1
Simulation of periodic nanostructures using spatially variable controls
One significant advantage of spatially variable controls is that it lends Monte Carlo
formulations naturally to thermal analysis of periodic nanostructures. Periodic nanostructures have attracted considerable interest in the field of thermoelectrics. Specifically, practitioners have been attempting to increase the figure of merit of thermoelectric materials through nanopatterning, by exploiting size effects for reducing the
thermal conductivity while increasing the electrical conductivity or at least keeping
the latter approximately constant.
These attempts required solution of the BTE
in representative unit cells of these materials, which has led to the development of
periodic boundary conditions [6,37, 381 for this purpose.
For this discussion, we will refer to the example geometry sketched in Figure 3-2,
where a nanoporous structure with cylindrical pores organized in a triangular lattice
is represented. Let us assume that the thermal conductivity in the direction ey is of
interest. The most effective methodology for creating a heat flow through the unit cell
53
in traditional Monte Carlo schemes requires 16, 37] deviations from local equilibrium
to be periodic, which is equivalent to requiring the phonon energy distributions at two
corresponding boundary points (represented by A and B in Figure 3-2 of the periodic
cell) to differ by the quantity eq
-
e'
6
. In other words, the boundary condition is
expressed by
e(xA,
, ,p) - e(xB, O, op) e e T
(3.25)
Implementation of condition (3.25) requires deletion of particles exiting the domain
at
XB
and introduction of particles at
x =
XA moving in the inward direction that
sample the distribution
e(xR, Q, U, p) + e
-
(3.26)
eq.
The most straightforward method to sample from this distribution begins [61 by collecting the pool of particles leaving the domain at
XB.
In traditional periodic bound-
ary conditions, this entire pool would be reintroduced at XA satisfying e(xA, Q2, w, p) =
e(xB,
, L, p); to satisfy (3.25) one needs to modify this pool by deleting particles ac-
cording to the distribution e" and introducing new particles that sample eq.
order to remove particles from e',
In
a new set of particles needs to be sampled from
this distribution and then the particles in the pool that are most similar to the newly
sampled particles can be deleted. Particles leaving the domain at
XA
can be treated
analogously.
This approach introduces an unknown amount of error because the particles crossing periodic boundaries will not correspond exactly to the particles that are sampled
to guide the deletion process. By contrast, the deviational formulation is naturally
suited to simulating (3.25) exactly, because it allows for negative particles. In addition, using a spatially variable control alleviates the burden of drawing particles from
spatially variable source terms at the boundaries, by replacing them by one uniform
source term which we detail thereafter.
In a linearized setting, we can generalize condition (3.25) with the following ex54
lattice.
Wq
W
pression:
e (xa, L, ,A) e (X, B, C4, A ) =(xa
- xB) - Vx Text
(X)
q
(3.27)
OT
where Text may be understood as an externally imposed temperature gradient. We
impose Text to be linear, thus its gradient is uniform.
ee
+ (Text(x) - T,,q)ae'
/T
Using the control ec.ontroi =
considerably simplifies the boundary condition on the
deviational distribution to
e d(xa, W, ,A) = d (xB, W, AO
(3.28)
For a Monte Carlo simulation, this means that when a particle encounters the boundary at point A, it is simply reintroduced at point B with unchanged properties (frequency, polarization and traveling directions). The heat flow is ensured by the presence of the anisotropic source term induced by the spatially varying control. In Figure
3-3, we show the two components of the heat flux field for the problem depicted in
Figure 3-2, calculated with the kinetic-type Monte Carlo method presented in this
chapter, and applying the particle termination criterion explained in section 3.4. Af-
55
026
0.2
0.4
~
x 107
3
X10
x1010
0
6
5
0.22
4
0
0.4
2
0
.63
0..
0.6
-1
0.8
-2
2
0.8
1
0
0.5
x(m)
a.
0
1
X 0-7
b.
1
0.5
X (m)
X10
7
Figure 3-3: x and y components of the heat flux in the elementary cell of a triangular
network of cylindrical pores subjected to a temperature gradient of 106 K m-' in the
y-direction. The pore surfaces are assumed diffusely reflective. Reproduced from 131.
ter solving the Boltzmann equation with these conditions, the thermal conductivity
may be found by dividing the heat flux by the norm of the temperature gradient.
A second case where spatially variable controls are of particular interest is as an
additional variance reduction tool: by allowing the control to vary as a function of
space it is possible [21,22,36] to track variations in local conditions, thereby minimizing the deviation from equilibrium that needs to be simulated and thus ultimately
minimizing variance. Sophisticated approaches that use asymptotic solutions of the
BTE as controls will be explained in Chapter 7.
3.4
Termination of particle trajectories
The kinetic-type algorithm calculates each trajectory as a sequence of segments separated by scattering events. At the latter, the frequency, polarization and traveling
direction of the particles are reset according to the post-scattering distribution. Since
scattering events strictly conserve the number of deviational particles, particle trajectories will continue indefinitely until stopped. In transient problems, this does not
pose a challenge since particle trajectories can be stopped at the final time of interest. In steady problems, however, other termination criteria need to be introduced
56
to prevent the simulation time extending to infinity (recall that no time parameter
exists in steady problems).
We start with a discussion of the type of typical boundary conditions encountered
in simulations.
i- Boundaries with prescribed temperature: these boundaries include an absorption term. Therefore, when the particle intersects the boundary, it is terminated.
In a system where collisions with such boundaries are frequent, termination will
be relatively efficient and complete particle trajectories may be computed.
ii- On the contrary, if encounters with prescribed temperature boundaries are infrequent (compared, for instance, to adiabatically reflective boundaries), then
the number of scattering events per particle trajectory before termination will
tend to be large and each particle trajectory will be computationally expensive.
In the most extreme case, the system may have no absorbing boundaries, and
the particle trajectory should theoretically be infinitely long. A typical instance
of this situation is periodic problems as introduced in section 3.3. In such cases,
specific criteria must be applied for terminating the particle trajectories. The
termination criterion presented in [5] for nanoporous periodic materials relies
on the fact that, as the number of "randomizing" collisions - i.e. encounters
with diffuse boundaries and scattering events - undergone by a given particle
grows, its individual contribution to the final estimate (in this case, the heat
flux) tends to decrease because of randomization effects. After a given number
of collisions, the contribution that would result from further computing the trajectory is smaller than the uncertainty associated with this contribution, and
therefore computing more segments of the trajectory would not provide valuable information. We illustrate this in Fig. 3-4(a) and (b), where we plot the
contribution from each segment against the associated standard deviation for
the nanoporous problem depicted in Fig. 3-2. In Fig. 3-4(b), the uncertainty is
evaluated by dividing the standard deviation per particle by the square root of
the number of particles used in the simulation. We observe that after approxi-
57
mately 11 randomizing events, the average added particle contribution becomes
smaller than the noise that is introduced. The approximation that results from
terminating the particle after such a number of collision events is therefore comparable to the statistical uncertainty. In other words, a particle trajectory can
safely be terminated after this number of collisions. The number of collisions
is different from the one reported in Ref. [51 because the present calculations
does not include the optical phonons. In section 4.1, we present an alternative
cancellation method which relies on particle signs to effect cancellation within
small regions of space (bins). This method is shown (in section 4.1) to be more
efficient, but also introduce numerical error due to the approximation associated
with cancellation at finite distance (within bins of finite size).
8
2.5 X 10
log I(q )Y1
- - -
2
1.5
6
1
5
0.5
-*
3\
0
-0.5
(a)
log (aJN)
1
2
3
4
5
6
7
Index of trajectory segment
21
0
8
j
(b)
5
10
20
15
Index of trajectory segment
j
Figure 3-4: (a) Average contribution of the j-th trajectory segment (defined as the
portion of trajectory before two successive randomizing events, namely scattering
event or diffuse collision with a boundary) to the final heat flux estimate in the ydirection for the problem sketched in Figure 3-2. The applied temperature gradient
is 106 K.m- 1 . (b) Logarithm of the absolute value and uncertainty of the average
contribution of the j-th trajectory segment, all expressed in W m-.
3.5
Validation
The proposed algorithm has been extensively validated using a number of test problems. The simple ID problem of section 3.2.1, for which the comparison to results
calculated with the timestep-based variance-reduced method shows excellent agree-
58
ment, is one of them. Here we further validate the method by calculating the heat
flux field in a thin film subjected to a longitudinal temperature gradient. We assume
that the thin film is delimited by diffuse reflecting boundaries at z = 0 and at z = d
and that the temperature gradient is imposed in the y-direction.
Such a thin film
can be simulated using the periodic boundary conditions and the spatially variable
control presented in section 3.3.
While the thin film problem could be simulated as a 2D model (similarly to the
problem depicted in Fig. 3-2), here we can arbitrarily reduce the distance between the
two periodic boundaries while keeping a constant temperature gradient (&Text/&y),
without modifying the result, since we are only interested in the heat flux in the direction along the film (y- direction). In other words, the formulation above is equivalent to simulating particles in a ID system (variations exist only in the z-direction),
where the effect of the temperature gradient is accounted for by the volumetric (linear)
source
ae,
TeoTq
CText
ay
D(a), p) V (w, p) cos(a)
47
(3.29)
where 0 here refers to the angle between the traveling direction of a particle and the ydirection. The signs of the particles are directly given by the sign of - cos(0)OTeq/&y.
Note that as usual, when drawing the traveling direction of the particle, the factor
sin(0), generally enclosed in the expression of the solid angle in spherical coordinate
d2 Q
=
sin(O)d0do, where
#
is the azimuthal angle, need to be accounted for (see [311).
Solution proceeds by using the algorithm explained in section 3.2 over the ID
domain 0 < z < d, with diffuse reflection at z = 0 and z = d. The issue of particle
cancellation discussed in section 3.4 can be treated simply by realizing that the expected value of the heat flux after the first encounter with a boundary or after the
first scattering event is zero because these two types of events randomize the direction of a particle. In other words, because the expected contribution of the trajectory
to the heat flux following a scattering event is zero, trajectories can be terminated
after their first scattering event. This is verified by Figure 3-5; Fig. 3-6 shows that
the resulting simulation method is in good agreement with the theoretical solution,
59
expressed by
ed(z,
- AU
&Text &e'rq
* Cos (0)
'P
ay
BTAe,'
,p, 0,
Z
i() zo()F
n1
sin(O) sin(O)
1- exp
(3.31)
(3.32)
< # < 27)
&Tex &e*q
*e
I" Cos ()
ay
(3.30)
1- exp
9"T
~
s-zo()(9 s ~Q5)
z-d
Ae,' sin(O) sin(O)
(3.33)
.
- A
< 7F) =
/
ed(z, w, p, 0, 0 <
where A,, is the mode-specific mean free path. For d = 100 nm, the proposed method
returns a thermal conductivity of 49.002 Wm-K- 1 with a statistical uncertainty
of +/- 0.002 Wm- 1 K- 1 . By comparison, the numerical integration of the thermal
conductivity (we used a rectangle method to calculate the integrals) returns a thermal
conductivity of 48.9996 Wm-'K1.
5
10
C4
.2
3
2
0
-1
2
1
3
4
5
index of trajectory segment
Figure 3-5: Average heat flux contributions between scattering j - 1 and scattering
j, for the calculation of the thermal conductivity along a thin film. After the first
collision, the expected heat flux is zero because of the symmetry of the problem.
3.6
Consistency and comparison with the solution to
the heat equation
In this section, we use the KMC-type method described in Chapter 3 to solve a
volumetric relaxation problem. The solution is compared to the solution of the heat
60
5 X10
5.
3:4.5
0
0.2
04
0.6
0.8
1
z (m)
x 10
Figure 3-6: Solid line: theoretical heat flux in a 100nm wide thin film. Dots: heat
flux computed with the proposed method
equation in order to verify the consistency of the results with results obtained from
applying the heat equation at late times, and also to illustrate the kinetic effects that.
arise at small time- and lengthscales.
3.6.1
Problem statement and solution of the heat equation
Let us imagine a system that is infinite in all directions. We use the material properties
of silicon at 300K. At t = 0, the material's temperature is increased by the amount
Ti (<
300 K) in a cubic region characterized by 0 < x 1 < L, 0
0
< L. The solution of the heat equation
< X3
OT = aV2T,
X2
< L and
(3.34)
at
where a is the thermal diffusivity, for t > 0, is given by:
AT(x, t) = Tig(xi, t)g(X 2 , t)g(X 3 , t)
(3.35)
with
g(xi, t) =
[erf
2 .h
viaon
)-
erf
.
vt 4
(3.36)
e w tt
and where A T(x, t) here refers to the deviational temperature with respect to 300 K.
61
We recall that the thermal diffusivity is the ratio of the thermal conductivity K and
the volumetric heat capacity C, which can be obtained from the material properties
K
j3,P
C =
)2(a, p, ,pVg(UP)T Teq)dW (3.37)
C
C,,p&d,
(3.38)
P
where
Cs,, = D
,p)(L,; (
aT
(3.39)
With these inputs, we may calculate analytically or numerically the average temperature in the cube [0, L] 3 at any time and compare it with the solution to the linearized
BTE.
3.6.2
Monte Carlo results
In Figures 3-7a and 3-8a, we show the average (deviational) temperature in the cube of
side L, for several different values of L, resulting from solution of the BTE. In the same
figure, we compare it with the solution of the heat equation. The calculation is done
for the single relaxation time model with
T
= 40 ps in Figure 3-7 and the frequency-
dependent relaxation time model in Figure 3-8. Although the results initially differ, as
expected, we do recover the solution to the heat equation for times that are long with
respect to the relaxation time (here, the diffusive behavior is retrieved after around
100 ns in the single relaxation time model, and 10 ps in the frequency dependent
model). We show a more relevant comparison in Figure 3-7b and 3-8b by plotting the
ratio
Tboltzmann/TFourier
between the two solutions. In both cases, the ratio converges
towards one, despite the relative statistical noise at very late times, which is due to
the very low signal available as the deviational temperature T goes to zero at late
times. This shows that, at late times, the linearized BTE is consistent with the heat
equation and this further confirms that the kinetic-type method solves the former
with no approximation. In addition, Figure 3-8 illustrates the fact that the diffusive
regime is only recovered at very late times (of the order of 10 ps) in silicon at ambient
62
temperature, meaning that using the heat equation to model physical phenomena of
interest is incorrect if the time scales that are involved are less than 10 pts.
2.2
100
28
36pm
2
\
-
4 pm
1.8
10 02
.6
2 pm
-1
-6
-
-6
-7
-8
-9
-10
1
10
10
10
v
3.6pm
_1 ________________
-1
10
pAm
2.8 m
2pm
~ ~ ~ ~~ ~
10
2
1
24
10
pm
log(t)
(b)
t (s)
(a)
Figure 3-7: (a) Solution of the Boltzmann equation (dots) and the heat equation (solid
lines) to the problem with cubic initial heating. (b) Ratio between the two solutions.
Arrows indicate solutions for different characteristic sizes (L). The solutions are
calculated in the single relaxation time approximation.
2
100
m
pm2
1.8-.-
3.6pm
10 2
1.6-
2 pm
1.42.8 pm
10
1.2
1.2
10
10
_
10
10
10
1
0
1
10-
3.6 pm
pm
-9
-8
-7
-6
-5
log(t)
(b)
t (s)
(a)
the heat equation (solid
and
(dots)
Figure 3-8: (a) Solution of the Boltzmann equation
lines) to the problem with cubic initial heating. (b) Ratio between the two solutions.
Arrows indicate solutions for different characteristic sizes (L). The solutions are
calculated using the frequency-dependent relaxation time model.
3.7
Application to transient thermoreflectance
As an application, we consider the transient thermoreflectance experiment presented
in [331 and used in [39-42] as a thermal conductivity spectroscopy technique. Note
63
that several versions of pump-probe thermoreflectance exist, all with their own advantages and shortcomings.
Using the algorithm described above, we simulate the
thermal response of a thin film of aluminum on a substrate of silicon (see Fig. 3-9)
after a laser pulse irradiates the surface and provides localized heating at t = 0; the
system is initially in equilibrium at To = 300 K, making this a convenient control
temperature, i.e.
Tq=
To. Heat transfer by electrons is neglected in this example.
Initial positions for the computational particles are determined from the experimentally derived initial temperature distribution
T - To = (Tz - To) exp (- pz) exp
2
Z> 0
(3.40)
where r represents the radial coordinate measured from the pulse center and z the
depth into the aluminum substrate; in fact, because in the linear approximation the
deviational temperature (T- TO) is proportional to the deviational energy, deviational
particles are drawn from a distribution that is proportional to the above temperature
distribution, in other words ed(x, w,
,
t = 0) ax (T(x)
-
To)Deeq
Teq
/0T. Also, since
the heating is impulsive in time, each particle's initial time is set to t = 0. Here, the
penetration depth
/-1
is taken to be 7nm, the characteristic radius RO is taken to be
15 microns and T1=301 K.
The Al/Si interface is treated as follows: the probability of a deviational particle
crossing or being reflected at the interface is calculated using the frequency dependent
interface condition detailed in [39,401. When a particle trajectory intersects the interface, the particle is transmitted or reflected by comparing the transmission /reflection
probability with a uniformly drawn random number. In case of crossing, a new traveling time until the next scattering event needs to be calculated using (3.11) and the
parameters of the new traveling medium. Reflection is treated as algorithm item [IVaJ
discussed above. More details on the model parameters, such as the transmission coefficient at the aluminum-silicon interface, can be found in [4] as well as appendix
F.
Figure 3-10 shows that the proposed method yields results that are in excellent
64
301K
300K
Figure 3-9: Schematic of a transient thermoreflectance experiment. Point 0 denotes
the center of the heating pulse, also taken to be the origin of the cartesian (x, y, z)
set of axes. The system is assumed semi-infinite in z > 0 direction and in the x - y
plane.
65
agreement with the deviational method presented in [4].
However, the additional
speedup due to the present algorithm allows us to calculate the response to a single
pulse up to 10 ps (see Fig. 3-11), which represents a two to three order of magnitude
improvement compared to the deviational method presented in
141
which could only
reach several nanoseconds. (For comparison purposes, we note that due to the small
temperature differences involved, simulation of this problem using standard Monte
Ultimately, we expect this improvement
Carlo methods is essentially intractable.)
to be invaluable towards the computational description of the phonon spectroscopy
experiment discussed in
1401.
300.2
Timestep-based variance-reduced method
Proposed method
-
300.15
300.11
300.05,
300
0
0.2
0.4
0.6
t (s)
0.8
1
x 10 8
Figure 3-10: Comparison of the surface temperature (average temperature in a cylinder of height 5nm and radius 2pm right beneath the surface), calculated by the
method explained in [41 and by the proposed algorithm. The two methods give essentially the same results.
3.8
Range of validity
The method presented in this chapter exploits a mathematical property of the linearized Boltzmann equation and introduces no approximation. In the limit where a
large number of particles is used, it is expected to solve the linearized Boltzmann
equation exactly (see proof in section 4.2), with no time or space approximation.
66
101
10
10
10
10
10-
10
Tne (s)
Figure 3-11: Surface temperature (here, average temperature in a cylinder of height
5nm and radius 10pm right beneath the surface) as a function of time for late times
(only possible with the proposed algorithm).
The validity of the method is therefore only dependent upon the validity of the linearization of the Boltzmann equation. The error associated with this approximation
may be assessed by calculating the difference between the non-linear post-scattering
distribution and the linearized one as a function of temperature.
Below, we present one such comparison by examining the behavior of the quantity
fZ hwu
D(,T
(eP
~1
-
(T
-
hwD
fE P 7(L,p,Tq) exp
-
Teq)T
DT
dcw
d
(
-k1
(3.41)
as a function of temperature.
Note that, in this measure, the difference between
the two deviational distributions is normalized by the non-deviational post-scattering
distribution.
Although this choice is, to some extent, arbitrary, it was made here
because it allows one to quantify the error with respect to the absolute temperature.
Alternatively, the error could be scaled by the deviational distribution. In the latter
case, the relative error would appear to be order 1 since the deviational distribution
itself is order 1.
Figure 3-12 verifies that the normalized error is second order in T - T.
This
analysis suggests that the error is still acceptable (less than 3%) up to temperature
differences of order 30K.
67
10
10
--
10
10-10-
0-15
10
-10- 3
10-6
100
AT/Teq
Figure 3-12: Normalized error between the linearized and exact collision operator,
in the case where T = 300 K. For AT/Teq = 0.1, i.e. for a temperature difference
AT = 30 K, the relative error, C, is about 2.2%. Reproduced from [3].
68
Chapter 4
Collision-induced sources and the
kinetic-type method
In the following section we use the concept of collision-induced sources to provide an
alternative method for terminating particle trajectories in steady simulations. In section 4.2 we use related theoretical arguments to provide an alternative justification for
decoupling computational (deviational) particle trajectories under linear Boltzmann
dynamics.
4.1
Termination of particle trajectories
In a number of practical cases, particle trajectories terminate when the particle exits
the system, usually through an absorbing boundary (prescribed temperature boundary). In periodic nanostructures, this cannot happen due to the periodic boundary
conditions; therefore other termination criteria need to be developed. In [5] and in the
previous chapter, we observed that the average contribution of particles to the heat
flux monotonically decreases in absolute value along the particle's trajectory (with
the number of scattering events) and that a particle trajectory can be terminated
when its contribution to the heat flux reaches the order of magnitude of the statistical uncertainty associated with each trajectory segment. Estimates of the heat flux
field resulting from this method are shown in Figure 3-3.
69
100 nm
100 nm
Particles incident
on the interface are
diffusely reflected
Figure 4-1: Fictitious material with diffusely reflective interfaces normal to the direction of the applied temperature gradient. Dashed lines denote the cell-grid used to
calculate Y(x), defined in section 4.1.
Here we discuss a different method for terminating particle trajectories that is
more efficient, but introduces some numerical error. This approach is inspired by the
work of Randrianalisoa et al. [43] who used this approach in the context of simulations of the Boltzmann equation but unrelated to our context, namely deviational
simulations. We illustrate this approach with the following example: we consider a
periodic nanostructure composed of parallel, adiabatic (diffusely reflective) plane surfaces normal to the applied temperature gradient; the spacing between the surfaces
is d=100 nm (see Figure 4-1). This example was chosen because the effective thermal
conductivity of this structure is exactly known (Keff = 0). This allows us to investigate
the error associated with the discretization in space introduced by this method but
also more easily illustrate the theoretical concept on which this algorithm is based.
The simulation proceeds in general terms as described in Section 3.2, but with one
major difference: Instead of simulating each particle to the end of their trajectory,
each particle is simulated until its first relaxation event. At this time, the local density
70
of energy "loss" events,
j
1d2
AI xEcell fw,fl 47r
1=
ddx,
(4.1)
is calculated by recording the number of trajectories terminating at each of the computational cells of length Ax in which the domain is discretized. For this a cell-grid
is used that is similar to the one in the cell-based algorithms described in Chapter 2
and in Refs. [4,29,311; for the particular one-dimensional problem discussed here, the
cell-grid is shown in Figure 4-1.
Energy conservation requires that energy relaxation events are followed by the
emission of new computational particles from the "gain" term of the Boltzmann equation, L(e')/7. The number of these particles is determined from energy conservation,
that requires their number be such that they carry the same amount of energy as the
net energy carried by particles whose trajectories have been terminated. The intermediate step of advancing all particles up to their first scattering event coupled with the
observation that the net amount of energy needs to be conserved, allows cancellation
between positive and negative particles within the same cell to take place, as was
utilized before in the timestep-based algorithm of Refs. [4,11].
Following this cancellation, the algorithm can proceed by noting that integration
of the particle trajectories up to the first relaxation event corresponds to solution of
the equation
d
VVxe
where
Q
(4.2)
=+
denotes any generalized source term, while 4' is a register of the amount of
particles to be generated in each cell from DL(e d)/T. This information can now be
used for solving
d
V9 *v,,e2 =
T
+
7
,(4.3)
where the new "source term" C(el)/T is approximated by generating i 1 Ax/Feff particles whose frequencies and polarizations are drawn from the post-scattering distribution D(w,p)C(e,)( ,p)/w(wpTeq); here, 6,ff is the particle effective energy rate
71
per unit area. The starting position for each particle is uniformly drawn within the
given cell, and the traveling direction is isotropically chosen since the post-scattering
distribution is isotropic in this model.
The process is repeated until the number of particles emitted by the new sources
is 0, that. is, until complete cancellation has been achieved. The latter is guaranteed
by the fact that the net energy rate emitted by the initial source is 0 number of positive and negative particles are involved in the simulation.
was not the case, i.e.
the same
(If this
if a non-zero energy rate was emitted, the problem would
be ill-posed since a steady state could not exist without absorbing boundaries.) In
practice, when the number of remaining particles become very low, the probability
for two particles to cancel may become very low as well and it may be preferable to
terminate the simulation even though there still are a few particles in the simulation.
The contributions from these few particles would be very small anyway.
This approach can now be explained as in section 4.2 by noting that it corresponds
to the decomposition ed
et + ej +
....
where step
j
> 1 in the above process solves
the equation
V
V e =
+
.
T
(4.4)
T
At each level, the number of particles to be generated from the source term DL(e )/T
is given by Vy. Step
j
= 1 corresponds to Equation (4.6) above. Summing over all
j
recovers the linearized Boltzmann transport equation
Vg ' VXed
-
C(ed)
T
ed +
Q.
(4.5)
From the above it follows that the track length estimator needs to be applied
to each segment of trajectory in order to calculate the corresponding heat flux contributions. Figure 4-2a shows the distribution of
'
and heat flux corresponding to
various steps in the process for the problem considered here. In this particular example, the heat flux at any level
j
(see part b of the figure) is non-zero. However, the
combined contribution of all steps results in a zero heat flux (within the numerical
approximation), as expected.
72
x
I
I
1015
201X
Density of first scattering
1ib
15
-
0.5
106
Heat flux from
initial source
S10
0
-
:~5
-0. 5
0
Density of fourth scattering
0
a.
Heat flux from
f(e,)/r and L(e )/T
0.2
0.4
0.6
04
0.8
1
X 10
X (m)
"O
0.2
0.4
7
X (m)
b.
0.6
0.8
1
X10 7
Figure 4-2: a. <Y and 144. b. Heat flux contributions resulting from q, C(e )/T and
These results were obtained using an "imposed temperature gradient" of 106
,(ed)/T.
K.m- 1
1e71
1e6
1e5
b.0
.S 1e4
S1e3
1e2
0
10
20
30
40
Index of collision-induced source
Figure 4-3: Using the termination technique based on spatial discretization, the number of particles in the system decreases exponentially.
73
- - --
1
-
i n-
102
10
--
=10--
10-5.-
104
10
AX (M)
Figure 4-4: Error in the effective thermal conductivity of the structure in figure 4-1
as a function of the cell-grid size Ax. The convergence order is quadratic.
The latter observation is used here to investigate the numerical error associated
with the introduction of a grid for the calculation of the source terms @). Figure 4-4
shows the difference between the numerical value of the heat flux and the exact result
as a function of the cell size (Ax). The results show that the discretization error is of
second order in Ax, in analogy with the second-order accuracy observed [34,44,451
in DSMC algorithms. Extension to 2D and 3D cases is straightforward.
In summary, while the cancellation approach outlined here is not needed in cases
where particle trajectories are made finite due to the existence of absorbing boundaries, combining it with the linearized and variance-reduced approach can benefit
the case of periodic nanostructures where absorbing boundaries may not be present.
Figure 4-3 shows that the number of particles decreases exponentially as a function
of
j.
This results in an appreciable speedup (factor of around 10) with respect to
the approach based on an empirical choice of the termination criterion [5], at the
expense of a second order approximation in Ax (see Figure 4-4) and a slightly more
complicated algorithm.
74
4.2
An alternative justification for independent particle trajectories
In this section, we provide an alternative theoretical justification for the claim that
algorithm described in section 3.2 satisfies the governing equation (3.3). The argument presented here is based on the method of collision-induced sources introduced
in the previous section. Here, however, we consider the (more general) unsteady case.
Let us consider again, a process by which all particles are simulated until its first
relaxation event. This corresponds to the solution of equation
ed
Oed
at+V-Ved
=
+Q
(4.6)
- the particle remains at the same spatial
Resampling each particle properties (Lp,
location) from the post-scattering distribution (3.10), amounts to sampling the source
,(el)/T with no approximation other than statistical (no time or space discretization).
Therefore, following the application of another advection step, we obtain the solution
(again, with no time or space approximation) of the equation
Oed
at
+V
2
9
, V ,e
2
=
ed
_
2
T(L, P, Te)
+
.
(4.7)
T(, p, Teq)
d
This process may be repeated recursively to provide solutions ej,
j > 2.
Let us consider the sum
ed . The convergence properties of this sum will not
be proved here, but may be expected in the time dependent case by recalling that,
given a fixed time, all particles must eventually "cross" this time horizon after a finite
number of collision events. In steady cases, the convergence is made possible thanks
to the presence of absorbing boundaries (prescribed temperature boundaries). In connection with the problem of particle trajectory termination discussed in sections 3.4
and 4.1, the case of systems with only non-absorbing boundaries is more problematic.
In that case, the convergence results from the progressive cancellation of positive and
negative deviational energy rates (note that a steady problem with non-absorbing
75
boundaries and a net positive or negative deviational energy creation rate would be
ill-posed). To conclude, assuming the sum converges, it is straightforward to verify
that ed -
e
is a solution of (3.3). We highlight that this solution comes with
no time or space approximation.
76
Chapter 5
Adjoint formulation and "backward"
simulation method
In this chapter, we present a formulation which again exploits linearity of the governing equation to gain computational advantages (speedup, simplicity), namely an
adjoint formulation based on the duality between the linearized Boltzmann equation
and its adjoint. This formulation is distinct from the one presented in chapter 3, but
at the same time complementary-that is, the computational advantages of the two
can be compounded (multiplicatively in the case of speedup).
This chapter is based on Ref. [46], where this material first appeared. In addition
to the present chapter, adjoint formulations are also discussed in section 7.2, where
schemes particularly suited to multiscale problems are developed and described.
5.1
Notation
We start by introducing the notation that will be useful for denoting the adjoint
formulation and the associated numerical technique. In what follows, the sum over
polarizations will be implied by the integral over frequencies L. In addition to the
above, we will also be using the following notation:
- The deviationaltemperature will be denoted by T (instead of T - Teq).
77
- The mode-dependent free path As, is defined as the product V(w, prLY
,
p, T,,).
- We define the mean free path as
(A)
=
SD e-!A, dd,
fe
fe, D
(5.1)
d
&L
- The frequency and polarization-dependent Knudsen number Kn ,, is defined as
A,,P/L, where L is the smallest characteristic lengthscale in the problem. The
Knudsen number based on the mean free path is defined as (Kn) = (A)/L.
- Let
D ( ,, p) OeeT-q
47r
OT
5.2
(5.2)
The adjoint Boltzmann equation
In this section we derive the adjoint formulation of the linearized Boltzmann equation
for phonon transport in the relaxation time approximation. To our knowledge such
formulation (for phonon transport) has not appeared before, although adjoint formulations have been developed in other domains of linear transport (e.g. radiation,
neutron transport)
5.2.1
[35,471
and have served as inspirations for this work.
Background
The adjoint formulation is best introduced in a framework where boundary and initial conditions are incorporated into the governing equation as (special) sources of
deviational particles. As in chapter 4,
define q
=
Q/(&e
Q
is the sum of all particle sources, and we
0/DT). From this definition, it follows that energy-based devia-
tional particles are emitted from (47r)-'D(aegq/T)q.
We also recall that, here, T
eq
denotes the deviational temperature. With these definitions in mind, the deviational
Boltzmann equation reads
OtT
+ Vg V(D
+ q
78
(5.3)
where D = ed(Oe2e~q /9T)-
1
and the linearized operator C can now be written as
()=((' j d
(5.4)
47 LPdL,
We also define the scalar product
TE dd 2 d3 xdt
(f, D) =
where the integration is over the whole phase space, and with respect to which
(5.5)
C/r
is self-adjoint; namely,
(,
C(fl)/7) =
(F)/T, ()
(5.6)
In addition to sources, Monte Carlo simulations and experimental setups are also
characterized by detectors, which sample phonons as a means of returning "measurements" of quantities of interest.
Mathematically, a detector is defined by its
characteristic function h; the quantity of interest, I, is then written as
I=
r
h-ed dLd2
d3xdt = Kh, @)
(5.7)
The function h prescribes both the type of quantity that is estimated (temperature,
heat flux...) and the location (in phase space, including time) over which the quantity
is averaged. For example, for the average deviational temperature within a volume
V at time t such that ti < t
< t2 ,
h is given by
1
CV(t 2 - t 1 )
where
1
ll.r2J
(5.8)
v refers to the indicator function of V, i.e. the function that takes the value
1 inside the volume V and 0 otherwise. For the temperature at a given time to, h
would instead be given by
h=
1
1v6(t - to)
CV
(5.9)
where 6(t - to) refers to the Dirac delta function centered in time on to. Although
79
these expressions might not always seem intuitive, they can be verified by considering
a thermalized system at (deviational) temperature T: in the linearized framework,
ed
-
T~e q/T;
5.2.2
substituting in equation (5.7) leads to I = T.
The fundamental relation
We now introduce the adjoint Boltzmann equation
04)*
-
77x
- V9 * V*=+
((D*)-(*
h
(5.10)
T
at
In this equation, particles simulating the adjoint distribution, D*, evolve backwards
in time and are emitted by the adjoint source, h, which is the function characterizing
the detector in the original problem.
The specification of the adjoint problem is
completed by using the source q as the adjoint detector, in the sense
1* =
qBG*d d 2 d3 xdt = (q, *)
(5.11)
The importance of the adjoint formulation can be summarized by the relation
I* = I
(5.12)
which we will refer to as the fundamental relation. In words, this relation implies that
any quantity of interest (of the form (5.7)) can be obtained by solving the adjoint
problem which uses the detector (of the original problem), h, as a source and the
source (of the original problem), q, as detector. Based on the observation that the
adjoint equation describes particles that move backwards in time, we will frequently
use the term "backward problem" to describe the adjoint problem defined by equations
(5.10) and (5.11); in analogy, we will use the term "forward" to describe the original
problem defined by equations (5.3) and (5.7).
80
To prove the fundamental relation we write
=
T
- V --2(T*)
-*
)
(5.14)
atT
(,
=
,
h) = 1
(5.15)
Obtaining expression (5.14) from (5.13) requires integration by parts and, depending
on the problem of interest, some manipulation.
We now discuss this integration for the term involving the time derivative. The
use of sources for imposing initial conditions allows us to extend the integration over
time from -oc to oc by taking iD(t < 0) = 0 and D*(t >
tfinai)
= 0 where
tfinal
denotes
the last detector instance. As a result:
a-]*dt = [@
*G
= --
a'-dt
-
ffD
dt
(5.16)
(5.17)
and therefore
)= (-
(5.18)
(Vg - V(, @*)
(5.19)
We now consider the term
which can be written in the form
Jt
V9n9* A*dwd29d 2xdt - (KVg V*) (5.20)
-
-
where n is the inward-pointing normal vector to the boundary &X. The above proof
requires the first term in (5.20) to vanish. This will be established for various boundary conditions of interest below. In the case where the spatial domain is unbounded,
one may proceed by assuming (as was done in this work) that the integral over the
boundary OX tends to zero when the latter is made infinitely large.
81
A sufficient
condition for this is bI* -- 0 sufficiently fast, as x -+ oc; this is expected to be satisfied by problems that can be simulated by the deviational Monte Carlo method. The
case of diffusely/ specularly reflective boundaries is treated in section 5.2.4; prescribed
temperature boundaries require a small modification of the Boltzmann equation and
are treated in Section 5.2.5.
Periodic boundary conditions are discussed in section
5.3.2.
5.2.3
Adjoint particle dynamics and simulation
Comparison of the adjoint BTE (5.10) and the original linearized BTE (5.3) reveals
strong similarities, suggesting that algorithms for performing forward simulations
could also be used for backward simulations with small modifications. As expected,
one difference between the two lies in the source term.
In the forward case the
energy-based particles are emitted from the distribution Eq. By analogy, the adjoint
particles must be emitted from the distribution Eh. In contrast to the forward case
where f
qdwd 2
d3 xdt has the unit of energy, f
hdwd 2 d3 xdt will not,
in general, have the unit of energy. Nonetheless, energy will be conserved provided the
number of computational particles is conserved during collisions, since this guarantees
f
E dd
dd2
Note that although the quantity E*
Efhd d2 q(
=
2q
3
(5.21)
xdt/N does not always
represent an energy, we will still refer to it as "adjoint effective energy".
The second difference can be found in the rules for calculating a particle trajectory.
The minus signs in the left-hand side of (5.10) means that:
- the time parameter of a particle monotonically decreases
- a particle with parameter
,
moves in the -f
direction.
In practice, the isotropy of the collision operator and typical boundary conditions
(e.g. diffuse reflection, prescribed temperature boundary) means that the "backward"
algorithm differs very little from the "forward" algorithm.
82
5.2.4
Reflecting boundaries
Let us consider a point
Xb
on a reflective boundary, whose inward pointing normal is
denoted by n. When a particle encounters a diffusely reflective boundary, it is reflected
back, and its traveling direction is randomized, such that the outgoing distribution
is isotropic. As a result, the distribution D, for a given frequency and polarization,
obeys the following relation at the boundary (x
4(P - n > 0)
= Xb):
- nd2 Q
*D
Ln< 0
-
(5.22)
Since particles subject to the adjoint Boltzmann equation travel backward in time,
the diffusely reflective boundary conditions for the adjoint distribution
n < 0) = -
*
Ln
7
* reads:
- nd2 Q
(5.23)
.-n>O
We may now use (5.22) and (5.23) to write
2
*d2Q =OQ- n@@*d2p +
- n-
foIO- n@ *d2f2
J
f
(5.24)
Qn>2
J<
-n<0
<0)1n<
+I
*
0) f.jn>
0
(5.25)
= -
7r(D*(Q - n < 0)-c(f2 - n > 0) + 7r(Of - n > 0)(D*(f2 - n < 0) = 0
(5.26)
which proves that the surface integral over the diffusively reflective boundary is zero.
For specular reflective walls, proving that
5 20
i
- nD@*d
0
(5.27)
follows by noticing that if both D and D* satisfy the specular reflection condition,
then so does their product.
83
5.2.5
Proof of the fundamental relation (5.12) for the prescribedtemperature boundary conditions
A boundary with prescribed (deviational) temperature Tb is modeled as a black body.
Any particle incident on the boundary is absorbed. At the same time, the boundary
emits particles from the equilibrium (Bose-Einstein) distribution with temperature
parameter Tb. The classical model consists of simply defining the boundary condition
by specifying the incoming distribution at the wall for incoming particles:
(Db(Wp, Xb, 0 - n
(5.28)
> 0) = Tb.
Here, following the general methodology developed in section 5.2.1, this boundary
condition is expressed in terms of a combination of source terms. Emission of particles
by the boundary can be represented by the source term:
(5.29)
qb = 6(X - Xb)H(Vg - n)Vg - nTb
where H is the Heaviside function defined by
I
for x > 0
0
for x < 0
H(x) =
(5.30)
In addition to this source term that is independent of D and which replaces the
thermalized region beyond the boundary, we need to use a source term that absorbs
particles incident on the boundary. Such source term can be written [48] in the form:
6(X -
(5.31)
Xb)H(-Vg - n)Vg - n
Since D appears explicitly in the above expression, we write the linearized BTE (5.3)
in the form:
atT + V
(3
-
a D(D)
- VD =
+ q+
6(X -- x)H(-V
84
- n)Vg - nD
(5.32)
where q includes qb and any other sources that do not depend on D. By analogy, the
adjoint BTE is given by:
((D*) - (D*
8@*
V - Vg =
-
+h - 6(x - Xb)H(Vg - n)Vg - n * (5.33)
T
We now repeat the integration by parts procedure of section 5.2.2 by writing
I*
V* -
(q,
*)
(
+ Vg
(5.34)
f f
)D
JxEOX
VG -
- 6(x
-
Xb)H(-Vg
T
Ot
V *n
*d2 dwdtd 2 x + (,
(D
-___
Ot
Jtwp,1n
-
n)Vg
- Vg
*, V*) (5.35)
.
-
T
(6(x - Xb)H(-Vg - n)Vg - nD, D*)
(5.36)
By noting that
J/ /
nEb*d2
Vg
dL;dtd 2 x
=
(3(x
-
Xb)H(-Vg - n)Vg - n@, V*
+(D, 6(x - Xb)H(Vg - n)Vg - n@*)
we obtain I*
5.3
I.
Applications
The adjoint formulation can provide a number of computational benefits, including
algorithmic simplicity and considerable computational speedup for certain classes of
problems. The latter can be described as problems in which the "detector is small",
that is, problems for which the outputs of interest are defined over small regions
of physical space, or more generally, phase space. An example of the former is the
transient thermoreflectance experiment discussed in the next section, in which the
quantity of interest is the temperature at the specimen surface (which, stricly speaking, has zero volume in three dimensions); an example of the latter is spectrally
85
resolving the contribution of individual phonon modes to the effective thermal conductivity (the detector extends over a small range of frequencies, or, in the worst
case, features a delta function in frequency).
In these problems, in the "forward"
Monte Carlo method, the probability for a particle to be found in the detector at a
given time is small. The adjoint formulation uses h as a source thus providing an
opportunity for alleviating this burden. If the source is larger than the detector, the
adjoint formulation ensures that the signal collected by the detector will be enhanced,
leading to improved signal (variance reduction). Clearly, the speedup will depend on
the size-ratio between the detector and the source; in cases where the detector features a delta function and the source does not, the speedup is theoretically infinite (in
practice the forward calculation would smear the delta function into a computational
bin in order to collect some samples, thus making the speed-up finite, but introducing
error in the process).
Examples of applications of the adjoint formulation are given in the following
sections. Note that although the adjoint formulation is indifferent to the numerical
implementation (i.e. timestep based, or KMC-type), here we will proceed to demonstrate these methods using the KMC-type method developed in [5] and described in
section 3.2.
5.3.1
Surface temperature in a transient thermoreflectance experiment
This section illustrates the adjoint formulation with the pump-probe thermoreflectance
experiment [41,49] depicted in Figure 3-9.
Background
Using the formalism introduced in this chapter, the initial deviational temperature
field is:
Ti(x) =
exp
86
(Z
- 2)
R2
(5.37)
where T is taken as 1 K. Like in section 3.7, the penetration depth )-
is taken to
be 7nm and the characteristic radius Ro is taken to be 15 microns.
This problem features only one source term (the initial condition), which can be
written as
q = T(x)6(t)
(5.38)
The quantity of interest is the surface temperature at time t,
function hj for the corresponding detector is
IdissO(z) 6 (t
j
=1,
...
, M.
The
- t); here, we consider the
slightly more general case of the temperature in a general and arbitrary volume V
hj = 1 v6(t - tj)
(5.39)
VC
because as shown below, the adjoint formulation lends itself to this generalization
naturally.
Note here that, for simplicity, we will use the same symbol V to denote
the region of interest and its volume.
Adjoint calculation
Let us consider here the case of one sampling time, namely tAI; extension to multiple
sampling times is discussed in section 5.3.1. A particle from the corresponding adjoint
source (forward detector) hM is emitted at time t m and travels backward in time. At
t = 0, the position
Xend
is noted, leading to
- 3 Zend -
IM,i = E*fft exp
as the contribution of particle
i
2"
(5.40)
to the estimate of the temperature at time tL. Here,
the weight of each particle, or adjoint effective energy, is given by
eff =
hd 2 d;d 3xdt
- I
1 1
=N VC1
(5.41)
1
,,,U
C, 1vdwd x
87
=
N
N
(5.42)
and is independent of the (forward) detector shape. The temperature is thus given
by
N
'v)I,i
T(t = t
(5.43)
i=1
Multiple sampling times
To treat other sampling times tj < t,
in principle we have to simulate new particles
starting at time t = tj and measure their position (and contribution) at time t = 0.
However, by noting that the evolution rules for particles emitted at tj are the same for
all
j
(only the "internal clock" of the particles differs), we may reuse the information
given by the trajectory of the particle emitted at time t m, by simply recording their
contributions at times tM - tj for all j.
Ultimately, this process amounts to setting t = 0 when the particle is emitted,
then to counting the time forward while computing the trajectory.
Contributions
can then be sampled at times t, exactly like in the "forward" Monte Carlo method.
Algorithmically, the only difference lies in the exchange of source and detector.
Computational results
Figure 5-1 shows the temperature variation as a function of time at three locations,
as measured by the distance p from the origin, on the sample surface (in this case
the forward detector is a point on the surface z = 0). We note here that the profiles
at p = 6pm and p = 12pm were obtained using the same particles as for the p = 0
calculation by exploiting the translational invariance of the problem. Namely, since
translation of the particle source in the x (or y) direction leaves the particle trajectory
unaltered, translation of the adjoint detectors should also result in equivalent results.
This implies that contributions to the temperature at distance p from the origin can
be calculated using data from particles from the original calculation using the "shifted"
detector
IM,i
=
exp
-3Zend - 2 (Xend
-
P)
end
(5.44)
Figure 5-2 shows the standard deviation in the temperature measurement in the
88
case of the forward method. In this method, the temperature on the surface is measured in a cylindrical bin of depth (measured from the surface) d. The figure clearly
shows that the statistical uncertainty (and thus the statistical accuracy of the results)
deteriorates as d is made smaller. On the other hand, d needs to be made as small
as possible to minimize deterministicerrors resulting from averaging over a finite volume (rather than strictly on the surface). In the limiting case where the detector is
simply a disk at the surface, calculating the surface temperature using a (forward)
Monte Carlo method becomes impossible (unless time discretization is introduced, in
which case it is "just" very expensive) since the probability that a particle hits the
interface at a specified time is 0. Using the adjoint makes such a calculation possible
by switching the source and the detector.
100
- =0
--- p=6RM
--- p=12 [m
102
T (K)
10-4--
106
161-12
10
- 10
10
10
-8
10
-6
-4
10
t (s)
Figure 5-1: Temperature as a function of time in a transient thermoreflectance experiment. Results shown for the pulse center, 6pm away from the pulse center, and
12pm away from the pulse center.
Discussion
Although the above example clearly highlights the computational gains made available by the adjoint method in principle, here we note that the magnitude of the
89
computational gain in this particularexperimental setup [33] is hard to quantify: the
physical detector is a laser probe which theoretically measures the surface temperature by relating it to the surface reflectivity.
Since the reflection of photons at a
surface involves a penetration to some (small) depth, a more accurate model of this
process would take into account that, the measured quantity uses a finite depth such
as d ~~2 nm (range of the optical skin depth for visible light in aluminum). Consequently, the benefit from using the adjoint should be determined by comparing the
size of the detector with the size of the source properly adjusted for the above effects.
From a broader perspective, physical detectors usually can only access the surface of a given system, whereas phonons are generated via mechanisms which are
inherently volumetric (Joule effect, electron-phonon interaction). For these reasons,
accurate and faithful description of physical experiments is, in general, expected to
strongly benefit from the adjoint formulation.
Finally, we note that drawing random particles from the distributions derived
from the detectors is usually easier than from the forward source terms. For example,
a temperature detector usually weighs all samples within a volume equally and thus
calls for the creation of a uniform distribution when used as an (adjoint) source. In
contrast, the distribution associated with the initial temperature field (5.37) in the
above example is a product of a decaying exponential and a Gaussian. Although this
distribution is invertible, this would not necessarily be the case with more general
initial conditions.
5.3.2
Highly resolved calculations of mode-specific
thermal
conductivity calculations
Another class of methods where the detector is "small" includes problems for which the
quantity of interest needs to be spectrally resolved. Previous work [50] has highlighted
the fact that the contribution of low frequency phonon modes is challenging to resolve
due to their small densities of states and very large free paths. Unfortunately, due
to their low density of states, the forward Monte Carlo technique tends to "under-
90
400
-
300 -
d=10nm..
d
--
- - -d=5nm-
'd=1nm
-
350
250 ---
200
-
T
150
,-
-
-
100-
0
-
50
0
0.5
1
t (s)
1.5
X10
Figure 5-2: Ratio between the particle standard deviation of the temperature, and
the temperature, for 3 cylindrical detectors with height 10nm, 5nm and Inm.
resolve" estimates of their heat flux contributions, while on the other hand it "overresolves" the contributions of phonons with the highest densities of states (see Fig.
5-5). Increasing the number of samples in order to reach the desired level of resolution
at low-frequencies will reduce the statistical uncertainty for every frequency, hence
wasting computational resources.
The adjoint formulation lends itself naturally to this situation.
The quantities
of interest here are the heat flux contributions from individual phonon modes (in
the isotropic relaxation time approximation, this corresponds to bins in phonon frequency).
To illustrate the method, we will study a nanostructure that has been considered
in recent work [6] and calculate the contribution of each phonon frequency to the
thermal conductivity. Specifically, we analyze a single period of the porous periodic
structure shown in Figure 5-3. The system is subjected to a temperature gradient,
and periodic boundary conditions are applied [6]. As explained in [3,5], applying a
spatially variable control with uniform gradient results in strictly periodic boundary
conditions for the deviational quantity ed and particles are emitted from the source
91
I
4n2
X2
Figure 5-3: Sketch of the nanoporous structure studied in section 5.3.2. The adjoint
method is used to accurately calculate the heat flux contributions from each "frequency bin". A spatially variable control temperature Teq(x), with uniform gradient,
is used, as in Ref. [5]. The dashed square represents the boundary of the computational domain, along which periodic boundary conditions are applied (see Ref. [5,6]).
term
Q
where q
= Eq
(5.45)
outside the pore
-V9 * VxTeq. To spectrally resolve the effective thermal conductivity, we
=
need to calculate the heat flux for a given frequency "bin"
[wO
- Aw/2, wo + Aw/2]
and a given polarization po. We are interested in the response in the direction of
the applied temperature gradient. In other words, the characteristic function for this
detector is
h
where
e1
= L[oO-w/2o+Aw/2]
6
ppoVg * 6i
(5.46)
is the unit vector in the direction of the applied temperature gradient-see
Figure 5-3. The adjoint approach is only valid if
/ J/
2
Vg * nD=D*dwd
d2 x = 0
(5.47)
where OX refers to the boundary of the square computational domain and the square
92
pore, and where n is the normal vector pointing inward. The diffuse reflective surface
of the pore was treated in 5.2.1. To show that (5.47) is true, we may simply notice
that the periodic boundary condition imposes ed(xi, W,p,
) = ed(x 2 , O, p,
0)
where
x, and x 2 are corresponding points of two opposite sides of the periodic boundary
condition. As a result, ed (x1,
, p,
)V
*ni
=
-ed (x 2 , W, p,
)Vg *n 2 , leading to the
desired result after integration over the boundary domain.
We introduce the adjoint equation and the adjoint source q* = h. Adjoint particles
are then emitted from
7
=q
kD
Oe'eq
(5.48)
.1L,-w2w+~11PPV
q [wo--Ao/2Lo+Aw/2]Sp,p 0 Vg
1(.8
and assigned the weight
1
-
N fxQu,
1 D(wo, po) 9ee
= Nq*dd2d3X
N
4
2
Vg(L.o, po)Aw.
OT W
(5.49)
We note once again that the resulting backward algorithm is nearly identical to the
forward one as explained in [3,
51 and section 5.2.3, with the main difference being
that the initial frequency/ polarization (and the resulting velocity) properties can now
be chosen by the practitioner (instead of being randomly drawn from the distribution
=q).
Figure 5-4 shows results calculated using the adjoint method for two different
pore sizes (25nm and 50nm).
These results were obtained using the method for
terminating particle trajectories described in [5]; trajectories were terminated after 30
scattering events. These results confirm that low frequency (large free path) phonons
may play a critical role in the design of nanostructures for efficient thermoelectric
materials. In Figure 5-5, we show the same result with the forward method using
the same overall number of particles. We clearly see how the quality of the results
deteriorates in the very low frequency (large free path) regime.
In other words,
obtaining the insights shown by Figure 5-4 with the forward method is significantly
more costly. For example, we found that the statistical uncertainty associated with
93
the contributions of particles with free paths of 10 tim, 100 pm and 1 mm were
10 times, 25 times and 70 times smaller, respectively, when calculated using the
adjoint method rather than the forward method. These uncertainties correspond to
speedup factors of approximately 100, 600 and 5000. In reality, the observed speedup
will be somewhat smaller due to the following considerations: first, because long free
paths-which are larger than the system periodicity and thus require more operationsare more frequently sampled in the backward calculation, for a given number of
particles, this method is approximately 5 times more expensive than the forward
method. Second, the quality of the solution using the backward method is worse than
that of the forward method for high frequencies. The latter can be rectified at small
computational cost by customizing the number of particles used for each particular
frequency range. This is possible in the backward case because the detector is no
longer frequency specific, ensuring that all particles emitted in a particular frequency
range will contribute.
The forward method does not allow such a flexibility. The
results in figure 5-4 were calculated using the same number of particles for each
frequency bin.
5.4
Discussion
In this chapter we discussed an adjoint formulation for the linearized Boltzman transport equation for phonons in the relaxation-time approximation. We showed that,
similarly to what is found in the fields of radiation, neutron transport, or computer
graphics, the adjoint approach is particularly suited to situations where the detector
is small and the source is large. In the case of phonons, this is not only often true
in a spatial sense, but also in a spectral sense. The free paths of phonons in semiconductors are known to cover a very broad range and, for this reason, the ability
to discriminate individual phonon-mode contributions, as shown in Figure 5-4, is a
very powerful feature of the adjoint framework. Although the precise speedup will
depend on the relative size of the detector and source, in the examples considered
here, speedups ranging from one to three orders of magnitude were observed.
94
We
1.
4
r
1.2Pore size: 25nm
1 Ak"
0.8-
Akbulk
0.60.4Pore size: 50nm
0.2-8
-7
-6
-5
-4
-3
-2
log(A) (m)
Figure 5-4: Frequency-resolved differential contribution to the thermal conductivity
acoustic
(measured heat flux per unit temperature gradient) from the longitudinal
(LA) modes in the problem defined in figure 5-3. Result is normalized by the corresponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using the adjoint method. Results shown for square pores of side
25nm and 50nm; the spacing between the pores is 2 microns in both cases. These
calculations used 28000 particles per frequency cell, for a total of 1399 frequency cells
(thus a total of approximately 40 million particles).
95
1.2
-
1
Pore size: 25nm
0.8-
Akuk0.
0.4-
-P o re0 .s iz e : 5 0 n rrn _
0.2
0-0.2
-8
-7
-6
-5
-4
-3
-2
log(A) (m)
Figure 5-5: Frequency-resolved differential contribution to the thermal conductivity
acoustic
(measured heat flux per unit temperature gradient) from the longitudinal
correthe
by
normalized
is
Result
5-3.
in
figure
defined
(LA) modes in the problem
sponding frequency-resolved differential contribution to the bulk thermal conductivity, and calculated using the forward method. Results shown for square pores of side
25nm and 50nm; the spacing between the pores is 2 microns in both cases. These
calculations used a total of 40 million particles.
96
note that, in consultation with the authors, Chengyun Hua and Austin Minnich [51]
applied the proposed adjoint formulation to the investigation of boundary scattering
in nanocrystalline materials. The method allowed them to show that low frequency
phonons, in spite of the nanocrystalline structure, still carry a significant proportion
of the heat and that, as a consequence, design of efficient thermoelectric materials
should account for such effects.
An additional strength of the adjoint approach is its simplicity: the forward linearized approach relies on a cell-based approach, where quantities need to be sampled
in computational cells of specific geometries. Sampling the contribution of a particle
trajectory requires to study the overlap between the cell geometry and the trajectory
geometry, which may be complicated. Unless the original source term is complicated
itself, the adjoint alleviates this problem.
We also note that the adjoint formula-
tion proposed here is sufficiently general to be applicable to timestep-based MC and
KMC-type algorithms.
One weakness of the adjoint, method is that each detector has to be replaced by
an adjoint source. As a result, the more detectors, the more complex and thus less
desirable the adjoint method becomes.
Although exceptions sometimes occur (for
instance, we saw in section 5.3.1 that multiple time detectors may be treated the
same way as in the forward problem; in section 3.6.2, the adjoint method was also
useful for comparing several source sizes with only one calculation), in practice, the
adjoint method is best suited to problems requiring high resolution (low statistical
uncertainty) in small regions of phase space.
One example is the use of adjoint
formulation to validate the jump coefficients of the asymptotic theory in [52,531 and
in the next chapter. These validations require a high level of accuracy for low Knudsen
numbers, which is made possible by the method outlined here.
Finally, we note that, in the field of neutron and gas transport, studies of the
adjoint BTE have yielded theoretical results whose application extended well beyond
Monte Carlo simulations [54].
97
98
Chapter 6
An asymptotic solution of the
Boltzmann transport equation in the
limit of small Knudsen number
The Monte Carlo method presented in Chapter 3, coupled with the adjoint formulation developed in Chapter 5, is very efficient and enables the solution of a large range
of phonon transport problems.
One limitation of this Monte Carlo method arises
in connection to the computational cost of each particle trajectory.
For problems
whose length scales are on the order of the mean free path, the cost per particle is
usually low enough for obtaining highly accurate solutions with a reasonable overall
computational cost. However in the Kn
< 1 limit, the number of scattering events
- and therefore the computational cost - per particle trajectory increases with the
square of the characteristic lengthscale. As a consequence, lengthscales for which the
Knudsen number is significantly smaller than 0.1 are difficult to simulate with high
accuracy, even though kinetic effects are still large enough for justifying a Boltzmann
description instead of a Fourier description.
In this chapter, we address some of these limitations but also seek to improve
our fundamental understanding of microscale transport processes by developing an
asymptotic method for solving the Boltzmann equation in the small mean free path
limit Kn < 0.1. In this regime, we expect that the classical Fourier description will
99
provide a good approximation to the solution, with kinetic effects becoming increasingly important as Kn increases.
The asymptotic analysis provides a method for
rigorously deriving the modifications required to the Fourier description so that the
latter can continue to provide solutions for continuum fields that are consistent with
the Boltzmann equation and thus can be used as a means of effectively solving the
Boltzmann equation much more efficiently than direct numerical methods of solution. Such expansions have been used by Sone and co-workers [55, 56] to derive the
continuum equations and boundary conditions describing rarefied gas dynamics.
6.1
Asymptotic analysis for the bulk
As in previous chapters, relaxation times and group velocities may depend on frequency and polarization. For this reason, the Knudsen number
(Kn) must be defined
in an average sense. For the sake of consistency with the mean free path definition
(5.1), we choose the following (arbitrary) definition
(Kn) =
fW
'
Cs,,Kne,,da
d
(6.1)
where
C,,= haLD(w, p)
and Kne,, = AW,,/L= V,(,p)r(,p, Teq)/L.
OT
Introducing the dimensionless coordi-
nate x = x'/L, we write the steady Boltzmann equation in the form
Q - VX ()= 'C(1)(6.3)
KnL,,
100
(6.2)
where D is defined as in Chapter 5. The usual macroscopic quantities of interest such
as temperature, energy density and heat flux can be calculated from
Tot= Teq + 47rC j
Etot
q"
Eeq
=
+
C,,d 2 Qdu;
j
C,,pV 9
= Teq +
T(x)
(6.4)
C,,, qd 2 Qdw
(6.5)
d 2Qdw
(6.6)
We will refer to T(x) as the deviational temperature, since it represents deviation
from the equilibrium temperature Teq.
6.1.1
Bulk solution
The asymptotic solution relies on a "Hilbert-type" 1571 expansion of the solution D of
the form
00
=
Z(iKn)"@n
(6.7)
n=O
Given the nature of the proposed solution, similar expansions can be written for the
temperature and the heat flux fields
00
T
(Kn)
=
Tn
(6.8)
(Kn)nq"
(6.9)
n=O
00
q=
n=O
In this section, we only consider solutions far from any boundary.
As will be
shown below, close to the boundary, kinetic effects become important due to the
incompatibility of the bulk solution with the kinetic (Boltzmann) boundary condition
and a separate, boundary layer analysis is required.
Therefore, we let (DG = Z(Kn)n DGn be the bulk solution, anticipating that D
4)G + @K, where
4
=
K represents kinetic boundary layer corrections that are zero in the
bulk and will be similarly expanded later.
101
When the expansion for (DG is inserted in the Boltzmann equation we obtain
00
n - VX
n
00
(Kn)
DG
Z= Kn)n [L Gn)- Gn
n
(6.10)
n=0
n=O
By equating terms of the same order ((Kn) 1 and higher powers) and assuming
that Kn ~ (Kn), we obtain the following relationship for all n > 0
Q - Vx4Gn
=
(n) [C(GGn+1)
Kn
-
(6.11)
(Gn+1]
In addition, considering the two terms of order 0 in the right hand side of (6.10),
we find that
(GO
is determined by the solution of the equation
(GO
where C
= f
Cep/Tdw.
= L
(4G)
oR
=
47,p
(6.12)
CT
The assumption Kn ~(Kn) is satisfied when the range
of free paths is relatively small (and is exactly satisfied in the single free path case
A.,,
= A = const), but becomes harder to justify in materials with wide range of free
paths. In the latter cases, it has the effect of reducing the value of (Kn) for which
the theory presented here is valid. This is further discussed and quantified in section
6.2.1.
From equation (6.12) we deduce that (GO
since this is the case for L(GO).
is a function that depends on x only,
We note here that any function that only depends
on x is a solution. Additionally, since
)GO =
GO(x), we find that the zeroth order
deviational bulk temperature is given by
TGO (X) =
4
47C I'R
G,,p( GO(x)d2 Qdw
At this stage, the spatial dependence of
(GO
=
GO (X).
is undetermined.
(6.13)
The additional
information needed will be inferred from the application of a solvability condition to
102
(DGI.
Using (6.11) we find the following expression for the order 1 solution
Gi
Kn
n
(6.14)
Vx
GO
This equation states that a necessary condition for
(G1
to be the order 1 solution is
that it is equal to the sum of -Kn(Kn)-
and a function that only depends
~
-
Q - Vx
on x. Since the temperature associated with
- TG1
=G1
GO
VxDGO is zero, we can write
l -
Kn(Kn)'-'
-
(6.15)
- VXTGO
Finally, order 2 may be derived following the same procedure. Equation
Kn
n Q - VxIGI
LC(
(Kn)
G2) -
(6-16)
(G2
implies
G2)
V7xT G 1
Kn 22
Kn
(Kn)2
Q.7x
(P VVxTGO)
In the following section, we show that the temperature associated with
DG2
(6.17)
is C( G 2
)
4G2 =.C(
Kn
(Kn)
TG2. We will show it by deriving the governing equation for TGO.
6.1.2
Governing equation for the temperature field
The solvability condition required to determine
(DGn
is a statement of energy conser-
vation, namely
L
1.((D) dw
T
C:-- Pdd2
dd2T
''~
=
(6.18)
Writing this relationship for each order results in
I' CCLLPI(IG,
C
-WTGn+1du42
l d
47-F
(6.19)
for any n and implies
"in
Cw,,PVg
-
x)GndW 2
103
(6.20)
Applying this relationship to <bG1, we obtain
CV9Q - VX
TG1
Kn)
(Kn)
-
' vxTGO
dwd 2 Q
0
(6.21)
which implies
(6.22)
=0
,2TGO
This concludes the proof that the 0-th order temperature field obeys the steady state
heat equation. Moreover, from (6.17) it follows that
Ku 2
(Kn) 22
Ku
(bG2 = TG2 -
7VxTG1
(Rn)
xV(Q 'VxTGO)-
(6.23)
In Appendix A we show that higher-order (in fact, possibly all order) terms similarly
obey the heat equation. In other words, for order 1 and 2, the highest order considered
here, the temperature field in the bulk,
TG1
=
TG,(x)
and TG 2 (X), obey
(6.24)
V TG 2 = 0
0,
Before we close this section, we note that although in the Laplace-type equations
derived above for the temperature the thermal conductivity does not appear, the
above asymptotic analysis still clearly predicts that in the bulk, the material constitutive relation (thermal conductivity) is equal to the "traditional"bulk value. This can
be seen from first-principles by inserting (6.15) into (6.6) to obtain
1l
47
1L
V7
CWf (9 - VxTO) dod2p = -KV
TO
(6.25)
,
=
where the second equality follows from recognizing the well known expression
j
Vg2 TC
104
,;d
(6.26)
Order 1 boundary layer analysis
6.2
In this section, we extend the asymptotic analysis of the previous section to the vicinity of boundaries, where as will be shown below, a boundary layer analysis is required
for matching the bulk solution of the previous section to the kinetic (BTE) boundary conditions of interest. Here we will consider two kinetic boundary conditions,
namely, those of prescribed temperature and diffuse adiabatic reflection. In this work
we assume that boundaries are flat; boundary curvature will be considered in a future
publication. Without loss of generality we assume that the boundary is located at
X 1 = 0 and with an inward normal pointing in the positive xi direction; X 2 and X 3 will
denote cartesian coordinates in the plane of the boundary. Moreover, we will use Q1,
Q2
(Xi,
and
2
to refer to the components of the unit vector 0 in the coordinate system
Q3
, X 3 ). In other words, Q1 = cos(O), Q2 = sin(O)
cos(o) and Q3 = sin(O) sin(#).
We now derive the general equation governing the boundary layer correction required in the boundary vicinity for matching the bulk solution to the kinetic (BTE)
boundary conditions. We introduce the boundary layer function
Hilbert expansion (Dn
= "DGn +
Kn)
with
(JKO
DK,
written as a
= 0 and insert it in the Boltzmann
equation, obtaining
r
&Q Ki
(IKn)
j
C(DKi) -
= ZKn)
(Ki
Kn
(6.27)
In the vicinity of the boundary, a new characteristic lengthscale, namely the distance from the boundary, becomes important.
Similarly to [23], we introduce a
"stretched" variable defined by 7 = xi/(Kn). Equation (6.27) can thus be written
in the form
(
?Kn,-1
Ki =
9
(Kn)
)K- 4Ki -- (Kn)i
.Kn
2 Ki 3 O&Ki
.09X
2
OX 3
(6.28)
By equating terms of the same order, we find that each boundary layer term is solution
105
to a ID (in space) Boltzmann-type equation. For (K1, this equation is
QOK1
=KfI(
K1) -
(Kn,
(6.29)
Kn
917
The equations for
(K1
n > 2 include "volumetric source" terms resulting from the
derivatives of the lower order boundary layer in the boundary tangential directions
X3).
Specifically, for each order i > 2:
Q
0
(Ki)
Ki
=
n
-
Q2 0
Ki
Kn
Ki-1 + Q3
OX2
Ki-1
Ox3
(6.30)
)
(x 2 and
The case i = 2 will be considered in the following section, where second-order boundary layer analysis is carried out.
6.2.1
Boundary conditions for prescribed temperature boundaries
As specified in Chapters 3 and 5, in the linearized case, the incoming distribution of
deviational particles from a prescribed temperature boundary at temperature
fA = T
Tb
of
is
6.31)
T
or simply, in terms of quantity D defined in section 5.2.1
6.32)
(bo =Tb
We note that
(GO
is isotropic and is thus able to match
TGO =Tb
(Db
provided we set
6.33)
at the boundary. Therefore, at order 0, the solution to the Boltzmann equation with
prescribed temperature boundaries is given by the heat equation complemented by
the usual Dirichlet boundary conditions and no boundary layer correction is required
106
(4KO=
0, which also implies that To = TGO).
G1 = TG1-Kn(Kn)-1&'.
This situation changes at order 1. The order 1 distribution
VXTG0 is not isotropic due to the gradient of TGO. As a consequence, there is a mismatch between the order 1 solution and the boundary condition (which has been
satisfied by
(GO
and is thus zero for all subsequent orders). This mismatch can be
corrected by introducing a boundary layer term
governed by equation (6.29) and
(K1
subject to boundary condition
(K1Mq=0
+ (G1
=
0,
(6.34)
which translates into the following relation
@K1 70=o
-TG1
Kn
9-
--
'
VxTGO .n=o
(6.35)
The term VxTG0 is known from the order 0 solution. The term TG 1
and determined by the fact that there exists only one value for
tends to 0 for I -
TG1
,=0
is unknown
,7=0 such that
DK1
oc [23]. This determination proceeds by decomposing 4DK1 into
three components
KI =
where each of
K1,i,
i
K1,i
+ (K1,2
(6.36)
+ 4K1,3
1, 2, 3 is the solution to an equation of the form (6.29) with
the associated boundary condition:
(K1,1
(K1,2
+
r-0 =
,0
=
(C
2
+
T
j
(Kn)
Knj
(Kn)
&X 1
2
Kn
OK1,3 q=0
Anticipating the values of
OK1,i
-C
3
+ T
(K/
)=0
(6-38)
OTGO
OX2
'q0
OTGO
G3
&x3
to scale with &TG01o/xi
(6.37)
(6.39)
170
=0
in the above equations
we have set OG11O1=0 = ci(TG0&-xi)=o. The constants c1 , c 2, c3 are uniquely
determined by the condition that
FK1,1, (K1,2
107
and
(KI,3
individually tend to zero for
o
.
r --
Under the above conditions,
)K1,2
and
and the associated constants c 2 and
(bK1,3
c 3 can be found analytically. One can easily verify that c 2 = C3
(DK1,2 =
exp
a
KnQ2
Kn
1X
(KnQ1
r7~O
0
=
0, with
for Q 1 > 0
(6.40)
for Q, <0
and
exp
3
KKn,3
(
(Kn)
q=o
X3
for Q 1 > 0
(6.41)
KnQ
0
for Q, <0
are solutions to (6.29) with boundary conditions (6.38) and (6.39), respectively. The
temperature field associated with these functions is zero. Here we note that the above
solutions have the property
(6.29) with the term
.C(DK1)
C(DK1,2)
= C(
= 0 and thus are also solutions of
K1,3)
removed. We will use this observation throughout this
chapter for obtaining analytical solutions to a number of boundary layer problems.
The problem for
(DK1,1
must be solved numerically. Given the boundary condition
it needs to satisfy, let us write
(K1,1
=
(OTGo/x) r=0 and solve for 91 K1,1-
'?K1,1
The numerical method developed and used for this purpose is explained in Appendix
B.
In the case of a Debye and gray material referred to here as the single free
path (Kn = (Kn) for all
fJj
Cs,,pKl,ldwd
2
,p), it yields ci = 0.7104, while the resulting
TK1, 1
Q/47 is plotted in Figure 6-1.
In summary, the boundary condition for the order 1 bulk temperature field is
TG1(Xl = 0)
c1
TGO
O
xi=0
(6.42)
or more generally
TGXb = cl
On
(6.43)
where OTGO/n refers to the derivative in the direction of the normal to the boundary
pointing into the material, n, and
xb
the boundary location.
In other words, the
boundary condition is of the jump type and the associated temperature jump is
108
0
-0.02-0.04
-0.06
-0.08-0.1-0.12
-0.14
0
0.5
1
1.5
Figure 6-1: Temperature profile associated with
single-free-path material.
proportional to the derivative of the
2
2.5
TK1,1/(TGO/OXl rI--O) for a
TK1,1
0th order solution in the direction normal to the
boundary.
The amplitude of the corrective boundary layer that is added near the wall is also
proportional to the normal derivative:
TK1,1
OT77
(6.44)
K 1,1
Note that although a non-zero temperature field is associated with
DK1,1,
sponding heat flux is zero. This is explained by the fact that
by construction,
DK1,1,
the corre-
tends to 0 at. infinity. Since the boundary layer problem is one-dimensional in space,
by energy conservation, the heat flux has to be constant in x 1 and is therefore zero
everywhere. We also note that although
4K1,2
and
DK1,3
do not contribute to the
temperature field, they do contribute in the heat flux q'jI in the direction parallel to
the boundary. Their contribution can be obtained by substituting (6.40) and (6.41)
into (6.6); the result is summarized in table 6.1.
Numerical solution for complex material models
Above we reported the value of the coefficient ci and boundary-layer function
DK1,1
in the single free path case. In this section we report results for two more realistic
material models. Specifically, we consider a material with realistic dispersion relation
109
and a single relaxation time, as well as a material with realistic dispersion relation
and frequency-dependent relaxation times. The dispersion relation in both cases is
taken to be that of the [100] direction in silicon. The single relaxation time is taken to
be 40ps. In the case of a variable relaxation time we use a slightly modified Born-von
Karman-Slack model [4,46,50] where the grain size used for boundary scattering is
0.27 mm instead of 2.7 mm. The reason for this approximation is that it facilitates the
verification of the order 1 behavior with Monte Carlo simulation. We do not consider
optical phonons in this work, but the method can be straightforwardly extended to
this case.
We find ci = 1.13 in the single relaxation time model and c1 = 32.4 in the Bornvon Karman-Slack model. The associated boundary layers are plotted in figures 6-2
and 6-3, respectively. It is important to note that:
- The values of coefficient ci and the function TK1,1 depends on the definition of
(Kn) or, equivalently, (A), which is rather arbitrary.
This however does not
influence the final result because the asymptotic temperature field, ultimately
(see (6.8)) depends on the products ci(Kn) and TKl,l(Kn) (see for instance
solution (6.112)).
- The boundary layer in the modified Born-von Karman-Slack model is particularly wide (on the order of millimeters). This observation, as well as the large
value of ci, is a manifestation of the stiffness (multiscale nature) of this problem,
resulting from the wide range of free paths present in this material; mathematically, it is due to the factor Kn/(Kn) that appears in (6.35) and which tends to
give more weight to modes with very large free paths and makes the assumption
Kn ~ (Kn) questionable. Since, by assumption, the sum of all bGn(Kn)n should
exist -which requires I'(Kn)"
<
1- this has the overall effect of limiting the
range of applicability of the asymptotic model to Knudsen numbers that are
lower than the nominal (Kn) < 0.1. It is important to note, however, that this
limitation is a result of the fundamental physics of the problem: even at "low"
Knudsen numbers given by
(Kn) < 1/ci, there exist modes with long free paths
110
(i.e. Kn
-
0(0.1)) introducing kinetic effects and making the TGO solution (of
V TGO = 0) inadequate.
-0.1T
K1, 1
-0.2
-0.3
-0.4
-n 5
0
0.2
0.4
0.6
0.8
( n)
c
1
x1
0
S
Figure 6-2: Temperature boundary layer function TK1,1 for a material with silicon
dispersion relation and a single relaxation time.
U
-5-10
TK1,1
-15
-20
-25
-30
-35
0
0.5
1
1.5
x10
x ()
Figure 6-3: Temperature boundary layer function
Karman model.
2
TK1,
3
for the modified Born-Von-
Validation
We validate our result using a one-dimensional problem, in which a modified Born-von
Karman-Slack material is placed between two boundaries at prescribed temperatures
and located at x, = -L
and x, = L, respectively.
The order 0 solution to this
problem is a linear temperature profile TO(x1 ) which yields a heat flux tsi-MATO/L,
111
where ATO is the temperature difference between the boundaries; here,
Nsi-M
denotes
the bulk thermal conductivity associated with the modified Born-von Karman-Slack
material. The temperature profile
TG1
is obtained by solving the Laplace equation
with jump conditions
TG1(l
(6.45)
- L) = c,
and
Tcl1(x 1
L) =
1
T-c
(6.46)
OX 1 xl=L
and yields the modified heat flux ssi-M(1 - c1(Kn))ATo/L. We note that when calculated from an order n temperature field, the heat flux is inherently an order n + 1
quantity; in other words the above result is correct to order 2. In Figure 6-4, we plot
the difference between the actual heat flux (q" , obtained using kinetic-type deviational Monte Carlo simulation presented in Chapter 3) and the asymptotic approximation, both normalized by /si-MATO/L, namely, c = q" L/ssi-M/AT - (1 - c (Kn)).
The observed asymptotic behavior is order 2 which validates the order 1 accuracy of
the asymptotic solution.
10-1
10
101
102
103
(Kn)
Figure 6-4: Difference between the order 1 asymptotic solution and a Monte Carlo
solution for the heat flux between two boundaries at different temperatures.
112
6.2.2
Boundary condition for a diffuse adiabatic boundary
The case of diffuse adiabatic boundaries can be treated through a similar approach,
where the mismatch between the bulk asymptotic solution and the boundary condition
is analyzed and corrected. We recall the boundary condition at the kinetic level [31]
q
Sj
for Q 1 > 0
Q'd
'i 2'
(6.47)
K 1<0
A major difference from the prescribed temperature boundary is that applying this
0th order bulk solution gives no information, because GGO satisfies
condition to the
(6.47) regardless of its value at the wall. The boundary condition for TGO is obtained
by analyzing the order 1 mismatch.
The order 1 boundary layer problem may be
defined by applying the boundary condition (6.47) to Di =
(DG1
+
K1-
It results in
the following condition:
TG1 ,,=qO -
1
-
VxTGO
=O
+
f(6.48)
(TGI
The isotropic term
VxTGO r=O +
r=0 -
TG1
section 6.2.1, we define
(D K1 ,q=O
K1 rq=o)
Qd 2 Q,
for Q > 0
readily cancels from both sides of the equality. Similarly to
=K1,1 + (K1,2
(K1
+ (K1,3
where each (K1,i
is associated
with the temperature gradient in direction i (as given by a right-handed set with x 1
being the direction normal to the boundary) and is a solution to the Boltzmann-type
equation (6.29) with boundary condition:
09TGO
-
Qi
Ox
;=
-
f
Ji< 0 ki
--
xi
/
OTGO
r1Tc +DK1,i+O
q
K1,i rr=o)
Q'd
for Q
170-/
(6.49)
We find that solutions (6.40) and (6.41) satisfy the above conditions for i = 2 and
i
=
3 respectively, and do not impose any condition over the tangential derivatives of
113
0
TGO. For I = 1, (6.49) results in
+
3
1
Ox1 'q=0
= -T DK1,1
10 - 2
jf
1
K1,1
0 Q'dQ,
for Q, > 0
'/<0
(6.50)
The only solution possible with this boundary condition is DK,
0. This can be seen by noting that if (&TGO/&xi) 1=0
#
tion by Q, and integrating over 0 < Q, < 1 yields f
q0 = (&TGO0x)
"0
0, multiplying the above equaCK1,1
qq=0idQ 1
0, which
is impossible (this can be seen by starting from the equation governing GK1,1-of the
type (6.29)-and integrating over 0
dition
4K,1(77
--
OC) -+
y < oc and -1
< Q1 < 1 and using the con-
0). We thus conclude that TGO must satisfy the boundary
condition
-0,
=TGO
(6.51)
OnXb
which agrees with the Neumann boundary conditions associated with adiabatic boundaries.
6.3
Order 2 boundary layer analysis
In the previous sections, we described the jump relations and boundary layers that
appear at order 1 for correcting the mismatch between the bulk solution and the
kinetic boundary conditions. We now apply a similar approach for obtaining the order
2 corrections in the case of prescribed temperature and diffuse adiabatic boundaries.
6.3.1
Order 2 analysis for prescribed temperature boundaries
The reasoning presented for order 1 is quite systematic and subsequent orders can be
treated in a similar way.
The second order correction
Q
(K2
must be solution of (6.30) for i
= (Kn) 'K2)-42
(I1
Kn
-
114
0K1
+
O _O
2
2, namely:
(6.52)
0
Ox3
with the boundary conditions
4K2q=O
- -G2
q=O =
+
-TG2#=o
n
(Kn)
~DT12
Kn~
x
a2TG
KnG
q
(Kn)2 ExiOx
=O
for Q 1 > 0
(6.53)
The derivatives of the first order boundary layer which appear in (6.52) as volumetric
source terms may be written as
8@K1Kn
=K1Kn
-
Q2
(Kn)
OX2
02TG0-(n
Go
2TGO
2TQ3
09X2
exp
OX2ax3
q
for Q 1 > 0
(Kn
Q,GKn
(6.54)
-
@K1
K(n
=(n)
(Kn)
09X3
[
0 2 TG0
0KQ2
+
+
9X309X2 q=0
0 2TcG
,_O
OX
(Kn)
1 Kn
Q1Kn
exp
ex
for Q 1 >
0
(6.55)
The boundary condition (6.53) includes three terms with first order partial derivatives
of TG1 and nine terms with second order derivatives.
in (6.54) and (6.55) introduce four source terms.
In addition, the terms given
We therefore introduce sixteen
constants such that the order 2 "temperature jump", TG2 ,=o, may be written as
+iJ0 Gi O- (6.56)
3
TG2OT=0
2
T
di 0 1
i-1
q=i
TG
3
'=0
ij=1
2
_
TGo
ij=2
We accordingly introduce sixteen boundary layer functions such that the total order
2 boundary layer may be written as:
3
K2
K
TG1
i=1
0 2TGO
3
K2,i
E
'=0
i
ij=1
0 2 TG6
3
3
K2,ij
1=0
ij=2
i
In equation (6.56) and (6.57), coefficients di and gij, and functions
qfK2,ij
(6.57)
K2,ij
rl=0
'FK2,i
and
are determined by boundary value problems of the same form as the ones
115
The problems that determine
discussed in section 6.2.1 satisfying equation (6.29).
the coefficients jij include the source terms from equations (6.54) and (6.55). In the
interest of brevity, we only discuss the ones associated with the source term (6.54).
The remaining two may be easily deduced from (6.55). Coefficient
1
=
Q,07 '
Knn
(C('4
\
2 ,2 2
)
-
P,7 = 0) + j22 = 0,
q'K2,22(,,
'1K2,22
+ (K
Kn)
eXP
2 x
Q2
(
22
j)1QKn
is solution to:
for Q, > 0
for Q1 > 0
lim VI'K2,22(0, L, p, 7) = 0
(6.58)
Coefficient
Q
23 is solution to:
D'I'K223
=
q'
(Ku)
\(XK2,23)
'I K2,23(Q, W, P 7 = 0)
lim
+ j23 =
-
Kn
(K) Q 2 Q3
PK2,23+
eXp
- (Kn)
Qfor
Q > 0
for Q1 > 0
0,
4f K2,23 (R LC,A 7) = 0
(6.59)
Two results can be obtained immediately:
.
- Coefficients dj, d 2 and d 3 are solutions to the same problems as ci, c 2 and c3
Consequently, they are equal and their associated boundary layers are the same,
provided TGO is replaced by TG 1 in Eqs. (6.40), (6.41), (6.43) and (6.44).
- Coefficients gij and
jj for i
$ j
are zero.
This is inferred from the same
argument that we used to conclude that c 2 and c 3 are zero. Specifically, we may
first solve analytically for an alternative problem from which the operator L is
removed. For instance, we replace (6.59) by:
QI
4
(KKn)
'K2,23
071 '
q1 'K2,23(,
=
89
-
Kn
, A 77 =
lim 'FK2,23(A, LL, A
+ Kn
K2,23 + 7Kn
(Kn)
0) + j23
=
0,
-jKn)
-(exp
2Q3 e
Qjfr n
0K
for Q, > 0
1
for Q, > 0
) = 0
(6.60)
116
It is readily verified that a solution to (6.60) is
Kn
=
exp
for Q, > 0
Kn
(Kn)j
Q ex
0 for Q 1 <0
5K2,23
with j23
23-
0. We then verify that
(6.61)
(IK2,23) = 0 and therefore qfK2,23 is the
solution of the original problem (6.59).
We are left with five undetermined coefficients, namely gi, 922,
933 , j22 and #33. These
can be determined using the numerical approach presented in Appendix B (suitably
modified in order to accommodate the volumetric source terms which appear in the
mathematical formulation).
Instead of following this approach, in Appendix D we
prove that
3
02G
g___O
i
i=1
92TG0
3
g
+
q=0
i
i=2
= 0
(6.62)
q=0
and that, therefore, the temperature jump associated with the second order derivative
is zero, while the boundary layer, although not zero, integrates into a zero temperature. In other words, the second order temperature jump is given by the condition
TTG1
an=c1 .(6.63)
TG2|q=
This result is quite convenient and lends itself particularly well to implicit application
of boundary conditions; this is discussed in section 6.4.4. The analogy to the order
one temperature jump extends to the temperature boundary layer that is given by
TK2,1
TK1,1
OTG1
0
r1=0
(6.64)
In addition to this temperature boundary layer, the analysis yields a second order
heat flux boundary layer. It may be calculated analytically by inserting expression
(6.57) for 4DK2 into
K'2
=
f"'I
4W'
C7
117
K2Vg 2QdL,
(6.65)
which can be written in terms of incomplete Gamma functions.
We validate these
results in section 6.4.5.
Order 2 analysis of a diffusely reflective boundary
6.3.2
In section 6.2.2, we resorted to an analysis of the order 1 boundary layers to obtain
the order 0 boundary condition, and showed the latter amounts to the well-known
Neumann boundary condition. Similarly, we here proceed with the order 2 analysis
in order to find the boundary condition for the order 1 temperature field.
Inserting (6.23) in (6.47) and introducing a boundary layer term yields, for Q 1 > 0
and for all frequency/ polarization modes:
Kn
TG2
xb -
(Kn
I'
7
'
'(T(Kn)
VXTG1
-
Xb +
n
Kn 2
(n2
V
VXTGO)
(Q
2
'7XTG1 1n
Q
Kn)2
Xb
(K2lXb
''Q
VXTGO
X b
+
K2 Xb'
2
(6.66)
Moving to the coordinate system (x 1 , x 2 ,X 3) and the stretched coordinate r/, we first
note that in (6.66), the derivatives
02To
TGo
(6.67)
axi(9xl 'q=0
are zero for i
=
2, 3 because (OTGO/xi) j=0 = 0.
Boundary layer
and
4K2,23.
(DK1,2
and
(K2
Components
(DK1,3
may be decomposed into 4 components,
DK2,2
and
4K2,3
(K2,1,
(bK2,2,
IK2,3
are similar to the order 1 boundary layers
(see expressions (6.40) and (6.41)), with the only difference being
that TGO is replaced by Tc 1 . Component
(K2,23
corrects the anisotropic mismatch
associated with the bulk term 2Q2Q 3 [( 2 TGo/(X20X 3)],=0. It is a solution to the ID
Boltzmann equation (6.29) with boundary condition
4K2,23
-7=0
-2Q
2 Q3
118
2
0 TGO
eOX 2 &X 3
(6.68)
qj-0
for Q 1 > 0, and 0 at infinity, and is therefore given by
Components
(K2,2,
= -2Q 2Q 3
and
(K2,3
&2TGO
19X20X3 q0
(K2,23
-- r/(Kn)
exp
QKn
HQ)(.9
(6.69)
H()
)
(K2,23
do not contribute to a temperature jump or
(temperature) corrective layer, but they do contribute to a heat flux boundary layer
as summarized in table 6.2.
The last component is solution to the following problem:
0
)K2,1
Q=
Kn
_(Kn)
0r7
Kn
(L(DK2,1)
K2,1
2K
(n
(Kn)
i-2
2
-r_n)
T
2 (iTGQ1eXp
r1=40Q1Kn
for Q 1 > 0
-
(
2
+ Q
3
Kn
Kn
Q2-1)]
+ (n)
(Kn)
2
_ Kn)
= -2j
lim
17-*00
IK2,1
(,,pTI)
02 TGO
+
Ir
Q1K2,1
K2,1
=0
_1=0
=OdQ1,
for Q 1 > 0 and all w, p
0
(6.70)
In the above, we anticipated a jump relation of the type
02TG
Go
19TG8TG1
1
Ox 1
o
(6.71)
D
0x
and factorized the second condition of problem (6.70) accordingly. In the interest of
simplicity, we also define
qfK2,1 = 4K2,1/(2TG/
1=-
Although we could solve problem (6.70) using the method described in Appendix
B, we will here directly find the value of
-y
without specifically calculating
TK2,1-
We first proceed by multiplying the boundary condition (second equation of problem
(6.70)) by Q 1 and integrating over the half sphere described by Q 1 > 0 to obtain
f
11K2,1=0dQ1
s3
We
=
2
(Kn)
ntn)
(6.72)
We also multiply the first equation of problem (6.70) by VgC1W,p and integrate it over
119
all frequencies and solid angles and 0 < rj < oc to obtain
2 (6.73)
Kn
2
1
CspVQOPK2,1 q-
C
-
2,i,
i/LQ4'P
,,pVg1K
-6
Since
(DK2,1
21
r=OdwdQ]
.
I-,P
VLPgTn
Cprd
tends to 0 at infinity, we deduce
3 f ,Kn 2 VgC pd(6
'
16 (Kn) f, KnVCspdi
which can be rewritten in the form
d
V3T
33 fo P
'gpg
pd. 2C (6.75)
2
16 (A) fp Vg TC,, dJ
In the single mean free path model, y = -3/16.
used for finding
layer
-'
Note also that the approach that we
may be used for finding the heat flux associated with the boundary
(DK2,1-
Validation of the order 1 boundary condition for diffuse reflective walls
In this paragraph, we validate the condition 6.71 by considering a 2D problem. The
system is infinite in the xi-direction and is defined by a diffuse reflective wall at x 2
=
0
and a prescribed temperature boundary with temperature Tb(xl) = T, cos(27x 1/3)
at x 2
=
1. The order 0 temperature field is readily given by
TGO(X1, x2) = T, cos 27x3
cosh(2wX 2 /3)
cosh(27/3)
(6.76)
Figure 6-5 shows the system configuration and the contour plot for TGO/Tw. Knowing
TGO,
we may calculate TG1 by solving the Laplace equation with the order 1 bound-
ary condition (6.43) at the prescribed temperature wall, and (6.71) at the diffusely
120
reflective wall. We obtain the solution by noticing that it is of the form
TG(x1, x2) = cos
(
2
7
)
B exp
27X2
+ B 2 exp (2732
(6.77)
and by finding B 1 and B 2 such that the order 1 boundary conditions at both walls
are satisfied. We recall that there is no order 1 boundary layer near the diffuse wall,
and we therefore readily obtain the value of the order 1 asymptotic solution at point
A represented in Figure 6-5. In Figure 6-6, we plot the difference C between the order
1 asymptotic solution at point A and the actual solution calculated using the adjoint
Monte Carlo method presented in Chapter 5, against the Knudsen number.
This
difference is clearly an order 2 quantity, which confirms the validity of the diffuse
order 1 boundary condition.
1
0.8
0~
000
&
0.6
0.4
CVC
" o
0.2
A-
0
'n
-1.5
-1
-0.5
0.5
''
1
1.5
X1
Figure 6-5: Contour plot of the solution of Laplace's equation in a thin film with sinusoidal Dirichlet boundary conditions (top boundary x 2 = 1) and Neumann boundary
conditions (bottom boundary x 2 = 0).
A note on the physical interpretation of (6.71)
At first glance, the boundary condition (6.71) seems to suggest that the net heat flux
at the boundary is not zero; in this section we show that, in fact, this condition ensures
121
1
10-2
10
Z
10
-2___I___
(Kn)
10
10
Figure 6-6: Absolute value of the difference between the order 1 asymptotic temperature solution and the Monte Carlo results, at point A of the problem depicted in
Figure 6-5. The difference is clearly an order 2 quantity.
that the net heat flux at the boundary is zero by balancing the heat flux contribution
of the first-order kinetic boundary layer. To this end, we consider a control volume
of size Ax
2
Figure 6-7.
along the boundary and I into the material (x1 direction), as shown in
Here, I is a distance of the order of the mean free path, namely large
enough for the boundary layer to be negligible, but small enough when compared to
the system size L. The bulk solution contribution to the heat flux exiting this control
volume is given by
q"
(x1 = l/L,x 2 ) = -
(Kn) 2
(A)
(Kn) DTGO
(A Ox 1
(A)
_TG1
OTG1
Oxi
xi=1l/L,X2
+ 0 ((Kn)3))
(6.78)
This heat flux may be written as
,,
1L,)
K n)
(A)
TGo
Ox
1
x
O,X2
+ (Kn)22
(A)
TGO
(9X2
1Ox
(1n)2 0T0 1
+ 0 ((Kn)3)
(A)
Ox1 x I=OX2
(6.79)
122
Using (6.51) and (6.71), as well as the fact that V2TGO = 0 we can rewrite this
equation in the form
(Kn)2
(
+ 0 ((Kn)3)
+
q" (x = l/L, x 2 ) =
(6.80)
Clearly this quantity is non-zero in the presence of temperature gradients along the
boundary.
In the presence of such temperature gradients the heat flux near the
boundary in direction x 2 , q" 2 , is also non-uniform and composed of the bulk heat flux
and the boundary layer correction (see table 6.2). As can be easily verified, q"2 can
be integrated on a normal line from x 1 = 0 to x=
f~=
q" 2 dx1 =
(Kn)
f1=0
+ (Kn)
=-j
(A)
J
2
OTGO
dx1
X2
=O,X2
g
--
Q2CVK1,2
2
ddx1 + 0 ((Kn)
, V322dw) (Kn) 2 &TGo
3
)
L
ff
=
+
q'2
l/L, yielding:
0
((Kn)3)
(6.81)
Assuming that no next heat flux crosses the boundary, energy conservation over the
control volume requires that
q'
- q' + q" Ax 2 = 0
(6.82)
which leads to expression (6.75) for y and verifies that, physically, boundary condition
(6.71) is required to balance the order (Kn) 2 net heat flux due to kinetic boundary
layers in the presence of a temperature gradient along the boundary. Note that these
equations can be simplified considerably by considering only TGi in the x 1 direction
and the boundary layers in the x 2 direction, because V2TGO = 0 ensures balance of
the other two terms.
123
)
q, (x,X2 + AX 2
(ql
AX2
X2),AX2
1_
Figure 6-7: Schematic of order two energy balance at a boundary along which a
temperature gradient exists.
6.4
Summary and discussion of results
We have derived the continuum equations and associated boundary conditions that
provide solutions equivalent to those of the Boltzmann equation up to second-order
in Knudsen number for steady problems. This derivation shows that the governing
equation in the bulk, up to at least second order in Knudsen number, is the steady
heat conduction equation with the bulk thermal conductivity. Kinetic effects, always
present at the boundaries due to the inhomogeneity introduced by the boundary
and the concomitant mismatch between the distribution introduced by the kinetic
(Boltzmann) boundary condition and the distribution function in the bulk, become
increasingly important (can be observed in larger parts of the physical domain) as the
Knudsen number increases. Fortunately, these kinetic effects can be systematically
described and incorporated into the continuum solution relatively straightforwardly.
Specifically, the solution for the temperature and heat flux fields can be written in
the form
00
T
=
(Kn) (TGi + TKi)
To +
(6-83)
i=1
and
00
=
Z(Kn)' (qG'
124
+ q'j:s)
(6.84)
respectively. Here, TGi denotes the ith order solution of the Laplace equation subject
to boundary conditions appropriate to that solution order (see below).
Similarly,
q" denotes the heat flux obtained from the above solution, via q'i = -x1V7TGi-1
where K is the unmodified bulk thermal conductivity and where we recall that x is
dimensionless. Additionally, TKi and q
are the boundary layer corrections to the
bulk fields that are important close to the boundaries.
We have studied two types of kinetic boundary conditions: prescribed wall temperature and diffuse reflection. The associated heat-equation boundary conditions and
boundary layer corrections are summarized in sections 6.4.1 and 6.4.2, respectively.
6.4.1
Prescribed temperature boundaries
At order 0, the Laplace equation V2TGO = 0 is supplemented by the traditional
Dirichlet boundary conditions; no corrective boundary layers are required.
The order 1 correction is obtained by solving VXTG1
=
0 subject to a
jump
condition of the form (6.43). A corrective boundary layer exists for the temperature
in the direction normal to the boundary (but no heat flux boundary layer) while a
corrective heat flux boundary layer exists for the directions parallel to the boundary
(but no temperature boundary layer).
The order 2 correction is obtained by solving V'TG 2 = 0 subject to a jump condition of the form (6.63). The second-order corrective boundary layers are proportional
to the second derivatives of the order 0 temperature field and result in heat flux
corrections but do not result in any temperature corrections.
These results are summarized in table 6.1.
6.4.2
Diffuse reflection
At order 0, the Laplace equation VSTGO
=
0 is to be solved subject to the Neumann
boundary condition (6.51), as expected. We note that this boundary condition was
derived by analyzing an order 1 boundary layer problem. The boundary layer correction arising from this analysis appears in the order one solution; it contributes only
125
Table 6.1: Boundary conditions and boundary layers for prescribed temperature boundaries,
up to order 2. Symbol ei refers to the unit vector corresponding to direction xi in a righthanded set where x, represents the coordinate normal to the boundary and pointing into the
material.
Order
Boundary condition
boTempry ayers
Heat flux boundary layers
0
TGO = Tb
None
None
1
TG1
2
TG2~1
xb
TKI = TK1,1
= Cl
Cl
,, J X Tan TK1 11 f PQ 4r -i 2 I a
T 21x,CI -C1 T ,=T ,1q/C an Xan K2 p 7 K L,~
H(Q)
q
Q
ei exp-
dd2
Table 6.2: Boundary conditions and boundary layers for diffusely reflective walls, up to order
1.
Order
Boundary condition
0
1
0=O
aTG1 Jo
an.1X
=h Xh AQ1a
None
None
None
K47
Heat flux boundary layers
boundary layers
q'k1 =
f
H(0 1 )
axT
2
e" expQ?
ei exp aC-xb).n-\ 2
dd
to the heat flux field (no temperature boundary layer at order one).
The order 1 bulk temperature field is obtained by solution of V7TGi = 0 subject to
the jump condition (6.71). The associated boundary layer analysis results in an order
2 boundary layer. Although the associated heat flux boundary layer can be calculated,
similarly to equation (6.70) by direct integration of (6.73), it is not presented in this
thesis.
These results are summarized in table 6.2.
6.4.3
A one-dimensional example
In this section we consider a simple ID problem as a means of illustrating the application of the asymptotic theory to problems of interest. We consider a silicon slab
of thickness L confined between two walls at different prescribed temperatures. Using dimensionless coordinates, the walls are located at x 1 = 0 and x=
deviational temperatures TL and TR, respectively.
126
1 and have
We recall that under the asymptotic analysis, the temperature field is given by
T(xi) = TGo(x 1 ) + (Kn)Ti(xi) + O((Kn) 2)
= TGO (xi) + (Kn) (TG1(xi) + TK1(xl)) + O((Kn) 2 )
(6.85)
(6.86)
The order 0 solution straightforwardly reads
TGO(X1) = TL + (TR
-
(6.87)
TL)X1
since it is the solution of the heat conduction equation subject to no-jump boundary
conditions. Therefore, the boundary conditions for the order 1 field are
TG1(hl = 0)
c,
&TGO
=
cl(TR - TL)
(6.88)
1
TG1(Xl
= 1)
-ci OTGO =
OXi
C(TL
- TR)
(6.89)
which results in
TG1(Xl)
Cl(TR -
TL) (I
-
2x,)
(6.90)
The boundary layer TK(xil/(Kn)) = (TR - TL)rK,(xil(Kn)) contributes to the
solution near the wall at x1 = 0, while the function (TL - TR)TK1,1((l - x1 )/(Kn))
contributes close to the wall at x, = 1. The resulting solution correct to order 1
(eq (6.86)) is plotted in figure 6-8 for (Kn) = 0.1 in the single free path model and
compared to our benchmark (adjoint Monte Carlo) result. The agreement is excellent;
we note in particular that even though the boundary layer correction is small at
this Knudsen number, the temperature jumps are considerable and are accurately
captured by the asymptotic solution.
If desired, calculation of T(xi) to second order in
heat conduction equation for
TG2
(Kn) proceeds by solving the
subject to the second order boundary conditions.
127
I1
__
-
-
0.9
SMonte
Carlo
0.1
0.2
0.8 H
0.7H
I-
cc 0.6
0.5
0.4
0.3
0.2
0.1
0
0.3
0.4
0.5
0.6
07
0.8
0.9
1
x1
Figure 6-8: Order 1 solution (plain line) compared to the solution computed by highly
resolved Monte Carlo simulation at Kn) = 0.1.
Applying (6.63) to this problem yields
TC1
TG2(X
TG2(XI
=
(X 1
=
0) =
10) =
O1TG1
1) = -c1
(X 1 =
-2c(TR
1) = 2c (TR
Oxi
-
TL)
(6.91)
-
TL)
(6-92)
with the solution
-
TL)(1 - 2x1
(6.93)
)
TG2(X1) = -2c (TR
In fact, in this particular problem where only first derivatives are non zero, the process
by which (6.93) was derived can be repeated for all orders without knowledge of the
higher order jump coefficients, leading to an asymptotic solution that is, in principle,
correct to all orders. In other words, for n > 1:
-
TL)(1
2x 1
-
)
TGn(xl) = (-2)n' c'(TR
(6.94)
Summing all orders (provided 2(Kn)ci < 1), we obtain:
TR -TL
+ 1
(Kn)cI
(1
-
2x 1
(6.95)
)
TG(xl) - TL
The boundary layer corrections of all orders can also be obtained (and summed) using
128
the same process. For example, for the boundary at x1
TK (x1)
TR-TL
TR - TL
1n
-
I + 2(Kn)ci
The second boundary layer (at x 1
=
0, we obtain
TK ,Kn)
1 (x1/(Kn)).
(6.96)
1) is obtained in an analogous fashion. This
=
solution is asymptotically accurate to all orders, meaning that the error converges to
0 faster than any power of (Kn); for a discussion on the error associated with the
asymptotic expansion see [581.
Figure 6-9, compares the order 1, infinite order and "exact" (Monte Carlo) solution
for (Kn) = 0.5. The infinite order solution is in very good agreement with the exact
solution, while the order 1 solution is clearly inadequate at this Knudsen number.
- -
0.8- -
-order 1 solution
infinite order solution
a Monte Carlo
0.70.60.5-
0.4-0.3
-
0.2
0
0.1
0.2
03
05
0.4
0.7
06
0.8
09
1
x1
Figure 6-9: Order 1 solution (dashed line) and "infinite order" solution (solid line)
compared to the solution computed by a finely resolved Monte Carlo simulation for
(Kn) = 0.5. At this Knudsen number the boundary layer contribution is clearly
visible (the solution is no longer a straight line).
6.4.4
"Implicit" boundary conditions
In the rarefied gas dynamics literature
159] jump
boundary conditions are frequently
imposed in a implicit fashion thus avoiding the solution procedure shown above where
the governing equation needs to be solved for each order of solution. For example, a
set of boundary conditions up to second order given by
TGo
x
129
= T
(6.97)
TG1
=
On
(6-98)
0
and
TG2
nXb
may be imposed by solving V 2 TG
2
+
(6.99)
TGO
Xb
0 subject to
+ /3(Kn) 2 &2 TG
TG 1X - Tb = o(Kn) OTG
On
On
(6.100)
Xb
One can show that these two forms are equivalent (to order (Kn) 2) by expanding
TG
and similarly for
Xb
(TGo + KKn)TG1 + (Kn) 2TG 2 +
and substituting into (6.100).
&TG/On x
-)
Xb
(6-101)
Equating terms of the
same orders of (Kn) we obtain equations (6.97), (6.98) and (6.99), at order zero, one
and two, respectively.
Clearly the implicit form relies on the jump coefficients (a, 3, etc) remaining the
same at each order (e.g. in (6.98) and (6.99)). If the above condition is satisfied, in
addition to requiring less solutions of the governing equation, the implicit form has
one more advantage: provided higher order derivatives (not included in (6.100)) do
not appear at higher order, the solution will be correct to all orders, since it is easy
to verify that (6.100) then implies that
OTGn+1
TGn+2
On
O 2 TGn
+
On Xb
(6.102)
O2Xb
for all n > 0.
This property can be illustrated with the example of section 6.4.3. In the present
case a = ci and 3 = 0. Solution (6.95) can be obtained directly by solving d 2 TG/dX2
0 subject to
TG
xb -
Tb = ci(Kn)
(6.103)
Although an infinite order solution is always welcome, we also need to keep in
130
mind that some fortuity was involved in this problem in which all higher derivatives
of the solution are zero. In the general case, given that ) = 0, we expect the implicit
condition (6.103) to provide solutions that are accurate at least to second order and at
most up to order m - 1 where m denotes the order of derivative featuring a non-zero
jump coefficient.
6.4.5
A two-dimensional example
In this section we use a two-dimensional example to illustrate the application as well as
convergence properties of the asymptotic solution theory just presented. Specifically,
we consider a slab of material of thickness 2L, with the dimensionless coordinate
x 2 defined such that x 2 = 0 describes the median plane of the slab. The material
boundaries at x 2
=
are at the prescribed (deviational) temperatures
1 and x 2 = -1
T, cos(27x 1 /3) and -Tw cos(27x 1 /3), respectively.
The order 0 solution for the temperature field is
TGO (X1, X 2 ) =Tw
27x,
COS
3
sinh ( 2 x)
sinh(6.104)
)sinh
(T3)
and is plotted in Figure 6-10. The order 1 bulk temperature field can be obtained by
solving the Laplace equation with the boundary conditions:
TG1(XI, X2 =
11) = Tc 1 (Kn)
OTGO
(6.105)
1)
(X 1 , x 2
OX2
X2)
27
Tw(Kn)ci -coth
-
n3n3
sinh (27
.
sinh (-
O2
S 27rx
cos
(6.106)
)
(Kn)TG1 (x,
)
resulting in
The order 2 bulk temperature field is then obtained by solving the Laplace equation
with the boundary conditions:
TG 2 (X 1 , X 2
=
1)
= Tc1
(Kn)
131
&TG1
aX 2
(X 1 , x 2
=
l)
(6.107)
Thus we obtain the second order term:
27
-)
1i
2 cs(27x
s (
3
sinh
3
( 27X2
(6.108)
sinh (-3
)
Kn)2C227
2
)2
- coth
ci
=Tw(Kn)
(Kn) 2 TG2 (IlX2)
G2,3
At point (XI = 0, X2 = 1), the boundary layer temperature terms must be added to
the bulk terms. The order 1 and order 2 boundary terms are, respectively:
(Kn) 2 TK 2 (Xl
where
T
=
27
0, X2
1)
27
-coth 3(
TwTK1,1(0)(Kn)
TTK1,1(0)c1(Kn)
(2 coth
(3
(3
2
))2
(
)
1)
2
)
(Kn)TK (xl = 0,
K1,1(0) is the value taken by the temperature boundary layer function at the
boundary.
The superposition of all these temperature terms should therefore give an order 2
accurate solution of the temperature at point B. In fact, as explained in the previous
section, a solution of a similar order can be achieved by directly looking for the
solution of the Laplace equation TG with boundary conditions:
TG(X 1 ,x 2 =1)
=c1(Kn)
e9T~
Ox2
(x1, X2 = i1)
(6.110)
This is the case here because as shown in section 6.3.1 second-order derivatives do not
appear in the jump conditions or the temperature boundary layer. Applying these
"implicit" boundary conditions, we obtain
TG(X12)
Cos(T 27xi
=
1 + c1(Kn)
coth (Lo)
3
( 2 2)
sinh (2,)
sinh
Adding the boundary layer -FTK,(0)Kn)OTG/&X2 (X1, X2 =
1) to this result, we
obtain the following expression for the temperature at point B:
1
TW a
TKll(0)(Kn) 2-coth
(()
o e
1+ C1( Kn L coth (2")
The accuracy of each of these solutions is compared in Figure 6-11,
132
(6.112)
which plots
T(xi = 0, X2 = 1) - Tasymptotic, for all 3 asymptotic solutions (first-order, second-order
and implicit) for the single-relaxation-time model defined in section 6.2.1. Here, the
benchmark T(xi
=
0, x 2
=
1) is a highly resolved solution obtained using the adjoint
that the
Monte Carlo method described in section 7.2.2 and Ref. [461. The figure shows
features
implicit formulation leads to an order 2 solution overall which additionally
As
slightly improved accuracy compared to the "simple" order 2 solution (6.108).
explained in section 6.4.4 and Ref. [521, the solution would be "infinite" order (meaning
that the order of accuracy is asymptotically faster than any power of the Knudsen
number) if no higher order derivative appeared in the jump boundary conditions.
that
From the second-order convergence observed for the implicit solution, it appears
a non-zero jump coefficient appears in front of the third-order derivative (m = 3).
B
1
0.5
d.2
X2
0-
0
A
0.
-0.5
-0
P.
-1
-1.5
N
-1
-0.5
_70I
0
0.5
1
1.5
Figure 6-10: Contour plot of the solution (7.28) of Laplace's equation in a thin film
with sinusoidal Dirichlet boundary conditions.
6.5
Extension to time-dependent problems
Although the analysis presented here has so far been limited to steady problems,
Hilbert
extension to unsteady problems is relatively straightforward. In fact, the
133
100
10
104
- -
implicit conditions
- order 1
- order 2
10~6
10 2
10-
(Kn)
Figure 6-11: Convergence of asymptotic temperature solutions at (x1
the two-dimensional example considered in section 6.4.5.
expansion has been extended to time-dependent problems by Sone
=
0, x 2
=
1) in
1231 and Takata
[60,611, who showed that, other than the additional time-derivative in the governing
equation, time dependence does not introduce any new physics up to order 1 in (Kn).
In this section we show that this is also true for phonon transport by introducing
the dimensionless time-dependent Boltzmann equation
St
= D)
VX
at
+
-
Kn
(6.113)
where t is a dimensionless time, defined by t -- t'/to, where to is a characteristic time
of variation and the Strouhal number is given by
StW
= St= L
(6.114)
Vgt
We analyze cases where (St) ~(Kn), where the average Strouhal number, (St), follows
from the definition of (Kn) in (6.1). The condition (St) ~(Kn) can be rewritten as
to ~ L 2 /'
~ (T)/(Kn) 2 , which implies that it amounts to an assumption of diffusive
scaling in time.
134
Expanding the time dependent function 4J as in Eq. (6.7) results in the same
forms for orders 0 and 1 (equations (6.12) to (6.14)). Differences appear at order 2.
Specifically, the form of the order 2 solution reads:
Kn~
' -xTG1 --2 V -x(Q
-VxTGO)
(6.115)
=
L(
G2) -
Kn 2
St~n &TGO
(Kn) 2
(Kn)
+G2
V
(Kn)2
at
Applying the solvability condition (6.20) results in
I
C
w
(St~n &T0
Kn2
Rn
, - VxTG 1 -
+
'I
2
-V
(Q - VXT)
= 0
d2 d
(6.116)
which, after integration, yields the heat equation for the order 0 temperature field:
OTGO
at'
=_,-
K
C
(6.117)
2VTG.
Applying the solvability condition to the order 3 solution would similarly yield the
heat equation for the order 1 temperature field. Although not strictly needed for our
purpose here, we may express
1
TG2
TG2
=
C
]I
C
aG
2d
4
,a47
TG
C"
47r
= I
C
2
(DGG2) in Eq. (6.115) by writing:
(6.118)
Q(L)
((DG2)
-
St~n &T0
(Kn) 2 at
(,,n
-- OT
Rn 2
VTo)
(Kn) 2 x (9 - VxTGO) d2 d;
+Kn2Q
(6.119)
yielding
L((G2)
=
TG 2 +
1
C(Kn)2
P
OsIPSt ndw
TGO
at
-
1"
1
C(Kn) 2
fP
Kn 2
3
2
duSV TGo
(6.120)
which in the general case differs from TG 2 . We note that .C(DG 2 ) = TG2 holds in the
case where the relaxation time does not depend on frequency and polarization.
The order 0 boundary condition was obtained in section 6.2.1 by noticing that
the order 0 distribution matches the distribution emitted by the boundary with no
boundary layer correction. Introducing time dependence does not modify this result.
135
Therefore the Dirichlet boundary condition (6.33) remains unmodified at order 0 in
the time-dependent case. At order 1, we showed that the jump boundary condition
emerges from the analysis of the boundary layer correction required by the mismatch
between the order 1 bulk distribution and the boundary emitted distribution. As
before, time-dependence does not modify the form of the order 1 bulk distribution.
Therefore, the order 1 jump condition (6.43) remains unmodified in the presence of
time dependence.
We now examine the case of diffuse reflective boundary conditions. We showed in
section 6.2.2 that an order 1 boundary layer analysis results in the order 0 condition,
stating that the- normal temperature derivative to the wall must be 0.
Since this
analysis only involves order 0 and order 1 terms, it remains unchanged in the timedependent case.
In section 6.3.2, the order 1 condition was derived by analyzing
an order 2 boundary layer problem.
Applying the analysis to the time-dependent
case first requires to adapt equation (6.66) in order to include the order 2 terms
introduced by the time-dependence.
However, as before, the terms TG 2
xb
cancel
in the time dependent case, and the analysis will therefore yield the same order 1
boundary condition (6.71), with the same coefficient
'}
defined by Eq. (6.71).
This shows that the theory developed in this chapter may be applied to (diffusely
scaling) time-dependent problems up to order 1, with the only change being that the
Laplace equation is replaced by the unsteady heat. equation (6.117).
6.5.1
Application to a transient problem
To illustrate the conclusion obtained in the previous section, we consider here a square
particle heated up to 301 K and placed in a thermal bath at 300 K. We assume that
the boundaries are well described by prescribed temperature walls at To = 300 K.
We also assume that the Knudsen number is small and we therefore calculate the
temperature field inside the particle by solving the heat equation
at
=
cxV2T
136
(6.121)
with the "jump-type" boundary conditions at the walls, written in an implicit sense
as explained in section 6.4.4
T(x
= Xb)
- To
=
c1(Kn)
(6.122)
On
In Figure 6-12, we show the results AT = T(x = (0, 0), t) -To
obtained by solving this
problem with a finite difference scheme and compare it with results obtained using the
adjoint Monte Carlo method presented in Chapter 5. We also show the solution obtained by solving the heat equation with Dirichlet boundary conditions. The timestep
and the discretization were chosen fine enough for the finite difference results to appear fully converged. The material model adopted here is that of silicon with single
relaxation time (c, ~ 1.13), and the Knudsen number is 0.1. Clearly, the asymptotic
solution achieves better accuracy than the solution of the heat equation with Dirichlet
boundary conditions. This example shows that, for time dependent problems with
Knudsen number of order 0.1, one may obtain solutions that are consistent with the
Boltzmann equation by solving the heat equation with appropriately modified boundary conditions. This method therefore provides a nuance to the relatively common
claim that size effects in nanoscale systems manifest themselves through a decreased
thermal conductivity [621: we showed here that the thermal conductivity is unmodified, but that the additional resistance to the flow of heat is caused by the modified
boundary conditions which slow down the thermal relaxation of the system.
6.6
Discussion
We have presented an asymptotic solution method for solving the Boltzmann equation
in the limit Kn < 1 for steady problems. The resulting solution provides governing
equations and boundary conditions that determine the continuum temperature and
heat flux fields in arbitrary three-dimensional geometries. Our results show that the
equation governing the temperature field is the steady heat equation. Although this
can be shown by the usual kinetic theory analysis [7,8] (expanding the distribution
137
(0,0)
301 K
300 K
-6
+S
b08
-10
(b)
(a)
-12 -
0
* Adjoint Monte Carlo
-Heat equation
Order 1solution
-
0.2
0.4
0.6
0.8
t(s)
1
1.4
1.2
x 1o
Figure 6-12: (a) The problem is a square particle with side L = 10(A). The temperature is calculated at the centre point (0,0) (b) Temperature at the center of the
square after initial heating.
function about the local equilibrium and giving no consideration to boundaries), the
novel contribution of the present chapter is the derivation of boundary conditions that
complement this equation so that the resulting solutions of this system are rigorously
consistent with solutions of the Boltzmann equation and explain the temperature
jumps at the boundaries previously observed and remarked upon [631.
The emergence of the heat equation as the governing equation is a manifestation
of the fact that, in the Kn < 1 limit, the bulk description remains unperturbed and
kinetic effects only appear at the boundaries. Specifically, one can show that, to first
order in Kn, the heat flux in the bulk
JG1
Tc:xTGV
-Pj
(6.123)
is given by the same constitutive law derived by standard kinetic theory [7,8]. Kinetic
effects near boundaries can be captured by solution of the heat equation subject to
jump boundary conditions at order 1 in (Kn) and higher, as well as the addition of
boundary layer functions to the temperature field and the heat flux.
Note that in these solutions as well as the solution presented here, there is no
evidence or justification for modifying the material constitutive relation (thermal
conductivity); in other words, in these descriptions the thermal conductivity is always
138
equal to the bulk value and the modified (reduced) transport rate associate with
the size effects is a result of the additional resistance associated with the boundary
effects as well as kinetic corrections that are to be linearly superposed to the final
heat conduction result.
Although the agreement of the first-order result at (Kn) = 0.1 with simulations
was excellent, this may have been a result of the simplicity of the problem considered
here.
The asymptotic procedure outlined here can be extended to higher order in
(Kn) to obtain, in principle, an even more accurate asymptotic description. Studies
in rarefied gas dynamics
[591 show that second-order asymptotic formulations are
reliable to engineering accuracy up to (Kn) ~ 0.4 and in some cases beyond.
139
140
Chapter 7
Two applications of the asymptotic
theory
In this chapter we present two applications of the asymptotic theory presented in
Chapter 6. In section 7.1, we show how the asymptotic theory can be used to calculate the Kapitza conductance (and temperature jump) associated with the interface
between two materials. In section 7.2, we show how the asymptotic solution for the
distribution function in the Kn < 1 limit can be used as a control for extending the
range over which deviational simulation methods remain efficient and thus making
them well suited to applications involving multiscale problems. This work is based
on Refs. [46,52,53], where this material appeared first.
7.1
Application to interfaces
The theoretical and numerical considerations presented in the previous chapter are
quite general and can be extended to a variety of problems where boundaries introduce "size effects" by injecting inhomogeneity into the problem. A classic example of
such a problem is the interface between two materials: the presence of the interface
results in a temperature jump, already shown in this work to be the signature of
kinetic correction required due to the inhomogeneity associated with the presence of
a boundary. In this section we show how the asymptotic theory enables us to rigor-
141
ously relate the
Kapitza conductance to the kinetic properties of the interface (e.g.
reflection transmission coefficients). Our aim here is not to conduct an exhaustive
study but rather to demonstrate the applicability of the ideas presented earlier. As
a result, we will focus on one specific transmission model and the single relaxation
time model. We assume the following:
- The interface separating the two media, denoted a and b, is sharp (infinitely
thin) and planar.
- When a phonon encounters the interface, it is either reflected or transmitted.
In either case, its traveling direction is randomized while it keeps the same frequency and polarization. We denote the transmission probability from material
a to material b by Xab, while Pab
1 -- Xab denotes the probability of reflection
at the interface while traveling from a to b. Similarly, Xba and Pba denote the
transmission and reflection probabilities for travel from b to a, respectively.
In what follows, we will use
Ta, C,,p,a,
Vg,a and
C,,p,b, Vg,b
Tb,
to denote the relaxation
time, frequency-dependent specific heat and magnitude of the (frequency and polarization dependent) group velocity in materials a and b, respectively. As before and
without loss of generality, let us align the interface with the x 2
-
x 3 plane (at xi
=
0)
and let the positive x 1 direction point from material a to material b. In this notation,
the kinetic boundary condition associated with the interface is given by
Vg,b
4
b
=
-+=
-
Xab
aIx=O-
Pba. q-|b+xi O
4,pa Vg,aQd 2
'
Vg,bQd 2
a<0O 47
(7.1)
+f< [0 ab 2 ;a+X I0- Cw,p~b V
1>0
yd 2 n
71
47
where superscript "+" (resp. "") refers to particles moving in the positive (resp.
negative) x, direction.
142
The order 0 solution in each material phase is solution to the Laplace equation
V7TG0 = 0 with the condition
Replacing 41a,=0-
TG0, aI0-- = TGob
and Db|x,=0+ by TGO
x 1 =0
1=
=
TGOxl=0 at the interface.
in (7.1) and performing the integrations,
we obtain:
Vg,bC,,p,bTGOxx=0 = XabTGO x1=0Cw,p,aVg,a + PbaTG0 x=0Q,p,b g,b
g,aCw,p,aTG0 x1=0 =XbaTG0 x=0CL,p,blg,b + PabTG
(7.2)
x1=0C0,p,a 1 g,a
which implies
0 = XabCw,p,aVg,a - XbaCw,,bVg,b.
(7.3)
The principle of detailed balance guarantees that, the above is true for all L, p. Note
that the condition
of (G0
xi=0 =
TGOa
TGO
21=0-
= TGo,bjx=0+ = TGo
does not determine the value
x,=o
21=o. The additional required condition is given by heat flux
continuity. In other words, the additional condition reads:
K TGO
a,
OTGO
xj=0-
0X 1
(74)
x1=0+
Following the procedure of section 6.2.1, we find that the order 1 solutions
{ 1,a =T1,
Rna
-Kn>"
-VTGO,a
(Kn
1)1,b
T1,b -
(Kn)
nb
(7.5)
V TGO,b
cannot satisfy condition (7.1) without the introduction of boundary layers.
Here
Kn denotes Vg,ir/L, while (Kn) is a "reference" Knudsen number calculated from
the properties of one of the two materials (results are independent of the chosen
reference).
143
We introduce two boundary layer functions T'Ka and
and
Cb,
XPKb, and
two constants
Ca
anticipating temperature jumps at the interface of the form
T1,a x1=0~
Ca1 T 0 a
-
(7.6)
OTO,b
Ti~ XJO+ b
nb
x,=O+
Limiting our analysis to variations only in the x 1 direction, we insert the order 1
solution (boundary layer included) in condition (7.1), to obtain
C'
Vgb '
(-KnbQl + Cb(Kn) + 'PKb(
Xab
-
0
2 ,p,a Vg,a
n))
xi=0+
T
Ob
Ox 1
xt0
-Kna Q' - ca(Kn) - qIKa(hn)) xi=0-
00 ,a
2Oxi
Vg,b (-KnbQ' + Cb Kn} + T Kb Kn)
Pba
-
x1=O-
'
=
+
1
1
0
(-KnaQi - ca(Kn) Xba
2 Vg,b (-KnbQl
''Vga
2aOXa
(-Kna
XIKa(Kn))
x- X
+ CbKKn) + 4'Kb(Kn))
i - ca(Kn) -
=O
Ob'd'
O'
x1=0+
I'Ka(Kn)) 1,0-
1=0+
Q'dQ'
T0 'a
1
1=
-
Vg,a
Q
0,b
xi=0+
-I
(7.7)
We then solve this boundary layer problem numerically, using the procedure outlined
in Appendix B, with a few key differences that we outline below:
-
The frequency range depends on the material type. Consequently, the number
of computational cells in each material differs. We refer to the number of cells
in material a as Na and the number of cells in material b as Nb.
-
The two boundary layers are coupled and solved for together. In their discretized
144
form, they are written as:
Na/2-1
FKa
Aiha,i exp(Aa,iT)
=
(7.8)
Nb/2-1
=
Bihb,i exp(Ab,i 1
)
Kb
i= 1
where Aa,i (resp.
Abi)
are the Na/2
-
1 (resp. Nb/2 - 1) strictly positive (resp.
negative) eigenvalues of the scattering operator in materials a (resp. b).
-
Inserting expressions (7.8) into the discretized version of (7.1) results in a system
of Na/2 + Nb/2 equations with Na/2 + Nb/2 unknowns. However, the matrix
of the linear system is not invertible because, due to identity (7.3), the two
columns corresponding to unknowns Ca and
This
are linearly dependent.
Cb
indeterminacy is corrected by noting that the temperature jump condition
(
T1,b - T,a
Ca
',a
OX 1
x,=0-
+
'Ti
Cb
xi=O+/
OX 1
KKn)
(7.9)
needs to be supplemented by the continuity of heat flux across the interface
OTGL,a
a
=
__G_,b
b
1 2=0Ox
I1
X
210+
(7.10)
In other words, the jump condition can be written as
T1,b - T,a =
with
= Ca/2a
C
CIa
x
XJ'=
(Kn)
(7.11)
+ cb/b being the only unknown coefficient. As a result, we
replace the column vectors corresponding to ca and Cb with a single column
vector in the unknown
2
and discard one of the equations and solve for the
system of Na/2 + Nb/2
-
1 equations with Na/2 + Nb/2 - 1 unknowns.
145
7.1.1
Validation
We test the asymptotic solution method outlined here on a simple one-dimensional
problem with the following features:
- The total length of the system is 2L. The two materials are aluminum (-1 <
x 1 < 0, hence, material a) and silicon (0 < xi 5 1, hence, material b).
We
use IA1 and Isi to denote the range of frequencies of the two material dispersion relations, respectively. To facilitate the verification of the accuracy of the
method, we use a constant relaxation time model in each material; specifically,
we take
Ta
= 10-11s in Al and
Tb
= 4 x 10-" s in Si.
- A temperature difference of 1K is applied across the system by imposing a
prescribed temperature of 301 K at x 1 = -1,
while the boundary at x=
1 is
maintained at 300 K.
- We define (Kn) as the ratio between the mean free path in the silicon phase
and L. We choose L such that (Kn) = 0.1.
- The phonon transmissivities at the interface x,
described in
0 are adapted from the model
=
1391, which given a "target" interface conductance G (as input),
predicts
2
JLCAI2?Si,P Cw,p,AIVg,AI
,Xab
+
1
TCaIAIPPAI~A
for frequencies in
'Al
n
'Si
1
+1
(.2
W7c7iv77
(0 otherwise). Coefficients Xba are deduced from the
principle of detailed balance.
Our numerical solution is shown in Figure 7-1. The figure compares the temperature profile obtained with the kinetic-type deviational Monte Carlo method to the
order 0, order 1 and "infinite" order asymptotic solution. The order 1 solution provides significant improvement with respect to order 0. Due to the one-dimensional
nature of the problem studied here and the absence of higher than first-order derivatives of temperature in either material, an "infinite" order solution is possible: it
146
can be obtained by solving the following system of four equations in four unknowns
(TAI(xil = -1),
TAI(xil
1 - TAI(Xl
=
0-), Tsi(xi
-1)
= CAl(Kn)
0+) and Tsi(xi = 1))
(TA(xl
=
Tsi( 1 = 0 ) - TA1(Xl = 0-)
&TA1
Aj
O
0) - TA(xl
-1)
c(Kn),'css i
OTAIa~siOx,
T~
Si O
=0+
1=0+(7.13)
Tsi(i = 1) = csi(Kn) (T(x 1 = 0+) - T(xi = 1))
After adding the corresponding boundary layer functions we find that this solution
agrees very well with the Monte Carlo result. Using this model, we obtain the actual
conductance value G = 108 MWm-2K-1, which is very close to the "target" value
110 MWm-2K-
1
used as input to the model described in [39]. Perhaps more impor-
tantly, we note that the MC simulation also predicts a conductance value (obtained
by extrapolating the bulk temperature profiles in order to calculate the temperature
difference at the interface) of 108 MWm- 2 K- 1, which is in perfect agreement with the
asymptotic result. By comparison, the diffuse mismatch model predicts an interface
conductance of G = 343 MWm-2K-1. This is consistent with the fact that the diffuse
mismatch model results in an upper bound for the interface conductance [64].
We note that the "infinite" order solution may not. be available in the general,
higher-dimensional case. Related treatments of "connection" problems associated with
different carriers have appeared in [65-68].
7.2
Implementing asymptotically-derived controls through
the adjoint approach
Previous work using spatially variable controls for improved variance reduction [21,
361 utilized the local equilibrium-based on real-time (cell-based) estimates of its
parameters-as a control. One drawback of this approach is that the resulting discontinuities in the control (at cell boundaries) require particle generation at cell
147
-- order 0
- - - order 1
-order
o
0 Monte Carlo solution
0.8
0.6
-0.4
0.2
-1
0
-0.5
X
0.5
1
1
Figure 7-1: Temperature profile in a 1D system with an Al/Si interface
boundaries, which becomes cumbersome in higher dimensions. Here, we introduce
a new approach which uses asymptotic solutions of the Boltzmann equation as controls and show how the adjoint formulation can make such approaches more efficient
as well as simpler to code. This section considers steady state problems only, although
extension to transient problems will be considered in future work.
7.2.1
Spatially variable control temperature in the adjoint
framework
We first consider a variable control of the form econtroi(x) = eq + To(x)&eq
/&T
which results in the following equation governing ed:
Oed
L(ed)
-
+ V9 * 17"ed =V
ed
_e(7
- 17XTO
(7.14)
""
As explained in Chapter 3, under these dynamics, particles are emitted from the
at the system boundaries.
Tbaeeqeq /T.
In the linearized setting, we may write
Thus, eb(xb) - econtrol(Xb) = (T - TO(xb))Oe'e~q /T.
-
econtrol(xb)
eb(xb) =
e'
+
distribution :Vg * VITo in the bulk and from the distribution eb(xb)
To simplify the
discussion, we will assume that, by choice, TO(x) obeys Dirichlet boundary conditions
for prescribed temperature boundaries and Von Neumann boundary conditions for
reflective boundaries.
This allows us to eliminate the boundary effects from the
148
present discussion, although extending the conclusions of this paragraph to more
general choices of To is straightforward.
Drawing particles from
Vg V-To can be a significant programming burden if the
spatial dependence of To is complicated. Let us explore the implications of applying
an adjoint approach to this situation; we consider here the steady state case. Particles
are emitted from the detector function, while the quantity of interest is given by
[-Vg - VTo(x)] Eb*dwd2
I =
d 3xdt.
(7.15)
The contribution of a particle, with weight 4f, to the final estimate can be written
as
edV9
'
VxT (x(t))dt
(7.16)
-TN =eff
/tend
t=0
-
where the time t is only used formally to parametrize the line integral (see Refs. [2,5]).
In other words, the value of the line integral does not depend of the direction of the
time parametrization, which is consistent with the fact that the time is absent from
the steady state adjoint equation. We may considerably simplify this expression by
introducing the particle coordinates at the scattering points xi. Since the trajectory is
a series of Nseg linear segments delimited by the points xi, expression (7.16) becomes
i=Nseg-1
4
IN
ff
(TO(Xi+1 )
E
-
TO(Xi)).
(7.17)
i=O
The fact that particles travel in the opposite direction of V9 is important for deriving
the above expression, since the line integral over a segment may be written as
j-V- VxTO (xi
- (t - ti)Vg) dt
-
[TO (xi - (t - ti)Vg)] +1
(7.18)
-
To(xi+1) - To (xi)
(7.19)
Equation (7.17) straightforwardly simplifies into
IN
=
4
ff
(TO (XNseg)
149
-
To(Xo))
(7.20)
This result seems powerful in the sense that, instead of drawing particles from the
source distribution as in the forward case, in the adjoint case we only need to calculate
the value of To at the emission and termination points (with the latter usually given
by boundary conditions); this represents a considerable simplification.
However, this result needs to be put into context by comparing with the case
of the adjoint algorithm with fixed control.
In the case of fixed control (and no
other sources-i.e. the same problem studied above) the source term only includes the
prescribed temperature boundaries. In other words, the same development as above
leads to
IN
=effTO(XNseg)
(7.21)
which only differs from (7.20) by the term To(xo). The latter term is usually fixed
(when the estimate is calculated at one point only).
This means that, although the adjoint formulation led to considerable simplification (removing the need to sample the source term in equation (7.14)), the statistical
uncertainty of the adjoint formulation with spatially variable control is not smallerin fact, it may be higher-than the statistical uncertainty of the adjoint calculation
with a fixed control. In other words, the adjoint formulation with the source term
esource
soc
=ej
=eeq + To(x)&e Teq /0T is not expected to provide improved variance reduc-
tion compared to the adjoint formulation with a fixed control, in contrast to forward,
time-step based algorithms were additional variance reduction is observed when a
(suitably chosen) spatially variable control is used [36]. On the other hand, the control
esource=
eq + To (X)aeCT /T
remains useful for imposing a temperature gradient
for effective thermal conductivity calculations
13].
Fortunately, significantly reduced variance is indeed possible with a spatially variable control within the adjoint formulation. In fact, as we show below, the improved
variance can be achieved while retaining the simplification resulting from avoiding the
generation of samples from complex distributions. Such formulations are discussed in
the following section.
150
7.2.2
Asymptotic control for multiscale problems
Let To(x) be the solution of Laplace's equation with ad hoc boundary conditions. We
know from Chapter 6 that
ed(x)
=
eq
[To(x) + (Kn) (TK(x) + TG1(x)) - TV, - VTo]
(7.22)
is a first-order asymptotic solution of the Boltzmann equation in the expansion parameter
(Kn)
(assumed small) and subject to arbitrary kinetic (Boltzmann) boundary
conditions.
Here, we adopt a heuristic approach which amounts to including the most readily
available first order term of (7.22) in the control, namely choose
econtrol = eo - VgT
VT
By noting that L(VgF - VTo) = 0, the BTE for ed
(7.23)
"
e - econtrol can now be written
=
V d L (ed) - ed +
Vg - Vxe=
Oe r
TV9 * V (Vg
- VTO(x))
Tq
OT
(
in the form
(7.24)
The source term that appears is composed of all the second order derivatives of To.
It can be explicitly written as the double sum
02T
V
2
T
.
ZQiQj
(7.25)
Drawing particles from such a distribution as is required in forward frameworks is
very challenging. In addition to this volumetric source, other source terms appear at
the boundaries, from the mismatch (anisotropy) between the control and the boundary condition.
For instance, for a prescribed temperature boundary, the modified
boundary condition reads:
e(,,T
-n >0) =V9
-V.T
(7.26)
151
On the other hand, in the case of the adjoint formulation, for the source given in
(7.25), and using the same procedure used for (7.15) to (7.20), the contribution of
trajectory (particle)
j
can be shown to be
i=Nseg-1
Ii =
eff
E
TiVg,
- (XTo(xi)
- VXTo(xi+1))
(7.27)
i-O
where
Ti
and Vg,i respectively refer to the characteristic relaxation time and the
velocity vector of the particle on segment i. A source term of type (7.26) is treated by
adding
VgT
-VT(xNseg)
to the contribution. This cancels the last term of expression
(7.27) for i = Nseg - 1 (all trajectories terminate at the boundaries).
Finally, we need to recall that the final estimate will be obtained by adding the
stochastic estimate to the deterministic value derived from the control. The deterministic value for the temperature is To. In the case of the heat flux, the deterministic
heat flux associated with the control is
7.2.3
kbuikVxT.
Validation and accuracy
In this section we validate the method described in the previous section 7.2.2, which
we will refer to as asymptotically controlled adjoint (ACA), using the two dimensional
problem already considered in the previous chapter and depicted in Figure 6-10. We
consider an infinitely long slab of material of thickness 2L.
We denote the non-
dimensionless coordinate in the infinite direction by x, and the coordinate in the
other direction x 2 . At x 2 = L the material is held at a prescribed (deviational)
temperature Tb
T cos(27x1/(3L); at x 2 = -L
given by Tb = -Tw cos(27x 1 /(3L)).
the deviational temperature is
By linearity, the discussions that follow do not
depend on Tw; all calculations were performed with Tw = 1 K.
The system was chosen because the analytical solution To(x) of Laplace's equation
can be obtained analytically:
To (xi,x 2 )
=
Tw cos
sinh
( 27x,3Lsinh
152
( 2,)
(7.28)
(2y)
This will allow us to focus our validation on the stochastic error only.
1.2
Uniform control
ACA method
1 0.8
Analytical solution of
Laplace's equation
0.6T(K)
-
0.4
0.2 -
-
'(Kn)
0.1
0
-0.2
(Kn) =0.5
0
0.6
0.4
0.2
0.8
1
X2
Figure 7-2: Deviational temperature profile along the segment AB in Figure 6-10.
Comparison between the solution to the BTE calculated using the adjoint method
of section 5.2.1, and the ACA method of section 7.2.2. The analytical solution of
Laplace's equation is also shown.
This phonon transport problem can be easily solved using either the forward
Monte Carlo method or the adjoint method. In Fig. 7-2, we show the temperature
calculated on 51 equispaced points of the line parametrized by x,
=
0 and 0 < x 2
<
1,
using both the adjoint method with uniform control, and the ACA method presented
above, for (Kn) = 0.5 and (Kn)
=
0.1. In the interest of simplicity, both calculations
used the single mean free path model (constant relaxation time and Debye model); in
other words, A,
=
VgT
=
A = constant. The agreement between the two methods
is excellent.
Figure 7-3, shows the statistical uncertainty associated with the calculation of the
x 2 component of the heat flux at point (0, 0).
It clearly reveals that, in the ACA
method, the standard deviation scales linearly with the Knudsen number, while in
the adjoint method with uniform control it is approximately constant. In appendix
E we provide a mathematical explanation for the scaling observed in the ACA case.
In particular, we highlight the major difference that arises, in terms of statistical
153
properties, when a temperature field other than the
0th order solution, To, is used as
control and show that using a solution of Laplace's equation is key to this result.
This result is of great importance for multiscale simulations because it means
that, for a fixed uncertainty, low Knudsen number systems (large lengthscales) can
be simulated using the ACA technique at a fixed computationalcost. This follows from
the fact that the cost of computing a single particle trajectory increases proportionally
to (Kn)-
2
as (Kn) -- 0, since transport follows a diffusive scaling in this regime. On
the other hand, the statistical uncertainty scales as
u/vHY, where a is the standard
deviation of particle contributions to the estimate and N is the number of particles
used; therefore for a fixed statistical uncertainty the scaling o- oc (Kn) requires N oc
(Kn) 2 . Since the overall cost per simulation scales with the product of the number of
particles times the cost of a single trajectory, we obtain a constant cost. In contrast,
the cost of methods which have a constant statistical uncertainty as a function of (Kn)
(such as traditional MC methods, as well as forward deviational methods) increases
as (Kn)-
2
in the (Kn) -- 0 limit, a manifestation of the kinetic description becoming
stiff in this limit. In other words, the ACA formulation overcomes this stiffness and
results in a computational method that can simulate large systems as efficiently as
small systems, a highly desirable feature of any multiscale method [22].
In fact,
for classes of problems for which the cost of computing a single trajectory scales
as (Kn)', such as the case of solving for the heat flux at a location close to the
boundary' discussed in the next section, the cost of the ACA formulation is expected
to scale as
(Kn) and therefore decrease as lengthscales increase.
We note that the above features pertain to problems for which the adjoint method
is primarily suited for, namely problems in which the solution of interest is the transport field in a small region of space (see section 5.4 for further discussion). It should
also be noted that the above scaling estimates for the cost refer to the case where
the acceptable uncertainty level is prescribed in an absolute sense. In some cases, for
instance when calculating the heat flux (which is formally a quantity that also scales
'This scaling may be arrived at by applying the optional stopping theorem to the martingale
representing the X2 -coordinate of the particle position 169]
154
with (Kn) [531), it may be more appropriate to consider the uncertainty in a relative
sense - namely, the ratio between the uncertainty and the calculated heat flux. A
multiscale method that features a constant cost with respect to the relative uncertainty would have to use a control which includes the higher order terms presented
in Chapter 6 and is a direct extension of the ideas developed in the present section.
Figure 7-4 shows the uncertainty associated with the calculation of the x 2 component of the heat flux at points (0, 0) and (0, L) when using a material model with
frequency-dependent free path (for a description of the material model see Appendix
F). This figure reveals that, when the variable free path model is used, the computational advantage associated with the ACA method is beneficial only for low Knudsen
numbers ((Kn) < 0.02 in 7-4a and even lower for 7-4b). The reason for the breakdown
of the efficiency for large Knudsen numbers lies in the fact that the contribution (7.27)
to the estimate is a sum of terms that scale with (Kn). While the variance reduction
stems in part from this fact for low Knudsen number, this becomes a hindrance in
the ballistic limit. We note that even at these small Knudsen numbers significant
discrepancies exist between the Boltzmann and Fourier solutions, and thus Boltzmann solutions are still necessary; this is further quantified in the following section,
which also lays out an approach for recovering and in fact enhancing some of the
computational benefits lost in the presence of widely variable free paths.
7.2.4
Using a "hybrid" control for models with widely variable
free paths
In section 5.3.2, we showed that the adjoint method is well suited to the case where
mean free paths cover a wide range and when we seek to calculate the contribution
of each individual mode to a given quantity of interest (typically, the heat flux or
the thermal conductivity). On the other hand, as was shown in Figure 7-4, in the
presence of a large variation in free paths, the benefits associated with the ACA
method become significant only for very low knudsen numbers. In this section we
show that this limitation can be overcome with a slight modification of the ACA
155
1010
0
00 3
0
100
10
-0- Uniform control
7
---ACA method
X2
.
10
10 -3
- I
10 0
10-1
10-2
(Kn)
Figure 7-3: Standard deviation o-,,2 of the particle contributions to the estimate of
the heat flux at point A of Figure 6-10 in the X2 direction, in the single mean free
path model. The standard deviation is proportional to (Kn) with the ACA method.
The latter outperforms the adjoint method with fixed control for (Kn) < 0.2.
formulation; in fact, the modified formulation, referred to here as hybrid, improves
the performance of the ACA formulation for all (Kn) at almost no additional cost.
4&,
In order to motivate the hybrid version, we first explain why the ACA formulation
fairs poorly as (Kn) increases. When this method is applied to a phonon model with
highly variable free paths, such as the one described in Appendix F, the terms that
compose the sum (7.27) span several orders of magnitude. The presence of phonon
modes with large free path (exceeding 100pm) tends to increase the variance of the
estimate because the prefactor V.7- becomes comparatively large. We can overcome
this limitation by adapting the expression of the control. Since the problem is caused
by the prefactor V9r which appears in (7.27), we propose a control which uses the
order 1 formulation for the small mean free path modes only, namely
econtrol = eo - 1Kn,,PcV r- - V.To.
OBeq
(7.29)
.,
In words, according to this definition, the "hybrid" control uses the order
modes with large free paths (A
> cL), while it introduces the order
156
0 solution for
term for small
free paths (As,, < cL). There is some degree of freedom in the choice of the constant
c, although some trial-and-error revealed that, for the model that we used and the
problem tested, c ~ 0.4 is close to optimal. Repeating the derivation procedure of the
previous section, we can show that, algorithmically, the adjoint routine stays nearly
the same, apart from the two following changes:
- The contribution of a particle
j
to the estimate is now:
i=Nseg-1
j;
= 4eff
Z
i=O
(7.30)
-Fi
where
{To(xi+i) - To(xi)
riV,, - (VXTo(xi) - VXTo(xi+1))
if A,, -> cL
(7.31)
if AW,, < cL
Similarly to the ACA approach, the second case of formula (7.31) must account
for the mismatch between the control and the boundary conditions by adding
effTNseg
1
Vg,Nseg1 VxTO(XNseg)
if the particle encounters the boundary with a
mode obeying the criterion As, < cL.
- The deterministic quantity associated with the final estimate needs to be calculated using the hybrid control.
We emphasize that implementing these changes only requires minor modifications
since the core of the algorithm, i.e. the calculation of particle trajectory, remains the
same. Only the values assigned to the estimates change. In fact, in the comparison
of the three approaches in Figures 7-3 and 7-4, all results were obtained using the
same random numbers (all three methods were evaluated using the same particle
trajectories).
These results show that the hybrid method outperforms the two other approaches
for all average Knudsen numbers. The amount of computational savings is however
problem dependent. The results presented in Fig. 7-4a correspond to a favorable case
where the heat flux is calculated at a surface point (point B in Figure 6-10). In this
case, the length of a trajectory - and therefore computational time per particle - is
157
proportional to
(Kn)- 1 enabling us to accurately resolve the asymptotic behavior of
the standard deviation of the solution all the way to (Kn) = 0.001 (in general, the
standard deviation of a given quantity, as a higher moment of the distribution, is
more expensive to resolve than the actual quantity).
At the crossing point of the constant-control and the ACA method, (Kn) ~~0.02,
the hybrid approach already reduces the standard deviation by a factor of 10, which
corresponds to a speedup of around 100.
Such a Knudsen number might appear
small in the sense that kinetic effects might be expected to be negligible at such
scales (10pm). As noticed in Chapter 6, this point of view would be incorrect since
the free paths of low-frequency modes, known to significantly contribute to the heat
flux [501, do not behave diffusively. Our calculations corroborate this claim; we find
that, at this scale, the normal heat flux near a boundary still differs from the heat flux
calculated using Fourier's Law by 30%. At (Kn) ~ 0.002, the speedup is close to a
factor 2000 (standard deviation improvement of almost 45) and, although the system
is quite close to the diffusive limit, we find a difference of almost 10% with respect to
the Fourier solution. In addition, kinetic effects near boundaries and ballistic effects
near interfaces [53] can not be captured by the Fourier description at any (Kn).
The results presented in Fig. 7-4b show a less favorable case in which the heat flux
is calculated in the middle of the domain (point A in Figure 6-10). In this case, the
length of particle trajectories is proportional to
(Kn)-2, making accurate resolution
of the standard deviation in the solution more expensive. As a result, we are unable
to study the asymptotic behavior of the standard deviation of the ACA method for
(Kn)
$
0.01. We also observe that the asymptotic behavior of the hybrid method for
(Kn)
-
0 has not reached the expected o- oc Kn. We attribute this to the presence of
a wide range a free paths which causes some phonons to behave ballistically even at
these small Knudsen numbers (at this Knudsen number, the discrepancy between the
Fourier solution and the simulation result is 13%) delaying the onset of the asymptotic
behavior. As Fig. 7-4a shows, the scaling o- oc (Kn) is still valid for the hybrid case
at a surface point (for a discussion of the effect of the dependence of trajectory length
on (Kn)-(Kn)- 1 vs (Kn)- 2 -as well as the validity of the mathematical justification
158
to the variable free path case, see Appendix B). We finally note that despite the
fact that the hybrid approach has not reached the asymptotic regime at the smallest
Knudsen number considered here, (Kn) ~ 0.001, the speedup provided by the hybrid
method is appreciable, namely a factor of 25. In Figure 7-5, we show similar plots
for artificial models where we deliberately decrease the maximum mean free path of
the model. This shows how the onset of the order 1 asymptotic behavior is delayed
as the maximum mean free path increases.
The hybrid approach, which takes advantage of the fact that the modes with small
free paths behave diffusively, shares connections with the work in [27], in which these
modes are indeed assumed diffusive. The key difference is that in our method, apart
from the inherent stochastic error arising from the statistical uncertainty, no further
approximation is made. To wit, the proximity of the diffusive regime for these modes
is used for switching between two modes of variance reduction (for solving the exact
same equation); diffusive behavior or particular modes of interaction between the
long- and short-free-path modes is at no point assumed.
We showed that by using a control inspired by asymptotic solution of the Boltzmann equation, steady state problems of arbitrarily low Knudsen number can be
treated at constant cost. This last feature results from an absolute statistical uncertainty that is proportional to (Kn) (see Appendix E). The associated quadratic
savings balance the quadratically increased cost caused by the calculation of longer
trajectories in the low Knudsen number limit. As a result, simulations of structures
or devices with lengthscales ranging from nanometers to hundreds of microns (see
Figure 7-3) are not only possible, but also efficient.
In this section, we only used the most readily available asymptotic solution of
the Boltzmann equation, namely, the order 0 temperature field and its gradient. We
expect that using higher order approximations, as derived in [531 and in Chapter
6, would contribute even further to reducing the cost. Including such higher-order
terms would be very complicated in a forward particle Monte Carlo.
It would be
close to straightforward in the adjoint framework since, as already demonstrated in
this section, only the values assigned to the particle contributions would be modified,
159
while the adjoint particle trajectories would remain unchanged.
1010
1011
1010
(
--Uniform control
- - - ACA method
Hybrid method
-
-0- Uniform control
- - -ACA method
Hybrid method
0i
100
10
10
10
(a)
-
10
-2
_
103
108
10
1)
(b)
(Kn)
10-2
10
1
100
(Kn)
Figure 7-4: (a) Standard deviation -ql of the particle contributions to the estimate
of the heat flux at point B of Figure 6-10 in the x 2 direction, in the variable mean free
path model. (b) Standard deviation O-q" of the particle contributions to the estimate
of the heat flux at point A of Figure 6-10 in the X2 direction, in the variable mean
free path model.
160
10
Max A =69 [tm
Max A = 690 pm
10
10-
10
A = 6.9 pm
7ax
-+-ACA method
Max A = 690 nm
10 0
103
-
10'
102
- - Hybrid method
100
(IKn)
Figure 7-5: Standard deviation aq/ of the particle contributions to the estimate of
the heat flux at point A of Figure 6-10 in the x 2 direction, with artificial material
parameters made such that the maximum free path corresponds to different orders
of magnitudes. While the onset of the order 1 behavior is clear in the problem with
largest mean free path 690nm, the onset is progressively delayed as the largest mean
free path is increased.
161
162
Chapter 8
Final remarks and future directions
8.1
Summary
The developments presented in this thesis take advantage of the linearization of the
Boltzmann equation for phonons to devise efficient multiscale methods for solving
heat transport problems. Methods have been developed both in the stochastic and
analytical (asymptotic) domain.
Advances have been achieved by exploiting the specific properties of the linearized
Boltzmann equation in the relaxation time approximation. The ability to use independent particles in MC simulations derives from energy conservation as well as the
observation that the post-scattering particle properties (frequency and polarization)
are always drawn from the same distribution, regardless of the local temperature.
Thanks to the independence of particle trajectories, one may simulate particles sequentially without introducing time or space discretization. The KMC algorithm is
not only multiscale in space, thanks to the deviational formulation which focuses
the calculation at locations where the deviation from equilibrium is the largest, it is
also multiscale in time in the sense that it automatically adapts to a wide range of
relaxation times, in contrast to methods based on a fixed timestep.
The adjoint formulation introduced in Chapter 5 exploits the linearity of the BTE
towards an alternative way of simulating the problem of interest in which sources and
detectors are exchanged, while introducing no approximation. This is particularly
163
useful for cases where accurate estimates are required in small regions of the phase
space, such as simulating actual experimental setups since physical detectors are
typically very localized in space.
We also showed that adjoint formulations lend
themselves naturally to the use of spatially variable controls resulting in formulations
that are particularly efficient in the (Kn) -+ 0 limit in which deterministic numerical
as well as traditional MC solution methods become stiff.
The last contribution of this thesis is the development of asymptotic techniques
for solving phonon transport problems featuring small Knudsen numbers.
For the
first time, we provided a rigorous explanation for the temperature jumps commonly
observed at material interfaces, while showing the connection between the continuum
heat conduction equation and associated boundary conditions and the Boltzmann
equation.
Specifically, we showed that solutions of the Boltzmann equation (in an
asymptotic sense) can be written as a superposition of solutions of the heat conduction
equation with jump boundary conditions and kinetic boundary layer corrections in
the vicinity of the boundaries.
We note that when the range of free paths in a given material is broad - as in,
for example, the case of silicon at room temperature, the largest free path can reach
several hundreds of microns - the range of applicability of the asymptotic theory is
restricted, since it technically requires characteristic lengthscales to be larger than
the largest free path. This restriction may be understood in three complementary
ways:
- Although at lengthscales smaller than or on the order of the largest free path the
asymptotic order of convergence may not be clearly manifested, it is reasonable
to expect that asymptotically derived solutions are still more precise than the
order 0 solution (Fourier's law with no jump conditions).
- The fact that large lenthscales are required for the asymptotic order of convergence to be observed is an indication of the fact that significant kinetic effects
are still present even at these relatively large scales.
The fact that numeri-
cal solution methods are particularly expensive in this limit makes asymptotic
164
approaches particularly useful.
- As demonstrated in Chapter 7, asymptotic solutions can be used as controls for
Monte Carlo solutions. In particular, we showed that the adjoint formulation
lends itself naturally to this task, yielding a multiscale method whose computational cost is independent of the Knudsen number. We expect that by using
higher order asymptotic solutions as controls, an algorithm whose cost decreases
as the Knudsen number decreases may be possible.
8.2
Future directions
The linearization of the phonon Boltzmann equation, and the associated Monte Carlo
algorithms exhibit very strong similarities with the methods applied in the fields of
neutron transport or radiation
1351. The major difference lies in the inherent physics
of phonons, most noticeably, the large number of phonon modes involved - in comparison, radiation calculations usually consider a single collisionless mode, while neutron
calculations frequently rely on "two-group calculations" involving fast neutrons and
slow neutrons [70]-, the associated post-scattering distributions, and the broad range
of free paths. The requirement of accurate solution methods is particularly important
in the field of neutron transport, due to the highly restrictive engineering requirements
associated with the safety of nuclear reactors. Thus, very elaborate variance reduction schemes, based on importance sampling [35], have been developed and are still
being explored nowadays.
Recently, statistical inference tools, such as neural net-
works, have been explored as a means of providing increased variance reduction for
neutron transport simulations 171]. Introducing and adapting these tools to the field
of phonon transport could further improve the techniques presented in this thesis.
One limitation of the work presented here is the use of the semi-empirical relaxation time model. While this model gives good results for most common materials, it
fails to capture some important phonon behavior, namely conservation of momentum
during normal three-phonon scattering events, which appears to be important in a
number of materials, and is required for explaining thermal transport in graphene.
165
Advances towards solving the Boltzmann equation with the "true" scattering operator, calculated ab-initio from Density Functional Theory, have been made in the
recent past. Broido et al. [72-74] developed and applied very efficient methods for
solving the homogeneous Boltzmann equation with ab-initio scattering operator and
obtained computational results showing good agreement with experimental thermal
conductivity measurements. Landon and Hadjiconstantinou [75j went a step further
by developing the first Monte Carlo method simulating phonon transport problems
with an ab-initio scattering operator. However, due to the complexity of the latter,
only 2D materials such as graphene could be efficiently treated.
One extension of
Landon's work and the present work would be the development of a timestepless
(KMC-type) method that would take full advantage of the linearity of the governing
equation in order to solve 3D problems in general geometries.
Finally, the methods presented here are both efficient and robust and can adapt to
a wide range of phonon transport problems. They have the potential to considerably
assist with the interpretation of experiments, such as the transient thermoreflectance
used as an example in Chapters 3 and 5.
At its current stage, the Monte Carlo
method presented in this thesis has the potential to contribute towards a resolution
of this inverse problem. Recent works [411 tend to confirm this view.
166
Appendix A
Derivation of the governing equation
for the order 1 and order 2 bulk
temperature fields
In this section, we show that TG1 and
TG2
are solution to the Laplace equation. We
start with the case of TGI. We apply the solvability condition (6.20) to
C 9 -x
Kn2
Kn
(TG2
(Kn)
2
*xTG1
(Kn)2
(G2
Vx (Q -VxTGO))
to obtain:
20A.
(A. 1)
removes terms containing odd powers of Qj, yielding
Cn
Q2.
d2 d
= 0
(A.2)
)
Integration over d2
from which we conclude that
(A.3)
V2TG1 = 0
To obtain the Laplace equation for
TG2,
we apply (6.20) to
(G3.
After carrying
out the angular integration and cancelling terms containing odd powers of Qj we are
167
left with
(Ln)
-2TX2
i?
Z
4
6
+
i3,k,l
LQirjX&kx
xixO
TG
/o
)
Kn 2
Kn
C"V
dwd 20
=
0.
k
(A.4)
Thus, in order to show that the Laplace equation holds for TG 2 , we need to show that
(E
ini,j,k,I
QiQjQkQ1
'0T _
OxL&Xj&XkOUI
d2 Q
(A.5)
=0.
Performing the angular integration, we obtain
is
QiGjGk
94 O TGO -d2
OXiOXjOXk&XI
-
47 04TGO
5
Ox Ox
=
xV TGO =
0
(A-6)
as desired.
It appears that this procedure can be applied to all higher order terms (TG3, TG4,
etc.).
168
Appendix B
Numerical solution of the kinetic
boundary layer problem
We seek to determine
(TI) and ci that satisfy the following problem statement
'K1,1
QPK1 1
Q1
Oil
(Kn)
(
Kn
Kn
PK1,1 (Ql, W, P,7
(L('K1,1)
0) + C 1 =
lim q'K1,1(Q1, wP, 7) = 0,
where
fK1,1
l=DK1,1/(OTG/O/xl)q=O
-
'K1,1)
IKn
(Kn)
Q1,
for Q1 > 0
(B.1)
for all Q 1, w, p
and q = x 1 /(Kn) is a stretched coordinate.
Boundary layer problems of this form have been studied in the context of asymptotic solutions of the Boltzmann equation describing rarefied gas flow [58].
In the
case of gases it was shown that there exists a unique solution for this problem and
the coefficient ci [76]. The mathematical considerations leading to this conclusion are
beyond the scope of this work. We will show below that this statement is consistent
with the well-posedness of the solution method we are proposing.
In the absence of an analytical solution, a numerical solution of (B.1) based on
some form of discretization must be pursued.
Here we develop a technique which
avoids discretization of the 71 coordinate. This improves computational efficiency by
reducing the number of discretized dimensions but also avoids the error associated
169
with the truncation of the infinite domain by a finite-domain approximation.
The
resulting computational benefits are substantial, especially in the numerically stiff
case of variable mean free path models where the term Kn/(Kn) varies by several
orders of magnitude (the largest may be up to 104 times larger than the smallest)
leading to a boundary layer that. extends many mean free paths before it becomes
negligibly small (see Figure 6-3 for an example).
The method requires discretization of the frequency and angular coordinates. In
this work we used the following discretization:
- Q 1 is discretized into 2v segments of equal size in the range [-1, 1]. The center
points of the segments will be denoted Q 1,j for i = 1,
...
, 2v.
- Although accounting for the different phonon polarization modes is necessary,
we can, without any loss of generality, treat the frequency discretization of the
different modes as one single discretized range, with N, cells of length Awy and
centered on Lc;j, for
j
= 1, ... , N,. The corresponding values of the density of
states, relaxation times and mean free paths are evaluated at
LjU
and denoted
respectively Dj, Tr, and Aj.
Let us denote Q(r/)
=
['l (q/), i 2(1),
approximation of function
qFK1,1
... , lr2N(7)]'
in the phase space, where 2N
(j+N L(i -1))
For conciseness, we will write
the vector representing the discrete
= PK 1,1 (r/, Llj,
=
N, x 2v and
(B. 2)
Q11
as Pjj. The discrete form of equation (6.29)
<(j+N,(i-1))
is
J8i- 077
=
A)
AjQi
(A
\l"r
D'
D m
C2,Tm
de'q
m
e
dT
-
(B.3)
ij
/Ar
We write equation (B.3) in the form
DY
Ox
= MY
170
(B.4)
where
A A
(D
i,
2KO 2 mdT
Wrn
j deq
-
jm
(B5)
(
1
M(ij),(I,m) = A
The general form of the solutions of equation (B.4) can be found by calculating the
eigenvectors of M and the associated eigenvalues.
Due to the problem symmetry,
we expect that if A # 0 is an eigenvalue of M, then -A
is also an eigenvalue (since
values of the parameter p appear in pairs tp). We therefore expect an even number of distinct eigenvalues and eigenvectors. We also know that 0 is an eigenvalue,
corresponding to the uniform solution.
As a consequence, we find at most N - 1
eigenvalues of a given sign. Here, we are interested in the eigenvectors corresponding
to the negative eigenvalues, since positive eigenvalues A lead to solutions of the form
exp(A77) which diverge for y -
oc. Similarly, the uniform solution is not considered
since the solution is assumed to converge to 0. In Appendix C, we will show that we
find exactly N - 1 independent eigenvectors for the negative eigenvalues.
Let us now assume that the N - 1 eigenvectors hi, i
-
1, ... , N -
1 and their
corresponding eigenvalues Ai are calculated. The solution can then be written as the
sum
N-1
Ashi exp(A q)
(B.6)
i=1
where Ai denote N - 1 unknowns, to be determined by using the boundary condition
at q = 0. In its discrete form, the boundary condition may be written as a set of N
equations
KnOi, + ci =
3Q
(Kn)
1
,i for Q1,j > 0 , z = V + 1, ... , 2v,
for the N components of V) (P(Q 1 ,j < 0)
=
=1, ... , Nw
(B.7)
0) . As a result, equation (B.7) yields
a system of N equations with N unknowns, including c1 . We show that this linear
system is well-posed in Appendix C.
171
172
Appendix C
Well-posedness of the discretized
boundary layer problem
In Appendix B, we described a numerical method for solving the order 1 boundary layer problem and determining the jump coefficient associated with a prescribed
temperature boundary. Here we present the method for finding the eigenvectors and
eigenvalues of the matrix associated with this solution and ip particular with in equa-
tion (B.4).
We start by ranking the result of the product AjQi,i in order of increasing value
(from -oc
to +oc).
Hence, for i < N,
[i
We then introduce the notation pi to denote the sorted list.
< 0, and for i > N + 1, pi = -
1
2N+1-i.
We now consider that
the components of < are reordered accordingly. With these conventions, the discrete
form of the ID BTE may now be written as:
E_ Z'joj -0i
Ox
We first highlight the property 1 oz
(C.1)
Pi
= 1. This is a direct consequence of energy
conservation. Therefore, a vector with all components identical and non-zero is an
eigenvector with eigenvalue 0. This simply corresponds to the uniform solution. If A
is an eigenvalue and if h is the associated eigenvector of matrix M, we have, for each
173
component. i of this eigenvector
Zah
= (1 + Api)hi
(C.2)
$
Let us first consider the case where E %vhj 0. In this case, we can, without loss
of generality, scale h such that E ahj = 1, yielding the following expression for the
components of the eigenvectors:
+(C.3)
1
hi =
.
Consequently, eigenvalues in this case satisfy:
g1(A) ==
C.1
(C.4)
.I + Apj
Eigenvalue determination
Let assume that all p are distinct. We will consider the complementary case later.
The function g is strictly decreasing on every open interval (-1/pki, -1/pttj1)
when
Pj > 0, and strictly increasing when pi < 0. If pj > 0, g(A) tends to +oc when A
tends to -1/p
and to -oc
when A tends to -1/p
when A tends to -1/p+.
If pj < 0, g(A) tends to -oo
and to +oo when A tends to -1/1p+1.
In both cases, this
shows that (C.4) has a unique solution on each interval (-I/pij, -I/pj+1),
for
j
= N (since in this case, (-i/lp, -I/pj+1)
therefore obtain 2N - 2 solutions. We use Aj,
except
does not define an interval).
We
j < N - 1, to denote the positive
solutions and (h)j the corresponding eigenvectors, whose components are given by
(C.2). By symmetry of g, A-j
=
-Aj
are the negative solutions, and we use (h)_j to
denote the associated eigenvectors. We have found a total of 2N - 1 eigenvectors. If
there were a 2N-th eigenvector with another eigenvalue A, then there would exist a
(2N + 1)-th eigenvector with eigenvalue -A,
which is impossible. We deduce that, M
only has 2N - 1 eigenvectors and is not diagonalizable. The 2N - 1 eigenvectors all
correspond to distinct eigenvalues and are therefore linearly independent. Equation
174
(C.4) can be easily solved numerically in each interval (-/Ipy, -1pj+i)
by using, for
instance, a bisection algorithm. In other words, eigenvalues and eigenvectors can be
found very efficiently.
We now consider the case where pj are not all distinct. This case can usually be
avoided, but it happens if the discretization is chosen such that there exists several
values of Q1,j and Aj such that the product Q1 ,iAj is the same. Let us assume that
there exists an index Jo and a positive integer
I such that Ij0 = pjo+l =
pj 0+'. The reasoning outlined above and based on the assumption that Y
...
=
ahj # 0
then only returns 2N - 1 - 21 eigenvectors and eigenvalues. We will refer to these
eigenvectors as "type 1 eigenvectors". The 21 remaining eigenvectors, which we will
refer to as "type 2
Z&
cigenvectors", may be found differently by examining the case
ajhj = 0. From this assumption we find that, for all components of the eigenvector
h,
h)h
hi = Ah.
(C.5)
lijo
Let us define I vectors (h)j,, (h)jo+1,..., (h)j,+,- 1 , as follows:
(hi)jo+m-i = 0 if i
(hjo)jo+m-l =
(hjo+m)jo+m-1
#
Jo and i # jo + m
1
I
(C.6)
ejo
= -
I
ajo+rn
where m E {1, 2,
... ,
l}. These vectors are eigenvectors with eigenvalues -1i.
In
total, we have 2N - 1 eigenvectors, as in the previous case.
C.2
Well-posedness
Let us now show the well-posedness of the problem by analyzing the invertibility
of the matrix P that describes the system of N equations with N unknowns. By
construction, each column of this matrix corresponds exactly to the (N + 1)-th to the
2N-th components of the N-I eigenvectors with negative eigenvalue. All components
175
of the N-th column are the same number (which we can normalize to 1). This actually
corresponds to the eigenvector with eigenvalue 0. We start with the case where all tj
are distinct. Although the linear independence of the eigenvectors is clear in this case
(since they correspond to different eigenvalues), we cannot directly conclude that P
is invertible, since P only contains the "truncated" eigenvectors". We here consider
the determinant of the matrix. We know that the coefficients of P are of the form:
Pi = 1
1 - ptiAj
where
AN
(C.7)
0-
Calculating this type of determinant is a classic (yet not straightforward) undergraduate exercise of linear algebra. We briefly recall its solution here for the interested
reader. Let us define a rational function R(X) as the following determinant:
1
1
1-pAI X
T-1-p1\2
1
R (X) =
1
1
..
_1-
...
N-1
1
1-A2X
I
I-p2.N-)
1
1
1
i-pNX
1
I-PNAN-1
-pN A2
We can develop R(X) with respect to the first column to obtain:
A
N
Z
R(X)
(C.9)
1I - pi X
or
R(X)
Q(X)
=
(C.10)
R1 (I - PiX)
where Q(X) is a degree N - 1 polynomial.
From the definition of the determinant R(X), we have R(A 2 )
R(AN) (we recall that AN
=
0). In other words, Aj, for j E {2, 3, ...
176
,
R(A 3 )
=
...
=
N} are the N - 1
roots of Q(X). Therefore:
R(X) =
(C. 11)
N i=2
Hj=1 (I -- PiX)
We deduce that R(A1 ) -/ 0.
where Qo is an undetermined non-null coefficient.
In
other words, P can be inverted and the problem is well-posed.
We finally need to consider the case where all
pi
are not distinct. We will limit
our analysis to the case where there exists an index jo and a positive integer 1 such
that -ti, =
Io+1= --- = pi,+,. Extension to the existence of other common values
is straightforward.
Given the expression of the eigenvectors in that case, we can
reorganize the lines and columns of P such that P is expressed as the block matrix
P =
S
0
U
V
(C.12)
where S is an (N - 1 - 1) x (N - 1) matrix whose coefficients are defined as in the
previous case (the "missing" line being that of the identical values of pi,. The matrix
V is an (I + 1) x I matrix defined as follows:
V =
Ck
1
1
i
ago
1
0
1
1
'i' C o
ai0
---
0
C01+1
0
0
(C.13)
- oi +2
0:
0
..
Matrix 0 is null, and matrix U is formed with coefficients (1 - piOAj)-
1
. All the lines
of U are therefore identical.
By multiplying the first line of V by ai0 , the second line by ai,+i and so on, and
by adding the resulting lines, we obtain a line of zeros. Therefore, by applying this
elementary combination of lines, we can transform P into
P
177
=
(C.14)
where S is an (N - 1) x (N - 1) matrix formed by expanding S with an additional line
containing the coefficients (1 -
ti 0Aj)
1
Zm
aio+,m. The matrix V is an I x I matrix
formed by removing the first line of V. It is a diagonal matrix with non-null zero
coefficients. The matrix
0
is an(N - 1) x / matrix, and
U
is formed by removing
the first line of U. Per the previous discussion, S is invertible. Since 0 is null and
V is diagonal, then P is invertible. Consequently, P is invertible and the system is
well-posed.
178
Appendix D
Proof of relation (6.62)
The remaining five coefficients in relation (6.62), can be determined by finding the
function 4' that satifies:
85
2
+n n Qi O
exp -i (K H(Q 1
)
0
077
Kn
' (,,p,7=
(Kn)3
((
-
0)=
Kn 22 3
Kn
Y
lim '(Q, L, p, 7)
=
E
K
-2 (Kn
2TGO
2
x
2
g(n
Q 1 Kn
_o
&2 T
,2=0' for Q
1>0
_
0
(D.1)
Let
ID
=5_ 1 X4,
where
X4,
k = 1, ...
,
5 correspond, respectively, to the five terms in
relation (6.62). We proceed with a strategy similar to the one used for showing that
C2
C3 = 0 in section 6.2.1, namely, solve for each 'Dk individually, and then evaluate
,C(Xk).
In the present case
results are shown in Table D.1.
Z(02 TGo/OX) '=
deduce that
=
Z L(4' k)
'
C(X k)
# 0 but
Y
_1
(Tk)
= 0.
Some intermediate
The last column of Table D.1 and the fact that
0 can be used to show that
= 0. This proves that Zk
f2 Z 4'/(4w)d2 p
4Ik
=0
and therefore
is the solution of (D.1) with the
specified source terms and boundary conditions, and that the resulting boundary layer
satisfies the boundary conditions without requiring a temperature jump correction,
that is, relation (6.62) is proved.
179
Table D.1: Source terms appearing in the second order boundary layer problem, and the
associated solutions.
fnqkd'f
k
47rk
Kn 2
1Kn2
Kn)
Q exp
H(
?Kn) H(_1)
(-i(Kn)'
e
21
_6(Kn2)
2, Knex
23
(Kn)2
ex1
+
I~
Q
Kn
=2
)4 U2q7
z 8x
)
Q 1)32
r7=0 - T) K=0
E(- -n)Q 2 Kn
-
180
1
K2Tn
2(Kn)
ex
(j ( I K
2
ep
e
exp
(n
K
(
(
2
__
6(Kxnz)
eXp (
Q
n
4(Kn Q10j1GKn
K
iKn
i= O 3
'' Kn2 2(Kn)
exp
& Q
82T) X]
K
Kn
Kn
3
))
dQ]22
xd
i=2
- '
7=
Appendix E
On the convergence rate of the ACA
method: mathematical justification
and discussion
In section 7.2.2, we find that using the spatially variable control
deTE
econtrol =
dT
q
(T
-Vg
(E.1)
- VTo)
yields estimates whose standard deviations scale with the Knudsen number (Kn),
provided that the temperature field To is a solution to Laplace's equation. In this
section, we provide a mathematical explanation for this assertion. In the interest of
simplicity, we consider here the case of a constant free path A,
= A. The case of
variable free path can be treated by simple extension of this approach and is expected
to yield similar results. This is further discussed below.
In the linearized algorithm, each particle, i, is associated with a contribution yi
which is a realization of a random variable Y such that, ultimately, the quantity
estimated is the average I* = Y yi/N = E(Y). The standard deviation of Y and 1*,
respectively
-y and o04, are related by
04*
= -Y/VK.
Showing that the standard
deviation of or scales with (Kn) amounts to showing that
-y scales in the same
manner (with (Kn)). The random variable Y is a sum of random variables, Y =
181
Z1 + Z 2 +
...
+
where each variable Zj corresponds to the contribution of a
ZNseg,
single segment of trajectory, as shown in section 7.2.2. Let us first recall that Zj is
given by
Zj = NS *Tj1Vg,_
except for
j
Nseg. Note that
- (V7To(x_
1
) - V To(xj))
(E.2)
NSE*ff is independent of (Kn).
t*
When (Kn) is
small, Zj may be written as
&3T
Al
Zj=
-Et*otAlj
where l (~
xQ mQn - Et2ot
6FT
&xm&xn
0
2 (9x,,OXnqxj
QrmnnQq+
h.o.t.
(E.3)
A) is the length of that segment of the trajectory and h.o.t. denotes higher
order terms. This can be rearranged in the form
I.
02T
TO
tEt(Kn)2_A &x',Ox'
3To
311tJ2(K2n
2 A 2 axI (,I DXI
m n QmKn)
+
Z
QrnQnQq
h.o.t.
(E.4)
EZ3 + 0 ( jKn)3)
(E.5)
where x' is the dimensionless coordinate defined by x' = xj/L. First we note that
due to the isotropy of the post-scattering distribution, E(Qi) = 0. Moreover, here we
are examining the case
V,To = 0. From these two observations it directly follows
that E(Z) = 0 and therefore
n
n
Z(E.6)
=
i 1
defines a martingale
1691
with (optional) stopping time
We therefore have Y =
+ ZNseg + (,
fl=
Nseg
-
1.
where ( represents the contribution of
Nseg - 1 order 3 terms and
ZNseg
=tot
(Kn)
0 TO
(Q
,
I XNseg
182
.)
(E.7)
The variance of Y is therefore:
Var(Y) = Var(Y) +Var(() +Var(ZNeg ) +2Cov(, ZNseg) +2Cov((,
ZNseg )
+2Cov(Y, ()
(E.8)
Below, we examine each of the terms of the above expression and show that they all
(Kn) 2
.
scale with
* Variance of Y: By applying the optional stopping theorem to the martingale
S
2-
E"_n
Var(Zi) (see [69]), which implies that E(S.) = 0, we obtain
Neg--
Var(fNeg -1)
Var(Zi))
= E
(E.9)
We note that, provided the second derivatives of To are bounded, we can find a
Zi are all smaller than M 1 (Kn) 4
.
positive constant Mi such that the variances of
It follows that:
Nseg-1
Var(Y) < E
(E.10)
Var(Y) < Mi(Kn) 4E(Nseg)
(E.11)
)
M1i (Kn)4
and therefore
Finally, since the average number of jumps is asymptotically proportional to
(Kn -2:
Var(Y) = 0 ((Kn) 2
)
(E.12)
* Variance of (:
The variance of ( is defined as E((2) - E(()2, where:
2)
-Neg-1
A2
(E.13)
(
E(( 2 ) =E
Nseg-1 Nseg -1
22jA
4L
E(W2) =E
j=1
183
(Kn)6
(E.14)
Wald's equation [69] applies to the latter expression and yields
Nseg -1
E((2)
Nseg1
=E
(E.15)
A2 (Kn)6) I)
j=1
We can find a positive constant AM 2 such that:
E((2) < A/12 E ((Nseg - 1)2) (Kn)6
(E.16)
E ((2) = 0 ((Kn)2)
(E.17)
In other words:
Also:
2
(=A
(Kn)3
(E.18)
)
(Nsef-10
)
E(() =E
E(() < M3E (Nseg - 1) (Kn) 3
(E.19)
E(() =0 ((Kn))
(E.20)
We finally find that Var(() = 0((Kn) 2 ).
* Variance of
ZNeg:
From the definition of ZNseg, we immediately find that Var (ZNAg) = 0((Kn) 2).
* Covariance of Y and ZNseg:
Cov
(,
ZNseg)
Cov
(,
ZNSeg)
Cov (,
ZNeg)
Cov
, ZNseg
=E
(VZN,)
IE(Y)E(ZNseg)
-
(E.21)
(E.22)
E (f Zeg)
=E
(ZNseg
=
-E (ZNseg
I
==Q
) Pf=
P(Y =Qd,d
The martingale central limit theorem for Y states that P(Y
184
(E.23)
(E.24)
Q) tends asymp-
totically to a Gaussian with standard deviation o =
isotropy associated with the scattering process E
Var(f).
=)
(ZNseQ
Also, due to
E
(ZNseg
0((Kn)). Hence:
Cov (ii, ZNseg) =0 ((Kn)
(f,ZNseg)
Cov (iY,
ZNseg)
* Covariance of Y and
exp
(E.25)
(~dQ)
=0 (KKn)o-)
(E.26)
=0 ((Kn)2)
(E.27)
(:
We note that the value of
( is obtained using the same random numbers as
.
Cov
>
We may still obtain an upper bound for the covariance using:
Cov (f, () =E (()
Cov
C)
(;,
- E(Y)E(()
=E (f()
(E.29)
Cov(VC) =JiE(CY
j
Cov (Y, C) =0 ((Kn)
where, since E(()
0
V2-7a
exp I
2a--
di)
(E.31)
(E.32)
2
~ VVar(C)
as of
0
(KKn)).
* Covariance of
( and
Coy (ZNseg,
C)
ZNseg:
=E
Cov (ZNseg, () =E
(ZNseg()
(0
-
(E.33)
E(ZNseg)E(C)
Elj1
Nseg --
N:e (
4
A3-(Kn)
n)2)
+ 0((Kn))
(E.34)
i=1I)
2
Coy (ZNseg, () =0 ((Kn)
(E.35)
)
=
(E.30)
1P(Y = ()dy
)
Cov (V, C) =0 ((Kn)
0 ((Kn)), we estimate E ( )
(E.28)
185
which is again obtained using Wald's equation.
In summary, this shows that
Var(Y) = O((Kn) 2 ),
(E.36)
which implies that the standard deviation associated with the estimate I*, o7, scales
linearly with (Kn).
We note here the following:
1 Even in cases where E[Nseg] - O((Kn)- 1 ) (and thus Var(Y) = O((Kn)3 )), such
as in the proximity of a boundary, the leading order term in Var(Y) is still of
O((Kn)2 ), yielding the same result.
2 In the above development, V2,To = 0 comes as a necessary condition for the
first-order scaling, (o-1
oc Kn), to be true. To see this, let us imagine that, in
a given region of space, this condition is not satisfied. Then, in such a region,
Y is no longer a martingale because the expected value is of Zj is no longer
0. As a consequence, the result (E.9) can no longer be used. More precisely,
if V2,To > 0 in a given region, then the values taken by Zi will be correlated
to each other in this region. This will cause the terms E(E Zi2) to introduce
a quantity of order
(Kn)0 in the expression of E(Y 2). More details would be
required to prove that this order 0 term is not compensated by other terms, but
this induces E(Y 2 ) to scale with (Kn) 0 . Thus, since E(Y) scales with (Kn),
.
then the variance scales with E(Y 2 ) oc (Kn) 0
To illustrate the second point, let us consider the example discussed in section
7.2.3, namely, using the adjoint method to find the X 2-component of the heat flux at
point (0, 0) of the problem depicted in Figure 6-10. Figures E-1 and E-2 show
"
Zi
as a function of the index n, where Z here corresponds to the heat flux contribution.
TO(XI, X2 ) = T, cos
27x,
3L
186
sinh
s
(2
sinh ()
)
Figure E-1 shows the result obtained using
(E.37)
as a control which is a solution of Laplace's equation with Dirichlet boundary conditions; Figure E-2 shows the result obtained using
To(xi, x 2 ) = T, cos (23L)
I - X).
(E.38)
as a control, which is not a solution of Laplace's equation. Figure E-i shows that
when To satisfies the Laplace equation, very little correlation between each segment
of a trajectory exists and, as a result, the standard deviation of the contributions is
proportional to (Kn), in agreement with (E.36). In contrast, Figure E-2 shows that
when To does not satisfy the Laplace equation, the strong correlation between different
segments of the trajectory makes the contribution of each particle independent of the
0
Knudsen number (proportional to (Kn) as discussed in item 2 above), leading to a
.
0
standard deviation that also scales as (Kn)
-3
0.1
8-
2-
0
-2
-0.05-
-4
234123456
-0.1
-
jump index
jump index
5
x10
x10
Figure E-1: Evolution of the contributions of particles to the final estimate, when the
adjoint method is used with the control (E.1) along with a temperature field which is
the solution of Laplace's equation for (Kn) = 0.01 (Left) and (Kn) = 0.001 (Right).
This calculation was performed using the single-mean-free-path model.
187
442--
0
0
-2-
0
1
2
3
jump index
4
0
6
5
x 10
1
2
3
jump index
4
6
5
x 10
Figure E-2: Evolution of the contributions of particles to the final estimate, when the
adjoint method is used with the control (E.1) along with a temperature field which is
not a solution of Laplace's equation, for (Kn) = 0.01 (Left) and (Kn) = 0.001 (Right).
This calculation was performed using the single-mean-free-path model.
188
Appendix F
Material models
Dispersion relations (and the resulting group velocities) and scattering rates for the
simulations in this thesis were calculated using the material model described in detail in
[4]; the description is reproduced here for convenience. We note that since
the primary focus of the present thesis is the development of numerical approaches,
this material model choice represents a balance between simplicity and fidelity. In
other words, although this material model is sufficiently realistic to capture a number of important features that have a large influence on the computational method
(e.g. wide range of free paths), it includes a number of simplifying assumptions (e.g.
isotropic dispersion relation [77]) that may need to be re-examined when used to
model transport in nanostructures.
Extension to more realistic material models of
varying complexity, including the ab-initio scattering operator, will be considered in
future work.
In the present model, dispersion relations are adapted from the experimentally
measured dispersion relation in the [100] direction ( [78] for Al, [39] for Si).
Note
that extension to more realistic dispersion relations such as the one presented in [79]
is straightforward, as long as the post-scattering traveling directions are assumed
isotropic, as assumed in [79]. From the dispersion relation, the density of states may
189
be derived using
D(, LA) =
D( ,TA)
1
o
2
2 w c(w, LA) 2X (w, LA)
(F.1)
(F.2)
W
=
2c(w, TA) 2 V(w, TA)
2r
where c(w, p) refers to the phase velocity (given by w/k, k being the wavenumber).
Note the absence of the factor 1/2 for the TA modes due to the presence of two such
modes which, in this model, share the same properties.
For aluminum in the TTR calculation, a constant relaxation time is used; it is
chosen to fit the desired lattice thermal conductivity (as in [39]) and is given by:
(F.3)
TAI = 1011S5
For silicon, the expressions are taken from [80], with constants from [39].
For
acoustic modes, these are
Z 1 = AL
phonon-phonon scattering, LA
phonon-phonon scattering, TA
TT'
2
T1. 49 exp
= ATJ2 T'-6 5 exp
impurity scattering
77-1 = AI,
boundary scattering
B 1 = Wb
(-p
(-T)
4
where the constants take the following values
Parameter
Value (in SI units)
AL
2.09 x 10-
AT
19/(27)
1.23 x 10- 1 9/(2)
2
2
0
A1
'Wb
80
3 x 10-45
1.2 x 106
For a given polarization we obtain the total relaxation time by applying the Matthiessen
rule
7-
1
190
71
(F.4)
For simplicity, we use Einstein's model to treat optical phonons and consider them
as immobile. Their behavior is therefore purely capacitive. Einstein's model states
that the contribution of optical phonons to the vibrational energy per unit volume in
a crystal is given by [7]
NPN'hWE
U =plL
V[exp(hwE/kBT) - 1]
(F.5)
where Np = 3 is the number of polarizations, N' = 1 is the number of optical states
per lattice point,
WE
is the Einstein radial frequency (WE
=
9.1
X
1013S- 1 [39]), V is the
volume of a lattice point (with a lattice constant a = 5.43, V = a 3 /4 = 4 x 10-
29
m 3).
For the relaxation time of optical phonons at 300K, we use the value [81]
To =
3 x 10- 1 2 s
191
(F.6)
192
Bibliography
[11 J.-P. M. P6raud, C. D. Landon, and N. G. Hadjiconstantinou, "Deviational
methods for small-scale phonon transport," Mechanical Engineering Reviews, 1,
FE0013, 2014.
[2] J.-P. M. P6raud and N. G. Hadjiconstantinou, "Deviational phonons and thermal transport at the nanoscale," The 2012 InternationalMechanical Engineering
Congress and Exposition, paper number IMECE2012-87547, 2012.
[3] J.-P. M. P6raud, C. D. Landon, and N. G. Hadjiconstantinou, "Monte Carlo
methods for solving the Boltzmann transport equation," Annual Review of Heat
Transfer, 17, 205-265, Begell House, 2014.
[41 J.-P. M. P6raud and N. G. Hadjiconstantinou, "Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo
formulations," Physical Review B, 84, 205331, 2011.
[5] J.-P. M. P6raud and N. G. Hadjiconstantinou, "An alternative approach to efficient simulation of micro/nanoscale phonon transport," Applied Physics Letters,
101, 153114, 2012.
[6]
Q. Hao, G. Chen, and M. S. Jeng, "Frequency-dependent Monte Carlo simulation of phonon transport in two-dimensional porous silicon with aligned pores,"
Journal of Applied Physics, 106, 114321, 2009.
[7]
G. Chen, Nanoscale Energy Transport and Conversion. New York, NY: Oxford
University Press, 2005.
[8] J. M. Ziman, Electrons and Phonons. Oxford, UK: Clarendon Press, 1960.
[91 K. F. Hsu, S. Loo, F. Guo, W. Chen, J. S. Dyck, C. Uher, T. Hogan, E. K. Polychroniadis, and M. G. Kanatzidis, "Cubic AgPbmSbTe2-m: Bulk thermoelectric
materials with high figure of merit," Science, 303, 818-821, 2004.
[10] B. Poudel,
Q.
Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan, D. Wang,
A. Muto, D. Vashaee, X. Chen, J. Liu, M. S. Dresselhaus, G. Chen, and Z. Ren,
"High-thermoelectric performance of nanostructured bismuth antimony telluride
bulk alloys," Science, 320, 634-638, 2008.
193
[11] J.-P. Phraud, "Low variance methods for Monte Carlo simulation of phonon
transport," Master's thesis, Massachusetts Institute of Technology, Cambridge,
MA, June 2011.
[12] A. J. H. McGaughey and J. M. Larkin, "Predicting phonon properties from equilibrium molecular dynamics simulations," Annual Review of Heat Transfer, 17,
Begell House, 2014.
[13] Y. Chalopin, A. Rajabpour, H. Han, Y. Ni, and S. Volz, "Equilibrium molecular
dynamics simulations on interfacial phonon transport," Annual Review of Heat
Transfer, 17, Begell House, 2014.
[141 S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, "Phonons and related crystal properties from density-functional perturbation theory," Reviews of
Modern Physics, 73, 515-562, 2001.
[15]
R. Peierls, Selected Scientific Papers of Sir Rudolf Peierls With Commentary,
ch. On the kinetic theory of thermal conduction in crystals (Annalen der Physik,
3, 1055-1101, 1929). World Scientific, 1997.
[161 R. Peierls, Quantum Theory of Solids. Oxford University Press, 1955.
[17]
L. Boltzmann, Vorlesungen iiber Gastheorie. (Lectures on Gas Theory). Barth.
Leipzig. English translation by S.G. Brush, 1964 (University of California Press,
Berkeley, CA), 1896.
[18] R. E. Marshak, "Theory of the slowing down of neutrons by elastic collision with
atomic nuclei," Reviews of Modern Physics, 19, 185-238, 1947.
[19] J. Duderstadt and L. Hamilton, Nuclear Reactor Analysis. Wiley, 1976.
[20] L. L. Baker and N. G. Hadjiconstantinou, "Variance reduction for Monte Carlo
solutions of the Boltzmann equation," Physics of Fluids, 17, 051703, 2005.
[21] T. M. M. Homolle and N. G. Hadjiconstantinou, "A low-variance deviational
simulation Monte Carlo for the Boltzmann equation," Journal of Computational
Physics, 226, 2341-2358, 2007.
[22] G. A. Radtke, J.-P. M. P6raud, and N. G. Hadjiconstantinou, "On efficient simulations of multiscale kinetic transport," PhilosophicalTransactions of the Royal
Society A, 371, 20120182, 2013.
[23] Y. Sone, Kinetic Theory and Fluid Dynamics. Boston: Birkhiuser, 2002.
[24] G. P. Srivastava, The Physics of Phonons. New York, NY: Taylor & Francis
Group, 1990.
[25] D. J. Ecsedy and P. G. Klemens, "Thermal resistivity of dielectric crystals due to
four-phonon processes and optical modes," Physical Review B, 15, 5957-5962,
1976.
194
[26] P. L. Bhatnagar, E. P. Gross, and M. Krook, "A model for collision processes
in gases. I. Small amplitude processes in charged and neutral one-component
systems," Physical Review, 94, 511-525, 1954.
[27] J. M. Loy, J. Y. Murthy, and D. Singh, "A fast hybrid Fourier-Boltzmann transport equation solver for nongray phonon transport," Journal of Heat Transfer,
135, 011008, 2013.
[281 T. Klitsner, J. E. VanCleve, H. E. Fisher, and R. 0. Pohl, "Phonon radiative
heat transfer and surface scattering," Physical Review B, 38, 7576-7594, 1988.
[29] R. B. Peterson, "Direct simulation of phonon-mediated heat-transfer in a Debye
crystal," Journal of Heat Transfer, 118, 815-822, 1994.
[30] G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows.
Oxford, UK: Clarendon Press, 1994.
[31] S. Mazumder and A. Majumdar, "Monte Carlo study of phonon transport in
solid thin films including dispersion and polarization," Journal of Heat Transfer,
123, 749-759, 2001.
[32] D. Lacroix, K. Joulain, and D. Lemonnier, "Monte Carlo transient phonon transport in silicon and germanium at nanoscales," Physical Review B, 72, 064305,
2005.
[331 A. J. Schmidt, X. Chen, and G. Chen, "Pulse accumulation, radial heat conduction, and anisotropic thermal conductivity in pump-probe transient thermoreflectance," Review of Scientific Instruments, 79, 114902, 2008.
[34] N. G. Hadjiconstantinou, "Analysis of discretization in the direct simulation
Monte Carlo," Physics of Fluids, 12, 2634-2638, 2000.
[35] J. Spanier and E. M. Gelbard, Monte Carlo Principles and Neutron Transport
Problems. Reading, MA: Addison-Wesley, 1969.
[361 G. A. Radtke and N. G. Hadjiconstantinou, "Variance-reduced particle simulation of the Boltzmann transport equation in the relaxation-time approximation,"
Physical Review E, 79, 056711, 2009.
[371 R. Yang and G. Chen, "Thermal conductivity modeling of periodic twodimensional nanocomposites," Physical Review B, 69, 195316, 2004.
[38] M. S. Jeng, R. Yang, D. Song, and G. Chen, "Modeling the thermal conductivity
and phonon transport in nanoparticle composites using Monte Carlo simulation,"
Journal of Heat Transfer, 130, 042410, 2008.
[391 A. J. Minnich, Exploring electron and phonon transportat the nanoscalefor thermoelectric energy conversion. PhD thesis, Massachusetts Institute of Technology,
Cambridge, MA, 2011.
195
[40] A. J. Minnich, G. Chen, S. Mansoor, and B. S. Yilbas, "Quasiballistic heat
transfer studied using the frequency-dependent Boltzmann transport equation,"
Physical Review B, 84, 23, 235207, 2011.
[41] A. J. Minnich, "Determining phonon mean free paths from observation of quasiballistic thermal transport," Physical Review Letters, 109, 205901, 2012.
[42] A. J. Minnich, "Measuring phonon mean free paths using thermal conductivity
spectroscopy," Annual Review of Heat Transfer, 16, Begell House, 2013.
[43] J. Randrianalisoa and D. Baillis, "Monte Carlo simulation of cross-plane thermal
conductivity of nanostructured porous silicon films," Journal of Heat Transfer,
130, 072404, 2008.
[44] F. J. Alexander, A. L. Garcia, and B. J. Alder, "Cell size dependence of transport
coefficients in stochastic particle algorithms," Physics of Fluids, 10, 1540-1542,
1998.
145] D. J. Rader, M. A. Gallis, J. R. Torczynski, and W. Wagner, "Direct simulation
Monte Carlo convergence behavior of the hard-sphere gas thermal conductivity
for Fourier heat flow," Physics of Fluids, 18, 077102, 2006.
146] J.-P. M. Praud and N. G. Hadjiconstantinou, "Adjoint-based deviational Monte
Carlo methods for phonon transport calculations," Physical Review B,
235321, 2015.
91,
147] K. M. Case and P. F. Zweifel, Linear Transport Theory. Reading, MA: AddisonWesley, 1967.
[48] R. Sanchez and N. J. McCormick, "Review of neutron transport approximations,"
Nuclear Science and Engineering, 80, 481-535, 1982.
[49] A. Schmidt, "Pump probe thermoreflectance," Annual Review of Heat Transfer,
16, 159-181, Begell House, 2013.
[50] F. Yang and C. Dames, "Mean free path spectra as a tool to understand thermal
conductivity in bulk and nanostructures," Physical Review B, 87, 035437, 2013.
[51] C. Hua and A. J. Minnich, "Importance of frequency-dependent grain boundary scattering in nanocrystalline silicon and silicon-germanium thermoelectrics,"
Semiconductor Science Technology, 29, 124004, 2014.
[52] J.-P. M. P6raud and N. G. Hadjiconstantinou, "On the equations and boundary conditions governing phonon-mediated heat transfer in the small mean free
path limit. An asymptotic solution of the Boltzmann equation," The 2014 International Mechanical Engineering Congress and Exposition, paper number
IMECE2014-36475, 2014.
196
[53] J.-P. M. P6raud and N. G. Hadjiconstantinou, "Extending the range of validity
of Fourier's law into the kinetic transport regime via asymptotic solution of the
phonon Boltzmann transport equation", arXiv:1508.04694, 2015.
[54] S. Takata, "Symmetry of the unsteady linearized Boltzmann equation in a fixed
bounded domain," Journal of Statistical Physics, 140, 985-1005, 2010.
[55] Y. Sone, "Asymptotic theory of flow of rarefied gas over a smooth boundary i,"
Proceedings of the Sixth InternationalSymposium on Rarefied Gas Dynamics, 1,
243-253, 1969.
[56] K. Aoki, S. Takata, and F. Golse, "A rarefied gas flow caused by a discontinuous
wall temperature," Physics of Fluids, 13, 2645-2661, 2001.
[57] H. Grad, "Asymptotic theory of the Boltzmann equation," Physics of Fluids, 6,
147-181, 1963.
[58] Y. Sone, Molecular Gas Dynamics:
Boston: Birkhduser, 2007.
Theory,
Techniques, and Applications.
[59] N. G. Hadjiconstantinou, "The limits of Navier-Stokes theory and kinetic extensions for describing small scale gaseous hydrodynamics," Physics of Fluids, 18,
111301, 2006.
[60] S. Takata and M. Hattori, "Asymptotic theory for the time-dependent behavior
of a slightly rarefied gas over a smooth solid boundary," Journal of Statistical
Physics, 147, 1182-1215, 2012.
[61] M. Hattori and S. Takata, "Second-order Knudsen-layer analysis for the generalized slip-flow theory I," Bulletin of the Institute of Mathematics, 2015.
[62] Y. Hu, L. Zeng, A. J. Minnich, M. S. Dresselhaus, and G. Chen, "Spectral mapping of thermal conductivity through nanoscale ballistic transport," Nature Nanotechnology, 2015.
[63] R. Yang, G. Chen, M. Laroche, and Y. Taur, "Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation," Journal of Heat Transfer, 127, 298-
306, 2005.
[64] T. Zeng and G. Chen, "Phonon heat conduction in thin films: impacts of thermal
boundary resistance and internal heat generation," Journal of Heat Transfer,
123, 340-347, 2001.
[65] P. Degond and C. Schmeiser, "Macroscopic models for semiconductor heterostructures," Journal of Mathematical Physics, 39, 4634, 1998.
[66] K. Aoki, P. Degond, L. Mieussens, S. Takata, and H. Yoshida, "A Diffusion Model
for Rarefied Flows in Curved Channels," Multiscale Modeling and Simulation, 6,
1281-1316, 2008.
197
[67] K. Aoki, P. Degond, S. Takata, and H. Yoshida, "Diffusion models for Knudsen
compressors," Physics of Fluids, 19, 117103, 2007.
[68] S. Takata, H. Sugimoto, and S. Kosuge, "Gas separation by means of the Knudsen
compressor," European Journal of Mechanics - B Fluids, 26, 155-181, 2007.
[69] G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 3rd
edition. Oxford University Press, 2001.
[70] B. Davison, Neutron transport theory.
physics, Clarendon Press, 1958.
International series of monographs on
[71] E. Dumonteil, "On a new variance reduction technique: neural network biasing
- A study of two test cases with the Monte Carlo code TRIPOLI4," Nuclear
Technology, 168, 793-798, 2009.
[72] L. Lindsay, D. A. Broido, and N. Mingo, "Flexural phonons and thermal transport
in graphene," Physical Review B, 82, 115427, 2010.
[73] J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. Li,
Z. Yao, R. Huang, D. Broido, N. Mingo, R. S. Ruoff, and L. Shi, "Twodimensional phonon transport in supported graphene," Science, 328, 213-216,
2010.
[74] N. Mingo, D. A. Stewart, D. A. Broido, L. Lindsay, and W. Li, "Ab initio thermal transport," Length-Scale Dependent Phonon Interactions, 137-173, Springer
New York, 2014.
[75]
C. Landon and N. G. Hadjiconstantinou, "Deviational simulation of phonon
transport in graphene ribbons with ab initio scattering," Journal of Applied
Physics, 16, 163502, 2014.
[76] C. Bardos, R. E. Caflisch, and B. Nicolaenko, "The milne and kramers problems
for the boltzmann equation of a hard sphere gas," Communications on pure and
applied mathematics, 39, 323-352, 1986.
[77] Z. Chen and Z. Wei and Y. Chen and C. Dames, "Anisotropic Debye model for
the thermal boundary conductance," Physical Review B, 87, 125426, 2013.
[78] R. Stedman and G. Nilsson, "Dispersion relations for phonons in Aluminum at
80 and 300K," Physical Review, 145, 492-500, 1966.
[79] K. Kukita and Y. Kamakura, "Monte Carlo simulation of phonon transport in
silicon using a realistic dispersion relation," Journal of Applied Physics, 114,
154312, 2013.
[801 A. S. Henry and G. Chen, "Spectral phonon transport properties of silicon based
on molecular dynamics simulations and lattice dynamics," Journal of Computational and Theoretical Nanoscience, 5, 1-12, 2008.
198
[81] P. G. Klemens, "Anharmonic Decay of Optical Phonons," Physical Review, 148,
845-848, 1966.
199