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Modeling and analysis of nanostructure growth process kinetics and variations for scalable nanomanufacturing

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Modeling and analysis of nanostructure growth process kinetics and variations for scalable nanomanufacturing

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Content MODELING AND ANALYSIS OF NANOSTRUCTURE GROWTH PROCESS KINETICS
AND V ARIATIONS FOR SCALABLE NANOMANUFACTURING
by
Li Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
Dissertation Committee
Dr. Qiang Huang (Chair) - Industrial and Systems Engineering
Dr. Yong Chen - Industrial and Systems Engineering
Dr. Jianfeng Zhang (Outside Member) - Math
December 2013
To my father, mother and Haiyue
ii
Acknowledgements
I would like to express my gratitude to my advisor Professor Qiang Huang, for his supervision and
encouragement throughout my research. Without his help, this dissertation would not have been
possible.
I would like to express my sincere gratitude to my other dissertation committee members,
Professor Yong Chen and Professor Jianfeng Zhang for their insightful advice and constructive
comments to improve my research. I would also like to thank Prof. Maged Dessouky, Prof. Joe
Qin and Prof. Sheldon Ross for their helpful feedbacks during my dissertation proposal.
I would like to thank my collaborators for their support and help, including Prof. Tirthankar
Dasgupta, Prof. Shekhar Bhansali, Prof. Praveen K. Sekhar, Dr. Li Zhu and Mr. Yu An.
I would also to thank all my colleagues and friends for their constructive comments in my
research, including Ms. Lijuan Xu, Mr. Jian Wu, Mr. Jizhe Zhang, Dr. Pai Liu and Dr. Tingting
Gang.
Contents
Dedication ii
Acknowledgements iii
Abstract xi
Chapter 1: Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Nanotechnology Background and Potentials . . . . . . . . . . . . . . . . . 1
1.1.2 Research Challenges in Nanomanufacturing Process Modeling . . . . . . . 2
1.2 State of Art on Nanomanufacturing Modeling . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Physical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Statistical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Physical-statistical modeling . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Application domains for different modeling approaches . . . . . . . . . . . 11
1.3 Research Tasks and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 2: Statistical Weight Kinetics Modeling 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Experimental Details and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Statistical Weight Kinetics Modeling and Maximum Likelihood Estimation . . . . 18
2.3.1 Postulating and fitting models with constant error variance . . . . . . . . . 19
iv
2.3.2 Generalizing to heteroscedastic model . . . . . . . . . . . . . . . . . . . . 28
2.4 A Bayesian Hierarchical Model For Estimation with Limited Observations . . . . . 31
2.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Proof for Transition Function Choice . . . . . . . . . . . . . . . . . . . . . . . . . 35
Chapter 3: Cross-Domain Model Building and Validation (CDMV) 37
3.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Related Physical and Statistical Models and Uncertainties . . . . . . . . . . . . . . 39
3.2.1 Candidate Physical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Candidate Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Cross-Domain Model Building and Validation Approach to Understand Nanoman-
ufacturing Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Growth Attributed to Direct Top Impingement . . . . . . . . . . . . . . . . 43
3.3.2 Growth Attributed to Side Absorption . . . . . . . . . . . . . . . . . . . . 44
3.3.3 Cross-Domain Modeling of Nanowire Growth . . . . . . . . . . . . . . . . 46
3.3.4 Uncover Growth Kinetics for Silica Nanowire Growth . . . . . . . . . . . 48
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Chapter 4: Bayesian Hierarchical CDMV for Characterizing Variations Among Nano
Experimental Runs under Larger Uncertainties 54
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Strategy to Model Variations among Nano Experimental Runs under Large Uncer-
tainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Incorporating Mixed-effects Modeling Under CDMV Framework . . . . . . . . . . 60
4.3.1 Review and Re-parameterization of CDMV Model . . . . . . . . . . . . . 61
4.3.2 Mixed Effects Modeling and Estimation . . . . . . . . . . . . . . . . . . . 63
4.3.3 Bayesian Hierarchical CDMV Model . . . . . . . . . . . . . . . . . . . . 66
4.3.4 Exploratory Individual Models for Each Parameter . . . . . . . . . . . . . 67
v
4.3.5 Physical Interpretation and Final Model Building . . . . . . . . . . . . . . 71
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 5: Conclusions and Future Research 75
5.1 Summary and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
References 78
vi
List of Tables
2.1 Weight growth data(mg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Parameter Estimation Comparison among Four Models . . . . . . . . . . . . . . . 29
2.3 Posterior Inference of Model 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1 Residual Sum of Squares of Logistic and Gompertz Model . . . . . . . . . . . . . 42
3.2 Initial Parameter Values for 1100 °C . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Bayesian Estimation for 1050 °C . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Estimation Comparison under 1100 and 1050 °C . . . . . . . . . . . . . . . . . . . 52
3.5 Logistic Growth Estimation under 1100°C and 1050 °C . . . . . . . . . . . . . . . 52
4.1 Experimental runs of silica nanowire weight change data (mg) under 1050°C . . . . 56
4.2 Estimation hierarchical model withW
0
as random effect . . . . . . . . . . . . . . 69
4.3 Estimation for hierarchical model withW

as random effect . . . . . . . . . . . . 70
4.4 Estimation for hierarchical model witha

as random effect . . . . . . . . . . . . . 71
4.5 Estimation for hierarchical model witha as random effect . . . . . . . . . . . . . . 71
4.6 Estimation for hierarchical model withW
0
andW

as random effect . . . . . . . . 73
vii
List of Figures
1.1 Yield strength of gold nanowire as a function of lateral dimension [20] . . . . . . . 3
1.2 Comparison between VLS and OAG growth mechanism [102] . . . . . . . . . . . 5
1.3 Experiment Strategy for [72] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Level plots of estimated interaction parameters together with top view SEM
images for measured ZnO nanowires [94]. White strips are nanowire bundles (a
type of defects) on top view SEM images. Dark color clusters correspond to local
peaks of nanowire interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Physics-driven hierarchical modeling of growth process at micro/nano scale [27] . . 11
1.6 Morphology-local decomposition of NW length field [27] . . . . . . . . . . . . . . 11
1.7 Modeling Strategies Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Different Nanostructure Growth Processes . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Open Tube Furnace with Pd Coated Si Sample on Support Si Substrate . . . . . . . 18
2.3 SEM Micrographs (25 KV , 20K) Illustrating the Morphology of the Pd Coated Si
Wafer Heated at 1100 °C for (A)15, (B)20, (C)60, and (D)90 mins . . . . . . . . . 20
2.4 Weight Changes of Silica NWs Over Time for Three Replicates . . . . . . . . . . . 21
2.5 Fitted Curve of Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
viii
2.6 Fitted Curve of Model 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Transition functionh(t;
) with different
(t
0
=80s) . . . . . . . . . . . . . . . . . 25
2.8 Fitted Curve of Model 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Comparison of models 3,4 and 5 near transition point . . . . . . . . . . . . . . . . 28
2.10 Fitted Curve of Model 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Different chains simulated using WinBugs . . . . . . . . . . . . . . . . . . . . . . 32
2.12 Posterior distributions of
1
;
2
;a andt
0
. . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Crossing Domain Model Building and Validation . . . . . . . . . . . . . . . . . . 38
3.2 Logistic and Gompertz Model under different temperatures: solid line for Logistic
model and dotted line for Gompertz model . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Schematic of growth from two sources . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Estimation under 1100°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Uncertainty under 1050°C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Growth data under 1050°C for different runs and results of separate fittings . . . . 57
4.2 Schematic of the proposed strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Schematic of nanowire growth from two sources [85] . . . . . . . . . . . . . . . . 61
4.4 Effects of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5 Posterior parameter distributions withW
0
(denotes asWs in WinBUGS program)
as random effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
ix
4.6 Trace plot for parameters withW
0
as random effects(
e
omitted due to space limi-
tation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.7 Comparison of residual plots for CDMV and Hierarchical CDMV . . . . . . . . . 74
x
Abstract
Nanomanufacturing is currently a major bottleneck that hinders the transformation of nanotech-
nology from laboratory to industrial applications. Due to both limited process understanding and
control, there are great challenges for reliable and cost-effective scalable production of nanostruc-
tures. As a result, despite of nanomaterials’ superior electrical, mechanical, chemical and biologi-
cal properties, their great potentials in high impact fields such as energy, medicine and information
technology have not been materialized.
In order to achieve scalable nanomanufacturing, the nanostructure fabrication process has to be
quantitatively modeled before subsequent improvement efforts such as diagnosis, monitoring and
control. This research will therefore focuses on characterizing nanostructure growth processes to
understand growth kinetics which is the law governing how growth behaviors change with time.
This dissertation will also investigate the among experimental runs variations and identify their
root causes for improved process modeling.
There are two fundamental research challenges in nanomanufacturing process modeling: (i)
There are only limited physical knowledge for nanostructure growth processes. Despite of numer-
ous studies in nanostructures growth area, we are still often largely uncertain about the physical
mechanisms driven the growth process under a certain condition. The complexity comes from the
fact that there are often multiple mechanisms at work in one growth process and it is very diffi-
cult to identify their relative contributions. (ii) Experimental observations are expensive and time-
consuming to obtain. Nanostructure growth experiments are usually costly due to expensive equip-
ment(such as metal-organic chemical vapor deposition (MOCVD) system) as well as requirement
xi
of constant human attention. Moreover, the measurement of nanostructures often involves equip-
ment such as scanning electron microscopy (SEM), transmission electron microscopy (TEM) and
atomic force microscope (AFM). Besides significant equipment cost, one can only view/measure
a tiny portion of the substrate at one time.
This dissertation will systematically model nanostructure growth process kinetics and its vari-
ation as part of greater effort towards achieving scalable nanomanufacturing. The focus of the
modeling work is on quantitative macro scale measurements (such as overall weight) of nanos-
tructure grown using ”bottom-up” instead of ”top-down” approach. Special emphasis is placed on
integrating physical knowledge and experimental data due to the challenges mentioned above in
the whole modeling process. The major research tasks includes: (i) statistical model building and
selection to gain initial understanding of nanowire weight growth kinetics; (ii) cross-domain model
building and validation(CDMV) to utilize all available information in both physical and statistical
domains and achieve greater confidence in process modeling; (iii) characterization of variations
among nano experimental runs under larger uncertainties. The first two tasks are more focused
on modeling the general growth kinetics while the last one adds among-runs variations model-
ing to improve prediction as well as provide guidance for future process improvement. Observed
nanowire weight growth data are used to demonstrate and validate proposed models.
In summary, this work tackles two central challenges in nanomanufacturing modeling, namely,
uncertain physical mechanism and limited data, by developing a cross-domain modeling frame-
work. The CDMV modeling framework developed here incorporates both physical knowledge
and experimental observations and achieves high modeling confidence. Moreover, this modeling
framework can be extended to characterize and identify potential variabilities sources and thus
reduce the process uncertainties.
For future work, one potential extension is to incorporate inherent nanowire growth variation
modeling. More specifically, instead of generalizing from one nanowire to the whole substrate and
using it to represent the general growth trend, one can model each nanowire individually. With
xii
such approach, one can further investigate the growth variabilities over each substrate which is not
possible with current macroscopic model.
xiii
Chapter 1
Introduction
This dissertation would model the nanostructure growth process kinetics and its variation. Such
models would serve as the basis for future process diagnosis, monitoring and control efforts and
thus helps to achieve reliable and cost-effective scale-up nanomanufacturing which is a key bottle-
neck in achieving the promises of nanomaterials.
The focus of the modeling work is on quantitative macro scale measurements, such as overall
weight, of nanostructure grown using ”bottom-up” instead of ”top-down” approach. The work
starts with a statistical model, and then extends to a general framework (Cross Domain Model
Building and Validation, CDMV) which incorporates physical knowledge and experimental obser-
vations. By integrating physical knowledge and experimental data, proposed models are partic-
ularly useful as there are large uncertainties in physical knowledges and limited data. With the
process model, one can further explore the among-runs variations and their sources.
In the following sections of this chapter, one will first introduce background, motivation and
challenges for nanomanufacturing modeling research. After an extensive review of nanomanufac-
turing modeling work, key research gaps would be identified. The research tasks and objectives
would be proposed at the end.
1.1 Background and Motivation
1.1.1 Nanotechnology Background and Potentials
Nanotechnology is the science and engineering about manipulating matters with scales ranging
from 1nm to 100nm (1 nm= 10
9
meters). The idea that one can manipulate individual molecules
1
and atoms originated from Dr. Feyman’s famous talk ”There is Plenty of Room at the Bottom”
[16]. The rapid development of nanotechnology only began after the scanning tunnel microscope
was developed in the 80s, however, as there was no way to see or control matters at this scale
before. Despite the field is relatively young, it has shown great potentials in high impact field
such as energy, medicine and information technology [47, 71, 78]. Better performance and new
functionalities are reported from laboratories around the world in past three decades [8, 23, 34, 76,
81,82,89]. For example, carbon nanotube, due to its superior physical and electrical properties, can
be used to build highly linear RF electronics and circuit [83]. Based on those exciting discoveries,
the world market for nanotechnology is projected between 1 and 3 trillions [32].
One reason behind the success of nanotechnology is that at nano-scale the quantum effects
begin to dominate the material properties. This provides opportunities to manipulating material
properties by controlling its scale. Taking gold nanowire as an example, its yield strength is shown
to be increasing with smaller scale and can increase up to more than 100 times compared with
macro-scale gold wires [20](Fig. 1.1). Another advantage of nano scale materials is their large
surface area comparing with bulk materials. A simple calculation will show that one cubic meter of
cubic nanoparticles can have total surface area of 6000 square kilometers which is close to the total
area of Delaware! Due to their large surface area, nanoparticles play an important role in catalyst
industry which counts for more than 20% of gross national product in industrial countries [49]. On
Nanowerk.Com, one can find a detailed list of nanotechnology, products and instruments.
1.1.2 Research Challenges in Nanomanufacturing Process Modeling
Despite the vast potentials and laboratory successes of nanotechnology, its applications are still
relatively limited. One vital bottleneck that hinders the transformation of nanotechnology from
laboratory to industrial success is nanomanufacturing: the typical process yield for nanodevices is
10% or less [42]. This low process yield, due to both limited process understanding and control,
poses great challenge for reliable and cost-effective production of nanostructures.
2
Figure 1.1: Yield strength of gold nanowire as a function of lateral dimension [20]
In order to achieve scalable nanomanufacturing, process kinetic modeling is essential. It pro-
vides a quantitative model to characterize the temporal growth trend and is the basis for further
variation modeling. Nanomanufacturing process modeling, in general, faces two major challenges:
• There are only limited physical knowledge for nanostructure growth processes.
Despite of numerous studies in nanostructures growth area, we are still often largely uncer-
tain about the physical mechanisms driven the growth process under a certain condition
[69, 70, 85]. The complexity comes from the fact that there are often multiple mechanisms
at work in one growth process and it is very difficult to identify their relative contribution.
Furthermore, physical model derived from first principles often involves large amount of
unknown parameters which makes them difficult to be directly applied in a manufacturing
setting.
• Experimental observations are expensive and time-consuming to obtain.
3
Nanostructure growth experiments are usually costly due to expensive equipment(such as
metal-organic chemical vapor deposition(MOCVD) system) as well as requirement of con-
stant human attention. Moreover, the measurement of nanostructures often involves equip-
ment such as scanning electron microscopy(SEM), transmission electron microscopy(TEM)
or atomic force microscope(AFM) [24, 41, 48]. Besides significant equipment cost, one can
only view/measure a tiny portion of the substrate at one time. As a result, it is very time-
consuming to obtain experimental observations.
Due to the limitation both in physical knowledge and experimental observation, traditional pure
physical or statistical modeling strategy often does not work well in the case of nanomanufacturing
applications and a novel strategy combining the understanding in both domains is needed.
1.2 State of Art on Nanomanufacturing Modeling
In this section, one will focus on the modeling works on one-dimensional nanowires growth as
they are among the most widely studied nanostructures due to their potential wide applications
[8, 23, 25, 29, 33, 56, 82, 84, 89]. Furthermore, as the study will focus on a Pd-catalyzed silica
nanowire growth process, those modeling works on nanowires are most relevant.
1.2.1 Physical modeling
Physical modeling represents the modeling approach that describes the relations among nanostruc-
ture features and growth conditions based on the physical/chemical laws governing the processes.
As physical models depend on the underlying growth mechanism, one will review modeling efforts
on two potential mechanisms relating to the silica nanowire growth experiment, namely, vapor-
liquid-solid(VLS) and oxide-assisted growth(OAG).
4
Figure 1.2: Comparison between VLS and OAG growth mechanism [102]
Physical models based on VLS mechanism
Before introduce the VLS growth mechanism, the concepts of absorption and diffusion should be
introduced. In general, absorption is a process in which atoms or molecules enter bulk liquid or
solid phase while diffusion is the process of materials move from one site to another following
thermodynamic laws.
VLS growth mechanism (Fig. 1.2) was first proposed in 1964 by Wagner and Ellis [77] to
explain Si whisker growth with the understanding that the process is driven by absorption. A
diffusion based growth mechanism was also proposed in literature [6, 64]. Later on, [22] argued
that the growth was mainly due to direct deposition instead of diffusion. But again in 2005, [35]
reported evidences for sidewall and surface diffusions for slow growth of Si NW with 70-200 nm
diameters. Therefore, neither absorption nor diffusion based models can be excluded for metal
catalyzed VLS growth.
5
The development of VLS growth modeling follows two tracks: theoretical modeling and exper-
imental description. In the theoretical aspect, Ruth and Hirth [64] proposed a diffusion-induced
NW growth rate model:
dL
dt
=
2
L(N
1
N
0
)
R
; if NW lengthLL
f
dL
dt
=
2
L
f
(J
N
0

)
R
; ifL>L
f
(1.1)
whereD is diffusion coefficient, is mean life-time of the adatom on the NW sidewall,
is atomic
volume of Si, J is impingement flux, R is NW radius, N
0
is adatom concentration at the liquid
alloy, andN
1
is adatom concentration at the base substrate.L
f
=
p
2D is diffusion length.
Roper et.al [62] proposed an absorption-induced model which incorporates surface energy of
different interfaces and was able to predict catalyst concentration and supersaturation in droplet
as well as solid-liquid interface shape modification mentioned in [22]. Schwarz and Tersoff [67]
use a continuum approach in which some realistic features appeared automatically such as the
tapered wire base. In general, in stead of predicting growth rate, those models are usually used to
understand various phenomenons observed in NW growth.
Dubrovskii et.al [15] proposed a model that unified absorption-induced, diffusion-induced
models, and compared their model with experimental data for Si and GaAs NWs:
dL
dt
=V
0
[(1 +
R
1
R cosh()
)( + 1) (1 +
R
2
R
tanh())( + 1)
1
V

dR
dt
] (1)V (1.2)
whereR
1
andR
2
are used to consolidate various parameters in diffusion induced contribution and
have no intrinsic physical meanings, R is the droplet radius, denotes supersaturation in nucle-
ation, denotes supersaturation of gaseous phase,V is deposition rate, is the ratio of whisker
lengthL to the adatom diffusion length on the side surface, V

is a kinetic parameter relating to
liquid and solid phase volume ratio, and is the relative difference between the deposition rate and
6
growth rate of the substrate surface. Moreover, in 2008, Schwalbach and V oorhees [66] calculated
and plotted the phase diagram for Au-Ge VLS growth system.
The experimental work focuses on the observation and measurement of VLS NW growth.
In [92], Wu and Yang reported direct observation of VLS growth of Si NW using in situ TEM
images. Kikkawa et al. [35] measured silicon NW growth under different temperature settings
and estimated the growth rate as well as activation energy. Kodambaka et al. [38] reported based
on in situ TEM that Si and Ge NW may have different growth mechanisms: VLS for Si, and
VLS/VSS for Ge. Sekhar [70] reported selective growth of amorphous silica NWs on a silicon
wafer deposited with Pt thin film and established VLS mechanism with PtSi phase acting as cata-
lyst.
Physical Models based on OAG mechanism
Besides metal catalyzed VLS mechanism, OAG is also proposed as a growth mechanism of semi-
conducting NWs. One can refer to Fig 1.2 for the comparison of the two. In [86], Wang et al.
proposed by thermal evaporation of a powder mixture of Si and SiO
2
, one can obtain Si NWs.
The mechanism is that SiO vapor generated from powder mixture would form a shell around
Si nanoparticles and resulted in a polycrystalline Si core and a SiO
2
shell. The advantage of
this growth mechanism is that metal catalyst impurities can be avoided. Zhang et.al [101] pro-
vided theoretical support for the mechanism by computing the energy of Si
n
O
m
structure. It is
claimed [102] that OAG is capable of producing large quantities of Si NWs and it can be utilized
for Ge, Carbon as well as III-V compound semiconducting NWs.
The fact that VLS and OAG may present in one growth setting and shifting of relative contribu-
tions of different mechanisms is shown in [58] and it is observed that for temperature higher than
1100 °C, VLS mechanism is dominant and with temperature lower than 1050 °C, OAG is domi-
nant. The coexistence of two mechanisms is also observed in [69] under 1100 °C with Pd catalyst.
As the two growth mechanisms’ coexistence is widely reported [36, 73, 97], imaging, structural,
and chemical analysis has been used to determine the mechanism but without much success [5].
7
1.2.2 Statistical modeling
Statistical techniques have been introduced into NW growth studies as they are suitable to charac-
terize inherent variabilities. Unlike physical modeling, statistical modeling treats the process as a
black box and focuses on inferencing the relationship between growth conditions and results using
observation data only. In methodology side, design of experiment (DOE) is popular in statistical
NW growth modeling due to the expensive and time consuming nature of NW growth studies. For
example, using 2
63
III
fractional factorial design (FFD), Shafiei et al. [72] identifies that thickness
of the gold layer, synthesis temperature and synthesis time are the key factors in determining the
diameters of VLS-synthesized ZnO NWs. After obtaining the main effects, Box-Behnken design
(BBD) was used to create a response surface. In summary, 25 experiment runs were needed: 8
for FFD effects screening and 17 for BBD response surface creating. The experiment strategy is
summarized in Figure 1.3.
Figure 1.3: Experiment Strategy for [72]
In another study, Xu et al. [96] find the optimal reaction settings for ZnO NW growth and
thus improves growth quality with pick-the-winner rule and one-pair-at-a-time main effect analy-
sis. Four control factors were changed in their experiment process: furnace temperature, time in
furnace, concentration of zinc precursor and amount of capping agent compared to zinc precursor
8
concentration. Using a seven-stage experimental sequence, they found that with precursor concen-
tration in the vicinity of 1 mmol/L, at reaction temperature around 80 °C, for about 30 hours, one
can obtain best quality ZnO NW with aspect ratio at 23.
Moreover, statistical modeling has the potential to optimize growth conditions while keeps
process robust to variations. In [11], Dasgupta et al. proposed a multinomial regression model to
compute probabilities of obtaining each of three one-dimensional CdSe nanostructures: nanosaws,
NWs and nanobelts. Using Monte Carlo simulation, they maximized the probability of obtaining
one specific structure while making sure that the variance is under control.
Studies mentioned above all focus on macroscale process optimization and variation reduction.
Besides them, statistical modeling was also developed to describe nanoscale variations among
neighboring nanowires which are barely understood in physics [93, 94].
Xu and Huang [94] innovatively established a Gaussian Markov random field model to extract
nanowire interaction patterns for any local region of interest on the substrate. Because of the
important roles played by nanowire interactions such as competing for source materials on local
morphology, their interaction analysis not only provides a metric for assessing nanostructure qual-
ity, but also enables a method to automatically detect defects and identify defect patterns. Figure
1.4 shows the estimated interaction patterns based on real collected lengths of ZnO nanowires [94].
From the figure one can see, local peaks of estimated interactions (dark color clusters) identify
nanowire bundles (white strips).
Later on, Xu and Huang [93] extended the interaction analysis in [94] from local regions to the
whole substrate. Their major efforts were to reduce the metrology requirements imposed by [94]
so that the interaction analysis could be performed with current nanostructure characterization
techniques e.g. SEM (Scanning Electron Microscopy) and under manufacturing relevant time
spans. Both a sampling strategy that selects which sites to measure over the substrate and an
analysis technique that estimates nanostructure interactions based on corresponding ”incomplete”
feature measurement were developed. Case studies in [93] showed that their developed approach
9
Figure 1.4: Level plots of estimated interaction parameters together with top view SEM images for
measured ZnO nanowires [94]. White strips are nanowire bundles (a type of defects) on top view SEM
images. Dark color clusters correspond to local peaks of nanowire interactions.
achieved comparable interaction estimation accuracy with significantly reduced metrology efforts.
It thus serves as a supporting tool for scale-up nanomanufacturing.
1.2.3 Physical-statistical modeling
The need for physical-statistical modeling becomes apparent as either physical knowledge or data
alone is sufficient. The combined information, however, is able to improve the understanding of
one growth mechanism. For example, Huang [27] proposed a Bayesian hierarchical framework
with consideration of scale effects (Figure 1.5). The time-space evolution of NWs at various sites
on a substrate is described by the growth model at each scale. Two major components of the
model are NW morphology which is characterized by growth kinetics and local variability which
is modeled by an intrinsic Gaussian Markov random field (Figure 1.6).
Physical-statistical model is also helpful in selection of growth mechanism among a few can-
didates. In [12], Dasgupta et al. successfully formulated a physical-statistical model consisted of
statistical Poisson regression model and physical diffusion model, and used it to identify correct
growth mechanism for a solution based ZnO NW growth process. Between the pore dominated
and diffusion dominated growth mechanisms, the estimation result from the physical-statistical
10
X(s,t) = η
k-1
(s,t) + φ(s) + ε
k, β
k-1
(s,t)
κ, ν
Model parameters
ξ

Kinetics-driven hierarchical model
for nanomanufacturing
Morphology-Local Decomposition
Process variables from growth kinetics
θ = (T, α, γ,…)
Figure 1.5: Physics-driven hierarchical modeling
of growth process at micro/nano scale [27]
X(s)
s
1
s
2
s
n
… … s
j
X(s
j
)
η(s)
φ(s
j
) + ε
NWs
Figure 1.6: Morphology-local decomposition of
NW length field [27]
model implied that the most growth with one polymer layer is pore dominated, but with 2 or more
polymer layers, the growth becomes diffusion dominated.
1.2.4 Application domains for different modeling approaches
These three nanostructure modeling strategies have their own suitable domains of application. As
illustrated in Fig. 1.7, with sufficient physical knowledge, physical modeling is readily a choice
even when experimental data is less accessible. When growth mechanisms are debatable, uncer-
tainties in the first principles will invalidate the purely physical modeling approaches. In the case
of MOVPE growth of GaAs nanowires, multiple physical models have been proposed based on
different mechanism such as diffusion induced growth [54] and absorption induced growth [33].
But there is no conclusive evidences for either of these two mechanisms.
On the other hand, statistical modeling faces issue of limited data in nanomanufacturing due
to costly growth experiment and structure characterization. Moreover, the collected data tends
to have large variations because of the poor in situ control of process variables such as growth
temperature gradient. As a result, a large pool of candidate models can statistically fit the data.
The data requirement for purely statistical modeling is rarely satisfied.
Physical-statistical modeling comes into play when physical knowledge or data alone is insuf-
ficient, but the combined information makes it feasible to improve the understanding of one growth
11
Amount of Data
Physical Knowledge

Statistical
Modeling

Physical Modeling

Physical-Statistical
Modeling
Cross-Domain
Modeling
Figure 1.7: Modeling Strategies Comparison
mechanism [27] or select among a few candidate mechanisms [12]. When the uncertainties in phys-
ical knowledge and data continue to increase and growth mechanisms still require to be derived or
investigated, a new modeling scheme, the cross-domain model building and validation (CDMV)
approach will be developed which would be discussed in details in Chapter 3.
1.3 Research Tasks and Objectives
This dissertation focuses on the modeling and and analysis of nanostructure growth kinetics and
variations. This work is aimed to provide a modeling framework to understand both the overall
growth kinetics and variations from different sources.
The major research tasks include:
1. Statistical weight kinetics modeling
This research task aims to understand and model the kinetic aspect or the change of substrate
weight over time in the selective growth of silica nanowires catalyzed through Pd thin film.
Six different weight kinetics models in predicting weight changes during growth would be
investigated. Model estimation and comparison would be conducted using both maximum
likelihood estimation (MLE) and Bayesian approaches. Owing to the embedded kinetics
12
information in the nonlinear growth models, we propose Bayesian hierarchical modeling
based on its ability to incorporate prior knowledge, which is highly desirable especially with
limited process data.
2. Cross-Domain model building and validation(CDMV)
Understanding nanostructure growth faces issues of limited data, lack of physical knowl-
edge, and large process uncertainties. These issues result in modeling difficulty because
a large pool of candidate models almost fit the data equally well. For this task, we will
derive the process models from physical and statistical domains, respectively, and reinforce
the understanding of growth processes by identifying the common model structure across
two domains. This cross-domain modeling framework would be illustrated by studying the
weight growth kinetics of silica nanowire data as in previous task but under two tempera-
ture conditions. Physical insights thus obtained can then be used for prediction and control
purposes.
3. Characterization of variations among nano experimental runs
Variations among experimental runs are commonly observed in the bottom-up growth of
nanostructures. Such variabilities, coupled with large physical and measurement uncertain-
ties, make it challenging task to achieve better understanding of the mechanisms of growth
processes. The CDMV approach was developed to model nanofabrication process under
large physical and statistical uncertainties. However, the variations among nano experimen-
tal runs was not consider therein. In this part we propose to further incorporate analysis of
such variations into the CDMV framework. Process model parameters varying or invariant to
experimental runs are treated as random effects or fixed effects respectively. The improved
analysis of variation sources is expected to provide better confidence in model prediction
under large uncertainties. Furthermore, the physical sources of variations among runs can be
identified, which leads to control opportunity for nanomanufacturing. The obtained physical
insights can be used as guidance for future process improvement.
13
This main body of this dissertation would be organized in three chapters which are written
as research papers. Chapter 2,3 and 4 would be focused on each of the aforementioned three
research tasks. Finally, Chapter 5 concludes the dissertation with a summary of the contributions
and discussion for future research, respectively.
14
Chapter 2
Statistical Weight Kinetics Modeling
In the bottom-up nanomanufacturing, the fabrication is usually achieved from the growth of the
nanostructures. By studying the data collected from such processes, general patterns emerge. As
one can observe from Figure 2.1 that both nanowire weight growth and graphene area growth
show similar growth kinetics. As a result, by modeling the growth kinetics of one process, one can
uncover insights that are not only useful for the specific process modeled but also instructive for
the study of other similar processes.
● ● ● ● ● ● ● ● 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time(min)
Coverage
(a) Graphene Area Growth [90]
●●●●●● ●●● ●●● ●●● ● ● ● ●●● ●●● ●●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ●●● ● ●● ●● ● ● ● ● 0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight(mg)
(b) Nanowire Weight Growth
Figure 2.1: Different Nanostructure Growth Processes
This chapter will propose a statistical model building and selection approach to characterize
nanowire growth weight kinetics. Using the weight growth of silica NWs using Pd thin film as
catalyst under 1100°C as an example, multiple statistical models are proposed and compared.
A heteroscedastic exponential-linear model is identified to have the ability to chracterize the the
growth data well with nice physical intepretation. Furthermore, bayesian hierarchical modeling
15
approach is demonstrated as an efficient tool with limited data if one is confident in the model
structure.
2.1 Introduction
Among one-dimensional nanostructures, silica nanowires (NWs) have been investigated in
recent years due to their photoluminescent properties and excellent biocompatibility. Different
approaches, such as laser ablation [98], oxide-assisted growth [102], sol-gel template method
[100], carbothermal reduction [91], and gas-phase growth [87] (i.e. vapour-liquid-solid (VLS),
solid-liquid-solid, vapor-solid growth) have been explored to synthesize these oxide NWs. The
ability to produce stable NWs at desired locations in a controlled manner on planar substrates is
of critical importance for fabricating NW-based devices such as nanoFETs, nanotransistors and
microphotonic nanosytems [76]. Sekhar et al. [70] reported selective growth of silica nanowires
in silicon using Pt thin film as a catalyst. The mechanism of nanowire growth was established to
follow the VLS model with the PtSi phase acting as the catalyst. The mechanism was validated by
careful selection of Pt thickness, growth temperatures, and Ar gas flow. They claimed the advan-
tages of the synthesis approach are (a) tunable size and distribution of the parent nanoclusters,
(b) its ability to form metal-dielectric hybrid configurations, and (c) a simple and cost effective
solution for growing large-scale arrayable nanowires.
Since the VLS mechanism was first introduced by Wagner and Ellis [77], both adsorption-
induced VLS models [15, 22, 77] and diffusion-induced VLS models [15, 35, 64] have been
developed. Ruth and Hirth (1964) [64] proposed a NW growth rate model: dL=dt =
2
=(N
1
N
0
)L=R; if NW lengthL diffusion lengthL
f
=
p
2D; anddL=dt = 2L
f

(J
N
0
=)=R; ifL > L
f
; whereD is diffusion coefficient, is mean life-time of the adatom on the
NW sidewall,
is atomic volume of Si, J is impingement flux, R is NW radius, N
0
is adatom
concentration at the liquid alloy, andN
1
is adatom concentration at the base substrate. Dubrovskii
16
et al. [15] developed a more complicated model to unify the two types of VLS models in molec-
ular beam epitaxy. In either case, the length of NWs exhibits a rapid exponential growth at the
beginning and a linear growth afterwards.
Extending the aforementioned kinetics models to this silica NW growth process in an open
CVD furnace could be challenging, as it has completely different boundary conditions. NWs
grown in open tube systems tend to curl and bundle which causes a metrology issue. Therefore, in
this study I investigate the substrate weight change process or weight kinetics during VLS growth
of silica NWs. One focuses on identifying the proper weight kinetics model under uncertainties.
The contribution is to obtain the engineering insight on weight changes during growth.
2.2 Experimental Details and Results
Prime grade 2´´ n-type silicon wafers were used as substrates for subsequent Pd deposition. Pd was
sputtered onto Si substrate with film thickness being 5nm. High purity Ar was chosen as carrier
gas and was set to 25 SCCM throughout the course of the experiment. Prior to placing samples,
the furnace was flushed with Ar for 10 minutes to minimize interference from gaseous impurities.
Then the sample was slowly introduced into the furnace along with a Si support wafer (Si Source),
and heated up to 1100 °C. Once the furnace reached 1100 °C, the heating process was timed. Prior
investigation [70] indicates the occurrence of melting process of Pd-Si during the ramp up to 1100
°C. Figure 2.2 shows the experimental setup for NW growth.
A high-precision microbalance (Sartorius R200D) was used to measure the weight of the
nanowire sample. The microbalance has a readout accuracy of 0.01mg with a capacity to weigh
up to 199.9999 g. The microbalance is typically calibrated with a known calibration standard
before the sample measurement. After calibrating the equipment, the silicon sample is placed on a
Whatman’s filter paper, whose weight is zeroed-out before loading the sample. The sample is then
covered by a plexiglass chamber to avoid any ambient noise causing spurious data. The sample
is typically left for 10 seconds on the microbalance to get a stable reading and then the sample
17
Figure 2.2: Open Tube Furnace with Pd Coated Si Sample on Support Si Substrate
weight is recorded. The initial mass of the Pd coated Si sample was weighted and compared with
the mass of the same sample after the NW growth for the desired time. A set of weight changes
were recorded from 0.5 to 210 minutes with three replicates at each time point. Scanning Electron
Microscopy (SEM, Hitachi S800) was used to image the sample surface after furnace treatment to
observe the structure and morphology. The weight growth data is presented in table 2.1.
Figure 2.3 shows the SEM micrograph of silica NWs observed at 15, 20, 60, and 90 minutes.
It is evident that NWs are not straight with large variations among them. Hence investigating
overall weight changes make more sense to characterize this particular growth process. Figure 2.4
illustrates the weight changes of Silica NWs during the growth period for three replicates. It is
interesting to notice that the initial growth experienced weight loss. This might be due to source
Si evaporation [37] and/or measurement error. In model fitting one still retains the data before 15
minutes.
2.3 Statistical Weight Kinetics Modeling and Maximum Like-
lihood Estimation
Let the random variabley(t) denote the NW substrate weight change at timet, and letE[y(t)] =
(t;). In this section, I shall consider different functional forms of (t;) and examine how
well they fit the data (t
i
;y
i
);i = 1;:::;n obtained from experiments. First, I shall consider a set
18
Table 2.1: Weight growth data(mg)
Time(min) Replicate 1 Replicate 2 Replicate 3
0.5 0.06 0.04 0.05
1 0.04 0.04 0.03
2 0.01 0.02 0.01
3 -0.02 -0.01 -0.01
4 -0.05 -0.06 -0.04
8 -0.24 -0.37 -0.27
10 -0.16 -0.2 -0.19
15 -0.15 -0.18 -0.14
20 0.01 0.013 0.012
30 2 1.33 1.352
45 2.56 2.42 2.45
60 5.16 6.26 5.95
75 8 7.66 7.82
90 8.84 9.42 9.21
100 9.72 9.67 9.59
120 9.91 9.96 9.97
150 10.1 10.3 10.3
180 10.4 10.4 10.3
210 10.5 10.3 10.4
of models for which the variance of y(t) does not change with time. After identifying the most
appropriate functional form (t;) that explains the mean of the response, I will consider more
generalized models with heteroscedasticity, that is, non-constant variances.
2.3.1 Postulating and fitting models with constant error variance
One now assumes the following additive model with constant error variance:
y(t) =(t;) +; (2.1)
19
A
C
D
500 nm
B
500 nm
500 nm 500 nm
Figure 2.3: SEM Micrographs (25 KV , 20K) Illustrating the Morphology of the Pd Coated Si Wafer Heated
at 1100 °C for (A)15, (B)20, (C)60, and (D)90 mins
where N(0;
2
), and postulate four possible functional forms for the mean term(t;). The
parameters of each model are estimated using by maximizing the likelihood function of the param-
eters (maximum likelihood estimation or MLE). The variance of each estimate is obatined from
the corresponding diagonal element of the inverse of the Fisher information matrix ( [74], Ch3),
which is computed numerically.
Model 1. Exponential model: The curves of weight changes in Fig. 2.4 may simply suggest the
following exponential model:
(t;) =
1
exp(
2
=t) (2.2)
with parameters = (
1
;
2
).
20
0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight Gain(mg)
●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Replicates
R1
R2
R3
Figure 2.4: Weight Changes of Silica NWs Over Time for Three Replicates
The log-likelihood function for this model is given by:
L
1
(;
2
) =
1
2
2
n
X
i=1
(y
i

1
exp(
2
=t
i
))
2
(2.3)

n
2
log(
2
)
The estimated parameters are:
^
= (^
1
; ^
2
) = (15:44; 59:85), ^
2
= 0:53 with estimated
standard deviations 0:49; 3:16 and 0:10 respectively. Akaike’s information criterion, (popularly
known as AIC and defined as2L + 2p, where L is the maximized likelihood and p is the
number of parameters), is computed as 132.03. It is evident from Fig. 2.5 that model 1 fails to
capture the turning point of the growth curve and clearly deviates from the data at the late stage.
21
0 50 100 150 200
0 2 4 6 8 10 12
Time(min)
Weight Gain(mg)
Replicates
R1
R2
R3
Figure 2.5: Fitted Curve of Model 1
Model 2. Exponential-linear model with unknown change point t
0
and continuity at the first order
derivative: Based on the weight change curves in Fig. 2.4 and previous grow rate models
[15, 27, 35, 64], I can postulate the following exponential-linear model with unknown change
pointt
0
:
(t;) =
8
>
>
<
>
>
:

1
exp(
2
=t); tt
0
at +b; t>t
0
(2.4)
22
with = (
1
;
2
;t
0
). Let
1
(t) =
1
exp(
2
=t), and
2
(t) =at +b. Assuming that the curve
is continuous and differentiable att
0
, it is easy to see that
a =
1
(1

2
t
0
)e

2
t
0
; and (2.5)
b =

1

2
t
2
0
e

2
t
0
(2.6)
so that model 2 is essentially a three-parameter model.
The log-likelihood function can be written as
L
2
(;
2
) =
n
2
log(
2
)
1
2
2
(2.7)
n
X
i=1
[I
tt
0
(y
i

1
exp(
2
=t
i
))
2
+I
tt
0
(y
i
at
i
b)
2
]
wherea andb satisfy (2.5)-(2.6). andI
A
(t) = 1; ift2A;A can be any set; and 0; otherwise.
The estimates of ^
1
, ^
2
and
^
t
0
are obtained as 15.43, 59.82 and 200 respectively. In this case,
the parameter estimation degenerates to pure exponential estimation, which is undesirable.
Model 3. Exponential-linear model with unknown change pointt
0
and continuity att
0
: This model
is the same as model 2, except for the fact that I do not impose continuity at the first order
derivative, which is deemed as a much stronger assumption. This is thus a four-parameter
model with = (
1
;
2
;t
0
;a) and log-likelihood function as:
L
3
(;
2
) =
n
2
log(
2
)
1
2
2
(2.8)
n
X
i=1
[I
tt
0
(y
i

1
exp(
2
=t
i
))
2
+I
tt
0
(y
i
at
i
b)
2
]
23
0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight Gain(mg)
Replicates
R1
R2
R3
Figure 2.6: Fitted Curve of Model 3
whereb =
1
exp(
2
=t
0
)at
0
.
The estimates of ^
1
, ^
2
, ^ a, and
^
t
0
are 32.11, 105.65, 0.0092, and 86.37 respectively. The error
variance is estimated as ^
2
= 0:086 and the AIC is computed as 31:62. The standard errors of
^
1
; ^
2
; ^ a; and
^
t
0
are estimated as 2:65; 5:29; 0:0016 and 2:16 respectively, while the standard
error of ^
2
is 0.016. One noticed from Fig. 2.6 that unlike models 1 and 2, the linear phase
is captured and the resulting fit is much better than either of these models. Yet the transition
between two phases seems not smooth.
24
Model 4. An exponential-linear model with a smooth transition function: In order to ensure a
smooth transition between the linear and exponential phases without imposing the strong restric-
tion of model 2 (that is, continuity at the first order derivative or
0
1
(t
0
) =
0
2
(t
0
)), I introduce a
transition functionh(t;
) (see [3, 80]) that joins the two phases.
(t) = [1h(tt
0
;
)]
1
(t) +h(tt
0
;
)
2
(t) + (2.9)
where = (
1
;
2
;t
0
;a;
); N(0;
2
), and h(t;
) is continuous and smooth functions
statisfying conditions listed in Appendex 2.6.
0 50 100 150 200
0.0 0.2 0.4 0.6 0.8 1.0
t
h
γ
0.1
0.5
1
2
Figure 2.7: Transition functionh(t;
) with different
(t
0
=80s)
Model 4 is a smooth approximation of(t), and the change point is included in a continuous and
differentiable functionh(tt
0
;
) so that it can be estimated together with other parameters.
25
Based onh(t;
) = 1 1=(1 + exp(t=
2
)) suggested in [80], I developed a transition function
that satisfies a slightly different set of conditions(please refer to Fig. 2.7 and Appendix 2.6),
given the nature of
1
(t) and
2
(t) in model 4:
h(t;
) =
exp (
t

2
) exp (
t
0

2
)
1 + exp (
t

2
)
(2.10)
Comparing to the original transition function, the new transition function makes sure that when
t approaching 0, the transition function’s value is approaching 0 and thus makes sure that there
is no transition function caused-bias around time 0.
The likelihood function is:
L
4
(;
2
;
) =
n
2
log (
2
)
1
2
2
n
X
i=1
(2.11)
fy
i
[1h(t
i
t
0
;
)]
1
(t
i
)
h(t
i
t
0
;
)
2
(t
i
)g
2
withh(t;
),
1
(t),
2
(t) defined above.
The estimates of ^
1
; ^
2
; ^ a, and
^
t
0
are 31.68, 104.85, 0.0091, and 86.77 respectively. The error
variance ^
2
is estimated as 0.086, and AIC is 31:71. The standard errors of ^
1
; ^
2
; ^ a, and
^
t
0
are
estimated as 2.57, 5.18, 0.0016, and 2.20 respectively, while the standard error of ^
2
is 0.016.
The estimated
is 0:604 with standard deviation 1.5, and this small ^
assures that the distortion
brought to the fitted curve by the transition function is very small. In fact, by checking Fig. 2.8,
one can hardly see the smoothing effect and the fitted curve is very close to model 3’s result.
If one magnifies the area around the transition point, as shown in the Fig. 2.9, one can see the
transition is smooth.
26
0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight Gain(mg)
Replicates
R1
R2
R3
Figure 2.8: Fitted Curve of Model 4
Model Comparison: The summary of MLE estimation is given in Table 2.2 (Note that Model
2 is degraded to Model 1). To compare models, one fits the first four models again using the first
two replicates as training data and the third one as validation data. Clearly, models 1 and 2 are
different (and much worse) as compared to models 3 and 4. The mean square errors (MSEs) are
0:537 for models 1 and 2, 0:0595 for model 3, and 0:0596 for model 4. Quite naturally, the question
arises whether the gain obtained by introducing the complicated model 4 is substantial. To answer
this question, one tests the hypothesisH
0
:
= 0 versusH
1
:
> 0 using the likelihood ratio
test (LRT). LRT statistics is constructed as 2 (L
4
L
3
), whereL
3
andL
4
are the log-likelihood
given by Eqs. (2.8) and (2.11). (When
* 0, the transition model actually becomes model 3, i.e.,
the reduced model.) The likelihood ratio statistic asymptotically follows a
2
distribution with 1
df. The statistics is computed as 0.12 with ap-value of 0.725, which means there is not enough
27
86 88 90 92 94
8.8 9.0 9.2 9.4 9.6 9.8
Time(min)
Weight Gain(mg)
Model 3
Model 4
Model 5
Figure 2.9: Comparison of models 3,4 and 5 near transition point
evidence to rejectH
0
in favor ofH
1
. Therefore, model 3 seems to be adequate to explain the mean
part of the curve. Table 2.2 summarizes the MLE results for different models.
2.3.2 Generalizing to heteroscedastic model
Having identified model 3 as the best functional form for(t;), one now tries to make the model
more realistic by considering the variance structure. The exponential phase in model 3 usually
lasts shorter compared to the linear phase. It is relatively difficult to observe the growth during the
initial period, making the inherent measurement error much larger during that period. To check
whether the data reflects this, I consider the following modification of Eq. (2.1):
28
Table 2.2: Parameter Estimation Comparison among Four Models
^
1
^
2
^ a
^
t
0
^
2
Model1 Avg 15.44 59.85 NA NA 0.53
Std 0.49 3.16 NA NA 0.10
Model3 Avg 32.11 105.65 0.009 86.37 0.086
Std 2.65 5.29 0.002 2.16 0.016
Model4 Avg 31.68 104.85 0.009 86.77 0.086
Std 2.57 5.18 0.002 2.20 0.016
Model5 Avg 28.59 99.75 0.007 92.49 0.11,0.013
Std 1.70 4.32 0.001 1.71 0.025,0.005
Model 5. Exponential-linear with unequal variances in two phases:
y(t) =
8
>
>
<
>
>
:

1
(t;) +
1
; tt
0

2
(t;) +
2
; t>t
0
(2.12)
where
1
(t) and
2
(t) respectively represent the exponential and the linear components of
E[y(t)] in model 3. As in model 3, I impose the continuity constraint
1
(t
0
;) =
2
(t
0
;),
so that one have = (
1
;
2
;t
0
;a). One assumes
1
N(0;
2
1
) and
2
N(0;
2
2
), where

2
1
>
2
2
.
The log-likelihood function for model 5 is:
L
5
(;
2
1
;
2
2
) =
N
X
i=1
fI
tt
0
[
1
2
log (
2
1
)

1
2
2
1
(y
i

1
exp(
2
=t
i
))
2
]
+I
t>t0
[
1
2
log(
2
2
)

1
2
2
2
(y
i
at
i
b)
2
]g (2.13)
The estimated parameters are:
^
= (^
1
; ^
2
; ^ a;
^
t
0
) = (28:59; 99:75; 0:0067; 92:49), ^
2
1
= 0:11,
^
2
2
= 0:013 with standard errors (
^
1
;
^
2
;
^ a
;
^
t
0
) = (1:70; 4:32; 0:00075; 1:71),
^
2
1
= 0:025,
29

^
2
2
= 0:005. The AIC is computed as 17:11. The fitted model is shown in Fig. 2.10. By comparing
the estimated transition point from Fig. 2.9, one can see model 5 has slightly different and more
accurate estimation, which is critical to pinpoint the change point of growth process.
0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight Gain(mg)
Replicates
R1
R2
R3
Figure 2.10: Fitted Curve of Model 5
To test the assumption of unequal variances, I conduct LRT of the hypothesisH
0
:
1
=
2
versus H
1
:
1
>
2
. The LRT statistic is obtained as 2 (L
5
L
3
) =9.86, which is very
significant with respect to a
2
distribution with 1 df. Therefore it is reasonable to consider a
model with unequal variances during the two phases.
30
2.4 A Bayesian Hierarchical Model For Estimation with Lim-
ited Observations
In the previous section, I identified model 5 as the most appropriate model that accounts for the het-
eroscedasticity. Model 5 has a nice and realistic physical interpretation. However, the frequentist
analysis suffers from the limitation that the standard errors of estimators are obtained by utilizing
the asymptotic normality of the MLE. The accuracy of such an approximation will be questionable
when limited observations are available (often the case in nanomanufacturing). The Bayesian hier-
archical model considers the structure of multiple parameters and reflects the dependence among
parameters in a joint probability model [63]. It not only incorporates all the features of model
5, but also utilizes available prior information about the parameters to address the concern of the
lacking of data. I thus postulate the following model:
Model 6. A hierarchical exponential-linear model:
Level 1 : yN(;
2

)
Level 2 : =I
tt
0

1
exp (
2
=t)+
I
t>t
0
(at +
1
exp (
2
=t
0
)at
0
) (2.14)

2

=I
tt
0

2
1
+I
t>t
0

2
2
Level 3 : Priorsfor = (
1
;
2
;t
0
;a);
and
2
1
;
2
2
:
The model at level 2 of the hierarchical framework is nothing but the model 5 discussed previ-
ously.
Assuming that one does not have any prior information before the experiment, one
chooses the following non-informative priors for estimation:
1
;
2
Unif(0; 100); t
0

Unif(10; 150);
1
N(10; 10
4
);
2
Unif(10; 300), andaN(0; 10
4
).
31
Table 2.3: Posterior Inference of Model 6
variable mean std 2:5% 97:5%
^
R

1
27.82 1.751 24.56 31.34 1.000067

2
97.75 4.514 89.08 106.8 1.00058
a 6.63E-3 9.67E-4 4.70E-3 8.52E-3 0.9999926
t
0
93.36 2.041 90.29 97.73 1.000031

1
.353 .041 .283 .444 0.9999975

2
.139 .0330 .0922 .219 0.999996
99750 99850 99950
20 25 30 35
convergence of alpha1
index
alpha1
99750 99850 99950
80 90 100 110 120
convergence of alpha2
index
alpha2
99750 99850 99950
0.003 0.005 0.007 0.009
convergence of a
index
a
99750 99850 99950
85 90 95 100 105
convergence of t0
index
t0
Figure 2.11: Different chains simulated using WinBugs
32
histogram of alpha1
alpha1
Density
20 30 40 50 60 70
0.00 0.05 0.10 0.15 0.20
histogram of alpha2
alpha2
Density
80 100 120 140 160
0.00 0.02 0.04 0.06 0.08
histogram of a
a
Density
0.000 0.005 0.010 0.015
0 100 200 300 400
histogram of t0
t0
Density
75 80 85 90 95 100 110
0.00 0.05 0.10 0.15 0.20
Figure 2.12: Posterior distributions of
1
;
2
;a andt
0
Markov Chain Monte Carlo (MCMC) simulation [46] through WinBUGs software [75] was
applied to draw from the posterior distributions of the parameters. One used 4 chains whose
initial values were widely separated. The length of each chain is 100000, but only the last 50000
simulated values were used for inference while the first 50000 observations were treated as draws
during the “burn-in” period. A thinning rate of 100 was adopted to alleviate the auto-correlation.
The convergence of different chains was confirmed by the Gelman-Rubin statistics
^
R which is
close to 1. The simulation results are shown in Table 2.3. Plots of different chains and histograms
33
from the posterior draws of all the parameters shown in Fig. 2.11 and Fig. 2.12 respectively. Both
these figures indicate convergence of the simulations.
2.5 Summary and Conclusion
In this chapter, one postulated six weight kinetics models based on the data collected under one
growth condition. All the models were embedded with growth kinetics information. Through
maximum likelihood estimation (MLE) and Bayesian hierarchical model estimation, one compared
all the models. The exponential-linear model with unequal variances in two phases fits the data
best.
Since the MLE utilizes the asymptotic normality, Bayesian hierarchical model is more suitable
when data is limited. Particularly when one obtained confidence in the model structure from ini-
tial experimental study, Bayesian approach was shown to be more efficient for investigating new
nanowire growth conditions.
34
2.6 Proof for Transition Function Choice
Given in [80],h(t;
) is continuous and smooth functions statisfying the following conditions:
8

>0
lim
t!+1
(1h(t;
))
1
(t) = 0
8

>0
lim
t!+1
(1h(t;
))
2
(t) = 0
8

>0
lim
t!1
h(t;
)
1
(t) = 0
8

>0
lim
t!1
h(t;
)
2
(t) = 0
8
t6=0
lim

!0
h(t;
) =
8
>
>
<
>
>
:
1; t> 0
0; t< 0
8

>0
h(t;
); h is non-decreasing in its domain.
In this study, one noticed that as time can only be positive, one should make the third and forth
conditions stronger by changing1 to 0.
35
In the following section,
1
(t) and
2
(t) are defined as in (2.4) andh(t;
) is defined in (2.10),
then one can verify that:
8

>0
lim
t!+1
(1h(t;
))
1
(t) =
8

>0
lim
t!+1

1
exp (

2
t
)
1 + exp (
t
0

2
)
1 + exp (
t

2
)
= 0
8

>0
lim
t!+1
(1h(t;
))
2
(t) =
8

>0
lim
t!+1
(at +b)[1 + exp (
t
0

2
)]
1 + exp(
t

2
)
= 0
8

>0
lim
t!0
h(t;
)
1
(t) =
8

>0
lim
t!0

1
exp(

2
t
)
exp (
t

2
) exp (
t
0

2
)
1 + exp (
t

2
)
= 0
8

>0
lim
t!0
h(t;
)
2
(t) =
8

>0
lim
t!0
(at +b)[
exp (
t

2
) exp (
t
0

2
)
1 + exp (
t

2
)
] = 0
8
t6=0
lim

!0
h(t;
) =
8
>
>
<
>
>
:
8
t6=0
lim

!0
exp(
t

2
)exp(
t
0

2
)
1+exp(
t

2
)
= 1; t> 0
8
t6=0
lim

!0
exp(
t

2
)exp(
t
0

2
)
1+exp(
t

2
)
= 0; t< 0
h(t;
) =
exp (
t

2
) exp (
t
0

2
)
1 + exp (
t

2
)
,
and it can be shown that partial derivatives are larger than 0 for allt and
. All conditions are met
with the choice of transition function.
36
Chapter 3
Cross-Domain Model Building and
Validation (CDMV)
Understanding nanostructure growth faces issues of limited data, lack of physical knowledge, and
large process uncertainties. These issues result in modeling difficulty because a large pool of
candidate models almost fit the data equally well. Through the Integrated Nanomanufacturing
and Nanoinformatics (INN) strategy, one derives the process models from physical and statistical
domains, respectively, and reinforce the understanding of growth processes by identifying the com-
mon model structure across two domains. This cross-domain model building strategy essentially
validates models by domain knowledge rather than by (unavailable) data. It not only increases
modeling confidence under large uncertainties, but also enables insightful physical understanding
of the growth kinetics. This method is presented by studying the weight growth kinetics of silica
nanowire under two temperature conditions. The derived nanowire growth model is able to provide
physical insights for prediction and control under uncertainties.
3.1 Motivating Example
In this chapter, from data of silica nanowire growth under another process condition, namely, 1050
°C compared with 1100 °C , it is observed that the growth behavior changed significantly (Fig.3.2)
and the previous statistical model cannot link the two process conditions. Besides different growth
behaviors observed under the two temperature settings, there is no theoretical investigation of
weight kinetics for nanowire growth. As a result, large uncertainties are in both physical and
statistical domains. In this case, a new modeling scheme called cross-domain model building and
37
validation (CDMV) approach is developed. The CDMV derives models from both physical and
statistical domains and cross validate the model from two domains against one another (Fig. 3.1)
which is challenging as it requires understanding in both domains.
With the CDMV model, successfully characterized the growth behavior under both 1100 °C
and 1050 °C. Moreover, the potential physical mechanism shifting for two process conditions
which explains the changing of growth behaviors is uncovered.
More importantly, the situations such that both physical knowledges and experimental data are
limited are common in nanomanufacturing and the CDMV approach would be particularly useful
in those situations.
Cross-domain Modeling
Physical
Knowledge
Uncertainty
Modeling
Physical Model Domain
Statistical Model Domain
Data Validation
Figure 3.1: Crossing Domain Model Building and Validation
In following sections, domain knowledge which can potentially be utilized to develop physical
and statistical models will be presented first. After that I will illustrate the detailed procedure of
developing CDMV approach for weight kinetics study of silica nanowire growth. Summary will
be presented in the last section.
38
3.2 Related Physical and Statistical Models and Uncertainties
3.2.1 Candidate Physical Models
There is great uncertainty regarding the understanding of growth mechanisms in silica nanowire
growth. According to [70], the mechanism of nanowire growth was VLS growth with silicide phase
acting as catalyst. The mechanism was experimentally validated by the selection of metal film
thickness, growth temperature and gas flow. However, [69] claimed evidences of oxide-assisted
growth. At present, the existences and relative importances of those two different growth mecha-
nisms are still open for discussion.
In general, existing modeling work for nanowire growth aim to understand length growth based
on VLS mechanism. Since it was first introduced by Wagner and Ellis [77], both adsorption-
induced VLS models [22] and diffusion-induced VLS models [64] have been developed. Ruth and
Hirth [64] proposed a NW growth rate model:
dL
dt
=
2
L(N
1
N
0
)
R
; if NW lengthLL
f
dL
dt
=
2
L
f
(J
N
0

)
R
; ifL>L
f
(3.1)
whereD is diffusion coefficient, is mean life-time of the adatom on the NW sidewall,
is atomic
volume of Si, J is impingement flux, R is NW radius, N
0
is adatom concentration at the liquid
alloy, and N
1
is adatom concentration at the base substrate. L
f
=
p
2D is diffusion length.
Dubrovskii et al. [15] developed a more complicated model to unify the two types of VLS models
in molecular beam epitaxy:
dL
dt
=V
0
[(1 +
R
1
R cosh()
)( + 1) (1 +
R
2
R
tanh())( + 1)
1
V

dR
dt
] (1)V
where R
1
and R
2
describing diffusion induced contribution, R is the droplet radius, denotes
supersaturation in nucleation, denotes supersaturation of gaseous phase,V is deposition rate,
39
is the ratio of whisker lengthL to the adatom diffusion length on the side surface,V

is a kinetic
parameter relates to liquid and solid phase volume ratio while is the relative difference between
the deposition rate and growth rate of the substrate surface.
As discussed above, the growth can also be attributed to oxide-assisted growth mechanism.
But there is no quantitative understanding due to the difficulty of measuring the curly and bundled
nanowires grown under this mechanism.
The complicated model formulation for VLS growth, coupled with the lack of quantitative
models for oxide assisted growth, makes the findings through pure physical modeling approaches
inconclusive.
3.2.2 Candidate Statistical Models
Growth process modeling has been under statistical investigation as well [4,68]. Examples include
growth of population [57], novel technology adaptation [17] , energy demand [2] as well as growth
of tumors [44]. The plethora of statistical models in this area can be categorized into two major
approaches: polynomial fitting with on consideration of subject knowledge, and sigmoid function
fitting based on assumptions of growth processes [68]. I only consider the second approach in this
paper as polynomial fitting does not provide any physical insight.
The sigmoid growth models, however, have difficulties in model selection. To illustrate this
point, I’ll use two popular sigmoid growth models, namely, logistic and Gompertz growth model
to fit the data and show the difficulty in choosing the appropriate model. Note that the nanowire
weight change data in this study has S-shape as well.
The logistic growth model assumes the growth rate is proportional to the current size and
remaining growth potential, and the growth is driven by autocatalytic reaction [4]. Logistic growth
has a increasing growth rate at first and declining growth rate with saturation afterwards. It provides
40
a simple interpretation when the growth rate increases with a growing population but is limited by
available resources. The parametrization I used here is due to Fletcher [18]:
W (t) =
W

1 +
W

W(0)
W(0)
exp(
4r
M
t
W

)
(3.2)
in whichW

is maximum growth weight,W (0) is initial weight andr
M
is the maximum growth
rate. It should be noted that for logistic growth, a non-zero initial weight is required to start growth
as the growth rate is proportional to the existing growth. In this case, it is assumed that the initial
weight represented initial nucleation.
●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 50 100 150 200 250 300
0.0 0.2 0.4 0.6 0.8 1.0
Time(min)
Weight(mg)
Losgistic
Gompertz
(a) 1050 degrees
●●● ●●● ●●● ●●● ●●● ● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● 0 50 100 150 200
0 2 4 6 8 10
Time(min)
Weight(mg)
Losgistic
Gompertz
(b) 1100 degrees
Figure 3.2: Logistic and Gompertz Model under different temperatures: solid line for Logistic model and
dotted line for Gompertz model
The Gompertz growth model has been applied to tumor [43] and population growth [39] studies
with the assumption that the relative growth rate decreases with logarithm of growth size. It has
relative slow growth at the beginning:
W (t) =W

exp( exp(k(tt
i
))) (3.3)
whereW

is maximum growth weight,k is a rate constant andt
i
is inflection point.
41
I fit those two models using nls() function in R with the silica nanowire data (Fig. 3.2). The
residual sum of squares of both models under two conditions are summarized in Table.3.1. As
one can see, it is difficult to identify which model is superior simply from the fitting results. Note
that the two models have different assumptions regarding the growth mechanisms. Uncertainties in
nanomanufacturing data make it difficulty to select among candidate models for purely statistical
models.
The analysis of existing physical and statistical modeling approaches show the need of a new
modeling approach to handle uncertainties in both physical and statistical domains for nanomanu-
facturing.
3.3 Cross-Domain Model Building and Validation Approach to
Understand Nanomanufacturing Processes
Cross-domain model building and validation is a strategy of Integrated Nanomanufacturing and
Nanoinformatics [26]. One derives the process models from physical and statistical domains,
respectively, and reinforces the understanding of growth processes by identifying the common
model structure across two domains. This cross-domain model building strategy essentially vali-
dates models by domain knowledge rather than by (unavailable) data.
The approach is demonstrated using the silica nanowire growth process studied in [28]. Based
on the existing physical understandings, the nanowire growth is largely attributed to material
absorption into nanowire surface. As shown in Figure 4.3, there are two sources contributing
to the nanowire growth: the direct impingement of silicon vapor to the Si-Pd droplet at the top of
Table 3.1: Residual Sum of Squares of Logistic and Gompertz Model
Temperatures Logistic Gompertz
1050 0.3326 0.3079
1100 6.1487 4.6114
42
the nanowire and source supply from nanowire side surface. One will analyze the weight change of
a single nanowire and then extends the model to the whole substrate through uniformity assump-
tion. Thus the behavior of a typical nanowire’s growth in the middle of the substrate is assumed
to represent the overall weight growth on the substrate. This assumption, although restrictive and
idealized, still captures the main behaviors of the growth process as one will show later.
h
2r
h
Constant
Radius
Constant Height
2r
h
2r
Side
Growth
Top
Impingement
Figure 3.3: Schematic of growth from two sources
One will first derive the growth models for top impingement and side growth separately based
on the available physical understanding.
3.3.1 Growth Attributed to Direct Top Impingement
The direct impingement of vapor Si particles are absorbed by the droplet formed of Pd catalyst
and Si, then nucleate and contributed to the nanowire growth [62, 77]. While the nanowire grows,
the Pd catalyst in the droplet, however, gradually diffuses into the silica nanowire. As a result,
the growth is fast at first when the catalyst is abundant and will slows down as the Pd catalyst is
consumed. When the Pd is exhausted, the growth stops. I postulate the rate of weight change as
dW
dt
/ (W

W ) (3.4)
43
W is the growth weight,t is the growth time andW

is the maximum possible weight.
The proposed model bases on assumptions that the growth rate is proportional to the catalyst
amount left. FurthermoreW

is a constant determined by the amount of Pd catalyst and should
not depended on growth timet nor weightW . I also assume that the Pd catalyst consumption is
proportional to the weight growth. The formulation results in desirable growth behavior observed
in experiments: large growth rate at first and steady decreasing growth rate with final stop. More
importantly, with current data, it would be impossible to fit more advanced physical models as
described in [62] due to multiple unknown physical coefficients and constants. The simple formu-
lation has the advantages of both estimability and easy intepretability.
Equation (3.4) perfectly matches the confined exponential growth mode in statistical literature
[4]:
dW
dt
=a(W

W ) (3.5)
wherea is a rate coefficient.
Remark: Model structures from both physical and statistical domains agree with each and
enhance one’s belief and understanding of nanowire growth driven by top impingement of mate-
rials. On the other hand, better insight of the underlying mechanism for statistical confined expo-
nential model is gained in the context of nanomanufacturing.
3.3.2 Growth Attributed to Side Absorption
The nanowire growth attributed to the absorption from side surface is harder to characterize as
there are more factors involved, namely, nanowire side area and space/resource competition among
different nanowires. The relationship between nanowire side area and growth rate depends on
the geometric shape of the nanowire and consider two growth scenarios: (1) constant diameter
nanowire, and (2) constant height nanowires.
When nanowires do not taper and keep a constant diameter during the growth process, all Si
absorbed on the side face diffuses to the top and contributes to the length growth. With assumed
44
constant concentration of Si vapor around the nanowires, the growth is proportional to the size of
absorbing area, thus I can deduce:
dW
dt
/S = 2rh (3.6)
W =r
2
h (3.7)
In (3.6) and (3.7),S is the side surface area of nanowire,r is the radius of nanowire,h is the
height and is the density. Asr is constant in this case,
dW
dt
/
2W
r
/W (3.8)
On the other extreme, the nanowires do not grow in height but grow laterally over the whole
nanowire uniformly. Based on (3.6) and (3.7), holdingh as a constant:
dW
dt
/S = 2rh =
2
p
Wh

/
p
W (3.9)
The derivation is based on the assumption that onlyr changes in this case and other parameters
can be treated as constants over time.
The real growth is somewhat between the two scenarios and the model can be generalized as:
dW
dt
/W

; 0:5 1 (3.10)
The growth from side surface, however, is also limited by the space for the nanowires and
I assume that the space/resource competition can be modeled as growth rate is proportional to
remaining space/resource:
dW
dt
/W

(W
0
W ) (3.11)
Equation (3.11) can describe the initial growth rate increase with growth side surface area as well as
the limiting behavior when the competition of space/resource for growth among nanowires makes
45
further growth impossible. W
0
is used to denote the maximum weight possible due to the space/
resource limitation.
Remark: The uncertainty regarding the side growth is clear in the physical domain. But the
logistic growth model in statistical domain is quite close to Equation (3.11) with = 1. The
similarity in the model structures from two domains greatly assist us to choose the S-shape logistic
growth model over the Gompertz growth model.
3.3.3 Cross-Domain Modeling of Nanowire Growth
Due to the complexity of nanowire growth under various conditions, the true growth model can be
(i) dominated by either top impingement or side absorption, or (ii) driven by both growth mecha-
nisms with varying percentage of contributions. One therefore propose a general model for overall
nanowire weight growth, which combines Equations (3.11) and (3.5):
dW
dt
=a

(W

W ) +aW

(W
0
W ) (3.12)
In this model, coefficients a

and a can be viewed as weights for top impingement and side
absorption mechanisms, respectively. The weights can be determined through statistical model
selection procedure for specific growth conditions. The corresponding growth mechanisms can
thus be uncovered within this modeling framework.
The model (3.12), even with = 1, however, poses challenges for parameter estimation as
there are four unknown parameters in a quadratic polynomial ofW on the right hand side of the
differential equation. As the quadratic expression can be uniquely determined by three parameters,
one of the four unknown parameters becomes inestimable. Furthermore, the physical knowledge
ofW

andW
0
are very limited.
46
The model structure given in (3.12) reminds us the combined model of confined exponential
and logistic model discussed in statistical literature [4]:
dW
dt
=a

(W

W ) +aW (1
W
W

) (3.13)
=
aW
2
W

+ (aa

)W +a

W

(3.14)
This formulation requires the consolidation of two parametersW
0
withW

in (3.12) into the
same parameterW

. What it means is that for the same growth process, the limiting weight, con-
tributed either from top impingement or from side absorption, should be the same. This assumption
is taken and it turns out that numerical result of estimation seems to support it.
To obtain a closed formed solution for the differential equation is possible. Letk(t) = exp((a+
a

)(t +C)), then:
W =
W

(a

+ak(t))
a(1 +k(t))
(3.15)
C is the constant associated with initial valueW
0
.
W
0
=
W

(a exp[(a +a

)C]a

)
a(1 + exp[(a +a

)C])
(3.16)
Or one can write:
C =
1
a +a

ln(
a

W

+aW
0
aW

aW
0
) (3.17)
The growth rate function would be:
dW
dt
=
W

(a +a

)
2
k(t)
a[1 +k(t)]
2
(3.18)
47
Based on the growth rate function, the inflection point would be:
t
i
=C;W
i
=
(aa

)W

2a
(3.19)
Considering the error of weight measurement, I assume there is a random error term which
follows a normal distributionN(0;
2
) with
2
unknown.
3.3.4 Uncover Growth Kinetics for Silica Nanowire Growth
In this section, one will apply the cross-domain modeling result to silica nanowire growth and
analyze the growth kinetics under two growth temperatures: 1100 °C and 1050 °C.
Nanowire model estimation for two growth conditions: Detailed descriptions of the process
conditions and data for 1100 °C (57 data points observed) can be found in the previous work [28].
To estimate the parameters = (a

;a;W

;C) in (4.1), the non-linear regression is performed by
using PORT library [19] implemented in nls() function in R language. As the algorithm does not
guarantee convergence to a global minimum due to the fact that W (t) is not convex, one starts
from five widely separated sets of initial values shown in Table 2.
Table 3.2: Initial Parameter Values for 1100 °C
Run Number a

a W

C
1 0.2 0.01 8 -80
2 0.1 0.005 6 -60
3 0.05 0.002 4 -40
4 0.02 0.001 2 -20
5 0.01 0.0005 1 -10
Five different sets of initial values give same estimation for = (a

;a;W

;C) for 1100 °C:

1100
= (0:0024 0:0005; 0:068 0:004;
10:24 0:08;57:04 0:73)
48
As one may notice from Fig. 3.4, the result of curve fitting is good considering the inherent
variability in the data.
●●● ●●● ●●● ●●● ●●● ● ● ● ● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● 0 50 100 150 200
0 2 4 6 8 10
Estimation for 1100 Degrees
Time
Response
Start 1
Start 2
Start 3
Start 4
Start 5
Figure 3.4: Estimation under 1100°C
The model estimation under 1050 °C, however, faces tremendous challenge. Although one has
31 data points, all those data are collected before growth saturation due to safety concerns. Since
one has only partial information, there are huge uncertainties in the data regarding the maximum
growth weight and the inflection point (Fig. 3.5, notice that I don’t have any data after 310 min).
In order to achieve best understanding with incomplete information, one proposes to use a
Bayesian estimation scheme to characterize the uncertainty involved. The idea is that each param-
eter has its own distribution and one can update the prior knowledge of those parameters by obser-
vation. The prior distributions for parameters are assumed to be normal distributions as follows:
a

N(0:068; 0:316
2
);aN(0:002; 0:141
2
);
W

N(6:5; 2
2
);CN(150; 316
2
);
U(0:05; 1)
49
●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 200 400 600 800 1000
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Uncertainty Under 1050 Degrees
Time
Response
Figure 3.5: Uncertainty under 1050°C
For rate parametera

anda, as one doesn’t have much knowledge about how they would change
with temperature, the prior distribution is centered at estimated value under 1100 °C with large
variability. For inflection pointC, I noticed that if I use a logistic model for growth under 1050 °C,
the estimation isC150 and I use it as the center of the prior distribution and I acknowledged
the fact that the information is limited by using a large variance. The maximum weightW

’s prior
distribution is based on possible maximum and minimum values from the data for two temperature
settings. The random noise is assumed to have a standard variation from 0.05 to 1 as observed
under two temperature settings.
With the prior distributions, one uses Markov chain Monte Carlo simulation [46] with Win-
BUGS [75] to draw posterior samples. Three widely separated chains are used and first half of
100,000 runs is discarded as burn-in. The estimation result is shown in Table 3.3. One of the inter-
esting result is thatC is much larger than 300 and one didn’t have any data points after growth
saturation.
50
Table 3.3: Bayesian Estimation for 1050 °C
Parameter Mean S.D 2.5% 97.5%
a

3.43E-4 1.25E-4 1.01E-4 5.75E-4
a 3.18E-3 1.67E-3 8.13E-4 6.82E-3
W

6.15 0.73 4.74 7.58
C -586.5 89.31 -743.5 -356.9
0.099 0.013 0.077 0.13
I checked the histograms and trace plots(omitted here) and all of them affirmed that conver-
gence had been achieved.
Uncover Growth Kinetics: With models established, I set to discover physical insights for the
two growth conditions. First, I aim to connect two models by recognizing model parameters are
temperature dependent. For instance, as rate coefficient, a

and a’s dependence on temperature
should follow Arrhenius equation:
a

/ exp(
E
a

kT
) (3.20)
a/ exp(
E
a
kT
) (3.21)
E
a
and E
a
are corresponding growth processes’ activation energy, k is the Boltzmann constant
andT is temperature.
The other two parametersW

andC are also related to temperature. The dependence, however,
is more complicated and one has no corresponding quantitative models. TakingW

as an example,
as a parameter associated with space/resource and catalyst limitation, it is depended on temperature
as the Si evaporation process and Pd diffusion process are temperature dependent. Another factor is
that with different temperatures, the number of initial Pd-Si droplets formulated may be different.
Due to the limited knowledge of those factors, W

andC’s temperature dependence will be left
for future investigation.
From Table. 3.4, it can be observed that a

decreases slower than a when temperature is
lowered from 1100 °C to 1050 °C. Asa

is associated with catalyzed growth from the top anda
51
Table 3.4: Estimation Comparison under 1100 and 1050 °C
a

a W

C
1100 0.0024 0.068 10.24 -57.04
1050 3.43E-4 3.18E-3 6.15 -586.5
Rate Ratio(1100/1050) 6.997 21.38 - -
is associated with growth from side, it can be assumed that the activation energy E
a
is smaller
than E
a
which is consistent with the observation. More specifically, one can calculate from the
Arrhenius equation that the ratio
E
a

Ea
= 0:635.
Table 3.5: Logistic Growth Estimation under 1100°C and 1050 °C
W (0) W

r
M
a
1100 0.0940 10.15 0.204 0.0788
1050 0.07403 1.185 0.00398 0.0132
If one compares the estimation of logistic growth model (Table 3.5, using parametrization in
(3.2) ) with CDMV model, and computed corresponding rate parametera, one would notice that
while logistic growth model is a good approximation of CDMV model under 1100 °C, the rate and
maximum growth parameters under 1050 °C is dramatically different from CDMV model. This
can be explained as side growth dominated the process under 1100 °C. With lower temperature,
however, top impingement becomes more important due to smaller activation energy. In this new
circumstance, logistic growth model based on side growth alone is no longer adequate and cross-
domain model should be used.
3.4 Conclusion
Establishing quantitative nanomanufacturing process models faces challenges of large uncertain-
ties in process physics and experimental observations. Current three modeling strategies, i.e.,
physical, statistical, and physical-statistical modeling, while successful in their specific domains
52
of application, may not be sufficient to address those challenges In order to utilize all available
information and achieve greater confidence in process modeling, one proposes a cross-domain
modeling and validation approach to integrate nanomanufacturing and nanoinformatics.
This new modeling strategy is demonstrated for silica nanowire growth process where both
physical growth mechanisms and measurement data contain large uncertainties. By deriving phys-
ical models and cross-validating with growth models in statistical domains, one is able to establish
a generic formulation of growth models suitable for statistical selection of growth mechanisms
under various conditions. Using both non-linear regression and Bayesian approach, models under
two growth conditions are characterized satisfactorily. More importantly, I am able to uncover
the dominant physical growth mechanisms for each growth conditions. The developed approach
provide an powerful alternative for nanomanufacturing process analysis and characterization.
53
Chapter 4
Bayesian Hierarchical CDMV for
Characterizing Variations Among Nano
Experimental Runs under Larger
Uncertainties
Variations among experimental runs are commonly observed in the bottom-up growth of nanos-
tructures. Such variabilities, coupled with large physical and measurement uncertainties, make
it challenging task to achieve better understanding of the mechanisms of growth processes. In
the previous chapter, the CDMV approach was developed to model nanofabrication process under
large physical and statistical uncertainties. However, the variations among nano experimental runs
was not consider therein. In this chapter I propose to further incorporate analysis of such varia-
tions into the CDMV framework. Process model parameters varying or invariant to experimental
runs are treated as random effects or fixed effects respectively. The improved analysis of varia-
tion sources is expected to provide better confidence in model prediction under large uncertainties.
Furthermore, the physical sources of variations among runs can be identified, which leads to con-
trol opportunity for nanomanufacturing. This approach would be demonstrated with the same
Pd-catalyzed nanowire growth process as in previous chapters. The obtained physical insights can
be used as guidance for future process improvement.
54
4.1 Introduction
It is widely known in the nano community that variations among experimental runs can be large
in the bottom-up growth of nanostructures. The outcomes from different runs can vary greatly
even under the same process parameter settings because of limited equipment control of actual
growth environments such as the temperature distribution or gas flow pattern within the growth
chambers. Considering the already limited physical knowledge and experimental data owing to
high experimental and measurement cost, variations among nano experimental runs add significant
challenges to the understanding of the process physics for nanomanufacturing.
The existing literature to uncover the process physics of nanostructure growth, as reviewed
in [95], can be roughly categorized into three main groups: physical modeling, statistical model-
ing, and physical-statistical modeling of growth processes. In case of limited data, lack of physical
knowledge, and large process uncertainties, a pool of candidate models could fit the data equally
well. To address this issued caused by large uncertainties, we proposed in [85] a Cross-Domain
Model building and Validation (CDMV) scheme, which we derived the process models from phys-
ical and statistical domains, respectively, and reinforced the understanding of growth processes
by identifying the common model structure across two domains. However, the CDMV approach
developed in [85] does not consider the variations among nano experimental runs. Most nanoman-
ufacturing modeling literatures [10, 14, 28, 38] focus on the characterization of the general growth
trend, with a few works on fine-scale growth variabilities within a single run [94]. Proper mod-
eling of variabilities among different runs would greatly enhance the understanding of the growth
processes under large uncertainties.
On the other hand, variations among runs have been considered in other areas such as medical
and ecological settings as the natural large variabilities among human/plant/animal subjects poses
similar challenges [52, 55, 61, 65]. For instance, pharmacokinetics studied the drug concentration
and its time-varying characteristics, which corresponds to our aim of understanding nanostructure
55
growth processes. The variations among different human subjects in pharmacokinetics also cor-
responds to our uncertainties among nano experimental runs. The methodology predominantly
applied to those medical and ecological studies are mixed-effects modeling [45, 50, 60, 88]. Yet
mixed-effect modeling alone is insufficient to handle the large uncertainties due to limited data
and lack of physical knowledge in nanomanufacturing.
Table 4.1: Experimental runs of silica nanowire weight change data (mg) under 1050°C
Growth Time(min) Weight Change (mg) Date Run
20 0.06 12/9/2009 1
30 0.05 12/9/2009 1
40 0.04 12/9/2009 1
60 0.12 12/9/2009 1
75 0.16 12/9/2009 1
90 0.24 12/9/2009 1
96 0.24 12/22/2009 1
96 0.21 12/22/2009 1
5 0 12/30/2009 1
5 0 12/30/2009 1
5 0.009 4/29/2010 2
10 0.016 4/29/2010 2
20 0.044 4/29/2010 2
30 0.114 4/29/2010 2
60 0.09 4/29/2010 2
90 0.321 4/29/2010 2
120 0.528 4/29/2010 2
5 0.005 5/4/2010 3
10 0.078 5/4/2010 3
20 0.086 5/4/2010 3
30 0.161 5/4/2010 3
60 0.195 5/4/2010 3
90 0.24 5/4/2010 3
120 0.443 5/4/2010 3
In this study we propose to incorporate mixed-effects modeling into our CDMV framework to
characterize variation among nano experimental runs under large uncertainties. In particular, we
illustrate the approach using a Pd-catalyzed silica nanowire growth process [69, 70]. Due to large
56
uncertainties, our initial study of the silica nanowire growth process lead to five candidate growth
models under growth temperature of 1100°C, and a heteroscedastic exponential-linear model was
chosen through model comparison [28]. With additional experimental runs under growth tempera-
ture of 1050°C, we [85] developed the CDMV approach to characterize the mean growth behaviors
and identify the temperature dependence of the growth rate parameters. But we did not explicitly
model the different degrees of variabilities observed under two growth conditions (1100°C and
1050°C).
0 50 100 150 200
0.0 0.2 0.4 0.6 0.8
Different Runs and Seperate Fittings
Time
Weight
● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● Run 1
Run 2
Run 3
Figure 4.1: Growth data under 1050°C for different runs and results of separate fittings
The experimental runs under 1100°C with three replicates at specific times are more consistent
among replicates, while the runs under 1050°C reveals large variabilities [28]. The potential reason
behind this inconsistency is that the experiments under 1100°C were conducted at approximately
the same time period while experimental runs under 1050 °C spread over one year. To check the
validity of this hypothesis, the observed data is divided into three runs based on the experimental
dates (Table 4.1). By tentatively fitting individual models (same parametric model as in [85]) for
57
each run, one can observe that the behaviors of growth process in different runs vary significantly
(Figure 4.1). Without directly tackling the clear variations among three runs, our ability to uncover
the process physics will be compromised.
Motivated by this specific application, we propose the strategy in Section 4.2 to consider vari-
ations among nano experimental runs under large uncertainties. In section 4.3, we incorporate
mixed-effects modeling into our CDMV framework to characterize variation among experimental
runs with example demonstrated for silica nanowire growth process. The estimation results and
their physical interpretations are also discussed. Conclusion is given in Section 4.4.
4.2 Strategy to Model Variations among Nano Experimental
Runs under Large Uncertainties
Before presenting our modeling strategy, it is necessary to distinguish the impacts of variations
among runs and ”large uncertainties”. Large uncertainties described in this study mean uncer-
tainties caused by the lack of data and physical understanding in nanomanufacturing processes.
This type of uncertainties impacts on the identification of model structures and subsequent under-
standing of physical mechanisms contributing to nanostructure growths. The variations among
experimental runs, on the other hand, generally affect the accuracy of estimating the model param-
eters associated with specific physical mechanisms. Therefore, variations among runs would be
better handled if the issue of large uncertainties can be addressed first.
As worthy of noting, there are extensive literatures on ”run-to-run” studies [51], which focus
on control of the manufacturing process inputs based on previous run inputs settings and results
to achieve optimized results. Our study is different from such studies as we don’t have a clearly
defined set of process parameters that can be well controlled and optimized. The among-runs vari-
ations modeling, instead, is more of an exploratory modeling stage and aims to identify potential
58
physical processes responsible for large among-runs variations, improve parameter estimation and
thus provide guideline for design of future manufacturing processes.
The proposed strategy can be illustrated by Fig. 4.2. The CDMV approach was developed
in [85] to address large uncertainties in the example of silica nanowire growth. For the purpose of
clarity, we briefly summarize its basic idea. The existing physical knowledge only attributes the
nanowire growth to the direct impingement of silicon vapor to the Si-Pd droplet at the top of the
nanowire and source supply from nanowire side surface. The experimental data does not provide
convincing support to competing hypotheses on growth mechanisms. The CDMV approach, using
top impingement modeling as an example, first derives the rate of weight gaindW=dt proportional
to the room left to gain extra, based on the physical knowledge that the growth rate is limited
by the amount of catalyst left on the top of nanowires. In the statistical domain, the confined
exponential model has been used to describe, e.g., how the population growth rate is limited by the
resource left. Therefore, although large uncertainties exist, through cross-domain model validation
by knowledge (not by data), we could adopt a confined exponential model to characterize the
nanostructure growth contributed from the top impingement. Contribution from side nanowire
surface can be derived accordingly and more details can be found in [85].
With candidate model structures (e.g., confined exponential model for the top impingement
process) identified by the CDMV approach, we propose to incorporate the mixed effects modeling
to the existing CDMV framework in order to characterize the variations among experimental runs.
As shown in Fig. 4.2, if one model parameter remains practically the same among different runs,
it is considered as a fixed effect. On the other hand, if one model parameter varies among different
runs due to, e.g., change of experimenters, one would assume that this parameter is sampled from
an underlying distribution, thus it is called a random effect. Besides the ability to characterize
the variabilities among different runs, mixed-effect modeling can help provide insights about the
source of variations: one can uncover the underlying physical reasons for large variability by
identifying proper random effects. For instance, Pinheiro and Bates in [61] identified that heights
of pine trees at age zero can be considered fixed while asymptotic heights and growth rate vary
59
Top$
Impingement$
Cross/domain$Modeling$
$
$
Physical$
Knowledge$
Uncertainty$
Modeling$
Physical)Domain)
$
$$$$Growth$limited$by$the$$
$$$$$$amount$of$catalyst$$
$$$$$$$$$$le?$on$the$top$$$
Sta0s0cal)Domain)
!
Confined$exponenBal$model$
for$populaBon$growth$
Data$ValidaBon$
dN
dt
= a(N
⇤ N)
dW
dt
/ (W
⇤ W)
dW
dt
=a(W
⇤ W)
Mixed/effects$Modeling$
$
Model$Parameters$(a,$W*)$
$
Fixed$effect$$Random$effect$
Figure 4.2: Schematic of the proposed strategy
among different trees. Therefore, our proposed strategy is expected to address variations among
runs under large uncertainties in nanomanufacturing. We will demonstrate this strategy in Section
3 using silica nanowire growth process studied in [85].
4.3 Incorporating Mixed-effects Modeling Under CDMV
Framework
The following section would be organized as follows: first we would summarize and re-
parameterize CDMV model developed in [85], then we’ll review the state of art in mixed effects
modeling and estimation. With the proper preparation, a hierarchical structure incorporating
CDMV and mixed effects modeling would be developed. Four exploratory models would be used
to identify random effects followed by discussion of the physical implications. Finally, we would
present the final hierarchical model and parameter estimation with comparison to CDMV model
without among-runs variations modeling.
60
4.3.1 Review and Re-parameterization of CDMV Model
In the previous publications [85], a cross-domain model for the same growth growth process was
developed. For the purpose of identifying corresponding physical parameters behind different
experimental runs, the model needs to be re-parameterized. Here we will briefly summarize the
CDMV model as well as perform the re-parametrization as preparation for further model building
efforts.
h
2r
h
Constant
Radius
Constant Height
2r
h
2r
Side
Growth
Top
Impingement
Figure 4.3: Schematic of nanowire growth from two sources [85]
As shown in Figure 4.3, there are two sources contributing to the nanowire growth: the direct
impingement of silicon vapor to the Si-Pd droplet at the top of the nanowire and source supply from
nanowire side surface. The direct impingement of vapor Si particles are absorbed by the droplet
formed of Pd catalyst and Si, then nucleate and contributed to the nanowire growth [62, 77]. The
rate for top growth is proportional to the catalyst left and becomes zero when it is used up. For the
side growth, Si absorbed on the side face diffuses to the top and contributes to the length growth.
The rate for side growth is more complicated to model as it changes for different scenarios, but
61
in our case it is proportional to the current growth and stops due to space limitation. CDMV
successfully includes both growth sources and identify their relationship with statistical growth
models, namely, confined exponential for top impingement and logistic for side growth, and cross-
validate them. The final CDMV model obtained is formulated withk(t) = exp((a +a

)(t +C)):
W =
W

(a

+ak(t))
a(1 +k(t))
(4.1)
In this model,a

anda are weight growth rate coefficients for top impingement and side absorp-
tion growth mechanisms, respectively,W

is maximum possible weight, andC is the negative of
growth inflection time. AsC is difficult to interpret physically, we re-parameterize it with initial
valueW
0
which represent the initial weight when the growth temperature achieved:
W
0
=
W

(a exp[(a +a

)C]a

)
a(1 + exp[(a +a

)C])
(4.2)
(4.1) can be re-parameterize usingW
0
instead ofC as:
W =
W

(a

+
W

a+W
0
a
W

W
0
exp[(a +a

)t])
a +
W

a+W
0
a
W

W
0
exp[(a +a

)t]
(4.3)
Takingg(t) =
W

a+W
0
a
a(W

W
0
)
exp[(a +a

)t]), (4.3) can be simplified as:
W =
W

(a

+ag(t))
a(1 +g(t))
(4.4)
With re-parameterized model, each model parameter has a clear physical interpretation and
by examining their variabilities among different runs, one can trace back to different physical
processes. The empirical effects of different random parameters can be seen in Fig. 4.4. The
curves are generated by changing the value of one parameter respectively while keeping other
parameters constant. As one may notice, the randomness in different physical parameters have
different effects on the overall growth curve shape. For example, if the maximum weightW

is the
62
0 500 1000 1500
0 2 4 6 8
Effects of W*
time
l0
0 500 1000 1500
−1 0 1 2 3 4 5 6
Effects of a*
time
l0
0 500 1000 1500
0 1 2 3 4 5 6
Effects of a
time
l0
0 500 1000 1500
0 1 2 3 4 5 6
Effects of W0
time
l0
Figure 4.4: Effects of parameters
random effect, the growth curves for each run would tend to split apart while the growth saturates.
Moreover, while thea

’s effects are most significant in the earlier growth stage, the effects ofa and
W
0
are clear for the whole growth period. As a result, if we can identify the responsible physical
parameter for the variation observed, we can further predict the growth behavior even if we don’t
have data for some growth periods.
4.3.2 Mixed Effects Modeling and Estimation
As we discussed before, mixed effects modeling are generally used when there may be large vari-
ations in model parameters with different subjects/runs.
63
A general nonlinear mixed effects model for the jth experiment in ith run (using notations
in [45]) can be formulated as(we put the corresponding quantities for our study in following paren-
theses when possible):
y
ij
=f(
i
;x
ij
) +e
ij
(4.5)
where y
ij
is the response (nanowire weight growth in our case) for jth experiment (each obser-
vation) in ith run (one of the three batches of experiments performed), x
ij
is the predictor for
jth experiment inith run,f is a nonlinear function (CDMV model) with parameter
i
ande
ij
is
noise term which usually is assumed to follow an i.i.d normal distribution with zero mean. The
parameter vector
i
(W
0
;W

;a

;a in our case) can be different in different runs:

i
=A
i
+B
i
b
i
,b
i
N(0;D) (4.6)
where is a vector of population parameters,b
i
is random effects associated with runi, andD is
covariance matrix.
Different ways have been proposed to estimate parameters in non-linear mixed effects models,
ranging from (restricted) maximum likelihood estimation to full bayesian hierarchical treatment
[30, 31, 40, 45, 79]. A brief summary of proposed approaches and the choice of method for this
study would be discussed in following parts. For an extensive literature review in this area, one
can refer to [13] and [61].
Maximum Likelihood Estimation
The algorithm for obtaining MLE for the parameter generally contains two iterative steps [40,
45]:
1. Obtain maximized likelihood estimation of and posterior mode ofb with known
2. Maximize over with known;b
The algorithm stops when some criteria for convergence are achieved.
64
The major challenge in applying MLE for non-linear mixed effects model is that due to the
random effects, the likelihood function in general is an intractable integral over the distribution of
those effects. More specifically, in order obtain maximize likelihood estimation of parameter,
one needs the marginal(integrate over random effects) density of responsey:
p(y) =
Z
p(yjb)p(b) db (4.7)
but asf is nonlinear inb, there are no closed-form expression for this integral in general case.
In order to approximate this integral, one way is to use first-order method, which approximates
the integral by using taylor expansions and thus eliminating the non-linearity [88, 99]. Another
approach is to maximize the likelihood directly using deterministic or stochastic approximation of
the integral: such as (adaptive) gaussian quadrature [60] for deterministic methods and importance
sampling as well as monte carlo integration [50] for stochastic methods.
The methods based on MLE often involve numerical integration or optimization and thus suffer
from the common problems such as local optimality and slow convergences due to complex non-
linear structures. In our study, due to the limited data volume, such issues actually prevent us
obtaining a meaningful estimation in initial studies(omitted here). Moreover, we already obtained
estimation for the CDMV model without random effects. Such prior knowledges, which could be
highly valuable, cannot be incorporated in the MLE approach systematically.
Bayesian Hierarchical Estimation
An alternative for MLE estimation is bayesian hierarchical approach [21]: the random effects can
be incorporated as an additional hierarchy while fixed effects have same distribution over different
runs. Such bayesian hierarchical approach is particularly appealing comparing to MLE based
approaches for our study: from a physical perspective, prior knowledge and constraints can be
naturally introduced by prior distributions; from statistical estimation perspective, the numerical
65
convergence difficulties due to complex non-linearity in a maximum likelihood approach can be
avoided.
As a result, we propose a Bayesian hierarchical approach for the among-runs variation model-
ing. The priors would come from previous overall estimation as well as domain knowledge of the
growth process. Different choice of random effects would be tested and compared.
4.3.3 Bayesian Hierarchical CDMV Model
With the preparation in both among-runs variations modeling and re-parameterized CDMV model,
we are able to propose the hierarchical CDMV model with additional ability to identify sources of
among-runs variations.
The hierarchical CDMV model would contain probability distribution of actual growth given
mean growth function, mean growth function given model parameters in each run, distribution of
model parameters in different runs and prior distributions. Those distribution are chosen based on
the physical constraints, for example, the maximum weight should be a non-negative value.
Level 1 : y
i
(t)N(g(t;
i
);
2

)
Level 2 :
i
N(

;

) (4.8)
Level 3 : Priorsfor

;

and
2

Hereg(t;) is the explicit solution for CDMV model (Equation 4.3),i denotes different runs and
= (W
0
;W

;a

;a).
With the model structure, the most important modeling task is to determine which physical
parameters should be modeled as random effects and which parameters should be modeled as
fixed effects. There are generally two approaches for this modeling issue: one based on prior
physical knowledge [31] and the other based on statistical analysis of the observed data [61]. As
our purpose is to determine which physical parameter is responsible for the variabilities among
runs, the first approach is unavailable to us. The statistical literatures on the choice of random
66
or fixed effects, however, concentrate on the linear mixed model due to the potential complex
interactions among parameters in non-linear models [1,7,9]. In [59], Pinheiro and Bates proposed
a method to determine random effects by first fitting a model with all effects as random and then
check if there is rank deficiency in the estimated covariance matrix and thus identify potential fixed
effects. This method, however, cannot be applied with very limited amount of data, as in our case,
due to ill-conditioned covariance matrix(a general 4 by 4 covariance matrix to estimate with only
24 data points).
Due to the limitation of the both physical and statistical approaches in selecting random effects,
we propose a two-stage approach: fit a hierarchical CDMV model for each ofW
0
,W

,a

anda
first. More specifically, in each of those models, only one parameter is considered as random
among three runs. Then by inspecting the results from those models, we would be able to see
which physical parameter is likely to be responsible for the among-runs variations. After checking
the physical meanings of the selected parameter/parameter group, we will then build a model with
the chosen random effects.
4.3.4 Exploratory Individual Models for Each Parameter
In this section, we’ll first detail the construction of the model for initial weightW
0
and as the other
three models are constructed similarly, only numerical setups and results would be included.
For the initial weight model, the priors for other three parameters are based on the estimations
of CDMV model without considering among-runs effects [85]. W
0
is assumed to follow a normal
distribution among different runs. The prior for the mean of W
0
is chosen as uniform in the 95
percentile. The variance parameter
W
0
is chosen as relatively non-informative due to our lack of
prior knowledge.
67
Symbolically, the hierarchical model can be written as:
Level 1 : y
i
(t)N(g(t;W
0;i
;W

;a

;a);
2
)
Level 2 : W
0;i
N(
W
0
;
2
W
0
) (4.9)
Level 3 :
W
0
U(0:2; 0:85);
W
0
U(0:005; 1)
;W

N(6:15; 0:73
2
);a

N(0:000343; 0:000125
2
)
;aN(0:00318; 0:00167
2
);U(0:005; 10)
To get the posterior distribution, we use Markov chain Monte Carlo simulation [46] with Win-
BUGS [75] to draw samples. Three chains with initial values sampled from prior distributions are
used and first half of 10,000 runs is discarded as burn-in. We also use thinning rate 100 to reduce
auto correlation. By checking the histograms (Fig.4.5) and trace plots (Fig.4.6), we confirmed that
good mixing had been achieved. The estimation results are shown in Table 4.2.
Figure 4.5: Posterior parameter distributions withW
0
(denotes asWs in WinBUGS program) as random
effects
68
Figure 4.6: Trace plot for parameters withW
0
as random effects(
e
omitted due to space limitation)
Table 4.2: Estimation hierarchical model withW
0
as random effect
Parameter Mean S.D 2.5% 97.5%
a

3.80E-4 2.42E-5 3.32E-4 4.38E-4
a 3.41E-3 5.07E-4 2.44E-3 4.43E-3
W

6.71 0.50 5.70 7.70

W
0
7.97E-3 0.034 -0.061 0.077
W
0;1
-0.024 0.017 -0.056 0.012
W
0;2
0.02 0.019 -0.016 0.057
W
0;3
0.029 0.02 -0.01 0.07

W
0
0.048 0.023 0.012 0.096
0.05 0.009 0.037 0.073
The hierarchical model for maximum weightW

can be set up similarly with priors:
W
0
U(0:2; 0:85);
W
N(6:15; 0:73
2
);
W
U(0:005; 10);
a

N(0:000343; 0:000125
2
);aN(0:00318; 0:00167
2
);U(0:005; 10)
69
The estimation results for maximum weight model are summarized in Table 4.3. As the his-
tograms and trace plot are similar to Fig. 4.5 and Fig. 4.6, they are omitted here and afterwards.
Table 4.3: Estimation for hierarchical model withW

as random effect
Parameter Mean S.D 2.5% 97.5%
a

3.65E-4 3.44E-5 2.96E-4 4.34E-4
a 1.83E-3 5.07E-4 8.29E-4 2.85E-3
W
0
1.23E-3 0.019 -0.03 0.049

W
6.45 0.70 5.06 7.78
W

1
6.12 0.80 4.52 7.81
W

2
9.16 1.13 6.78 11.02
W

3
8.21 1.73 2.46 10.56

W
3.44 2.07 0.74 8.69
0.05 0.011 0.035 0.077
The hierarchical model for top impingement ratea

is also set up similarly with priors(notice
the priors for variance parameter corresponds to the scale of the parameter):
W
0
U(0:2; 0:85);W

N(6:15; 0:73
2
);
a
N(0:000343; 0:000125
2
);

a
U(0:00001; 0:00005);aN(0:00318; 0:00167
2
);U(0:005; 10)
The estimation results for top impingement rate model are summarized in Table 4.4.
The hierarchical model for side growth ratea is set up similarly with priors:
W
0
U(0:2; 0:85);W

N(6:15; 0:73
2
);a

N(0:000343; 0:000125
2
);

a
N(0:00318; 0:00167
2
);
a
U(0:0001; 0:0005);U(0:005; 10)
The estimation results for side growth rate model are summarized in Table 4.5.
70
Table 4.4: Estimation for hierarchical model witha

as random effect
Parameter Mean S.D 2.5% 97.5%

a
3.72E-4 2.93E-5 3.11E-4 4.25E-4
a

1
3.76E-4 3.48E-5 3.31E-4 4.65E-4
a

2
3.61E-4 3.92E-5 3.29E-4 4.31E-4
a

3
3.85E-4 3.59E-5 3.40E-4 4.52E-4

a
3.80E-5 9.21E-6 1.54E-5 4.96E-5
a 3.39E-3 5.21E-4 2.42E-3 4.43E-3
W
0
2.56E-5 0.014 -0.027 0.028
W

6.99 0.55 5.98 8.11
0.05 0.009 0.035 0.071
Table 4.5: Estimation for hierarchical model witha as random effect
Parameter Mean S.D 2.5% 97.5%
a

3.83E-4 3.67E-5 3.13E-4 4.70E-4

a
3.39E-3 4.92E-4 2.36E-3 4.33E-3
a
1
3.26E-3 5.83E-4 2.11E-3 4.38E-3
a
2
3.53E-3 5.91E-4 2.35E-3 4.71E-3
a
3
3.45E-3 5.93E-4 2.24E-3 4.56E-3

a
3.10E-4 3.67E-5 1.11E-4 4.92E-4
W
0
2.44E-3 0.016 -0.029 0.034
W

6.70 0.60 5.60 8.00
0.06 0.01 0.044 0.082
4.3.5 Physical Interpretation and Final Model Building
We noticed that from Table 4.2-4.5 by comparing the estimation of

and

, that the among-runs
behavior of the parameters show two different patterns: for growth rate variablesa

anda, there
is no significant differences among three runs; for initial weight W
0
and maximum weight W

,
however, it seems that there are among-runs variabilities. This observation indicates the potential
source of among-runs variations and focus for future improvement:
• Physically, the rate parametersa

anda correspond with the growth temperature. The con-
sistency among different runs suggests that the either the growth temperature is precisely
controlled or the growth rate is relatively insensitive to small temperature variations. Either
71
way, there is not much to improve on the temperature side if one wants to reduce the among-
runs variations.
• The initial weight W
0
depends on both the sample preparation and the temperature ramp-
up before the growth temperature is achieved. The variations due to it should be further
investigated and thus achieve better control of the growth process.
• The maximum weightW

also depends on the sample preparation as the nucleation of cata-
lyst would determine the number and size of nanowires to be grown. The large variabilities
shown inW

offers opportunities in improving among-runs consistency by focusing on the
sample preparation.
Based on the discussion above, we propose the final hierarchical CDMV model:
Level 1 : y
i
(t)N(g(t;W
0;i
;W

i
;a

;a);
2
)
Level 2 : W
0;i
N(
W
0
;
2
W
0
);W

i
N(
W
;
2
W
)
Level 3 :
W
0
U(0:2; 0:85);
W
0
U(0:005; 1) (4.10)
;
W
N(6:15; 0:73
2
);
W
U(0:005; 10);
;a

N(0:000343; 0:000125
2
);aN(0:00318; 0:00167
2
)
;U(0:005; 10)
The MCMC simulation details are still the same as for the exploratory model forW
0
only with
the estimation results in Table. 4.6.
The final model estimation shows the large among-runs variations inW
0
andW

thus further
confirmed the results obtained in exploratory individual models. Moreover, the MSE of the hierar-
chical CDMV model(use the mean of posterior distributions as parameter estimations) is 0.00131
which is only 11 percent of the MSE for old model which is 0.0116. The improvement of accu-
racy brought by the additional modeling of among-runs variation can be better visualized by the
72
Table 4.6: Estimation for hierarchical model withW
0
andW

as random effect
Parameter Mean S.D 2.5% 97.5%
a

3.53E-4 3.59E-5 2.84E-4 4.24E-4
a 3.36E-3 5.29E-4 2.33E-3 4.39E-3

W
6.43 0.69 5.06 7.75
W

1
6.16 1.04 4.25 8.33
W

2
9.50 1.54 6.69 12.69
W

3
7.22 1.27 4.93 9.93

W
3.32 2.10 0.51 8.67

W
0
3.10E-3 3.27E-2 -0.063 0.073
W
0;1
-8.89E-3 0.018 -0.046 0.027
W
0;2
-0.014 0.024 -0.059 0.033
W
0;3
0.03 0.026 -0.019 0.081

W
0
0.045 0.024 7.40E-3 0.095
0.043 0.008 0.031 0.061
residual plots as in Fig. 4.7. As one may observe, the introduction of random effects resulted in
residuals with no clear bias or other patterns compared to original CDMV model.
4.4 Conclusion
Precise among-runs control is essential to scale-up manufacturing of nanostructures. The CDMV
model developed in Chapter 3, while well characterize the mean growth behavior even with large
uncertainties in both physical and statistical domains, lacks ability to model the among-runs vari-
abilities. In this research, we propose to use a hierarchical CDMV model to describe and further
identify the physical sources of the among-runs variabilities.
It is demonstrated that the hierarchical CDMV model is able to identify physically meaningful
sources of among-runs variabilities using a Pd-catalyzed silica nanowire growth data. By showing
that the among-runs variabilities is likely due to the sample preparation process, this analysis offers
guidance for future finer control of such growth processes. Moreover, the final hierarchical CDMV
model improves the estimation accuracy significantly and eliminated the patterns in residuals.
73
20 40 60 80
−0.2 −0.1 0.0 0.1 0.2
CDMV
Time
Residual
Run 1
Run 2
Run 3
20 40 60 80
−0.2 −0.1 0.0 0.1 0.2
Hierarchical CDMV
Time
Residual
Run 1
Run 2
Run 3
Figure 4.7: Comparison of residual plots for CDMV and Hierarchical CDMV
74
Chapter 5
Conclusions and Future Research
5.1 Summary and Contributions
The large uncertainties in both physical and statistical domains pose great challenges in the nanos-
tructure fabrication process modeling. This dissertation contributes to the nanomanufacturing pro-
cess modeling by developing a cross-domain modeling framework which is flexible to incorporate
different variability sources. This major contribution is developed in three steps:
1. Statistical weight kinetics modeling provides an initial understanding of the weight kinet-
ics. Furthermore, bayesian hierarchical modeling approach’s effectiveness is identified espe-
cially with limited data.
2. Cross-Domain model building and validation(CDMV), as a framework for nanomanufa-
turing process modeling, is developed. By reviewing traditional, statistical and physical-
statistical models, it is noticed that in typical nanomanufacturing environment, due to large
variabilities in both physical and statistical domains, none of them can achieve satisfactory
performance. The CDMV framework, however, by cross-domain model building and vali-
dating, can make efficient use of both the existing physical knowledge and limited data and
achieve credible modeling.
3. Characterization of variations among nano experimental runs is a further extension of
CDMV modeling framework. By incorporating analysis of widely observed among-runs
variations in nanomanufacturing into the CDMV framework, the improved analysis of vari-
ation sources can provide better confidence in model prediction under large uncertainties.
75
Moreover, the physical sources of variations among runs can be identified, which leads to
potential control opportunity for nanomanufacturing.
5.2 Future Research
There are several natural extensions of the current research that can be explored. Here are some
examples.
1. Inherent Nanowire Growth Variation Modeling Using Stochastic Differential Equation:
In Chapter 3, the CDMV model is developed by generalizing physical model from one
nanowire to the whole substrate. While such an approach can capture the main growth
kinetics, the inherent randomness of model parameters over the substrate is overlooked. For
example, the initial weightW
0
, varies from nanowire to nanowire due to the variations in Pd
catalyst sputtering and temperature gradient on the substrate.
Those inherent randomness can be explicitly modeled if one allows the growth model param-
eters for each nanowire have their distributions instead of a constant representing average
growth. As a result, the overall growth kinetics can be represented as:
dW
i
dt
=
a
i
W
2
i
W

i
+ (a
i
a

i
)W
i
+a

i
W

i
(5.1)
With each parameter as defined in Chapter 3 but only for individual nanowires. What needs
to be noticed is that each parameters here is no longer a constant but a random variable. Thus
the overall weight growth can be written as:
W =
K
X
i=1
W
i
(5.2)
K is the overall number of nanowires and is a random variable itself.
76
2. Among-Runs Variation Control:
In Chapter 4, it is noticed that the sample preparation process is likely to be the source of
underlying among-runs variations. To reduce such variations, one can further study such
processes and identify controllable inputs. Once a process model is established(through
design of experiments, for example), instead of only modeling such variabilities, one can
actually better control the preparation process and thus achieve more stable nanostructure
fabrication.
3. Other Applications of CDMV Methodology:
One of the advantages of CDMV is its versatility. All the studies in this dissertation focus
on a specific Pd-catalyzed silica nanowire growth process while CDMV can be used in
many other circumstances, for example, MOCVD nanostructure growth processes. More-
over, other advanced manufacturing areas such as additive manufacturing may also have
similar challenges due to large uncertainties in both physical and experimental domains, and
CDMV can be readily applied in those circumstances.
77
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Abstract (if available)

Abstract Nanomanufacturing is currently a major bottleneck that hinders the transformation of nanotechnology from laboratory to industrial applications. Due to both limited process understanding and control, there are great challenges for reliable and cost-effective scalable production of nanostructures. As a result, despite of nanomaterials' superior electrical, mechanical, chemical and biological properties, their great potentials in high impact fields such as energy, medicine and information technology have not been materialized. ❧ In order to achieve scalable nanomanufacturing, the nanostructure fabrication process has to be quantitatively modeled before subsequent improvement efforts such as diagnosis, monitoring and control. This research will therefore focuses on characterizing nanostructure growth processes to understand growth kinetics which is the law governing how growth behaviors change with time. This dissertation will also investigate the among experimental runs variations and identify their root causes for improved process modeling. ❧ There are two fundamental research challenges in nanomanufacturing process modeling: (i) There are only limited physical knowledge for nanostructure growth processes. Despite of numerous studies in nanostructures growth area, we are still often largely uncertain about the physical mechanisms driven the growth process under a certain condition. The complexity comes from the fact that there are often multiple mechanisms at work in one growth process and it is very difficult to identify their relative contributions. (ii) Experimental observations are expensive and time-consuming to obtain. Nanostructure growth experiments are usually costly due to expensive equipment(such as metal-organic chemical vapor deposition (MOCVD) system) as well as requirement of constant human attention. Moreover, the measurement of nanostructures often involves equipment such as scanning electron microscopy (SEM), transmission electron microscopy (TEM) and atomic force microscope (AFM). Besides significant equipment cost, one can only view/measure a tiny portion of the substrate at one time. ❧ This dissertation will systematically model nanostructure growth process kinetics and its variation as part of greater effort towards achieving scalable nanomanufacturing. The focus of the modeling work is on quantitative macro scale measurements (such as overall weight) of nanostructure grown using ""bottom-up"" instead of ""top-down"" approach. Special emphasis is placed on integrating physical knowledge and experimental data due to the challenges mentioned above in the whole modeling process. The major research tasks includes: (i) statistical model building and selection to gain initial understanding of nanowire weight growth kinetics

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Asset Metadata

Creator Wang, Li (author)

Core Title Modeling and analysis of nanostructure growth process kinetics and variations for scalable nanomanufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Publication Date 11/20/2013

Defense Date 11/20/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Bayesian hierarchical model,Nano,OAI-PMH Harvest,statistical modeling

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Huang, Qiang (committee chair), Chen, Yong (committee member), Zhang, Jianfeng (committee member)

Creator Email wang40@usc.edu,wangfan8@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-347602

Unique identifier UC11295873

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Legacy Identifier etd-WangLi-2165.pdf

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Document Type Dissertation

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

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