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ATTAINING NANOSCALE PATTERNS FOR RENEWABLE ENERGY APPLICATIONS
By
JUSTIN CHET-MUN WONG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2016
© 2016 Justin Chet-Mun Wong
To my grandmother, the toughest person I know
4
ACKNOWLEDGEMENTS
I would like to first thank my family for their encouragement and understanding,
not only in graduate school, but throughout my life. I never would have made it as far as
I have today without them.
Thank you to my advisor, Dr. Kirk Ziegler, for his wonderful guidance and support
during my time in graduate school. Over these past several years you have been a
great mentor and I could not have asked for a better advisor. I only hope I am not the
last Justin to be a part of your lab.
Thanks to my committee members, Dr. Peng Jiang, Dr. Jason Weaver, and Dr.
Charles Cao, and all of the other professors who I have had the honor of working with
during my time at UF. Thank you for being sources of knowledge and inspiration.
I would like to thank my friends here at UF and elsewhere, for providing an ever-
important buffer from work. Your comradery and humor kept me going through many
tough hours.
Lastly, to Katherine: thank you for always being my biggest supporter.
5
TABLE OF CONTENTS page ACKNOWLEDGEMENTS ............................................................................................... 4
LIST OF TABLES ............................................................................................................ 7
LIST OF FIGURES .......................................................................................................... 8
LIST OF ABBREVIATIONS ........................................................................................... 10
ABSTRACT ................................................................................................................... 11 CHAPTER 1 INTRODUCTION AND BACKGROUND .................................................................. 13
Thermoelectric Devices ........................................................................................... 14
Thermoelectric Figure of Merit ................................................................................. 14 Thermoelectric Thermopower .................................................................................. 15
Increasing Figure of Merit ........................................................................................ 17 Improving Phonon Scattering .................................................................................. 19 Bismuth Telluride Thermoelectric Nanostructures ................................................... 20
Anodized Aluminum Oxide Templates ..................................................................... 21
Electrodeposition ..................................................................................................... 24
Carbon Nanotubes ................................................................................................... 25 Chapter Summary .................................................................................................... 26
2 UNIFORM ELECTRODEPOSITION OF BISMUTH TELLURIDE NANOWIRES ..... 40
Electrodeposition Methods ...................................................................................... 40 Materials and Methods ............................................................................................ 42
AAO Templates .................................................................................................. 42 Electrodeposition into AAO Templates ............................................................... 43 Focused Ion Beam ............................................................................................. 43
Bismuth Telluride Nanowire Deposition ............................................................. 44
Results and Discussion ........................................................................................... 45 Chapter Summary .................................................................................................... 47
3 GROWTH OF CATALYZED CARBON NANOTUBES IN AAO TEMPLATES ......... 55
Materials and Methods ............................................................................................ 56 Catalyst Deposition in AAO Templates .............................................................. 56 Catalyst PVD in AAO Templates ........................................................................ 57 Complex Multilayer Substrates ........................................................................... 58
6
AAO Mask Creation and Transfer ...................................................................... 58 RIE Pattern Transfer .......................................................................................... 59 Anodization of CMS Pre-Patterned Surfaces ..................................................... 60
CNT Growth ....................................................................................................... 60 Results and Discussion ........................................................................................... 60 Chapter Summary .................................................................................................... 63
4 CREATION OF AAO PATTERNED THERMOELECTRIC NANOSTRUCTURES ... 71
Materials and Methods ............................................................................................ 72
AAO Template Transfer ..................................................................................... 72 RIE Pattern Transfer .......................................................................................... 73
Metal Nanoparticle Arrays .................................................................................. 73 Results and Discussion ........................................................................................... 74 Chapter Summary .................................................................................................... 76
5 CNT GROWTH MECHANISM IN AAO TEMPLATES ............................................. 81
Results and Discussion ........................................................................................... 81
Outside of Pore Growth Rate ............................................................................. 82 Concentration Profile Inside Pore ....................................................................... 85
Inside Pore Growth Rate .................................................................................... 87 Chapter Summary .................................................................................................... 89
6 CONCLUSIONS ...................................................................................................... 99
Chapter Summary .................................................................................................... 99 Future Work ........................................................................................................... 100
Barrier Layer Thinning and Pore Opening to Create AAO Nanofluidics ........... 100 Fabrication of Thermoelectric Devices ............................................................. 100
APPENDIX: CALCULATIONS .................................................................................... 102
Growth Rate Outside Pore ..................................................................................... 102 Concentration Profile ............................................................................................. 104 Growth Rate Inside Pore ....................................................................................... 106
Substitution of Concentration Profile in Growth Rate ............................................. 108
LIST OF REFERENCES ............................................................................................. 110
BIOGRAPHICAL SKETCH .......................................................................................... 115
7
LIST OF TABLES
Table page 2-1 Bismuth and tellurium atomic percentages versus deposition potentials. ............ 54
8
LIST OF FIGURES
Figure page 1-1 Thermoelectric device using the Seebeck Effect (power-generation). ................. 28
1-2 Temperature range versus ZT of common n-type thermoelectric materials ......... 29
1-3 ZT and its constituent variables versus increasing carrier concentration ............. 30
1-4 Schematic of a cobalt-antimony skutterudite unit cell .......................................... 31
1-5 Schematic of a superlattice configuration of SiGe ............................................... 32
1-6 Thermoelectric figure of merit, ZT, versus decreasing critical dimension length .. 33
1-7 Schematic of an electrochemical cell ................................................................... 34
1-8 Top down SEM image of AAO ............................................................................. 35
1-9 Schematic of pore propogation versus current response ..................................... 36
1-10 Schematic of individual pore growth .................................................................... 37
1-11 Mechanism of dissolution of aluminum oxide into solution................................... 38
1-12 AAO template pore wall structure ........................................................................ 39
2-1 Schematic of milling process ............................................................................... 48
2-2 Metal nanowire deposition in AAO templates ...................................................... 49
2-3 Bottom of AAO template pore and barrier layer ................................................... 50
2-4 SEM image of bottom versus top side pore opening with AuPd substrate ........... 51
2-5 Plot of measured atomic percentages of bismuth versus applied potential .......... 52
2-6 Bismuth telluride electrodeposition in AAO template ........................................... 53
3-1 Schematic of CMS creation ................................................................................. 64
3-2 Nickel catalysts deposited at bottom of AAO template......................................... 65
3-3 CNT growth from AAO templated catalysts ......................................................... 66
3-4 Schematic of completed CMS .............................................................................. 67
9
3-5 SEM image of CMS and the resulting AAO ......................................................... 68
3-6 Pattern transfer onto aluminum ............................................................................ 69
3-7 Pre-patterned anodization on CMS ...................................................................... 70
4-1 Comparison of pattern transfer on silicon germanium.......................................... 77
4-2 High magnification SEM image of 60 and 75 minute pore opening ..................... 78
4-3 Plot of confining dimension thickness vs pore opening time ................................ 79
4-4 SEM image of Au nanoparticle deposition in dimpled surfaces ........................... 80
5-1 Top down SEM image of nickel-catalyzed CNTs grown from AAO template ....... 91
5-2 High magnification SEM image of extended nickel-catalyzed CNT growth .......... 92
5-3 Wide field SEM image of extended nickel-catalyzed CNT growth ....................... 93
5-4 Schematic of catalyzed CNT tip growth ............................................................... 94
5-5 Schematic of flux into AAO channel for catalyzed CNT growth ........................... 95
5-6 Plot of concentration versus height for various kw. ............................................... 96
5-7 Plot of 𝑑ℎ/𝑑𝑡 versus the normalized height of CNT in pore for varying kw ........... 97
5-8 Plot of 𝑑ℎ/𝑑𝑡 versus the normalized height of CNT in pore for varying keff .......... 98
10
LIST OF ABBREVIATIONS
AAO
CMS
Anodized Aluminum Oxide
Complex Multilayer Substrate
CNT Carbon Nanotube
CVD
FIB
ICP
MAIC
NRF
PECVD
PS
PVD
RIE
SCE
SEM
TEM
ZT
Chemical Vapor Deposition
Focused Ion Beam
Inductively Coupled Plasma
Major Analytical Instrumentation Center at UF
Nanoscale Research Facility at UF
Plasma Enhanced Chemical Vapor Deposition
Polystyrene
Physical Vapor Deposition
Reactive Ion Etching
Saturated Calomel Electrode
Scanning Electron Microscopy
Tunneling Electron Microscopy
Figure of Merit
11
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
ATTAINING NANOSCALE PATTERNS FOR RENEWABLE ENERGY APPLICATIONS
By
Justin Chet-Mun Wong
August 2016
Chair: Kirk J. Ziegler Major: Chemical Engineering
The global demand for energy is ever increasing. Fossil fuels comprise much of
the current energy supply, but they are a finite and exhaustible source. Therefore,
additional sources of energy must be brought to widespread commercial usage, and
many are already currently available. These alternative energy supplies take many
forms, from solar cells to thermoelectric devices to batteries. However, most alternative
energy devices are not yet commercially viable and device efficiency must be improved
before these devices are widely adopted. Research in this area has focused on the use
of nanomaterials to increase the efficiency of alternative energy devices. Nanomaterials
have the potential to increase device efficiency due to unique material properties which
become dominant only at nanometer length scales. Many methods exist for the
fabrication of nanostructures, but one of the most promising methods is the use of
nanoporous aluminum oxide templates due to its extremely high density and self-
ordering through a hexagonal, close packed pattern. This nanoporous aluminum oxide
can be used as a template for the creation of highly dense, highly ordered
nanostructures.
12
In this work, I demonstrate how to effectively utilize the unique advantages of
aluminum oxide templates for a variety of applications. The first section will focus on the
electrodeposition of uniform nanowire arrays, which may be suitable for thermoelectric
devices, onto the native aluminum substrate. Second, I introduce a method to transfer
aluminum oxide templates to a complex multilayer substrate for a variety of applications,
including the growth of highly-ordered, catalyzed carbon nanotubes. Next, I expand off
of this method to pattern the nanoporous aluminum oxide hexagonal structure directly
into a desired material. Using this process, I am able to pattern highly-ordered
nanostructured thermoelectric materials with dimensions below 15 nanometers. Finally,
I develop a simple model to describe the growth of catalyzed carbon nanotubes in
nanoporous aluminum oxide templates. This model can give a deeper understanding of
the underlying mechanisms present in catalyzed carbon nanotube growth in
nanoporous aluminum oxide. This work highlights an effective approach for creating the
highly-dense, highly-ordered nanostructures required for many energy conversion
applications.
13
INTRODUCTION AND BACKGROUND
One of the largest challenges facing the world today is its ever increasing need
for energy. As the global population increases and more people gain access to modern
technology, world energy consumption is expected to increase from 505 quadrillion Btu
in 2008 to 770 quadrillion BTU in 2035 [1]. Because of this ever increasing need the
world’s fossil fuel energy sources are expected to be depleted in the range of 100 years,
with coal as the only remaining major fossil fuel after the year 2042 [2]. It is prudent to
reduce fossil fuel usage in the near future for several reasons. First, with instability and
uncertainty in several key regions around the world, it is important to move towards
energy independence. Second, the usage of fossil fuels releases undesirable pollutants
into the atmosphere [3]. Finally, it is important to ensure a continuous supply of energy
so reducing dependence on fossil fuels before depletion is key. Therefore, it is
imperative that alternative forms of energy generation are presently developed to
reduce and eventually eliminate the world’s future fossil fuel usage.
Fortunately, there has been a recent increase in alternative energy usage. It is
calculated that renewable energy sources made up 19% of the total world energy
generation in 2008 and is estimated to grow to 23% in 2035 [1]. While heartening that
the percentage growth of renewable energy sources is expected to outpace the growth
rate of other energy sources, investment in current renewable energy technology alone
is not enough to replace fossil fuels as the world’s major energy source. Much more
effort must be devoted to the research of advanced alternative energy sources if the
world is to avoid a critical energy shortage in the future.
14
Thermoelectric Devices
While there are many forms of alternative and renewable energy sources, one of
the most intriguing are thermoelectric devices, which convert thermal energy to
electrical energy. Similarly, thermoelectric devices can be used for cooling by applying
an electrical current to cause a cooling effect. Thermoelectric devices have many uses,
from the automotive industry [4, 5] to more directed uses such as biothermal batteries
for pacemakers [6]. NASA currently uses thermoelectric devices for solid state cooling
as the absence of moving parts allows for increased reliability [6]. It is predicted that
thermoelectric devices could be used in many localized cooling applications and even
replace traditional compression refrigerators [5]. However, the largest advantage of
efficient thermoelectric devices would be the capture and conversion of low grade waste
heat, which is heat produced by other processes and devices that do work. Since no
process is 100% efficient, much of the energy that is not converted into usable work is
expended as waste heat. Thermoelectric devices have great potential for recapturing
much of this waste heat and turning it back into a usable form of energy. Even the ability
to recapture a small fraction of waste heat that is generated worldwide would be a huge
step towards decreasing reliance on fossil fuels.
Thermoelectric Figure of Merit
The major challenge in the adaptation of thermoelectric devices is creating highly
efficient devices. Thermoelectric device efficiency can be expressed as the
dimensionless thermoelectric figure of merit, ZT. A ZT of 3 or higher is considered
necessary to create commercially viable devices [7], while current laboratory-scale bulk
material devices have only achieved a ZT of 1.4 [4]. It is clear that the ZT of
thermoelectric devices must be increased in order to achieve widespread adaptation.
15
One promising method to increase the ZT is through the decoupling of the electrical and
thermal conductivity of the thermoelectric material. In bulk materials, the electrical and
thermal conductivity are typically coupled; high electrical conductivity in a material
corresponds to a high thermal conductivity. However, a higher ZT can be achieved if the
electrical conductivity or thermal conductivity can be altered separately.
Thermoelectric Thermopower
The basis of thermoelectric devices is found in a relationship known as the
Seebeck Effect, which describes the relationship between the temperature gradient and
electric potential of a thermoelectric material. The ratio of the temperature gradient and
electrical potential is known as the Seebeck coefficient, given as:
𝑆 =∆𝑉
∆𝑇
(1-1)
where S is the Seebeck coefficient, ∆V is the electric potential, and ∆T is the
temperature gradient.
Alternatively, a current can be passed through a thermoelectric device to cause a
temperature gradient. This is known at the Peltier Effect. The Peltier coefficient and
corresponding heat rejected is given as:
𝛱 = 𝑆 ∙ 𝑇 (1-2)
𝑄𝑃 = 𝛱 ∙ 𝐼 (1-3)
where T is the temperature in Kelvin at the surface of the Peltier cooler, I is the current,
and QP is the rejected heat. These cells can be connected to produce electrical power
as shown in Figure 1-1.
Many current thermoelectric devices use a combination of n-type and p-type
materials. An n-type material is one that has free electrons, while a p-type material is
16
one that lacks electrons. N-type and p-type materials are commonly referred to as
electron rich or electron deficient materials, respectively. In the p-type material, the lack
of extra electrons creates electron vacancies, or holes. Electrons and holes are
collectively referred to as free charge carriers. A common method for creating n-type
and p-type materials is doping, where a small amount of impurities are introduced to the
material to provide the desired quality. The choice of dopant determines whether the
material is n-type or p-type. In the power generation mode, the thermal energy is the
driving force. Since the heat source provides more thermal energy, the charge carriers
are more mobile and therefore the free charge carriers tend to aggregate towards the
heat sink. This directional motion of charge carriers creates a current through the device
that is used to generate electrical power. In the refrigeration mode, the applied electric
potential is the driving force. The charge carriers are forced to the heat rejection surface
due to the electric potential, carrying thermal energy with them. This causes a
temperature drop at the active cooling surface.
Thermoelectric devices are measured using the thermoelectric figure of merit,
ZT. It is a dimensionless number incorporating several properties of the material, which
describes the efficiency of a thermoelectric device. The ZT is given for a single material
as:
ZT =𝑆2𝜎𝑇
𝜅
(1-4)
where T is the average temperature of the material, 𝜎 is the electrical conductivity, and
𝜅 is the thermal conductivity.
One prominent method of increasing ZT in thermoelectric devices is the use of
nanostructures [8, 9]. Nanowires, in particular, show promise because of their one-
17
dimensional structure, where phonon scattering becomes significant enough to disrupt
the thermal conductivity while the electrical conductivity is still promoted along the
vertical axis of the nanowire. Successful nano-structuring of thermoelectric materials
would provide an important step towards increasing the ZT of devices [8]. Current
experimental nanowire thermoelectric efforts have reached a ZT of 2.4 in Bi2Te3/Sb2Te3
superlattice nanowire structures [10] while models predict much higher ZT with
decreasing nanowire diameter, surpassing a ZT of 3 in the sub-10 nanometer range [8].
While the efficiency of thermoelectric devices is often explained with the ZT for a
single material, in reality, most nanowire-based thermoelectric devices are constructed
with a combination of p and n type materials. Therefore, it is more useful to define ZT in
terms of p-type and n-type materials as shown:
𝑍𝑇 =(𝑆𝑝 − 𝑆𝑛)
2𝑇
[(𝜌𝑛𝜅𝑛)12 + (𝜌𝑝𝜅𝑝)
12]
(1-5)
where ρ is the electrical resistivity.
Thermoelectric materials exhibit a local ZT maximum versus temperature, which
is specific to a given material. A plot of some of the most common thermoelectric
materials showing effective temperature ranges is shown in Figure 1-2. Perhaps the
most widely studied thermoelectric material is bismuth telluride and its alloys [11].
However, there has been much recent research on the use of different materials,
including silicon germanium and silicon germanium alloys [12].
Increasing Figure of Merit
Increasing ZT is possible through adjustment of the constituent variables. The
Seebeck coefficient and temperature are inherent properties of the material and system,
respectively, so given a particular application (set output temperature) it is impractical to
18
suggest increasing ZT through modification of these variables. Instead, increasing ZT
by increasing the electrical conductivity or reducing the thermal conductivity is the ideal
method. Increasing the electrical conductivity is possible by increasing the carrier
concentration but, in most thermoelectric materials, the charge carriers move quite
readily through the structure. This free charge carrier movement facilitates both
electrical and thermal conductivity; as one increases, so does the other. This
relationship is expressed in the Wiedemann-Franz law:
𝜅
𝜎= 𝐿 ∙ 𝑇 (1-6)
where L is the Lorentz number (2.44 x 108 WΩK-2). Therefore, an increased carrier
concentration can actually be detrimental to the performance of thermoelectric devices,
as ZT exhibits a local maximum versus increasing carrier concentration, as seen in
Figure 1-3.
Because of the difficulties involved in increasing ZT through manipulation of the
electrical conductivity, the thermal conductivity is the most promising variable to
manipulate. The total thermal conductivity is a sum of the thermal carrier contributions
from electrons and lattice wave vibrations, otherwise known as phonons. By disrupting
the travel of either electrons or phonons within the material, the thermal conductivity
decreases. However, since the electron contribution is proportional to the electrical
conductivity, disrupting electron travel would also decrease the electrical conductivity.
Therefore, phonon transport must be disrupted without interrupting electron transport in
order to decrease overall thermal conductivity for increased ZT. This idea is commonly
referred to as a thermoelectric material behaving as an electron crystal, phonon glass
[13], where the material behaves as a crystal with regards to electron mobility and as a
19
glass with regards to phonon mobility. Crystalline materials transfer electrons readily,
preserving a high electrical conductivity, while in glassy materials the phonon mean free
path is on the order of angstroms, resulting in heavy phonon scattering and reduced
thermal conductivity.
Improving Phonon Scattering
While there are many strategies for increasing phonon scattering, the three major
categories are the use of dopants, complex lattice structures, and nano-structuring. In
doping, a base thermoelectric material such as Bi2Te3 is chosen and the dopant is
incorporated through either a substitution or interstitial mechanism. The dopant acts as
a phonon scattering site, thereby reducing the thermal conductivity. Some experiments
involve the use of other elements as the dopant [14, 15], while others add a dopant to a
ternary system [16]. Still others have shown that adding a dopant to a quaternary
system has led to an increase in ZT [17].
The second strategy to increase phonon scattering consists of complex lattice
structures, such as skutterudites and superlattice structures. A skutterudite is a class of
minerals that forms a cubic structure consisting of CoP3, CoAs3, CoSb3, RhP3, RhAs3,
RhSb3, IrP3, IrAs3, or IrSb3 [18]. An example unit cell is shown in Figure 1-4. Many
times, other elements, such as Fe, are substituted in varying concentrations in a
strategy similar to the dopant one above; the resulting mismatches in lattice structure
can disrupt phonon travel and reduce the overall thermal conductivity. In a superlattice
structure, shown in Figure 1-5, the basic unit cell is replaced by alternating nanometer-
scale elemental layers. This layering introduces grain boundaries, which act as phonon
scattering sites to reduce the thermal conductivity.
20
The third category to increase phonon scattering is nano-structuring of
thermoelectric materials; decreasing the feature size of the thermoelectric material to a
low-dimensional structure, such as nanowires. Nanowires are an ideal candidate for
reduced dimensions because of their large surface area to volume ratio, which provides
a high surface area for phonon surface scattering. The scattering on the surface walls of
the nanowires reduces the mean free path of phonons within the material, decreasing
the overall thermal conductivity. Simultaneously, the electrical conductivity is decreased
due to confined dimensions, but the effect on electron mobility is minimal compared to
the effect on phonon mobility due to the larger phonon mean free paths and directed
electron transport. Nanostructures, such as nanowires, provide a way to effectively
decouple the thermal conductivity from the electrical conductivity, thereby increasing
ZT. This effect is shown in a pioneering paper by in the field of reduced dimension
thermoelectric materials: given a nanostructure, such as nanowires, it is predicted that
the ZT increases exponentially as the confining dimension is reduced to zero [8]. The
conclusion of the work is shown in Figure 1-6.
To obtain a ZT of 3 to produce a commercially viable thermoelectric device, the
characteristic dimension must be reduced to the order of 10 nanometers. Much work
has been conducted on shrinking the characteristic dimension of nanostructures for
thermoelectric applications [19, 20]. Additionally, others have combined multiple
strategies in attempts to increase ZT, such as the creation of superlattice nanowires [21,
22].
Bismuth Telluride Thermoelectric Nanostructures
There are a diverse range of methods that have been used to nanostructure
bismuth telluride. Bismuth telluride is one of the most commonly used thermoelectric
21
materials with a bulk ZT value around 1 at low temperatures relative to many other
thermoelectric materials [5]. This makes bismuth telluride a valuable thermoelectric
material as it is one of the most viable options to capture low-grade waste heat emitted
from other industrial processes. Typically, this waste heat is dissipated through cooling
methods, but if thermoelectric devices could capture even a small fraction, it would be a
huge advancement in the field of alternative energy.
However, any nanostructures used for this purpose not only must be easily
tunable, but must also be easily produced on an industrial scale. One common method
is the catalytic vapor-liquid-solid (VLS) growth of Bi2Te3 nanostructures using gold
nanoparticles as the catalyst material [23, 24]. VLS growth is advantageous because
nanostructure size can be controlled through catalyst particle size, but the resulting
nanowire density is low compared to other methods, which is undesirable for high
efficiency devices. A second method is a solution-based method in which Te nanowires
are first synthesized using a Te precursor solution [25, 26]. Bi precursor solution is then
added to the Te nanowire solution, which reacts to form Bi2Te3 nanowires. This
produces a higher density of nanowires than VLS growth, but does not control the
alignment and sizing of the nanowires, making them difficult to incorporate into devices.
A third method is the electrodeposition of Bi2Te3 into nanostructured templates, which
can be done with a variety of electrodeposition techniques and template materials [27-
29]. Electrodeposition is a cheap method for creating nanostructures, which is also
easily scalable to larger processes.
Anodized Aluminum Oxide Templates
Perhaps the single most important aspect in the creation of nanostructures is the
density at which they are grown. Highly ordered, highly dense nanostructures provide
22
distinct advantages over unorganized nanostructures of similar dimensions. Anodized
aluminum oxide (AAO) is a highly-ordered, high-density hexagonal porous array, which
is formed with nanometer scale pores through electrochemical anodization [30-32].
Using AAO as a template for creating nanostructures offers several advantages over
other fabrication methods. AAO templates provide an extremely high density of a
vertically-aligned pattern. Additionally, anodization is a cheap and easy method that
allows for control of several key nanowire parameters, such as pore diameter through
temperature, applied voltage, and pH [32]. This precise control is necessary if the
desired nanowire diameters and aspect ratios for high ZT are to be achieved as well as
the high density necessary for carbon nanotube-based alternative energy devices.
Aluminum can be anodized into AAO for use as a template through an
electrochemical reaction, such as the one shown in Figure 1-7. By applying an
overpotential to high purity aluminum immersed in an acidic anodization solution, a
close-packed hexagonal structure is formed as shown in Figure 1-8. With the initial
applied potential, a dense barrier layer type oxide forms. Upon continued anodization
small dimples are formed at the surface of the oxide and continue to propagate into the
aluminum for the duration of the anodization. The continued anodization helps to form
the close-packed, hexagonal pattern as the initial irregularities on the surface of the
aluminum are smoothed. The electrochemical response and corresponding pore
structure is shown in Figure 1-9, in a four-step process described by Hill [33].
The acidic solution also simultaneously helps form and etch the aluminum oxide,
creating a porous upper layer and compact barrier layer between the porous layer and
aluminum substrate [30, 34]. Pore growth is propagated due to the oxygen anions from
23
solution diffusing through the oxide layer and combining with the aluminum cations from
the metal substrate as shown in Figure 1-10. However, this growth is counteracted by
the dissolution of the aluminum oxide into solution as shown in Figure 1-11.
The growth and dissolution rates are determined by a complex interaction of
many parameters, but it is estimated that 30% of aluminum ions are dissolved into
solution [35]. However, precise control over the pore diameter and length of the pores
can be achieved by manipulating key anodization conditions, such as type of electrolyte,
pH, temperature, and applied voltage [32].
Various materials can be deposited into the pores of the template to fabricate
vertical arrays of nanowires on the substrate, provided the barrier layer at the bottom of
the pores is removed to connect the nanowires to the aluminum substrate. Therefore,
two important steps must be taken to expose the underlying aluminum substrate,
without destroying the highly ordered AAO structure. The first is barrier layer thinning, in
which the anodizing voltage is stepped down to thin the barrier layer at the bottom of the
pores. Since the barrier layer is proportional to the anodizing overpotential, reducing the
overpotential produces a thinner barrier layer relative to the walls of the pores as the
anodization reaction happens through the barrier layer. The second part of the process
is pore opening, in which the barrier layer is etched to expose the aluminum substrate.
Because the etching is isotropic, the pore walls are simultaneously etched and precise
timing and control are necessary to completely etch the barrier layer without completely
removing the pore walls. The pore opening is monitored using chronoamperometry, in
which a small overpotential is applied to the AAO template and the resulting current is
monitored. Once the barrier layer is etched, the current increases dramatically as the
24
conductive aluminum substrate is exposed at the bottom of the pores. These two
processes result in an AAO template attached to a conductive aluminum substrate as
shown in Figure 1-12.
Electrodeposition
Using electrochemical reactions has proved useful not just for the creation of
AAO, but also for the creation of nanowires for the same advantages listed above: a
cheap and scalable method. In an electrochemical cell, there are two reactions that take
place, a reduction and an oxidation reaction, commonly referred to as a redox reaction.
These occur at the surface of two electrodes, the working electrode and counter
electrode. The working electrode is the electrode that is designated as the desired final
product, while the counter electrode provides the balancing reaction. Depending on the
desired product, the working electrode can either be in a reduction or oxidation reaction.
For example, the creation of AAO is an oxidation of the Al at the working electrode into
AAO, while the creation of metallic nanowires is a reduction at the working electrode, as
positive metal ions are deposited into pure atomic form. Figure 1-7 shows a typical
electrochemical cell.
Bismuth telluride nanowires can be created using an electrodeposition method.
Stoichiometric bismuth telluride, Bi2Te3, is the most useful for thermoelectric devices
and much of the work in this area has focused on the conditions necessary to deposit
the ideal 2:3 ratio in AAO templates [29]. Others have focused on the characterization of
Bi2Te3 nanowires grown from electrodeposition [28]. However, there has been little
focus on the uniformity of the deposited Bi2Te3 nanowires. Most work with the Bi2Te3
nanowires are still embedded in the AAO template or mechanically polish the top of the
AAO template to produce uniform length. While useful for studying essential properties
25
for nanowire-based bismuth telluride thermoelectric devices, non-uniform arrays are
more difficult to incorporate for device fabrication.
Carbon Nanotubes
In addition to nanowire-based thermoelectric devices, carbon nanotubes (CNTs)
are a promising candidate for nanostructured alternative energy applications. CNTs
have a wide range of potential applications due to their excellent mechanical and
electrical properties. Perhaps one of the most exciting applications is the use of CNTs
for energy storage. It has been shown that CNTs could be used to store hydrogen for
use in hydrogen fuel cells through physisorption into the centers of the CNTs [36].
Similarly, CNTs can be used as the anodes of lithium ion batteries, offering
improvements over currently available lithium ion batteries by increasing the energy
density by a factor of 3 [37]. Yet, even with so many of these promising applications, the
cost-effective growth of CNTs remains a challenge. Various methods of CNT growth
exist, but most involve high energy processes that are not cost-effective and therefore
not suitable for industrial scale. Therefore, exploring cost-effective growth methods is a
vital area in the world of CNT research.
CNTs can be created through a wide variety of methods, but some of the most
promising involve the growth of CNTs within an AAO template. The template provides a
structured growth guide for the CNTs, which can be grown using either catalyzed or
uncatalyzed growth [38, 39]. For catalyzed growth, metals, such as cobalt, iron, and
nickel, are commonly used as catalysts [40, 41]. Chemical vapor deposition (CVD) is
used for catalyzed, template-based growth, which produces an extremely high density
of CNTs when compared to other growth methods.
26
Growth of freestanding CNTs has been widely studied and growth mechanisms
of CNTs have been separated into two general categories: tip growth and base growth.
In tip growth, the catalyst particle sits at the top of the CNT, with new growth originating
at the tip of the CNT. In base growth, the catalyst particle remains on the substrate
surface and the CNT is grown from the base. There are many theories that attempt to
explain the differences between tip and base growth, but there is no general consensus
on the subject. It is hypothesized that base growth occurs when the catalyst particles
have a higher attraction to the substrate, while tip growth occurs when there is less
interaction between the catalyst particle and substrate [42-44]. In addition, the number
of walls in CNTs is believed to be dependent on both the catalyst particle size and the
growth conditions [43].
Chapter Summary
In summary, highly-ordered, highly-dense AAO templates can be created using
electrochemical methods. These templates are ideal candidates for use in creating
nanostructured materials, which can be incorporated into alternative energy devices,
such as thermoelectrics, batteries, and solar cells. However, many challenges remain
with the creation of these nanostructures, and much of the present work addresses
these important issues.
In this dissertation, I create a wide variety of nanostructures which can be
incorporated in the creation of highly efficient alternative energy devices. First, I show
that AAO templates can be used as an easy, scalable method for creating tunable
thermoelectric nanowires. I utilize a backfill method as an alternative to other common
electrodeposition methods and address several issues present in these common
methods. Second, I create silicon germanium nanostructures with the potential for
27
enhanced ZT. This is done through a unique AAO transfer method which can be used to
create highly ordered nanoparticles for anisotropic etching of the silicon germanium,
directly transferring the highly ordered, highly dense AAO pattern to the silicon
germanium substrate. Third, I extend this process to create pre-patterned aluminum,
which I use to create complex multilayered nanostructures. I use these nanostructures
as a catalyst support and growth template for high density CNTs. Fourth, I describe a
predictive model for the growth mechanisms and growth regimes of AAO templated
CNTs, which has no theoretical model to date. Finally, I conclude with directions for
future work.
28
Figure 1-1. Thermoelectric device using the Seebeck Effect (power-generation). The p and n refer to p-type and n-type materials and their respective charge carriers and current is generated with a temperature gradient as the driving force. Drawing courtesy of author.
29
Figure 1-2. Temperature range and resulting ZT of several common n-type thermoelectric materials. Notice the difference in local maxima for various materials. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials [5], 2008.
30
Figure 1-3. ZT and its constituent variables versus increasing carrier concentration. A common alternative representation of the Seebeck coefficient is α. Additionally, α2σ is known as the power factor, a representation of the potential of the material as a thermoelectric generator. Reprinted by permission from Macmillan Publishers Ltd: Nature Materials [5], 2008.
31
Figure 1-4. Schematic of a cobalt-antimony skutterudite unit cell. Reprinted from Nano Energy, Vol. 2, Alam, H. and Ramakrishna, S., A review on the enhancement of figure of merit from bulk to nano-thermoelectric materials, Pages No. 23, Copyright 2012, with permission from Elsevier [18].
32
Figure 1-5. Schematic of a superlattice configuration of SiGe. Instead of a standard unit cell structure, Si and Ge layers are alternated on a nanometer scale, creating many grain boundaries for phonon scattering Reprinted from L. Shi, J. Jiang, G. Zhang, and B. Li, "High thermoelectric figure of merit in silicon-germanium superlattice structured nanowires," Applied Physics Letters, vol. 101, p. 233114, 2012, with the permission of AIP Publishing [12].
33
Figure 1-6. Thermoelectric figure of merit, ZT, as a function of decreasing critical dimension length. A commercially viable ZT of 3 occurs near the critical dimension length of 100 Å or 10 nanometers Reprinted from L. D. Hicks, T. C. Harman, and M. S. Dresselhaus, "Use of quantum-well superlattices to obtain a high figure of merit from nonconventional thermoelectric materials," Applied Physics Letters, vol. 63, p. 3230, 1993, with the permission of AIP Publishing [9].
34
Figure 1-7. Schematic of an electrochemical cell. In an anodization process, electrode 2 is the working electrode. In an electrodeposition process, electrode 1 is the working electrode. A third reference electrode can be added to the system, especially in electrodeposition processes where the overpotential is small, to maintain a consistent baseline. Drawing courtesy of author.
35
Figure 1-8. Top down SEM image of AAO. Notice the close-packed hexagonal structure. Photo courtesy of author.
36
Figure 1-9. Schematic of pore propogation in relation to the current response for a typical anodization process. Drawing courtesy of author.
37
Figure 1-10. Schematic of individual pore growth. The growth of the aluminum oxide is driven by the applied potential, which causes ion migration through the barrier layer. Aluminum ions originate from the aluminum substrate while oxygen ions diffuse from solution. Drawing courtesy of author.
38
Figure 1-11. Mechanism of dissolution of aluminum oxide into solution. The dissolution is carried out by the excess hydrogen ions and water molecules in solution. Drawing courtesy of author.
39
Figure 1-12. AAO template pore wall structure. The jagged edges are due to the fracturing necessary to view sidewalls. Photo courtesy of author.
40
UNIFORM ELECTRODEPOSITION OF BISMUTH TELLURIDE NANOWIRES
Electrodeposition is a widely used process in industries ranging from decorative
plating to computer chip fabrication. It is so widely used because it is relatively cheap
compared to other methods of deposition, such as physical vapor deposition (PVD).
Additionally, it is easy to modify electrodeposition parameters to obtain desired
deposition characteristics. Because of the precise control necessary for creating
nanostructured materials, electrodeposition remains a popular choice for deposition into
AAO templates. In the following work, I discuss the work I conducted to investigate
electrodeposition into AAO templates, specifically for bismuth telluride nanowires in
AAO templates suitable for device fabrication, and some of the challenges associated
with this method.
Electrodeposition Methods
There is no general consensus on the best electrodeposition method with which
to deposit nanowires into AAO templates, let alone the best method for depositing
thermoelectric materials into AAO templates. The traditional method is a potentiostatic
method, where the voltage is set to a determined deposition potential [29]. This
approach is the most common form seen in bulk electroplating, due to the constant
driving force of the applied potential. Alternatively, a galvanostatic method is used to set
the deposition rate constant, as the current is an indication of the amount of material
deposited, through the total amount of charge passed [45]. A third method is pulsed
deposition, in which the applied voltage is modulated between a deposition potential
and a rest potential [46]. This allows the ions in solution time to replenish depleted
areas. A fourth method used is alternating current deposition, in contrast to the direct
41
current methods above [47]. All of these methods have been extensively utilized in the
deposition of nanowires in AAO templates, but potentiostatic deposition was chosen for
much of the depositions performed in this work. However, the other electrodeposition
methods were explored at various points as well to determine the most suitable method
for each deposition.
Part of my study was to characterize specific aspects of the nanowire creation
process. While AAO is a widely used material in nano-structuring, barrier layer thinning
remains an overlooked aspect. In barrier layer thinning, the voltage is stepped down 5 V
every 5 minutes until no appreciable current is measured, which results in a staircase-
like function. Instead, I incorporated a ramped down thinning method, which
demonstrated great control over the barrier layer thinning process. However, imaging
such resulting structures is extremely difficult due to the mechanical properties of AAO.
Typical methods used to image side views of AAO, such as fracturing, cleaving, and
dicing, are not feasible unless the template has been through pore opening so that the
pore walls are thin enough to fracture along well-defined fault lines. With only barrier
layer thinning, the walls are too thick to fracture cleanly and the bottom pore structure is
either deformed or destroyed. However, pore opening etches the barrier layer and
surrounding walls, removing the area of interest. Therefore, imaging becomes a non-
trivial task and a unique approach was developed to capture images of the barrier layer.
Overall, this approach allowed me to achieve better control of the barrier layer thinning
and subsequent pore opening processes, which led to higher uniformity nanowire
depositions in the resulting bismuth telluride nanowire study.
42
Materials and Methods
AAO templates
AAO templates were created using a two-step anodization process. High purity
aluminum film (99.999% purity or greater, 150 µm thickness) was purchased from
Goodfellow. The aluminum was cleaned with three separate solvents: soapy water,
acetone, and ethanol, respectively, by sonication for 15 minutes each. The aluminum
was then electrically isolated on one side either with nail polish or through the use of a
sample holder that exposed only one side of the aluminum with the other side
electrically connected to a lead. Samples were submerged in an anodization bath and
an overpotential was applied specific to the desired final pore size. In a typical case, 40
V in a 0.3 M oxalic acid was used to obtain 40 nanometer pores. The bath temperature
was kept constant at 15 ºC and the first anodization was carried out for a minimum of 8
hours. This was to allow sufficient time for the first anodization to smooth out any
surface irregularities that were inherent in the starting aluminum surface. After the first
anodization, the AAO was removed by submerging the sample in an aqueous mixture of
6 wt. % phosphoric acid and 1.8 wt. % chromic acid. The acid mixture was kept at a
temperature of 60 ºC and the etching continued until no AAO remained on the surface
of the sample. The time for the etching varied depending on the thickness of the AAO
from the first anodization. After complete removal of the AAO, a patterned surface of
dimples existed on the surface of the remaining aluminum. The sample was then placed
back into the anodization bath for a second anodization. The second anodization was
carried out with the same conditions as the first anodization, with the anodization
stopped based on desired AAO template thickness (observed growth rate for the
conditions listed above was approximately 5 µm/h). Additional annealing and
43
electropolishing steps were occasionally included after cleaning and electrical isolation,
respectively, to produce the highest ordered samples.
To prepare the sample for electrodeposition, the aluminum oxide barrier layer at
the bottom of each pore was removed through barrier layer thinning and pore opening.
At the end of the second anodization, the overpotential was ramped down from the
anodizing voltage to 0 V over a period of 15 minutes. Pore opening was achieved with
an aqueous solution of 5 wt. % phosphoric acid. The solution was kept at room
temperature and the pore opening process was monitored through chronoamperometry.
A potential of 0.05 V was applied and the sample was rinsed and removed from the
phosphoric acid solution when the current hit a maximum, indicating the complete
etching of the barrier layer. Combining these processes produced an AAO template with
an underlying conductive aluminum substrate.
Electrodeposition into AAO templates
Electrodeposition cells were constructed in a similar fashion to the cell depicted
in Figure 1-7, with the addition of a reference electrode for reproducibility of applied
potentials. Deposition conditions varied based on the desired nanowire composition, but
a typical deposition was that of gold nanowires. These depositions were carried out in a
potentiostatic regime at +0.65 V vs AgCl using a Technigold RTU gold plating bath from
Technic, Inc. The deposition was halted once the first sign of overgrowth was visually
confirmed as a golden color change on the surface of the AAO template.
Focused ion beam
The focused ion beam (FIB) Dual Beam Strata DB235 at the Major Analytical
Instrumentation Center (MAIC) was used to achieve the structure needed to image the
barrier layer at the bottom of the AAO pores, in a process similar to the liftoff technique
44
for transmission electron microscopy (TEM) sample preparation. An AAO template was
created using the method described above, with barrier layer thinning the last step
before being mounted on an imaging stub. A platinum protective layer was sputtered
onto a small section of the sample. The ion beam was then used to mill a wedge which
was adjustable based on required imaging parameters, such as tilt and angle. The ion
beam power was altered to provide an imaging feature size down to the 10 nanometer
range, with little to no roughness seen in traditional mechanical methods of sidewall
pore imaging. The milling process is shown in Figure 2-1.
Bismuth telluride nanowire deposition
Bismuth telluride is difficult to electrodeposit because tellurium does not readily
dissolve in solution. Initially, electrodeposition solutions were made by heating a 1 M
nitric acid solution before adding tellurium. The solution was then cooled to room
temperature, when much of the dissolved tellurium precipitated out of solution. In many
cases in literature, the molarity is calculated without regard to the precipitate. However,
the precipitate actually contains a majority of the added tellurium so it was vital to filter
the precipitate and calculate the tellurium remaining in solution. A less tedious method
was incorporated by dissolving the tellurium in a 5 M nitric acid solution at room
temperature, in which the tellurium dissolved more readily, and then diluted to a 1 M
nitric solution. After the tellurium solution was made, bismuth was added to complete
the deposition solution. All deposition solutions for the following work were 7.5 mM
bismuth and 10 mM tellurium dissolved in 1 M nitric acid.
Depositions into AAO templates were conducted on two substrates, the native
aluminum substrate and a 50:50 gold:palladium film sputtered onto the top of the
template. To create the Au:Pd electrode, 100 nanometers of 50:50 Au:Pd was sputtered
45
onto the top of the AAO template. The aluminum substrate was then etched away using
an aqueous solution of copper chloride and hydrochloric acid with concentrations of 100
g/L copper chloride in a 4 M hydrochloric acid solution. After aluminum removal, pore
opening was performed using a 5 wt. % phosphoric acid aqueous solution on the pore
bottoms (as the electrode was now located on the top of the AAO template).
Potentiostatic control was used for all depositions, held at -50 mV vs SCE. The
temperature of the deposition solution was maintained at 4 ºC to prevent dissolution of
the AAO template during the deposition process and to control the rate of the
deposition. Depositions were carried out over a period of several hours due to the lower
solution temperatures.
Results and Discussion
Initial depositions of gold nanowire arrays showed a lack of uniformity, and a
clear periodic growth front was observed in many depositions. This subsequently led to
an uneven filling of the AAO pores shown in Figure 2-2. A secondary problem was the
poor adhesion between the deposited nanowires/AAO template complex and the
aluminum substrate. There were many instances of delamination of the AAO template
with the embedded nanowires, even due to normal processes, such as drying the
sample after deposition. It was determined that the nucleation of nanowires was being
hindered by a part of the nanowire creation process. These results led to the
investigation of a key step in the process, barrier layer thinning. Through several
deposition trials, it was determined that a ramped down thinning process was superior
to stepped down thinning. A method was developed to image the resulting pore
structure after barrier layer thinning. This process accomplished two objectives: imaging
the barrier layer and confirming the predicted ramped structure of the pore bottoms. The
46
images of the ramped down barrier layer thinning are shown in Figure 2-3. It is
hypothesized that the ramped down thinning led to a more consistent pore opening
because the voltage could be taken down to near zero before a lack of resulting current,
as opposed to a stepped down procedure which did not produce any current below 15
V. A lower voltage indicated a thinner barrier layer, which leads to less variation in pore
opening.
For the bismuth telluride nanowire depositions, the Au:Pd top electrode provided
several benefits over the native aluminum substrate. Aluminum is a poor material for
electrodeposition and for use in device fabrication. Additionally, the Au:Pd electrode
minimized pore widening compared to pore widening with the native aluminum
substrate, as shown in Figure 2-4.
Depositions were carried out at room temperature, but multiple problems were
observed. The nitric acid would attack the AAO template, destroying the porous
structure. Additionally, higher temperatures introduced dendritic growth of bismuth
telluride compounds. Therefore, a lower temperature was instituted to minimize these
issues and a summary of electrodeposition voltages vs atomic percentages is listed in
Table 2-1 and plotted in Figure 2-5. The deposition voltage for stoichiometric conditions
was identified as -110 mV vs SCE through interpolation of these trials.
Using the interpolated deposition potential as a basis, bismuth telluride
nanowires were deposited in AAO templates. This was done using a three electrode
system, with an SCE reference electrode and platinum counter electrode. As shown in
Figure 2-6, depositions showed a much greater degree of uniformity compared to other
depositions discussed earlier.
47
The results show good correlation with the expected deposition potentials seen in
other works. However, the inclusion of ramped down barrier layer thinning, along with
added parameters, such as the Au:Pd electrode, temperature control, and accurate
molarity determination, resulted in uniform bismuth telluride depositions, which are not
prioritized in previous works.
Chapter Summary
In this work, uniform bismuth telluride nanowire arrays were created after careful
investigation of the processes involved in creating AAO templates as well as the
examination of numerous electrodeposition conditions. Stoichiometric bismuth telluride
was achieved in several cases, but electrodeposition of nanowire arrays, especially in
multi-component systems, can prove to be tricky. Many parameters, such as
temperature, solution concentration, and electrode surfaces, can vary minutely and
have unproportioned effects on final deposition characteristics, such as the uniformity
studied here, as well as the composition and crystallinity. Therefore, other nano-
structuring methods are explored in Chapter 3 and Chapter 4 that utilize the advantages
of AAO templates, without many of the complexities that occur in electrodeposition.
48
Figure 2-1. Schematic of milling process. A platinum protective layer is placed on top of the AAO template. Then the FIB mills an area next to the protective layer, which exposes the underlying pore structure and barrier layer. Drawing courtesy of author.
49
Figure 2-2. Metal nanowire deposition in AAO templates. A) Side view of nanowire deposition in AAO template and B) top-down view of nanowires deposited in AAO template. Notice that some pores remain partially unfilled as indicated by the circled area. Photos courtesy of author.
50
Figure 2-3. Bottom of AAO template pore and barrier layer. The higher contrast area is indicative of the compact barrier layer as opposed to the less dense aluminum oxide which makes up the pore walls. Photo courtesy of author.
51
Figure 2-4. SEM image of A) bottom side pore opening with AuPd substrate vs B) top side pore opening with aluminum substrate. The bottom side becomes the new side exposed to the deposition solution since the Au:Pd electrode is sputtered on the opposite side. Photos courtesy of author.
52
Figure 2-5. Plot of measured atomic percentages of bismuth versus applied potential. The interpolated applied potential for stoichiometric Bi2Te3 is -110 mV vs SCE.
0
10
20
30
40
50
60
70
80
90
100
-200 -150 -100 -50 0 50 100 150
Ato
mic
Per
cen
tage
Applied Potential (mV)
Bismuth Atomic Percentage versus Applied Potential
53
Figure 2-6. Bismuth telluride electrodeposition in AAO template. Side view of bismuth telluride nanowires in AAO template with insert of low magnification side view. Discontinuous wires were due to the fracturing of the AAO template. Photo courtesy of author.
54
Table 2-1. Bismuth and tellurium atomic percentages at various deposition potentials.
Overpotential (mV) Bismuth (at. %) Tellurium (at. %)
-170 30 ± 1.0 70 ± 1.0 50 57 ± 2.8 43 ± 2.8 80 86 ± 4.7 14 ± 4.7 100 92 ± 2.9 8 ± 2.9 120 88 ± 4.0 12 ± 4.0
55
GROWTH OF CATALYZED CARBON NANOTUBES IN AAO TEMPLATES
Electrodeposition remains a popular method for creating nanostructures in AAO
templates, but in addition to nanowires, CNTs have garnered much interest in recent
years. AAO can be used as a template for CNT growth, not only as vertical guide
channels for growth, but also for the high density patterning of the catalyst nanoparticles
for catalyzed CNT growth. Therefore, the first half of the work described here focuses
on the growth of catalyzed carbon nanotubes with electrodeposited catalysts in AAO
templates.
Alternatively, AAO can be formed on a wide range of substrates, provided
aluminum can be deposited on the given substrate through physical vapor deposition
(PVD) or other means. This eliminates the need to transfer nanowires arrays off the
native aluminum substrate, as nanowires and other nanostructures can be grown
directly on a desired substrate [33, 48-50]. However, much of the work in this area is
focused on aluminum-on-silicon. Direct anodization of aluminum results in a random
ordering of pores, instead of the highly ordered porous structure seen in traditional two-
step anodization methods. To address this, some have deposited thicker layers of
aluminum for use as a sacrificial first anodization layer [51]. This provides better
ordering, but can be costly and impractical; if a high degree of ordering is required,
aluminum in excess of 10 µm must be deposited. Instead, complex multilayer substrates
(CMS) were developed to address these issues. The CMS described in this work can
provide tunable substrates which are tailored for any specific application. Using the top
AAO layer as a nano-structuring template, the underlying layers can be aligned to best
suit a particular application.
56
The work in Chapter 2 along with concurrent electrodeposited catalyst work
described in this work led in turn to the development of the CMS, which take advantage
of the properties of AAO, without the need for electrodeposition. In this way, materials
can be carefully controlled with physical methods instead of solution-based processing.
In creating CMS, PVD tools such as sputter deposition and e-beam deposition are
extensively used so that the thickness of each layer can be controlled down to the
Angstrom level.
In this work, a comprehensive study of AAO templated, catalyzed CNT growth is
described, which led to the investigation for creating CMS for greater flexibility in nano-
structuring applications. In support of CMS, a process was investigated to transfer the
high-density honeycomb patterning of AAO to any desired substrate, giving several
advantages over contemporary methods for CNT growth.
Materials and Methods
Catalyst deposition in AAO templates
AAO templates were created using a two-step anodization process similar to the
conditions described in Chapter 2. High purity aluminum film was cleaned by sonication
in soapy water, acetone, and ethanol, respectively, for 15 minutes each. The aluminum
was then electrically isolated on one side with nail polish. Samples were submerged in a
0.3 M oxalic acid anodization bath. To obtain 40 nanometer pores, 40 V was applied
and kept for 16 hours. After the first anodization, the AAO was removed by etching in 6
wt. % phosphoric acid and 1.8 wt. % chromic acid at 60 ºC. After complete removal of
the AAO, the sample was placed back into the anodization bath for a second
anodization. The second anodization was carried out with the same conditions as the
57
first anodization, with the anodization stopped based on desired AAO template
thickness. Observed growth rate for the conditions listed above was 5 µm/hr.
To prepare the sample for electrodeposition, the aluminum oxide barrier layer at
the bottom of each pore was removed through barrier layer thinning and pore opening.
At the end of the second anodization, the overpotential was ramped down from the
anodizing voltage to 0 V over a period of 15 minutes. Pore opening was achieved with
an aqueous solution of 5 wt. % phosphoric acid. The solution was kept at room
temperature and the pore opening process was monitored through chronoamperometry.
A voltage of 0.05 V was applied and the sample was rinsed and removed from the
phosphoric acid solution when the current hit a maximum, indicating the complete
etching of the barrier layer. Combining these processes produced an AAO template with
an underlying conductive aluminum substrate. The catalyst nanoparticles were then
electrodeposited at the bottom of the channels using a three electrode cell. The AAO
template was attached as the working electrode with a carbon counter electrode and a
silver chloride reference electrode. Several deposition regimes were explored, and a
pulsed deposition process was determined to be the best for catalyst deposition. Typical
pulsed deposition regimes were -0.3 V for 10 ms and 0 A for 1 s as a rest period to
allow replenishment of ions. The deposition solution was vigorously stirred at room
temperature.
Catalyst PVD in AAO templates
Nickel catalyst was deposited using the e-beam evaporator located at NRF. The
e-beam conditions were specified by NRF based on the desired material. 10
nanometers of nickel was deposited from the top of the AAO template, with an
annealing period afterwards to allow the nickel to migrate to the bottom of the pores.
58
Complex multilayer substrates
Samples were prepared by first cleaning the silicon substrate using sonication in
soapy water, acetone, and ethanol for 15 minutes each. Desired materials were
deposited with Angstrom level precision using the e-beam evaporator located at the
Nanoscale Research Facility at UF. Deposition conditions differed based on the
deposited materials, with ramp rates and overall power differing for each material.
Typical deposition rates were 2 Angstrom/second. The e-beam evaporator contained 4
pockets which maintained vacuum in between depositing materials. After all layers were
deposited, the backside of the sample was electrically isolated and then anodized at 40
V in oxalic acid at 3 ⁰C, to reduce the rate of the anodization.
AAO mask creation and transfer
The AAO mask was created in much the same way that AAO templates for
electrodeposition were created. High purity aluminum foil disks were cleaned by
sonication in soapy water, acetone, and ethanol for 15 minutes each. The aluminum
was isolated by placing in a sealed holder with only one side open to the anodization
solution, then anodized at 40 V in 0.3 M oxalic acid for 16-24 hours. The oxide layer
was etched away using an aqueous solution of 6 wt. % phosphoric acid and 1.8 wt. %
chromic acid at 60 ⁰C for one hour. The resulting pre-patterned aluminum was then
prepared for second anodization as an AAO mask, which was where the process
diverged from traditional AAO templates. For the AAO mask, second anodization was
carried out for 3 minutes, with no barrier layer thinning. This created a thin AAO layer
that was approximately 200 nanometers thick. The AAO was then dried with nitrogen
gas before a polystyrene (PS) layer was spin-coated to the top of the mask using a 5 wt.
59
% PS solution dissolved in toluene. An even coating was applied on the top of the AAO
mask, and spin-coating was carried out for 1 minute at 10,000 rpm. After this, the AAO
mask/polystyrene complex was left to dry overnight. A solution of copper chloride in
hydrochloric acid was used to remove the aluminum substrate (100 g/L copper chloride
in 4 M hydrochloric acid). The PS layer acted as a protective layer to keep the AAO
mask intact during the aluminum removal. The AAO mask was transferred to a 5 wt. %
phosphoric acid solution for pore opening. The AAO mask was thin enough to float on
the surface of the solutions, which simplified the transfer of AAO from the copper
chloride solution to the phosphoric acid solution. After 1 hour, the AAO mask was
removed from the phosphoric acid solution, rinsed and transferred to the aluminum-on-
silicon substrate. The transfer was performed by lifting the AAO mask off the surface of
the rinse water using the aluminum-on-silicon substrate. It was important to achieve a
clean lift off; otherwise the AAO mask would wrinkle on the substrate. The resulting
sample was left to dry before the RIE process.
RIE pattern transfer
RIE is a dry etching process which utilizes plasma for chemical as well as
physical etching of the target substrate. For etching the transferred AAO on
Al/catalyst/Si substrate, the PS protective layer was first removed with oxygen plasma.
The conditions for this process were 40 sccm oxygen gas, 10 Torr total pressure, and
150 W power for 5 minutes. The aluminum was then etched using a Cl2/BCl3/Ar mixture
for 3 minutes. These conditions selectively etch the aluminum, while leaving the AAO
mask intact. All etching was done using the RIE machines at NRF.
60
Anodization of CMS pre-patterned surfaces
After the pre-patterning of the surface, the AAO mask was removed using scotch
tape and acid etching. The CMS was then anodized to propagate the porous pattern
through the aluminum layer. For anodization, the backside of the CMS was electrically
insulated with nail polish. The anodization was conducted at 3 ⁰C in oxalic acid, lower
than the 15 ⁰C anodization conditions of the AAO mask fabrication, to reduce the rate of
the anodization for better ordering and stability. The anodization was carried out until
the aluminum was completely oxidized as indicated by the current response dropping to
zero. The CMS creation process is represented in Figure 3-1.
CNT growth
CNT growth was carried out in a tube furnace. After the sample was loaded into
the quartz tube, the ends were sealed and vacuum is pulled on the chamber to the
mTorr range. Argon gas was introduced to the system as the carrier gas and the
chamber was heated to 650 ºC. Once the chamber reached the growth temperature, the
ethylene feed and hydrogen gases were introduced into the chamber. Growth times
ranged from 15-60 minutes. Afterwards, ethylene and hydrogen were turned off while
the argon continued to flow until the chamber had cooled enough to remove the
samples.
Results and Discussion
The electrodeposition of catalysts for CNT growth exhibited some success with
nickel deposition as seen in Figure 3-2. However, there was variation of the catalyst
heights across individual samples, with many of the pores containing nanostructures
that were too big for catalyzing CNT growth. The variation could be due to a complex
combination of variables, but perhaps the most likely culprit is the differences in surface
61
energies inherent in the nano-patterned aluminum substrate. These inconsistencies led
to the investigation of PVD methods of catalyst deposition. The AAO template was used
to pattern the catalyst material and to provide a growth channel for the CNTs. The e-
beam provided a greater degree of control over the size and uniformity of the catalyst
material than electrodeposition. Growth from these catalysts is shown in Figure 3-3. The
observed growth is extremely dense, but not well-aligned. While the AAO channels
provide a growth guide inside the pores, once the CNT leaves the pores the growth
direction is random. This blocks access to surrounding pores and can limit the overall
number of CNTs. The specific growth mechanism utilizing AAO templates is discussed
in detail in Chapter 5.
Another important aspect of CVD growth of CNTs was the hydrogen gas. It acted
as a reducing agent for the metal catalysts which may have oxidized in ambient
atmosphere. The hydrogen helped to scrub the amorphous carbon from the top of the
template, promoting CNT growth. With these improvements, the CNT growth was
evident out of the pores, but lacked the desired vertical alignment throughout the entire
growth regime. Additionally, the AAO template is believed to hinder growth due to
reduced diffusion through to the bottom of the pores. This resulted in the type of
unordered growth seen in Figure 3-3. Because of this, the AAO template needed be
used as a patterning agent, but remain under 500 nanometers to eliminate any diffusion
concerns. With these constraints in mind, a complex multilayer substrate (CMS) was
designed which addressed each issue. A typical CMS is shown in Figure 3-4. The top
aluminum layer can be anodized, which resulted in an AAO layer on a CMS, as shown
in Figure 3-5. Previous efforts in literature have anodized aluminum directly on silicon,
62
which produces a similar result to what is seen in Figure 3-5. However, CMS provided
two distinct advantages: improved adhesion to the substrate and an integrated
functional layer which can be adjusted for a given application. However, one issue with
the direct anodization of the aluminum layer is the relative disorder of the resulting
porous structure. To address this, a pattern transfer method was developed to pre-
pattern the surface of the aluminum top layer. This pre-patterning can be seen in Figure
3-6. The AAO template was partially removed to show the pattern transfer in the first
image, while the template was removed completely in the second image to reveal the
pre-patterned aluminum surface on CMS. The resulting anodization pattern is shown in
Figure 3-7, with the topes of the pores opened to highlight the 1:1 pattern transfer and
resulting pore creation. The use of a pattern transfer method eliminates the need for a
first anodization, instead pre-patterning the surface for a highly-ordered second
anodization. The pre-patterning is done with RIE, which is an etching technique that
uses both a physical etching through bombardment and chemical etching through
reactions at the material surface. RIE initiates a plasma utilizing gases specific to the
desired etching process. Because of this, RIE is an extremely tunable process, with
changes in types of gases, power, flow rates, and time affecting the etching
characteristics and selectivity. Therefore, the final pores and resulting CMS can be
tuned to fit a wide range of parameters.
The creation of CMS solves several issues which were encountered in previous
experiments. The first is the elimination of the aluminum substrate on which AAO is
typically grown. Since aluminum is a poor substrate for electrodeposition, it was
necessary to replace the aluminum with a desirable substrate for many of the potential
63
nano-structuring applications. The second addressed the adhesion and transfer issues
of electrodeposited nanowires directly on aluminum substrates. In addition to being a
poor substrate for electrodeposition, aluminum is not the desired substrate for many
alternative energy applications. By tailoring the CMS to a particular application, many of
the processing steps can be eliminated from the overall fabrication process.
Chapter Summary
In this work, the creation of CMS and pattern transfer methods are described.
This unique structure and technique has many promising nano-structuring applications.
One such application for CMS is the catalyzed growth of CNT. There have been many
attempts at growing uniform CNT through both catalyzed and uncatalyzed processes
using AAO. However, AAO templated CNT growth is a complex endeavor, with subtle
variations in catalyst deposition, gas diffusion, and pore channels creating vast
differences in the final product. Indeed, many results in the field are erroneously claimed
to be catalyzed growth, and vice versa. The combination of CMS and AAO pattern
transfer method addresses many of the shortcomings in current CNT growth methods.
Advanced growth techniques such as plasma enhanced CVD (PECVD) could also be
used to further improve the alignment of the CNTs. In addition, the pattern transfer
technique developed in this work can be utilized for a wide variety of exciting
applications, and one such application is described in Chapter 4 for the creation of
nanostructured thermoelectric materials.
64
Figure 3-1. Schematic of CMS creation after the initial metal layers were deposited using e-beam evaporation. Processes involved include the AAO pattern transfer and anodization on multilayer substrates. Drawing courtesy of author.
65
Figure 3-2. Nickel catalysts deposited at bottom of AAO template. A pulsed deposition regime was used: -0.3 V for 10 ms followed by 0 A for 1 s to allow for replenishment of ions. Photo courtesy of author.
66
Figure 3-3. CNT growth from AAO templated catalysts. Notice that not all pores have CNTs growing out of the pore; some remain hidden within the pore. Photo courtesy of author.
67
Figure 3-4. Schematic of CMS. In this example, silicon was the substrate, with gold as the active layer, and aluminum as the top layer for creation of the AAO template. Titanium served as a blocking and adhesion layer. Drawing courtesy of author.
68
Figure 3-5. SEM image of CMS and the resulting AAO. Notice the lack of ordering due to the first anodization process, compared to the order porous surface after pattern transfer. Photo courtesy of author.
69
Figure 3-6. Pattern transfer onto aluminum. This pattern transfer pre-patterned the aluminum surface, much like a first anodization, but without the sacrificial aluminum layer. A) The AAO mask/aluminum boundary is shown. B) The pre-patterned aluminum surface is shown with the AAO mask removed. Photos courtesy of author.
70
Figure 3-7. Pre-patterned anodization on CMS. Each dimpled surface produces at least one pore. In this case, RIE was used twice: once to pre-pattern the aluminum surface and once after the anodization to open the top of the dimpled structure to observe the underlying pore propagation. Photo courtesy of author.
71
CREATION OF AAO PATTERNED THERMOELECTRIC NANOSTRUCTURES
The pattern transfer process detailed in Chapter 3 alleviates many of the current
issues seen in the creation of CNT by effectively utilizing the unique advantages of AAO
while drastically reducing the growth barriers imposed by the templates. The pattern
transfer method is an extremely versatile method which can also be used for
thermoelectric applications. This work highlights this application of the AAO pattern
transfer process and its benefits compared to contemporary thermoelectric nano-
structuring applications.
Recall from Figure 1-6 that ZT increases exponentially as the confining
dimension approaches zero. Much research has been focused towards reducing the
confining dimension in nano-structuring applications, particularly using metal assisted
etching [52]. However, many times the confining dimension remains greater than 50
nanometers, which limits the ZT enhancement. It is not until sub 20 nanometers that
significant improvements to ZT are realized, with the inflection point even smaller, at 10
nanometers. Some work has attempted to address this issue with a similar pattern
transfer method and metal assisted etching, but there are many areas yet to be
explored and much of the challenges still remain in achieving consistent sub 20
nanometer dimensions [53, 54].
Therefore, in this work I present a scalable and effective route for sub 15
nanometer nanostructured materials for thermoelectric applications. This is done
through a pattern transfer method, which reproduces the honeycomb pattern seen in
AAO templates on thermoelectric substrates. Unlike contemporary pattern transfer
methods, this method preserves the pore walls as the confining dimension instead of
72
preserving the dimension of the pores themselves. RIE is utilized to produce an
anisotropic etch into the thermoelectric substrate, which provides a straightforward and
scalable solution for sub 15 nanometer confining dimensions for thermoelectric
applications.
Materials and Methods
AAO template transfer
AAO templates were made in the usual two step anodization method described
in Chapter 2. High purity aluminum foil disks were cleaned by sonication in soapy water,
acetone, and ethanol for 15 minutes each. The aluminum was placed in a sealed holder
with only one side open to the anodization solution, then anodized at 40 V in 0.3 M
oxalic acid for 16-24 hours. The oxide layer was etched away using an aqueous solution
of 6 wt. % phosphoric acid and 1.8 wt. % chromic acid at 60 ⁰C for one hour. The
resulting pre-patterned aluminum was then prepared for second anodization to create
an AAO mask. For the AAO mask, second anodization was carried out for 3 minutes,
with no barrier layer thinning. This created a thin AAO layer that was approximately 200
nanometer thick. The AAO was then dried with nitrogen gas before a polystyrene (PS)
layer was spin-coated to the top of the mask using a 5 wt. % PS solution dissolved in
toluene. An even coating was applied on the top of the AAO mask, and spin-coating
was carried out for 1 minute at 10,000 rpm. After this, the AAO mask/polystyrene
complex was left to dry overnight. A solution of copper chloride in hydrochloric acid was
used to remove the aluminum substrate (100 g/L copper chloride in 4 M hydrochloric
acid). The PS layer acted as a protective layer to keep the AAO mask intact during the
aluminum removal. The AAO mask was transferred to a 5 wt. % phosphoric acid
solution for pore thinning. The AAO mask was thin enough to float on the surface of the
73
solutions, which simplified the transfer of AAO from the copper chloride solution to the
phosphoric acid solution. After removal from the phosphoric acid solution, the AAO
mask was transferred to the substrate by lifting the AAO mask off the surface of the
rinse water using the aluminum-on-silicon substrate. It was important to achieve a clean
lift off; otherwise the AAO mask would wrinkle on the substrate. The resulting sample
was left to dry before the RIE process.
The pattern transfer process was easily tuned for a wide range of substrates, and
in this particular work a silicon germanium substrate was used to demonstrate the
viability of this process for thermoelectric applications. Silicon germanium wafers were
purchased from MTI Corporation, 2% Ge content, p-type, and <100> orientation. Prior
to transfer, the silicon germanium substrate was cleaned through sonication in soapy
water, acetone, and ethanol baths for 15 minutes each and dried with nitrogen gas in
between each solvent, just as the other substrates were cleaned in Chapter 3.
RIE pattern transfer
The samples were etched using much of the same procedure as the one used in
Chapter 3. Etching was done using the Trion RIE/ICP at NRF. The PS protective layer
was first removed with oxygen plasma using 40 sccm oxygen gas, 10 Torr total
pressure, and 150 W power for 5 minutes. The silicon germanium substrate was then
etched using an SF6 mixture for 4 minutes to etch the pattern transfer of the AAO mask
onto the substrate.
Metal nanoparticle arrays
To create high density, AAO templated metal nanoparticle arrays, gold or silver
was deposited onto the nanostructured silicon germanium samples through e-beam
deposition. The deposition was performed using the e-beam evaporator at NRF, with
74
preset conditions which corresponded to specific metals. Samples were annealed at
600 ⁰C for gold and 250 ⁰C for silver for one hour to allow the particles to settle at the
bottom of the etched cavities.
Results and Discussion
Initially, all of the pattern transfer was done using AAO templates with 60 minutes
of pore thinning. However, the pattern etching resulted in highly anisotropic conditions,
which allowed for a reduction in the pore wall thickness. The time of the pore thinning in
the phosphoric acid etching solution is the single most important factor to consider when
controlling pore wall thickness. The longer the pore thinning process, the smaller the
resulting confining dimension of the transferred pattern will be. A study of pore thinning
times was conducted, with 30 minutes, 60 minutes, and 75 minutes used as the pore
thinning times. The results of the pattern transfer are shown in Figure 4-1 and Figure
4-2 for 60 minutes and 75 minutes and the compiled results are shown in Figure 4-3. It
is clear that the pore thinning time correlated strongly to the thickness of the resulting
nanostructures, which indicated a highly anisotropic etching process with no
degradation of the silicon germanium nanostructures during the pattern transfer
process.
At 75 minutes, the upper limit of pore thinning times studied, the confining
dimension is approximately 14 nanometers. This reduction of the confining dimension
confirms the viability of the pattern transfer method for creation of sub 15 nanometer
nanostructures for thermoelectric applications and is an improvement over other etching
methods. In addition, it is a positive step towards sub 10 nanometer dimensions as it is
likely that the trend line would be valid through the 10 nanometer limit. This prediction is
75
extrapolated from the recorded data and is included in Figure 4-3 as a dotted line.
However, careful control must be taken to ensure that the pore opening process is
consistent, as sub 10 nanometer dimension masks would likely be difficult to keep
consistent due to variations in etching uniformity.
In addition to the creation of these thermoelectric nanostructures, the longer
order propagation of the AAO template pattern is necessary for creating suitable
thermoelectric devices. The plasma etching of silicon germanium shows excellent
pattern transfer of the AAO patterning, but the limits of the Trion RIE constrain the depth
of the nanostructures to a few microns. In order to create the necessary nanostructures
tens to hundreds of microns deep for a practical device, further propagation of the AAO
patterning must be considered. One option is to utilize metal assisted etching. This is
done by depositing a thin metal film onto the nanostructured silicon germanium and
annealing the sample so the metal can settle into the nanocavities. This nanoparticle
array could then be the basis for a metal assisted etching procedure to propagate the
porous channels through the substrate. Conditions would vary for different substrates,
but in the case of silicon germanium, etching solutions would closely follow silicon
etching recipes, which are widely known. An example of the nanoparticle array is shown
in Figure 4-4, utilizing a nanostructured silicon germanium substrate which was
described above.
Another method for propagation of the nanostructures developed in this work
would be deep RIE (DRIE). DRIE differs from RIE because it utilizes a cyclical, two step
etching process to alternately etch the bottom and passivate the sidewalls of the desired
features. Through multiple cycles, this process produces nearly vertical sidewalls with
76
channels that can be up to 500 microns deep. This process would be well suited for
thermoelectric materials such as silicon germanium because this process is extensively
utilized for silicon based devices and would mirror the procedures described for the RIE
pattern transfer.
Chapter Summary
In this work I presented an effective method for creating nanostructured materials
for thermoelectric applications using highly dense AAO templates. The nanostructures
had an average confining dimension of 14 nanometers, lower than that seen in other
works which use AAO templates as etching masks. The pattern transfer method
produces a 1:1 replica of the AAO honeycomb patterning, resulting in a highly ordered,
highly dense thermoelectric nanomaterial. In addition, practical routes towards long
order propagation of these nanostructures were shown through proof of concept high
density nanoparticle arrays. Perhaps most exciting is the scalability of the pattern
transfer method, which provides a viable path towards sub 10 nanometer
nanostructured thermoelectric materials.
77
Figure 4-1. Comparison of pattern transfer on silicon germanium. A) The pore opening time was 60 minutes and B) 75 minutes. Notice the well-defined patterning on the surface of the silicon germanium, which indicated a highly selective pattern transfer process. Photos courtesy of author.
78
Figure 4-2. High magnification SEM image comparison of A) 60 minute pore opening and B) 75 minute pore opening. The confining dimension thickness is shown on each. Photos courtesy of author.
79
Figure 4-3. Plot of confining dimension thickness vs pore opening time. Standard error is included in the plot for each thickness: 63.1 ± 3.9 nm, 29.6 ± 2.4 nm, and 13.8 ± 2.1 nm.
0
10
20
30
40
50
60
70
20 30 40 50 60 70 80 90Co
nfi
nin
g D
ime
ns
ion
(n
m)
Pore Opening Time (mins)
Thickness of Confining Dimension vs Pore Opening Time
80
Figure 4-4. SEM image of Au nanoparticle deposition in dimpled surfaces for metal assisted etching. Photo courtesy of author.
81
CNT GROWTH MECHANISM IN AAO TEMPLATES
The work in Chapter 3 showed that the growth of CNTs using an AAO templated
method was a complicated process with many variables affecting the end growth
results. Perhaps most puzzling was the role which the AAO template itself played in the
growth. Initially it was believed that the AAO would be an inert material in the process,
neither affecting nor hindering the growth of CNTs. However, it was apparent that the
CNT growth was not a 1:1 ratio with the AAO template. Yet, there is almost no literature
which discusses the growth of CNTs in AAO templates and no comprehensive review of
AAO templated CNT growth. This work addresses some of the current deficiencies in
the literature concerning AAO templated CNT growth.
All of the CNT growth seen in Chapter 3 has been tip growth. This has been in
line with what literature would suggest, given the relatively large catalyst particle size
and reaction conditions. However, what has been intriguing is that the CVD growth of
AAO templated CNTs has produced uneven CNT growth. In these experiments, some
CNTs grew quickly out of the pores, while others lagged behind. This is highlighted in
Figure 5-1. While some of the CNTs were grown out of the pores, others could be seen
just below the surface of the AAO template. The desired result would be a uniform
growth front of CNTs, but given the apparent randomness of CNT growth, a closer
inspection was needed and what follows in this work is a proposed mechanism for the
growth of CNTs in AAO templates.
Results and Discussion
Based on the work in Chapter 3, there were two growth regimes for AAO
templated CNT growth: inside the pore and outside the pore. It was clear that CNTs did
82
not grow evenly throughout the AAO template. This results in a lower CNT density and
is shown in Figure 5-2, with Figure 5-3 showing the continued unaligned growth of
CNTs. There have been many mechanisms proposed for the catalyzed growth of CNTs,
one of which is that by Puretzky [55]. However, there is very little literature on the
growth of CNTs inside AAO templates, and the main objective of this work is to highlight
the differences between the two regimes and provide an effective model for the growth
of CNTs inside the pores. Both will be discussed in detail, beginning with the simpler
case of growth outside the pore.
Outside of pore growth rate
In the case of a CNT growing outside of the pore, the growth is governed by the
kinetics of the system. Feed gas flows to the surface of the catalyst, is broken down and
absorbed into the catalyst, and growth is propagated through the catalyst after
supersaturation is reached. Each of these processes has a specific rate constant, which
is a representation of the speed at which the carbon atoms transition from one area of
the catalyst to another. In this case the growth can be represented by a kinetics based
model, which is a modified version of that proposed by Puretzky [55]. The main liberty
taken with this model versus the model proposed by Puretzky is the exclusion of the gas
phase pyrolysis term. In the relatively low reaction temperature of the AAO templated
CNT growth, it is assumed that there are no high temperature byproducts which would
poison the catalytic activity. Therefore, in the model detailed here, catalytic activity stops
only when the deactivation layer covers the entire surface of the catalyst. A schematic
of the process is shown in Figure 5-4 and the rate equations are shown:
𝑑𝑛𝑐𝑑𝑡
= 𝑁𝑎𝑧𝐴𝑐 (1 −𝑛𝑑
𝛼𝐴𝑐𝜌𝐴) − (𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑛𝑐
(5-1)
83
𝑑𝑛𝑑𝑑𝑡
= 𝑘𝑐𝑑𝑛𝑐 − 𝑘𝑑𝑏𝑛𝑑 (5-2)
𝑑𝑛𝑏𝑑𝑡
= 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 − 𝑘𝑏𝑡𝑛𝑏 (5-3)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 (5-4)
where Nx is the moles of a given area, Naz is the molar flux, kxy is the rate constant
where state x precedes state y, Ac is the surface area of the catalyst, α is the number of
monolayers of adsorbed carbon as an estimate of the number of walls in the CNT, and
ρA is the surface density of carbon atoms in a CNT. Assuming steady state, 𝑑𝑛𝑥
𝑑𝑡= 0 and
the flux remains constant so the equations can be solved to give the following solutions
for nx:
𝑛𝑐 =∆ − 𝑣𝑛𝑑𝑘′
𝑤ℎ𝑒𝑟𝑒 𝑁𝑎𝑧𝐴𝑐 = ∆
𝑁𝑎𝑧𝛼𝜌𝐴
= 𝑣
𝑘𝑐𝑏 + 𝑘𝑐𝑑 = 𝑘′
(5-5)
𝑛𝑑 =𝑘𝑐𝑑∆
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′
(5-6)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘
′∆
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
=
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)]
(5-7)
84
If the rate of carbon transferred to the deactivation layer is assumed to be much
slower than the adsorption of the hydrocarbon, kcd<<kcb, then the above equation can
be simplified:
𝑑𝑛𝑡𝑑𝑡
=𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
(5-8)
𝐺𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑝𝑜𝑟𝑒 =𝑑𝑧
𝑑𝑡= √
𝐴𝑐𝜋
𝑁𝑎𝑧
(𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴)
𝑤ℎ𝑒𝑟𝑒 𝑘𝑐𝑑
𝑘𝑑𝑏𝑘𝑐𝑏= 𝑘𝑒𝑓𝑓
(5-9)
To obtain the growth rate, the conversion 𝑑𝑛𝑡
𝑑𝑧= 2𝜋𝑟𝑐𝛼𝜌𝐴 is used, relating the number of
moles consumed for the CNT to the height of the CNT through a mole balance. In this
relation, 𝐴𝑐 = 4𝜋𝑟𝑐2, assuming that the radius of the catalyst, rc, is a suitable
approximation of the radii of the CNT walls and h is the height of the CNT.
Therefore, the growth rate of the CNT is a function of the flux to the catalyst and
the rate constants. Since the flux is constant, the rate constants determine the overall
growth rate. While the exact catalysis process is still debated, there are several works
that examine specific aspects of ethylene decomposition on nickel which allow for an
estimation of the rate constants used in this present study. Using a combination of
experimental and calculated values within the Arrhenius rate equation, values of kcb are
calculated within an order of magnitude of 0.0001 s-1, depending on the specific bond
dissociation studied [56]. To estimate keff, it is reasonable to assume that kcd and kdb are
equivalent since 𝑑𝑛𝑑
𝑑𝑡 is at steady state and therefore, the rates in and out of the bulk are
balanced. Then, 𝑘𝑒𝑓𝑓 ≈1
𝑘𝑐𝑏 and can be estimated as 10,000 s.
85
Concentration profile inside pore
The experimental results suggested a hindrance of CNT growth inside some of
the AAO pores. One possibility is that the pores presented a diffusion barrier to the feed
gas. Another possibility is the presence of AAO as a competing catalyst. In many cases,
AAO is used as the sole catalyst for CNT growth, so it is likely that the AAO
preferentially catalyzes at least a portion of CNT growth. Perhaps most likely is a
combination of the two, which can favor the growth of AAO catalyzed CNTs. A
schematic for the growth system in a single AAO pore is shown in Figure 5-5. By
considering diffusion in one dimension with no convection, a concentration profile was
developed for an AAO pore channel to determine the role which diffusion played in the
growth, taking into account the simultaneous reaction with the pore sidewall.
−𝐷𝐴𝐵𝑑2𝐶𝑎𝑑𝑧2
+ 𝑘𝑤𝐶𝑎 = 0
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠: Г =𝐶𝑎𝐶𝑎0
𝑎𝑛𝑑 𝜁 =𝑧
𝐿
(5-10)
𝑑2Г
𝑑𝜁2−𝑘𝑤𝐿
2
𝐷𝐴𝐵Г = 0
𝐺𝑟𝑜𝑢𝑝𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠: 𝛷 = √𝑘𝑤𝐷𝐴𝐵
𝐿2
(5-11)
𝑑2Г
𝑑𝜁2− 𝛷2Г = 0
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: Г = 𝐶1 cosh(𝛷𝜁) + 𝐶2 sinh(𝛷𝜁) [57]
𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠
(5-12)
𝐶𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑡 𝑡𝑜𝑝 𝑜𝑓 𝑝𝑜𝑟𝑒 {𝐶𝑎 = 𝐶𝑎0, 𝑧 = 𝐿Г = 1, 𝜁 = 1
1 = 𝐶1 cosh(𝛷) + 𝐶2 sinh(𝛷)
86
𝐶1 =1 − 𝐶2 sinh(𝛷)
cosh(𝛷)
𝑁𝑜 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑎𝑡 𝑏𝑜𝑡𝑡𝑜𝑚 𝑜𝑓 𝑝𝑜𝑟𝑒
{
𝑑𝐶𝑎𝑑𝑧
= 0, 𝑧 = 0
𝑑Г
𝑑𝜁= 0, 𝜁 = 0
𝑑Г
𝑑𝜁=1 − 𝐶2 sinh(𝛷)
cosh(𝛷)𝛷 sinh(𝛷𝜁) + 𝐶2𝛷 cosh(𝛷𝜁)
𝑑Г
𝑑𝜁= 0 =
1 − 𝐶2 sinh(𝛷)
cosh(𝛷)𝛷 sinh(0) + 𝐶2𝛷 cosh(0)
𝐶2 = 0
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠
Г =cosh(𝛷𝜁)
cosh(𝛷)
(5-13)
The calculated concentration profile is not dependent on time because the time
scale for diffusion for these systems is on the order of milliseconds, much less than the
time scale of growth in the pore, which is on the order of minutes. Therefore, the steady
state concentration profile for this particular case is solely dependent on the reaction
rate at the AAO sidewall. The value of kw can be estimated through a study of ethylene
decomposition on aluminum oxide catalyzed systems as 0.05 s-1, using a similar
Arrhenius rate equation approach as the calculation of keff in the previous section [58].
In addition, the diffusion coefficient and pore length are well-known parameters and
these values can be used to calculate the aggregated constant, Φ, to obtain a
dimensionless relationship between the feed gas concentration versus 𝜁. This
relationship is shown in Figure 5-6 for a range of kw. In many cases, the concentration is
virtually constant throughout the length of the pore. However, diffusion can still be an
87
important consideration in other cases, specifically if kw is large or if there are other
resistances to diffusion, such as carbon buildup on the sidewall.
Inside pore growth rate
The growth rate inside a pore can be modelled much in the same process that
the growth rate outside the pores is modelled. To account for the possibility of AAO
catalyzed growth inside the pores, an additional reaction term was added to the outside
of pore equations. Recall the schematic for the catalyzed tip growth in an AAO pore
shown in Figure 5-4 and the equations are as follows:
𝑑𝑛𝑐𝑑𝑡
= 𝑁𝑎𝑧𝐴𝑐 (1 −𝑛𝑑
𝛼𝐴𝑐𝜌𝐴) − (𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑛𝑐 + 𝑘𝑎𝑐𝑛𝑎
(5-14)
𝑑𝑛𝑑𝑑𝑡
= 𝑘𝑐𝑑𝑛𝑐 − 𝑘𝑑𝑏𝑛𝑑 (5-15)
𝑑𝑛𝑏𝑑𝑡
= 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 − 𝑘𝑏𝑡𝑛𝑏 (5-16)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 (5-17)
One consequence of the concentration profile is that it is valid to also assume
steady state for the inside pore case, as the concentration is virtually constant
throughout the pore. This yields the following solutions for nx:
𝑛𝑐 =𝛽 − 𝑣𝑛𝑑𝑘′
𝑤ℎ𝑒𝑟𝑒 𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎 = 𝛽
𝑁𝑎𝑧𝛼𝜌𝐴
= 𝑣
𝑘𝑐𝑏 + 𝑘𝑐𝑑 = 𝑘′
(5-18)
𝑛𝑑 =𝑘𝑐𝑑𝛽
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′
(5-19)
88
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘
′𝛽
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
=
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)]
𝐴𝑔𝑎𝑖𝑛 𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑘𝑐𝑑 ≪ 𝑘𝑐𝑏 𝑎𝑛𝑑 𝑑𝑛𝑡𝑑ℎ
= 2𝜋𝑟𝑐𝛼𝜌𝐴 𝑤𝑖𝑡ℎ 𝐴𝑐 = 4𝜋𝑟𝑐2
(5-20)
𝑑𝑛𝑡𝑑𝑡
=(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
(5-21)
𝐺𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑖𝑛𝑠𝑖𝑑𝑒 𝑝𝑜𝑟𝑒 =𝑑𝑧
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴]
𝑤ℎ𝑒𝑟𝑒 𝑘𝑐𝑑
𝑘𝑑𝑏𝑘𝑐𝑏= 𝑘𝑒𝑓𝑓
(5-22)
As in the out of pore case, the growth rate is dependent on the rate constants.
Note that for inside the pore, the concentration was solved with respect to z, as
referenced from the bottom of the pore. The concentration can be related to the flux
through Fick’s First Law to combine into the growth rate. Recall:
Г =𝐶𝑎𝐶𝑎0
=cosh(𝛷𝜁)
cosh(𝛷)
(5-13)
𝑁𝑎𝑧 = 𝐷𝐴𝐵𝑑𝐶𝑎𝑑𝑧
= 𝐷𝐴𝐵𝐶𝑎0𝑑
𝑑𝑧[cosh(𝛷𝜁)
cosh(𝛷)]
(5-23)
𝑁𝑎𝑧 =𝐷𝐴𝐵𝐶𝑎0𝛷
𝐿 cosh(𝛷)[sinh(𝛷𝜁)]
(5-24)
𝐼𝑛𝑠𝑖𝑑𝑒 𝑝𝑜𝑟𝑒 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑑ℎ
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴]
(5-22)
89
The kacna term represents the additional adsorption of carbon from the AAO
sidewall to the catalyst surface. This can be approximated as a linear function of z so
that 𝑘𝑎𝑐𝑛𝑎 = 𝑘"𝑧; as z increases, there is additional buildup of carbon on the sidewall
which yields more adsorbed sidewall carbon into the catalyst. Substituting the flux into
the growth rate equation yields:
𝑑𝜁
𝑑𝑡=(𝐴 sinh(𝛷𝜁) + 𝐵𝜁)
[𝐶 sinh(𝛷𝜁) + 𝐷]
𝑤ℎ𝑒𝑟𝑒 𝐷𝐴𝐵𝐶𝑎0𝛷√𝐴𝑐
cosh(𝛷)√𝜋= 𝐴
𝐿2
√𝜋𝐴𝑐𝑘" = 𝐵
𝐷𝐴𝐵𝐶𝑎0𝛷𝑘𝑒𝑓𝑓
𝐿 cosh(𝛷)= 𝐶
𝛼𝜌𝐴 = 𝐷
(5-25)
The initial concentration, 𝐶𝑎0, is determined using ideal gas law under typical
growth conditions given in Chapter 3. Using this and the other values for the constants,
the growth rate can be plotted as a function of 𝜁 and is shown in Figure 5-7 for a range
of kw. This plot confirms the dependency of CNT growth in AAO pores on the rates of
the carbon adsorption and sidewall reaction. For varying kw, if kw is small then the
growth rate is linear, as the rate is dominated by the rate of carbon adsorption. As kw
becomes larger, then the diverted carbon reduces the overall growth rate as the
competing rates of kw and keff slow the overall growth rate. Similar dependencies are
shown with varying keff in Figure 5-8.
Based on this model, it is clear that the sidewall reactions of AAO catalyzed
growth can affect the overall growth of CNTs in individual pores. This is especially true
90
in cases where the AAO sidewall reaction is relatively fast compared to the absorption
of carbon into the catalyst particle. Thus, minor variations in initial conditions such as
different localized activation energies or catalyst particle surface energies could have
significant effects on the growth of a CNT through the entire pore.
Chapter Summary
The growth of catalyzed CNTs is a complex process that involves many
variables. Utilizing AAO as a nano-structuring template adds an additional layer of
complexity which can cause inconsistencies in the final growth results. In addition to the
competing catalytic issues, it is possible that the CNT overgrowth hinders the overall
diffusion, reducing the flux within the pores and causing variations in the available
carbon for growth. In addition, it is possible that a catalyst nanoparticle is partially
deactivated prior to the growth phase and therefore has a higher energy barrier to
overcome to begin growth. These small individual deviations can combine to cause
significant deviations in the overall growth front of CNTs and make the uniform growth
of templated CNTs a complex endeavor. However, with a thorough understanding of the
underlying mechanisms, many of the issues seen with AAO templates can be better
understood and addressed.
Therefore, in this work I proposed a simple and effective mechanism to explain
the catalyzed growth of CNTs in AAO templates. This mechanism explains two distinct
growth regimes: outside the pores and inside the pores. For each regime, a growth rate
is derived from initial rate equations. These growth rates can be used to approximate
the growth of an AAO templated CNT system as well as explain many of the
phenomena seen in previous experimental work.
91
Figure 5-1. Top down SEM image of nickel-catalyzed CNTs grown from AAO template. Continued growth eventually covered the surface of the template, but did not contain a CNT protruding per pore. Photo courtesy of author.
92
Figure 5-2. High magnification SEM image of extended nickel-catalyzed CNT growth. Notice the absence of 1:1 ratio of CNTs to pores. Photo courtesy of author.
93
Figure 5-3. Wide field SEM image of extended nickel-catalyzed CNT growth. Photo courtesy of author.
94
Figure 5-4. Schematic of catalyzed CNT tip growth. Kinetic rate constants for each reaction arrow were represented by kxy, as in the carbon transitions from x to y. For outside the pore growth, process 6 is excluded, but for inside the pore growth, all processes are included. Drawing courtesy of author.
Legend
1. Naz, molar flux (𝑚𝑜𝑙
𝑚2𝑠)
2. nc, adsorbed hydrocarbon on catalyst surface 3. nd, deactivated catalyst surface carbon 4. nb, absorbed carbon in bulk catalyst 5. nt, carbon in CNT 6. na, carbon adsorbed from AAO to catalyst
95
Figure 5-5. Schematic of flux into AAO channel for catalyzed CNT growth. Drawing
courtesy of author.
Legend
1. Naz, molar flux (𝑚𝑜𝑙
𝑚2𝑠)
2. z, changing growth front 3. kw, reaction rate of carbon with AAO sidewall 4. nb, absorbed carbon in bulk catalyst 5. Ca0, concentration of Ca at z=0 (bulk) 6. S, Cross sectional area of pore
96
Figure 5-6. Plot of concentration, Γ, versus height, ζ, for various kw.
97
Figure 5-7. Plot of 𝑑ℎ
𝑑𝑡 versus the normalized height of CNT in pore (h/L). As kw
increases, the growth rate decreases as more of the available carbon catalyzes on the AAO sidewall.
98
Figure 5-8. Plot of 𝑑ℎ
𝑑𝑡 versus the normalized height of CNT in pore (h/L). As keff
increases, the growth rate decreases as kcb decreases and less feed gas is absorbed into the nickel.
99
CONCLUSIONS
Chapter Summary
Nanotechnology is a broad and evolving field with the potential to revolutionize
our world. In fact, it is virtually inevitable, as the world moves increasingly towards a
technology driven world. At the heart of this drive is the research which constantly
redefines the boundaries of nanotechnology by pushing the limits of materials while
simultaneously innovating new methods for the creation of these nanostructures.
Nanotechnology has drastically changed our lives and our understanding of the world
for the better, from more efficient car catalytic converters to the vast improvements of
computer chips.
It is in this spirit that I began my doctoral work and attempted to contribute to this
important field, specifically the use of nanotechnology in alternative energy devices.
While I originally started with the intention of doing my research in solar cells, I quickly
realized that the true story of alternative energy is a comprehensive view of a wide
variety of alternative energy applications. Solving the world’s energy crisis will take
more than a single energy source, and therefore it was more practical to study
nanostructures which could be incorporated into a range of alternative energy
applications. The key to this was the AAO template, an extremely versatile tool for the
fabrication of nanostructures. From this single material, I was able to create uniform
nanowire arrays, grow CNTs, and fabricate thermoelectric nanostructures. The results
of this work provide a scalable and effective path for the creation of these
nanostructures, but perhaps more importantly, allowed me to understand the properties
of materials at the nanoscale and gave me an appreciation for the unique challenges
100
associated with nanomaterial fabrication. With these in mind, there are a couple of
directions which I believe would be useful future areas to study, based on the results I
have shown in this work.
Future Work
Barrier layer thinning and pore opening to create AAO nanofluidic channels
An emerging area of research is that of nanofluidics. As with material properties
at the nanoscale, transport properties at the nanoscale can vastly differ. In this work,
AAO templates were used to create various nanostructures, but the templates could
also be used as the basis for nanofludics studies. The characterization work I performed
on the barrier layer thinning and pore opening of AAO templates opens up numerous
opportunities to create nanochannel arrays of tunable diameters and lengths. These
nanochannels could be used to study a wide range of interesting topics, from transport
forces characteristic at the nano length scale to applications such as the separation of
molecules and bioscreening.
Fabrication of thermoelectric devices
There are two major challenges of nano-based alternative energy applications:
creating suitable nanostructures with appropriate critical dimensions and the
incorporation of those nanostructures into working devices. While I addressed the first
challenge in my work, the second challenge is equally as daunting. Many
nanostructures, while important for the nanotechnology field of study, are not
necessarily suitable for device fabrication. This could be due to scaling issues. In this
work, the nanostructures created were fairly short, which limits the overall effectiveness
in alternative energy applications. Techniques mentioned in Chapter 4 such as HF
etching or DRIE, for example, would require extensive study for the successful
101
application into suitable nanostructures. In addition, recent work has shown that pore
sizes of AAO templates can be reduced below 10 nanometers, which is a significant
improvement over the ubiquitous 40 nanometer templates currently in use. Therefore, it
is a nontrivial task to develop methods to incorporate nanostructures into alternative
energy devices, just as important as it is to understand and fabricate these
nanostructures in the first place.
102
APPENDIX CALCULATIONS
In this appendix are the extended calculations for Chapter 5. Equations shown in
Chapter 5 are cross referenced here.
Growth Rate Outside Pore
𝑑𝑛𝑐𝑑𝑡
= 𝑁𝑎𝑧𝐴𝑐 (1 −𝑛𝑑
𝛼𝐴𝑐𝜌𝐴) − (𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑛𝑐
(5-1)
𝑑𝑛𝑑𝑑𝑡
= 𝑘𝑐𝑑𝑛𝑐 − 𝑘𝑑𝑏𝑛𝑑 (5-2)
𝑑𝑛𝑏𝑑𝑡
= 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 − 𝑘𝑏𝑡𝑛𝑏 (5-3)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 (5-4)
𝐿𝑒𝑡 𝑁𝑎𝑧𝐴𝑐 = ∆
𝐿𝑒𝑡 𝑁𝑎𝑧𝛼𝜌𝐴
= 𝑣
𝐿𝑒𝑡 𝑘𝑐𝑏 + 𝑘𝑐𝑑 = 𝑘′
𝐴𝑠𝑠𝑢𝑚𝑒 𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒
𝑇ℎ𝑒𝑛 𝑑𝑛𝑐𝑑𝑡
= 0 = ∆ − 𝑣𝑛𝑑 − 𝑘′𝑛𝑐
𝑛𝑐 =∆ − 𝑣𝑛𝑑𝑘′
(5-5)
𝑑𝑛𝑑𝑑𝑡
= 0 = 𝑘𝑐𝑑 (∆ − 𝑣𝑛𝑑𝑘′
) − 𝑘𝑑𝑏𝑛𝑑
0 =𝑘𝑐𝑑∆
𝑘′−𝑘𝑐𝑑𝑣𝑛𝑑𝑘′
− 𝑘𝑑𝑏𝑛𝑑
(𝑘𝑐𝑑𝑣
𝑘′+ 𝑘𝑑𝑏)𝑛𝑑 =
𝑘𝑐𝑑∆
𝑘′
103
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘
′
𝑘′)𝑛𝑑 =
𝑘𝑐𝑑∆
𝑘′
𝑛𝑑 =𝑘𝑐𝑑∆
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′
(5-6)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 = 𝑘𝑐𝑏∆ − 𝑣𝑛𝑑𝑘′
+ 𝑘𝑑𝑏𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏∆
𝑘′−𝑘𝑐𝑏𝑣𝑛𝑑𝑘′
+ 𝑘𝑑𝑏𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏∆
𝑘′+ (𝑘𝑑𝑏 −
𝑘𝑐𝑏𝑣
𝑘′)𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏∆
𝑘′+ (
𝑘𝑑𝑏𝑘′ − 𝑘𝑐𝑏𝑣
𝑘′) (
𝑘𝑐𝑑∆
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏∆
𝑘′+𝑘𝑐𝑑∆(𝑘𝑑𝑏𝑘
′ − 𝑘𝑐𝑏𝑣)
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏∆(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘
′) + 𝑘𝑐𝑑∆(𝑘𝑑𝑏𝑘′ − 𝑘𝑐𝑏𝑣)
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝑘𝑐𝑑𝑣∆ + 𝑘𝑐𝑏𝑘𝑑𝑏𝑘
′∆ + 𝑘𝑐𝑑𝑘𝑑𝑏𝑘′∆ − 𝑘𝑐𝑑𝑘𝑐𝑏𝑣∆
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝑘𝑑𝑏∆ + 𝑘𝑐𝑑𝑘𝑑𝑏∆
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘
′∆
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
(5-7)
𝑅𝑒𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)]
𝐴𝑠𝑠𝑢𝑚𝑒 𝑘𝑐𝑑 ≪ 𝑘𝑐𝑏
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘𝑐𝑏𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏𝑘𝑐𝑏]
104
𝑑𝑛𝑡𝑑𝑡
=𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
(5-8)
𝑑𝑛𝑡𝑑𝑧
= 2𝜋𝑟𝑐𝛼𝜌𝐴
𝑤ℎ𝑒𝑟𝑒 𝐴𝑐 = 4𝜋𝑟𝑐2
𝑑𝑛𝑡𝑑𝑧
= √𝜋𝐴𝑐𝛼𝜌𝐴
√𝜋𝐴𝑐𝛼𝜌𝐴𝑑𝑧
𝑑𝑡=
𝑁𝑎𝑧𝐴𝑐
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
𝑑𝑧
𝑑𝑡=
𝑁𝑎𝑧√𝐴𝑐
√𝜋 [𝑘𝑐𝑑𝑁𝑎𝑧𝑘𝑑𝑏𝑘𝑐𝑏
+ 𝛼𝜌𝐴]
𝐿𝑒𝑡 𝑘𝑐𝑑
𝑘𝑑𝑏𝑘𝑐𝑏= 𝑘𝑒𝑓𝑓
𝐺𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑝𝑜𝑟𝑒 =𝑑𝑧
𝑑𝑡= √
𝐴𝑐𝜋
𝑁𝑎𝑧
(𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴)
(5-9)
Concentration Profile
𝑆ℎ𝑒𝑙𝑙 𝑏𝑎𝑙𝑎𝑛𝑐𝑒: 𝑆𝑁𝑎𝑧|𝑧 − 𝑆𝑁𝑎𝑧|𝑧+∆𝑧 − 𝑆𝑘𝑤𝐶𝑎∆𝑧 = 0
𝑑𝑁𝑎𝑧𝑑𝑧
+ 𝑘𝑤𝐶𝑎 = 0
𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛, 𝑛𝑜 𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛
−𝐷𝐴𝐵𝑑2𝐶𝑎𝑑𝑧2
+ 𝑘𝑤𝐶𝑎 = 0 (5-10)
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝑜𝑓 𝑉𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝐿𝑒𝑡 Г =𝐶𝑎𝐶𝑎0
𝐿𝑒𝑡 𝜁 =𝑧
𝐿
105
𝐶𝑎0𝐿2𝑑2Г
𝑑𝜁2−𝑘𝑤𝐶𝑎0𝐷𝐴𝐵
Г = 0
𝑑2Г
𝑑𝜁2−𝑘𝑤𝐿
2
𝐷𝐴𝐵Г = 0
𝐿𝑒𝑡 𝛷 = √𝑘𝑤𝐷𝐴𝐵
𝐿2
(5-11)
𝑑2Г
𝑑𝜁2− 𝛷2Г = 0
(5-12)
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: Г = 𝐶1 cosh(𝛷𝜁) + 𝐶2 sinh(𝛷𝜁)
𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 1, 𝐶𝑎 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑖𝑛 𝑏𝑢𝑙𝑘
𝐵𝐶1 {𝐶𝑎 = 𝐶𝑎0, 𝑧 = 𝐿Г = 1, 𝜁 = 1
1 = 𝐶1 cosh(𝛷) + 𝐶2 sinh(𝛷)
𝐶1 =1 − 𝐶2 sinh(𝛷)
cosh(𝛷)
𝐵𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 2, 𝑁𝑜 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑜𝑛 𝑎𝑡 𝑏𝑜𝑡𝑡𝑜𝑚 𝑜𝑓 𝑝𝑜𝑟𝑒
𝐵𝐶2
{
𝑑𝐶𝑎𝑑𝑧
= 0, 𝑧 = 0
𝑑Г
𝑑𝜁= 0, 𝜁 = 0
𝑑Г
𝑑𝜁=1 − 𝐶2 sinh(𝛷)
cosh(𝛷)𝛷 sinh(𝛷𝜁) + 𝐶2𝛷 cosh(𝛷𝜁)
𝑑Г
𝑑𝜁= 0 =
1 − 𝐶2 sinh(𝛷)
cosh(𝛷)𝛷 sinh(0) + 𝐶2𝛷 cosh(0)
𝐶2 = 0
Г =cosh(𝛷𝜁)
cosh(𝛷)
(5-13)
106
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛: 𝐶𝑎𝐶𝑎0
=cosh(𝛷𝜁)
cosh(𝛷)
Growth Rate Inside Pore
𝑑𝑛𝑐𝑑𝑡
= 𝑁𝑎𝑧𝐴𝑐 (1 −𝑛𝑑
𝛼𝐴𝑐𝜌𝐴) − (𝑘𝑐𝑏 + 𝑘𝑐𝑑)𝑛𝑐 + 𝑘𝑎𝑐𝑛𝑎
(5-14)
𝑑𝑛𝑑𝑑𝑡
= 𝑘𝑐𝑑𝑛𝑐 − 𝑘𝑑𝑏𝑛𝑑 (5-15)
𝑑𝑛𝑏𝑑𝑡
= 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 − 𝑘𝑏𝑡𝑛𝑏 (5-16)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 (5-17)
𝐿𝑒𝑡 𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎 = 𝛽
𝐿𝑒𝑡 𝑁𝑎𝑧𝛼𝜌𝐴
= 𝑣
𝐿𝑒𝑡 𝑘𝑐𝑏 + 𝑘𝑐𝑑 = 𝑘′
𝐴𝑠𝑠𝑢𝑚𝑒 𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒
𝑑𝑛𝑐𝑑𝑡
= 𝛽 − 𝑣𝑛𝑑 − 𝑘′𝑛𝑐
𝑛𝑐 =𝛽 − 𝑣𝑛𝑑𝑘′
(5-18)
𝑑𝑛𝑑𝑑𝑡
= 0 = 𝑘𝑐𝑑 (𝛽 − 𝑣𝑛𝑑𝑘′
) − 𝑘𝑑𝑏𝑛𝑑
0 =𝑘𝑐𝑑𝛽
𝑘′−𝑘𝑐𝑑𝑣𝑛𝑑𝑘′
− 𝑘𝑑𝑏𝑛𝑑
(𝑘𝑐𝑑𝑣
𝑘′+ 𝑘𝑑𝑏)𝑛𝑑 =
𝑘𝑐𝑑𝛽
𝑘′
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘
′
𝑘′)𝑛𝑑 =
𝑘𝑐𝑑𝛽
𝑘′
107
𝑛𝑑 =𝑘𝑐𝑑𝛽
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′
(5-19)
𝑑𝑛𝑡𝑑𝑡
= 𝑘𝑏𝑡𝑛𝑏 = 𝑘𝑐𝑏𝑛𝑐 + 𝑘𝑑𝑏𝑛𝑑 = 𝑘𝑐𝑏𝛽 − 𝑣𝑛𝑑𝑘′
+ 𝑘𝑑𝑏𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝛽
𝑘′−𝑘𝑐𝑏𝑣𝑛𝑑𝑘′
+ 𝑘𝑑𝑏𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝛽
𝑘′+ (𝑘𝑑𝑏 −
𝑘𝑐𝑏𝑣
𝑘′)𝑛𝑑
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝛽
𝑘′+ (
𝑘𝑑𝑏𝑘′ − 𝑘𝑐𝑏𝑣
𝑘′) (
𝑘𝑐𝑑𝛽
𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝛽
𝑘′+𝑘𝑐𝑑𝛽(𝑘𝑑𝑏𝑘
′ − 𝑘𝑐𝑏𝑣)
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝛽(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘
′) + 𝑘𝑐𝑑∆(𝑘𝑑𝑏𝑘′ − 𝑘𝑐𝑏𝑣)
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝑘𝑐𝑑𝑣𝛽 + 𝑘𝑐𝑏𝑘𝑑𝑏𝑘
′𝛽 + 𝑘𝑐𝑑𝑘𝑑𝑏𝑘′𝛽 − 𝑘𝑐𝑑𝑘𝑐𝑏𝑣𝛽
𝑘′(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑐𝑏𝑘𝑑𝑏𝛽 + 𝑘𝑐𝑑𝑘𝑑𝑏𝛽
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘
′𝛽
(𝑘𝑐𝑑𝑣 + 𝑘𝑑𝑏𝑘′)
(5-20)
𝑅𝑒𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑒 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏(𝑘𝑐𝑏 + 𝑘𝑐𝑑)]
𝐴𝑠𝑠𝑢𝑚𝑒 𝑘𝑐𝑑 ≪ 𝑘𝑐𝑏
𝑑𝑛𝑡𝑑𝑡
=𝑘𝑑𝑏𝑘𝑐𝑏(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧𝛼𝜌𝐴
+ 𝑘𝑑𝑏𝑘𝑐𝑏]
𝑑𝑛𝑡𝑑𝑡
=(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
(5-21)
108
𝑑𝑛𝑡𝑑𝑧
= 2𝜋𝑟𝑐𝛼𝜌𝐴
𝑤ℎ𝑒𝑟𝑒 𝐴𝑐 = 4𝜋𝑟𝑐2
𝑑𝑛𝑡𝑑𝑧
= √𝜋𝐴𝑐𝛼𝜌𝐴
√𝜋𝐴𝑐𝛼𝜌𝐴𝑑𝑧
𝑑𝑡=(𝑁𝑎𝑧𝐴𝑐 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧
𝑘𝑑𝑏𝑘𝑐𝑏𝛼𝜌𝐴+ 1]
𝑑𝑧
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑐𝑑𝑁𝑎𝑧𝑘𝑑𝑏𝑘𝑐𝑏
+ 𝛼𝜌𝐴]
𝐿𝑒𝑡 𝑘𝑐𝑑
𝑘𝑑𝑏𝑘𝑐𝑏= 𝑘𝑒𝑓𝑓
𝐺𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑖𝑛𝑠𝑖𝑑𝑒 𝑝𝑜𝑟𝑒 =𝑑𝑧
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴]
(5-22)
Substitution of Concentration Profile in Growth Rate
Г =cosh(𝛷𝜁)
cosh(𝛷)
Г =𝐶𝑎𝐶𝑎0
(5-13)
𝐶𝑎𝐶𝑎0
=cosh(𝛷𝜁)
cosh(𝛷)
𝑁𝑎𝑧 = 𝐷𝐴𝐵𝑑𝐶𝑎𝑑𝑧
= −𝐷𝐴𝐵𝐶𝑎0𝑑
𝑑𝑧[cosh(𝛷𝜁)
cosh(𝛷)]
(5-23)
𝑁𝑎𝑧 =𝐷𝐴𝐵𝐶𝑎0𝐿 cosh(𝛷)
𝑑
𝑑𝜁[cosh(𝛷𝜁)]
𝑁𝑎𝑧 =𝐷𝐴𝐵𝐶𝑎0𝛷
𝐿 cosh(𝛷)[sinh(𝛷𝜁)]
(5-24)
𝐼𝑛𝑠𝑖𝑑𝑒 𝑃𝑜𝑟𝑒 𝐺𝑟𝑜𝑤𝑡ℎ 𝑅𝑎𝑡𝑒: 𝑑𝑧
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘𝑎𝑐𝑛𝑎)
[𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴]
109
𝑘𝑎𝑐𝑛𝑎 = 𝑘"𝑔(𝑥) 𝑤ℎ𝑒𝑟𝑒 𝑔(𝑥) 𝑐𝑎𝑛 𝑏𝑒 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑑 𝑎𝑠 𝑎 𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑧
𝑘𝑎𝑐𝑛𝑎 = 𝑘"𝑧
𝑑𝑧
𝑑𝑡=
1
√𝜋𝐴𝑐
(𝐴𝑐𝑁𝑎𝑧 + 𝑘"𝑧)
[𝑘𝑒𝑓𝑓𝑁𝑎𝑧 + 𝛼𝜌𝐴]
𝑑𝑧
𝑑𝑡=
(√𝐴𝑐
√𝜋[𝐷𝐴𝐵𝐶𝑎0𝛷𝐿 cosh(𝛷)
(sinh(𝛷𝜁))] +𝐿𝑘"
√𝜋𝐴𝑐𝜁)
[𝑘𝑒𝑓𝑓 [𝐷𝐴𝐵𝐶𝑎0𝛷𝐿 cosh(𝛷)
(sinh(𝛷𝜁))] + 𝛼𝜌𝐴]
𝑑𝑧
𝑑𝑡=
(𝐷𝐴𝐵𝐶𝑎0𝛷√𝐴𝑐𝐿 cosh(𝛷)√𝜋
(sinh(𝛷𝜁)) +𝐿𝑘"
√𝜋𝐴𝑐𝜁)
[𝐷𝐴𝐵𝐶𝑎0𝛷𝑘𝑒𝑓𝑓𝐿 cosh(𝛷)
(sinh(𝛷𝜁)) + 𝛼𝜌𝐴]
𝑑𝑧
𝑑𝑡= 𝐿
𝑑𝜁
𝑑𝑡=
(𝐷𝐴𝐵𝐶𝑎0𝛷√𝐴𝑐𝐿 cosh(𝛷)√𝜋
(sinh(𝛷𝜁)) +𝐿𝑘"
√𝜋𝐴𝑐𝜁)
[𝐷𝐴𝐵𝐶𝑎0𝛷𝑘𝑒𝑓𝑓𝐿 cosh(𝛷)
(sinh(𝛷𝜁)) + 𝛼𝜌𝐴]
𝑑𝜁
𝑑𝑡=
(𝐷𝐴𝐵𝐶𝑎0𝛷√𝐴𝑐cosh(𝛷)√𝜋
(sinh(𝛷𝜁)) +𝐿2𝑘"
√𝜋𝐴𝑐𝜁)
[𝐷𝐴𝐵𝐶𝑎0𝛷𝑘𝑒𝑓𝑓𝐿 cosh(𝛷)
(sinh(𝛷𝜁)) + 𝛼𝜌𝐴]
𝑙𝑒𝑡 𝐷𝐴𝐵𝐶𝑎0𝛷√𝐴𝑐
cosh(𝛷)√𝜋= 𝐴
𝐿2𝑘"
√𝜋𝐴𝑐= 𝐵
𝐷𝐴𝐵𝐶𝑎0𝛷𝑘𝑒𝑓𝑓
𝐿 cosh(𝛷)= 𝐶
𝛼𝜌𝐴 = 𝐷
𝑑𝜁
𝑑𝑡=(𝐴 sinh(𝛷𝜁) + 𝐵𝜁)
[𝐶 sinh(𝛷𝜁) + 𝐷]
(5-25)
110
LIST OF REFERENCES
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BIOGRAPHICAL SKETCH
Dr. Justin Chet-Mun Wong graduated with a Doctor of Philosophy in chemical
engineering from the University of Florida in 2016. He received his Bachelor of Science
in 2011 from the University of Maryland, where he also studied chemical engineering.
During his time at the University of Florida, Dr. Wong was actively involved in the
student chapter of the Electrochemical Society. He has presented his work at numerous
national conferences including those of the Electrochemical Society and American
Institute of Chemical Engineers.
Dr. Wong’s dissertation, Attaining Nanoscale Patterns for Renewable Energy
Applications, was supervised by Dr. Kirk Ziegler.
