Directed Self-Assembly:
A Study of the E.ect of Electric Fields on Silica
Monolayers
Matthew O. Withers
A thesis presented in partial ful.llment of
the requirements for the degree of
Bachelor of Science
Department of Physics and Engineering
Washington and Lee University
Lexington, VA, USA
May 19, 2020
Abstract
In this thesis, we describe, analyze, and extend ionic self-assembly of monolayers (ISAM), a bottom-up nanostructure production technique designed to coat surfaces in uniform layers of charged nanoparticles. Using mean .eld theory, we develop cooperative sequential adsorption with evaporation (CSAE) models of the assembly process, designed to predict the particle coverage density of ISAM samples. We simulate the particle assembly process via the Monte Carlo technique, and we evaluate our CSAE models primarily by comparing them to these simulated results. Finally, aided by scanning electron microscopy, we analyze experimental ISAM samples. This experimental approach provides us with information about the time scale of assembly, as well as the relationship between our CSAE models and particle suspension concentration. Our approach considers ISAM under no external in.uence, as well as ISAM conducted under constant and oscillating electric .elds. Assembly under electric .elds represents a type of directed self-assembly of monolayers (DSAM), an emerging technique designed to control particle coverage density using an external in.uence.
Acknowledgments
I would .rst and foremost like to thank my thesis adviser, Dr. Dan Mazilu, for all of his assistance, not only in terms of the writing of this thesis, but also in terms of the academic, professional, and personal guidance he has o.ered throughout my undergraduate career. His advice in the laboratory and on trips to conferences both at home and abroad have had a signi.cant impact on my development as a scientist and communicator.
I would also like to thank my academic adviser, Dr. Irina Mazilu, for all the assistance she has o.ered in terms of planning my academic trajectory. I would also like to thank her for taking the time to mentor me in teaching and scienti.c communication. These skills will be invaluable throughout the rest of my career. Furthermore, this thesis would not have been possible without the countless hours of theoretical discussions we shared.
In the laboratory, I would like to thank Ms. Emily Falls for her assistance with scanning electron microscope training and certi.cation, as well as Mr. David Pfa. for answering any question I had concerning the equipment in the IQ Center. I would also like to thank Mr. Chris Compton for all of his laboratory expertise, especially with respect to the acquisition of new, quality laboratory equipment.
Finally, I would like to thank all of my student partners in the nanoscience laboratory for providing a fun and collegial work environment. Partners over the years include Elise Baker, Gillen Beck, Laura Bruce, Emily Hassid, Cory Morris, Mitchell Roberts, Ben Zeman, and Nolan Zunk.
Some portions of the research contained in this thesis was completed as a part of Washington and Lee University’s Summer Research Scholars Program. My stipend for the program was provided by the E. A. Morris Charitable Foundation. I presented much of this research at conferences, including the APS March Meeting (2019) in Boston, MA, and the 8th International Conference on Mathematical Modeling in Physical Sciences in Bratislava, Slovakia. Travel to these conferences was funded, in part, by the Johnson Scholarship Program.
Contents
Abstract
Acknowledgments
1
Introduction
and
Background
1.1
Introduction
to
Nanoscience
iii v 1
................................ 1
1.2
Self-AssemblyofMolecularMonolayers.......................... 5
1.3
ApplicationsofSelf-Assembly
............................... 9
1.4
ImagingTechniques
.................................... 11
1.4.1
ScanningElectronMicroscopy(SEM)
...................... 11
1.4.2
AtomicForceMicroscopy(AFM)
......................... 14
2
Self-Assembly
under
No
External
Field
(ISAM)
17
2.1
ISAMOverview....................................... 17
2.2
CSAEModelingTechniquesandResults
......................... 18
2.2.1
CSAE-TL...................................... 19
2.2.2
CSAE-NN...................................... 23
2.3
SimulationTechniquesandResults
............................ 27
2.4
ExperimentalTechniquesandResults
.......................... 33
3
Self-Assembly
under
External
Electric
Fields
(DSAM)
39
3.1
Overview
of
Electric
Fields,
Electric
Potentials,
and
Capacitors
. . . . . . . . . . . . 39
3.2
PerpendicularElectricFields
............................... 41
3.2.1
DSAMforPerpendicularFields
.......................... 42
3.2.2
CSAEModelingTechniquesandResults
..................... 43
3.2.3
SimulationTechniquesandResults
........................ 49
3.3
ParallelElectricFields
................................... 53
3.3.1
DSAMforParallelFields
............................. 53
3.3.2
CSAEModelingTechniquesandResults
..................... 54
3.3.3
SimulationTechniquesandResults
........................ 57
3.4
OscillatingElectricFields
................................. 59
3.4.1
OverviewofOscillatingFields........................... 59
3.4.2
CSAEModelingTechniquesandResults
..................... 59
3.5
ExperimentalTechniques
................................. 65 4
Discussion:
Comparing
CSAE
Models
and
Simulated
Data
67 5
Ising
Model
Approach
to
ISAM
77 6
Conclusion
83 Bibliography
85 A
Mean
Field
Theory
89 B
Python
Tools
93
B.1
odeint
............................................ 93
B.2
ArtistAnimation
..................................... 95
C
Code
Samples
97
C.1
CSAE-TLNumericalSolutions
.............................. 97
C.2
CSAE-NNNumericalSolutions
.............................. 98
C.3
CSAE-TLSimulations
................................... 99
C.4
CSAE-NNSimulations
................................... 102
C.5
SimulationVideoOutput
................................. 106
Chapter 1
Introduction and Background
1.1 Introduction to Nanoscience
On December 29, 1959, renowned physicist Richard Feynman addressed members of the American Physical Society at their annual gathering. His talk, entitled “There’s Plenty of Room at
the
Bottom”
[1],
considered
a
world
in
which
scientists
could
carefully
manipulate
individual
molecules and atoms to store information on the smallest scales. For nearly two decades, his thoughts went mostly unnoticed. However, by the 1980s, they had become an important tool for members of the newly emerging nanoscience community, which used Feynman’s ideas to both inspire new areas of research and justify their belief that meaningful physics could be reasonably conducted at the nanoscale.
Feynman’s exploration of what would become nanoscience begins with an unusual question: is there enough room on the head of a pin to print the entirety of the Encyclopaedia Britannica? He answers this question by stating that the head of a pin would need to be magni.ed roughly 25,000 times before its area would equal the area of all the pages in the encyclopedia. Naturally, therefore, one could print the entire encyclopedia on a pin head if he or she could simply reduce the text of the encyclopedia by 25,000 times. Feynman argues that, even at such a small scale, the smallest discernible dots contained in a standard printing of the encyclopedia would contain around 1000 atoms—more than enough to ensure that the information could be preserved for a long period of time without corruption. Furthermore, Feynman notes that the electron microscopes available in 1959 could certainly read text of that size, and an inverted electron microscope lens could be used to focus ions onto the pin’s surface to engrave the text. Once one copy of the text had been created, further copies could easily be produced via a plastic mold of the original.
Feynman then considers the physical storage of information via a binary code. Reasoning that each letter within the Latin alphabet would require between six and seven bits of data for unique storage, he suggests using cubes of 5 × 5 × 5 = 125 atoms to encode the information within (not just on the surface of) a pin head. For one binary state, called a dot, one type of atom would be used, while for the other state, called a dash, another type of atom would su.ce. Given these parameters, Feynman calculates that “all the books in the world can be written in this form in a cube of material one two-hundredth of an inch wide—which is the barest piece of dust that can be made out by the human eye.” Herein lies Feynman’s central argument: not only does the nanoscale provide su.cient room to store vast amounts of information, but also it appears to provide plenty of room.
After demonstrating the vast storage capabilities of the nanoscale, Feynman continues his argument by taking up a discussion of the electron microscope. While avoiding a discussion of any practical engineering details, Feynman suggests that the electron microscope of his day could be improved to the point of being able to discern individual atoms. His reasoning, he states, relies upon the fact that the wavelength of an electron is only around 1/20 of an angstrom, much smaller than the width of an atom. The results of improving the resolution of the electron microscope to even 100 times its 1959 resolution would have numerous consequences for the scienti.c community. Biologists would be able to probe the structures of DNA, RNA, amino acids, proteins, and microsomes directly. Chemists, rather than using an extensive set of reactions to determine the atomic makeup of a complicated molecule, could simply look at the molecule under the electron microscope to determine its structure and constituent parts.
Inspired by the ability of biological systems to complete complex tasks at the molecular level, Feynman continues by considering a variety of miniaturized machines and devices. He .rst suggests a miniature computer, perhaps one with wires 10 to 100 atoms in diameter, which would allow for circuits thousands of angstroms in width. From Feynman’s point of view, developing miniaturized computer components is an important step in creating computers powerful enough to complete human-like tasks, such as (what we now call) machine learning and image recognition. In 1959, standard computers took up entire rooms. Building a computer with enough processing power to learn from past experience and recognize images would take up a space roughly the size of the Pentagon. A computer of this size would encounter a vast array of problems besides simply being unwieldy. First, it would require too much material—not just in terms of cost but, more importantly, in terms of availability on Earth. Second, it would produce too much heat. Third, it would consume as much power on a yearly basis as is produced annually by the Tennessee Valley Authority. Finally, because information would need to be passed between components separated by vast distances, such a computer would be prohibitively slow.
Feynman considers the production of other miniaturized machines via a variety of methods. First, he suggests evaporating alternating layers of conductors and insulators onto a surface, thus converting the surface into a viable circuit containing components as complicated as coils, capacitors, and transistors. He also suggests that the mechanical devices of our world (e.g., cars and engines) could be scaled down directly to nanosize. Doing so would require considerations of force scaling and material strength. Furthermore, any electrical systems in the scaled-down device would need to be redesigned since magnetic properties change at such a size. However, scaled-down devices would have a variety of bene.ts, the chief of which would be the elimination of a need for lubricating agents and heat sinks (devices on the order of 10-9 m would dissipate heat much too quickly to require standard cooling techniques). To produce such scaled down devices, Feynman imagines a chain of tiny hands, each a quarter of the size of the one before it, which eventually produce the “tools” necessary to manufacture on the nanoscale. The engineering feasibility of this idea is less important than its spirit: Feynman imagines a world in which we produce versions of macroscopic tools which function on the nanoscale, a concept which would allow for advancements in .elds as far reaching as medicine and electrical engineering.
Feynman ends his discussion of the nanorealm by considering the potential to produce synthetic molecules directly. If an electron microscope could (at least theoretically) view individual atoms, why could some other tool not be used to move them around and arrange them in a desired structure? Such a tool would have a profound impact on the .eld of chemistry. No longer would chemists have to rely upon long reaction chains just to produce a sample of a molecule that inevitably contains impurities. Instead, if a chemist desired a particular molecule, he or she could simply put it together piece by piece.
The impact of Feynman’s speech on the .eld of nanoscience is perhaps best measured by the number of real-world advancements it has inspired. Electron beam lithography
[2,
3,
4,
5]
is an excellent example. In electron beam lithography, a beam of electrons is focused onto the surface of an electron-sensitive .lm known as a resist. When the beam makes contact with the resist, it changes its properties (“cures” it). Later, a solvent can be applied to the surface of the resist, causing either the cured or uncured portion to fall away. In this way, intricate patterns and structures can be drawn on the resist’s surface. The resist is then easily used as a template for etching onto a more solid material (substrate) such as a silicon wafer. Electron beam lithography is very similar to Feynman’s suggestion that text from the Encyclopaedia Britannica could be written onto the head of a pin by reversing the optics of an electron microscope. Perhaps even more directly inspired
is
nanostructure
etching
via
focused
ion
beams
[2,
6,
7].
A focused ion beam is simply a steady stream of ions focused via an electromagnetic optical system until powerful enough to cut away a surface. Rather than relying upon resists and solvents, focused ion beam etching cuts nanostructures into a substrate directly, allowing for rapid prototyping of complex nanostructures.
Feynman’s discussion also predicted the development of nanostructure stamping technology [2,
8,
9,
10].
During
his
discussion
of
the
encyclopedia,
Feynman
suggests
creating
plastic
molds
of
its miniaturized version so that copies could be easily produced for students and researchers all over the
world.
George
Whitesides’
nanostamping
technology
[8],
in
which
silicone
rubber
is
allowed
to
cure on top of an already-produced nanostructure, is almost identical to Feynman’s vision.
While three-dimensional, high-density storage of the type Feynman envisioned when discussing his attempt to compress every known book into a single speck of dust has not been realized, nanoscientists have managed to utilize three dimensions in the construction of nanostructures. Perhaps
the
best
known
example
of
a
three-dimensional
nanostructure
is
the
integrated
circuit
[11,
12].
Incorporating potentially thousands of transistors, resistors, capacitors, coils, and other electrical components, integrated circuits require more space than the simple two-dimensional plane to be realized.
Feynman’s suggestions about imaging in biology and chemistry have ultimately been realized via
the
atomic
force
microscope
[13],
which
routinely
produces
images
of
individual
atoms
[14].
The
atomic
force
microscope
can
even
be
used
to
detect
the
atomic
structure
of
molecules
[15]
and
move
atoms
around
[16],
allowing
for
the
production
of
fully-synthetic
molecules.1
Furthermore, his ideas concerning the production of nanoscale computers and machines, particularly his comments on
nanostructure
production
via
evaporation,
have
found
life
via
the
study
of
biological
motors
[17]
and
the
development
of
molecular
beam
epitaxy
[18].
The techniques for producing nanostructures in today’s research environment are generally divided into one of two categories. Top-down approaches, as the name suggests, involve the selective removal of material from a surface until
all
that
remains
is
the
desired
nanostructure
[2].
On the other hand, bottom-up approaches involve the use of chemical processes to deposit materials sequentially
on
a
surface,
eventually
building
up
a
nanostructure
layer
by
layer
[2].
It
is
often
easy
to think of top-down approaches like the work of a sculptor. The sculptor begins with a large block of material and removes portions little by little until a masterpiece emerges. In contrast, bottom-up approaches are similar to the work of a painter, who layers paint on a canvas until a clear, structured image appears.
1This method of molecule production is still nowhere near as e.cient as typical chemical synthesis methods.
While the methods generally classi.ed as top-down approaches are especially diverse, most laboratories use procedures that include similar steps. Almost every procedure includes an oxidation step, in which a protective layer of silicon dioxide is placed on the surface of a silicon substrate. During masking, photolithographic methods are used to etch features into the freshly deposited silicon layer. With the desired features etched, thus exposing portions of the silicon substrate, the implantation phase commences. In implantation, charged ions are delivered to the exposed portions of the substrate. This leads to doping, which changes the exposed areas’ electrical properties relative to the rest of the substrate. After implantation, etching occurs. During etching, the manufacturer removes the protective silicon dioxide layer and any undesired potion of the silicon substrate. Metallization involves the use of evaporative or electromechanical methods to deposit metals on the surface of the nanostructure, allowing for electrical contacts to form between critical components. Finally, manufacturers use lift-o. to deposit a photoresist on the surface of the nanostructure, which, when selectively cured and placed in a dissolving bath, allows for control over the selective removal
of
material
[2].
Well-established nanostructure production processes that make use of top-down approaches include
photolithography
[19],
electron
beam
lithography
[2,
3,
4,
5],
reactive
ion
etching
[2,
6,
7],
molecular
beam
epitaxy
[18],
self-assembled
masks
[20],
focused
ion
beam
milling
[21],
and
stamp
technologies
[8].
Utilizing
di.erent
combinations
of
the
methods
explained
above,
these
manufac
turing processes o.er nanoscientists a high level of control over the nanostructure’s construction.
While each step of nanostructure production via top-down approaches is carefully controlled by the experimenter, bottom-up approaches use a much more hands-o. approach, in which the random collisions of suspended particles facilitate the assembly of a structure. Using random collisions as a means of assembly implies that bottom-up approaches rely heavily on the enthalpy and entropy of the system. If the enthalpy is too low, the nanoparticles will not have enough energy to (weakly) bond and form a structure. If the entropy is too low, incorrectly formed nanostructures (often dubbed “erroneous” nanostructures in the literature) will be unable to disassemble, thus ruining the sample. Conversely, if the entropy is too high, any correctly formed nanostructures will be
unstable.
This
instability
will
result
in
their
quick
dissolution
[2].
Ideas from traditional chemical synthesis and techniques from contemporary experimental chemistry are also important in bottom-up nanostructure production. For example, the theory of chemical synthesis helps to predict the amount of reactants needed to drive an irreversible reaction. Similar analysis helps nanoscientists determine the amount of nanostructure components to add to a solvent to facilitate the self-assembly of a nanostructure. Concepts from surface chemistry are also especially important in nanostructure production, as nanostructures commonly have high surface to volume ratios, and their surface atoms are often unable to bond traditionally with atoms of
the
same
type
[2].
Contemporary examples of bottom-up nanostructure production include organic synthesis [22],
vesicle
[23]
and
micelle
[24]
production,
self-assembled molecular monolayers [25,
26,
27],
kinetic
control
of
growth
[28,
29]
(nanowire
[30]
and
quantum
dot
[31,
32,
33]
production),
and
DNA
nanotechnology
[34,
35].
In each case, the nanoscientist sacri.ces direct control over the construction of the assembly in order to take advantage of self-assembling characteristics.
1.2 Self-Assembly of Molecular Monolayers
One of the most versatile nanostructure production methods currently in use is the self-assembly of molecular monolayers. This method, in which charged particles arrange themselves into particle sheets of uniform thickness, is used to coat objects in nanoparticles, thus changing the objects’ surface properties. For example, self-assembly of molecular monolayers has been used to apply thin .lms to sheets of glass, thus producing the thin-.lm interference necessary for an anti-re.ective
coating
[36,
37,
38].
For
the
remainder
of
this
thesis,
we
discuss
self-assembly
of
molecular
monolayers in detail, in particular the experimental techniques necessary to produce high quality monolayers. We also develop and evaluate modeling and simulation techniques designed to predict monolayer properties before the assembly process begins.
A functional method of achieving self-assembly of molecular monolayers was .rst proposed by Iler
[25]
in
1966.
Building
upon
the
work
of
Langmuir,
whose
1941
patent
[26]
states
that
adsorbing
ions will form a single layer until every available occupation site is .lled, Iler demonstrated how particles of colloidal size can adsorb onto a surface of the opposite charge. Iler’s advancement was particular important to nanoscience because most of the particles that scientists wish to employ in the self-assembly process (e.g., SiO2, TiO2, etc.) are of colloidal size.
Iler’s method began with the selection of an assembly surface. While he stated that any anionic surface is su.cient, his experiments made use of a sheet of clean, hydrophilic glass.2
The siliceous nature of the glass ensured that it would remain anionic throughout the experiment. With the assembly surface selected, Iler then coated it with a cationic layer. In his case, the cationic layer was a 0.25% aquasol of colloidal boehmite alumina containing 5-6µm (diameter) .brils of AlOOH. He followed the cationic layer with a 2% aqueous sol of colloidal silica. Each silica particle was essentially spherical with a diameter of approximately 100 nm. Attracted by the cationic aluminum layer, the silica adsorbed onto the surface of the glass, forming a uniform layer, or thin .lm. Iler found that, after rinsing the assembled layer of silica, he could repeat this process to form additional layers. This permitted to creation of silica .lms of varying thicknesses.
Iler made two important observations while developing this assembly method. First, he noted that the alternation of cationic and anionic layers was essential to the development of multiple layers of silica. Silica attempting adsorption after a uniform layer had formed would be repelled by the charge of already deposited silica. By placing an aluminum layer on top of the adsorbed silica, the suspended silica would be attracted to the resulting cationic charge, thus permitting the formation of an additional layer. Second, Iler observed that .lms with thicknesses over 50 nm would become visible in re.ected light. This visibility is due to the thickness of the .lm approaching the wavelength of visible light.
Iler also noted a number of factors a.ecting the formation of layers. First, he found that rinsing the sample with distilled water between the deposition of cationic and anionic layers was essential to ensuring that the next layer would be uniform. Second, he observed that the concentration of the cationic and anionic colloidal solutions used could also be important. As a general rule, he suggested using concentrations less than 0.5% for small particles (de.ned as particles with speci.c surface areas on the order of several hundred square meters per gram) and 3-5% for larger particles (e.g., the 100 nm silica used in his experiment). Third, the pH of the solutions was critical. In particular, Iler found that the adsorption of this silica particles took place most e.ciently (i.e., rapidly and completely) in the low pH range (pH 2-4). Finally, suspensions that contained
2Iler selected black-tinted glass to aid in the analysis of the optical properties of the assembled layers.
supercolloidal aggregates or a gel phase would produce layers with signi.cant irregularities. These irregularities would often prevent the formation of a layer capable of producing interference, thus rendering the layer invisible when observed under visible light.
Iler’s
methods
were
built
upon
by
Lvov,
Ariga,
Onda,
Ichinose,
and
Kunitake
[27]
in
1997.
Using a quartz crystal microbalance (QCM), scanning electron micoscopy (SEM), and atomic force micoscopy (AFM), their team completed more thorough analysis of monolayers produced using Iler’s method.
In their experiments, Lvov et al. used poly(diallyldimethylammonium chloride) (PDDA), sodium poly(styrenesulfonate) (PSS), and poly(ethyleneimine) (PEI) as their bonding agents. They tested a variety of anionic nanoparticles, including silicon dioxide (SiO2), titanium dioxide (TiO2), and cesium dioxide (CeO2). Their primary investigation involved the use of the quartz crystal microbalance technique to detect changes in the mass of the assembly. This mass value could then be converted into a measurement of the thickness (d) of the assembled layer. Their team was interested in determining whether the thickness of the .lm increased linearly as additional particle layers were deposited. For the particular QCM their team used, the change in quartz frequency depended directly upon the mass (M) and inversely upon the surface area (A) of the sample:
M
8
.F =(-1.83 × 10) . (1.1)
A Furthermore, the sample thickness (d) was related to this frequency shift via
d (nm) = 0.022(-.F (Hz)). (1.2)
Lvov et al. assembled a thin .lm layer by layer, measuring the change in frequency it produced within the QCM after each layer (either cationic or anionic) had fully formed.3
Their results demonstrated several interesting properties. First, the deposition of the bonding agent (PDDA, PSS, PEI, etc.) contributed minimally to the thickness of the .lm. This behavior is shown by the nearly constant .F at
odd
adsorption
steps
(see
Figure
1.1).
Meanwhile, the adsorption of nanoparticles produced a consistent frequency shift, which corresponds to a consistent increase in
.lm
thickness,
no
matter
the
layer
being
produced
(see
even
adsorption
steps
in
Figure
1.1).
Furthermore, the magnitude of the frequency shift (i.e., the magnitude of the increase in .lm thickness) depended upon the the concentration of nanoparticles but not on the size of the particle used
(see
Figure
1.2).
Speci.cally,
an
increase
in
concentration
corresponded
to
an
increase
in
the
size of the growth step. These results were particularly signi.cant because they indicated that the experimenter can closely control the thickness of a .lm simply by determining the appropriate concentration of particles and number of nanoparticle layers.
Lvov et al. also observed that SiO2, an anionic nanoparticle, could not adsorb onto the anionic PSS bonding agent. Additionally, SiO2 could not be adsorbed onto another SiO2 layer without .rst depositing a cationic bonding agent between the two layers. These results con.rmed Iler’s observation that the alternation of cationic and anionic layers is necessary for successful self-assembly of molecular monolayers. The
team
also
investigated
a
claim
published
in
[39],
which
stated that the rinsing of the sample between deposition steps would result in the desorption of some of the colloidal particles. However, Lvov et al. found that their results were reproducible regardless of the number of rinsing steps used. This result indicates that little to no desorption occurs during rinsing.
3They used a deposition time of around 15 minutes.
Figure 1.1: Frequency shift in Lvov et al.’s quartz crystal microbalance per adsorption step for several di.erent particle suspension concentrations. The thickness of the adsorbed .lm varies directly with the frequency shift. Notice that deposition of the bonding agent (odd adsorption steps) produces minimal changes in frequency, while particle deposition (even adsorption steps) increases
the
frequency
at
a
consistent
rate.
(Reproduced
from
[27].)
Figure 1.2: Frequency shift in Lvov et al.’s quartz crystal microbalance versus particle size for several di.erent particle suspension concentrations. Notice that the frequency shift, and, consequently, the .lm thickness, depends upon nanoparticle concentration but not nanoparticle size. (Reproduced
from
[27].)
1.3 Applications of Self-Assembly
Self-assembling nanoparticles play an important role in a variety of engineering applications. Perhaps the most direct application of self-assembled bilayers is the production of anti-re.ective coatings
[40].
In
an
anti-re.ective
coating,
light
is
incident
upon
two
surfaces.
The
.rst
is
the
surface
of a thin-.lm (the coating), and the second is the surface of the coated substrate. At each point of incidence, some of the light refracts while the remainder re.ects. We depict this arrangement in
Figure
1.3.
An anti-re.ective coating exploits the physical geometry of the thin .lm and the
di.erence between the indices of refraction of the .lm and substrate to cause one of the re.ected light rays of experience a p-phase shift with respect to the other.4
This produces destructive interference between the two re.ected beams, which, in turn, prevents light from re.ecting o. of the coated material. Therefore, all of the incident light passes through the coated material. Anti
4A p-phase shift corresponds to an optical path length di.erence which is an integer multiple of ./2, as we show in
Figure
1.3.
re.ective coatings are often applied to eyeglasses, increasing the amount of light that can pass through the glass and reducing glare. As
described
in
[36,
37,
38],
bilayers
of
silica
and
cationic
macromolecules (like PDDA) can be used to produce anti-re.ective coatings, with the number of bilayers applied and layer separation a.ecting the re.ectance.
Another application of nanoparticle assembly is drug delivery. As
explained
in
[41],
nanopar
ticles are often used to protect drug molecules until they reach a particular part of the body, creating a type of targeted drug delivery. This process is often completed using dendrimers, or tree-like assemblies of nanoparticles. Figure
1.4
is
a
geometric
diagram
of
a
dendrimer.
Notice
how the nanoparticles (white circles) form branches, which enclose cavities called dendric boxes that hold the drug molecules (gray octagons). When the dendrimer reaches the targeted area, the
branches
open,
destroying
the
dendric
boxes
and
releasing
the
drug
molecules.
[42]
provides
a
description of one investigation which uses self-assembling particles in this way.
In electrical engineering, nanoparticle self-assembly serves as an important tool in the development of nano-scale circuitry. [43]
describes
the
use
of
self-assembly
techniques
to
arrange
electrochemically synthesized nanowires into viable electrical circuits. Nano-scale circuits, whether constructed using more traditional top-down approaches or via bottom-up approaches like stochastic assembly, have made possible massive advances in computing capabilities. For this reason, we anticipate further work in this area.
Finally, nanoparticle self-assembly also facilitates more controlled nano-construction techniques like nanoprinting. As
explained
in
[44],
nanoprinting
via
self-assembly
allows
scientists
to
easily copy complicated nanostructures. In many cases, these techniques permit construction times that are far less than required by more traditional techniques, including electron beam lithography and scanning probe lithography. Nanoprinting’s ability to revolutionize the reproducibility of more complicated structures makes this a particularly active area of research.
1.4 Imaging Techniques
1.4.1 Scanning Electron Microscopy (SEM)
Scanning
electron
microscopy
[2],
the
primary
technique
used
throughout
our
investigation
for the imaging of nanoparticle monolayers, is a class of experimental methods which use the wavelike properties of electrons to produce an image, just as the wave-like properties of photons are used
to
produce
an
image
in
an
optical
microscope.
SEM
was
pioneered
by
Ernst
Ruska
[45],
who
in 1931 demonstrated that the image of a grid could be magni.ed if the grid were placed after a converging electron beam’s focal point. Ruska’s work noted that the magni.cation m produced by his electron beam behaved according to the rules of geometric optics, which state that
'
s
m = - , (1.3)
s
'
where sis the image distance (the distance between the lens and the image) and s is the object distance
(the
distance
between
the
sample
and
the
lens)
[2].
Optical
microscopes
are
limited
by
the
.nite wavelength of a photon (i.e., the cannot image any object that is smaller than the wavelength of light used to illuminate it). This same limitation exists for electrons, which according to de Broglie, also have an associated wavelength .. However, de Broglie’s formula
h
. = , (1.4)
mv
where . is the wavelength of the particle, h is Planck’s constant, m is the particle mass, and v is the magnitude of the particle velocity, predicts that electrons (moving at speeds typical of those that can be produced in an electron microscope) possess wavelengths anywhere from 0.08 °
A to 0.03
°
A. With most atoms having radii on the order of a few angstroms, these values indicate that SEM techniques can, in principle, achieve atomic resolution. For imaging on the nanoscale, SEM is more than su.cient.
Most
SEM
imaging
is
achieved
according
to
the
following
method
[2].
First, an electron beam is produced using either thermionic emission or .eld emission. In thermionic emission (the method used by our SEM), a .lament is heated, increasing the energy of internal electrons until they overcome an energy barrier, denoted by a work function f (see
Figure
1.5)
[12].
Electrons that have overcome this energy barrier can then be collected into an electron beam. The current density of the produced electron beam is given by Richardson’s law
4pme f
2
J = exp- , (1.5)
3 (kBT )
hkBT
where m is the mass of an electron, e is the fundamental charge, h is Planck’s constant, kB is Boltzmann’s constant, T is the temperature of the .lament, and f is the work function of the .lament
[2].
Alternatively,
in
.eld
emission,
a
strong
external
electric
.eld
is
used
to
encourage
the
electrons
in
a
source
material
to
tunnel
through
the
energy
barrier
(see
Figure
1.6)
[12].
For
this
method, the current density of the produced electron beam is proportional to the tunneling rate, given by the Fowler-Nordheim equation
v
3/2
42mf
2
J . Eexp - , (1.6)
3e. E
where E is the magnitude of the electric .eld used to induce tunneling, m is the mass of the electron, e is the fundamental charge, . is the reduced form of Planck’s constant, and f is the work function of
the
source
material
[2].
While thermionic emission is more typical, .eld emission is capable of higher resolution due to a higher consistency in electron wavelength. Furthermore, the aberrations in electron lenses have less of a negative e.ect on an electron beam produced via .eld emission than a beam produced via thermionic emission.
After a beam of electrons has been produced, a series of magnetic lenses are used to focus the beam on the surface of the sample. The path of an electron de.ected by electric and magnetic .elds is given by
2
dr
m = -ev × B - eE, (1.7)
2
dt
where m is the mass of the electron, r is the electron’s position vector, e is the fundamental charge, v is the velocity of the electron, B is the magnetic .eld vector, and E is the electric .eld vector. This
equation
predicts
a
motion
similar
to
that
depicted
in
Figure
1.7,
which
shows
a
series
of
two
magnetic lenses being used to focus a diverging electron beam originating at point O onto an image plane at point I. The vector product between v and B forces the electrons to spiral around the magnetic
.eld
[2].
Once the beam has been focused into a small spot on the sample, magnetic de.ection of the beam is used to raster across the sample’s surface. The image is then produced in one of several ways, all of which depend upon the interaction between the electron beam and the sample at their point of contact. For especially thin samples, the electrons striking the surface are collected directly using a sensor placed below the sample. In this scenario, often called scanning transmission electron microscopy (STEM), the transmitted current is used to produce the image. Thicker samples often rely upon the collection of elastically backscattered electrons, or even inelastically scattered electrons. SEMs that can achieve especially high electron beam energies can even excite and detect X-ray
emissions
from
the
electrons
in
the
sample
[2].
1.4.2 Atomic Force Microscopy (AFM)
Another emerging method for the imaging and manipulation of nanoscale structures is atomic force
microscopy
(AFM).
Developed
by
Binnig,
Quate,
and
Gerber
[13]
in
1986
to
address
several
limitations of scanning tunneling microscopy (another imaging method), AFM measures the de.ections of a cantilever tip to produce a reliable image of a surface. Advantages of this method include the ability to directly manipulate atoms or molecules on a sample as well as the option of producing three dimensional images and sample height pro.les. AFM techniques are also commonly used to measure the electrical properties of samples.
As
depicted
in
Figure
1.8,
AFM
imaging
begins
with
a
laser
source,
which
re.ects
o.
of
a
cantilever arm and is detected by a two segment photodiode. As the cantilever arm rasters across
the sample, atoms within the sample de.ect the tip, causing the entire cantilever arm to vibrate at a frequency given by
1 k
f0 = 2p m , (1.8)
where k is the e.ective spring constant of the arm and m is its mass. The image is ultimately
produced using information about the de.ection of the cantilever arm (dz), which is proportional
to the quotient of the di.erence and sum of the two photodiode currents (iA and iB):
iA - iB
dz . . (1.9)
iA + iB
AFM can be performed in several di.erent modes, including contact mode, in which the tip is allowed to make direct contact with the sample, non-contact mode, in which the tip is held a set distance above the sample and de.ected by electrostatic repulsion, and vibrating mode, in which the
tip
is
allowed
to
vibrate
on
top
of
the
sample
[2].
Chapter 2
Self-Assembly under No External Field (ISAM)
2.1 ISAM Overview
Building on the work of Iler and Lvov, our models, simulations, and experiments concern thin .lms constructed from uniform layers of nanoparticles. To produce these layers, we use the ionic self-assembly
of
monolayers
process
(ISAM).
Figure
2.1
details
the
ISAM
process
when
used
to
adhere
negatively charged particles to a .at glass surface such as a microscope slide. Due to the presence
of silica in glass, the surface of any standard glass slide immersed in water has an innate negative charge
[47].
This is indicated by the negative signs on the surface of the Clean Slide in Figure
2.1.
We .rst dip the slide in a suspension of poly(diallyldimethylammonium chloride) (PDDA), which is a standard polycation. Its positively charged molecules easily adhere to the surface of the negatively charged slide, forming a single monolayer of positively charged ions. This is indicated by
the
positive
signs
on
the
Polycation
Monolayer
Slide
shown
in
Figure
2.1.
After
producing
the
polycation monolayer, we then dip the slide in a suspension of silicon dioxide (SiO2) nanoparticles, which is anionic. Thus, the electrostatic forces present between the cationic PDDA and the anionic SiO2 cause the SiO2 to adhere to the surface of the slide. We now have a single bilayer formed from individual monolayers of PDDA and SiO2, as indicated by the Polycation/Anion Bilayer Slide in Figure
2.1.
By
repeating
the
ISAM
process,
we
can
produce
thin
.lms
with
any
number
of
bilayers.
2.2 CSAE Modeling Techniques and Results
Physicists and engineers naturally desire a means of predicting the properties of a .lm produced via ISAM. For this purpose, a variety of techniques have been developed using principles from statistical physics to model the nanoparticle assembly process. By connecting the assembly process to emergent properties such as particle coverage density, it is possible to accurately predict the optical properties of a thin .lm produced via ISAM.
One of the primary modeling techniques for nanoparticle assembly is the use of a cooperative sequential
adsorption
with
evaporation
(CSAE)
model
[41,
48].
CSAE
models
imagine
the
assembly
surface
as
a
grid,
as
shown
in
Figure
2.2.
Each
location
on
the
grid
is
called
a
site
and
is
indicated
mathematically by a site number i. Each site can exist in one of two states, indicated by a state value ni. Sites
that
contain
particles
(colored
black
in
Figure
2.2)
are
considered
occupied
and
are
denoted mathematically by ni =
1.
Sites
that
do
not
contain
particles
(colored
white
in
Figure
2.2)
are considered unoccupied and are denoted mathematically by ni = 0. Sites transition between states according to a transition rate c(ni . (1 - ni)). The rules of the transition rate depend upon the particular CSAE model used. CSAE models di.er in their complexity, which corresponds to the amount of physical detail they encompass. Our work considers two di.erent CSAE models: the total lattice model (CSAE-TL) and the nearest neighbors model (CSAE-NN). We consider the construction and use of each model in detail in the following sections.
2.2.1 CSAE-TL
The
transition
rate
for
the
CSAE-TL
model
[41]
is
given
in
Equation
(2.1):
n i=1 ni
cTL(ni . (1 - ni)) = ni. + µ(1 - ni)1 - . (2.1)
N
When a site is occupied (i.e., ni = 1), the .rst term, ni., is active. This term is called the evaporation term because it allows the particle to detach from the surface of the slide (a transition cTL(1 . 0)) with a probability . . [0, 1]. For this reason, . is called the evaporation coe.cient.
n i=1 ni
When a site is unoccupied (i.e., ni = 0), the second term, µ(1 - ni)1 -, is active.
N
This term is called the deposition term because it allows a particle to attach to the surface of the slide at the unoccupied site (a transition cTL(0 . 1)). The probability of deposition relies upon the deposition coe.cient µ . [0, 1]. Unlike the evaporation coe.cient ., µ changes as the number of particles on the slide changes. This behavior is important because charged nanoparticles experience electrostatic repulsion when in proximity to one another. The CSAE-TL model considers the
previously
deposited
particles
as
one
single
charge
screen
(see
Figure
2.3).
Thus,
the
probability
of deposition, which begins at µ when no particles have been deposited, decreases as the fraction
n i=1 ni
of occupied sites N increases. (N indicates the total number of sites on the lattice.)
While the CSAE-TL model fails to account for the distribution of charge on the surface of the slide, it produces an analytically solvable equation for steady state particle coverage density. This makes the CSAE-TL modeling method particularly useful in situations where exact charge distribution detail is not necessary or when numerical solution methods are not available.
Converting the CSAE-TL transition rate into an equation for coverage density requires the use of mean .eld theory, a collection of approximative methods from statistical physics (see Appendix A for more detail). By assuming that edge e.ects are negligible in the interior of the slide and that particles tend to distribute themselves evenly across the slide, we reason that particle coverage density, ., does not depend upon location on the slide. Thus, mean .eld theory’s assumption that each site will have approximately the same number of occupied neighbor sites and feel the same e.ect is valid. This allows us to replace the local ‘.eld’ felt by each particle with an overall mean .eld that is felt by the entire lattice.
Mathematically, we use the mean .eld approximation as follows. We begin with the partial
di.erential equation
. ..
. ni .t = -. ni + µ (1 - ni) 1 - n i=1 ni N (2.2)
where ni indicates the mean individual site occupation. Because we assert that neighboring sites are uncorrelated, we can assume that the ensemble average of nearest neighbor correlations is approximated by the product of the mean individual site occupations. Mathematically, this means that
ninj = ni nj , (2.3)
where ni and nj are any two neighboring sites. This equality reduces our equation as follows:
n. ni i=1 ni
= -. ni + µ(1 - ni )1 - . (2.4)
.t N Technically, each site ni has
a
di.erent
rate
equation
like
the
one
expressed
in
Equation
(2.2).
However, if we assume every site is relatively the same, we can reason that ni = n . (2.5) This assumption removes the site-speci.c nature of our rate equations, allowing us to apply the same equation for all sites. Finally, we de.ne particle density . as ni
. =. (2.6)
N
This gives us the following partial di.erential equation for particle coverage density:
..
2
= -.. + µ(1 - .). (2.7)
.t
s
38.2%.
Equation
(2.7)
suggests
that
. changes over time. Using the odeint function from Python’s SciPy package (see Appendix B.1 for a description of odeint and Appendix C.1 for a detailed look at
our
program),
we
can
produce
plots
depicting
the
coverage
density
over
time.
Figure
2.4
is
an
example of such a plot. In this case we use . = µ =0.5. Notice how the particle coverage density (which we have denoted as a percentage of slide area) rapidly increases until a steady state is achieved. The particle coverage density at steady state .can be easily calculated without the use
s of numerical solving methods like odeint by simply setting .. = 0
in
Equation
(2.7)
and
solving
.t
for .:
2µ ± .(4µ + .)+ .
.= . (2.8)
s
2µ
When . = µ =0.5, we calculate .˜ 38.2%, which matches the steady state achieved in Figure
s
1
2.4.Figures
2.5
and
2.6
show
how
.varies as . and µ change, respectively. From these plots, we
s
make three primary observations. First, the steady state coverage density .decreases steadily as
s
the evaporation coe.cient . increases. This mathematical behavior mirrors what we would expect physically: a higher chance of particle detachment corresponds to a lower steady state coverage density. Second, .increases as the deposition coe.cient µ increases. This result also makes sense
s
physically, as a higher likelihood of particle deposition should correspond to a higher steady state
1The second solution, .s ˜ 262% is discarded because it describes a non-physical scenario.
s
alters the value of .but not the shape of the response curve.
s
s
alters the value of .but not the shape of the response curve.
s
coverage density. Third,
we
note
that
the
slopes
of
the
curves
in
both
Figure
2.5
and
Figure
2.6
are steeper for lower values of . and µ, respectively. This non-linear behavior indicates that small changes in . or µ have a greater impact when these values are lower on the lower end of their domain than when they are on the higher end.
2.2.2 CSAE-NN
The transition rate for the CSAE-NN2
model
[41,
48]
is
given
in
Equation
(2.9):
.
cNN (ni . (1 - ni)) = ni. + (1 - ni)aß. (2.9)
When a site is occupied (i.e., ni = 1), the .rst term, ni., is active. Like in the CSAE-TL model, this term is called the evaporation term because it allows the particle to detach from the surface of the slide (a transition cTL(1 . 0)) with a probability . . [0, 1]. For this reason, . is once again called
.
the evaporation coe.cient. When a site is unoccupied (i.e., ni = 0), the second term, (1-ni)aß, is active. This deposition term allows a particle to attach to the surface of the slide at the unoccupied site (a transition cTL(0 . 1)); however, it functions quite di.erently from the deposition term found in the CSAE-TL model. The probability of deposition is now governed by three di.erent parameters: a, ß, and .. a . [0, 1] describes the probability of deposition when no nanoparticles are present on the slide surface. Like the decrease from µ in the CSAE-TL model, the probability of deposition will decrease from a as more particles occupy the slide. However, the CSAE-NN model takes a more detailed approach, decreasing the deposition probability from a according to the number of deposited particles neighboring the site in question (hence the nearest neighbors model name). This approach ensures that local variations in the electrostatic force created by previously deposited
nanoparticles
are
taken
into
account
(see
Figure
2.7).
Two parameters are needed to describe this variable deposition probability. ß = 0 describes the strength of the electrostatic force which causes individual SiO2 particles to repel one another. . = j.NN nj quanti.es the number of sites neighboring site ni. As the number of occupied neighboring sites increases, the strength of electrostatic repulsion increases exponentially. Thus, the CSAE-NN model describes a scenario in which particle deposition becomes increasingly di.cult as a particular region of the slide becomes more occupied with nanoparticles
Like with the CSAE-TL model, the transition rate for the CSAE-NN model can be converted into a partial di.erential equation describing the change in particle coverage density over time. Once again, the mean .eld approximation is integral to this conversion. We begin with the following rate equation:
.ni.
= -.ni + (1 - ni)aß. (2.10)
.t
Because the application of mean .eld theory allows us to approximate the correlations between
As a reminder to the reader, CSAE-NN stands for cooperative sequential adsorption with evaporation using the
neighboring sites as
ninj = ni nj , (2.11)
Equation (2.10) reduces to
. ni .t = -. ni + (1 - ni )a ß. . (2.12)
nearest neighbors approach.
By performing a Taylor series expansion on ß. , we can show that, to a .rst approximation, . (.)
ß= ß:
1
..lnß 2
ß= e =1+ .lnß +(.lnß)+ ···(2.13)
21
=1+ . lnß + .2 ln2ß + ··· (2.14)
2 1
=1+ . lnß + . 2ln2ß + ··· (2.15)
2
(.)lnß
= e(2.16)
(.)
= ß. (2.17)
Equation
(2.12)
now
becomes
.ni (.)
= -.ni + (1 - ni )aß. (2.18)
.t
Like with the CSAE-TL model, we eliminate the need for site-speci.c equations by assuming that ni = n . For the CSAE-NN model, this also means that . = zn , where z is the mean number of the nearest neighbors for each site.3
De.ning the particle coverage density . once again as
ni
. = , (2.19)
N
3For a square lattice, z = 4.
we arrive at
..
z.
= -.. + (1 - .)aß. (2.20)
.t
Using odeint,
we
can
produce
plots
describing
the
numerical
solution
to
Equation
(2.20).
Figure
2.8
is
an
example
of
such
a
plot,
produced
when
a = . =0.5, ß = 1, and z = 4. Once again,
s
the coverage density increases rapidly until steady state is achieved. Unlike with the CSAE-TL model, however, the CSAE-NN model does not have an analytic solution for the coverage density at steady state .s. This does not mean that an analytic approximation is not possible. To .nd such an approximation, we begin by setting .. =
0
in
Equation
(2.20):
.t z.
s
0= -..+ (1 - .)aß. (2.21)
ss
This yields the transcendental equation
z.
aßs
.= . (2.22)
s z.
. + aßs
This equation can be solved graphically. Analytically, however, we can take the Taylor series expansion about ß = 1:
..
s
.= .(ß = 1)+(ß - 1) + ··· . (2.23)
ss
.ß
ß=1
In our approximation, we keep only the constant and linear terms. If we assume that .(ß = 1) =
s
a
.+a , we arrive at
2
a aa
.= - (1 - ß)4 1 - . (2.24)
s
. + a. + aa + .
For
our
example,
Equation
(2.24)
predicts
a
particle
coverage
density
of
.˜ 50%, which is ap
s
proximately
the
same
as
the
steady
state
predicted
by
our
numerical
solution
(Figure
2.8).
Figures
2.9
and
2.10
demonstrate
how
.changes as . and ß change, respectively. Like with
s
s
ß alters the value of .but not the shape of the response curve for ß = 1. When ß> 1 (e.g.,
s
the orange curve) the .response exhibits a maximum. The orange (ß =2.0) curve represents an
s
over-packing scenario, which occurs when previously deposited particles attract particles that have not yet been deposited.
Figures
2.5
and
2.6,
these
plots
show
a
decrease
in
particle
coverage
density
.as the evaporation
s
coe.cient . increases. Conversely, .increases as the deposition coe.cient ß increases. These
s
results mirror our physical expectation, as a higher evaporation tendency should reduce the slide’s coverage density while a higher deposition tendency should increase coverage density.
In
Figure
2.9
we
include
an
extra
curve
plotted
when
ß =2.0. This helps to demonstrate the physical correspondence of the ß factor and its distinction from a. While a describes the probability of particle deposition before any particles have attached to the slide (identical to µ in the CSAETL model), ß describes the interactions between particles on the slide and particles attempting to deposit. If ß< 1
(red,
green,
and
blue
curves
in
Figure
2.9),
particles
present
on
the
slide
repel
Figure 2.10: The ß response of steady state particle coverage density calculated using our CSAENN model. Notice that a higher ß value corresponds to a higher .value. Also notice that changing
s
. alters the value of .but not the shape of the response curve. The response curves are linear
s
because we only include the .rst two terms in our Taylor series expansion.
depositing particles, with values closer to ß = 1 corresponding to a lower repulsive force. If ß = 1, there is not interaction between particles on the slide and depositing particles. Finally, if ß> 1 (orange
curve
in
Figure
2.9),
particles
on
the
slide
attract
depositing
particles.
This
attraction
not
only allows a coverage density greater than 100%, a scenario we call over-packing, but also produces a maximum particle coverage density. While the particles used in our experiment do not exhibit ß> 1 behavior, the model remains signi.cant in several other applications, including extensions to voter models and other scenarios investigated in the social sciences.
Of
note
in
Figure
2.10
is
the
linear
behavior.
This
is
the
result
of
our
use
of
only
the
constant
and linear terms in our Taylor series approximation for .. We will discuss the limits of including
s
only these terms in Chapter 4, where we compare our model and simulation results.
2.3 Simulation Techniques and Results
The use of the mean .eld approximation to convert our CSAE-TL and CSAE-NN transition rates into numerically solvable partial di.erential equations for particle coverage density depends upon several assumptions. Perhaps the most important is the supposition that each constituent particle experiences the same overall average electric .eld (i.e., a mean .eld). To verify this method, we must compare our models’ results to experimental data. As we will explore in Chapter 2.4, physical samples of nanoparticle bilayers produced via ISAM are necessarily time consuming to produce. Thus, one e.cient way to test the validity of our use of the mean .eld approximation is via computer simulation.
As
originally
published
in
[46],
we
produce
two
Python
simulations
of
the
ISAM
process,
one
that
performs
site
transitions
according
the
CSAE-TL
transition
rate
(Equation
(2.1))
and
one
that
performs
site
transitions
according
the
the
CSAE-NN
transition
rate
(Equation
(2.9)).
The
syntax
governing each Python program is shown in detail in Appendix C.3. Here, we discuss the general methodology surrounding our simulation technique, which is based on the ubiquitous Monte Carlo method.
Throughout our simulation e.orts, we attempt to replicate attachment and detachment on a slide with (m×n) attachment sites. Each simulation begins by collecting values for the appropriate evaporation and deposition coe.cients (., µ, a, ß, ., etc.). The program then creates an (m +2 ×
n + 2) matrix:
.
.
......
···
d0,0 d0,1 d0,n d0,n+1
···
d1,0 d1,1 d1,n d1,n+1
.. ..
.
. ... .
.
.. ..
dd··· dd
m,0 m,1 m,n m,n+1
dd··· dd
m+1,0 m+1,1 m+1,n m+1,n+1
......
D =
(2.25)
.
The entries found in submatrix
.
.
..
···
d1,1 d1,n
..
.
. . . . . .
dm,1 · · · dm,n
..
Dsub
=
(2.26)
represent the sites (ni) found on the simulated slide. Meanwhile, the exterior entries {d0,0, d0,1, ··· , d0,n, d0,n+1}, {dm+1,0, dm+1,1, ··· , dm+1,n, dm+1,n+1}, {d0,0, d1,0, ··· , dm,0, dm+1,0}, and {d0,n+1, d1,n+1, ··· , dm,n+1, dm+1,n+1} represent nanoparticles within the SiO2 suspension that are in the
same plane as the surface of the slide.4
Before simulation begins, every entry in Dsub is assigned a value 0, while every exterior entry is assigned a value of 1. This attempts to replicate the initial state of the system: when the slide is exposed to the nanoparticle suspension, it is completely unoccupied (every entry di,j = 0), while, to a .rst approximation, the suspension is concentrated enough to be completely occupied (every site in the suspension holds a value of 1). Before every simulated time step, the program .rst sets every exterior entry to a value of 1.5
It then cycles through every entry in Dsub. At each entry, it .rst generates a random number R . [0, 1]. Next, it determines whether the entry is occupied (di,j = 1) or unoccupied (di,j = 0). Then, using the appropriate evaporation and deposition coe.cients and either the CSAE-TL or CSAE-NN transition rate, the simulation calculates a value P . [0, 1]. When P = R, the site changes state, representing either evaporation or deposition depending on its original state. When P >R, the site maintains its original state. At the end of each time step, the simulation program calculates the simulated slide’s coverage density by dividing the number of occupied entries in Dsub by the total number of entries in Dsub.
4While tracking nanoparticles from the suspension which occupy the same plane as the slide is not particularly important when simulating ISAM, it becomes especially important in our simulations of nanoparticle assembly under parallel electric .elds (Chapter 3.3). For this reason, we include these elements in our simulation.
5Again, this becomes important under parallel electric .elds, when the exterior entries can change state. We assume that the suspension is concentrated enough to ensure that there is always a nanoparticle available on the exterior of the slide.
Our simulations produce several di.erent outputs. The .rst is a plot of the coverage density versus time step, an example of which is shown in
Figure
2.11.
Notice how the shape of this
simulation output is similar to the shape of the numerical solutions we produced using our mean .eld approximation-derived rate equations. The slide’s particle coverage density rapidly increases until steady state is achieved. However, the steady state in the simulation result shows a slight variance around an average steady state value. This presumably indicates that the simulation is better at demonstrating small .uctuations in particle coverage density as evaporation and deposition continue to occur on the slide’s surface after steady state has been achieved.
The second output type is a static image of the slide’s state at the end of any time step. These images are produced using MatPlotLib’s imshow function, which converts entries in a matrix into
an
image.
Figure
2.12
is
an
example
of
one
of
these
images.
The
blue
cells
represent
sites
that
are occupied by a nanoparticle (di,j = 1), while the white cells represent unoccupied sites (di,j = 0). Note that, like with the matrix D, only the interior cells represent the slide’s surface. The exterior cells (all blue/occupied in our example image) represent particles in the suspension.
The third output type is an animation showing how the slide’s occupation state changes over time. We produce this animation using the ArtistAnimation function from MatPlotLib’s animation class. While this method is described in more detail in Appendix C.5, we note here that the function simply collects individual static images and plays them in succession. Thus, by producing a static image of the simulated slide’s state at the end of each time step, ArtistAnimation allows us to easily produce an animation of the entire adsorption process. An example of one of Figure 2.12: A sample (20 × 20) grid produced via a simulation of the CSAE-NN transition rate. Notice that all sites on the outer edges (the suspension) are occupied (blue), while internal sites (the
slide
surface)
are
either
occupied
or
unoccupied
(white).
(Reproduced
from
[46].)
our animations can be found at https://youtu.be/nQiVCYc1epk.
Figures
2.13
and
2.14
demonstrate
how
the
steady
state
particle
coverage
density
.changes
s
as . and µ vary
for
our
simulation
of
the
CSAE-TL
model,
while
Figures
2.15
and
2.16
demonstrate
how .changes as . and ß vary for our simulation of the CSAE-NN model. We carried out all
s
simulations on a 100×100 grid over 1000 time steps. In Chapter 4, we compare in detail the results of
our
models
(Figures
2.5
-2.6
and
2.9
-2.10)
and
our
simulations.
2.4 Experimental Techniques and Results
Using ISAM to produce thin .lms in the laboratory setting is a time consuming process. Before sample production, we .rst prepare suspensions of PDDA and SiO2 nanoparticles by mixing stock suspension with de-ionized (DI) water. We determine the ideal concentration for each investigation via a series of qualitative experiments, in which we image samples produced using a wide variety of PDDA and SiO2 concentrations. We select the combination of concentrations which result in the most uniform distribution of particles. We allow every suspension to stir for at least 24 hours before deposition. This stirring helps to ensure a uniform distribution of the stock (solute-like material) throughout the DI water (solvent-like material).
After preparing our suspensions, we clean our glass microscope slides using the AcetoneMethanol-Isopropol
Alcohol
(AMI)
method
[49].
We place each slide in a staining jar .lled with acetone and allow it to soak under sonication for .ve minutes. We then move the slides to a staining jar .lled with methanol for another .ve minutes of sonication, followed by a staining jar of isopropol alcohol for yet another sonication cycle. The now-cleaned slides .nally move to a DI water bath, where they remain until they are dried under a gentle .ow of N2 gas. Drying under a .ow of N2 gas prevents spotting, as well as the introduction of dust particles from the atmosphere.
Sample production then proceeds as follows: for each sample, we .ll one staining jar with unused PDDA suspension, one with unused SiO2 suspension, and two with unused DI water. We dip a cleaned slide in the PDDA suspension, allowing it to soak. We then quickly move the slide to one of the two DI water jars, where we gently stir it. Following this rinsing step, we dip the slide in the SiO2 jar, where it again soaks. We rinse the slide in the remaining DI water jar, leaving it in the DI water until we are ready to dry the sample under a gentle .ow of N2 gas. As during the cleaning procedure, the N2 gas prevents spotting as well as the deposition of particulates from the atmosphere.
Figure
2.1
shows
this
process
schematically.
The time periods used for each dipping and rinsing step vary depending upon the requirements of each investigation.
Determining the steady state particle coverage density of each sample requires the production of several images of the sample’s surface. Throughout our investigation, we use scanning electron microscopy (SEM) as our imaging method of choice. SEM requires conductive samples. Therefore, before we image our samples, we .rst coat them in gold using a Cressington 108 plasma sputter coater. Under vacuum, the sputter coater bombards a thin sheet of gold with argon atoms, producing a plasma which uniformly coats the sample.
We coat each of our samples two times for one minute. We .nd that this coating procedure produces clear, analyzable SEM images of our particles. We use carbon tape to mount the sample onto the SEM stage, ensuring that a portion of the tape wraps around to the surface we intend to image. Wrapping the tape onto the imaging surface provides a conductive path for electrons that often build up on the surface of the slide. This buildup of electrons, often called “charging” of the sample, re.ects electrons released from the SEM’s electron gun, obscuring the image.
We image each gold coated sample using a EVO MA-15 scanning electron microscope, manufactured by Zeiss. Our images are produced via the detection of secondary electrons in a high vacuum. In
particular,
we
use
the
speci.cations
listed
in
Table
2.1.
To ensure that we capture a representative sample of particle coverage densities across the entire slide, we often perform a strati.ed random sample, in which we divide the slide into 16 individual strata and use a random number generator to determine which strata to image.
EHT Target 20.00 kV
Aperture Size 20.00 µm
Collector Bias 300 V
Signal SE1
Spot Size 150
Table 2.1: SEM settings used throughout our experimental investigation.
Figure
2.17
is
one
of
the
many
SEM
images
we
produced
throughout
our
investigation.
Notice
how the nanoparticles in this image exhibit high contrast with the background, a characteristic that will aid in our calculation of the particle coverage density. To determine particle coverage density, we use ImageJ, a Java-based image processing and analysis suite originally designed by the National Institutes of Health. Due to the high contrast in our SEM images, we can easily use ImageJ’s Make Binary process to convert our image into a black and white equivalent without the loss
of
signi.cant
detail
(see
Figure
2.18).6
The program’s Analyze Particles process then easily identi.es the particles and calculates the particle coverage density. Figure
2.19
is
an
example
of
the visual output provided by the Analyze Particles process.
6We also Invert the image to ensure that the particles are represented by black regions while the slide surface is represented by white regions.
The results of our experiments are published
in
[50].
Our .rst investigation focuses on determining the relationship between a slide’s steady state particle coverage density and the concentration of the nanoparticle suspension. We produce and image samples for each suspension concentration
using
the
methods
outlined
above.
Table
2.2
provides
an
overview
of
the
experimen
tal
conditions,
and
Figure
2.20
summarizes
our
results.
SiO2 Nanoparticles SNOWTEX ST-20L from Nissan Chemical
Suspension pH 10.3
Suspension Temperature 21 .C
PDDA Concentration 10 mM
Dipping Time (per step) 10 min
Rinsing Time (per step) 1 min
Table 2.2: Experimental conditions during our investigation of the relationship between particle coverage density and suspension concentration.
By
.tting
the
CSAE-TL
model
to
our
experimental
results
(see
red
curve
in
Figure
2.20),
we
.nd that we can express the deposition coe.cient µ in terms of the evaporation coe.cient . and concentration C as follows:
-./2C
µ = e. (2.27)
This equation maintains its validity when . is low (. ˜ 0.1). It generally performs better at low concentrations.
Our second investigation seeks a thorough description of the time dependence of the deposition process. While
previous
investigations
[27,
36]
thoroughly
describe
the kinematics of nanoparticle deposition, they do not provide an adequate investigation of the time scales necessary to achieve steady state. Theoretically, electrostatic screening causes the deposition process to divide into two distinct regimes: a rapid Langmuir-type adsorption followed by a slow approach to the maximum particle coverage density. [27]
suggests
a
90%
saturation
of
a
PDDA
monolayer
with adsorbed silica in 10 s. However, because these results were collected over longer time periods (on the order of seconds), they failed to capture information about how full saturation is achieved. By imaging particle coverage densities that occur during deposition times less than 2 s, we hope to characterize this portion of the deposition process, especially the amount of time needed to transition from Langmuir-type adsorption to slow adsorption.
To obtain this characterization, we begin by dipping cleaned slides in a 10 mM suspension of PDDA for 10 min. We follow the PDDA dip with a 1 min rinse in DI water and then dry the slides under a stream of N2 gas. Each slide is pre-marked with small, regularly spaced reference dots. As we dip them into the SiO2 suspension, we .lm the process with a high-speed camera at 1000 frames/s. This allows us to review the footage and determine the exact time each pre-marked point spends in the suspension, to a precision on the order of milliseconds. We collect 11 data points, ranging in exposure time from 0.058 s to 0.639 seconds. We image each using the SEM and calculate the coverage density using ImageJ.
Figure
2.21
shows
our
experimental
results,
along
with
the
predictions
of
the
CSAE-TL
model
(µ =0.7 and . =0.2). The results show that approximately half of the particles are deposited
in the .rst 0.058 s. This corresponds with the rapid Langmuir-type adsorption. Between 0.058 s and 0.639 s, adsorption continues slowly, and the system eventually transitions smoothly to steady state. This time interval corresponds to the expected slow approach to maximum particle coverage density. Overall,
the
system
achieves
steady
state
much
faster
than
in
[27].
We
believe
this
occurs
because their results were produced using a nanoparticle suspension containing NaCl, while our suspension lacked NaCl. When NaCl is present, the attractive force between the PDDA and SiO2 molecules is reduced since counterions can form. This reduced attractive force should slow down the deposition process.
Chapter 3
Self-Assembly under External Electric Fields (DSAM)
While the ability to model, simulate, and produce thin .lm samples via ISAM is of great importance to nanoscientists, there remains much to be learned about the e.ect of external bias on the assembly process. Particles could undergo self-assembly in a variety of environments, including those with strong electric .elds, strong magnetic .elds, or even intense mechanical vibrations. In this chapter, we explore attempts to model, simulate, and produce thin .lms formed under the in.uence of electric .elds. In doing so, we develop a new production technique, dubbed Directed Self-Assembly of Monolayers, or DSAM, in which the particle coverage density of a thin .lm sample is controlled by the strength of an applied electric .eld. The ability to easily control particle coverage density promises to be of great importance to researchers and manufacturers alike.
3.1 Overview of Electric Fields, Electric Potentials, and Capacitors
Before discussing two distinct electric-.eld driven DSAM techniques, we .rst brie.y review the
physics
of
electric
.elds
and
potentials
[51],
as
well
as
parallel
plate
capacitors
[52].
Like
any
classical .eld, an electric .eld pervades all space. It originates from any object that possesses electric charge, typically denoted with the variable Q. Likewise, only an object with an electric charge may experience an electric .eld. Mathematically, the electric .eld is perhaps best understood in relation to the electric force F. In general, we understand that a particle with charge Q experiences a force
e
F= QE (3.1)
e
when subjected to an electric .eld E. The electric .eld itself may be calculated one of two ways. If originating from a collection of point-like charges, the principle of superposition states that the .eld may be calculated by adding the .eld vectors of each individual charge. Mathematically, this technique is written as
n
1 qi
E(r) =
i, (3.2)
2
4pe0
i
i=1
where e0 is the permittivity of free space, qi is the charge of point-like charge i, and
= r - ri,
i
as
depicted
in
Figure
3.1.
If
originating
from
a
continuous
distribution
of
charge,
the
.eld
may
be
calculated via integration:
E(r) = 1 4pe0 1 2 ˆdq. (3.3)
dq can be written as
'
dq . .dl (3.4)
'
. sda (3.5)
'
. .dt , (3.6)
where . is the charge-per-unit-length, s is the charge-per-unit-area, and . is the charge-per-unit
'' '
volume, for linear (dl ), surface (da ), and volume (dt ) charge distributions, respectively.
Since the curl of an electrostatic .eld E is always 0 (i.e., V× E = 0), the electrostatic .eld is conservative and possesses a potential function V (r). We de.ne the electric potential as
r
V (r) =-E · dl, (3.7)
O
where O is some prede.ned reference point. Using this de.nition of electric potential, we say that there exists a potential di.erence between two points a and b:
b
V (b) - V (a)= -E · dl. (3.8)
a
The potential di.erence plays an important role in the calculation of the .eld of a parallel plate capacitor, the primary tool we use to produce electric .elds during DSAM. Additionally, we can easily calculate the electric .eld produced by an electric potential via
E = -VV. (3.9)
Equation
(3.9)
suggests
that
electric
.eld
lines
are
always
perpendicular
to
the
contours
produced
by the potential.
One common method of producing a constant electric .eld E is the use of a parallel plate capacitor.
As
depicted
in
Figure
3.2,
a
parallel
plate
capacitor
produces
a
uniform,
linear
electric
.eld, directed from the plate with higher electric potential to the plate with lower electric potential.1
The magnitude of an electric .eld E produced by a parallel plate capacitor is given by
V
E = , (3.10)
d where V is the potential di.erence between the plates and d is the distance between the plates.
3.2 Perpendicular Electric Fields
When applying an electric .eld during the ISAM process, the .eld’s orientation is critical to determining the modeling, simulation, and experimental techniques used. While any orientation is theoretically possible, we limit our discussion to .elds that are perpendicular and parallel to the surface of the glass slide used during the deposition process. In this section, we describe our e.orts to model, simulate, and produce experimentally samples that are subjected to perpendicular electric
.elds.
The
modeling
portion
of
our
treatment
is
published
as
[53].
This
scenario
is
by
far
the simplest, as negatively charged particles will be forced directly onto the slide when subjected
1Near the edges of the plates, the .eld is not truly uniform. However, since we only consider e.ects near the center of the capacitor throughout our work, we will treat the .eld as if it were perfectly uniform.
to a .eld oriented out of the surface of the slide. Conversely, they will be forced away from the slide when subjected to a .eld oriented toward the slide’s surface.
3.2.1 DSAM for Perpendicular Fields
The process for achieving directed self-assembly under a perpendicular electric .eld is similar to
the
traditional
ISAM
process
(see
Figure
2.1).
We again begin by dipping a clean microscope slide, which possesses and innate negative surface charge, in the standard polycation PDDA. This gives the slide a positive surface charge, which will allow negatively charged SiO2 nanoparticles to adhere to its surface. Just before dipping in the SiO2 suspension, however, we power a parallel plate capacitor, which is placed on the outside of the staining jar holding the suspension. As depicted in Figure
3.3,
the
plates
of
the
capacitor
are
oriented
parallel
to
the
surface
of
the
glass
slide.
This
orientation produces an electric .eld that is perpendicular to the slide’s surface. For a .eld that is directed away from the slide’s surface, which encourages negatively charged particles to travel toward the glass slide, the negative terminal of the power supply is attached to the plate closest to the surface to be examined. Conversely, for a .eld that is directed toward the slide’s surface, which encourages negatively charged particles to travel away from the glass slide, the positive terminal of the power supply is attached to the plate closest to the surface to be examined. We leave the capacitor powered until the end of the deposition process, ensuring that depositing/evaporating nanoparticles experience the e.ects of the uniform .eld at all times.
3.2.2 CSAE Modeling Techniques and Results
Like with ISAM, we once again use cooperative sequential adsorption with evaporation (CSAE) models to represent the assembly process mathematically. To account for the presence of the electric .eld, however, we alter our transition rates by replacing the evaporation term ni. with a new set of terms: .1(1 - ni)+ .2(ni). The .1(1 - ni) term represents particles being driven onto the surface of the slide (with probability .1) by the electric .eld. Similarly, the .2(ni) term describes particles being driven away from the surface of the slide (with probability .2) by the electric .eld. Both .1 and .2 can be scaled to the strength of the applied electric .eld (typically given in V/m) using experimental data, with higher values of .1 and .2 corresponding to stronger electric .elds.
With our new terms implemented, the transition rate for the CSAE-TL model becomes
n i=1 ni
cTL(ni . (1 - ni)) = .1(1 - ni)+ .2(ni)+ µ(1 - ni)1 - . (3.11)
N
Equation
(3.11)
can
be
described
via
the
partial
di.erential
equation
n
.ni
i=1 ni
= -.2 ni + .1(1 - ni )+ µ (1 - ni)1 - . (3.12)
.t N
Applying the mean .eld approximation, in which higher order correlations are approximated as
ni
ninj = ni nj and . = , yields
iN
..
2
= -.2. + .1(1 - .)+ µ(1 - .), (3.13)
.t
the mean .eld equation for the CSAE-TL model under the in.uence of perpendicular electric .elds. The steady state solution .to this equation, found when .. = 0, is then
s .t
22
± .1 +2.1.2 +4.2µ + .2 + .1 + .2 +2µ .= (3.14)
s
2µ
when µ .= 0 and
.2
.= (3.15)
s
.1 + .2
when µ = 0 and .1 + .2 .
= 0.
Likewise, the transition rate for the CSAE-NN model becomes
.
cNN (ni . (1 - ni)) = .1(1 - ni)+ .2(ni) + (1 - ni)aß. (3.16)
Equation
(3.16)
can
be
described
via
the
partial
di.erential
equation
.ni.
= -.2 ni + .1(1 - ni )+ (1 - ni)aß. (3.17)
.t
Using
the
mean
.eld
approximation,
Equation
(3.17)
becomes
..
z.
= -.2. + .1(1 - .) + (1 - .)aß. (3.18)
.t
Like with ISAM, the steady state equation for CSAE-NN is transcendental. We can again use a Taylor series expansion about ß = 1 to .nd an approximate solution, yielding
a2 - a.12
a + .1 a + .1
.= - (1 - ß)4 1 - . (3.19)
s
a + .1 + .2 a + .1 + .2 a + .1 + .2
Like
with
ISAM,
Equations
(3.13)
and
(3.18)
can
be
solved
numerically
using
odeint. Figures
3.4
and
3.5
show
how
steady
state
particle
coverage
density
changes
with
respect
to
µ (for the CSAE-TL case) and ß (for the CSAE-NN case) at several di.erent values of .2 .
= 0 when .1 = 0. This con.guration of .-values represents a .eld directed into the slide, which encourages negatively charged nanoparticles to detach. Figures
3.6
and
3.7
show
similar
information
when
.1 .0 and
= .2 = 0. This con.guration of .-values represents a .eld directed away from the slide, which encourages negatively charged nanoparticles to attach.
For both CSAE-TL and CSAE-NN, we see the steady state coverage density .increase as
s
the deposition coe.cients µ and ß increase. Like in our modeling of assembly under no electric .eld, these results mirror reality, in which a higher tendency to deposit corresponds to a higher steady state coverage density. In
Figures
3.4
and
3.5,
where
.1 = 0 and .2 .0, we see a higher
= .2 value correspond to a lower .value. This is because this scenario re.ects a .eld oriented so as
s
to encourage detachment. In
Figures
3.6
and
3.7,
where
.1 .0 and .2 = 0, both models predict
= that a 100% coverage density will be achieved for all values of µ. In this case, a 100% coverage density is achieved because .2 = 0 ensures that no particles can detach. A scenario in which a .eld encouraging detachment is superimposed over a system that also experiences evaporation would require a non-zero .2 in addition to the non-zero .1.
The
linear
behavior
in
Figure
3.5
is
once
again
the
result
of
our
Taylor
series
approximation
for ., in which we take only the constant and linear terms. Like with the no-.eld case, we will
s
discuss the limits of this approximation in Chapter 4.
Figure 3.4: The µ response of steady state particle coverage density calculated using our CSAETL model. Here, particles assemble under a perpendicular .eld encouraging particle detachment (.1 = 0, .2 .value. Also notice that
= 0). Notice that a higher µ value corresponds to a higher .
s
changing . alters the value of .but not the shape of the response curve.
s
Figure 3.5: The ß response of steady state particle coverage density calculated using our CSAENN model. Here, particles assemble under a perpendicular .eld encouraging particle detachment (.1 = 0, .2 .value. Also notice that
= 0). Notice that a higher ß value corresponds to a higher .
s
changing . alters the value of .but not the shape of the response curve. The response curves are
s
linear because we only include the .rst two terms in our Taylor series expansion.
Figure 3.6: The µ response of steady state particle coverage density calculated using our CSAETL model. Here, particles assemble under a perpendicular .eld encouraging particle attachment (.1 = 0, .2 = 0). Notice that the system achieves a steady state particle coverage density of .= 100% for all .1 values since a .eld of any strength will continue to drive particles onto the
s
slide surface until 100% coverage is achieved. The red and green curves for .1 =0.1 and .1 =0.5 are hidden under the blue curve representing .1 =0.9.
Figure 3.7: The ß response of steady state particle coverage density calculated using our CSAENN model. Here, particles assemble under a perpendicular .eld encouraging particle attachment (.1 .= 0, .2 = 0). Notice that the system achieves a steady state particle coverage density of .= 100% for all .1 values since a .eld of any strength will continue to drive particles onto the
s
slide surface until 100% coverage is achieved.. The red and green curves for .1 =0.1 and .1 =0.5 are hidden under the blue curve representing .1 =0.9.
3.2.3 Simulation Techniques and Results
The simulation technique for particles assembling under a perpendicular electric .eld is identical to the Monte Carlo procedure used for particles assemblying under no .eld. The only change in the simulation program is the use of the perpendicular electric .eld CSAE-TL and CSAE-NN transition rates for calculation of the value P , which we use to determine whether a particle site should transition states via comparison to a random number R. Figures
3.8
and
3.9
show
steady
state coverage densities across a wide range of µ and ß values, respectively, from simulations on a 100 × 100 grid over 1000 time steps when .1 = .0. These values represent a .eld
0 and .2 = oriented
so
as
to
encourage
particle
detachment.
Figures
3.10
and
3.11
show
steady
state
coverage
densities across of a wide range of µ and ß values, respectively from simulations on a 100 × 100 grid over 1000 time steps when .1 .0 and .2 0. These values represent a .eld oriented so as
== to encourage particle deposition. In Chapter 4 we compare these simulation results to our CSAE models
for
nanoparticle
self-assembly
under
perpendicular
electric
.elds
(Figures
3.4
-3.7).
= 0).
Figure 3.9: The simulated ß response (points) overlaid with the model-predicted ß response (curves) for CSAE-NN. Here, particles assemble under a perpendicular .eld encouraging particle detachment (.1 = 0, .2 .
= 0).
Figure 3.10: The simulated µ response (points) overlaid with the model-predicted µ response (curves) for CSAE-TL. Here, particles assemble under a perpendicular .eld encouraging particle attachment (.1 .
= 0, .2 = 0). The red points and curve for .1 =0.1 and the green points and curve for .1 =0.5 are hidden under the blue points and curve representing .1 =0.9.
Figure 3.11: The simulated ß response (points) overlaid with the model-predicted ß response (curves) for CSAE-NN. Here, particles assemble under a perpendicular .eld encouraging particle attachment (.1 .
= 0, .2 = 0). The red points and curve for .1 =0.1 and the green points and curve for .1 =0.5 are hidden under the blue points and curve representing .1 =0.9.
3.3 Parallel Electric Fields
The other natural orientation for our applied electric .eld is parallel to the surface of the slide. While the perpendicular orientation simply altered the evaporation term by forcing us to consider particles being driven into and away from the slide, the parallel orientation instead forces us to consider di.usion (i.e., the translational motion of particles on the slide surface) in addition to deposition and evaporation. In this section, we consider this scenario from the perspective of mathematical models, computer simulations, and experimental production methods, building on the
work
found
in
[46].
3.3.1 DSAM for Parallel Fields
The DSAM method for electric .elds parallel to the surface of the slide is almost identical to the DSAM method for electric .elds perpendicular to the slide’s surface. Like in the perpendicular case, as well as with ISAM, we begin by dipping a clean slide in the polycation PDDA. The now positively charged glass can attract negatively charged SiO2 nanoparticles. Immediately before dipping in the nanoparticle suspension, we power the parallel plate capacitor, which is now oriented perpendicular to the slide’s surface. This orientation creates a uniform electric .eld parallel to the surface
of
the
slide
(see
Figure
3.12).
3.3.2 CSAE Modeling Techniques and Results
Modeling the deposition, evaporation, and di.usion of particles assembling under a parallel electric
.eld
requires
reconsideration
of
the
grid
model
discussed
in
Chapter
2.2
(see
Figure
2.2).
While deposition and evaporation continue to follow the same rules as described by CSAE-TL and CSAE-NN for ISAM, di.usion demands that we consider the state of particle site immediately adjacent to the site under consideration. Figure
3.13
shows
why the consideration of adjacent particles is necessary. Under the in.uence of the electric .eld E, each negatively charged particle has
a chance of di.using to the neighboring site in the up-.eld direction. 2
This behavior is demonstrated by the red particle, which is attempting to move up-.eld to the immediately adjacent site. However, di.usion is forbidden if the site in the up-.eld direction is occupied since the electrostatic repulsion between the two particles will prevent the di.using particle from moving. This rule is demonstrated by the green particle, which cannot di.use into the adjacent site in the up-.eld direction due to the presence of another particle. Assuming di.usion occurs with a probability . (also scalable to the strength of the applied .eld using experimental data) that depends upon the strength of the
2In the case of positively charged particles, di.usion would occur in the down-.eld direction.
electric .eld, we describe this di.usive behavior mathematically using the transition term
.ni(1 - ni±1). (3.20)
Like in our other transition terms, ni represents the transitioning site. Thus, di.usion can only occur if ni = 1, which indicates that the site is occupied. ni±1 represents the immediately adjacent site in the up-.eld direction, with the ± sign indicating that the exact direction depends upon the orientation of the .eld. Thus, if the immediately adjacent site is occupied (ni±1 = 1) the di.usion term goes to 0 is and di.usion cannot occur. Conversely, if the immediately adjacent site is unoccupied (ni±1 = 0) the di.usion term goes to . and di.usion is permitted.
Adding our di.usion term to the transition rate for the CSAE-TL model yields
n i=1 ni
cTL(ni . (1 - ni)) = ni. + µ(1 - ni)1 - + .ni(1 - ni±1). (3.21)
N
This transition rate can be expressed as the partial di.erential equation
n
.ni
i=1 ni
= -.ni + µ (1 - ni)1 - + .ni (1 - ni±1 ), (3.22)
.t N
which under the mean .eld approximation becomes
..
22
= -.. + µ(1 - .)+ .(. - .). (3.23)
.t
Equation
(3.23)
represents
the
rate
equation
for
the
CSAE-TL
model
under
parallel
electric
.elds.
It suggests a steady state particle coverage density (.) of
s
22
.- 2.. +4.µ + .± . . . ± 2µ .= . (3.24)
s
2. - 2µ
when . .= µ and
µ
.= (3.25)
s
. + µ
when . = µ. If we add our di.usion term to the transition rate for the CSAE-NN model, we .nd
.
cNN (ni . (1 - ni)) = ni. + (1 - ni)aß+ .ni(1 - ni±1). (3.26)
As a partial di.erential equation, this transition rate becomes
.ni.
= -.ni + (1 - ni)aß+ .ni (1 - ni±1 ). (3.27)
.t
Under
the
mean
.eld
approximation,
Equation
(3.27)
becomes
..
z. 2
= -.. + (1 - .)aß+ .(. - .), (3.28)
.t
the rate equation for CSAE-NN under parallel electric .elds. Once again, the equation for the steady state particle coverage rate (.) is transcendental; however, we can approximate a value for
s
.by performing a Taylor series expansion about ß = 1. Using this method we .nd
s
a.(ß = 1) a
s
.= .(ß = 1) - (1 - ß)4 1 - , (3.29)
ss
a + . + ..(ß = 1) - .a + . + ..(ß = 1) - .
ss
where
2
-(. + a - .) ± (. + a - .)+4.a .(ß = 1) = . (3.30)
s
2.
odeint can
again
be
used
to
solve
both
Equation
(3.23)
and
(3.28).
In
Figures
3.14
and
3.15,
we show the change in steady state particle coverage density as µ (for CSAE-TL) and ß (for CSAE-NN) at several . values. Steady state coverage density once again increases as µ and ß increase, mirroring physical reality. Raising the di.usion coe.cient ., which corresponds to increasing the strength of the parallel electric .eld, seems to suggest elevated .values in both
s
models. Nevertheless, the .curve maintains its overall shape no matter the . value used. Like our
s
models for assembly under no electric .eld .eld and perpendicular electric .elds, the linear behavior of .under
the
CSAE-NN
model
(see
Figure
3.15)
is
a
result
of
our
Taylor
series
approximation,
s
which we discuss further in Chapter 4.
ss
response curve.
ss
response curve.
3.3.3 Simulation Techniques and Results
Like in our simulations of nanoparticle assembly under no electric .elds and perpendicular electric .elds, we use a Monte Carlo-based method to simulate nanoparticle assembly under parallel electric .elds. The algorithm contains one major di.erence, however. If the comparison of our test value P and random value R leaves a particular cell unchanged (i.e., if neither deposition nor evaporation occurs), the site can still change state via di.usion. If the unchanged cell is occupied, the program checks the cell in the up-.eld direction to determine its state. If this neighboring cell is occupied, no di.usion occurs. If it is not occupied, the program determines whether di.usion occurs by comparing the random value R to our di.usion probability .. If R = ., di.usion occurs, and the program changes the states of the unchanged cell and the neighboring cell in the up-.eld direction. If the unchanged cell is unoccupied, the program checks the cell in the down-.eld direction to determine its state. If this neighboring cell is unoccupied, no di.usion occurs. If it is occupied, the program determines whether di.usion occurs by comparing the random value R to our di.usion probability .. If R = ., di.usion occurs, and the program changes the states of the unchanged cell and the neighboring cell in the down-.eld direction.3
Figure
3.16
and
3.17
show
steady state particle coverage densities for simulated particle assembly following the CSAE-TL and CSAE-NN transition rates using a variety of . values. Each simulation was completed on a 100 × 100 grid over 1000 time steps. In Chapter 4, we compare
3Note that this description is appropriate for a simulation of negatively charged particles. All instances of “up.eld” and “down-.eld” would switch for a simulation of positively charged particles.
these
results
to
our
modeling
results
(Figures
3.14
and
3.15).
3.4 Oscillating Electric Fields
3.4.1 Overview of Oscillating Fields
Thus far, our treatment of electric .eld-driven DSAM has been limited to the use of constant .elds produced by a capacitor attached to a direct current (DC) power source. However, assembling particles can also experience oscillating .elds, easily produced by attaching a capacitor to an alternating current (AC) power source. For brevity and to remind the reader of their experimental origins, we refer to constant .elds as “DC .elds” and oscillating .elds as “AC .elds” throughout the following sections.
We
touch
on
the
consequences
of
assembly
under
AC
.elds
brie.y
in
[53].
In
this
case,
the
strength and orientation of the .eld changes over time. Thus, assembling nanoparticles will oscillate back and forth as the .eld changes direction. We can apply AC .elds to a system under assembly in any orientation. Furthermore, we can also superimpose AC .elds and DC .elds. For the sake of simplicity, our mathematical and experimental treatment focuses on four main scenarios:
1.
Perpendicular AC Field
2.
Parallel AC Field
3.
Perpendicular AC Field; Parallel DC Field
4.
Parallel AC Field; Perpendicular DC Field
Scenarios (3) and (4) are especially important because there is reason to believe that the initial kinetic energy imparted to assembling nanoparticles by a small AC .eld can aid in a DSAM process that is primarily driven by a DC .eld.
3.4.2 CSAE Modeling Techniques and Results
Like with our CSAE models for DSAM under DC .elds, we produce models for our selected AC .eld cases by constructing transition rates, converting them to partial di.erential equations, and applying the mean .eld approximation. This provides di.erential equations describing the particle coverage density . over time. We can solve these equations numerically for all times. In the steady state, we can also use algebraic manipulation (CSAE-TL) or Taylor series expansions (CSAE-NN) to solve for the approximate steady state particle coverage density .without the use
s
of numerical solving tools.
To introduce an oscillating .eld, our transition rates must include a new coe.cient, equivalent in nature to the evaporation (.), deposition (µ, a, and ß), and di.usion (.) coe.cients we have already discussed. As a reminder, each coe.cient represents the probability that a nanoparticle will attach/detach from location on a grid representing the slide surface. However, this AC-.eld coe.cient must be able to vary its strength in the same way that the voltage of the AC power source varies. For a sinusoidal power source, we have
.cos(.t + f), (3.31)
where . is the maximum value of the AC-.eld coe.cient (scaled to the amplitude of the power source), . is the angular frequency of the sinusoidal source, and f is the phase shift of the sinusoidal source. When the AC-.eld is not oriented perfectly perpendicular or parallel to the slide surface,
this coe.cient resolves into two components, which we call ..cos(.t + f) for the perpendicular component and ..cos(.t + f) for the parallel component.4
For clarity throughout our derivations, we use .. and .. explicitly.
For scenario (1), in which we only apply a perpendicular AC .eld, the transition rate for
5
CSAE-NNis
.
c(ni . (1 - ni)) = ni. + (1 - ni)aß+ ni..cos(.t + f). (3.32)
This equation assumes that the .eld only oscillates the state of a site that is already occupied. More complicated models could also consider oscillations of non-occupied sites. As a di.erential equation, this transition rate becomes
.ni.
= - ni . + (1 - ni)aß+ ni ..cos(.t + f). (3.33)
.t
The mean .eld approximation then produces
..
z.
= -.. + (1 - .)aß+ ... cos(.t + f). (3.34)
.t
Figure
3.18
shows
a
numerical
solution
for
this
equation
across
a
wide
range
of
times.
We
see
that
4We select . for the perpendicular .eld case because the coe.cient behaves like a time-varying evaporation coe.cient and . for the parallel .eld case because the coe.cient behaves like a time-varying di.usion coe.cient.
5The CSAE-TL model for this case is constructed by simply replacing the NN deposition term with the TL deposition
term
(see
Equation
(2.1)).
the system rapidly approaches steady state, just like in the no-.eld and DC .eld cases. However, because of the presence of the perpendicular AC .eld, the particle coverage density oscillates sinusoidally around an average steady state particle coverage density (in this case, .˜ 40%).
s
These are the results we expect, for, as the AC .eld oscillates, particles alternate between being driven into and away from the face of the slide, altering the coverage density periodically even in the steady state regime.
Scenario (2) describes the application of a parallel AC .eld. The CSAE-NN transition rate is
.
c(ni . (1 - ni)) = ni. + (1 - ni)aß+ ni(1 - ni-1)(1 - ni+1)..cos(.t + f). (3.35)
Like in the perpendicular case, we only permit oscillations of sites that are occupied. Furthermore, the two neighboring sites (ni-1 and ni+1) must also be unoccupied. If ni-1 and ni+1 were occupied, the electrostatic repulsion produced by the particles in these sites would inhibit oscillation. As a di.erential equation, this transition rate becomes
.ni.
= - ni . + (1 - ni)aß+ (1 - )(1 - )..cos(.t + f). (3.36)
ni ni-1 ni+1.t
After applying the mean .eld approximation, we have
..
z. 2
= -.. + (1 - .)aß+ .(1 - .)..cos(.t + f). (3.37)
.t
Figure
3.19
is
a
representative
numerical
solution
of
this
equation.
These
results
are
similar
to
the
results we obtained for the perpendicular AC .eld: the system rapidly approaches steady state, at which point the AC .eld forces the system to oscillate about an average steady state particle coverage
density
(in
this
case,
like
in
Figure
3.18,
.˜ 40%). The main distinction between the
s
parallel and perpendicular case is the amplitude of the coverage density oscillations. Even when we examine scenarios where .. = .., the amplitude is always smaller in the parallel .eld case. This is because, under a parallel .eld, particle coverage density can only increase or decrease as particles di.use onto or o. the slide at its edges. Under a perpendicular .eld, particles can be driven onto or away from the slide at any point, greatly increasing the degree to which the coverage density can change in steady state.
Scenario (3) includes both a perpendicular AC .eld and a parallel DC .eld. We construct the transition rate as follows:
.
c(ni . (1 - ni)) = ni. + (1 - ni)aß+ ni(1 - ni±1). + ni..cos(.t + f). (3.38)
Here, ni(1 - ni±1). represents the parallel DC .eld, which is only active for an occupied site with an unoccupied neighbor in the up-.eld direction. ni..cos(.t + f) represents the AC .eld and is only active for an occupied site. As a di.erential equation, we have
.ni.
= - ni . + (1 - ni)aß+ ni (1 - ni±1 ). + ni ..cos(.t + f). (3.39)
.t
The mean .eld approximation then gives us
..
z.
= -.. + (1 - .)aß+ .(1 - .). + ...cos(.t + f). (3.40)
.t
Figure
3.20
is
a
representative
numerical
solution
of
this
model.
These results are very similar to
Scenario
(1)
(see
Figure
3.18),
except
that
the
presence
of
the
parallel
DC
.eld
increases
the
average steady state particle coverage density value.
Finally, scenario (4) proposes a parallel AC .eld superimposed with a perpendicular DC .eld. The required transition rate is
.
c(ni . (1 - ni)) = (1 - ni).1 + ni.2 + (1 - ni)aß+ ni(1 - ni-1)(1 - ni+1)..cos(.t + f).
(3.41)
The terms (1 - ni).1 + ni.2 describe the perpendicular DC .eld, with the orientation determined by whether .1 or .2 is non-zero. The term ni(1-ni-1)(1-ni+1)..cos(.t+f) describes the parallel AC .eld. As a di.erential equation, we have
.ni.
= (1 - ni ).1 - ni .2 + (1 - ni)aß+ ni (1 - )(1 - )..cos(.t + f).
ni-1 ni+1
.t
(3.42)
The mean .eld approximation then produces
..
z. 2
= (1 - .).1 - ..2 + (1 - .)aß+ .(1 - .)..cos(.t + f). (3.43)
.t
Figure
3.21
and
3.22
are
numerical
solutions
for
this
equation
when
.1 = 0 and .2 = 0, respectively. Figure
3.21
shows
behavior
identical
to
Figure
3.19
since
the
perpendicular
DC
.eld,
which
is
driving
particles
away
from
the
slide,
has
the
same
e.ect
as
evaporation.
However,
Figure
3.22
shows
that,
when a parallel AC .eld is combined with a perpendicular DC .eld (driving particles into the slide) that is strong enough to achieve .= 100%, oscillations about .fail to appear.
ss
3.5 Experimental Techniques
Thus far, our e.orts to produce bilayers under the in.uence of perpendicular and parallel electric .elds in the laboratory have focused on experiment design. In general, we use the same steps
outlined
in
Figure
2.1
and
described
in
detail
in
Chapter
2.1:
(1)
we
dip
a
clean
slide
into
a
suspension of PDDA, (2) we rinse the PDDA-coated slide in DI water, (3) we dip the rinsed slide into a suspension of SiO2 nanoparticles, (4) we rinse again in DI water, and (5) we dry under a gentle .ow of N2 gas. However, for the application of an electric .eld during deposition, we must modify the staining jar holding the nanoparticles. By placing a sheet of copper tape on the two faces parallel to the slide, we form a capacitor which delivers a uniform electric .eld perpendicular to the slide face. By placing a sheet of copper tape on the two faces perpendicular to the slide, we form a capacitor which delivers a uniform electric .eld parallel to the slide face. This is shown schematically
in
Figures
3.3
and
3.12.
Figure
3.23
shows
one
of
our
laboratory
beakers,
which
contains copper tape sheets on all sides. Thus, we can select a perpendicular or parallel .eld arrangement by simply attaching our electrical connections to the appropriate pair of sides.
The two strips of tape form a capacitor when attached to a DC power source. Just before we dip the slide into the SiO2 suspension, we activate the DC power source, which we previously set to the desired voltage level. Activating the DC power source just before deposition begins is important, for, if the capacitor is charged long before deposition, a gradient of SiO2 nanoparticles can form within the suspension. Such a gradient would obscure our results.
Currently, we are investigating the electric .eld strength needed to a.ect the particles and produce DSAM. As we discuss in Chapter 3.1, the strength of the electric .eld is given by
V
E = . (3.44)
d
Thus, we can increase the electric .eld by either increasing the voltage V applied to the capacitor or decreasing the separation d between the capacitor plates. We have made preparations to take both approaches. To increase the applied voltage, we have acquired a BT-GP-10N30 power supply from Advance Energy, which produces DC voltages between 0 and 10,000 V. With our standard staining jar, which has a width of 39.3 mm, this supply can produce a uniform .eld between 0 and 254 kV/m. To decrease the plate separation, we have produced several specialty staining jars using a 3D printer.
Applying AC electric .elds during the assembly process is identical to applying a DC .eld. However, instead of a DC power source, we connect a function generator (SRS DS335) to the capacitor plates. Most commercially available function generators do not produce a peak voltage with enough strength to encourage particle oscillations. This problem can be recti.ed by introducing an ampli.er (Trek 2205), which increases the peak voltage of the function generator without altering the angular frequency and shape of the signal. Like with deposition under DC .elds, our e.orts to produce nanoparticles bilayers under the in.uence of an AC .elds is currently in the experiment design stage.
Chapter 4
Discussion: Comparing CSAE Models and Simulated Data
The utility of our CSAE models depends upon their ability to predict the coverage densities of simulated and experimentally produced bilayer samples. Because our experimental work is ongoing, we consider here only the .t of our models to simulated data. Furthermore, we focus primarily on no-.eld and DC-.eld cases, as, to date, these have received the most attention. We conduct our
comparison
using
a
graphical
approach
which
relies
upon
residuals
[54].
For
any
variable
x (in our case, µ for CSAE-TL models and ß for CSAE-NN models), the residual ei is simply a vector containing the di.erences between the simulated/experimental values yi and the values produced by the model f(xi):
ei = yi - f(xi). (4.1)
By plotting the residual vector against xi on a scatter plot, we can examine the resulting shape to determine the closeness of the model .t.
Figures
4.1
-4.8
are
the
residual
scatter
plots
for
our
models.
A
perfect
model
should
produce
a residual of zero for every data point comparison. The residual scatter of a strong, predictive model should demonstrate no structure; rather, the points should be distributed stochastically around zero. Finally, the residual of a poor model shows a non-stochastic mathematical structure. This structure indicates the model has failed to capture some aspect of the examined variable’s behavior. Visual inspection of the residual scatter plots of our models shows that the CSAE-TL models for no electric .eld and perpendicular electric .elds have strong predictive capabilities across all values of µ. There is a slight positive trend (i.e., stochastic behavior appears to be centered slightly above
-4 -3
zero); however, the order of magnitude (10-10) makes this discrepancy insigni.cant. Our CSAE-TL model for parallel .elds and CSAE-NN models demonstrate a distinct structure. This suggests that the mean .eld approximation has failed to capture every aspect of the behavior of µ (for CSAE-TL) of ß (for
CSAE-NN).
We
expect
this
behavior
because,
according
to
[55],
mean
.eld theory is only capable of capturing the qualitative behavior of a many-particle, multi-state system. A full treatment, which includes interactions between particles, is necessary to develop an quantitatively accurate model. In the case of particle assembly, analytical models of this type are impossible. Thus, we must be satis.ed with models of the type we have developed, which, as
shown
in
Figures
2.14,
2.16,
3.8
-3.11,
and
3.16
-3.17,
can
predict
the
general
trend
of
the
data (e.g., whether it is increasing or decreasing; the response of secondary variables like ., .1, .2, and .; etc.). For calculations of particle coverage density that do not require high precision, this qualitative agreement suggests that our models are useful, especially when simulation techniques are not available or are prohibitively time consuming.
Further
visual
inspection
of
Figures
2.14,
2.16,
3.8
-3.11,
and
3.16
-3.17
seems
to
suggest
that our CSAE models conform more closely with simulated data for higher values of µ and ß (i.e., for µ, ß = 0.6). To test this hypothesis, we generate new residual scatters for µ, ß = 0.6 only
(Figures
4.9
-4.16).
These
plots
show
a
slight
improvement
in
the
performance
of
our
models
at higher values of µ and ß. While purely stochastic behavior fails to appear, especially in the CSAE-NN models, the order of magnitude of the residuals falls. Therefore, we can con.dently say that our models, while still qualitative in nature, perform better when µ, ß = 0.6.
There are a number of potential improvements that could be made to our models. First, including higher order terms in the Taylor series expansions of our CSAE-NN models has the potential to produce mathematical descriptions that better capture the response of .to ß. Second,
s
we could consider a more sophisticated version of mean .eld theory, such as the Bethe approximation [55].
The
most
accurate
description
of
a
many-particle,
multi-state
system
allows
each
particle
to
interact with every other particle in the system. This is especially true of ISAM models since every particle experiences a di.erent electrostatic force from every other particle in the system during assembly. Mathematical treatments that include every interaction are typically unsolvable. Simple mean .eld theories, such as the one we have employed, assume that each individual particle experiences an e.ective .eld produced by the other particles instead of individual interactions (see Appendix A for more detail). While this description is mathematically tenable, it eliminates a signi.cant amount of physical information because interactions between individual particle are not considered. The Bethe approximation attempts to .nd a middle ground between these two approaches. Each particle interacts directly with its z nearest neighbors, forming a particle cluster. The remainder of the particles then form an e.ective .eld. While still taking advantage of the mean .eld’s ability to produce solvable mathematics, the Bethe approximation captures more detail, which could drastically improve the predictive abilities of our models.
Chapter 5
Ising Model Approach to ISAM
While CSAE models are particularly useful in modeling and simulating the ionic self-assembly of charged nanoparticles, there are many other mathematical frameworks capable of describing two-state systems. Another useful framework is the Ising model, originally developed by Heinrich Lenz
[56]
and
solved
by
Ernst
Ising
[57]
in
an
attempt
to
describe
how
phase
transitions
in
ferromagnets
emerge from a collection of individual magnetic spins. From equilibrium statistical physics, we know that the probability density of any system exchanging heat with a heat reservoir of temperature T is described by the canonical distribution:
-H(s)/kBT
e
Peq(s)= , (5.1)
Z where H(s) is the Hamiltonian of the system, kB is Boltzmann’s constant, and Z is the partition function, which normalizes the probability distribution. Lenz and Ising suggested that, for a system comprised of N interacting two-state particles in an external .eld B, the Hamiltonian is given by
N
H = -Jsisj - Bsi. (5.2)
i,j.NN i=1
In this equation, si = -1, 1 represents the state of particle i . [1,N]. In the case of ferromagnetism, the two possible states would be spin-up (1 ~.) and spin-down (-1 ~.); however, any two state system is possible, as we will see in our model, where si will come to represent the occupation state of site i on a square lattice. J is a coupling constant, which describes how two particles
1
within the system, si and sj, interact with one another.The coupling constant is applied to all pairs of nearest neighbors by the .rst sum. Finally, B is the strength of the external .eld.2
For ferromagnetism, the .eld is a magnetic .eld; however, other .elds, such as electric .elds, can also be used. The second sum ensures that each particle in the system feels the e.ects of the external .eld.
In time independent (i.e., equilibrium) systems, the properties of this model can be investigated
by
simple
substitution
into
Equation
(5.1).
However, for time dependent (i.e., non-equilibrium) systems, such as ISAM, the master equation approach
of
R
J
Glauber
[58]
is
neces
sary. Glauber begins with a master equation which ensures that the con.gurational properties of
1Generally speaking, J could be variable, indicating that di.erent pairs of particles have di.erent mutual interactions. This behavior is not necessary in our model; therefore, we assume J is constant.
2Like J, B could be variable. However, for our purposes, a constant B will su.ce.
transitioning particles are conserved:
dP (s, t)
'' '
= {c(s . s)P (s ,t) - c(s . s )P (s, t)}. (5.3)
dt
s
Here, we see a system in which the probability of a particle existing in state s at time t depends upon
'
the transfer of probability into state s from state s (gain), as well as the transfer of probability from
''
state s into state s (loss). In spin systems, a transfer from state s to state s would be represented by a spin .ip. In ISAM, such a transfer is represented by a particle attachment site transitioning from occupied to unoccupied (or vice versa) due to evaporation or deposition.
At
equilibrium,
Equation
(5.3)
becomes
'' '
0= {c(s . s)P(s,t) - c(s . s )P(s, t)}. (5.4)
eqeq
s
Glauber’s approach requires the selection of transition rates which satisfy the detailed balance condition:
'' '
c(s . s)Peq(s )= c(s . s )Peq(s). (5.5)
If we substitute
the
canonical
distribution
(Equation
(5.1))
into
the
detailed
balance
condition
(Equation
(5.5)),
we
.nd
that
'
c(s . s) .H/kB T
= e, (5.6)
'
c(s . s )
'
where .H = H(s ) - H(s) is the change in energy which occurs any time a particle transitions from one state to the other.
We now work to translate the CSAE-NN model into a the mathematics of the Ising model, following
the
method
outlined
in
[41,
59].
As a reminder to the reader, CSAE models imagine the .at surface upon which nanoparticle monolayers form as a square lattice. Each site i can be either occupied by a particle (ni = 1) or unoccupied (ni = 0). Empty sites can received particles
.
(a c(0 . 1) transition) at a rate aß, where . = j.NN nj, via the deposition process. Occupied sites can lose particles (a c(1 . 0) transition) at a rate .. This produces the transition rate
.
c(ni . (1 - ni)) = .ni + (1 - ni)aß. (5.7)
We can express the site states ni =0, 1 from our CSAE models in terms of the spin states si = -1, 1 from the Ising model via
1+ si s is any any of the 2N possible con.gurations of the N spins contained within the system. Mean
ni = 2 . (5.8)
Our transition rate is now
c(si) = 1 + si 2 . + 1 - si 2 aß j.NN 1+sj 2 . (5.9)
If we de.ne K = J kB T and h = B kB T , the detailed balance condition becomes
c(s) c(s ' ) = Peq(s ' ) Peq(s) = e -Ksi e Ksi NN sj -hsi NN sj +hsi . (5.10)
'
while, s describes the same state with exactly one of its spins .ipped. From this point, we seek K and h values appropriate to our particle assembly scenario. With these values in hand, we will know both J and B from
Equation
(5.2),
giving
us
the
Ising
model
description
of
ISAM.
We begin our search for K and h by supposing that each particle has z = 2 nearest neighbors. This describes a one-dimensional system which we will later extend into a full two-dimensional grid (z =
4).
In
this
case,
Equation
(5.9)
becomes
1+ si 1 - si 1+ si-1+si+1
c(si)= . + aß 2 . (5.11)
22 Substituting
into
the
detailed
balance
equation
(Equation
(5.10))
yields
si-1+si+1
1+si 1-si 1+ -Ksi NN sj -hsi
. + aß 2 e
22
.. = (5.12)
si-1+si+1 Ksi NN sj +hsi
1-si 1+si 1+ 2 e
. + aß
22
si-1+si+1
1+si 1-si 1+ -Ksi(si-1+si+1)-hsi
. + aß 2 e
22
.. = . (5.13)
si-1+si+1 Ksi(si-1+si+1)+hsi
1-si 1+si 1+ e
. + aß 2
22
We can write this equation eight times, once for each case of the set (si,si+1,si-1):
... ... ... ... ... ... ... ... .
The resulting system of eight independent equations provides enough information to .nd 1
K = ln(ß) (5.14)
4and 1 aß
h = ln . (5.15)
2 .
Generalizing these results to a system with z nearest neighbors gives us 1
K = ln(ß) (5.16)
4and
2ßz
1 a
h = ln 2 . (5.17)
4
.
Using z =
4
and
the
results
in
Equations
(5.16)
and
(5.17),
the
Hamiltonian
for
the
ISAM
grid
is
kBTa2ß4 N
H = - ln(ß) sisj + ln si . (5.18)
2
4
.
i,j.NN i=1
If we de.ne the magnetization of this system as M = si , as is normal in an Ising model description of a collection of spins, we .nally .nd that the particle coverage density is given by
1+ M
. = . (5.19)
2
For nanoparticle assembly under perpendicular electric .elds, we begin with the transition rate
.
c(ni . (1 - ni)) = (1 - ni).1 + ni.2 + (1 - ni)aß. (5.20)
To .nd the transition rate in terms of spin value si,
we
substitute
in
Equation
(5.8).
This
yields
1 - si 1+ si 1 - si j.NN 1+sj
c(si)= .1 + .2 + aß 2 . (5.21)
222
When z = 2, this transition rate produces the following detailed balance equation:
si-1+si+1
1-si 1+si 1-si 1+ -Ksi(si-1+si+1)-hsi
2 .1 + 2 .2 + 2 aß 2 e
.. = . (5.22)
si-1+si+1 Ksi(si-1+si+1)+hsi
1+si 1-si 1+si 1+ 2 e
.1 + .2 + aß
222
The eight possible combinations of (si-1, si, si+1) produce a system of eight equations, which we solve to .nd
1 .1 + aß2
K = ln (5.23)
4 .1 + aß
and
1 .1 + aß
h = ln . (5.24)
2 .2
Generalized for any z, K and h become
1 .1 + aßz
K = ln (5.25)
z/2
2z
.1 + aß
and
z/2
1 .1 + aß
h = ln . (5.26)
2 .2
Notice that when .1 = 0 and .2 = . (the
no-.eld
case)
Equations
(5.23)
-(5.26)
reduce
to
our
previous
results
(Equations
(5.14)
-(5.17)),
as
expected.
Using
z = 4 and the results of Equations
(5.25)
and
(5.26),
the
Hamiltonian
for
DSAM
under
perpendicular
electric
.elds
is
42 N
kBT 1 .1 + aß.1 + aß
H = - ln sisj + ln si . (5.27)
2
24 .1 + aß.2
i,j.NN i=1
For nanoparticle assembly under parallel electric .elds, we begin with the transition rate
.
c(si)= ni. + (1 - ni)aß+ ni(1 - ni±1).. (5.28)
In terms of spin states, this transition rate becomes
1+ si (1 + si)(1 - si±1)
c(si)= . +1 - si aß j.NN 1+2 sj + .. (5.29)
22 4
For the remainder of our treatment, we will assume that the parallel .eld is oriented so as to encourage particle di.usion in the positive direction. Thus, si±1 becomes si+1. 3
The transition rate is now
1+sj
1+ si 1 - si (1 + si)(1 - si+1)
j.NN 2
c(si)= . + aß + .. (5.30)
22 4 When z = 2, this transition rate produces the following detailed balance equation:
si-1+si+1
1+si 1-si 1+ (1+si)(1-si+1) -Ksi(si-1+si+1)-hsi
. + aß 2 + .e
22 4
.. = . (5.31)
si-1+si+1 Ksi(si-1+si+1)+hsi
1-si 1+si 1+ 2 (1-si)(1-si+1) e
. + aß + .
22 4
Using the eight possible combinations of (si-1, si, si+1), we produce a system of eight equations with solutions
1
K = ln(ß) (5.32)
4and
1 aß
h = ln . (5.33)
2 . + .
For a generalized z, K and h become
1
K = ln(ß) (5.34)
4and
z/2
1 aß
h = ln . (5.35)
2 . + .
For . =
0
(the
no-.eld
case),
Equations
(5.34)
and
(5.35)
become
Equations
(5.16)
and
(5.17),
as
expected. Using z =
4
and
the
results
of
Equations
(5.34)
and
(5.35),
the
Hamiltonian
for
DSAM
under a parallel electric .eld is
kBT 1 aß2 N
H = - ln(ß) sisj + ln si . (5.36)
22. + .
i,j.NN i=1
3If the .eld were oriented so as to encourage particle di.usion in the negative direction, si±1 would become si-1.
Chapter 6
Conclusion
In this thesis, we presented modeling, simulation, and experimental techniques for the production of thin .lms of SiO2 nanoparticles. We examined cases in which particles assembled under no external .eld (ionic self-assembly of monolayers, or ISAM). We also explored thin .lms formed under the in.uence of perpendicular and parallel DC and AC .elds (directed self-assembly of mono-layers, or DSAM).
Our modeling approach focused on two types of cooperative sequential adsorption with evaporation (CSAE) models, which imagine the deposition surface as a discrete grid with occupied and unoccupied sites. The total lattice model (CSAE-TL) limited particle deposition according to the total number of particles deposited. The nearest neighbors model (CSAE-NN) limited particle deposition according to the total number of nearest neighbors interacting with a particular site. Both models were converted to a di.erential equation predicting the time dependency of the deposition surface’s particle coverage density. Numerical solutions to these di.erential equations predicted that assembling systems would rapidly approach steady state.
Our simulation approach used the Monte Carlo method to alter the state of each site on an (m ×n) grid. At each site, the program would select a random number and compare it to the result of the applicable transition rate (CSAE-TL or CSAE-NN). The results of this comparison would determine whether the site’s state should change or remain the same. Like our models, simulations of ISAM and DSAM predicted a rapid approach to steady state. Residual analysis demonstrated strong quantitative agreement between our CSAE-TL models for the no .eld case and perpendicular .eld case and our simulations. Our CSAE-TL model for the parallel .eld case and our CSAE-NN models failed to agree quantitatively with our simulations. However, they agreed qualitatively, indicating that CSAE modeling via the mean .eld approximation is useful when high precision is not required or when simulation tools are unavailable. Future work could include using more accurate versions of mean .eld theory, including the Bethe approximation, to improve agreement between our models and simulations. Approaches of this nature generally require use of the Ising model, which we have also described for the reader in the context of thin .lm production.
Our experiments show that, under no electric .eld, the particle coverage density of a thin .lm can be controlled via the concentration of the nanoparticle suspension used during the ISAM process. When scaled properly, our theoretical models match experimental results for this investigation. Additionally, our investigation of the time dependence of particle coverage density shows that the assembling system does, in fact, rapidly approach steady state, just as predicted by our models and simulations. Furthermore, the increase in particle coverage density during that rapid approach can be divided into two stages, a Langmuir-type adsorption and a smooth transition to steady state. Experimental investigations of the in.uence of external .elds on nanoparticle assembly have progressed through the experiment design stage and are ongoing. We anticipate, in light of the predictions of our models and simulations, that electric .elds can be used to control particle coverage density. Future experimental work will include e.orts to con.rm this prediction.
Bibliography
[1] Feynman R P 1960 Engineering and Science 23 22-36
[2] Lindsay S M 2010 Introduction to Nanoscience (Oxford: Oxford University Press)
[3] Vieu C, Carcenac F, P´epin A, Chen Y, Mejias M, Lebib A, Manin-Ferlazzo L, Couraud L, Launonis H 2000 Appl. Surf. Sci. 164 111-117
[4] Tseng A A, Chen K, Chen C D, Ma K J 2003 IEEE Trans. Electron. Packag. Manuf. 26 141-149
[5] Broers A N, Hoole A C F, Ryan J M 1996 Microelectron. Eng. 32 131-142
[6] Lee R E 1979 J. Vac. Sci. Technol. 16 164
[7] Gloersen P G 1975 J. Vac. Sci. Technol. 12 2835
[8] Kumar A, Whitesides G M 1993 Appl. Phys. Lett. 63 2002-2004
[9] Loo Y, Willett R L, Baldwin K W, Rogers J A 2002 J. Am. Chem. Soc. 124 7654-7655
[10] Huang Y, Wu J, Yang S 2011 Microelectron. Eng. 88 849-854
[11] Kilby J S 1976 IEEE Trans. Electron Devices 23 648 -654
[12] Taylor J R, Za.ratos C D, Dubson M A 2015 Modern Physics for Scientists and Engineers (Mill Valley, CA: University Science Books)
[13] Bennig G, Gerber C, Stoll E, Albrecht T R, Quate C F 1987 Europhys. Lett. 3 1281
[14] Ohnesorge F, Binnig G 1993 Science 260 1451-1456
[15] Gross L, Mohn F, Moll N, Liljeroth P, Meyer G 2009 Science 325 1110-1114
[16] Custance O, Perez R, Morita S 2009 Nat. Nanotechnol. 4 803-810
[17] Montemagno C, Bachand G, Stelick S, Bachand M 1999 Nanotechnology 10 225-231
[18] Arthur J R 2002 Surf. Sci. 500 189-217
[19] Nall J R, Lathrop J W 1957 1957 International Electron Devices Meeting 117
[20] Clark N A, Douglas K, Rothschild K R 1988 Self-Assembled Nanometer Lithographic Masks and Templates and Method for Parallel Fabrication of Nanometer Scale Multi-Device Structures
4728591
[21] Volkert C A, Minor A M 2007 MRS Bull. 32 389-399
[22] Mercuri F, Maldoni M, Sgamellotti A 2012 Nanoscale 2 369-379
[23] Cha J N, Birkedal H, Euliss L E, Barti M H, Wong M S Deming T J, Stucky G D 2003 J. Am. Chem. Soc. 125 8285-8289
[24] Letchford K, Burt H 2007 Eur. J. Pharm. Biopharm. 65 259-269
[25] Iler R K 1966 Journal of Colloid and Interface Science 21 569-594
[26] Langmuir I 1941 Method of Substance Detection 2232539
[27] Lvov Y, Ariga K, Onda M, Ichinose I, Kunitake T 1997 Langmuir 13 6195-6203
[28] Luo T M, MacDonald J C, Palamore G T R 2004 Chem. Mater. 16 4916-4927
[29] Steinhart M, Goring P, Dernaika H, Prabhukaran M, Gosele U, Hempel E, Thurn-Albrecht T 2006 Phys. Rev. Lett. 97 027801
[30] Yin X, Wu J, Li P, Shi M, Yang H 2016 Chem. Nano. Mat. 2 37-41
[31] Xu S, Dadlani A L, Acharya S, Schindler P, Prinz F 2016 Appl. Surf. Sci. 367 500-506
[32] Gorbachev I A, Goryacheva I Y, Glukhovskoy E G 2016 J. Bionanosci. 6 153-156
[33] Achermann M, Petruska M A, Crooker S A, Klimov V I 2003 J. Phys. Chem. B 107 1378213787
[34] Seeman N C 1998 Annu. Rev. Biophys. Biomol. Struct. 27 225-248
[35] Seeman N C 2007 Mol. Biotechnol. 37 246-257
[36] Yancey S E, Zhong W, He.in J R and Ritter A L 2006 J. Appl. Phys. 99 034313
[37] Ibn-Elhaj M and Schadt M 2001 Nature 410 796-799
[38] Xi J Q, Schubert M F, Kim J K, Schubert E F, Chen M, Lin A, Liu W and Smart J A 2007 Nat. Photon. 1 1769
[39] Kotov N A, Dekany I, Fendler J H 1995 J. Phys. Chem. 99 13065
[40] Hecht E 2017 Optics, Fifth Edition (New York: Pearson Education, Inc.)
[41] Mazilu D A, Mazilu I and Williams H T 2018 From Complex to Simple -Interdisciplinary Stochastic Models (San Rafael, CA: Morgan & Claypool Publishers)
[42] Silva N P, Menacho F P and Chorilli M 2012 J. of Pharm. 2 23-30
[43] Mallouk T E and Kovtyukhova N I 2002 Chem. Eur. J. 8 4355-4363
[44] Zheng Y, Lalander C H, Thai T, Dhuey S, Cabrini S and Bach U 2011 Agnew. Chem. Int. Ed. 50 4398-4402
[45] Ruska E 1987 Rev. Mod. Phys. 59 627-638
[46] Withers M O, Baker E, Mazilu D A, Mazilu I 2019 J. Phys. Conf. Ser. 1391 012006
[47] Behrens S H, Grier D G 2001 J. Chem. Phys. 115 6716-6721
[48] Cook L J, Mazilu D A, Mazilu I, Simpson B M, Schwen E M, Kim V O, Seredinski A M 2014 Phys. Rev. E 89 062411
[49] Wolden C, Collins R 2015 Photolithography Procedure [online] Available at: http://inside.
mines.edu/impl/photo.html
[Accessed 8 May 2020]
[50] Mazilu I, Mazilu D A, Melkerson R E, Hall-Mejia E, Beck G J, Nshimyumukiza S, da Fonseca C M 2016 Phys. Rev. E. 93 032803
[51] Gri.ths D J 1999 Introduction to Electrodynamics, Third Edition (Upper Saddle River, NJ: Prentice Hall)
[52] Reese R L 2000 University Physics (Paci.c Grove, CA: Brooks/Cole Publishing Company)
[53] Baker E, Withers M O, Aldrich E, Sha.rey I, Pusztay J, Mazilu D A, Mazilu I 2019 J. Phys. Conf. Ser. 1391 012007
[54] Croarkin C, Tobias P 2012 NIST/SEMATECH e-Handbook of Statistical Methods (Washington, DC: National Institute of Standards and Technology) [online] Available at: https:
//www.itl.nist.gov/div898/handbook/index.htm
[Accessed 6 May 2020]
[55] Gould H, Tobochnik J 2010 Statistical and Thermal Physics: With Computer Applications (Princeton, NJ: Princeton University Press)
[56] Lenz W 1920 Z. Phys. 21 613-615
[57] Ising E 1925 Z. Phys. 31 253-258
[58] Glauber R J 1963 J. Math. Phys. 4 294-307
[59] Schwen E M, Mazilu I, Mazilu D A 2015 Eur. J. Phys. 36 025003
[60] Schollwock U n.d. Advanced Statistical Physics Lecture Notes (Munich: Ludwig-Maximilians-Universitat Munchen) [online] Available at: https://www.physik.uni-muenchen.de/lehre/
vorlesungen/sose_14/asp/lectures/ASP_Chapter5.pdf
[Accessed 6 May 2020]
[61] Kinder J M, Nelson P 2018 A Student’s Guide to Python for Physical Modeling, Updated Edition (Princeton, NJ: Princeton University Press)
[62] Hunter J, Dale D, Firing E, Droettboom M 2020 Matplotlib.Animation.Artistanimation -Matplotlib 3.2.1 Documentation [online] Available at: https://matplotlib.org/api/_as_gen/
matplotlib.animation.ArtistAnimation.html
[Accessed 6 May 2020]
[63] Hogg D W 2020 Python matplotlib.animation.ArtistAnimation() Examples -Example 4 [online] Available at: https://www.programcreek.com/python/example/96642/matplotlib.
animation.ArtistAnimation
[Accessed 6 May 2020]
Appendix A
Mean Field Theory
The development of our CSAE models depends heavily upon the use of mean .eld theory. Throughout Chapters 2 and 3, we use mean .eld theory’s assertion of a lack of correlation between deposition sites. This allows us to approximate the average correlation between neighboring particles as the product of the mean individual site occupations:
ninj = ni nj . (A.1)
Ultimately, this approximation makes it possible to express our original di.erential equations in terms of the particle coverage density .. Here we consider mean .eld theory more generally and in its original context: the Ising model.
As
discussed
in
[60],
mean
.eld
theory
depends
upon
the
Bogolyubov
inequality,
which
is
derived as follows. First, we assume we have a system with a classical Hamiltonian H. We now decompose this Hamiltonian into two parts:
H = H0 + H1. (A.2)
While the Hamiltonian can be broken into any two parts, the approximation requires that we include all the physical information that can be solved exactly in H0. Similarly, we should include the more di.cult parts of the Hamiltonian in H1. The partition function of H is given by
-ßH(s)
Z = e (A.3)
{s}
while the partition function of H0 is given by
-ßH0(s)
Z0 = e. (A.4)
{s}
If we decompose H correctly, we should be able to .nd Z0 analytically. Dividing
Equation
(A.3)
by
(A.4)
yields
-ß(H0+H1)
Z {s} e
-ßH1(s) -ßH1
== p0(s)e = e (A.5)
-ßH0
Z00.
{s} e {s}
Thus, by expressing the ratio of Z and Z0 in terms of the probability distribution p0(s) associated
-ßH1
with H0, we .nd that Z/Z0 is the expectation value of e with respect to p0(s). The convexity
' ''
inequality states that any function f(x) with f> 0 and f> 0 obeys
f( x p) = f(x) p. (A.6)
p refers to the probability density p(x) that we use to calculate the expectation value. Applying the convexity inequality gives us
-ßH1 -ß(H1)0
e 0 = e (A.7) Z -ß(H1)0
= e. (A.8)
Z0
Taking the logarithm of this function and using G = -kBT lnZ, the de.nition of Gibbs free energy, .nally gives us the Bogolyubov inequality:
G = G0 + H10. (A.9)
When splitting our Hamiltonian, we use a control parameter .. The mean .eld approximation arises when we minimize the Gibbs free energy with respect to this parameter. We begin with
(.)
H0 = -. si (A.10)
i
and
H(.) 1 = -J . sisj + (. - H) . si. (A.11)
{i,j} i
(.)
Since we consider a two state ising model, si = ±1. This means that the partition function for H
0
becomes
N
-ß. ß.
Z0 =e + e (A.12)
= (2cosh(ß.))N (A.13)
(.)
where N is the number of particles in the system. The Gibbs free energy associated with His
0
The expectation values of the spins with respect to Hare independent of one another. This
G0 = -kBT ln(Z0) (A.14)
= -NkBT ln(2cosh(ß.)). (A.15)
(.)
0
means that
si 0 = tanh(ß.) (A.16) for all values of index i. Independence also produces the following result for the expectation value
(.)
of H:
1
(.)2
H= - NJztanh(ß.)+ N(. - H)tanh(ß.), (A.17)
1
2
where z is the number of nearest neighbors. Plugging
Equations
(A.15)
and
(A.17)
into
Equation
(A.9)
tells us that we should minimize
1
G(.)= N - ß-1ln(2cosh(ß.)) - Jztanh2(ß.)+(. - H)tanh(ß.) . (A.18)
2
Computing dG/d. = 0 produces = Jztanh(ß.min), (A.19)
.min - H
where .min is the value of . at which the Gibbs free energy is minimized. The minimized free energy is
2N(.min - H)
G = -NkbT ln(2cosh(ß.min)) + . (A.20)
2zJ
The magnetization of the system is given by
1dG
m = - (A.21)
N dH
1 .G .G ..
= - + (A.22)
N .H .. .H
.min
Since the free energy is minimized, .G = 0 and . = .min. We now have
..
.min - H
m = . (A.23)
Jz
Substituting
in
Equation
(A.19)
.nally
yields
m = tanh ß(Jzm + H) . (A.24)
According
to
[55],
this
is
an
example
of
a
self-consistent
equation
for
m. In other words, the mean .eld that produces m depends upon m .
When
H
=
0,
which
represents
a
system
in
which
there
is
no
external
magnetic
.eld1,
Equation
(A.24)
becomes
m = tanh(ßJzm). (A.25)
Stable solutions to this transcendental equation exist only when the system is above a critical temperature Tgiven by
c
Jz
T= . (A.26)
c
kB
3
When T is close to T, m is necessarily small. This allows us to Taylor expand (tanhx ˜ x-x/3+···
c
when x << 1)
Equation
(A.25)
to
.nd
1
3
m = ßJzm - (ßJzm)+ ··· . (A.27)
3
1Each spin still experiences the mean magnetic .eld produced by all the other spins.
The solutions are m(T >T) = 0 (A.28)
c
and
1/2
3
1/2
m(T