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Scalable dielectrophoresis of single walled carbon nanotubes

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Scalable dielectrophoresis of single walled carbon nanotubes
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Fitzhugh, William A. ( author )
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Carbon nanotubes ( lcsh )
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Carbon nanotubes ( fast )
Dielectrophoresis ( fast )
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Single Walled Carbon Nanotubes (SWNTs) have attracted much attention as a candidate material for future nano-scale 'beyond silicon' devices. However industrial scale operations have been impeded by difficulties in separating the metallic and semiconducting species. This paper addresses the use of highly inhomogeneous alternating electric fields, dielectrophoresis, to isolate SWNT species in scaled systems. Both numerical and experimental methods will be discussed.
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Thesis (M.I.S) University of Colorado Denver
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Includes bibliographic references,
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Integrated Sciences Program
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by William A. Fitzhugh.

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Full Text
SCALABLE DIELECTROPHORESIS OF SINGLE WALLED CARBON
NANOTUBES
by
WILLIAM A. FITZHUGH
B.S., University of Colorado Denver, 2012
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado Denver in partial fulfillment
of the requirements for the degree of
Master of Integrated Sciences
Integrated Sciences
2015


This thesis for the Master of Integrated Sciences degree by
William A. Fitzhugh
has been approved for the
Integrated Sciences Program
by
Masoud Asadi-Zeydabadi, Chair
Jan Mandel
Randall Tagg
July 24, 2015
n


Fitzhugh, William A. (M.I.S.)
Scalable Dielectrophoresis of Single Walled Carbon Nanotubes
Thesis directed by Associate Professor Masoud Asadi-Zeydabadi
ABSTRACT
Single Walled Carbon Nanotubes (SWNTs) have attracted much attention as a
candidate material for future nano-scale beyond silicon devices. However indus-
trial scale operations have been impeded by difficulties in separating the metallic and
semiconducting species. This paper addresses the use of highly inhomogeneous alter-
nating electric fields, dielectrophoresis, to isolate SWNT species in scaled systems.
Both numerical and experimental methods will be discussed.
The form and content of this abstract are approved. I recommend its publication.
Approved: Masoud Asadi-Zeydabadi


DEDICATION
This thesis is dedicated to my parents, for their never ending support.


ACKNOWLEDGMENT
This thesis would not have been possible without the generous support of Dr. Asadi,
Dr. Tagg, Dr. Geyer, Dr. Golkowski, and Ryan Jacobs. Special thanks to Dr. Maron,
Dept, of Chemisty, Dr. Mandel and Dr. Langou, Dept, of Applied Mathematics,
and Dr. Huber and Kris Bunker, Dept, of Physics.


TABLE OF CONTENTS
Tables..................................................................... viii
Figures ..................................................................... ix
Chapter
1. Introduction............................................................... 1
1.1 Graphene............................................................ 1
1.2 Single Walled Carbon Nanotubes...................................... 4
1.3 Nanotube Absorption Spectroscopy ................................... 6
1.4 Dielectrophoresis................................................... 7
1.5 SWNT Dielectrophoresis ............................................. 8
1.6 Rectangular Waveguides and Resonators ............................. 10
1.6.1 Transverse Electric.......................................... 11
1.6.2 Transverse Magnetic.......................................... 11
1.6.3 Resonators................................................... 12
2. Theory.................................................................... 13
2.1 Accounting for Drag................................................ 13
2.2 Steady State Solution.............................................. 14
2.3 Boundary and Initial Conditions.................................... 15
2.4 Temporal Evolution................................................. 16
2.5 Slowest Decaying Eigencouple ...................................... 18
3. Numerical Methods......................................................... 20
3.1 Finite Differences ................................................ 20
3.1.1 Reshaping Tensors............................................ 20
3.1.2 Laplacian Matrix............................................. 21
3.1.3 Derivative Matrix ........................................... 21
3.2 Boundary Extrapolation ............................................ 22
3.2.1 Boundary Conditions.......................................... 24
vi


3.3 Forming A......................................................... 24
4. Numerical Results........................................................ 26
4.1 Mesh Size and Stabilty............................................ 26
4.2 T and $ in SI Units............................................... 27
4.3 Error in a Known Case............................................. 27
4.4 Parallel Plate Dielectric Waveguide .............................. 30
4.4.1 Boundary Conditions......................................... 32
4.4.2 {cp} Approximations......................................... 32
4.4.3 Forcing Exponential Decay .................................. 32
4.5 Conclusion........................................................ 34
5. Laboratory Methods....................................................... 37
5.1 Dispersion........................................................ 37
5.2 Experimental Setup................................................ 37
5.2.1 Field Intensity Distributions............................... 40
5.3 Secondary Suspension.............................................. 40
6. Experimental Results..................................................... 42
6.1 Pre and Post DEP.................................................. 42
6.2 Analysis ......................................................... 43
6.3 Conclusion........................................................ 43
6.3.1 Future Work................................................. 44
References.................................................................. 47
Appendix
A. DEP Finite Difference Code............................................... 49
A.l Two Dimensions.................................................... 49
A.2 Three Dimensions.................................................. 53
vii


TABLES
Table
6.1 Tight binding predicted photonic energies and wavelengths for nanotube
Van Hove transitions.................................................... 43
vm


FIGURES
Figure
1.1 Honeycomb lattice of graphene. Image from [If].......................... 2
1.2 Reciprocal Lattice of Graphene. Image from [15]......................... 3
1.3 Comparison of the tight binding approximation with ab initio calculations
for the brand structure of graphene. Image from [15].................. 3
1.4 Demonstration of the graphene rolling process for a (5, 3) nanotube. Image
from [4].................................................................. 4
1.5 Density of states for metallic and semiconducting SWNTs. Van Houe
transitions are illustrated by arrows. Image from [8]................. 6
1.6 Absorption spectra of a thin film of SWNTs. Image from [11]........... 7
4.1 Approximated and known eigenualues for the scaled Laplacian operator. . 28
4.2 Log-log plot of minimum real eigenualue component us the maximum field
intensity................................................................ 33
4.3 Log-log plot of minimum real eigenualue component us the maximum field
intensity................................................................ 34
4.4 Steady state concentration profiles for (a) 6.5 mm and (b) 3.5 mm dielec-
tric parallel plate umueguides stimulated at 2-45 [GHz]................ 35
5.1 Arrangement of needles in fused quartz capillaries (4mm interior diameter
on the left and 8mm diameter on the right.)............................ 38
5.2 4mm capillary with gold platted needles filled with SWNT SDS solution. 39
5.3 8mm capillary with gold platted needles filled with SWNT SDS solution. 40
5.4 Time auerage intensity field distributions (Avgxy) for the (a) 8mm and
(b) 4mm quartz tubing.................................................... 41
6.1 Bottom needle from 4mm capillary after four fiue minute exposures to
rnicrowaues.............................................................. 44
IX


6.2 Extinction spectra for SWNT SDS solution (a) before and (b) after DEP
exposure
45
x


1. Introduction
Consisting of a single cylindrical layer of graphene, Single Walled Carbon Nan-
otubes (SWNTs) were first discovered in the early 1990s at the NEC Fundamental
Research Laboratory in Tsukuba, Japan, by Sumio Iijima [9]. Since that time SWNTs
have been the subject of much attention as a candidate material for future nano-scaled
engineering. Many cutting edge applications have been theorized to take advantage
of their remarkable physical and electrical properties. However, SWNTs exist in both
semiconducting and metallic species, and all currently known manufacturing proce-
dures result in a heterogeneous mixture of both types [18]. Given that the majority
of SWNT based device designs require a specific electronic type, and that separation
remains a highly expensive process, commercialization of these remarkable materials
has been slow.
1.1 Graphene
Graphene consists of a planar array of sp2 hybridized carbon atoms. This lattice
structure cannot be represented by a Bravais lattice, which would require every carbon
to be located at a vector, originating at any other carbon, given by:
R = mdi + nda (1.1)
For some lattice vectors dj, 02 and integers m,n. Instead, the graphene lattice
must be represented as a lattice decorated with a basis. The unit cells of graphene
are given by the Hexagonal lattice, which in two dimensional Cartesian coordinates
has the following lattice vectors (for some lattice constant a).
di = a( 1 /2, v/3/2) (1.2)
a2 = a(- l/2,V3/2) (1.3)
1


As depicted in Figure 1.1, the solid dots correspond to the locations of the vectors
R = md[ + nd2 for the lattice vectors above.
Figure 1.1: Honeycomb lattice of graphene. Image from [Iff.
Two carbon atoms are then added to each unit cell, creating the hexagonal shape
in figure 1.1. The location of each carbon atom relative to the associated lattice point
is given by the vectors [14]:
* = a(0iwi)
(1.4)

(1.5)
The final result is known as the honeycomb lattice. Using the identity bj
2nSij, the reciprocal lattice vectors, b\ and fo, can be found to be:
S^fa.-L) (i.6)
£ = -(-1.4) (1-7)
a v3
Figure 1.2 presents a graph of the reciprocal lattice of graphene in this basis.
The shaded area represents the first Brillouin zone and the K, K', T, and M points
designate particular discrete points of important symmetry.
2


Figure 1.2: Reciprocal Lattice of Graphene. Image from [15].
The band structure of graphene is well approximated using a tight binding model
[4]. In this approach, the electron wave function is calculated by the superposition
of a core and nearest neighbor atomic wave functions. The bandstructure along the
reciprocal space triangle YKM is depicted in figure 1.3 for both ab initio and tight
binding models.
Figure 1.3: Comparison of the tight binding approximation with ab initio calculations
for the brand structure of graphene. Image from [15].
3


1.2 Single Walled Carbon Nanotubes
Single walled carbon nanotubes (SWNTs) consist of graphene rolled into hollow
cylinders. The structure of SWNTs is expressed via the chiral vector:
Ch = ma[ + na2 (1.8)
Where a[ and a2 are the lattice vectors of graphene and m and n are integers. The
chiral vector is one such that it circumvents the nanotube and returns to its original
location. Figure 1.4 shows this process for a (5, 3) nanotube. A and A! represent the
same point on the (5, 3) SWNT surface.
Figure 1.4: Demonstration of the graphene rolling process for a (5, 3) nanotube. Image
from Iff.
Blochs theorem, rip(r) = u(r)exp(ik r), tells us that the electron wavefunctions
in graphene must be a plane wave modulated by a periodic function with a periodicity
given by the graphene lattice (ie. d[ and a2). Given that Ch = md[ + nap.
tfj(r + Ch) = u(r + Ch)exp{ik (r + Ch)) (1.9)
4


ip(r + Ch) = u(r)exp(ik (r + Ch))
ip(r + Ch) = t/j{r)exp(ik Ch)
(1.10)
(1.11)
Additionally, the azimuthal continuity of a SWNT requires that:
ip(r + Ch) = r (1.12)
Comparing equations 1.11 and 1.12, we see that the only wave vectors that are
allowable under both of these boundary conditions are those such that:
expiik Ch) = 1
(1.13)
For the SWNT to behave metallically, the K wave vectors must be allowed. It
has been shown [14] that, in the basis described by equations 3.10 and 3.11:
K = ifcl -b2
3 3
(1.14)
44 = 4 (1,0,0)
6a
(1.15)
K Ch = (1,0, 0) [ma(l/2, y/3/2, 0) + na(-l/2, V6/2,
6a
(1.16)
K Ch= thl.0.0) dm n, V3(m + n).0)
6a 2
(1.17)
K Ch = (m n)
For equation 1.13 to hold for K, then for some integer l = 0,1,2, 3,
(1.18)
_> 2tt
K Ch= (m n) = 2irl
3
(1.19)
5


(m n) = 31
(1.20)
Accordingly, if the chiral vector (n, m) is such that n and m differ by an integer
multiple of 3, then the K wave vectors will be allowed and the nanotube will behave
metallically. Otherwise, the SWNT will behave as a semiconductor.
1.3 Nanotube Absorption Spectroscopy
absorbed by a specimen as a function of frequency. In the case of single walled carbon
nanotubes, as well as most crystalline solids, this behavior is dominated by Van Hove
singularities.
The density of states (DOS) is defined to be the number of states, or wave func-
tions, that exist within a given energy interval. Van Hove singularities refer to sharp,
non-smooth, peaks in the DOS. The differences in the DOS for metallic and semicon-
ducting SWNTs are represented in figure 1.5.
Figure 1.5: Density of states for metallic and semiconducting SWNTs. Van Hove
transitions are illustrated by arrows. Image from [8],
Absorption spectroscopy is the quantification of the relative amount of radiation
6


Each of the Van Hove transitions illustrated above (Mu, Sn, S22, S33) corre-
sponds to a particular change in energy. As such, each transition is associated with a
radiation frequency such that AE = hu, where h is Planks constant. Given that more
electrons exist with in the Van Hove singularities, the absorption of SWNTs peak at
those frequencies. Figure 1.6 depicts experimental results of absorption data for a
thin him of SWNTs coated on quartz. The peak at 650nm corresponds to metal-
lic nanotubes and the peaks at 950mn and 1700mn correspond to semiconducting
nanotubes.
Figure 1.6: Absorption spectra of a thin film of SWNTs. Image from [11].
1.4 Dielectrophoresis
Dielectrophoresis is the processes of exploiting inhomogeneous electric fields to
drive separation of particles in a solution based on the variable polarizabilities. For
the remainder of this paper, we will consider a set of particles denoted with subscript
p dispersed in a liquid denoted with subscript l.
Taking the dielectric function to be = ep>i iap>i/u>, where e is the electric
permittivity, <7 is the conductivity, and oj is the applied angular frequency, the effective
7


dipole moment of a dispersed spherical particle can be expressed as [3]:
p = 47rr3q Re()E
1 y2e* + f*J
(1.21)
Given an oscillatory electric field, this produces a time average DEP force.
Fdep 47rr q Re(
* *
3 TD ( eP ~ el \T7P2
2t*i + e*
)VE2
(1.22)
FDep = 4vrr3Q Re(CMF)VE2 (1.23)
Where CMF is the Clausius Mossotti function. For ellipsoidal particles, a signifi-
cantly more complicated expression for the time average DEP force emerges. However,
this function can be simplified under the assumption that the 3 axes of the particle
a,b,c are related by a b = c (the rod-like particle assumption).
1.5 SWNT Dielectrophoresis
Single walled carbon nanotube dimensions justify the rod-like particle assumption
and the resulting time averaged force is:
F
ivcPl
DEP
-q Re

VE2
(1.24)
e*i + (£p e*)L
Where d is the SWNT diameter, 1 is the length, and L is a depolarization factor
that is roughly 10 5 [3]. The high and low frequency limits of this expression yield
dominating terms of the dielectric function:
lim ^ (1.25)
w-S-0 (Jl
lim Q (1.26)

For metallic SWNTs both of these limits are positive as tp > q and ap > ai,
resulting in a positive DEP force, that is a force in the direction of the highest
8


inhomogeneity of the electric held intensity. For semiconducting SWNTs, however,
the opposite is true, tp < q and av < 07, resulting in a negative DEP force [3] [12].
Experimental results verify three of these four cases. [12] used a microelectrode
array and a 20 volt peak-to-peak (10[E]) AC signal over a 10 micron gap at varying
frequencies to determine the DEP forces acting on SWNTs dispersed in solution using
various concentrations of the surfactant SDBS. In the case of metallic SWNTs, their
results confirmed the theoretically predicted positive DEP forces at all frequencies. In
the semiconducting SWNTs, they found that the surfactant induced a small surface
charge that enabled positive DEP forces at low frequency. However, at high fre-
quency the semiconducting SWNT/SDBS micelles reverted back to the theoretically
predicted negative DEP force.
The frequency at which semiconducting SWNTs transition from positive to neg-
ative DEP force was termed the critical frequency. For SDBS concentrations of 0.1%
by weight, this frequency was found to be roughly 1 [MHz\. In summary, alternating
electric fields with frequencies less than 1 [MHz\ caused positive DEP forces in both
metallic and semiconducting SWNTs and accordingly separation did not occur. As
the applied held frequency increased above 1 [MHz\, the metallic SWNTs continued
to experience positive DEP forces while the semiconducting SWNTs transitioned to
negative DEP forces.
The maximum electric held strength, which occurred directly between the mi-
croelectrodes, needed to cause separation was determined by energetically comparing
the DEP force with Brownian motion. Electrohydrodynamical forces were not consid-
ered as the SWNT size would indicate that Brownian motion would be the primary
competing force [12].
Denoting the potential energy change between maximum electric held and mini-
mum held at far distances, the DEP escape potential was found to be:
9


Udep = fi Re{(CMf)}C (1 27)
Setting this equal to the thermal energy yields a maximum electric held of [12]:
/ gr.71
- V vrd2/e) Re{(dMF)} "" (L28)
1.6 Rectangular Waveguides and Resonators
Here we will derive the eigenmodes for rectangular waveguides and, by exten-
sion, rectangular resonators. Waveguides are designed such that propagating waves
destructively interfere at boundaries which would otherwise become sources of atten-
uation. Maxwells equations are used to derive the wave equations for both electric
and magnetic fields.
d2 ->
(c2v2-^)S = 0 (L29)
d2 -*
(C2v2-)H = 0 (1.30)
For metallic wavegudies, the boundary conditions are that of any conductor: the
tangential component of the electric held and the normal component of the magnetic
held must be zero.
E(r, t) x n = 0
(131)
H(r,t) n = 0 (1-32)
If we assume that the electromagnetic wave is traveling in the z direction, with a
wavefront modulation factor to ensure proper boundaries (E(f, t) = E0F(x,
and H(f,t) = H0G(x,y)et(-kzZ~UJt"1), then subsitution into equations 1.29 and 1.30
yeilds:
10


,d2Fx d2K
d2F, d2F,
dx2
_______1.2______________
dy2 z,dx2 dy2
UJ
kZ7) 2 {Fx,Fy,0)
(1.33)
,d2Gx d2G.
dx2
________1.2 ______
dy2 * dx2
d2G d2G
UJ
dy2
~ kz, 0) w(Gx, Gy, 0)
(1.34)
The solutions to this system, for a waveguide of width a on the x-axis and height
b on the y-axis, can be shown to be [1] [10]:
1.6.1 Transverse Electric
mr .rmr mr u^t-k z)
Ex = zE0cos(---x)sm(y)e i' t kz
b a b
(1.35)
. rmr rmr mr i( t_k z)
Ey = zE0---sm{----x)cos(y)e [ t kz
a a
(1.36)
TT TT rmr rmr mr n^t-k z)
Hx = H0----sm(-----x)cos(y)e *- z }
a a b
(1.37)
TT TT ^ Tm7r \ TU7r \ i(wt-k Z)
Hy = H0cos(-----x)sm(y)e *- z }
b a b
(1.38)
k2 + uj2/c2 .rmr .mr i(.,t h .
Hz = Ho~-cos(---------x)cos(y)e kz a b
Here m and n represent the mode number in the x and y directions, respectively.
kz = -\J^ (^)2 (pf)2 is the effective wave number in the z direction.
1.6.2 Transverse Magnetic
urn /nun . .run .
Ex = E0---cos(----x)sm(y)e { t kz > (1-40)
a a b
nir ,rmr .mr z) -
Ey = E0sin{------x)cos(y)e { t kz > (1.41)
11


(1.42)
k2 + uj1 /c2 rrm rm i(u,t_k z)
Ez = E0~ sm(--------x)szn(y)el[UJt kzZ)
rCr 0
Hx
rm /rrm
iH0 sin(-----
b a
x)cos(^-y)el{0Jt-kzZ)
(1.43)
H
. TT rrm .rrm
iH o---cos{----
a a
x)sm(-^y)el(-UJt-kzZ)
b
(1.44)
1.6.3 Resonators
Resonators are constructed by capping either end of the waveguide such that
reflections buid standing waves. The TE/TM mode structure is preserved, while the
ei{ujt^kzz) terms become 2cos(ujt)cos(kzz)
12


2. Theory
As was shown in section 1.5, an energetic approach to calculating the electric held
strength necessary for successful sorting of SWNTs predicted that the held would need
to peak at a value on the order of 106[^]. This was found by determining a DEP
escape potential that was larger than the thermal energy that the solution could
provide. Clearly, this is a sufficient condition for DEP to occur. However, in the case
of high drag systems, this condition may not be necessary as non-conservative forces
can dominate.
A fluidic force model of the system would provide more insight into the held
requirements. Dehning the high frequency time average DEP force, Brownian diffusive
force, and drag force to be:
7T cPl

8
Fdif
Fr
drag
(eP Q)V£2 (2.1)
\s> > 1 ^ i (2.2)
6tt arjv (2.3)
Where NA is Avogadros number, ( is the molar chemical potential, a is the
Stoke-Einstein radius, rj is the viscosity of the medium, and v is velocity.
2.1 Accounting for Drag
Taking advantage of the known ()-polarity of each force, the time averaged form
of Newtons second law gives:
-gy( q)V£2 6ttaVv -^V( = ma (2.4)
8 1\A
Equation 2.4 can be further simplihed by assuming that the particles quickly
achieve a terminal drift velocity:
13


RT
xp
ivcPl
-> Vt
8
67t arj
o \s> > 1 ^ (2.5)
to i ?h <1 g-v (2.6)
Finally using the relation between chemical potential and mole fraction, V(
Vxp, and mole fraction and molar concentration profile, Cp = xpC.
-+Vt= [TVE2 vcy (2.7)
^p
Where T = 4g^ (ep q) and $ = Using the relation between diffusive
flux, molar concentration profile, and velocity, J = CpWt, and the continuity equation,
^ + V-J = 0
+ TV (CPVE2) W2CP = 0 (2.8)
This equation is the quasi-linear heat, or the diffusion-advection, equation. In
the limit that the electric held disappears, this reduces to the linear homogenous heat
equation, a common model for noninteracting diffusive processes.
2.2 Steady State Solution
The steady state solution can be found by setting ^4 = 0. Clearly this leads to
the identity:
[CPVE2] = V2CP
Using the product rule on a scalar vector product: VfVdJ
and the relation V VV = V2Vg this further simplifies to:
(2.9)
V-0 A + -0V A,
vcp vu2 + CPV2E2 = ^V2UP
(2.10)
14


Guessing the solution Cp(r) = Coe$£'2^, the concentration profile Laplacian
becomes:
V2C = V = tv (CPVE2)
(2-11)
Clearly, as a result of equation 2.11, the guess satisfies equation 2.9 and accord-
ingly the solution holds for any adequately continuous electric held intensity.
2.3 Boundary and Initial Conditions
For a given containment chamber described by the domain Q with rigid walls,
dQ, the boundary condition(s) are of the Neumann type. Specifically, the boundaries
inhibit any concentration flux to pass through them. In terms of the normal vector,
n, this is expressed as:
Substituting in equation 2.7 expresses this condition as a requirement of the
concentration profile.
^ Csteady state (?) CoG (2.12)
n J(r) = Cp(r)(n vt{f)) = 0 : [VfedQ]
(2.13)
[CpYVE2 $VCP] -h = 0
(2.14)
(2.15)
Where 11 = n-'Vf. Considering the steady state solution determined in the last
section, it is clear that this boundary condition is innately satisfied.
(2.16)
15


t i / f 9E ^dCsteady state /0 -r
^ 1 steady state vd^ )
Any physically realizable initial condition could be used with this approach. How-
ever, for the remainder of this paper, only initial concentration profiles of fully dis-
persed SWNTs will be considered.
Cp(r,t = 0) = Ci : Vrdl (2-18)
Lastly, conservation of mass gives the final condition on the system. For a system
consisting of Nswnt number of SWNTs dispersed in the solution:
F(t) = f C(r,t)dV = Nswnt Vt (2.19)
J n
In summary, the system has been reduced to the partial differential equation and
conditions:
-N- + TV (CPVE2) V2Cp = 0 (2.20)
Cp(f,t = 0) = Cinitial : Vrdl (2.21)
Cp(r,t > oo) = : Vrdl (2.22)
F(t) = ( Cp{f\t)df = Nswnt ' Vt (2.23)
Jn
2.4 Temporal Evolution
In order to evaluate the temporal evolution of this system, an operator A must
be constructed such that:
dC_
~dt
= -AC
16
(2.24)


-> AC = TV (CPVE2) $V2CP (2.25)
The construction of such an operator will be numerically considered in the sub-
sequent chapter. Intermittently, consideration of the eigencouples of the operator,
(Xi,vn), allows for an expansion of equation 2.24 in an eigenbasis.
C = ae~At (2.26)
-> C =J2aivie~Xit (2.27)
i
For some set of constants {a*}. Given the presence of first order derivatives in
equation 2.20, A is not guaranteed to be Hermitian. As a direct result of this, complex
eigenvalues must be considered.
A j = aj + i[3j (2.28)
-> C = ^2 aie~ite~il3itvi (2.29)
i
The values of cq are known to be non-negative, as a negative value would lead to
exponential growth and a violation of the conservation of mass. Clearly, the decay
of the solution towards any eigenfunction is dominated by the term a,ie~ait. Unfortu-
nately, the values of are not given by the inner product as the eigenfunctions are
not known to be orthogonal:
a% (ClVi)
(2.30)
However, the values of the coefficients {a*} can be expressed as linear functions
of the function C. In other words, 3Wi such that:
17


(wi\C)
Cli
(2.31)
2.5 Slowest Decaying Eigencouple
In order to determine the long term behavior of equation 2.20, attention must be
focused on the eigenvalue with the smallest real part. This stems from it decaying at
the slowest rate and therefore dominating as time goes to infinity.
Considering the expression of the concentration in the N dimensional eigenbasis
of A:
N N
C ='Y^aivi = 'Y^'viai (2.32)
i i
-> C = [vi,v2, ,vn\[ai,a2, , aN]T = Ua (2.33)
Where U is defined to be the matrix made from columns of the eigenfunctions
and a is a column vector with components {iq}. Substituting in equation 2.25 for the
elements of a, equation 2.27 is expressed in terms of a matrix W* with row vectors
given by w*.
C = UW*C (2.34)
~^W = (U*)~1 (2.35)
While having the relation in equation 2.35 is extremely useful, direct calculation
of the inverse of the conjugated eigenfunction matrix is hugely expensive computa-
tionally. Instead, focusing only on obtaining the dominant term oq and therefore W\
can greatly reduce this cost.
W[l, 0,..., 0]T = {U*)~le\ = w\
18
(2.36)


> U*w\ = e\
(2.37)
Restricting consideration to the first m eigen-functions (the eigenfunctions corre-
sponding to the m eigenvalues with lowest magnitude real component), this can be
approximated in terms of , v%, ,
U*wx = e
(2.38)
Of course, N dimensional vector w\ cannot be found uniquely via equation 2.38.
However an m dimensional approximation, w\ = Umz, can be found by solving:
U*mUmz = eT
(2.39)
In conclusion, the eigen-expanded decay rates of the concentration towards the
steady state solution can be bounded termwise as follows.


ciie e
Vi
(2.40)
aite %Pitv < e ait ||u.
Vi W,
p Q-it I I r> i
* r*|
m^i I
(2.41)
19


3. Numerical Methods
In chapter 2, the problem of solving for the rate of approach of concentration to-
wards the steady state was discussed in detail using the eigenexpansion of the solution
in terms of an operator A such that ^ = AC. In this chapter, A will be constructed
using the finite difference method for interior points and polynomial extrapolation on
the boundaries. This method is outlined in [6]. Additionally upwinding methods can
be used to enhance stability [5].
3.1 Finite Differences
The method of finite differences consists of approximating the continuous con-
centration function as a discrete mesh. For a mesh in which there are m,n,p mesh
points in the x, y, and z directions, this is represented by (assuming a unitary distance
between the mesh points):
C(x, y, z) ~ Cijtk : 1 < i < m, 1 < j < n, 1 < k < p (3.1)
3.1.1 Reshaping Tensors
Given that both concentration, C, and electric held intensity, E2, tensors are
three dimensional, operators acting on these tensors must be four dimensional. To
avoid added complications, these m x n x p tensors are reshaped as mnp x 1 column
vectors.
Cijp ^
(r \
Cm,1,1
Cl 2,1
c
1 ,n,p
\Cmnp y
20
(3.2)


3.1.2 Laplacian Matrix
The matrix representation of the Laplacian operator, V2, can be found via a
central difference approximation. Equation 3.3 gives the five point central difference
formula for an equidistant mesh with step size h = 1.
d 2A(i) I,,. , 4 ... . 5 .... 4 ... . 1 .. . . .
qx2 ~ 2) + 1) 2^ + 3^ + ^ 12^ + 2 (^-3)
The following MATLAB code constructs the three dimensional Laplacian matrix
for application onto a reshaped mxnxp tensor. The kronQ function is the Kronecker
tensor product used to do block matrix multiplications. A five point formula is used
for fourth order accuracy.
LX = spdiags(ones(m,1)*[-1/12 4/3 -5/2 4/3 -1/12],-2:2,m,m);
LY = spdiags(ones(n,1)*[-1/12 4/3 -5/2 4/3 -1/12],-2:2,n,n);
LZ = spdiags(ones(p,1)*[-1/12 4/3 -5/2 4/3 -1/12] ,-2:2 p, p) ;
LY_expanded = kron(LY,speye(m,m)) ;
LZ_expanded = kron(LZ,speye(m*n,m*n));
L = (kron(speye(n*p),LX) + kron(speye(p,p),LY.expanded) + LZ_expanded) ./ (h~2)
3.1.3 Derivative Matrix
A similar approach can be used to find the derivative matrices, dxA, dyA, dzA:
dA(i) 1 . 2 , 2 , 1
qx ~ 2) + 1) + 1) + + 2) (3-4)
X = spdiags(ones(m,1)*[-1/12 2/3 0 -2/3 1/12],-2:2,m,m);
21


Y = spdiags(ones(n,1)*[-1/12 2/3 0 -2/3 1/12],-2:2,n,n);
Z = spdiags(ones(p,1)*[-1/12 2/3 0 -2/3 1/12] ,-2:2 p, p) ;
Y_expanded = kron(Y,speye(m,m)) ;
Z_expanded = kron(Z,speye(m*n,m*n));
DX = kron(speye(n*p),X);
DY = kron(speye(p,p),Y_expanded);
DZ = Z_expanded;
3.2 Boundary Extrapolation
The above expressions for the Laplacian and derivative matrices suffer from the
need of ghost points to be well defined at boundaries (ie. d^il(O) ~ Tb-Th-1)^
These ghost points can be eliminated by the use of forward/backwards difference
approximations.
dA(i)
dx
H--2Aii) + -2A{il) -A{i2))
(3.5)
% Fourth order central difference approximation.
X = spdiags(ones(m,1)*[1/12 -2/3 0 2/3 -1/12],-2:2,m,m);
Y = spdiags(ones(n,1)*[1/12 -2/3 0 2/3 -1/12],-2:2 n, n);
Z = spdiags(ones(p,1)*[1/12 -2/3 0 2/3 -1/12],-2:2,p,p);
% Boundary extrapolation (second order forward/central approximation).
X (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
X(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
Y (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
Y(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
Z (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
Z(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
22


Y_expanded = kron(Y,speye(m,m));
Z_expanded = kron(Z,speye(m*n,m*n));
DX = kron(speye(n*p),X);
DY = kron(speye(p,p),Y_expanded);
DZ = Z_expanded;
Forwards and backwards difference approximations are also be used in the first,
second, second to last, and last rows of the Laplacian matrices.
dxxA(xi) ~ 2Ai 5Ai+i + hb+3 (3-6)
dxxA(xi) ~ 2Ai hAi-i + 4ffj_2 A^3
(3.7)
-5/2 4/3 -1/12 0 ^ 1 2 -5 4 ... 0
4/3 -5/2 4/2 -1/12 0 2-5 4..
->
-1/12 4/3 -5/2 4/3 4-520
0 -1/12 4/3 5/2 y 1 ...4-5 2
(3.8)
LX = spdiags(ones(m,1)*[ >1/12 4/3 -5/2 4/3 -1/12],-2:2,m,m);
LY = spdiags(ones(n,1)*[ >1/12 4/3 -5/2 4/3 -1/12],-2:2,n,n);
LZ = spdiags(ones(p,1)*[ >1/12 4/3 -5/2 4/3 -1/12] ,-2:2,p,p);
LX(1:2,1:4) = [2 -5 4 0; 0 2 -5 4];
LX(end-1:end,end-3:end) =[4-52 0; 04-5 2];
LY(1:2,1:4) = [2 -5 4 0; 0 2 -5 4];
LY(end-1:end,end-3:end) =[4-52 0; 04-5 2];
LZ (1:2,1:4) = [2 -5 4 0; 0 2 -5 4];
23


LZ(end-1:end,end-3:end) =[4-52 0; 04-5 2];
LY_expanded = kron(LY,speye(m,m));
LZ_expanded = kron(LZ,speye(m*n,m*n));
L = kron(speye(n*p),LX) + kron(speye(p,p),LY.expanded) + LZ_expanded;
3.2.1 Boundary Conditions
It is important to note that the use of forwards/backwards difference methods
at the boundaries is not alternatives to boundary conditions. The general case is
complicated by the fact that the boundaries of the electric field and the boundaries of
the concentration are not the same. As an example, consider an electric field passing
through a transparent container. The walls of the container would enforce a Neumann
condition on the concentration, but not on the electric field. The boundary conditions
on the electric field would be determined by some setup external to the container.
For a known electric field, with boundaries outside the chamber, the above code
would allow for approximations of the derivatives and Laplacian using only interior
points. However, boundary conditions, Dirichlet or Neumann, must be specified for
the concentration profile. These conditions are addressed on a case-by-case basis.
3.3 Forming A
Given the vectors resulting from reshaping the concentration and electric field
intensity tensors, C and E2, the operator can be represented using the Laplacian and
derivative matrices:
AC = TVG VE2 + TCV2E2 $V2G (3.9)
AC = F(dxE2dxC + dyE2dyC + dzE2dzC) + FCV2E2 $V2G (3.10)
A Y(D QxE2/S.x + DdyE2Ay + DqzE 2 A2) + TDLE2 24
(3.11)


Where Dqxe2, DdyE2, Dqze2 are the diagonalized vectors resulting from the ap-
plication of the central difference derivative matrices on the electric held intensity
vector, Ax,Ay,Az are the up-winding derivative matrices, DLE2 is the diagonalized
vector from the central difference Laplacian applied to the electric held intensity, and
L is the up-winding Laplacian matrix.
25


4. Numerical Results
The MIT Electromagnetic Equation Propagation FDTD software, or MEEP, nu-
merically solves electromagnetics problems in which dielectrics and conductors are
well defined [17]. The output (ie. E,H,E2,H2) is returned on an equidistant mesh
of the set spatial dimension (1-3D) plus one temporal dimension. This output is
particularly well constructed for use in the method outlined in chapter 3.
4.1 Mesh Size and Stabilty
If we consider the DEP diffusion equation in terms of units of voltage (Volts -
[V]), time (Seconds [S]), concentration (arbitrary [C]), and distant (unity [D]).
BC
= $'V2C TVE2 VC TCV2E2 (4.1)
dt v '
[C]^]-1 = m[D]-2[C] [T] [D]-l[V]2[D}-2[D}-l[C] [T][C}[D]-2[V]2[D]-2 (4.2)
- [$] = [.D]2[Sr1 (4.3)
- [r] = [DnV]-2^]-1 (4.4)
Equation 4.1 specihes the diffusion and convection coefficients to be a = $ and
bv = YdvE2 respectively, where v is x, y, or z. The stability requirement for mesh
size is given by [13] to be:
hv (4.5)
bv
Given the equidistant mesh hx = hy = hz = h equation 4.5 reduces to the
requirement:
h
$ 1
T maxv dUE2
26
(4.6)


Expressing this as a requirement on the number of nodes for rectangular domain
with sides given by Lx,Ly,Lz:
[n, m,p\
Lx Ly Lz
h h h
I Vx 'by I iZ
1
h
[L
X j Ly j -Z/J
(4.7)
r i rr t t i r max* dEf
-> [n,m/p] > [Lx, Ly, Lz\-------- (4.8)
4.2 T and $ in SI Units
Here the values of T and $ will be calculated for a metallic nanotube of diameter,
d ~ 1.2[nm], and length, l ps 1000[nm\ (note that these dimensions are based on
the manufactures claims of the SWNTs used in the experimental section). Metallic
nanotubes have been estimated [2] [12] to have an effective relative dielectric constant
on the order of 103 104. To ensure stability, tr = 104 will be used as it corresponds
to a large T and therefore smaller step size. The Stokes-Einstien radius for a SWNT
was given in terms of d and l as [16]:
a
1 l
2 ln(l/d) + 0.32
~ 85 [nm\
(4.9)
r
7Td2l tmet ~ tH20
8 Ganr]
3.49E
23
m
U25
(4.10)

RT
NA^anr)
w 2.85E
(4.11)
4.3 Error in a Known Case
In the case of a constant electric held, the DEP finite difference operator A reduces
to the scaled negative Laplacian, A = TV2. Resulting in well known eigenvalues.
Equation 4.13 gives the eigenvalues for the two dimension Laplacian on a rectangular
domain with Neumann boundary conditions. Note that the DEP boundary condition
is in fact the zero flux Neumann condition: VC = -jj)CVE2 = 0.
27


(4.12)
A -a -TV2
A = (mr/Lx)2 + [nm/Ly)2 : n,m = 0,1, 2, 3,... (4-13)
Using the two dimensional code given in appendix A, with the Laplacian boundary
conditions set as described above, the driver constant-test given below results in the
smallest 20 real eigenvalue components. Together, these values have a less than one
percent relative error when compared to the known values resulting from equation
4.13. The results are given in figure 4.1.
Smallest 20 real eigenvalue components
Figure 4.1: Approximated and known eigenvalues for the scaled Laplacian operator.
% Tests dep_fd_2d when intensity is constant and the DEP equation reduces
% to the scaled Laplace equation: AC = Phi Laplacian(C)
28


% Known solution is lambda_i = Phi"-1 * ((n*Pi/L)'2 + (m*Pi/L)~2
% for n,m =0,1,2,..., N-l
% Test is on a N x N mesh with step size h (length, L = h*N)
N = 30; % Default mesh size
h = 0.01; % Default step size
gamma = 3.49E-23;
phi = 2.85E-12;
% Find dep_fd_2d estimated eigenvalues
A = dep_fd_2d(ones(N,N), gamma, phi, h) ;
D = eigs(A, 20, 'sr');
R = real(D);
% Calculate known eigenvalues;
L = [];
Li = [ ];
for i = 0:N-l
lambda = (i*pi/(N*h))"2;
Li = [Li lambda];
end
for 1=1:N
L = [L (LI(i)*ones(1,N) + LI)];
end
L = sort(L)';
L = phi.*L;
L = L(1:N~2);
SE = norm(R L(1:20))/norm(R);
29


fprintf('Relative error in the smallest 20 components: %8.2E \n', SE)
X = linspace (1,20,20) ;
figure
plot(X, L(1:20),X,R, '*')
title('Smallest 20 real eigenvalue components')
legend('Known values.','Approximated values.','Location','Northwest');
constant-test
Relative error in the smallest 20 components: 2.71E-03
4.4 Parallel Plate Dielectric Waveguide
Consider a dielectric parallel plate waveguide with wall gap a that is designed to
propagate in the z axis. Choosing a such that single mode, TEW, propagation occurs
for frequencies around 2Ah[GHz] would result in a = 45A9 ~ 6.5[mm] for the
dielectric value of the DEP solution. The following MEEP code simulates this system
under the excitation of an internal microwave line source.
(set! geometry-lattice
(make lattice (size 9.5 53 no-size))
(set! geometry
(list
(make block
(center 0 0)
(size infinity infinity infinity)
(material air)
30


(make block
(center 0 0)
(size 6.5 infinity infinity)
(material (make dielectric (epsilon 80)))
(set! sources
(list
(make source
(src (make continuous-src (frequency 0.0082)))
(component Ez)
(center 6.5 0)
(set! pml-layers
(list
(make pml
(thickness 1.0)
(set! resolution 10)
(run-until 1000
(at-beginning output-epsilon)
(to-appended "power" (at-every 1 output-dpwr))
31


4.4.1 Boundary Conditions
For the TEW propagation mode, the electric held takes the form given by equation
1.35. Using the double angle identity results in the electric held intensity of equation
Clearly, as a result of equation 4.16, the boundary condition for the concentration
prohle is again the Neumann condition = 0.
4.4.2 {cq} Approximations
The MEEP intensity output can be scaled to hnd, using dep-fdJ2d (given in
Appendix A), the minimum real eigenvalue component for various intensity maximum
magnitudes. The results of varying the intensity peak from 10-5^- to lO10^- are
plotted in figure 4.2.
4.4.3 Forcing Exponential Decay
As opposed to propagating waves, reducing the wall separation to below the
distance corresponding to the cutoff of 2.45 [GHz] leads to exponentially decaying
helds in the z direction. The only change to the MEEP code to rehect this is given
below. Again, the results of varying the intensity peak from lO5^- to 1010^ are
plotted in figure 4.3.
(set! geometry
4.15.
(4.14)
(4.15)
dxE2\f0,a) = E0 sin{x) = 0
a a
(4.16)
32


Minimum Real Eigenvalue Component vs Intensity Magnitude for 6.5 mm Waveguide
Figure 4.2: Log-log plot of minimum real eigenvalue component vs the maximum field
intensity.
(list
(make block
(center 0 0)
(size infinity infinity infinity)
(material air)
(make block
(center 0 0)
(size 3.5 infinity infinity)
(material (make dielectric (epsilon 80)))
(set! sources
(list
(make source
33


(src (make continuous-src (frequency 0.0082)))
(component Ez)
(center 3.5 0)
Minimum Real Eigenvalue Component vs Intensity Magnitude for 3.5 mm Waveguide
Figure 4.3: Log-log plot of minimum real eigenvalue component vs the maximum field
intensity.
4.5 Conclusion
Figures 4.2 and 4.3 clearly demonstrate that the slowest (largest) time constant,
given by tsiow = becomes progressively faster as the intensity max increases in
magnitude. Accordingly, those systems with higher intensity maximum are expected
to decay to the steady state faster than a system with the same geometry but weaker
max. Further, the slight increase in the eigenvalues of figure 4.3 over figure 4.2 indicate
that a sharper gradient (in this case, resulting from an exponential decay) does cause
an increase in the rate of decay to steady state.
34


(a) Above Cutoff
Steady State Solution of Stimulated Waveguide Above Cut-off
8000
7000
6000
5000
4000
3000
2000
1000
0 50 100 150 200 250 300 350 400 450 500
(b) Below Cutoff
Figure 4.4: Steady state concentration profiles for (a) 6.5 mm and (b) 3.5 mm. dielec-
tric parallel plate waveguides stimulated at 2.45 [GHz],
Additionally, see figure 4.4, the steady state solution in the case of an expo-
nentially decaying electric held intensity has a much sharper peak then that of the
non-decaying case. Given that the exponential decay obtains a higher peak concen-
tration and achieves it faster than the non-decaying case, it can be concluded that
intensity geometries that approach singularities are superior in driving SWNT DEP.
Lastly, linearly extrapolating the line in figure 4.3 to the breakdown intensity of
air, approximately lO13^- (which would have violated equation 4.6 to solve directly),
would suggest a minimum real eigenvalue component of roughly a
TflVfl KT2. This
35


results in a time constant on the order of just a few minutes.
36


5. Laboratory Methods
The fluidic force nature of dielectrophoresis of Carbon Nanotubes was investi-
gated using a three step process. First, super purified plasma discharge (> 95%w/v
SWNT by manufacturers claim) nanotubes were purchased from Nanotintegris and
suspended in a 0.25%w/v aqueous solution of SDS. This concentration is just above
critical micelle concentration. After successful dispersion, the aqueous SWNT-SDS
solution was placed in a cylindrical quartz capillary effectively forming a dielectric
waveguide. Propagation modes were excited using gold coated field concentrators and
a 2.45 [GHz] source. Lastly, the deposited nanotubes were resuspended and optically
evaluated.
5.1 Dispersion
An aqueous solution of the anionic surfactant sodium dodecyl sulfate (SDS) was
prepared at a concentration of 0.25%w/v. The critical micelle concentration of SDS
ranges from 6-8 mM or roughly 0.17 0.24%w/v. Single wall carbon nanotubes were
dispersed in this solution at a concentration of 0.06 mg/mL.
The SWNT/SDS (aq) mixture was initially bath sonicated twenty-three times
with each repetition consiting of 8 minutes, totalling 3 hours and 4 minutes. The
bath sonication was immediately followed by thirty minutes of horn sonication.
After a small reference sample was taken from this initial dispersion, the remain-
der was centrifuged in a Hemle Z360k at 13,000 rpm for 90 minutes. The top 80% of
the supernatant was carefully decanted and stored.
5.2 Experimental Setup
An electromagnetic single mode resonance chamber was constructed using a
shorted aluminum waveguide designed for 2.45 [GHz] waves. At one end, a mag-
netron aperture was placed at a quarter wavelength from the back wall. This allowed
backwards travelling waves to move a quarter wavelength to the back wall, undergo
a half wavelength reflection, and travel back a quarter wavelength, ending in a full
37


wavelength path length allowing constructive interference with the forward travelling
wave. A second aperture was placed at a half wavelength from the forward wall.
Given that this location is at an antinode of the generated standing wave, maximum
electric held amplitude is experienced at this aperture.
The magnetron, by design, operates as a diode so a voltage doubler is used to
maximize the efficiency. A relay is used to pulsate power to the magnetron and hence
control the time average power delivered to the resonant cavity.
The SWNT SDS (aq) dispersion is placed in a fused quartz capillary along with
two axial gold plated nickel needles creating a small gap. This is displayed in the
figure below.
Figure 5.1: Arrangement of needles in fused quartz capillaries (4mm interior diameter
on the left and 8mm diameter on the right.)
Given the high relative dielectric permittivity of the solution, the arrangement
creates a dielectric waveguide. The needles concentrate any applied held to a region
much smaller than the wavelength of the applied held. This is known to create an
approximately spherical scattered wave in turn, exciting any possible modes within
38


Figure 5.2: J^mm capillary with gold platted needles filled with SWNT SDS solution.
the capillary.
Unlike rectangular waveguides, with the simple propagation modes given in chap-
ter 1, the effective wavefronts of a cylindrical waveguide are dependent on Bessels
functions. Despite the complexity of the general case, the cutoff frequency for the
lowest order modes (TEn) are given by [1]:
_ 1.8412 FT c 2ttt V pt (5.1)
-> /c4mm 4.88 [GHz\ (5.2)
-> flmm 2AA[GHz] (5.3)
As a result of these cutoff frequencies, when exposed to microwave 2A5[GHZ]
frequencies, modes are excited in the 8mm capillary, whereas the waves exponentially
decay in the 4mm capillary.
39


Figure 5.3: 8mm capillary with gold platted needles filled with SWNT SDS solution.
Each capillary is inserted into the secondary aperture and subjected to pulsed
microwaves. The magnetron is powered in 25pS pulses at a 13[kHz\ repetition rate.
5.2.1 Field Intensity Distributions
MEEP was used to simulate this setup and resulting electric held intensity profiles
are depicted in figure 5.4. The higher contrast in the subfigure (b) indicates that the
4[mm] capillary experienced a sharper gradient than that of the 8 [mm] capillary.
5.3 Secondary Suspension
After four cycles, each five minutes in duration, of pulsed microwaves, the gold
needles are removed from the solution. Any SWNTs deposited on the needles are
removed by placing the needles in a small plastic tube with aqueous sodium dodecyl
sulfonate and bath sonicated for 24 minutes.
A sample of the resulting solution was taken and centrifuged in a similar method
to the initial dispersion. This sample then underwent absorption spectroscopy to eval-
uate metallic enrichment. A dual beam spectrophotometer is used with an aqueous
SDS background to remove any absorption not due to the nanotubes.
40


(a) Intensity Profile in 8[mm] Quartz
Avg
sxy
(b) Intensity Profile in 4[mm] Quartz
Figure 5.4: Time average intensity field distributions (Avgxy) for the (a) 8mm and
(b) 4mm quartz tubing.


6. Experimental Results
The optical absorption spectra of Single Walled Carbon Nanotubes can be pre-
dicted using a tight binding model as outlined in [19]. It was determined that the
spacing of the Van Hove singularities was given by the ratio 1 : 2 : 3 for metallic
SWNTs and 1 : 2 : 4 for semiconducting SWNTs. Further, the lowest order Van
Hove singularity for each species was found to be:
Where y0 ~ 2.9[eV] is the energy value of the hopping interaction, acc ~
the SWNT diameter range (according to the manufacturer). These relations lead to
the transition energies given in the table 6.1.
The Nn transition lies outside of the range of most spectrophotometers and the
M22 and S33 have significant overlap. As a result, the Mu and N22 are the most
interference of higher energy transitions below ~ 600 [nm], attention will be focused
on the range 600 900[nm].
Given the data in table 6.1, the Mn is expected to dominate in the region of
approximately 600 650[nm]. Conversely, the N22 transition should dominated in the
850 900[nm] region. The intermittent region, 650 850[nm], will be influenced by
both of these transitions.
6.1 Pre and Post DEP
After the initial sonication and centrifugation as was described in chapter 5, the
absorption data in figure 6.2 (a) was taken.
After the four five minute DEP cycles, the needles were removed from the solution
670 acc
d
270 flee
d
(6.1)
(6.2)
0.143[nm] is the nearest neighbor carbon carbon distance, and d = 0.9 1.7[nm] is
well suited for analysis. Given the 900[nm] limit of the spectrophotometer and the
and allowed to dry. No visible deposition had occurred on the needles placed in the
42


Table 6.1: Tight binding predicted photonic energies and wavelengths for nanotube
Van Hove transitions.
Transition Formula Energy Wavelength
jpM -Mi 670a.cc d 1.464 2.765[eE] 448 847 [nm]
jpM M2 1270dec d 2.928 5.529[eC] 224 423 [nm]
Ef; 27oacc d 0.488 0.922[eC] 1344 2541 [nm]
ES22 4'Yo cicc d 0.976 1.844[eC] 672 1270[nm]
Ei3 8joo.ee d 1.952 3.688[eC] 336 635 [nm]
8mm capillary; however, a clearly periodic deposition pattern occurred in the 4mm
capillary (figure 6.1). Note that spherical waves propagate as Bessels functions, not
sinusoids, so an increasing wavelength is expected as the wave moves from the point
towards the base of the needle.
The needles were cut and placed in a bath of aqueous sodium dodecyl sulfonate
and bath sonicated until the deposited had visibly been removed from the surface
of the needles (24 minutes). The needles were then discarded and the solution was
further bath sonicated for 22 eight minute cycles (totalling 3 hours and 4 minutes).
Optical extinction data was then taken for this post-DEP sample (figure 6.2).
6.2 Analysis
The post-DEP drop in the S22 and S33 into the negative absorption range indicates
that the free micelles in the background solution were able to absorb more in that
region that the remaining semiconducting SWNTs. The absence of the Mu peak from
448nm to roughly 725nm is believed to be the result of the smaller diameter, and
therefore more dense, SWNTs being filtered out during the centrifugation processes.
6.3 Conclusion
Comparison of the data presented in figure 6.2 shows a sharp decrease in the
semiconducting dominating region (850 + [nm]), a noticeable decrease in the 650
43


Figure 6.1: Bottom needle from 4mm capillary after four five minute exposures to
microwaves.
850[nm] region corresponding to both Mn and S22 transitions, and only a small
decrease in the Mn dominated region of 600 650[nm]. In accordance with Beers law,
this indicates that the semiconducting concentration was decreased in more drastic
manner than the metallic.
6.3.1 Future Work
The work demonstrated here clearly represents a significant step towards large
scale SWNT separation, however many difficulties still remain. Deposition did not
increase substantially with longer exposure times. Further, while the content of the
deposited nanotubes was highly metallic, the remaining solution was not of signifi-
cantly increased semiconducting content. This indicates that only a small percentage
of the metallic nanotubes were deposited.
Given that the solution had pre-centrifuge total SWNT concentration of 0.06^,
that approximately one third of nanotubes are metallic, and that the 4mm capillary
held roughly lmL of solution, it can be assumed that < 20pg (possibly much less due
to the centrifuge) of metallic nanotubes were in the system to begin with. From here
44


600 900 nm Extinction Data for Pre DEP Sample
(a) Pre DEP
600 900 nm Extinction Data for Post DEP Sample
(b) Post DEP
Figure 6.2: Extinction spectra for SWNT SDS solution (a) before and (b) after DEP
exposure.
45


it is obvious that if only a fraction of the metallic nanotubes are deposited, then the
yield of this system is quiet low.
Finite differences was used in order to gain insight into the behavior of the system
in the presence of fields approaching singularities. Given the inherent weakness of
finite differences to describe singularities, further numerical study of the system would
greatly benefit from more apt models such as finite element.
46


REFERENCES
[1] C.A. Balanis, Advanced Engineering Electromagnetics, ed. 2, Wiley, 2012.
[2] L. Benedict, S. Louie, and M Cohen, Static polarizabilities of single-wall
carbon nanotubes, Phys. Rev. B, 52, 8541, 1995.
[3] S. Blatt, Dielectrophoresis of Single Walled Carbon Nanotubes, Dissertation
Kasrlsruhe Institute of Technology, 2008.
[4] J.C. Charlier, X. Blase, S. Roche, Electronic and Transport Properties of
Nanotubes, Rev. of Mod. Phy., 79, 2007.
[5] R. Courant, E. Isaacson, M. Rees, On the solution of nonlinear hyperbolic
differential equations by finite differences, Comm. Pure Appl. Math. 5, 243 255,
1952.
[6] F. Gibou, R. Fedwik, A Fourth Order Accurate Discretization for the Laplace
and Heat Equations on Arbitrary Domains with Applications to the Stefan Prob-
lem, J. Comp. Phy. 202.2, 577-601, 2005.
[7] H.T. Ham, Y.S. Choi, I.J. Chung, An Explanation of Dispersion States of
Single-walled Carbon Nanotubes in Solvents and Aqueous Surf act Solutions Using
Solubility Parameters, J. Col. Int. Sci. 286, 216-223, 2005.
[8] S. A. Hodge, M. K. Bayazit, K. S. Coleman, and M. S. P. Shaffer,
Unweaving the rainbow: a review of the relationship between single-walled carbon
nanotube molecular structures and their chemical reactivity, Chem. Soc. Rev.,
41, 2012.
[9] S. Iijima, Helical Microtubules of Graphitic Carbon, Nature 543.6348, 1991.
[10] C. Johnk, Engineering Electromagnetic Fields and Waves, ed. 2, Wiley, 1988.
[11] A. Hartschuh, H. N. Pedrosa, J. Peterson, L. Huang, P. Anger, H.
Qian, A. J. Meixner, M. Steiner, L. Novotny, and T. D. Krauss,
Single Carbon Nanotube Optical Spectroscopy, ChemPhysChem, 6, 2005.
[12] R. Krupke, F. Hennrich, M. M. Kappes, and H. V. Lohneysen, Surface
Conductance Induced Dielectrophoresis of Semiconducting Single-Walled Carbon
Nanotubes, Nano Letters, Vol. 4, No. 8, 2004.
[13] E. Majchrzak and L. Turchan, The finite difference method for transient
convection-diffusion problems, Scientific Research of the Institute of Mathematics
and Computer Science, Vol. 11, No. 1, 2012.
47


[14] M.P. Marder, Condensed Matter Physics, John Wiley & Sons, 2010.
[15] A.N. Mina, A.A. Awadallah, A.H. Phillips, and R.R. Ahmed, Simula-
tion of the Band Structure of Graphene and Carbon Nanotube, J. Phys.: Conf.
Ser., 343, 2012.
[16] N. Nair, W. Kim, R. Braatz, and M. Strano, Dynamics of Surfactant-
Suspended Single-Walled Carbon Nanotubes in a Centrifugal Field, Langmuir,
24, 1790-1795, 2008.
[17] Ardavan F. Oskooi, David Roundy, Mihai Ibanescu, Peter Bermel,
J. D. Joannopoulos, and Steven G. Johnson, MEEP: A flexible free-
software package for electromagnetic simulations by the FDTD method, Com-
puter Physics Communications 181, 687702, 2010.
[18] J. Ouellette, Building the Nanofuture with Carbon Tubes, Industrial Physicist,
Vol. 8, Iss. 6, 2002.
[19] M. Ouyang, J. Huang, and C. Lieber, Fundamental Electronic Properties
and Applications of Single-Walled Carbon Nanotubes, Acc. Chem. Res., 35 (12),
1018-1025, 2002.
[20] B.R. Priya and H.J. Byrne, Inuestigation of Sodium Dodecyl Benzene Sul-
fonate Assisted Dispersion of Debundling of Single-Wall Carbon Nanotubes, J.
Phys. Chem. C, 112, 332-337, 2008.
48


APPENDIX A. DEP Finite Difference Code
The code listed here was used throughout this thesis to find the operator A such
that the DEP equation 2.20 holds. It is provided in both two and three spatial
dimensional coordinate systems.
A.l Two Dimensions
function [A] = dep_fd_2d(intensity, gamma, phi,h)
% Uses method of finite differences to build the 2D DEP operator:
% dC/dt = -AC
0,
0
% INPUT: intensity 2D E"2 tensor
% gamma Advection constant
% phi Diffusion constant
% h intensity tensor step-size (scale)
0.
0
% OUTPUT: A operator such that dC/dt + AC = 0
[m,n] = size(intensity); % Save tensor dimensions
intensity = reshape(intensity, [ ],1); % Form vector from 3D tensor
% Laplacian matrix
L = laplacian_2d(m,n,h);
% Derivative Matrice
[DX,DY] = derivative_matrices_2d(m,n,h);
% Find electric intensity derivative and laplacian vectors;
EDXE = DX*intensity;
EDYE = DY*intensity;
49


ELE = L*intensity;
% Diagonalize
DEDXE = spdiags(EDXE,0,m*n, m*n);
DEDYE = spdiags(EDYE,0,m*n, m*n) ;
DELE = spdiags(ELE,0,m*n, m*n) ;
% Neumann BC Derivative Matrix
[CDX,CDY] = derivative_matrices_2d(m, n, h) ;
% X boundary conditions
for i = 0: (n-1)
j = 1 + i*m;
k = m + i*m;
% Clear boundary rows
CDX ( j, :) = zeros (1, m*n);
CDX(k,:) = zeros(1, m*n) ;
% Load no flux requirement
CDX ( j,j) = EDXE ( j);
CDX(k,k) = EDXE(k);
end
% Y boundary conditions
for i = 1:m
j = m*n i + 1;
% Clear Boundary Rows
CDY(i,:) = zeros (1, m*n);
CDY(j, :) = zeros (1, m*n);
% Load Data
CDY(i,i) = EDYE(i);
CDY(j,j) = EDYE(j);
50


end
% Calculate operator
A = gamma.*((DEDXE*CDX)+(DEDYE*CDY)+DELE) phi.*L;
function [DX,DY] = derivative_matrices_2d(m, n,h)
% Returns the mn x mn derivative matrices in cartesian coordinates.
% Uses 4th order central difference approx in rows 3:end-2, 2nd order
% central difference approx in rows 2 and end 1, 2nd order forward
% difference in row 1, and second order backwards difference in row end.
if m<5 || n<5
error('m,n, must be greater than or equal to 5')
end
X = spdiags(ones(m,1)*[1 -808 -1],-2:2,m,m);
Y = spdiags(ones(n,1)*[1 -808 -1],-2:2,n,n);
%Boundary extrapolation (2nd order forward/central)
X (1: 2,1: 4) = [-18 24 -6 0; -6 0 6 0] ;
X(end-1:end,end-3: :end) = [0 -6 0 6; 0 6 -24 18] ;
Y (1: 2,1: 4) = [-18 24 -6 0; -6 0 6 0] ;
Y(end-1:end,end-3: lend) = [0 -6 0 6; 0 6 -24 18] ;
Y_expanded = kron(Y,speye(m,m));
DX = (kron(speye(n),X))./(12*h) ;
DY = (Y_expanded)./(12*h);
end
51


\end{listing}
\begin{lstlisting}
function [L] = laplacian_2d(m, n, h)
% Returns the mn x mn Laplacian matrix in cartesian Coordinates
% Uses 4th order central difference approx in rows 3:end-2, 2nd order
% central difference approx in rows 2 and end 1, 2nd order forward
% difference in row 1, and second order backwards difference in row end.
if m<5 | | n<5
error('m,n must be greater than or equal to 5')
end
LX = spdiags(ones(m,1)*[-1/12 4/3 -5/2 4/3 -1/12],-2:2,m,m);
LY = spdiags(ones(n,1)*[-1/12 4/3 -5/2 4/3 -1/12],-2:2,n,n);
%LX = spdiags(ones(m,1)*[1 -2 1],-1:1,m, m) ;
%LY = spdiags(ones(n,1)*[1 -2 1] ,-1:1, n, n) ;
% Upper Boundary Extrapolation
LX (1 : : 2,1: 5) = [-1 1 0 0 0; 1 -2 10 0] ;
LY (1 : : 2,1: 5) = [-1 1 0 0 0; 1 -2 10 0] ;
% Lower Boundary Extraptolation
LX(end-1:end,end-4:end) = [0 0 1 -2 1; 0 0 0 1 -1];
LY(end-1:end,end-4:end) = [0 0 1 -2 1; 0 0 0 1 -1];
LY_expanded = kron(LY,speye(m, m)) ;
L = (kron(speye(n),LX) + LY_expanded)./(h*2);
end
52


A.2 Three Dimensions
function [A] = dep_fd_3d(intensity, gamma, phi, h)
% Uses method of finite differences to calculate a steady state concentration profile
% resulting from a known electric field.
0,
0
% INPUT: intensity 3D E"2 tensor
% gamma Advection constant
% phi Diffusion constant
% h Step size (resolution) of intensity tensor.
0,
0
% OUTPUT: A operator such that dC/dt + AC = 0
[m,n,p] = size(intensity); % Save tensor dimensions
intensity = reshape(intensity,[],1); % Form vector from 3D tensor
% Laplacian matrix
L = laplacian_3d(m,n,p,h);
% Derivative Matrice
[DX,DY,DZ] = derivative_matrices_cartesian(m,n,p,h);
% Find electric intensity derivative and laplacian vectors;
EDXE = DX*intensity;
EDYE = DY*intensity;
EDZE = DZ*intensity;
ELE = L*intensity;
% Diagonalize
DEDXE = spdiags(EDXE,0,m*n*p, m*n*p);
53


DEDYE
= spdiags(EDYE,0,m*n*p, m*n*p);
DEDZE = spdiags(EDZE,0,m*n*p, m*n*p);
DELE = spdiags(ELE,0,m*n*p, m*n*p) ;
% Concentration Derivative Matrices (Neumann Boundary)
CX = sparse(DX);
CY = sparse(DY);
CZ = sparse(DZ);
% X boundary conditions
for i = 0:(n*p-l)
j = 1 + i*m;
k = m + i*m;
% Clear boundary rows
CX(j :) = zeros (1, m*n*p);
CX(k,:) = zeros(1, m*n*p);
% Load no flux requirement
CX(j,j) = EDXE(j);
CX(k,k) = EDXE(k);
end
% Y boundary conditions
for i = 0:(p-1)
for j = 1:m
k = j + i*m*n;
CY(k, :) = zeros (1, m*n*p);
CY(k,k) = EDYE(k);
end
for j = (1 + m*(n-1)) :m*n
k = j + i*m*n;
CY(k, :) = zeros (1, m*n*p);
CY(k,k) = EDYE(k);
% Clear Boundary Row
% Load Data
% Clear Boundary Row
% Load Data
54


end
end
% Z boundary conditions
for i = 1:m*n
j = m*n*p i;
% Clear Boundary Rows
CZ(i,:) = zeros (1, m*n*p);
CZ(j,:) = zeros (1, m*n*p);
% Load Data
CZ (i,i) = EDZE (i);
CZ(j, j) = EDZE ( j) ;
end
% Calculate operator
A = (gamma*((DEDXE*CX)+(DEDYE*CY)+(DEDZE*CZ)+DELE)) - phi*L;
end
function [DX,DY,DZ] = derivative_matrices_cartesian(m,n,p,h)
% Returns the mnp x mnp derivative matrices in cartesian coordinates.
% Uses 4th order central difference approx in rows 3:end-2, 2nd order
% central difference approx in rows 2 and end 1, 2nd order forward
% difference in row 1, and second order backwards difference in row end.
% All approximations use step size h.
if m<5 | n<5 | p<5
error('m,n,p must be greater than or equal to 5')
end
X = spdiags(ones(m,1)*[1/12 -2/3 0 2/3 -1/12],-2:2,m,m);
55


Y = spdiags(ones(n,1)*[1/12 -2/3 0 2/3 -1/12],-2:2,n,n);
Z = spdiags(ones(p, 1)*[1/12 -2/3 0 2/3 -1/12],-2:2,p,p);
% Boundary extrapolation (2nd order forward/central)
X (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
X(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
Y (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
Y(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
Z (1: 2,1: 4) = [-3/2 2 -1/2 0; 0 1-2 1];
Z(end-1:end,end-3:end) = [1-21 0; 0 1/2 -2 3/2];
Y_expanded = kron(Y,speye(m,m));
Z_expanded = kron(Z,speye(m*n,m*n)) ;
DX = (kron(speye(n*p),X))./h;
DY = (kron(speye(p,p),Y_expanded))./h;
DZ = (Z_expanded) ./h;
end
function [L] = laplacian_3d(m, n, p,h)
% Returns the mnp x mnp Laplacian matrix in cartesian Coordinates
% Uses 4th order central difference approx in rows 3:end-2, 2nd order
% central difference approx in rows 2 and end 1, 2nd order forward
% difference in row 1, and second order backwards difference in row end.
% All approximations use step size h.
if m<5 | n<5 | p<5
error('m,n,p must be greater than or equal to 5')
end
56


LX = spdiags(ones(m,1)*[ >1/12 4/3 -5/2 4/3 -1/12],-2:2,m,m);
LY = spdiags(ones(n,1)*[ >1/12 4/3 -5/2 4/3 -1/12],-2:2,n,n);
LZ = spdiags(ones(p,1)*[ >1/12 4/3 -5/2 4/3 -1/12] ,-2:2,p,p);
% Upper Boundary Extrapolation
LX (1: :2, 1 : :5) = = [0 0 0 0 0; 1 -2 1 0 0] ;
LY (1 : :2, 1 : :5) = = [0 0 0 0 0; 1 -2 1 0 0] ;
LZ (1 : :2, 1 : :5) = = [0 0 0 0 0; 1 -2 1 0 0] ;
% Lower Boundary Extraptolation
LX(end-1:end,end-4:end) =[001-21; 00000];
LY(end-1:end,end-4:end) =[001-21; 00000];
LZ(end-1:end,end-4:end) = [0 0 1 -2 1; 0 0 0 0 0];
LY_expanded = kron(LY,speye(m,m));
LZ_expanded = kron(LZ,speye(m*n,m*n));
L = (kron(speye(n*p),LX) + kron(speye(p,p),LY.expanded) + LZ_expanded)./(h"2)
end
57


Full Text

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SCALABLEDIELECTROPHORESISOFSINGLEWALLEDCARBON NANOTUBES by WILLIAMA.FITZHUGH B.S.,UniversityofColoradoDenver,2012 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoDenverinpartialfulllment oftherequirementsforthedegreeof MasterofIntegratedSciences IntegratedSciences 2015

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ThisthesisfortheMasterofIntegratedSciencesdegreeby WilliamA.Fitzhugh hasbeenapprovedforthe IntegratedSciencesProgram by MasoudAsadi-Zeydabadi,Chair JanMandel RandallTagg July24,2015 ii

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Fitzhugh,WilliamA.M.I.S. ScalableDielectrophoresisofSingleWalledCarbonNanotubes ThesisdirectedbyAssociateProfessorMasoudAsadi-Zeydabadi ABSTRACT SingleWalledCarbonNanotubesSWNTshaveattractedmuchattentionasa candidatematerialforfuturenano-scale'beyondsilicon'devices.Howeverindustrialscaleoperationshavebeenimpededbydicultiesinseparatingthemetallicand semiconductingspecies.Thispaperaddressestheuseofhighlyinhomogeneousalternatingelectricelds,dielectrophoresis,toisolateSWNTspeciesinscaledsystems. Bothnumericalandexperimentalmethodswillbediscussed. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:MasoudAsadi-Zeydabadi iii

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DEDICATION Thisthesisisdedicatedtomyparents,fortheirneverendingsupport.

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ACKNOWLEDGMENT ThisthesiswouldnothavebeenpossiblewithoutthegeneroussupportofDr.Asadi, Dr.Tagg,Dr.Geyer,Dr.Golkowski,andRyanJacobs.SpecialthankstoDr.Maron, Dept.ofChemisty,Dr.MandelandDr.Langou,Dept.ofAppliedMathematics, andDr.HuberandKrisBunker,Dept.ofPhysics.

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TABLEOFCONTENTS Tables........................................viii Figures.......................................ix Chapter 1.Introduction...................................1 1.1Graphene.................................1 1.2SingleWalledCarbonNanotubes....................4 1.3NanotubeAbsorptionSpectroscopy..................6 1.4Dielectrophoresis.............................7 1.5SWNTDielectrophoresis........................8 1.6RectangularWaveguidesandResonators...............10 1.6.1TransverseElectric........................11 1.6.2TransverseMagnetic.......................11 1.6.3Resonators............................12 2.Theory......................................13 2.1AccountingforDrag...........................13 2.2SteadyStateSolution..........................14 2.3BoundaryandInitialConditions....................15 2.4TemporalEvolution...........................16 2.5SlowestDecayingEigencouple.....................18 3.NumericalMethods...............................20 3.1FiniteDierences............................20 3.1.1ReshapingTensors........................20 3.1.2LaplacianMatrix.........................21 3.1.3DerivativeMatrix........................21 3.2BoundaryExtrapolation........................22 3.2.1BoundaryConditions.......................24 vi

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3.3FormingA................................24 4.NumericalResults................................26 4.1MeshSizeandStabilty.........................26 4.2)-326(andinSIUnits...........................27 4.3ErrorinaKnownCase.........................27 4.4ParallelPlateDielectricWaveguide..................30 4.4.1BoundaryConditions.......................32 4.4.2 f i g Approximations.......................32 4.4.3ForcingExponentialDecay...................32 4.5Conclusion................................34 5.LaboratoryMethods..............................37 5.1Dispersion................................37 5.2ExperimentalSetup...........................37 5.2.1FieldIntensityDistributions...................40 5.3SecondarySuspension..........................40 6.ExperimentalResults..............................42 6.1PreandPostDEP............................42 6.2Analysis.................................43 6.3Conclusion................................43 6.3.1FutureWork...........................44 References ......................................47 Appendix A.DEPFiniteDierenceCode..........................49 A.1TwoDimensions.............................49 A.2ThreeDimensions............................53 vii

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TABLES Table 6.1 Tightbindingpredictedphotonicenergiesandwavelengthsfornanotube VanHovetransitions. ............................43 viii

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FIGURES Figure 1.1 Honeycomblatticeofgraphene.Imagefrom[14]. .............2 1.2 ReciprocalLatticeofGraphene.Imagefrom[15]. .............3 1.3 Comparisonofthetightbindingapproximationwithabinitiocalculations forthebrandstructureofgraphene.Imagefrom[15]. ...........3 1.4 Demonstrationofthegraphenerollingprocessfora ; 3 nanotube.Image from[4]. ....................................4 1.5 DensityofstatesformetallicandsemiconductingSWNTs.VanHove transitionsareillustratedbyarrows.Imagefrom[8]. ...........6 1.6 AbsorptionspectraofathinlmofSWNTs.Imagefrom[11]. ......7 4.1 ApproximatedandknowneigenvaluesforthescaledLaplacianoperator. .28 4.2 Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. ...................................33 4.3 Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. ...................................34 4.4 Steadystateconcentrationprolesfora6.5mmandb3.5mmdielectricparallelplatewaveguidesstimulatedat2.45[GHz]. ..........35 5.1 Arrangementofneedlesinfusedquartzcapillariesmminteriordiameter ontheleftand8mmdiameterontheright. ................38 5.2 4mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. 39 5.3 8mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. 40 5.4 Timeaverageintensityelddistributions Avg xy forthea8mmand b4mmquartztubing. ............................41 6.1 Bottomneedlefrom4mmcapillaryafterfourveminuteexposuresto microwaves. ..................................44 ix

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6.2 ExtinctionspectraforSWNT-SDSsolutionabeforeandbafterDEP exposure. ...................................45 x

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1.Introduction Consistingofasinglecylindricallayerofgraphene,SingleWalledCarbonNanotubesSWNTswererstdiscoveredintheearly1990sattheNECFundamental ResearchLaboratoryinTsukuba,Japan,bySumioIijima[9].SincethattimeSWNTs havebeenthesubjectofmuchattentionasacandidatematerialforfuturenano-scaled engineering.Manycuttingedgeapplicationshavebeentheorizedtotakeadvantage oftheirremarkablephysicalandelectricalproperties.However,SWNTsexistinboth semiconductingandmetallicspecies,andallcurrentlyknownmanufacturingproceduresresultinaheterogeneousmixtureofbothtypes[18].Giventhatthemajority ofSWNTbaseddevicedesignsrequireaspecicelectronictype,andthatseparation remainsahighlyexpensiveprocess,commercializationoftheseremarkablematerials hasbeenslow. 1.1Graphene Grapheneconsistsofaplanararrayof sp 2 hybridizedcarbonatoms.Thislattice structurecannotberepresentedbyaBravaislattice,whichwouldrequireeverycarbon tobelocatedatavector,originatingatanyothercarbon,givenby: ~ R = m~a 1 + n~a 2 .1 Forsomelatticevectors ~a 1 ~a 2 andintegers m n .Instead,thegraphenelattice mustberepresentedasalatticedecoratedwithabasis.Theunitcellsofgraphene aregivenbytheHexagonallattice,whichintwodimensionalCartesiancoordinates hasthefollowinglatticevectorsforsomelatticeconstanta. ~a 1 = a = 2 ; p 3 = 2.2 ~a 2 = a )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 ; p 3 = 2.3 1

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AsdepictedinFigure1.1,thesoliddotscorrespondtothelocationsofthevectors ~ R = m~a 1 + n~a 2 forthelatticevectorsabove. Figure1.1: Honeycomblatticeofgraphene.Imagefrom[14]. Twocarbonatomsarethenaddedtoeachunitcell,creatingthehexagonalshape ingure1.1.Thelocationofeachcarbonatomrelativetotheassociatedlatticepoint isgivenbythevectors[14]: ~v 1 = a ; 1 2 p 3 .4 ~v 2 = a ; )]TJ/F15 11.9552 Tf 18.402 8.087 Td [(1 2 p 3 .5 Thenalresultisknownasthehoneycomblattice.Usingtheidentity ~a i ~ b j = 2 ij ,thereciprocallatticevectors, ~ b 1 and ~ b 2 ,canbefoundtobe: ~ b 1 = 2 a ; 1 p 3 .6 ~ b 2 = 2 a )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1 p 3 .7 Figure1.2presentsagraphofthereciprocallatticeofgrapheneinthisbasis. TheshadedarearepresentstherstBrillouinzoneandthe K K 0 ,,and M points designateparticulardiscretepointsofimportantsymmetry. 2

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Figure1.2: ReciprocalLatticeofGraphene.Imagefrom[15]. Thebandstructureofgrapheneiswellapproximatedusingatightbindingmodel [4].Inthisapproach,theelectronwavefunctioniscalculatedbythesuperposition ofacoreandnearestneighboratomicwavefunctions.Thebandstructurealongthe reciprocalspacetriangle)]TJ/F19 11.9552 Tf 136.544 0 Td [(KM isdepictedingure1.3forbothabinitioandtight bindingmodels. Figure1.3: Comparisonofthetightbindingapproximationwithabinitiocalculations forthebrandstructureofgraphene.Imagefrom[15]. 3

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1.2SingleWalledCarbonNanotubes SinglewalledcarbonnanotubesSWNTsconsistofgraphenerolledintohollow cylinders.ThestructureofSWNTsisexpressedviathechiralvector: ~ C h = m~a 1 + n~a 2 .8 Where ~a 1 and ~a 2 arethelatticevectorsofgrapheneand m and n areintegers.The chiralvectorisonesuchthatitcircumventsthenanotubeandreturnstoitsoriginal location.Figure1.4showsthisprocessfora ; 3nanotube. A and A 0 representthe samepointonthe ; 3SWNTsurface. Figure1.4: Demonstrationofthegraphenerollingprocessfora ; 3 nanotube.Image from[4]. Bloch'stheorem, ~r = u ~r exp i ~ k ~r ,tellsusthattheelectronwavefunctions ingraphenemustbeaplanewavemodulatedbyaperiodicfunctionwithaperiodicity givenbythegraphenelatticeie. ~a 1 and ~a 2 .Giventhat ~ C h = m~a 1 + n~a 2 : ~r + ~ C h = u ~r + ~ C h exp i ~ k ~r + ~ C h .9 4

PAGE 15

~r + ~ C h = u ~r exp i ~ k ~r + ~ C h .10 ~r + ~ C h = ~r exp i ~ k ~ C h .11 Additionally,theazimuthalcontinuityofaSWNTrequiresthat: ~r + ~ C h = ~r .12 Comparingequations1.11and1.12,weseethattheonlywavevectorsthatare allowableunderbothoftheseboundaryconditionsarethosesuchthat: exp i ~ k ~ C h =1.13 FortheSWNTtobehavemetallically,the K wavevectorsmustbeallowed.It hasbeenshown[14]that,inthebasisdescribedbyequations3.10and3.11: K = 1 3 ~ b 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 ~ b 2 .14 K = 4 3 a ; 0 ; 0.15 K ~ C h = 4 3 a ; 0 ; 0 [ ma = 2 ; p 3 = 2 ; 0+ na )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 ; p 3 = 2 ; 0].16 K ~ C h = 4 3 a ; 0 ; 0 a 2 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n; p 3 m + n ; 0.17 K ~ C h = 2 3 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n .18 Forequation1.13toholdfor K ,thenforsomeinteger l =0 ; 1 ; 2 ; 3 ;::: : K ~ C h = 2 3 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n =2 l .19 5

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m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n =3 l .20 Accordingly,ifthechiralvector n;m issuchthat n and m dierbyaninteger multipleof3,thenthe K wavevectorswillbeallowedandthenanotubewillbehave metallically.Otherwise,theSWNTwillbehaveasasemiconductor. 1.3NanotubeAbsorptionSpectroscopy Absorptionspectroscopyisthequanticationoftherelativeamountofradiation absorbedbyaspecimenasafunctionoffrequency.Inthecaseofsinglewalledcarbon nanotubes,aswellasmostcrystallinesolids,thisbehaviorisdominatedbyVanHove singularities. ThedensityofstatesDOSisdenedtobethenumberofstates,orwavefunctions,thatexistwithinagivenenergyinterval.VanHovesingularitiesrefertosharp, non-smooth,peaksintheDOS.ThedierencesintheDOSformetallicandsemiconductingSWNTsarerepresentedingure1.5. Figure1.5: DensityofstatesformetallicandsemiconductingSWNTs.VanHove transitionsareillustratedbyarrows.Imagefrom[8]. 6

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EachoftheVanHovetransitionsillustratedabove M 11 S 11 S 22 S 33 correspondstoaparticularchangeinenergy.Assuch,eachtransitionisassociatedwitha radiationfrequencysuchthat E = h ,where h isPlank'sconstant.Giventhatmore electronsexistwithintheVanHovesingularities,theabsorptionofSWNTspeakat thosefrequencies.Figure1.6depictsexperimentalresultsofabsorptiondatafora thinlmofSWNTscoatedonquartz.Thepeakat650 nm correspondstometallicnanotubesandthepeaksat950 nm and1700 nm correspondtosemiconducting nanotubes. Figure1.6: AbsorptionspectraofathinlmofSWNTs.Imagefrom[11]. 1.4Dielectrophoresis Dielectrophoresisistheprocessesofexploitinginhomogeneouselectriceldsto driveseparationofparticlesinasolutionbasedonthevariablepolarizabilities.For theremainderofthispaper,wewillconsiderasetofparticlesdenotedwithsubscript p dispersedinaliquiddenotedwithsubscript l Takingthedielectricfunctiontobe p;l = p;l )]TJ/F19 11.9552 Tf 12.588 0 Td [(i p;l =! ,where istheelectric permittivity, istheconductivity,and istheappliedangularfrequency,theeective 7

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dipolemomentofadispersedsphericalparticlecanbeexpressedas[3]: ~p =4 r 3 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l 2 l + p ~ E .21 Givenanoscillatoryelectriceld,thisproducesatimeaverageDEPforce. F DEP =4 r 3 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l 2 l + p r E 2 .22 F DEP =4 r 3 l Re CMF r E 2 .23 Where CMF istheClausiusMossottifunction.Forellipsoidalparticles,asignicantlymorecomplicatedexpressionforthetimeaverageDEPforceemerges.However, thisfunctioncanbesimpliedundertheassumptionthatthe3axesoftheparticle a;b;c arerelatedby a>>b = c therod-likeparticleassumption. 1.5SWNTDielectrophoresis Singlewalledcarbonnanotubedimensionsjustifytherod-likeparticleassumption andtheresultingtimeaveragedforceis: F DEP = d 2 l 8 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l + p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l L r E 2 .24 WheredistheSWNTdiameter,listhelength,andLisadepolarizationfactor thatisroughly10 )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(5[3].Thehighandlowfrequencylimitsofthisexpressionyield dominatingtermsofthedielectricfunction: lim ! 0 = p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l .25 lim !1 = p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l .26 FormetallicSWNTsbothoftheselimitsarepositiveas p > l and p > l resultinginapositiveDEPforce,thatisaforceinthedirectionofthehighest 8

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inhomogeneityoftheelectriceldintensity.ForsemiconductingSWNTs,however, theoppositeistrue, p < l and p < l ,resultinginanegativeDEPforce[3][12]. Experimentalresultsverifythreeofthesefourcases.[12]usedamicroelectrode arrayanda20voltpeak-to-peak 10[ V ]ACsignalovera10microngapatvarying frequenciestodeterminetheDEPforcesactingonSWNTsdispersedinsolutionusing variousconcentrationsofthesurfactantSDBS.InthecaseofmetallicSWNTs,their resultsconrmedthetheoreticallypredictedpositiveDEPforcesatallfrequencies.In thesemiconductingSWNTs,theyfoundthatthesurfactantinducedasmallsurface chargethatenabledpositiveDEPforcesatlowfrequency.However,athighfrequencythesemiconductingSWNT/SDBSmicellesrevertedbacktothetheoretically predictednegativeDEPforce. ThefrequencyatwhichsemiconductingSWNTstransitionfrompositivetonegativeDEPforcewastermedthecriticalfrequency.ForSDBSconcentrationsof0 : 1% byweight,thisfrequencywasfoundtoberoughly1[ MHz ].Insummary,alternating electriceldswithfrequencieslessthan1[ MHz ]causedpositiveDEPforcesinboth metallicandsemiconductingSWNTsandaccordinglyseparationdidnotoccur.As theappliedeldfrequencyincreasedabove1[ MHz ],themetallicSWNTscontinued toexperiencepositiveDEPforceswhilethesemiconductingSWNTstransitionedto negativeDEPforces. Themaximumelectriceldstrength,whichoccurreddirectlybetweenthemicroelectrodes,neededtocauseseparationwasdeterminedbyenergeticallycomparing theDEPforcewithBrownianmotion.ElectrohydrodynamicalforceswerenotconsideredastheSWNTsizewouldindicatethatBrownianmotionwouldbetheprimary competingforce[12]. Denotingthepotentialenergychangebetweenmaximumelectriceldandminimumeldatfardistances,theDEPescapepotentialwasfoundtobe: 9

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U DEP = )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(d 2 l 8 l Re f CMF g E 2 max .27 Settingthisequaltothethermalenergyyieldsamaximumelectriceldof[12]: E max s 8 kT d 2 l l Re f CMF g 10 6 [ V=m ].28 1.6RectangularWaveguidesandResonators Herewewillderivetheeigenmodesforrectangularwaveguidesand,byextension,rectangularresonators.Waveguidesaredesignedsuchthatpropagatingwaves destructivelyinterfereatboundarieswhichwouldotherwisebecomesourcesofattenuation.Maxwell'sequationsareusedtoderivethewaveequationsforbothelectric andmagneticelds. c 2 r 2 )]TJ/F19 11.9552 Tf 15.265 8.088 Td [(@ 2 @t 2 ~ E =0.29 c 2 r 2 )]TJ/F19 11.9552 Tf 15.264 8.088 Td [(@ 2 @t 2 ~ H =0.30 Formetallicwavegudies,theboundaryconditionsarethatofanyconductor:the tangentialcomponentoftheelectriceldandthenormalcomponentofthemagnetic eldmustbezero. E ~r;t ^ n =0.31 H ~r;t ^ n =0.32 Ifweassumethattheelectromagneticwaveistravelinginthe z direction,witha wavefrontmodulationfactortoensureproperboundaries ~ E ~r;t = E 0 ~ F x;y e i k z z )]TJ/F20 7.9701 Tf 6.587 0 Td [(!t and ~ H ~r;t = H 0 ~ G x;y e i k z z )]TJ/F20 7.9701 Tf 6.587 0 Td [(!t ,thensubsitutionintoequations1.29and1.30 yeilds: 10

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h @ 2 F x @x 2 + @ 2 F x @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; @ 2 F y @x 2 + @ 2 F y @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; 0 i = )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(! 2 c 2 h F x ;F y ; 0 i .33 h @ 2 G x @x 2 + @ 2 G x @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; @ 2 G y @x 2 + @ 2 G y @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; 0 i = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [(! 2 c 2 h G x ;G y ; 0 i .34 Thesolutionstothissystem,forawaveguideofwidth a onthe x -axisandheight b onthe y -axis,canbeshowntobe[1][10]: 1.6.1TransverseElectric E x = iE 0 n b cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .35 E y = )]TJ/F19 11.9552 Tf 9.298 0 Td [(iE 0 m a sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .36 H x = H 0 m a sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .37 H y = H 0 n b cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .38 H z = H 0 k 2 z + 2 =c 2 k z cos m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .39 Here m and n representthemodenumberinthe x and y directions,respectively. k z = q 2 c 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( m a 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( n b 2 istheeectivewavenumberinthe z direction. 1.6.2TransverseMagnetic E x = )]TJ/F19 11.9552 Tf 9.299 0 Td [(E 0 m a cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .40 E y = )]TJ/F19 11.9552 Tf 9.299 0 Td [(E 0 n b sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .41 11

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E z = E 0 k 2 z + 2 =c 2 k z sin m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .42 H x = iH 0 n b sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .43 H y = )]TJ/F19 11.9552 Tf 9.298 0 Td [(iH 0 m a cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .44 1.6.3Resonators Resonatorsareconstructedbycappingeitherendofthewaveguidesuchthat reectionsbuidstandingwaves.TheTE/TMmodestructureispreserved,whilethe e i !t k z z termsbecome2 cos !t cos k z z 12

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2.Theory Aswasshowninsection1.5,anenergeticapproachtocalculatingtheelectriceld strengthnecessaryforsuccessfulsortingofSWNTspredictedthattheeldwouldneed topeakatavalueontheorderof10 6 [ V m ].ThiswasfoundbydeterminingaDEP escapepotentialthatwaslargerthanthethermalenergythatthesolutioncould provide.Clearly,thisisasucientconditionforDEPtooccur.However,inthecase ofhighdragsystems,thisconditionmaynotbenecessaryasnon-conservativeforces candominate. Auidicforcemodelofthesystemwouldprovidemoreinsightintotheeld requirements.DeningthehighfrequencytimeaverageDEPforce,Browniandiusive force,anddragforcetobe: F DEP = d 2 l 8 p )]TJ/F19 11.9552 Tf 11.956 0 Td [( l r E 2 .1 F dif = )]TJ/F15 11.9552 Tf 15.675 8.088 Td [(1 N A r .2 F drag = )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 av .3 Where N A isAvogadro'snumber, isthemolarchemicalpotential, a isthe Stoke-Einsteinradius, istheviscosityofthemedium,and v isvelocity. 2.1AccountingforDrag Takingadvantageoftheknown -polarityofeachforce,thetimeaveragedform ofNewton'ssecondlawgives: d 2 l 8 p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 av )]TJ/F15 11.9552 Tf 18.331 8.088 Td [(1 N A r = m a .4 Equation2.4canbefurthersimpliedbyassumingthattheparticlesquickly achieveaterminaldriftvelocity: 13

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d 2 l 8 p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 av t )]TJ/F15 11.9552 Tf 18.331 8.087 Td [(1 N A r =0.5 v t = [ d 2 l 8 p )]TJ/F19 11.9552 Tf 11.956 0 Td [( l r E 2 )]TJ/F17 7.9701 Tf 17.405 4.707 Td [(1 N A r ] 6 a .6 Finallyusingtherelationbetweenchemicalpotentialandmolefraction, r = RT x p r x p ,andmolefractionandmolarconcentrationprole, C p = x p C v t =[)]TJ/F22 11.9552 Tf 22.992 0 Td [(r E 2 )]TJ/F15 11.9552 Tf 15.492 8.087 Td [( C p r C p ].7 Where)-347(= d 2 l 48 a p )]TJ/F19 11.9552 Tf 12.285 0 Td [( l and= RT N A 6 a .Usingtherelationbetweendiusive ux,molarconcentrationprole,andvelocity, J = C p v t ,andthecontinuityequation, @C p @t + r J =0 @C p @t +)]TJ/F22 11.9552 Tf 19.076 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p =0.8 Thisequationisthequasi-linearheat,orthediusion-advection,equation.In thelimitthattheelectricelddisappears,thisreducestothelinearhomogenousheat equation,acommonmodelfornoninteractingdiusiveprocesses. 2.2SteadyStateSolution Thesteadystatesolutioncanbefoundbysetting @C p @t =0.Clearlythisleadsto theidentity: )]TJETq1 0 0 1 266.836 204.283 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 266.836 193.093 Td [( r [ C p r E 2 ]= r 2 C p .9 Usingtheproductruleonascalar-vectorproduct: r [ ~ A ]= r ~ A + r ~ A andtherelation rr = r 2 ,thisfurthersimpliesto: r C p r E 2 + C p r 2 E 2 = )]TJ/F22 11.9552 Tf 9.08 8.201 Td [(r 2 C p .10 14

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Guessingthesolution C p ~r = C 0 e )]TJETq1 0 0 1 317.913 714.45 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 317.913 709.332 Td [( E 2 ~r ,theconcentrationproleLaplacian becomes: r 2 C p = r )]TJETq1 0 0 1 269.048 639.093 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 269.048 627.904 Td [( C 0 e )]TJETq1 0 0 1 298.426 643.033 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 298.426 637.915 Td [( E 2 ~r r E 2 = )]TJETq1 0 0 1 367.773 639.093 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 367.773 627.904 Td [( r C p r E 2 .11 Clearly,asaresultofequation2.11,theguesssatisesequation2.9andaccordinglythesolutionholdsforanyadequatelycontinuouselectriceldintensity. C steadystate ~r = C 0 e )]TJETq1 0 0 1 368.337 540.637 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 368.337 535.519 Td [( E 2 ~r .12 2.3BoundaryandInitialConditions Foragivencontainmentchamberdescribedbythedomainwithrigidwalls, @ ,theboundaryconditionsareoftheNeumanntype.Specically,theboundaries inhibitanyconcentrationuxtopassthroughthem.Intermsofthenormalvector, ^ n ,thisisexpressedas: ^ n J ~r = C p ~r ^ n ~v t ~r =0:[ 8 ~r@ ].13 Substitutinginequation2.7expressesthisconditionasarequirementofthe concentrationprole. [ C p )]TJ/F22 11.9552 Tf 7.315 0 Td [(r E 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( r C p ] ^ n =0.14 )]TJ/F19 11.9552 Tf 7.314 0 Td [(C p @E 2 @ ^ n = @C p @ ^ n .15 Where @f @ ^ n ^ n r f .Consideringthesteadystatesolutiondeterminedinthelast section,itisclearthatthisboundaryconditionisinnatelysatised. r C steadystate = )]TJETq1 0 0 1 320.597 101.887 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 320.597 90.697 Td [( C steadystate r E 2 .16 15

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! )]TJ/F19 11.9552 Tf 7.314 0 Td [(C steadystate @E 2 @ ^ n = @C steadystate @ ^ n .17 Anyphysicallyrealizableinitialconditioncouldbeusedwiththisapproach.However,fortheremainderofthispaper,onlyinitialconcentrationprolesoffullydispersedSWNT'swillbeconsidered. C p ~r;t =0= C i : 8 ~r .18 Lastly,conservationofmassgivesthenalconditiononthesystem.Forasystem consistingof N SWNT numberofSWNT'sdispersedinthesolution: F t = Z C ~r;t dV = N SWNT : 8 t .19 Insummary,thesystemhasbeenreducedtothepartialdierentialequationand conditions: @C p @t +)]TJ/F22 11.9552 Tf 19.076 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p =0.20 C p ~r;t =0= C initial : 8 ~r .21 C p ~r;t !1 = C 0 e )]TJETq1 0 0 1 343.84 257.292 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 343.84 252.174 Td [( E 2 ~r : 8 ~r .22 F t = Z C p ~r;t d~r = N SWNT : 8 t .23 2.4TemporalEvolution Inordertoevaluatethetemporalevolutionofthissystem,anoperator A must beconstructedsuchthat: @C @t = )]TJ/F19 11.9552 Tf 9.298 0 Td [(AC .24 16

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! AC )]TJ/F22 11.9552 Tf 7.314 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p .25 Theconstructionofsuchanoperatorwillbenumericallyconsideredinthesubsequentchapter.Intermittently,considerationoftheeigencouplesoftheoperator, n ;v n ,allowsforanexpansionofequation2.24inaneigenbasis. C = ae )]TJ/F20 7.9701 Tf 6.587 0 Td [(At .26 C = X i a i v i e )]TJ/F20 7.9701 Tf 6.586 0 Td [( i t .27 Forsomesetofconstants f a i g .Giventhepresenceofrstorderderivativesin equation2.20,AisnotguaranteedtobeHermitian.Asadirectresultofthis,complex eigenvaluesmustbeconsidered. j = j + i j .28 C = X i a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i .29 Thevaluesof i areknowntobenon-negative,asanegativevaluewouldleadto exponentialgrowthandaviolationoftheconservationofmass.Clearly,thedecay ofthesolutiontowardsanyeigenfunctionisdominatedbytheterm a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t .Unfortunately,thevaluesof a i arenotgivenbytheinnerproductastheeigenfunctionsare notknowntobeorthogonal: a i 6 = h C j v i i .30 However,thevaluesofthecoecients f a i g canbeexpressedaslinearfunctions ofthefunction C .Inotherwords, 8 a i ;v i ; 9 w i suchthat: 17

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a i = h w i j C i .31 2.5SlowestDecayingEigencouple Inordertodeterminethelongtermbehaviorofequation2.20,attentionmustbe focusedontheeigenvaluewiththesmallestrealpart.Thisstemsfromitdecayingat theslowestrateandthereforedominatingastimegoestoinnity. Consideringtheexpressionoftheconcentrationinthe N dimensionaleigenbasis of A : C = N X i a i v i = N X i v i a i .32 C =[ v 1 ;v 2 ;:::;v n ][ a 1 ;a 2 ;:::;a N ] T = Ua .33 Where U isdenedtobethematrixmadefromcolumnsoftheeigenfunctions and a isacolumnvectorwithcomponents f a i g .Substitutinginequation2.25forthe elementsof a ,equation2.27isexpressedintermsofamatrix W withrowvectors givenby w i C = UW C .34 W = U )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 .35 Whilehavingtherelationinequation2.35isextremelyuseful,directcalculation oftheinverseoftheconjugatedeigenfunctionmatrixishugelyexpensivecomputationally.Instead,focusingonlyonobtainingthedominantterm a 1 andtherefore w 1 cangreatlyreducethiscost. W [1 ; 0 ;:::; 0] T = U )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 e 1 = w 1 .36 18

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! U w 1 = e 1 .37 Restrictingconsiderationtotherst m eigen-functionstheeigenfunctionscorrespondingtothe m eigenvalueswithlowestmagnituderealcomponent,thiscanbe approximatedintermsof U m =[ v 1 ;v 2 ;:::;v m ]: U m w 1 = e m 1 .38 Ofcourse, N dimensionalvector w 1 cannotbefounduniquelyviaequation2.38. Howeveran m dimensionalapproximation, w 1 = U m z ,canbefoundbysolving: U m U m z = e m 1 .39 Inconclusion,theeigen-expandeddecayratesoftheconcentrationtowardsthe steadystatesolutioncanbeboundedtermwiseasfollows. C = X i a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i .40 a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t jj v i jjjj w i jj e )]TJ/F20 7.9701 Tf 6.586 0 Td [( i t jj v i jjjj U m z i jj .41 19

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3.NumericalMethods Inchapter2,theproblemofsolvingfortherateofapproachofconcentrationtowardsthesteadystatewasdiscussedindetailusingtheeigenexpansionofthesolution intermsofanoperator A suchthat @C @t = AC .Inthischapter,Awillbeconstructed usingthenitedierencemethodforinteriorpointsandpolynomialextrapolationon theboundaries.Thismethodisoutlinedin[6].Additionallyupwindingmethodscan beusedtoenhancestability[5]. 3.1FiniteDierences Themethodofnitedierencesconsistsofapproximatingthecontinuousconcentrationfunctionasadiscretemesh.Forameshinwhichthereare m n p mesh pointsinthex,y,andzdirections,thisisrepresentedbyassumingaunitarydistance betweenthemeshpoints: C x;y;z C i;j;k :1 i m; 1 j n; 1 k p .1 3.1.1ReshapingTensors Giventhatbothconcentration, C ,andelectriceldintensity, E 2 ,tensorsare threedimensional,operatorsactingonthesetensorsmustbefourdimensional.To avoidaddedcomplications,these m n p tensorsarereshapedas mnp 1column vectors. C i;j;k 0 B B B B B B B B B B B B B B B B B B B B B @ C 1 ; 1 ; 1 . C m; 1 ; 1 C 1 ; 2 ; 1 . C 1 ;n;p . C m;n;p 1 C C C C C C C C C C C C C C C C C C C C C A .2 20

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3.1.2LaplacianMatrix ThematrixrepresentationoftheLaplacianoperator, r 2 ,canbefoundviaa centraldierenceapproximation.Equation3.3givesthevepointcentraldierence formulaforanequidistantmeshwithstepsize h =1. @ 2 A i @x 2 )]TJ/F15 11.9552 Tf 26.039 8.088 Td [(1 12 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2+ 4 3 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(5 2 A i + 4 3 A i +1 )]TJ/F15 11.9552 Tf 16.077 8.088 Td [(1 12 A i +2.3 ThefollowingMATLABcodeconstructsthethreedimensionalLaplacianmatrix forapplicationontoareshaped m n p tensor.The kron functionistheKronecker tensorproductusedtodoblockmatrixmultiplications.Avepointformulaisused forfourthorderaccuracy. LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded./h; 3.1.3DerivativeMatrix Asimilarapproachcanbeusedtondthederivativematrices, @ x A @ y A @ z A : @A i @x )]TJ/F15 11.9552 Tf 26.04 8.088 Td [(1 12 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2+ 2 3 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 A i +1+ 1 12 A i +2.4 X=spdiagsonesm,1 [-1/122/30-2/31/12],-2:2,m,m; 21

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Y=spdiagsonesn,1 [-1/122/30-2/31/12],-2:2,n,n; Z=spdiagsonesp,1 [-1/122/30-2/31/12],-2:2,p,p; Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X; DY=kronspeyep,p,Y expanded; DZ=Z expanded; 3.2BoundaryExtrapolation TheaboveexpressionsfortheLaplacianandderivativematricessuerfromthe needof'ghost'pointstobewelldenedatboundariesie. @ x A A )]TJ/F20 7.9701 Tf 6.586 0 Td [(A G )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 Theseghostpointscanbeeliminatedbytheuseofforward/backwardsdierence approximations. @A i @x 1 )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(3 2 A i +2 A i 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 A i 2.5 %Fourthordercentraldifferenceapproximation. X=spdiagsonesm,1 [1/12-2/302/3-1/12],-2:2,m,m; Y=spdiagsonesn,1 [1/12-2/302/3-1/12],-2:2,n,n; Z=spdiagsonesp,1 [1/12-2/302/3-1/12],-2:2,p,p; %Boundaryextrapolationsecondorderforward/centralapproximation. X:2,1:4=[-3/22-1/20;01-21]; Xend-1:end,end-3:end=[1-210;01/2-23/2]; Y:2,1:4=[-3/22-1/20;01-21]; Yend-1:end,end-3:end=[1-210;01/2-23/2]; Z:2,1:4=[-3/22-1/20;01-21]; Zend-1:end,end-3:end=[1-210;01/2-23/2]; 22

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Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X; DY=kronspeyep,p,Y expanded; DZ=Z expanded; Forwardsandbackwardsdierenceapproximationsarealsobeusedintherst, second,secondtolast,andlastrowsoftheLaplacianmatrices. @ xx A x i 2 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 A i +1 +4 A i +2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A i +3 .6 @ xx A x i 2 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 A i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 +4 A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 .7 0 B B B B B B B B B B B @ )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 12 ::: 0 4 = 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 12 ::: . ::: )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 124 = 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 3 0 ::: )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 124 = 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(5 = 2 1 C C C C C C C C C C C A 0 B B B B B B B B B B B @ 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(54 ::: 0 02 )]TJ/F15 11.9552 Tf 9.299 0 Td [(54 ::: . ::: 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(520 0 ::: 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(52 1 C C C C C C C C C C C A .8 LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; LX:2,1:4=[2-540;02-54]; LXend-1:end,end-3:end=[4-520;04-52]; LY:2,1:4=[2-540;02-54]; LYend-1:end,end-3:end=[4-520;04-52]; LZ:2,1:4=[2-540;02-54]; 23

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LZend-1:end,end-3:end=[4-520;04-52]; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded; 3.2.1BoundaryConditions Itisimportanttonotethattheuseofforwards/backwardsdierencemethods attheboundariesisnotalternativestoboundaryconditions.Thegeneralcaseis complicatedbythefactthattheboundariesoftheelectriceldandtheboundariesof theconcentrationarenotthesame.Asanexample,consideranelectriceldpassing throughatransparentcontainer.ThewallsofthecontainerwouldenforceaNeumann conditionontheconcentration,butnotontheelectriceld.Theboundaryconditions ontheelectriceldwouldbedeterminedbysomesetupexternaltothecontainer. Foraknownelectriceld,withboundariesoutsidethechamber,theabovecode wouldallowforapproximationsofthederivativesandLaplacianusingonlyinterior points.However,boundaryconditions,DirichletorNeumann,mustbespeciedfor theconcentrationprole.Theseconditionsareaddressedonacase-by-casebasis. 3.3FormingA Giventhevectorsresultingfromreshapingtheconcentrationandelectriceld intensitytensors, C and E 2 ,theoperatorcanberepresentedusingtheLaplacianand derivativematrices: AC =)]TJ/F22 11.9552 Tf 19.74 0 Td [(r C r E 2 +)]TJ/F19 11.9552 Tf 19.076 0 Td [(C r 2 E 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( r 2 C .9 AC =\050 @ x E 2 @ x C + @ y E 2 @ y C + @ z E 2 @ z C +)]TJ/F19 11.9552 Tf 26.285 0 Td [(C r 2 E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C .10 A =\050 D @ x E 2 x + D @ y E 2 y + D @ z E 2 z +)]TJ/F19 11.9552 Tf 26.284 0 Td [(D LE 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( L .11 24

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Where D @ x E 2 ;D @ y E 2 ;D @ z E 2 arethediagonalizedvectorsresultingfromtheapplicationofthecentraldierencederivativematricesontheelectriceldintensity vector, x ; y ; z aretheup-windingderivativematrices, D LE 2 isthediagonalized vectorfromthecentraldierenceLaplacianappliedtotheelectriceldintensity,and L istheup-windingLaplacianmatrix. 25

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4.NumericalResults TheMITElectromagneticEquationPropagationFDTDsoftware,orMEEP,numericallysolveselectromagneticsproblemsinwhichdielectricsandconductorsare welldened[17].Theoutputie. E H E 2 H 2 isreturnedonanequidistantmesh ofthesetspatialdimension-3Dplusonetemporaldimension.Thisoutputis particularlywellconstructedforuseinthemethodoutlinedinchapter3. 4.1MeshSizeandStabilty IfweconsidertheDEPdiusionequationintermsofunitsofvoltageVolts[V],timeSeconds-[S],concentrationarbitrary-[C],anddistantunity-[D]. @C @t = r 2 C )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F22 11.9552 Tf 7.314 0 Td [(r E 2 r C )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F19 11.9552 Tf 7.314 0 Td [(C r 2 E 2 .1 [ C ][ S ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 =[][ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 [ C ] )]TJ/F15 11.9552 Tf 11.946 0 Td [([] [ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 [ V ] 2 [ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 [ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 [ C ] )]TJ/F15 11.9552 Tf 11.945 0 Td [([][ C ][ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 [ V ] 2 [ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 .2 []=[ D ] 2 [ S ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 .3 []=[ D ] 4 [ V ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 [ S ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 .4 Equation4.1speciesthediusionandconvectioncoecientstobe a =and b =)]TJ/F19 11.9552 Tf 20.881 0 Td [(@ E 2 respectively,where is x y ,or z .Thestabilityrequirementformesh sizeisgivenby[13]tobe: h << a b .5 Giventheequidistantmesh h x = h y = h z = h equation4.5reducestothe requirement: h<< )]TJ 37.177 16.289 Td [(1 max i; @ E 2 i .6 26

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Expressingthisasarequirementonthenumberofnodesforrectangulardomain withsidesgivenby L x ;L y ;L z : [ n;m;p ]=[ L x h x ; L y h y ; L z h z ]= 1 h [ L x ;L y ;L z ].7 [ n;m;p ] [ L x ;L y ;L z ] )-167(max i @E 2 i .8 4.2 )]TJ/F16 11.9552 Tf 11.797 0 Td [(and inSIUnits Herethevaluesof)-274(andwillbecalculatedforametallicnanotubeofdiameter, d 1 : 2[ nm ],andlength, l 1000[ nm ]notethatthesedimensionsarebasedon themanufacture'sclaimsoftheSWNTsusedintheexperimentalsection.Metallic nanotubeshavebeenestimated[2][12]tohaveaneectiverelativedielectricconstant ontheorderof10 3 -10 4 .Toensurestability, r =10 4 willbeusedasitcorresponds toalarge)-323(andthereforesmallerstepsize.TheStokes-EinstienradiusforaSWNT wasgivenintermsof d and l as[16]: a = 1 2 l ln l=d +0 : 32 85[ nm ].9 )-278(= d 2 l 8 met )]TJ/F19 11.9552 Tf 11.955 0 Td [( H 2 O 6 a 3 : 49 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(23 m 3 V 2 S .10 = RT N A 6 a 2 : 85 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(12 m 2 S .11 4.3ErrorinaKnownCase Inthecaseofaconstantelectriceld,theDEPnitedierenceoperator A reduces tothescalednegativeLaplacian, A = )]TJ/F15 11.9552 Tf 9.299 0 Td [( r 2 .Resultinginwellknowneigenvalues. Equation4.13givestheeigenvaluesforthetwodimensionLaplacianonarectangular domainwithNeumannboundaryconditions.NotethattheDEPboundarycondition isinfactthezerouxNeumanncondition: r C = )]TJETq1 0 0 1 369.539 76.661 cm[]0 d 0 J 0.478 w 0 0 m 6.116 0 l SQBT/F17 7.9701 Tf 369.539 69.55 Td [( C r E 2 =0. 27

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A !)]TJ/F15 11.9552 Tf 24.575 0 Td [( r 2 .12 = n=L x 2 + m=L y 2 : n;m =0 ; 1 ; 2 ; 3 ;::: .13 UsingthetwodimensionalcodegiveninappendixA,withtheLaplacianboundary conditionssetasdescribedabove,thedriver constant test givenbelowresultsinthe smallest20realeigenvaluecomponents.Together,thesevalueshavealessthanone percentrelativeerrorwhencomparedtotheknownvaluesresultingfromequation 4.13.Theresultsaregiveningure4.1. Figure4.1: ApproximatedandknowneigenvaluesforthescaledLaplacianoperator. %Testsdep fd 2dwhenintensityisconstantandtheDEPequationreduces %tothescaledLaplaceequation:AC=-Phi LaplacianC 28

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% %Knownsolutionislambda i=Phi-1 n Pi/L+m Pi/L %forn,m=0,1,2,...,N-1 %TestisonaNxNmeshwithstepsizehlength,L=h N N=30;%Defaultmeshsize h=0.01;%Defaultstepsize gamma=3.49E-23; phi=2.85E-12; %Finddep fd 2destimatedeigenvalues A=dep fd 2donesN,N,gamma,phi,h; D=eigsA,20,sr; R=realD; %Calculateknowneigenvalues; L=[]; L1=[]; fori=0:N-1 lambda=i pi/N h; L1=[L1lambda]; end fori=1:N L=[LL1i ones,N+L1]; end L=sortL; L=phi. L; L=L:N; SE=normR-L:20/normR; 29

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fprintfRelativeerrorinthesmallest20components:%8.2E n n,SE X=linspace,20,20; figure plotX,L:20,X,R, titleSmallest20realeigenvaluecomponents legendKnownvalues.,Approximatedvalues.,Location,Northwest; >> constant test Relativeerrorinthesmallest20components:2.71E-03 4.4ParallelPlateDielectricWaveguide Consideradielectricparallelplatewaveguidewithwallgap a thatisdesignedto propagateinthe z axis.Choosing a suchthatsinglemode, TE 10 ,propagationoccurs forfrequenciesaround2 : 45[ GHz ]wouldresultin a = 1 2 p 2 : 45 E 9 6 : 5[ mm ]forthe dielectricvalueoftheDEPsolution.ThefollowingMEEPcodesimulatesthissystem undertheexcitationofaninternalmicrowavelinesource. set!geometry-lattice makelatticesize9.553no-size set!geometry list makeblock center00 sizeinfinityinfinityinfinity materialair 30

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makeblock center00 size6.5infinityinfinity materialmakedielectricepsilon80 set!sources list makesource srcmakecontinuous-srcfrequency0.0082 componentEz center6.50 set!pml-layers list makepml thickness1.0 set!resolution10 run-until1000 at-beginningoutput-epsilon to-appended"power"at-every1output-dpwr 31

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4.4.1BoundaryConditions Forthe TE 10 propagationmode,theelectriceldtakestheformgivenbyequation 1.35.Usingthedoubleangleidentityresultsintheelectriceldintensityofequation 4.15. ~ E = E 0 sin a x ~e x .14 E 2 = E 0 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(cos 2 a x .15 @ x E 2 j ;a = E 0 2 a sin 2 a x =0.16 Clearly,asaresultofequation4.16,theboundaryconditionfortheconcentration proleisagaintheNeumanncondition @C @~n =0. 4.4.2 f i g Approximations TheMEEPintensityoutputcanbescaledtond,using dep fd 2 d givenin AppendixA,theminimumrealeigenvaluecomponentforvariousintensitymaximum magnitudes.Theresultsofvaryingtheintensitypeakfrom10 )]TJ/F17 7.9701 Tf 6.586 0 Td [(5 V 2 m 2 to10 10 V 2 m 2 are plottedingure4.2. 4.4.3ForcingExponentialDecay Asopposedtopropagatingwaves,reducingthewallseparationtobelowthe distancecorrespondingtothecutoof2 : 45[ GHz ]leadstoexponentiallydecaying eldsinthe z direction.TheonlychangetotheMEEPcodetoreectthisisgiven below.Again,theresultsofvaryingtheintensitypeakfrom10 5 V 2 m 2 to10 10 V 2 m 2 are plottedingure4.3. set!geometry 32

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Figure4.2: Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. list makeblock center00 sizeinfinityinfinityinfinity materialair makeblock center00 size3.5infinityinfinity materialmakedielectricepsilon80 set!sources list makesource 33

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srcmakecontinuous-srcfrequency0.0082 componentEz center3.50 Figure4.3: Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. 4.5Conclusion Figures4.2and4.3clearlydemonstratethattheslowestlargesttimeconstant, givenby slow = )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 min ,becomesprogressivelyfasterastheintensitymaxincreasesin magnitude.Accordingly,thosesystemswithhigherintensitymaximumareexpected todecaytothesteadystatefasterthanasystemwiththesamegeometrybutweaker max.Further,theslightincreaseintheeigenvaluesofgure4.3overgure4.2indicate thatasharpergradientinthiscase,resultingfromanexponentialdecaydoescause anincreaseintherateofdecaytosteadystate. 34

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aAboveCuto bBelowCuto Figure4.4: Steadystateconcentrationprolesfora6.5mmandb3.5mmdielectricparallelplatewaveguidesstimulatedat2.45[GHz]. Additionally,seegure4.4,thesteadystatesolutioninthecaseofanexponentiallydecayingelectriceldintensityhasamuchsharperpeakthenthatofthe non-decayingcase.Giventhattheexponentialdecayobtainsahigherpeakconcentrationandachievesitfasterthanthenon-decayingcase,itcanbeconcludedthat intensitygeometriesthatapproachsingularitiesaresuperiorindrivingSWNTDEP. Lastly,linearlyextrapolatingthelineingure4.3tothebreakdownintensityof air,approximately10 13 V 2 m 2 whichwouldhaveviolatedequation4.6tosolvedirectly, wouldsuggestaminimumrealeigenvaluecomponentofroughly min 10 )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 .This 35

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resultsinatimeconstantontheorderofjustafewminutes. 36

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5.LaboratoryMethods TheuidicforcenatureofdielectrophoresisofCarbonNanotubeswasinvestigatedusingathreestepprocess.First,superpuriedplasmadischarge > 95% w=v SWNTbymanufacturer'sclaimnanotubeswerepurchasedfromNanotintegrisand suspendedina0 : 25% w=v aqueoussolutionofSDS.Thisconcentrationisjustabove criticalmicelleconcentration.Aftersuccessfuldispersion,theaqueousSWNT-SDS solutionwasplacedinacylindricalquartzcapillaryeectivelyformingadielectric waveguide.Propagationmodeswereexcitedusinggoldcoatedeldconcentratorsand a2.45[GHz]source.Lastly,thedepositednanotubeswereresuspendedandoptically evaluated. 5.1Dispersion AnaqueoussolutionoftheanionicsurfactantsodiumdodecylsulfateSDSwas preparedataconcentrationof0 : 25% w=v .ThecriticalmicelleconcentrationofSDS rangesfrom6-8mMorroughly0 : 17 )]TJ/F15 11.9552 Tf 11.254 0 Td [(0 : 24% w=v .Singlewallcarbonnanotubeswere dispersedinthissolutionataconcentrationof0.06mg/mL. TheSWNT/SDSaqmixturewasinitiallybathsonicatedtwenty-threetimes witheachrepetitionconsitingof8minutes,totalling3hoursand4minutes.The bathsonicationwasimmediatelyfollowedbythirtyminutesofhornsonication. Afterasmallreferencesamplewastakenfromthisinitialdispersion,theremainderwascentrifugedinaHemleZ360kat13,000rpmfor90minutes.Thetop80%of thesupernatantwascarefullydecantedandstored. 5.2ExperimentalSetup Anelectromagneticsinglemoderesonancechamberwasconstructedusinga shortedaluminumwaveguidedesignedfor2.45[GHz]waves.Atoneend,amagnetronaperturewasplacedataquarterwavelengthfromthebackwall.Thisallowed backwardstravellingwavestomoveaquarterwavelengthtothebackwall,undergo ahalfwavelengthreection,andtravelbackaquarterwavelength,endinginafull 37

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wavelengthpathlengthallowingconstructiveinterferencewiththeforwardtravelling wave.Asecondaperturewasplacedatahalfwavelengthfromtheforwardwall. Giventhatthislocationisatanantinodeofthegeneratedstandingwave,maximum electriceldamplitudeisexperiencedatthisaperture. Themagnetron,bydesign,operatesasadiode-soavoltagedoublerisusedto maximizetheeciency.Arelayisusedtopulsatepowertothemagnetronandhence controlthetimeaveragepowerdeliveredtotheresonantcavity. TheSWNT-SDSaqdispersionisplacedinafusedquartzcapillaryalongwith twoaxialgoldplatednickelneedlescreatingasmallgap.Thisisdisplayedinthe gurebelow. Figure5.1: Arrangementofneedlesinfusedquartzcapillariesmminteriordiameter ontheleftand8mmdiameterontheright. Giventhehighrelativedielectricpermittivityofthesolution,thearrangement createsadielectricwaveguide.Theneedlesconcentrateanyappliedeldtoaregion muchsmallerthanthewavelengthoftheappliedeld.Thisisknowntocreatean approximatelysphericalscatteredwave-inturn,excitinganypossiblemodeswithin 38

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Figure5.2: 4mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. thecapillary. Unlikerectangularwaveguides,withthesimplepropagationmodesgiveninchapter1,theeectivewavefrontsofacylindricalwaveguidearedependentonBessel's functions.Despitethecomplexityofthegeneralcase,thecutofrequencyforthe lowestordermodes TE 11 aregivenby[1]: f c = 1 : 8412 2 r r 1 .1 f 4 mm c 4 : 88[ GHz ]5.2 f 8 mm c 2 : 44[ GHz ]5.3 Asaresultofthesecutofrequencies,whenexposedtomicrowave2 : 45[ GHZ ] frequencies,modesareexcitedinthe8 mm capillary,whereasthewavesexponentially decayinthe4 mm capillary. 39

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Figure5.3: 8mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. Eachcapillaryisinsertedintothesecondaryapertureandsubjectedtopulsed microwaves.Themagnetronispoweredin25 S pulsesata13[ kHz ]repetitionrate. 5.2.1FieldIntensityDistributions MEEPwasusedtosimulatethissetupandresultingelectriceldintensityproles aredepictedingure5.4.Thehighercontrastinthesubgurebindicatesthatthe 4[ mm ]capillaryexperiencedasharpergradientthanthatofthe8[ mm ]capillary. 5.3SecondarySuspension Afterfourcycles,eachveminutesinduration,ofpulsedmicrowaves,thegold needlesareremovedfromthesolution.AnySWNT'sdepositedontheneedlesare removedbyplacingtheneedlesinasmallplastictubewithaqueoussodiumdodecyl sulfonateandbathsonicatedfor24minutes. Asampleoftheresultingsolutionwastakenandcentrifugedinasimilarmethod totheinitialdispersion.Thissamplethenunderwentabsorptionspectroscopytoevaluatemetallicenrichment.Adualbeamspectrophotometerisusedwithanaqueous SDSbackgroundtoremoveanyabsorptionnotduetothenanotubes. 40

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aIntensityProlein8[ mm ]Quartz bIntensityProlein4[ mm ]Quartz Figure5.4: Timeaverageintensityelddistributions Avg xy forthea8mmand b4mmquartztubing. 41

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6.ExperimentalResults TheopticalabsorptionspectraofSingleWalledCarbonNanotubescanbepredictedusingatightbindingmodelasoutlinedin[19].Itwasdeterminedthatthe spacingoftheVanHovesingularitieswasgivenbytheratio1:2:3formetallic SWNT'sand1:2:4forsemiconductingSWNT's.Further,thelowestorderVan Hovesingularityforeachspecieswasfoundtobe: E M 11 = 6 0 a cc d .1 E S 11 = 2 0 a cc d .2 Where 0 2 : 9[ eV ]istheenergyvalueofthehoppinginteraction, a cc 0 : 143[ nm ]isthenearestneighborcarbon-carbondistance,and d =0 : 9 )]TJ/F15 11.9552 Tf 11.764 0 Td [(1 : 7[ nm ]is theSWNTdiameterrangeaccordingtothemanufacturer.Theserelationsleadto thetransitionenergiesgiveninthetable6.1. The S 11 transitionliesoutsideoftherangeofmostspectrophotometersandthe M 22 and S 33 havesignicantoverlap.Asaresult,the M 11 and S 22 arethemost wellsuitedforanalysis.Giventhe900[ nm ]limitofthespectrophotometerandthe interferenceofhigherenergytransitionsbelow 600[ nm ],attentionwillbefocused ontherange600 )]TJ/F15 11.9552 Tf 11.955 0 Td [(900[ nm ]. Giventhedataintable6.1,the M 11 isexpectedtodominateintheregionof approximately600 )]TJ/F15 11.9552 Tf 10.776 0 Td [(650[ nm ].Conversely,the S 22 transitionshoulddominatedinthe 850 )]TJ/F15 11.9552 Tf 11.982 0 Td [(900[ nm ]region.Theintermittentregion,650 )]TJ/F15 11.9552 Tf 11.982 0 Td [(850[ nm ],willbeinuencedby bothofthesetransitions. 6.1PreandPostDEP Aftertheinitialsonicationandcentrifugationaswasdescribedinchapter5,the absorptiondataingure6.2awastaken. AfterthefourveminuteDEPcycles,theneedleswereremovedfromthesolution andallowedtodry.Novisibledepositionhadoccurredontheneedlesplacedinthe 42

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Table6.1: Tightbindingpredictedphotonicenergiesandwavelengthsfornanotube VanHovetransitions. TransitionFormulaEnergyWavelength E M 11 6 0 a cc d 1 : 464 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 : 765[ eV ]448 )]TJ/F15 11.9552 Tf 11.955 0 Td [(847[ nm ] E M 22 12 0 a cc d 2 : 928 )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 : 529[ eV ]224 )]TJ/F15 11.9552 Tf 11.955 0 Td [(423[ nm ] E S 11 2 0 a cc d 0 : 488 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 922[ eV ]1344 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2541[ nm ] E S 22 4 0 a cc d 0 : 976 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 844[ eV ]672 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1270[ nm ] E S 33 8 0 a cc d 1 : 952 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 : 688[ eV ]336 )]TJ/F15 11.9552 Tf 11.955 0 Td [(635[ nm ] 8 mm capillary;however,aclearlyperiodicdepositionpatternoccurredinthe4 mm capillarygure6.1.NotethatsphericalwavespropagateasBessel'sfunctions,not sinusoids,soanincreasingwavelengthisexpectedasthewavemovesfromthepoint towardsthebaseoftheneedle. Theneedleswerecutandplacedinabathofaqueoussodiumdodecylsulfonate andbathsonicateduntilthedepositedhadvisiblybeenremovedfromthesurface oftheneedlesminutes.Theneedleswerethendiscardedandthesolutionwas furtherbathsonicatedfor22eightminutecyclestotalling3hoursand4minutes. Opticalextinctiondatawasthentakenforthispost-DEPsamplegure6.2. 6.2Analysis Thepost-DEPdropinthe S 22 and S 33 intothenegativeabsorptionrangeindicates thatthefreemicellesinthebackgroundsolutionwereabletoabsorbmoreinthat regionthattheremainingsemiconductingSWNTs.Theabsenceofthe M 11 peakfrom 448 nm toroughly725 nm isbelievedtobetheresultofthesmallerdiameter,and thereforemoredense,SWNTsbeinglteredoutduringthecentrifugationprocesses. 6.3Conclusion Comparisonofthedatapresentedingure6.2showsasharpdecreaseinthe semiconductingdominatingregion+[ nm ],anoticeabledecreaseinthe650 )]TJ0 g 0 G/F15 11.9552 Tf -212.555 -18 Td [(43

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Figure6.1: Bottomneedlefrom4mmcapillaryafterfourveminuteexposuresto microwaves. 850[ nm ]regioncorrespondingtoboth M 11 and S 22 transitions,andonlyasmall decreaseinthe M 11 dominatedregionof600 )]TJ/F15 11.9552 Tf 9.688 0 Td [(650[ nm ].InaccordancewithBeer'slaw, thisindicatesthatthesemiconductingconcentrationwasdecreasedinmoredrastic mannerthanthemetallic. 6.3.1FutureWork Theworkdemonstratedhereclearlyrepresentsasignicantsteptowardslarge scaleSWNTseparation,howevermanydicultiesstillremain.Depositiondidnot increasesubstantiallywithlongerexposuretimes.Further,whilethecontentofthe depositednanotubeswashighlymetallic,theremainingsolutionwasnotofsignicantlyincreasedsemiconductingcontent.Thisindicatesthatonlyasmallpercentage ofthemetallicnanotubesweredeposited. Giventhatthesolutionhadpre-centrifugetotalSWNTconcentrationof0 : 06 mg mL thatapproximatelyonethirdofnanotubesaremetallic,andthatthe4 mm capillary heldroughly1 mL ofsolution,itcanbeassumedthat < 20 g possiblymuchlessdue tothecentrifugeofmetallicnanotubeswereinthesystemtobeginwith.Fromhere 44

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aPre-DEP bPost-DEP Figure6.2: ExtinctionspectraforSWNT-SDSsolutionabeforeandbafterDEP exposure. 45

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itisobviousthatifonlyafractionofthemetallicnanotubesaredeposited,thenthe yieldofthissystemisquietlow. Finitedierenceswasusedinordertogaininsightintothebehaviorofthesystem inthepresenceofeldsapproachingsingularities.Giventheinherentweaknessof nitedierencestodescribesingularities,furthernumericalstudyofthesystemwould greatlybenetfrommoreaptmodelssuchasniteelement. 46

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REFERENCES [1] C.A.Balanis AdvancedEngineeringElectromagnetics,ed.2 ,Wiley,2012. [2] L.Benedict,S.Louie,andMCohen Staticpolarizabilitiesofsingle-wall carbonnanotubes ,Phys.Rev.B,52,8541,1995. [3] S.Blatt DielectrophoresisofSingleWalledCarbonNanotubes ,Dissertation KasrlsruheInstituteofTechnology,2008. [4] J.C.Charlier,X.Blase,S.Roche ElectronicandTransportPropertiesof Nanotubes ,Rev.ofMod.Phy.,79,2007. [5] R.Courant,E.Isaacson,M.Rees Onthesolutionofnonlinearhyperbolic dierentialequationsbynitedierences ,Comm.PureAppl.Math.5,243-255, 1952. [6] F.Gibou,R.Fedwik AFourthOrderAccurateDiscretizationfortheLaplace andHeatEquationsonArbitraryDomainswithApplicationstotheStefanProblem ,J.Comp.Phy.202.2,577-601,2005. [7] H.T.Ham,Y.S.Choi,I.J.Chung AnExplanationofDispersionStatesof Single-walledCarbonNanotubesinSolventsandAqueousSurfactSolutionsUsing SolubilityParameters ,J.Col.Int.Sci.286,216-223,2005. [8] S.A.Hodge,M.K.Bayazit,K.S.Coleman,andM.S.P.Shaffer Unweavingtherainbow:areviewoftherelationshipbetweensingle-walledcarbon nanotubemolecularstructuresandtheirchemicalreactivity ,Chem.Soc.Rev., 41,2012. [9] S.Iijima HelicalMicrotubulesofGraphiticCarbon ,Nature543.6348,1991. [10] C.Johnk EngineeringElectromagneticFieldsandWaves,ed.2 ,Wiley,1988. [11] A.Hartschuh,H.N.Pedrosa,J.Peterson,L.Huang,P.Anger,H. Qian,A.J.Meixner,M.Steiner,L.Novotny,andT.D.Krauss SingleCarbonNanotubeOpticalSpectroscopy ,ChemPhysChem,6,2005. [12] R.Krupke,F.Hennrich,M.M.Kappes,andH.V.Lohneysen Surface ConductanceInducedDielectrophoresisofSemiconductingSingle-WalledCarbon Nanotubes ,NanoLetters,Vol.4,No.8,2004. [13] E.MajchrzakandL.Turchan Thenitedierencemethodfortransient convection-diusionproblems ,ScienticResearchoftheInstituteofMathematics andComputerScience,Vol.11,No.1,2012. 47

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[14] M.P.Marder CondensedMatterPhysics ,JohnWiley&Sons,2010. [15] A.N.Mina,A.A.Awadallah,A.H.Phillips,andR.R.Ahmed SimulationoftheBandStructureofGrapheneandCarbonNanotube ,J.Phys.:Conf. Ser.,343,2012. [16] N.Nair,W.Kim,R.Braatz,andM.Strano DynamicsofSurfactantSuspendedSingle-WalledCarbonNanotubesinaCentrifugalField ,Langmuir, 24,1790-1795,2008. [17] ArdavanF.Oskooi,DavidRoundy,MihaiIbanescu,PeterBermel, J.D.Joannopoulos,andStevenG.Johnson MEEP:AexiblefreesoftwarepackageforelectromagneticsimulationsbytheFDTDmethod ,ComputerPhysicsCommunications181,687702,2010. [18] J.Ouellette BuildingtheNanofuturewithCarbonTubes ,IndustrialPhysicist, Vol.8,Iss.6,2002. [19] M.Ouyang,J.Huang,andC.Lieber FundamentalElectronicProperties andApplicationsofSingle-WalledCarbonNanotubes ,Acc.Chem.Res.,35, 1018-1025,2002. [20] B.R.PriyaandH.J.Byrne InvestigationofSodiumDodecylBenzeneSulfonateAssistedDispersionofDebundlingofSingle-WallCarbonNanotubes ,J. Phys.Chem.C,112,332-337,2008. 48

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APPENDIXA.DEPFiniteDierenceCode Thecodelistedherewasusedthroughoutthisthesistondtheoperator A such thattheDEPequation2.20holds.Itisprovidedinbothtwoandthreespatial dimensionalcoordinatesystems. A.1TwoDimensions function[A]=dep fd 2dintensity,gamma,phi,h %Usesmethodoffinitedifferencestobuildthe2DDEPoperator: %dC/dt=-AC % %INPUT:intensity-2DEtensor %gamma-Advectionconstant %phi-Diffusionconstant %h-intensitytensorstep-sizescale % %OUTPUT:A-operatorsuchthatdC/dt+AC=0 [m,n]=sizeintensity;%Savetensordimensions intensity=reshapeintensity,[],1;%Formvectorfrom3Dtensor %Laplacianmatrix L=laplacian 2dm,n,h; %DerivativeMatrice [DX,DY]=derivative matrices 2dm,n,h; %Findelectricintensityderivativeandlaplacianvectors; EDXE=DX intensity; EDYE=DY intensity; 49

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ELE=L intensity; %Diagonalize DEDXE=spdiagsEDXE,0,m n,m n; DEDYE=spdiagsEDYE,0,m n,m n; DELE=spdiagsELE,0,m n,m n; %NeumannBCDerivativeMatrix [CDX,CDY]=derivative matrices 2dm,n,h; %Xboundaryconditions fori=0:n-1 j=1+i m; k=m+i m; %Clearboundaryrows CDXj,:=zeros,m n; CDXk,:=zeros,m n; %Loadnofluxrequirement CDXj,j=EDXEj; CDXk,k=EDXEk; end %Yboundaryconditions fori=1:m j=m n-i+1; %ClearBoundaryRows CDYi,:=zeros,m n; CDYj,:=zeros,m n; %LoadData CDYi,i=EDYEi; CDYj,j=EDYEj; 50

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end %Calculateoperator A=gamma. DEDXE CDX+DEDYE CDY+DELE-phi. L; function[DX,DY]=derivative matrices 2dm,n,h %Returnsthemnxmnderivativematricesincartesiancoordinates. %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. ifm < 5 jj n < 5 errorm,n,mustbegreaterthanorequalto5 end X=spdiagsonesm,1 [1-808-1],-2:2,m,m; Y=spdiagsonesn,1 [1-808-1],-2:2,n,n; %Boundaryextrapolationndorderforward/central X:2,1:4=[-1824-60;-6060]; Xend-1:end,end-3:end=[0-606;06-2418]; Y:2,1:4=[-1824-60;-6060]; Yend-1:end,end-3:end=[0-606;06-2418]; Y expanded=kronY,speyem,m; DX=kronspeyen,X./ h; DY=Y expanded./ h; end 51

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n end f listing g n begin f lstlisting g function[L]=laplacian 2dm,n,h %ReturnsthemnxmnLaplacianmatrixincartesianCoordinates %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. ifm < 5 jj n < 5 errorm,nmustbegreaterthanorequalto5 end LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; %LX=spdiagsonesm,1 [1-21],-1:1,m,m; %LY=spdiagsonesn,1 [1-21],-1:1,n,n; %UpperBoundaryExtrapolation LX:2,1:5=[-11000;1-2100]; LY:2,1:5=[-11000;1-2100]; %LowerBoundaryExtraptolation LXend-1:end,end-4:end=[001-21;0001-1]; LYend-1:end,end-4:end=[001-21;0001-1]; LY expanded=kronLY,speyem,m; L=kronspeyen,LX+LY expanded./h; end 52

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A.2ThreeDimensions function[A]=dep fd 3dintensity,gamma,phi,h %Usesmethodoffinitedifferencestocalculateasteadystateconcentrationprofile %resultingfromaknownelectricfield. % %INPUT:intensity-3DEtensor %gamma-Advectionconstant %phi-Diffusionconstant %h-Stepsizeresolutionofintensitytensor. % %OUTPUT:A-operatorsuchthatdC/dt+AC=0 [m,n,p]=sizeintensity;%Savetensordimensions intensity=reshapeintensity,[],1;%Formvectorfrom3Dtensor %Laplacianmatrix L=laplacian 3dm,n,p,h; %DerivativeMatrice [DX,DY,DZ]=derivative matrices cartesianm,n,p,h; %Findelectricintensityderivativeandlaplacianvectors; EDXE=DX intensity; EDYE=DY intensity; EDZE=DZ intensity; ELE=L intensity; %Diagonalize DEDXE=spdiagsEDXE,0,m n p,m n p; 53

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DEDYE=spdiagsEDYE,0,m n p,m n p; DEDZE=spdiagsEDZE,0,m n p,m n p; DELE=spdiagsELE,0,m n p,m n p; %ConcentrationDerivativeMatricesNeumannBoundary CX=sparseDX; CY=sparseDY; CZ=sparseDZ; %Xboundaryconditions fori=0:n p-1 j=1+i m; k=m+i m; %Clearboundaryrows CXj,:=zeros,m n p; CXk,:=zeros,m n p; %Loadnofluxrequirement CXj,j=EDXEj; CXk,k=EDXEk; end %Yboundaryconditions fori=0:p-1 forj=1:m k=j+i m n; CYk,:=zeros,m n p;%ClearBoundaryRow CYk,k=EDYEk;%LoadData end forj=+m n-1:m n k=j+i m n; CYk,:=zeros,m n p;%ClearBoundaryRow CYk,k=EDYEk;%LoadData 54

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end end %Zboundaryconditions fori=1:m n j=m n p-i; %ClearBoundaryRows CZi,:=zeros,m n p; CZj,:=zeros,m n p; %LoadData CZi,i=EDZEi; CZj,j=EDZEj; end %Calculateoperator A=gamma DEDXE CX+DEDYE CY+DEDZE CZ+DELE-phi L; end function[DX,DY,DZ]=derivative matrices cartesianm,n,p,h %Returnsthemnpxmnpderivativematricesincartesiancoordinates. %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. %Allapproximationsusestepsizeh. ifm < 5 j n < 5 j p < 5 errorm,n,pmustbegreaterthanorequalto5 end X=spdiagsonesm,1 [1/12-2/302/3-1/12],-2:2,m,m; 55

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Y=spdiagsonesn,1 [1/12-2/302/3-1/12],-2:2,n,n; Z=spdiagsonesp,1 [1/12-2/302/3-1/12],-2:2,p,p; %Boundaryextrapolationndorderforward/central X:2,1:4=[-3/22-1/20;01-21]; Xend-1:end,end-3:end=[1-210;01/2-23/2]; Y:2,1:4=[-3/22-1/20;01-21]; Yend-1:end,end-3:end=[1-210;01/2-23/2]; Z:2,1:4=[-3/22-1/20;01-21]; Zend-1:end,end-3:end=[1-210;01/2-23/2]; Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X./h; DY=kronspeyep,p,Y expanded./h; DZ=Z expanded./h; end function[L]=laplacian 3dm,n,p,h %ReturnsthemnpxmnpLaplacianmatrixincartesianCoordinates %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. %Allapproximationsusestepsizeh. ifm < 5 j n < 5 j p < 5 errorm,n,pmustbegreaterthanorequalto5 end 56

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LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; %UpperBoundaryExtrapolation LX:2,1:5=[00000;1-2100]; LY:2,1:5=[00000;1-2100]; LZ:2,1:5=[00000;1-2100]; %LowerBoundaryExtraptolation LXend-1:end,end-4:end=[001-21;00000]; LYend-1:end,end-4:end=[001-21;00000]; LZend-1:end,end-4:end=[001-21;00000]; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded./h; end 57