Control of Micro- and Nano- Particles With

Electric and Magnetic Fields

A dissertation submitted to the Faculty of the Graduate School of Arts and Sciences

of Georgetown University in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Physics

Vincent Paul Spinella-Mamo, M.S.

Washington, DC

Dec 16, 2008

Control of Micro- and Nano- Particles With Electric and Magnetic Fields

Vincent Paul Spinella-Mamo, M.S.

Thesis Advisor: Makarand Paranjape, Ph.D.

Abstract

This dissertation is presented in two sections. The first section encompasses the con-

trol of microparticles via the application of externally applied electric fields. It has

previously been shown how electric fields can be employed to control and manipulate

matter. Here, I show that cells can be controlled via electric fields using a mathe-

matical description. Cells were chosen as the microparticle, as this phenomena can

be employed in a device to manipulate and transfect cells. Once described mathe-

matically, I have modeled the motion of these cells using a finite time step algorithm.

Once the mathematical and physical modeling is complete, I have designed, fabri-

cated, and tested one part of a device which will be employed for the specific purpose

of manipulation and transfection of cells.

Magnetic fields have been shown to interact with matter in a similar manner. Using

a similar construct as described above, the manipulation has been described mathe-

matically and then modeled by introducing novel search and optimization algorithms.

These techniques were applied to a ferrofluid system, which is a paramagnetic fluid

comprised of nanoparticles of iron oxide surrounded by a surfactant. A possible ap-

plication of ferrofluids under controlled magnetic fields relies on using the resulting

ii

local deformations in the paramagnetic fluid to alter the path of light for switch-

ing and optical projection techniques. This device has been designed, fabricated, and

tested as well. In addition, a consequence of the device testing led to the development

of a secondary application of the phenomenon. Another device has been developed

and demonstrated that can produce variable channel widths using the ferrofluid and

variations in the applied magnetic field.

iii

Acknowlegements

I would like to thank my advisor, Makarand Paranjape, for his oversight and guidance

on these projects and for giving me the opportunity to see my own ideas through to

fruition. He, his wife Shruti, and son Nikhil have also been good friends throughout

this journey, and I thank them for that. Furthermore, I would like to thank my

committee, Edward Van Keuren, Amy Liu, and YuYe Tong. I’ve bothered them

more than they will probably care to admit. Yet, they continued to help despite me

nearly blowing up entire laboratories through soldering. I would also like to thank

my colleagues and friends at the Applied Physics Laboratory, who provided me with

invaluable laboratory and life skills, especially Ann Darrin, Robert Osiander, Keith

Rebello, Chris Deihl, Stergios Papadakis, Jennifer Sample, and Sean Murphy. Special

thanks to Leon Der of the Georgetown machine shop for always suggesting a method

of solving a problem. As I continued here at Georgetown, I was helped by friends

here to stay sane as I toiled away in underground laboratories and windowless offices.

If it weren’t for them, I would have gone crazy(er) long ago. I would especially like

to thank Russel Ross who has been an invaluable addition to my life. If it weren’t

for him, I would have been on the streets. In addition, I would have never made it

through without Kristen Bloschock and her husband, Leo, Simon Hale, Brad Burns,

Yogesh Kashte, Kevin Pushee, Nadjib and Jianyun Zhou. I would also like to thank

the Shin-Etsu Company for providing Georgetown with a sample of their wonderful

iv

product, SiPR-M-7126. This has made my fabrication life a breeze. Finally, and

most importantly, I would like to thank my family. My mother for teaching me

science from a young age and doing experiments with me and always doing things for

me. For Angie, Johnny, and Jimmy. For my grandmothers Grace and Carmella and

my Uncle Joe. You were there when I needed you most. I only regret that my father

and grandfather were not around to see this.

v

Dedication

This work is dedicated to the memories of my father and grandfathers, who all passed

too soon. I hope this work makes them proud.

vi

Contents

Abstract ii

Acknowledgements iv

Dedication vi

1 Control of Microparticles with Electric Fields: Application to Cell

Transfection 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Controlling Dielectric Particles with Electric Fields . . . . . . . . . . 4

1.2.1 An introduction to the dielectrophoretic force . . . . . . . . . 4

1.2.2 Cell Transfection and Device Overview . . . . . . . . . . . . . 6

1.2.3 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . 12

1.3 Modeling Micro-particles in an Electric Field . . . . . . . . . . . . . . 14

1.4 Cell Transfection Device . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.2 Control Parameters . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4.3 Proof of Principle . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii

2 Control of Nanoparticles with Magnetic Fields 38

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Magnetism of Materials . . . . . . . . . . . . . . . . . . . . . 38

2.1.2 Ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.3 Magnetofluidics . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.2 Simulation of Ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.1 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.2.4 Field due to a current loop . . . . . . . . . . . . . . . . . . . . 57

2.2.5 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3 Ferrofluid Projection and Lithography . . . . . . . . . . . . . . . . . 63

2.3.1 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.3.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3.3 Array Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.4 Ferrofluid Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.5 An Unintended Consequence [Variable Channel for Microfluidics] . . . 80

3 Conclusion 86

A Needle and SU-8 Component Fabrication 89

B DEP Trapping Model 91

C Ferrofluid Genetic Algorithm 102

D Optical Ray Trace Program 119

E Ferrofluid Control Program 124

viii

F Variable Channel 129

ix

Chapter 1

Control of Microparticles with

Electric Fields: Application to Cell

Transfection

1.1 Introduction

This dissertation explores the manipulation of matter with electromagnetic fields.

While this is by no means a novel venture, the techniques and size scales employed

have widened the realm of manipulation, ultimately lending itself to several specific

new applications. In this chapter, which focuses on electric field control of micropar-

ticles, the physical principles and modeling of the phenomena will be discussed before

showing that the principles may be employed for the specific use in a cell transfection

device (here the cell, on the order of 10µm, will be the microparticle being controlled

with electric fields).

In order to achieve scientific advancement in the area of controlling microparticles

1

(and smaller), it is necessary to have the ability to manipulate these particles at will.

While methods such as electrophoresis, chromatography, and field flow fractionation

allow for the separation of specific particles based on certain parameters, they do not

offer the ability to guide the particle to specific points for subsequent manipulation.

In the past it has been shown that particles can be controlled via a variety of methods.

As a specific example, in the field of mass spectrometry, the principle that a charged

particle will curve in a magnetic field according to it’s charge and mass will obey the

equation shown below[1]:

mv2

rr = q~v × ~B (1.1)

|r| = m|v|q|B|sin(θ)

(1.2)

Here r is the radius of the charged particle, m is the mass of the particle, q is the net

charge on the particle, ~v is the velocity at which the particle enters a constant and

uniform magnetic field, ~B, which is oriented at an angle θ relative to the velocity of

the particle.

Continued technological and scientific innovation led to the development where op-

tics can be used for particle control. In 1970, Ashkin of Bell Labs published a paper

stating that particles could be manipulated using radiation pressure from visible laser

light[2]. Initial experimental observations showed that the electromagnetic force im-

parted by radiation pressure of a single beam could cause a particle to move towards

the beam axis and then in the direction of propagation of the beam. This yielded the

idea that using two TEM00 (a transverse electromagnetic wave, which has no elec-

tromagnetic field in the direction of propogation[1]) Gaussian beams opposing each

other, a complete optical “trap” can be formed, providing the ability to manipulate

2

the position of a particle at will.

Similarly, it is also possible to achieve the same level of control using an acoustic

source, instead of an optical one. The force caused by acoustic radiation on a parti-

cle, Fac, is given by[3]:

Fac = −4

3πR3kEacAsin(2kx) (1.3)

In the above equation R is the particle radius, k is the wavenumber of the acoustic

source, x is the distance from a node, Eac is the average acoustic energy density,

and A is a constant relating the density, speed of sound, and compressibility of the

medium and those of the particle in the medium. In this work, the authors were able

to apply the acoustic source vertically and balance the force with that of gravity to

establish the equilibrium positions of the particles to be[3]:

x =1

2ksin−1

[(ρ− ρ′)gkAEac

](1.4)

where g is the acceleration due to gravity and ρ and ρ′ are the densities of the medium

and particle, respectively.

From the equation, it should be noted that this equation yields control of particles

not by their size, but their respective acoustical properties.

3

1.2 Controlling Dielectric Particles with Electric

Fields

1.2.1 An introduction to the dielectrophoretic force

It is clear that electric fields can be used to manipulate charged particles. However,

even when a particle has no net charge, it may still be susceptible to forces created

by electric fields. When a dielectric particle is placed in an electric field, it becomes

polarized. If the electric field near the positive end of the dipole is different from that

near the negative end, there is a net force on the particle as illustrated in figure 1.1.

The direction and magnitude of this dielectrophoretic force depends on the geometry

of the particle and the dielectric properties of both the particle and the surrounding

medium.

A spherical particle in a uniform applied electric field, ~E0, has a polarization[1]:

~P = 3ε∗mε∗p − ε∗mε∗p + 2ε∗m

~E0 (1.5)

where ε∗p and ε∗m are the permittivities of the particle and the medium, respectively.

The factor K(ω), called the Claudius-Mossotti factor[4], is defined in equation 1.6.

K(ω∗) =ε∗p − ε∗mε∗p + 2ε∗m

(1.6)

The force on the dielectric particle can then be derived:

~p = 43πr3 ∗ 3ε∗pK(ω) ~E (1.7)

4

U = −(V

[3ε∗m(ω)K(ω) ~E

]· ∇

)~E (1.8)

~p · ∇ ~E = 4πr3ε∗pK(ω) ~E · ∇| ~E| (1.9)

~F = 124πr3ε∗m(ω)K(ω)∇| ~E|2 (1.10)

Here, the factor of 12

in the last equation comes from the fact that ~E ·∇ ~E = 12∇| ~E|2 [5].

For AC fields, ~E is the rms value. It should be noted that the imaginary component

of the Clausius-Mossotti factor determines characteristics about how fast charges can

rearrange on a particle. Low frequencies (less than 1kHz) are typically associated with

surface charges moving. As the frequency is increased, the factor becomes associated

with the effective conductivity and then the permeability[4]. An illustration of the

DEP effect can be seen in figure 1.1 as the polarized particle migrates towards the

region of higher electric field concentration. Furthermore, since the Clausius-Mossotti

factor is frequency dependent, dielectrophoretic forces can be selectively tuned to filter

out specific particles based on their frequency responses to the applied fields.

Figure 1.1: Dielectrophoretic force on a spherical particle[6]

5

1.2.2 Cell Transfection and Device Overview

One possible application of this technology comes in the study of biological processes,

specifically gene expression and protein synthesis via transfection. Protein synthesis,

or translation, within a cell is a complicated procedure which requires a variety of

molecules (DNA, tRNA, mRNA, ribosomes, etc)[7]. Translation occurs in three steps:

initiation, elongation, and termination. Genetic coding stored in the DNA sequence

is transferred outside of the nucleus via a molecule called mRNA. The mRNA at-

taches to a ribosome (a cellular molecule) inside the cell at a specific site (a sequence

of three amino acids, known as a codon, with the sequence AUG)[7]. From here,

each subsequent codon on the mRNA signals a tRNA molecule to bring and bind a

specific amino acid in the elongation phase. This terminates when the stop codon

(UGA, UAG, or UAA) is reached[7]. This entire sequence can be seen in figure 1.2

In order to study the mechanisms of genetic regulation and protein function, scientists

need a technique in which certain genetic sequences can be created, and ultimately, ex-

pressed via introduction into a eukaryotic cell (a cell which contains a nucleus). Viral

methods (infection) can accomplish this, however this method can only be employed

with certain sequences and also introduces other genetic material[9]. Transfection is

the technique by which nucleic acids are introduced into cells by non-viral methods[9].

Furthermore, once transfection is accomplished, it can be either transient (genetic in-

formation is only expressed for several days) or stable (genetic expression occurs over

a long-term)[9]. A transition between transient and stable transfection may be accom-

plished through several weeks of selection processes, cloning and characterization[9].

Eukaryotic cells, specifically, have the ability to pick up exogenous fragments of DNA

and incorporate them into their chromosome[7].

6

Figure 1.2: Cellular protein synthesis[8]. As shown in the initiation phase the pinkportion represents one side of the ribosome which attaches to the mRNA, red strip.The tRNA, in green, brings the first amino acid (yellow circle) to the mRNA site.At this point the ribosome is closed with the other larger pink portion. As shown inthe elongation phase, when amino acids are connected, the corresponding tRNA isreleased from the ribosome.

7

Transfection can be accomplished via several mechanisms. Chemically, the desired

amino acid complex can be placed in a reagent that creates a condition for endocy-

tosis, ultimately bringing the genetic material into the nucleus (endocytosis is the

means by which a cell is able to enclose foreign objects and bring them in the cellular

membrane[7]. These particles are generally too large to permeate through the cellular

membrane[7]). The first example of this is the use of DEAE-dextran in 1965[10]. This

polymer associates closely with the negative nucleic acid. The cationic potential of

the polymer then allows the complex to come into close contact with the negatively

charged cellular membrane, at which point endocytosis takes place. This form of

transfection is predominantly transient[10]. More recently, the popular method of

chemical transfection is through the use of calcium phosphate[11]. Use of calcium

chloride, however, only leads to a one in 106 stable transfections[7].

Figure 1.3: Chemical cell transfection mechanisms[9]

Physically, the most effective technique is direct microinjection, in which a single

cell at a time is held and injected with a micro-needle full of the desired genetic

8

material[9]. As expected, this method does not lend itself to experiments requiring

large amounts of transfected cells. Electroporation attempts to solve this problem by

creating micro-sized pores via perturbation of the cell membrane brought about by

an electrical pulse. Here, the problem lies in the fact that the required electricity to

perform electroporation may exceed the maximum level for the cell, ultimately de-

stroying it[9]. Finally, one other physical method is biolistic particle delivery, where

the nucleic acids are turned into microprojectiles that move at a high enough velocity

to be introduced into the cell[9].

Recently, a novel technique called magnetofection has been introduced to automating

cell transfection. This method relies on the fact that the biological vectors introduced

are paramagnetic and will quickly move in the direction of the gradient of an applied

magnetic field (this force will be discussed in further chapters). One group has used

this method to show that cells can be transfected this way within minutes by using a

neodymium-iron-boron magnet[12].

As previously noted, there is a lack of an effective method for ensuring that a large

number of cells are quickly transfected without lethal damage. A novel device is being

developed to perform the task. Instead of using electric fields in order to damage the

cell walls (as in electroporation), we propose the use of dielectrophoretic trapping

to localize cells onto an array of microneedles. The cells are then held in place by

pressure and transfected. This should allow automation of the process and enable a

much higher rate of transfection than previously established. By first localizing the

cell with DEP, it becomes possible to directly inject it. Further exploitation of the

capabilities of MEMS fabrication allows for the incorporation of microfluidics (for

enhanced control) and microneedles, creating a completely automated process.

9

The prototype device is in a 12x12 array configuration such that 144 cells can be

introduced and transfected at a time. At each site, a series of four pillars will localize

the cell. The traps allow cells that are not help to be flushed out of the microneedle

system, ensuring a 1:1 cell:needle correspondence. Once localized, the cell will be

brought down onto a microneedle made of silicon via a suction produced by several

microfluidic channels that sit above the surface of the needle. Once the cell is affixed

onto the needle, genetic information can be passed through and the cell can be re-

leased. The proposed schematic for an individual site is shown in figure 1.4. The four

pillar geometry was chosen as similar geometries have been shown to be successful

in other experiments. In addition, the symmetry of the configuration allows for an

easily repeated pattern, maximizing utilized area of the device, while not restricting

fluid flow.

10

Figure 1.4: Exploded view of single transfection element (of 144 total). Bottom

leftmost component is the top side of a silicon microneedle, above which are the SU-8

bilayer used to create the vacuum channels. Top rightmost component represents the

four metallic pillars attached to a substrate that will be used to generate the DEP

force.

Figure 1.5: Close up of the entire DEP section of the device in (a) isometric and(b) top views. Diagonal lines represent metallic traces on the substrate for electricalinterconnection, while the rectangular prisms indicate the pillars themselves.

11

1.2.3 Finite Difference Method

Before attempting to fabricate a device that will use these principles to manipulate

a particle, it is important to first model the expected behavior. While the specific

forces due to a proposed geometry can be calculated explicitly, numerical methods

can be used to simulate the physical response in more complex configurations, as well

as provide dynamic tracking of particle motion. A particularly useful set of numerical

techniques used to solve differential equations is the finite difference method. These

algorithms discretize a continuous variable in order to break the physical problem

into a finite number of pieces. One class of finite difference methods is based on the

forward Taylor approximation[13]. Here, the value of the at the point xi+1 based on

values of the function and its derivatives at the point xi:

f(xi+1) = f(xi) + f ′(xi)∆x +1

2!f ′′(xi)(∆x)2 +

1

3!f 3(xi)(∆x)3 + . . . (1.11)

Here the subscript ‘i’ refers to the discrete positions of the continuous variable x.

Truncating the series at the second term yields the first order approximation to the

series. By rearranging this equation, it is possible to approximate the first derivative

as:

f ′(xi) =1

∆x[f(xi+1)− f(xi)] + O(∆x) (1.12)

The last term in the above equation indicates that the terms ignored due to the trun-

cation of equation 1.11 are on the order of ∆x.

This is known as the forward difference method[13]. The series may also be solved

for in reverse by determining the derivative at position xi based on the value at xi−1,

12

as shown below:

f(xi−1) = f(xi)− f ′(xi)∆x + . . . (1.13)

f ′(xi) =f(xi)− f(xi−1)

∆x(1.14)

Combining the forward and reverse schemes, a third approximation to the first deriva-

tive is acquired. This is known as the center difference method, and is accurate to a

higher order in ∆x, namely[13]:

f ′(xi) =f(xi+1)− f(xi−1)

∆x−O((∆x)2) (1.15)

Expanding the center difference method to the second derivative yields the following

approximation[13]:

f ′′(xi) =f(xi+1)− 2f(xi) + f(xi−1)

(∆x)2+ O((∆x)2) (1.16)

This scheme is not limited to defining derivatives on the spatial scale. The Taylor

forward approximation can also be applied to the time domain, as follows:

x(ti+1) = x(ti) + x′(ti)∆t + O((∆t)2) (1.17)

This approximation is known as the finite time step, since the forward value is acquired

by “stepping forward” in time[13]. Mohebi et al use this method to solve for the

movement (and structuring) of ferrofluid nanoparticles. In general, the equation of

motion of a particle moving through a fluid is given as follows[14]:

md2~ri

dt2+ D

d~ri

dt= ~Fi (1.18)

13

Here, the subscript refers to a specific particle, i, D is the drag coefficient, m is the

mass, ~r is the position and ~F is the sum of the external forces on that particle.

Integration yields:

∆~ri =~Fi

D∆t +

m

D

(~Fi

D− ~V0i

)e−D∆t/m (1.19)

By making the assumption that the time step, ∆t is relatively large (yet within the

realm of the finite time step scheme), the second part of the above equation can

be neglected, as the exponential term vanishes in time (m/D u 10−8s for the mass

of the nanoparticle and the drag term of the carrier fluid[14], so ∆t on the order of

microseconds is large enough)[14]. This means that the change in position is effectively

given as:

∆~ri =~Fi

D∆t (1.20)

Carrying out this iteration over multiple time steps defines how the particles move

over a specified period of time. This scheme will be discussed in the next section.

1.3 Modeling Micro-particles in an Electric Field

As discussed previously, the dielectrophoretic force will be directed toward (or away

from) the gradient in an electric field. In order to create a force which localizes a

particle, several geometries have been proposed[15, 16, 4]. For this project, a geometry

consisting of four metallic pillars was chosen, with the same potential applied to each

pillar. Cylindrical pillars were chosen, as they lend themselves to the symmetric

nature of repeated devices, as well as yielding a large enough space for fluid input.

The dimension of the pillars and spacing was chosen such that in an x-y plane a

particle on the order of 10µm could be confined at the center of a 50x50µ2 area. By

14

defining the height of the pillar to be significantly large in comparison to the particle

diameter, it can then be assumed that the electric potential is independent of the

height, and hence, a two dimensional model can be used. For the purposes of a cell

trap, pillars should be on the order of the 1-10 times that of the diameter, or namely

10-100µm. Since there are no other charges in the system, it is possible to use the

Laplace equation to determine an analytic solution, as derived below:

∇2φ = 1ρ

∂∂ρ

[ρ∂φ

∂ρ

]= 0 (1.21)

φ = A log ρ + C (1.22)

As shown above, the Laplacian is expanded in cylindrical coordinates (ρ being the

radial variable and assuming no dependence on height and φ having an angular sym-

metry), where φ is the electric potential A and C are constants of integration.

The boundary condition is that the potential on the surface of each pillar φ equals the

applied potential. Since the electrostatic potential is a scalar, the potential created

from each pillar may be summed to obtain the total potential. The electric potential

was calculated by a commercially available finite-element program, FEMLAB, and

compared to the analytic solution (figure 1.6). FEMLAB is a powerful modeling tool

based on finite-element modeling. While this software is capable of rendering the

corresponding electric potential and field distributions, it is not able to determine the

trajectory of a dielectric particle in such a configuration. It can be used, however, to

verify that the results obtained by the program to be correct. Once electric potential

has been tabulated on a grid, it is easy to generate the dielectrophoretic force. The

electric field, which is the gradient of the potential, was calculated using a center

15

Figure 1.6: Potential field generated by a) FEMLAB and b) C routines. Color inboth models indicates the value of the electric potential. Darker shades of blueindicate lower potential with the red and yellow being the highest potential in theFEMLAB and C model, respectively. While the value of potential is not important,the distribution is.

difference approximation:

Ex(x, y) =1

2[φ(x + 1, y)− φ(x− 1, y)]

1

∆x(1.23)

Ey(x, y) =1

2[φ(x, y + 1)− φ(x, y − 1)]

1

∆y(1.24)

Figure 1.7 shows the resulting | ~E|2. The center difference method can be applied again

to determine the gradient | ~E|2, which is proportional to the dielectrophoretic force.

A slice of | ~E2| along the center line of the array is shown in figure 1.8 (using pillar

diameter of 10 µm, pillar spacing of 40 µm and an applied voltage of 1V). Purely from

inspecting the potential energy configuration, it can be seen that there is a barrier

to entry for the particle (i.e. if the particle is not imparted with enough velocity, it

will be reflected). Similarly, if the particle enters with too high of a velocity, it will

pass over the trap entirely, yielding an optimal range for velocity. At this point, it

is possible to simulate a particle moving in this field by a simple time step method.

16

Figure 1.7:∣∣∣ ~E

∣∣∣2

formed by pillar geometry in a) FEMLAB and b) C routines. Darker

shades of blue indicate lower values of | ~E|2, while reds indicate larger values. Onceagain, the values are not important, only the similarity in form should be noted inorder to apply the finite time step with confidence. While the value immediately nextto the pillar may differ, the important area is the center and regions betweem pillars,which determines the effective potential, as shown in figure 1.8

Given an initial condition of a particle (position and velocity) the trajectory can be

determined by integrating the equation of motion using a finite time-step (including

a Stokes’ drag term):

∆~r = 12

~F−b~vm

τ 2 + ~vτ (1.25)

∆~v =~Fm

τ (1.26)

A value of 0.18 was used for bm

(assuming that the dynamic viscosity of water to be

10−3Pa · s, the radius to be 5µm, and the density of the particle to be approximately

that of water, 1000kg/m3), −5.07 × 105 C2

N kgwas used for the DEP force coefficient

(using the relative permittivity of a polystyrene particle and water to be 2.5 and 80,

respectively).

17

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

6

7x 10

−3

Distance (um)

Effe

ctiv

e po

tent

ial e

nerg

y |E

2 | (V

2 /um

2 )

Figure 1.8: E2 midway between two pairs of pillars (which a dielectric particle wouldexperience travelling down the center of the device). As this is proportional to thepotential energy, the reader should note the entry and exit barriers

These values were taken from [15], and correspond to materials used in physical

experiments that will be discussed in detail in further sections.

The trajectory of the particle in the potential region defined by four pillars is shown

in figure 1.9. Iterations were stopped when the particle stopped moving (position at

time t+1 was equal to that at time t). As can be seen, the particle comes to rest

at the center of the trap. All simulations were run with the initial position of the

particle at the leftmost center point. For the example shown above, the following

parameters were used in the computational analysis: initial velocity = 75µ/s, array

unit spacing = 1µ, time step = 0.01s. With the aforementioned parameters, the total

number of timesteps for a complete trap (including all oscillations) was 24,774 steps.

It should also be noted that a significant number of these iterations were from the

18

Figure 1.9: Final position of particle using finite time step (indicated by blue dot)shown with respect to electric potential distribution. Height, and corresponding color,of potential distribution are in relative units. All initial conditions and parametersdescribed in text

final termination of the particle as it moved extremely slowly between two adjacent

array points in the center of the trap. Simultaneously, as expected, when the velocity

of the particle was too low, it did not enter the trap. When the velocity exceeded

a certain value, the particle passed over the trap. These velocity values are, neces-

sarily, defined by the geometry, particle parameters, and applied voltage. For this

particular case, effective trapping occurs between 70 and 82 µ/s, for lower and upper

bounds, respectively. It should be noted that there will be an array of these 4-pillar

arrangements. The repetitive potentials, when activated, will cause all particles at

rest to move to the center of the nearest “trap”. Therefore, by flowing the cells in

and then applying the potential to the pillars, it may be possible to overcome the

velocity restriction. This has, in fact, been demonstrated and will be shown in future

sections.

19

1.4 Cell Transfection Device

1.4.1 Fabrication

As derived in section 1.2.1, the functional portion of the DEP device is the geometry

of the electrical components. While DEP has been shown to work in various two-

dimensional configurations[4, 15], creation of a more powerful trap for particles in a

moving fluid requires a three dimensional geometry. In previous work, groups have

created metallic pillars necessary for DEP trapping via electroplating into a mold of

SU-8[16]. While effective, the fabrication technique proves difficult in its final steps

during the SU-8 removal process. The removal of SU-8 includes a DMSO-based photo-

stripper, an ashing to remove leftover SU-8, and a Nanostrip treatment[16, 17, 18].

The inert nature of SU-8 means that the procedure is not necessarily guaranteed to

remove all SU-8.

In an attempt to circumvent this difficult step, several other avenues were attempted

and will be described below. The functional part of the DEP device relies on the

ability to create a uniform electric potential on the outermost face of vertical pillars.

Since this is the only requirement necessary for generation of the intended electric

field for DEP, it should be possible to alter the structural portion of the pillar, while

retaining the necessary boundary conditions (of a conducting surface). This led to

the idea of using the SU-8 as the structural material for the pillar, instead of the mold

material into which, metal would be electroplated.

A conductive photopolymer that can attain high aspect ratios would be ideal, as it

would satisfy the requirements for both the structural and surface conduction. SU-8,

a photopolymer with which we have considerable experience, has been made conduc-

20

tive via several mechanisms, one of which is the incorporation of silver nanoparticles

before curing. This technique has been shown to increase the conductivity of SU-8

to levels of 106S/m[19]. A trial was attempted with 100-500nm silver particles from

InfraMat Advanced Materials (catalog #47MR-01C) mixed in SU-8 100. A major

problem arose since in order to create a conductance of that order requires 30% vol-

ume fraction of silver[20], the resulting silver content rendered the SU-8 completely

opaque and increased its viscosity considerably. The opacity limits the ability to poly-

merize and align subsequent fabrication layers. In order to overcome this limitation,

one group has suggested introducing HCl doped polyaniline (a conductive polymer)

introduced into SU-8[21]. Other groups have chemically altered SU-8 surfaces to elec-

trolessly plate metals to increase surface conductivity[22, 23]. These other methods

were not tested.

Instead of altering the SU-8 itself, another process was attempted. In place of using

the SU-8 to create a mold for electroplating, it was used to create pillars, on top of

which metal would then be deposited. This would yield pillars with a conductive

surface. The steps would be as follows (and illustrated in figure 1.10):

1. Pattern electrodes

2. Spin sacrificial resist layer to protect electrodes from metalization

3. Pattern SU-8 pillars

4. Deposit metal (to cover SU-8 Pillars)

5. Lift-off all excess metal

As the SU-8 layer is designed to be 100 microns in height, the sacrificial resist layer

should be patterned first to minimize the height between the mask and substrate dur-

21

Figure 1.10: Process flow for conductive SU-8 pillars as described in text. Light bluesubstrate is glass, brown is metallic electrodes and subsequent deposition, and darkblue is SU-8

22

ing photolithography (a relatively large separation between the mask and substrate

introduces lithographic anomalies, such as diffusion). The procedure was attempted

in this order, only to find out that Shipley, the positive sacrificial resist used, dissolves

in the developer for SU-8. To overcome this difficulty, steps 2 and 3 were swapped,

patterning the SU-8 before the Shipley.

While this method was tested and worked for larger features that were further spread

out (geometric configurations 10-50 times larger than those to be used in the intended

device), the configuration of pillars needed created a problem for this specific device

(pillars are on the order of 30µm in width with 160µm from center to center). Upon

spinning of the sacrificial resist layer, the close spacing of the pillars prevented the

resist from attaining a uniform spread by getting “stuck” at the base of the pillars, as

shown in figure 1.11. The resist build-up around the pillars prevented a uniform ex-

posure, as some areas were thicker than others. This, in addition to any lithographic

anomalies due to not being at the optimal distance for exposure, created a situation

in which lift-off could not be performed. Several trials were run on the metal coated

SU-8 pillars which had successful lift-off. No DEP effects were observed. This is be-

lieved to be because of the poor coverage of metal due to the high aspect ratio of the

pillars, yielding an open circuit between the electrical connections and the pillars. In

order to perform a good lift-off, metal deposition must be relatively vertical (allowing

an avenue for the sacrificial resist to dissolve). Coating the high-aspect ratio pillars,

however, requires a non-vertical deposition. These two requirements work in compe-

tition with each other for creating a successful device. Abandoning the attempt to

coat SU-8 pillars with metal and returning to the idea of electroplating pillars led

us to a new ultra-thick positive resist called SiPR (specifically the SiPR-7126M-20),

produced by Shin-Etsu MicroSi. This new material makes plating much easier than

23

Figure 1.11: SEM image taken after spin of sacrificial resist layer expose and devel-opment. As seen in the image on the right, resist build-up around the base of thepillars prevented deposition of metal to connect the pillar surface with the electricalconnections on the substrate

SU-8, as it provides the same thickness and aspect ratio as SU-8, but it can be re-

moved in acetone[24]. The optimized fabrication procedure is detailed below (and

illustrated in figure ??):

1. Clean microscope slides with acetone, isopropyl alcohol, de-ionized water, and

N2 dry

2. Magnetic sputter 250A of Cr@15W (1.3 A/s)

3. Magnetic sputter 2500A of Au@25W (4.2 A/s)

4. Spin Shipley 1813 45s@4000rpm

5. Pre-Expose Bake for 1hr@90oC

6. Expose for 10s@16mW/cm2

7. Develop in CD-30 for 2min

8. De-ionized water rinse, N2 dry

24

9. Au etch - 1 TFS: 2 De-ionized water (TFS is a Transene Chemicals silver type

etchant)

10. Acetone rinse to remove Shipley

11. Pre-Spin SiPR 7123 for 1s@300rpm followed by Spin for 45s@650rpm

12. Pre-Expose Bake 20min@120oC

13. Expose for 6.5min@16mW/cm2

14. Develop in CD-30 for 30min under agitation

15. De-ionized water rinse, N2 dry

16. Electroplate Ni @5mA for 2hrs Ni plating solution was obtained from Caswell,

Inc

17. Acetone bath with slight agitation to remove SiPR

18. Cr etch - 1 TFN : 7 De-ionized water (TFN is a Transene Chemicals nichrome

type etchant)

19. De-ionized water rinse, N2 dry

To minimize the number of electrical contacts, the pillars were interconnected via

a zig-zag pattern with all alternate rows ultimately connecting to the same applied

voltage. A close-up of the pillar electrical foundation is shown in figure 1.13. While

this geometry does function as intended, in order to minimize any asymmetric electric

field effects do to the electrical traces, an extended portion may be included (shown

in figure 1.13b)[15]. The above procedure yielded metallic pillars up to 80µm, as

shown by the Dektak (surface profiler) image in figure 1.14. A gold seed layer for

25

Figure 1.12: (1) Shows the Cr/Au plated electrode (silver and brown colors) on aglass substrate (light blue). (2) represents the spinning and patterning of Shipley(dark blue). (3) Gold etch and acetone bath. (4) Spin and pattern SiPR (shown inpurple). (5) Nickel plating onto exposed gold substrates (in gray). (6) SiPR removalin acetone and chrome etch yielding pillars.

26

Figure 1.13: (a) Electric trace for application of voltage to pillars. Squares are100µmx100µm and the horizontal lines are 30µm wide. Spacing in between squares is160µm.(b) Pad configuration to minimize asymmetric electric field effects. By plac-ing the configuration as a trapezoid, as opposed to a square, the input electric fieldbarrier is also decreased with respect to the exit electric field barrier[15]

electroplating is needed, as gold does not build up an oxide layer, which inhibits the

electrochemical reaction. The gold seed layer was placed on top of a layer of chrome

in order to adhere to the glass substrate (gold does not adhere well to glass).

27

Figure 1.14: Dektak of electroplated alignment mark and image of electroplated pil-

lars. Pillars are approximately 100µmx100µm and are 100µm tall above the 30µm

trace electrical interconnects.

After the pillars were created, it was then necessary to mate them to the hydraulic

side of the device. When the cells are localized using the dielectrophoretic trapping

mechanism, the device will then create a vacuum drawing them onto a silicon micro-

needle. Once on the needle, DNA material can be pumped through the other side.

A side view of the hydraulic component is shown in figure 1.15. This part was fab-

ricated and optimized by Yogesh Kashte and detailed in the procedure described in

appendix A. The aforementioned procedure was used to create the profile, as shown

in figure 1.16. A view of the fabricated needles is shown in figure 1.17. The TMAH

(tetramethylammonium hydroxide) etched backside allows for the introduction of ge-

netic material into the cell. The remaining part of the device is the channels that

allow for suction to be applied to the fluid to bring the cell onto the microneedle

itself. This is included as part of the procedure described in appendix A.

Attempts to simplify the procedure included making a large PDMS gasket, instead

28

Figure 1.15: Cross-sectional view of hydraulic component. As can be seen, the cellis anchored onto the microneedle (shown in gray) due to the pressure created in thechannels of the SU-8 bilayer (blue). At this point foreign material (shown as the smallcircles underneath) can be injected by applying a force via another SU-8 component(lowermost component)

29

Figure 1.16: Profile of silicon micro-needle

of the intricate second SU-8 layer. This, however, proved to be futile, as the upper

SU-8 layer was not rigid enough to stand on its own. Therefore, the original fabrica-

tion procedure was reinstated. In order to adhere the SU-8 component to the silicon

component, another thin layer of SU-8 must be applied. Since the features do not

allow for a spin coat, it was applied via a tertiary substrate. By spinning a layer of

SU-8 2 onto a glass slide and then pressing the 25 µm SU-8 bilayer onto this layer,

a sufficient amount of the polymer was transfered. Once the bilayer had SU-8 on it,

it was possible to align it to the silicon portion under a mask aligner and then press

it onto the substrate. Adhesion was completed via a UV exposure and subsequent

heating cycle.

In order to attach the DEP section of the device to the hydraulic section, a spacer

needed to be created to preserve the integrity of the metallic pillars. This was ac-

30

Figure 1.17: SEM image of fabricated microneedle

complished by creating a rectangular mold in SU-8 100. Once the 100µ high well

was created, it was used to cast PDMS (a thermally curable polymer). This gasket

layer was affixed to the DEP side via aquarium sealant. The sealant also possesses

the characteristic that it will permeate the PDMS membrane and enable adhesion to

the other side. This allows the conjunction of both devices to take place in one step,

while ensuring that the channels are not blocked with sealant. An illustration of the

position of the gasket is seen in figure 1.19.

1.4.2 Control Parameters

To operate such a device, the control parameters must be determined before operation.

These include the voltage applied to the device and flow rate of the microparticles

into the device. It has been shown in previous sections that an applied potential

can generate a trap for certain particles; however, there are certain limitations that

are imposed on the voltages that can be utilized (whether limited by particle size

or the fact that certain voltages can damage a cell). While a dielectrophoretic force

can be generated through a direct current[5], in order to minimize the energy that is

31

Figure 1.18: SU-8 Channels for applying suction to the fluid

transfered to the cell an alternating source must be used[15]. This lower amount of

energy ensures that the cell is not damaged by the applied potential. The imposed

transmembrane potential on a cell is given as[25]:

|V | = 1.5| ~E|R√1 + (ωτ)2

(1.27)

where V is the voltage across the membrane, R is the cell radius, ~E is the electric

field, τ is the time constant (a relationship between the cellular capacitance, conduc-

tance, and resistivity), and ω is the applied frequency of the field. This is obtained

by modeling the cell as a membrane covered solid sphere, using parameters for an

HL-60 (human promyelocytic leukemic cell line) cell. As can be seen, operating at

higher frequencies imposes a far lower voltage stress on the cell than at lower ones[25].

32

Figure 1.19: Full cross section of assembly. PDMS gasket is shown in green. Thetop side of shows the location of the DEP device with light blue being the glasssubstrate and gray being the metallic pillars and interconnects. The underside showsthe microneedles made out of silicon (shown in blue) and SU-8 bilayer (yellow).

Furthermore, it is imperative to estimate the maximum viable flow rate. This maxi-

mum occurs for two reasons. First, if the flow rate is too large, the force due to the

applied voltage may be insufficient to trap the cells (as described previously in the

simulation). Second, the shear stress on a cell due to the flow of the surrounding fluid

can also cause damage[25]. The shear stress may be described as[25]:

τSS = 4Ucη/h =4

h

(3Q

2A

)η (1.28)

Here, Uc is the centerline velocity, Q is the volume flow rate, A is the cross sectional

area of the cell, η is the viscosity, and h is the height of the chamber. Therefore, the

flow rate must be chosen such as not to damage the cell wall, but also to be within

the limits of the trapping mechanism formed by the applied voltages.

33

Figure 1.20: DEP section and microfluidic sections of the micro-cell transfectiondevice

1.4.3 Proof of Principle

Both sides of the device have been successfully fabricated and are shown next to

each other in figure 1.20. In order to test the dielectrophoretic trapping mecha-

nism, polystyrene beads were suspended in water and introduced into the active

region under an applied voltage of 30V. The relative permittivity of polystyrene is

2.5[15]. Diameters of the polystyrene spheres ranged between 10 and 50µm (approxi-

mately the same size as human cells). Polystyrene also has a similar density to water

(1.06g/cc)[15], and can be suspended easily. The fluid sample was pippetted directly

over the metal pillar array, requiring no entrance barrier to overcome. Furthermore,

the polystyrene beads are not effected by the shear stress limitations that cells are

subject to, as there is no delicate membrane to maintain. For all of these reasons,

polystyrene beads serve as an optimal test particle. As can be seen in figure 1.21, the

34

mechanism works well at trapping dielectric particles.

Figure 1.21: Trapped polystyrene particles under 30V. Red circles were added toemphasize the locations of trapped cells.

1.4.4 Discussion

Simulations have given an optimal input range for the introduction of fluid into the

device. Furthermore, fabrication and experimentation have shown that it is a viable

geometry to trap spherical particles. While the relative permittivity of polystyrene

and a human cell are orders of magnitude different (ε ≈ 120[26]), voltages can be

manipulated to obtain the same trapping force.

The proof of principle shows that the device is capable of localizing a microparti-

cle under the presupposed conditions. The device had not yet been tested on human

cells, as work with human samples requires access to a BSL-2 (biohazard safety level

35

2) lab, where all of the electronics and control equipment could be set up. Testing

on human cells requires special training for handling procedures of human biohazard

material. A human buffy coat (centrifuge layer of white blood cells) has been pur-

chased and is awaiting proper test set-up. Polystyrene and other synthetic particles

should, however, be sufficient to complete all levels of testing.

In addition to testing on human cells, the next process in the sequence is to success-

fully bond the electric control side with the silicon microneedle array. One possible

means of attaching the two sides is through the use of a gasket made out of poly-

dimethyl siloxane (PDMS). This PDMS gasket could be used to both contain the

cells, as well as serve as a separation layer to create space for the metallic pillars.

Perhaps the most challenging part of this part of the procedure will come in during

the alignment of the two sides of the device. As silicon side is opaque and metallic

traces prevent determination of the center of the metallic pillar array with respect to

the silicon microneedle and SU-8 bilayer through-hole. One advantage of the mecha-

nism of the device, however, is that the hold does not necessarily have to be directly

aligned under the center of the metallic pillar array. If it is possible to create signifi-

cant pressure to draw the particle down within a certain range, more leeway is given

to the alignment of the two sides.

Consideration must also be taken for all of the connections necessary to operate

device. One port must be made available to flow cells into the PDMS gasket, and

another to drain them from the control area. In addition, a vacuum must be attached

to the SU-8 bilayer to create the suction necessary to affix the cells onto the micronee-

dles. At this point, a mechanism (perhaps a nanoliter pump) can be attached to the

36

underside of the needles to introduce the foreign material.

Once the device is properly assembled, it can be tested by introducing a certain

tagging agent into the transfected cells. An example of such an agent is GFP (green

fluorescent protein), which emits light at a specific frequency when excited[27]. This

protein can be used to identify properly transfected cells and determine the overall

efficacy of the device.

37

Chapter 2

Control of Nanoparticles with

Magnetic Fields

The information presented in this section represents background research and tech-

nology pursuant to US provisional patent application number SPV1224001

2.1 Introduction

Using similar physical principles, the manipulation of nanoparticles using magnetic

fields will be explored in this chapter. While manipulation of nanoparticles via mag-

netic fields has been demonstrated in the past, a novel application is proposed, namely,

the control of iron oxide nanoparticles suspended in a carrier fluid (a ferrofluid), for

use in optical projection technology based on microfabricated devices.

2.1.1 Magnetism of Materials

Apart from the mathematical perspective of magnetism, it is important to know the

physical phenomena that are involved in magnetism. From a materials perspective,

38

magnetism takes on one of five distinct forms[28]:

1. Ferromagnetism

2. Antiferromagnetism

3. Ferrimagnetism

4. Paramagnetism

5. Diamagnetism

All forms are associated with the ordering of atomic magnetic moments that are

generated by unpaired electron spin and orbital motion. Perhaps the most recognized

form is that of ferromagnetism. This occurs in materials in which the atomic magnetic

moments order themselves in such a way as to produce a net magnetic field[29]. Due

to thermal fluctuations, this ordering, and hence magnetic field, is lost above what

is known as the Curie temperature[29]. In antiferromagnetic materials, the magnetic

moments align themselves such that there is no net magnetic field produced, yet

maintain a very specific ordering[29]. Like antiferromagnetism, magnetic moments in

ferrimagnetism align themselves to reduce the total magnetic field. The net field is

non-zero due to the fact that neighboring moments are not equal[29]. These three

examples of magnetism can be seen in figure 2.1. Materials which have a positive

susceptibility, χ > 0, are said to be paramagnetic, while those that respond negatively

are diamagnetic[29]. Paramagnetism can be found in conductors due to the internal

spin of electrons. Since half of the conduction electrons exist in the up-state and

the other half in the down-state, when an external magnetic field is applied, the

energy of the electrons aligned with the field is decreased, while the energy of the

electrons anti-parallel with the field increase in energy. Close to the Fermi energy,

39

these higher energy electrons can flip their spin, yielding a net magnetization[30].

The susceptibility of these metals is given as:

χp u µ0µ2Bg(EF ) (2.1)

Here, µB is the Bohr magneton and EF is the density of electron states at the Fermi

energy[30]. The temperature dependence of this response can also be noted as[30]:

g(EF ) =3N

2EF

(2.2)

χp u3

2χ

T

TF

(2.3)

Conductors also exhibit diamagnetism since the conduction electrons move in preces-

sion in the presence of an externally applied magnetic field. This is known as Larmor

precession and is described via the Bloch equation, where ~M is the magnetization, ~B

is the applied field, and γ is the gyromagnetic ratio[28]:

d ~M

dt= γ ~M × ~B (2.4)

This creates a dipole which is exactly in the opposite direction of the applied field[30].

It should be noted that the paramagnetic response is due to electron spin while the

diamagnetic response is due to the orbital interaction with the applied field[30].

2.1.2 Ferrofluids

A ferrofluid is a colloidal suspension of nanoparticles (on the order of 10nm) of iron

oxide (either Fe2O3 or Fe3O4) surrounded by a surfactant, which acts to prevent

particle agglomeration. The iron oxide particles themselves are ferro- (Fe3O4) and

40

Figure 2.1: Examples of atomic magnetic moment ordering in (a) Ferro- (b) Antiferro-and (c) Ferrimagnetism

ferrimagnetic (Fe2O3), and as such, each particle has a fixed magnetic dipole moment.

However, the particles are small enough that in the absence of a magnetic field their

orientation is governed by thermal fluctuations. Therefore, the fluid responds para-

magnetically as a whole[31]. Another class of magnetic fluids are magnetorheological

fluids, which contain larger magnetic particles that are too big to be suspended in the

carrier fluid by thermal fluctuations[32]. Since the magnetization of the nanoparticles

is much larger than that of the individual atoms, the magnetic susceptibility of fer-

rofluids is large, meaning a significant magnetic response can be observed in relatively

weak fields. Such nanoparticle based paramagnets are often called superparamagnetic.

The governing equations of these fluids can be used to predict their topography in a

given magnetic field. Starting from the equations of motion, it is possible to express

the pressure in a ferrofluidic system by using a modified Bernoulli equation. These

41

equations can also provide analytic expressions for the configuration of the fluid under

certain simple cases. These expressions will be used in later sections for comparison

with computational techniques. Looking at a steady state (∂v∂t

= 0) infinitesimal

amount of fluid oriented in a streamline that is at an angle α to the force of gravity,

the following Newtonian relationship holds (here, a is the cross sectional area, ds is

the infinitesimal length along the streamline, p is the pressure, v is velocity, M is the

magnetization, and H is the applied magnetic field)[32]:

ρ a ds vdv

ds= −a

dp

dsds− ρ a ds g sin α + µ0aM

dH

dsds. (2.5)

The left hand side is the mass times the acceleration of the infinitesimal volume. This

is equal to the sum of the forces on the volume, namely the forces due to pressure,

gravity, and magnetic fields. Rearranging these terms yields:

dp

ds+ ρv

dv

ds+ ρg

dh

ds− µ0M

dH

ds= 0, (2.6)

where, h is the height of the fluid. This is exactly the differential form of the Bernoulli

equation, which when integrated along two points of a stream line yields[32]:

1

ρ

∫ 2

1

dp +v2

2 − v21

2+ g(h2 − h1)− µ0

ρ

∫ 2

1

MdH = 0, (2.7)

or explicitly at the two distinct points as:

p1 +1

2ρv2

1 + ρgh1 − µ0

∫ H1

0

MdH = p2 +1

2ρv2

2 + ρgh2 − µ0

∫ H2

0

MdH. (2.8)

42

This can be written in a more expressive notation by:

p∗ +1

2ρv2 + ρgh− µ0MH = constant (2.9)

where M is the averaged magnetization (M = 1H

∫ H

0MdH). In switching from an

integration along a streamline to a pressure at a specific point, p∗ is introduced as the

combination of the thermodynamic pressure, p, the magnetostrictive pressure (ps):

ps = µ0

∫ H

0

ν

(∂M

∂ν

)

H,T

dH, (2.10)

where ν is an infinitesimal volume unit, and the fluid-magnetic pressure, pm:

pm = µ0

∫ H

0

MdH = µ0MH, (2.11)

at a specific point.

Here, in this ferrohydrodynamic Bernoulli equation (FHB), the elements of the orig-

inal Bernoulli equation can clearly be seen. The only modification is the addition

of the magnetostrictive pressure, which arises from the volume dependence of the

magnetization. The fluid-magnetic pressure is associated with the energy density of

the applied field in the fluid itself. When there are no viscous forces present, p∗ can

be defined by the surface of the fluid as:

p∗ = −pn + pc + p0 (2.12)

Here, p0 is the thermodynamic pressure, pn is the magnetic normal pressure at the

surface pn = 12µ0M

2n (which arises from the magnetization perpendicular to the sur-

43

face evaluated at the surface), and the capillary pressure pc = 2σH (which is purely

a function of the geometry of the surface, where H is the average of the principal

curvatures of the surface and σ is the surface tension).

Ferrofluids are also susceptible to instabilities that arise from the interplay between

fluid dynamics and magnetic response. Rosensweig instabilities[32] are one example,

where a flat horizontal surface of a ferrofluid becomes unstable when an applied mag-

netic field surpasses a critical value, as shown in figure 2.2.

Figure 2.2: Normal-field (Rosensweig) Instability[33]

These instabilities can be calculated analytically beginning with the governing equa-

tions of motion of small surface perturbations. The continuity equation for a fluid

demands that ~∇ · ~v = ~0. Coupled with the magnetic conditions ~∇ × ~H = ~0 and

~∇ · ~B = 0, the equations for the magnetic fluid become[32]:

ρ(∂~v

∂t+ ~v · ∇~v = −∇Π (2.13)

Π = p0 + ps + ρgz (2.14)

44

If there is a small perturbation to the surface, the velocity and pressure can be rewrit-

ten as: ~v = ~v0 + ~v1 and Π = Π0 +Π1. Assuming that for steady state approximations

~v0 = ~0:

ρ

(∂ ~v1

∂t+ ~v1 · ∇~v1

)= −∇Π0 −∇Π1 (2.15)

To first order in ~v1, this is reduced to[32]:

ρ∂ ~v1

∂t= −∇Π1 (2.16)

Taking the divergence of both sides yields:

∇2Π1 = 0 (2.17)

Π1 = AekzRe[ei(ωt−kxx−kyy)

], (2.18)

where ω and k are calculated based on initial conditions. From here, it is possible

to obtain the velocity in the z direction from equation number 2.16. It is also easy

to see that the surface perturbations can be calculated as a time integration of the

velocity. This yields the result that[32]:

z0 =Ak

ρω2Re

[ei(ωt−kxx−kyy)

](2.19)

As can be seen in this equation, the surface of the ferrofluid will form a sinusoidal

network of peaks and valleys. While the surface energy and gravitational potential

energy favor a flat surface, the introduced magnetic field will prefer an altered sur-

face. Once the field surpasses a critical value, the ferrofluid prefers a configuration

described by the peaks and valleys of the above equations.

45

This super-paramagnetic liquid has been used in a variety of applications. Ferroflu-

ids have been used as a dampening agent in stepper motors, controllable ink for ink

jet printers, inclinometers, galvanometers, and as a sealant[34]. It has been used as

a controllable means of thermal transport[35]. Recently, ferrofluids have also been

incorporated into microfabrication techniques. For example, the Rosensweig instabil-

ities have been used as molds for PDMS (polydimethyl siloxane)[36]. Another group

has used the fact that ferrofluids will reconfigure under an external field to selectively

block out ultraviolet light during photopolymerization[37].

2.1.3 Magnetofluidics

In order to design the proposed ferrofluidic device (which will be discussed in future

sections), it is necessary to determine the surface profile and maximum height of

a ferrofluid in the presence of an externally applied magnetic field. One case that

can be solved analytically is that where a constant and uniform field is applied to a

specific region of an infinitely flat region containing ferrofluid. The ferrohydrodynamic

Bernoulli is used to calculate the height and is described below:

P ∗ +1

2ρv2 + ρgh− µ0MH = P0, (2.20)

where the left side of the equation above is evaluated at a point on the surface inside

the region of applied field and the right side is from a point on the surface outside

of the applied field. P ∗ can also be determined based on the magnetostrictive and

46

capillary pressures at that point as:

P ∗ +µ0

2M2

n = P0 + 2Hσ (2.21)

For a given surface under an applied field, z = z(x, y) is the height of the surface and

H = 12∇· n is the average of the principal curvatures. Since the fluid is originally flat

and motionless, we take the average of principal curvatures, H, and the velocity, v,

to be zero. This condition also yields Mn = χH. We also take M = χH2

. Substituting

these into the previous equations yields:

P ∗ = P0 − µ0

2χ2H2 (2.22)

ρgh =µ0

2χH2(χ + 1) =

µ

2χH2 (2.23)

Solving for the height as a function of the applied magnetic field (B = χM),

h(B) =χB2

2ρgµ(2.24)

2.2 Simulation of Ferrofluids

2.2.1 Genetic Algorithms

Genetic algorithms are a relatively new, yet powerful search technique for problem

solving and optimization. Perhaps one of the leading authorities on genetic algo-

rithms, David Goldberg, states several reasons why genetic algorithms have an ad-

vantage over other search and optimization algorithms. Most explicitly are the facts

that calculus based methods are highly localized in scope and rely on the existence

of the gradient of a smooth function (which may not exist), while enumerative and

47

random search schemes lack efficiency[38]. This would lend itself naturally to mini-

mization of the total energy of the ferrofluid system.

Genetic algorithms are adaptive search schemes that are based on the mechanics

of natural selection. In nature, natural selection is the process of reproduction of se-

quences of genetic material that have been altered by mating and mutations[39]. This

genetic material gets passed on proportionally, based on the fitness of the individ-

ual (how well the individual operated in a specific environment). Genetic algorithms

mimic these relations by replacing the genetic sequences with sequences of characters

as a string[38]. These strings are, in essence, “artificial chromosomes”. They are then

subject to the same processes involved with natural selection. Fitness for the string

is the value associated with a user prescribed test function. The new generation of

strings is created proportionally based on the fitness of the parents. The strings are

then randomly mated and crossed at random points[38].

The very first genetic algorithms were not very complex. They did not contain incor-

porate mating and recombination, but simply used mutation. To explore a parameter

set for a function space, the artificial chromosome was subject to several mutations.

The test function was then evaluated with the new string. If the new string was more

fit, it became the parent and produced a new offspring. Otherwise, the child would

“die” and the parent would produce a new offspring. Again, no form of mating is

involved in the production of the “child”. This is known as the (1 + 1) Evolutionary

Strategy, as there is only one parent and one child at any given time[40]. As these

algorithms became more sophisticated, they gained ability to have multiple parents,

finite lifespans, multiple offspring, and finite ancestors. These evolution strategies

are designated as (µ, κ, λ, ρ), respectively[40]. In fact, another modeling technique,

48

known as simulated annealing, is a special case of this evolutionary strategy[41].

Genetic algorithms are perhaps more easily demonstrated than described. The fol-

lowing example is based on an example provided in [38]. Suppose the function

f(x) = 100 − (x − 7)2 is to be maximized on the domain [0,15], that is, f(x)max

would occur for x = 7. The variable ‘x’ is represented as a binary string of length

four (i.e. 0110, 1010, etc) to account for all possibilities of x. A fixed number of ran-

dom strings are selected for the initial population. For this example, the population

size is n = 4 and the initial members are shown in table 2.2.1 and the fraction fi

Σfito

represent the relative fitness of the individual with respect to the population[38]. The

intermediate generation is created by selecting a string from the initial population

at random, with probability of being chosen equal to the relative fitness. Selection

repeated until the number of strings in the intermediate population is equal to the

population size, n. The strings chosen for the intermediate step can be selected mul-

tiple times as represented by the variable “Count” in table 2.2.1. As can be seen in

Table 2.1: First Generation Genetic Algorithm for Test Function f(x) = 100−(x−7)2

String Number String Value fifi

ΣfCount

1 0110 6 99 0.324 22 0101 5 96 0.314 13 1111 15 36 0.118 04 0010 2 75 0.245 1

this first generation, the most relatively “fit” string is 0110, which means it has the

highest probability of reproducing. The least fit strings tend to “die off”, as they

will have a lower probability for selection to produce. In the mating process, two (or

more) strings of the intermediate step are randomly selected with probability pro-

portional to fitness. A random site, c, within the string is chosen for crossover, or

exchange of data. All information is swapped between the two mated strings after

49

the cth digit from the left of the string. For example, in the population shown in

table 2.2.1, strings 1 and 4 cross at point 2, meaning the first two digits of the child

string correspond to those of parent 1 string while the second two come from parent

2. Since only the n best fit children are carried forward, the fitness of the popu-

lation should rise. Multiple numerical modeling techniques have been employed to

Table 2.2: Population and Fitness After One Generation

Mating Pool Mate Crossover Site New Population Value Fitness0110 4 2 0110 6 990110 3 3 0111 7 1001111 2 3 1110 14 510010 1 2 0010 2 75

characterize ferrofluid structures under the influence of magnetic fields. Gollwitzer et

al have computed the topography of a ferrofluid using an iterative scheme involving

finite-element method and fixed point iterations to solve the coupled magnetostatic

and Young-Laplace problem [42], while Xuan et al have used the Lattice-Boltzmann

method to solve for the ferrofluid distribution [43]. Yet another group, Mohebi et al,

have shown that it is possible to recreate ferrofluid structure by performing a molecu-

lar dynamics simulation which includes dipole-dipole and drag forces on the particles

in a ferrofluid[14]. Other modeling techniques for ferrofluid structures involve Monte

Carlo simulations [44], and solving for specific parameters via analytically calculated

energy minimization [45, 46].

In more complex applied fields (such as those generated by loop currents), it be-

comes impossible to implement some of the aforementioned schemes. This is because

the topography in the z-direction can no longer be described as a simple function

(z = z(x, y)). Where this is not a limiting constraint to a particular numerical

scheme, for example with finite time step and Lattice-Boltzmann, a non-uniform field

50

introduces a much higher level of complexity and, hence, computational time. The

implementation of a genetic algorithm to solve the more complex configuration prob-

lem is not limited by these constraints or computational resources.

Genetic algorithms have been successfully implemented in other physics problems.

One example is the optimization of point-charges on a sphere by energy-minimization[47].

Here, the authors attempted to implement genetic algorithms to minimize the energy

of a distribution of point charges on a spherical surface by manipulating their con-

figuration. Not only were the authors able to reproduce all known results, they were

also able to extend the computation to much more complex configurations that had

previously been unsolvable by other techniques. In addition, a group has used genetic

algorithms to determine local configurations of up to 10 ferrofluid nanoparticles under

externally applied fields[48] (not yielding a general topography). This was a further

reassurance that the method of genetic algorithms for modeling ferrofluids may reveal

other interesting results including topographical characterization.

2.2.2 Model

It is possible to use an energy-variational technique to solve for the optimal surface

configuration of a ferrofluid in an externally applied magnetic field[32]. By minimizing

the total potential energy, Ut, with respect to z, where z is a function of position (x,

y) in the horizontal plane, it is possible to compute the topography of the ferrofluid

system.

Ut = Us + Ug + Um (2.25)

51

Us = σ∫ ∫ √

1 +(

∂z∂x

)2+

(∂z∂y

)2

dx dy (2.26)

Ug = 12

∫ ∫ρgz2 dx dy (2.27)

Um = −12µ0

∫ ∫ ∫MH0 dx dy dz + 1

2µ0

∫ ∫ ∫H2

0 dx dy dz (2.28)

In the above equations, Us is the surface energy, Ug is the gravitational energy, and

Um is the magnetic energy of the system. This energy minimization approach to the

problem naturally lends itself to a genetic algorithm as the “fitness” of the system

would be the total energy of the system. An offspring would be considered more fit

if the total energy of the system is lower than that of the parent.

The potential energy is a function of the configuration of the fluid. If space is dis-

cretized on a Cartesian grid, the configuration can be encoded as a binary array of

ferrofluid occupation states, i.e. cijk = 1 for a position (i, j, k) containing an finite

volume of ferrofluid and cijk = 0 for an empty position. Since it is assumed that there

is a fixed volume of ferrofluid in the problem, this leads to the additional constraint

that∑

cijk = constant.

For simplicity, the problem will be reduced to two dimensions here. The total energy

of the system is determined by the sum of the differential area units of energy, namely:

Utotal =∑

i,j (dUs + dUg + dUm)i,j (2.29)

dUs = 12σ∆

∑1a=−1

∑1b=−1((ci,j + ci+1,j+b) modulo 2) (2.30)

dUg = cijρgj(∆)3 (2.31)

dUm = −cij12µ0χH2

0 (∆)2 + 12µ0H

20 (∆)2 (2.32)

52

Above, ∆ is length of a single array element, σ is the surface tension, H0 is the applied

field, ρ is the density of the ferrofluid, and g is the acceleration due to gravity. In

equation 2.30 above, the 12

is used in the calculation of the surface area to eliminate

double counting (which is introduced by the particular summation algorithm imple-

mented). An example of a 4x4 array is shown in figure 2.3, where black pixels take

the value cij = 1, otherwise cij = 0.

Figure 2.3: 4x4 Occupation Array Example

The assumption is made that the “pixel” size is small enough that the magnetization

is uniform throughout the pixel. It is also assumed that the magnetic field applied is

low enough that it acts linearly, as M = χH. Since the purpose here is to demon-

strate that genetic algorithms can be used to solve for topographies, the magnetic

field is not recalculated after every iteration. Neglecting this yields a zeroth order

approximation, in which no Rosensweig instabilities will be recovered. While these

small deviations will not be seen, it will provide the correct average shape of the fluid.

To maintain the constraint condition of constant fluid volume, mutations can only

occur by swapping two pixels at a time. The easiest method of implementing this is

the (1 + 1) evolutionary strategy. At any given time, two configurations are com-

53

pared, the initial (parent) and a mutated version (the child). The fitness would then

be based on the total energy (lower energy = more fit). The more fit of the two

solutions survives to mutate again (becoming or remaining the parent), while the less

fit of the two is discarded. Furthermore, as the total energy is an arithmetic sum

of the differential energy of each pixel, it is then possible to implement a “guided

mutation”. That is to say that instead of randomly swapping pixels, it is possible to

swap the occupied pixel with the highest differential energy with that of the lowest

unoccupied (the energy of an unoccupied pixel is calculated as what the energy would

be if the pixel were occupied). This also leads to a natural terminal condition, where

Uchild > Uparent.

Before using the genetic algorithm to solve for unknown topographies, it is infor-

mative and beneficial to test it against an example where the solution is known. It

is possible to solve the aforementioned ferrohydrodynamic Bernoulli equation (FHB)

for the fluid topography for certain simple cases. One such case is when a constant,

uniform, magnetic field is applied normal to the plane of ferrofluid (as shown in figure

2.4). In this example, it becomes easy to determine an analytic solution to the to-

pography. Since the curvature is zero, the capillary pressure must also be zero. The

magnetic dipoles will align with the field, which is perpendicular to the surface, hence

pn =(

12µ0χ

2H2). Applying the equation at a point on the surface of the fluid both

inside and outside the region of magnetic field yields:

p∗1 + ρgh1 = p∗2 + ρgh2 − µ0MH (2.33)

54

Figure 2.4: Expected ferrofluid topography in a normal uniform magnetic field [32].Magnetic field is opposite gravity. Ferrofluid is shown in black and N and S representmagnetic poles. Height is indicated by the raised ferrofluid in the presence of theexternal magnetic field.

p* at either point is:

p∗1 = p0 (2.34)

p∗2 = p0 − pn = p0 − 1

2µ0χ

2H2. (2.35)

Substituting the boundary conditions into the FHB equation, the following result is

obtained for the change in height, as shown in previous sections:

∆h =µχH2

2ρg. (2.36)

All variables used in the program can be found in table 2.3. In order to test a

range of applied fields, the algorithm was implemented from 6667A/m to 28,333A/m,

in steps of 1667A/m. The field was applied uniformly in the middle third of the sim-

ulation box (i.e. all points on the grid, ci,j between i = 33 and i = 66 had the same

55

value of magnetic field). It should be noted that since the field was not recalculated

after every generation, edge effects will not be seen. These effects can be excluded

from height calculations, as the height is compared from the center of the grid with

the height at the edge.

Table 2.3: Variable List (all physical data from [42])

ρ χ ∆ Array Size Loop Radius σ

1236 kgm3 1.172 6e-3m 100x100 0.15m 30.5mN/m

2.2.3 Results

As shown in figure 2.5, the genetic algorithm performs well against the analytic solu-

tion in the uniform field case. Numerical values are given in table 2.4. The number

of generations required for termination was approximately 1600 for all fields, indi-

cating that it is not a function of applied field. Since the topography is encoded

as occupation sites of the array, the genetic algorithm has the ability to depict the

“phenotype” of the solution (i.e. the physical form of the ferrofluid). This is shown

in the case of the of the constant, uniform field in figure 2.6. It should be noted that

as the field is increased, there is a higher deviation from the analytic value. This

was explored by decreasing ∆ and, therefore, increasing the size of the array in an

attempt to have a more precise representation. This yielded a negligible change in

the height as computed via the algorithm. Further investigation shows that perhaps

this breakdown is due to the fact that the critical magnetic field is approximately

10kA/m. As can be seen, the simulation begins to deviate from the analytic solution

between 10-20kA/m.

56

0.5 1 1.5 2 2.5 3 3.5

x 104

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Magnetic Field (A/m)

Hei

ght (

m)

Figure 2.5: Genetic algorithm solution to ferrofluid height in a uniform magneticfield. Blue line indicates analytic solution based on the ferrohydrodynamic Bernoulliequation. Red line indicates genetic algorithm solution. Error bars indicate plus andminus one unit of division of the discretized occupation space as the division lengthis what limits the precision of the solution as determined by the genetic algorithm.

2.2.4 Field due to a current loop

As demonstrated in the above table and figures, the genetic algorithm that has been

implemented is able to reproduce results for known cases of uniform applied fields.

However, the field produced by a current in a loop of wire is non-uniform, as shown

in equation 2.37 below.

Bz = B01

π√

Q

[E(k)

1− α2 − β2

Q− 4α+ K(k)

](2.37)

Br = B0γ

π√

Q

[E(k)

1 + α2 + β2

Q− 4α−K(k)

](2.38)

57

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

10

20

30

40

50

60

70

80

90

100

Figure 2.6: Occupied (black) states and vacant (white) states for the “artificial chro-mosome” with 10% of spaces occupied in a uniform field of 20kA/m at a) initial stateb) 200 generations c) final state. Solution took 913 generations to terminate.

Here, ‘a’ is the radius of the current loop, ‘r’ is the distance from the center, B0 = I2a

,

γ = xr, α = r

a, β = x

a, Q = [(1 + α)2 + β2], and k =

√4αQ

. E(k) and K(k) refer to

E-type and K-type elliptic integrals. Implementing this field in the genetic algorithm

yields the ferrofluid structure shown in figure 2.7.

To have a reference for comparison, our group has attempted to experimentally map

the topography of the ferrofluid by cooling gelatin around the fluid in order to create

a mold of the configuration, which could later be measured. In an initial attempt, the

ferrofluid was frozen in liquid nitrogen, at which point gelatin was added to create

the mold. This form of freezing was even able to preserve all Rosensweig instabilities

created by an externally applied magnetic field. To form the mold, gelatin was poured

58

Table 2.4: Constant, Uniform Magnetic Field Ferrofluid Height

Generations H(A/m× 104) Comp. Height Analytic Height

1588 0.6667 0 0.0058651599 0.8333 0.012 0.0091641623 1.0000 0.012 0.0131961568 1.1667 0.012 0.0179611562 1.3333 0.024 0.0234591613 1.5000 0.024 0.029691630 1.6667 0.036 0.0366541594 1.8333 0.036 0.0443521617 2.0000 0.048 0.0527821606 2.1667 0.048 0.0619461570 2.3333 0.06 0.0718431597 2.5000 0.072 0.0824721598 2.6667 0.084 0.0938351589 2.8333 0.096 0.105931605 3.0000 0.096 0.118761585 3.1667 0.108 0.13232

into the container with the frozen ferrofluid. Unfortunately, the temperature of the

liquid gelatin instantly melted the frozen ferrofluid structure. In an attempt to cir-

cumvent this problem, the ferrofluid was placed in the freezer, along with a solenoid,

current source, and gelatin in hopes that the gelatin would set around the ferrofluid

structure. Even when the field was applied in a freezer, the current used to create

the instabilities was great enough to heat the ferrofluid / gelatin mixture enough to

prevent setting. Mapping of the topography was also attempted by passing an acous-

tic source and sensor over the ferrofluid surface and measuring the delay between the

signal output and acoustic detection to calculate the height. This method was also

abandoned, since it was unable to produce the resolution necessary to detect any

surface deformations, leaving purely qualitative forms of the surface for comparison.

While there is no empirical data to affirm the precise characteristics of the fluid, it

does reproduce the general structure seen in simple laboratory experiments, as shown

59

in figure 2.8. As can be seen from the genetic algorithm solutions, the fluid will shape

10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

10 20 30 40 50 60 70 80 90 100

5

10

15

20

25

30

Figure 2.7: Ferrofluid in magnetic field produced by loop current of 6000A with a)2%and b) 5% of the array filled. Image c) corresponds to 5% volume fill with 60000A.Axes correspond to occupation array indices, which are the discretized Cartesiancoordinates of the sample

itself according to the applied magnetic field and available amount of fluid in the sys-

tem. These topographies are easily produced using a solenoid with a ferrite core and

a power supply. Varying the current and fluid amounts in the laboratory setting has

resulted in the same structures as those generated in the simulations (for low enough

magnetic fields such that no instabilities are present). These solutions make sense. As

the field strength is increased for a specific volume of ferrofluid, the fluid moves from

a flat configuration to be drawn to the regions of highest field. As more fluid is added

60

to a region of constant field strength, the additional fluid must find the position of

minimum energy, naturally filling in as close to the source as possible. This occurs

until the field strength of the source is negligible compared to gravitational potential

energy and surface energy, at which point the configuration is once again flat. This

is shown in the configurations in figure 2.7.

Figure 2.8: Ferrofluid response to a solenoid. Ferrofluid was placed on top of aninverted plastic petri dish to hold it above the solenoid. Current was varied inthe solenoid to produce the image above using approximately 5mL of ferrofluid and300mA on a solenoid of unknown induction.

2.2.5 Comparison

All simulations were run on a Toshiba laptop with a Centrino Duo 1.7GHz proces-

sor with 2Gb of RAM. Calculating the ferrofluid configuration in a 100x100 array

took only 4.375 seconds for each of 16 different constant and uniform applied fields

(with an approximate 1600 generations per solution). This works out to be less than

61

3ms of computational time per generation. The number of generations for this case

is comparable to the number of time steps to obtain solutions generated in the 2D

Lattice-Boltzmann Equation for a similar sized array[43]. Using genetic algorithms,

however, the complexity involved to perform each step is drastically reduced (as a sin-

gle swap of “pixels” replaces updating forces and densities for every array point for a

single iteration). Similarly, the molecular dynamics simulation[14] takes 2ms at time

steps on the order of 10−7s, thereby taking several thousands to tens of thousands of

iterations to generate solutions (simulation had comparable number of data points,

478[14]). These steps also include complex calculations and storage of multiple data

entities. In the case of the more complex field (with 5% volume fill), the solution took

62.5ms.

Use of the genetic algorithm has proven to be an effective method for determining

the topography of a ferrofluid system. The computational and analytic agreement for

the known case of a constant and uniform magnetic field has been shown to be high,

both in numerical value and overall form. It is due to this that the algorithm was

applied with confidence to the more complex field. This yielded a topography which

agrees with observable experiments. It has also been shown that when compared to

other simulation methods, it requires less computational and numerical steps and can

be used to generate quick solutions of ferrofluid topographies.

62

2.3 Ferrofluid Projection and Lithography

2.3.1 Function

Projection of images onto screens has been a technology that has been around since

the turn of the 20th century with Thomas Edison [49]. With the advent of microtech-

nology, Texas Instruments introduced their DLP chip[50]. This chip contains over

two-million mirrors, each on the micron scale[50]. A beam of light is incident on the

DLP chip, which reflects the light and passes it through optics to enlarge the beam to

the desired size. By electrostatically activating the individual mirrors, pixel control is

established. A deflected mirror will divert the incident light from reaching the screen.

In order to include color, the technology may be used in either a 1-chip or 3-chip con-

figuration. In the one chip configuration, a color wheel spins such that the image is a

composite of all three colors (red, blue, and green). In the three chip configuration,

individual DLP chips are used for each color[50]. This is shown in figure 2.9.

Recently, ferrofluids have been used as reflective surfaces[51]. It should then be pos-

sible to use the idea of a ferrofluid reflective surface to create an array of micron

scale (30-50 micron) inductors to locally deform a ferrofluid for light manipulation.

Reflecting light off of these local ferrofluid surface perturbations leads to a more ro-

bust projection technique than DLP (requiring less lithographic steps and no moving

parts) as each array index can be addressed as a pixel of an image. If, instead of

white light, ultraviolet light is used, the device could be implemented in a complete

photolithographic system by loading the image data onto the inductor array. Fur-

thermore, the costs involved in photolithography can be very high. Masks can cost

several hundred to several thousand dollars and take several days to several weeks

to prepare[52]. Using the method of a programmable ferrofluidic display and pho-

63

Figure 2.9: Close-up of DLP chip and 1-chip and 3-chip implementations. DMDindicates the position of the Digital Micromirror Device[50]

tolithographic system would remove the majority of cost and time from the process.

In addition, changes that need to be made to a mask could be applied instantaneously.

The drawback, at the moment, is the achievable resolution and response time (which

will be discussed later).

The proposed device is based on the following principle of operation. When a col-

limated light source is incident on a plane surface, all of the incoming light that is

reflected will propagate in the same direction. Change in the direction of reflected

64

light can be accomplished via manipulation of the surface. In order to control the

state of the surface, a ferrofluid will be placed above an array of single loop induc-

tors. These inductors will be able to create a local magnetic field, which will, in turn,

create a local disturbance of the surface, as shown in the cross-section of figure 2.13.

A significant portion of the light incident at these locations will not reach the final

projection surface. By increasing the field to an appropriate amount (via the current

supplied), it is also possible to create Rosensweig instabilities [32], which should min-

imize the amount of light reaching the final surface.

Patterns can be created on the ferrofluid surface via a computer input to the ac-

tive region of the device. Using a UV light source, the final destination of the light

can be a UV-curable polymer substrate. This would lend itself to the application of

the device to photolithographic masking. If white light is used in conjunction with

correct optics for magnification, the device could be used as a projection system.

Prior to further discussion of the device, some issues related to feasibility should

be considered. In both projection technologies, a significant amount of light must

be incident on the final surface (typical photopolymerization for lithography requires

10 − 20mW/cm2[53]). Since the reflectivity of a ferrofluid is roughly 4% between

350nm to 750nm [54], the light source must be nearly twenty times stronger, or

exposures must last twenty times longer. The reflectivity can be increased by the

incorporation of metal like liquid films (MELLFs), which have shown to improve re-

flectivity of ferrofluids up to 50%[55]. Another consideration is the response time for

the ferrofluidic structures when the field is changed. The relaxation of the fluid is

65

governed by the Brownian relaxation time, as shown[55]:

τ =V η

kT(2.39)

where η is the viscosity of the carrier, V is the volume of the particle, T is the tem-

perature and k is the Boltzmann constant. Using the properties described in [56], the

relaxation time is determined to be approximately 6 microseconds. Most television

and computer monitors operate in the 60Hz regime, or on the order of 10−2s, being

much slower than the response time of the ferrofluid structures.

For a simple proof of concept, the idea was explored via simple ray-tracing. A pro-

gram was written to investigate the effect of the surface disturbances on incoming

collimated light rays. As a test case, an array of circular “bumps” was created to

simulate the height of the fluid under an applied magnetic field:

z(x, y) = a√

((x− xc)2 + (y − yc)2), (2.40)

where, ‘a’ is the desired radius of the bump and (xc, yc) is the center of each desired

bump (in arbitrary units). In order to perform ray tracing of incoming light, it is

necessary to calculate the normal vector to the surface, which can be done as follows

[57, 58]:

~n = ∇ (z − z(x, y)) =< − ∂z∂x

,−∂z∂y

, 1 > (2.41)

∂z

∂x= −1

2[z(x + 1, y)− z(x− 1, y)] (2.42)

∂z

∂y= −1

2[z(x, y + 1)− z(x, y − 1)] (2.43)

n = ~n|~n| (2.44)

66

While this can be calculated analytically, it is much simpler to do this numerically

for an array of such distortions. For the purposes of demonstration, the light rays

were taken to have the incoming vector form of:

~Lin =< 1, 0,−1 > (2.45)

The incoming ray is then broken down into components perpendicular and parallel

to n:

~Lin−parallel =(~Lin · n

)n (2.46)

~Lin−perpendicular = ~Lin −(~Lin · n

)n (2.47)

Upon reflection, the component of the wave vector along the surface normal is re-

versed, while the component perpendicular to the normal is conserved:

~Lout = ~Lin − 2(~Lin · n)n (2.48)

Propagating these rays to a final surface leads to the image shown in figure 2.12,

where the blue circles represent the positions on the final surface of the incident light.

As can be seen in figure 2.12, virtually all of the light that is incident on the de-

formed ferrofluid does not reach the final surface, as indicated by the absence of blue

circles. This is because it is reflected to other directions. The light that does reach

the final surface is minimal. The spherical shape of the whole should be reproduced

for all incident angles of light, as long as viewed from the perspective of a plane whose

normal is parallel to that of the direction of propagation. This simple test encouraged

further research of the proposed device.

67

Figure 2.10: Light ray reflection off of a ferrofluid deformation

Before fabrication and testing of the proof-of-principle in a micron scale device, a

macro scale proof of concept model was created. This model used a 3x3 array of

RC-1 inductors purchased through Triad Magnetics, which have an inductance of

5.6mH each and can withstand a current of up to 0.250A. They were controlled via

a standard parallel port on a PC. Since the array was only a 3x3, no software de-

coding was necessary. The parallel port is unable to supply an adequate amount of

current that is required for powering the array, so the output from the computer was

sent directly to the base of a TIP31C transistor (an npn transistor with a maximum

collector-emitter voltage of 100V and collector current of 3A). The inductor was con-

nected to the emitter and a power supply capable of 2A at 32V was connected to the

collector with no resistor. The program (attached as appendix E) was used to control

the fluid by controlling the on/off state of the transistor to supply the needed current

for the “pixel” in the array in such a way as to create the desired response. Since no

collimated light source or lenses were used for proof of concept, only crude images

were formed upon reflection off the substrate. In order to have a more definitive test

of the process of control, a 10x10 matrix is desired.

68

0 5 10 15 20 25 30 35 40 45 50050

0

5

10

15

20

25

30

35

40

45

50

0

10

20

30

40

50

0

10

20

30

40

500

10

20

30

40

50

Figure 2.11: Side and angle view of a single hemisphere for ray tracing. This imagecan be repeated to investigate more complex reflections.

The creation of highly localized fields requires the highest resolution photolithogra-

phy possible. This happens to be the 2µm Manhattan geometry that is offered by UC

Berkeley. Geometric and mask fabrication techniques limit the minimum diameter

of the square inductor loops to be 40µm. As shown in figure 2.14, 2 columns of 5

loops, forms a unit cell, which can be placed 40µm apart from each other (minimum

resolution of the feature is 2µm, so 10 loops with electrical trace connections of 2µm

in width and 2µm of space between each trace would require 2µm× 2× 10 = 40µm

spacing in between nearest inductors). The diameter was chosen to be the same as

the spacing in between columns, hence 40µm. Unfortunately, this mask was unable

to be processed, as it was too complex for the fabrication machines at UC Berkeley..

To carry on with the production of a prototype device, a new mask was created, using

a 30µm resolution. With this technology, a 4x4 array was created, as shown in figure

2.15.

Using the ferrohydrodynamic Bernoulli equations and Ampere’s law, it is possible

69

105

110

115

120

1250 5 10 15 20 25 30 35 40 45 50 105

110115

120125

0

10

20

30

40

50

85

90

95

100

105

Figure 2.12: End points of propagated light rays off of spherical disturbance calculatedvia ray tracing program shown from a perspective (a) parallel to reflected light and (b)arbitrary to the reflected light to emphasize scattering due to a single hemisphericaldisturbance. End points are shown where they intersect a plane whose normal isparallel to that of the propagation of the light. Axis are in arbitrary units.

to determine a crude approximation for the current needed to create local distur-

bances, of height ∆h. For the values for ρ, g, µ, and χ used in chapter 3,

∆h =µχH2

2ρgu 13.59B2. (2.49)

Using the value of the field at the center of a loop carrying current, i, we estimate

the height of the disturbance, in meters, to be:

∆h(i, z = 0) u 0.0238i2 (2.50)

To create a deflection on the order of the loop size (30µ) would require i = 35.5mA,

as the height would be equal to the diameter of the loop.

The ferrofluid that was used is the EFH1 by FerroTec[56]. Since this ferrofluid is

oil based, the fluid can be placed directly onto the electrodes without worrying about

70

Figure 2.13: Cross-section of activated device. Small deformations in the ferrofluid(black layer) are caused by applying current to local positions on the array corre-sponding to pixels of an image

short circuiting the electrical traces. Aluminum was used to make the conducting

features, as it is cheap, readily available, and is a good conductor (resistivity of

2.82x10−8Ωm [59]) for the purposes of this device.

Application of 43mA (the measured current output of the flip-flop) of current yielded

the ferrofluid configuration shown in figure 2.16. Surprisingly, instead of flowing to

the region of highest magnetic field, the ferrofluid reacted in such a was as to move

away from the field source (namely the wires of the array). The physics of this phe-

nomenon will be discussed in section 2.5. While this behavior was unfavorable for the

desired function of the device, it inspired another idea, which will also be described

in section 2.5. In order to avoid this problem in the current device, a 100µm well

was created with the use of SU-8 100. This well confined the ferrofluid to the region

above the array of the current loops, preventing the noted splitting down the applied

current lines. A cross section of the proposed device with SU-8 well is shown in figure

2.13.

71

Figure 2.14: Center of original 10x10 MEMS Device for Ferrofluid Control

2.3.2 Fabrication

The final fabrication for the device was carried out as follows (and illustrated in figure

2.17:

1. Clean standard microscope slide with acetone, isopropyl alcohol, and de-ionized

water then nitrogen dry

2. Thermally evaporate 1500A of aluminum onto slides

3. Spin Shipley 1813 45s@4000rpm

4. Pre-Expose Bake 90oC for 1hr

5. Expose for 10s@16mW/cm2

6. Develop in CD-30 for 2min

7. Place in aluminum etch for 3-5min (8H3PO4 : 1HNO3 : 1H2O)

8. Remove Shipley with acetone, isopropyl alcohol rinse, de-ionized water, N2 dry

72

Figure 2.15: 4x4 Mask for ferrofluid control. The image on the left depicts a fullview of the device, which shows all electrical contacts. Leftmost central square is thegrounding contact and all others follow traces to the individual loops. The image onthe right is a close-up of the center of the device, partial squares are the loops formagnetic field generation. The four electrical traces that lead up and down from theimage should provide the reader with the geometry for the minimum spacing betweenloops.

9. Pre-spin SU-8 100 for 10s@500rpm followed by Spin for 30s@3000rpm

10. Pre-Expose Bake on Hot Plate 65oC for 15min → 95oC for 30min → Room

Temperature (Ramp at 600oC/hr)

11. Expose 20s@16.4mWcm2

12. Post-expose bake on Hot Plate 65oC for 1min → 95oC for 30min → Room

Temperature (Ramp at 600oC/hr)

13. Develop in SU-8 Developer for 15min

2.3.3 Array Control

Once the device is fabricated, the next step is to implement a control scheme for

display. A variety of methods for display control exist and will be discussed in this

73

Figure 2.16: Ferrofluid splitting with applied current prior to development of SU-8100 well to localize the fluid. Black regions are ferrofluid, which are located abovethe conducting wires of the device. The image on the left shows the response whencurrent was applied from the a lead from the top of the device to the ground (farright). The image on the right was taken slightly after a current was applied from theleft after leads at the top and bottom were already supplied with a current source.

section. A relatively new and innovative way to control large arrays of LEDs (and

subsequently other I/O) is known as Charlieplexing. Charlieplexing is a method that

is used to control multiple switches with minimal connections. This method relies on

the use of quickly sending on and off signals to specific LED’s based on voltage drops.

By switching these LEDs on and off quickly enough, it gives the illusion that they are

constantly on[60]. While charlieplexing may be suitable for application in projection

and television systems, this method cannot be implemented for photolithography,

as a continuous stream of UV light is required for photoinitiation[61]. In order to

circumvent this requirement, a technique using flip-flops was used. This schematic

was designed to control the 10x10 ferrofluidic display, however it will function on any

nXm array, where n and m are positive integers less than 11. The method can be

expanded to include larger arrays, but more complex circuitry design is required.

74

Figure 2.17: Process flow for ferrofluid projection device. (1) shows aluminum (gray)thermally deposited on a glass substrate (light blue). (2) depicts the spinning andpatterning of Shipley (a positive resist). (3) The aluminum is selectively etchedand Shipley removed in acetone. Once the electrical traces have been established inaluminum, the SU-8 can be spun and patterned, as indicated in (4).

Figure 2.18: Final fabricated version of the ferrofluid control device. The image onthe left shows a close-up of the inductor loops. The image on the right shows theinductor loops as well as the interior of the SU-8 well.

75

The control circuit involves multiple octal D-type flip/flop devices, as well as sev-

eral three-to-eight converters. The job of the flip-flop is to store a value, in this case

a high or low signal, until reset[62]. For this particular circuit, the MM74HCT octal

D-type flip-flops were used from Fairchild Semiconductors. These chips were used

because they have a DC output current value of +−35mA[63], which is expected to

create ferrofluid distortions of the desired height. Measured output values from the

pins registered 43mA, however. This eliminates the need to introduce transistors to

the circuitry. In the event that higher currents are needed, 100 KSC2710YBU transis-

tors may be implemented, just as in the larger prototype. These transistors are NPN

type available through Fairchild Semiconductor and can withstand a throughput of

500mA, greatly increasing the expected deformations[64]. The 3-to-8 converters act

as decoders for the eight distinct signal inputs that are gathered from the parallel

port from the computer. While this is fairly complex to be represented in a com-

plete circuit diagram, the repetition of components allows for a simplification to be

depicted (and described) in figure 2.19.

Once the hardware has been constructed, the only thing that remains is the soft-

ware decoding of data. A program has been written in C (see appendix E) which can

decode a bitmap image and output it to the correct pins. In order to demonstrate

the process, LEDs were placed where the active input to the device is. Pictures of

the control circuitry and demonstrations can be seen in figures 2.20 and 2.21.

2.4 Ferrofluid Lithography

The function of the device was tested under two different conditions. In both tests,

the system was set up as shown in the ray trace of figure 2.22. As can be seen, a

76

Figure 2.19: Simplified circuit diagram of the hardware controller. Numbers 1 through14 indicate the corresponding pin-outs of a parallel port. Numbers I0 through I7indicate the outputs 1Q through 8Q of the primary flip-flop. An H indicates a highsignal (5V) and an L a low signal (0V). Each output of the top 3-8 converter (Y0through Y7) should be connected to the clock of a separate secondary flip-flop andis indicated by M. Each output of the bottom 3-8 converter (Y0 through Y7) shouldbe connected to the clock of a separate secondary flip-flop and is indicated by N.All outputs of the outputs (1Q through 8Q) of the secondary flip-flops should beconnected drectly to the ferrofluid device electrodes for a total of 2 3-8 converters x8 flip-flops controlled by each 3-8 converter x 8 outputs per flip-flop = 128 possibleconnections.

77

Figure 2.20: Screenshot of Control Software

light source was reflected off of a plane mirror onto the substrate. The reflected light

was altered via a second mirror, such that the output was parallel to initial incoming

rays. This was recorded by a CCD camera. As a preliminary test, white light from a

fiber optic source was used. A current source was then used to apply 30mA to each of

the eight electrodes. Initial reflection and deviated reflection is shown in figure 2.23.

As can clearly be seen, the application of current to the system generated enough

magnetic field in order to deform the ferrofluid in a local region enough to cause scat-

tering of the incoming light. The deformation was noted to be in the correct location.

In order to perform a more precise experiment, the white fiber optic light source

78

Figure 2.21: Actuated LED from bitmap image in figure 2.20

was replaced with a helium-neon laser source. This system was modeled in OSLO, a

freely available ray tracing software program. OSLO was used since it is capable of

incorporating a much broader range of optical geometries and performing more com-

plex ray tracing calculations in comparison with the simple program written to test

the proof of concept. A reflective spherical surface was placed where the ferrofluid is

located in figure 2.22. The radius of curvature was calculated using equations derived

in previous sections and assuming that the deformation was spherical and that the

diameter corresponds to the relative size of the distortion to beam size (1/10). The

spherical bump was placed off-center to model the position of the applied current and

79

Figure 2.22: Schematic of experimental set up for ferrofluid lithography test. Herethe light source is highly collimated (having a field angle of 0.0000572957795o)

local deformation. The resultant wavefront map is shown in figure 2.24. All angles

and distances used in the optical ray trace software were recreated in the laboratory

environment. Shown in figure 2.25 is the displacement of the incoming wave caused

by a 30mA current. This shows high congruence with the model given by OSLO.

2.5 An Unintended Consequence [Variable Chan-

nel for Microfluidics]

As noted in section 2.3.1, before the SU-8 well was constructed on the ferrofluidic

device, the ferrofluid actually split away from the activated wire. This phenomena

80

Figure 2.23: White light reflected at an angle off of ferrofluid surface before and afterapplication of 30mA to a single electrode. Camera captured images directly abovethe ferrofluid surface

Figure 2.24: The free ray-tracing software OSLO was used to track the wavefront ofa Gaussian beam as shown in figure 2.22 with a beam radius of 1 and wavelength of0.632, in relative units. The image on the left is the wavefront generated with a flatreflective surface. The image on the right is the wavefront calculated from a sphericaldeformation in the “ferrofluid” surface. Relative distances were input to match thosein the experimental setup. Compare with experimental images of laser spot in figure2.25

81

Figure 2.25: Laser spot captured on CCD at position indicated by camera in figure2.22. Image on the left is the laser spot while no current was applied, while the imageon the right is that of the laser spot after application of 30mA to single electrode. Ascan be seen, the shift in center follows that as calculated from the ray tracing simu-lation: figure 2.24. Intensity looks similar, as the CCD camera normalized intensityof all images

led to the idea and development of a variable width channel for microfluidic applica-

tions. The basis of the idea is that if ferrofluid is constricted to a channel above a

current carrying wire, the fluid will separate along the wire in response to the amount

of current provided. This split is perpendicular to the direction of the flow of current,

enabling the control of a variable width channel. A representative schematic of the

device is shown in figure 2.26. The device was fabricated via the procedure located

in appendix F.

Ferrofluid was introduced into the channel by placing a small drop at one of the ends.

A solenoid was placed below the device, such that the center was aligned with the

center of the microscope slide. When the solenoid was activated, by running a current

through it, the ferrofluid was drawn into the channel to the middle of the slide. Once

the fluid was in place, the current to the solenoid could be switched off.

82

Figure 2.26: Top view and side view of ferrofluid in-line particle filter / variablechannel

At this point, current was sent through the electrodes of the device itself, causing

the fluid to part. The current was slowly ramped up to 100mA, at which point the

fluid began to separate along the wire. The ferrofluid itself was slow to respond, tak-

ing upwards of 45 seconds to part. As the current was increased, the fluid completely

shifted to one side of the channel. This is believed to be due to the fact that the

wire is not exactly aligned in the center of the channel, creating a state in which it is

favorable for the fluid to move to one side. As the current was increased to 150mA,

the ferrofluid moved further to one side. Past 150mA the fluid showed little to no

movement under the camera. After cutting off the current supplied, the fluid began

to return slowly to its original state. This could be expedited by sending current

through the solenoid again.

What is most interesting about this device is that it is based on a ferrofluid response

that is counter to intuition. As discussed in previous sections, we assumed that as

the current through the wire was increased, the fluid would move toward the wire

83

Figure 2.27: a) Empty channel with wire underneath and b) ferrofluid filled channelafter 150mA was applied

itself, where the magnetic field is largest, thereby reducing the total energy of the

system. What happened was the exact opposite (ferrofluid migrated away from the

region of higher magnetic field). Speculation to the cause of this phenomena yielded

no satisfactory explanation as yet. When the situation was modeled with the genetic

algorithm, the fluid aggregated to be closest to the wire (not in accordance with phys-

ically observed results). This would indicate some other form of physical interaction

taking place. One hypothesis was that the dipole-dipole interactions between fluid

on opposite sides of the current source, which have opposite magnetization, created

an unfavorable energy configuration. This had to be rejected as the fluid migrated

away from the current source regardless of whether there was fluid on both sides of

the current source or not. Furthermore, it was postulated that heat from the wire

played a larger role in the dynamics of the fluid than the magnetic field. This theory

was discarded as well, as heat was shown to evaporate all of the carrier fluid, leaving

a black film, as opposed to causing migration of the fluid away from the heat source.

In addition, increases in current repelled the fluid further from the current source,

which would be typical of a diamagnetic response. Further investigation of the cause

84

of this phenomenon is needed.

85

Chapter 3

Conclusion

A MEMS device for cell transfection and other applications has been proposed and

investigated. The device uses dielectrophoretic forces to trap particles on a micronee-

dle array. The dielectrophoretic trapping component device was successfully tested by

trapping polystyrene beads. This component was also modeled via finite element and

finite difference techniques to visualize the motion of the trapped particles. The SU-8

bilayer that enables the suction mechanism for holding the cells (or microparticles)

has also been fabricated, tested, and bonded to the silicon microneedles. These two

components create the hydraulic components of the cell transfection device. When

combined with the dielectrophoretic side of the chip, a complete method for the au-

tomated cell transfection is established.

While this device was originally conceived for application such as measuring pro-

tein expression, more generally, the method allows more generally for the insertion

of foreign material into a dielectric particle, so application could extend beyond cell

transfection. Future work would entail the use of this device in successful cellular

transfection and other schemes (such as introduction of MRI contrast agents or par-

86

ticle tagging) and show that it works where other methods fail.

It has been shown that it is possible to create a device on the 100-micron scale which

is capable of altering the path of visible light through the use of local deformations in

a ferrofluid under the influence of an externally applied magnetic field. This enables

the device to function as a means to create a projected image onto a screen as well as

an instantaneous mask creator for photolithography. Furthermore, in developing this

device, a novel means for calculating the ferrofluid height using genetic algorithms

was introduced. These algorithms proved to be extremely useful, quick, and efficient

in solving the topographies in complex fields, in some instances outperforming other

algorithms.

Future directions for this work would include the use of MELLFs[55] on the ferrofluid

in order to enhance the reflection at specific wavelengths, as well as the possibility

of creating a mask that has a higher resolution. In addition, exploration of alternate

geometries for the inductors should be investigated.

As an additional, and unexpected, result of the work, a ferrofluid based filter was

developed. While the basic function of the device has been demonstrated, several

avenues are still open for exploration. Foremost is the question of why the ferrofluid

moves away from, instead of toward, the source of the current. The only reasonable

explanation is that the ferrofluid is exhibiting some sort of diamagnetic response.

Further experiments and testing are necessary to ascertain the exact nature of this

phenomena. In addition, operational parameters of the device itself must be quan-

tified, such as flow rates that can be used while maintaining the integrity of the

ferrofluid walls, as well as the particle sizes that can be filtered. Additional optimiza-

87

tion of the device may lead to creating precise fluid flow rates using the Bernoulli

equations to determine precise widths and areas for microfluidic applications.

88

Appendix A

Needle and SU-8 Component

Fabrication

1. DRIE etch holes in wafers 1.25µm in diameter and 40µm deep (done by Dr.

Yeubin Ning of Micralyne, Inc., in Canada)

2. Grow 500nm of oxide at 1100oC

3. Pattern back side of wafer with Shipley 1813 for creating holes

4. Spin coat front of wafer with Shipley 1813 for protection

5. TMAH etch back side of wafer

6. Place the wafer in an HF bath to remove all oxide

7. Regrow 100nm of oxide at 1100oC

8. Spin coat back with Shipley 1813 for protection

9. Pattern front with Shipley 1813 with 28µm disks at needle sites

89

10. Place wafer in HF bath to remove selected oxide regions

11. RIE etch with 10%02 and 90%SF6 at 150W for 30min to etch 20µm

12. Remove remaining oxide layer with HF bath

13. Spin Teflon onto glass slides 2500rpm for 25s

14. Bake 20min at 90oC

15. Spin and pattern top layer of SU-8 to 12.5µm

16. Spin and pattern bottom layer of SU-8 to 12.5µm

17. Batch develop both layers of SU-8 in SU-8 developer

18. Apply adhesive to SU-8 layer

19. Align to silicon microneedles and release from Teflon layer

90

Appendix B

DEP Trapping Model

/∗ This program uses a f i n i t e time s t ep t echn i que to t rackthe p o s i t i o n o f a

p o l a r i z a b l e p a r t i c l e in an e x t e r n a l l y app l i e d e l e c t r i c f i e l d .The f i e l d i s

gener taed by app l y ing a p o t e n t i a l to four e q u a l l y spacedp i l l a r s ∗/

#define dt 0 .01 // s i z e o f the time step , d t#define drag 0 .18#define s i g 1000#define dep coe f −5.07 e5

#include <s t d i o . h>#include <s t d l i b . h>#include <math . h>

struct vect int x ; int y ; f loat vx , vy ; c e l l ;

f loat ∗ g r id ;

f loat tx , ty ;

FILE ∗out ;FILE ∗pos ;FILE ∗ e f i e l d ;FILE ∗ t ruth ;

int l ;

91

f loat phi ( int x , int y , int d)

// func t i on re turns the a n a l y t i c a l s o l u t i o n o f the p o t e n t i a la t a s p e c i f i c po in t

f loat r = sq r t ( x∗x+y∗y ) ;f loat p ;i f ( (2∗ r /d) <= 1)

p = 1 ;else

p = 1 + log (d / (2∗ r ) ) / l og (2 .71828183) ;return p ;

void a s s i gn ( int x , int y , int s ide , f loat val , f loat ∗a )

// func t i on a s s i gn s a s p e c i f i c va lue to a predetermined areain memory

y ∗= s id e ;y += x ;∗( a + y) = va l ;

f loat r e t r i e v e ( int x , int y , int s ide , f loat ∗a )

// func t i on r e t r i e v e s a s p e c i f i c va lue l o c a t e d at apredetermined area in memory

return ∗( g r i d + x + l ∗y ) ;

void z e r o a l l ( f loat ∗a , int s )

// i n i t i a l i z e a l l array po in t s to zero

int x ;

92

for ( x = 0 ; x < l ∗ l ; x++)∗( g r i d + x) = 0 ;

void i n i t ( int d , int s , f loat r )

// i n i t i a l i z e s memory space f o r array and s e t s a l l v a l u e s o ft h i s space to zero , g enera t e s f i l e s f o r e xpo r t i n g

// data , and s e t s i n i t i a l p o s i t i o n and v e l o c i t y v e c t o r s o fincoming p a r t i c l e

out = fopen ( ”temp . out” , ”w” ) ;t ruth = fopen ( ” truth . out” , ”w” ) ;pos = fopen ( ” c e l l . out” , ”w” ) ;e f i e l d = fopen ( ” e f i e l d . out” , ”w” ) ;c e l l . y = 0 ;c e l l . x = 2 − s ;c e l l . vy = 0 ;c e l l . vx = r ;g r i d = ( f loat ∗) mal loc ( s izeof ( f loat )∗ l ∗ l ) ; // 4∗ s∗ sz e r o a l l ( gr id , s ) ;

void f i l l ( int s , int d , int t )

// maps the p o t e n t i a l ( s o l v e d a n a l y t i c a l l y ) to the genera tedarray

int countx , county ;int x , y ;f loat r , p , va l ;for ( countx = −s ; countx <= s ; countx++)

for ( county = −s ; county <= s ; county ++)

va l = r e t r i e v e ( countx + s , county + s , l , g r i d ) ;x = countx + t ;y = county + t ;r = sq r t ( x∗x + y∗y ) ;i f ( ( r <= (( f loat )d/2) ) | | ( va l == 1) )

a s s i gn ( countx + s , county + s , l , 1 , g r i d ) ;

93

elsea s s i gn ( countx + s , county + s , l , phi (x , y , d ) + val ,

g r i d ) ;

va l = r e t r i e v e ( countx + s , county + s , l , g r i d ) ;x = countx + t ;y = county − t ;r = sq r t ( x∗x + y∗y ) ;i f ( ( r <= (( f loat )d/2) ) | | ( va l == 1) )

a s s i gn ( countx + s , county + s , l , 1 , g r i d ) ;else

a s s i gn ( countx + s , county + s , l , phi (x , y , d ) + val ,g r i d ) ;

va l = r e t r i e v e ( countx + s , county + s , l , g r i d ) ;x = countx − t ;y = county + t ;r = sq r t ( x∗x + y∗y ) ;i f ( ( r <= (( f loat )d/2) ) | | ( va l == 1) )

a s s i gn ( countx + s , county + s , l , 1 , g r i d ) ;else

a s s i gn ( countx + s , county + s , l , phi (x , y , d ) + val ,g r i d ) ;

va l = r e t r i e v e ( countx + s , county + s , l , g r i d ) ;x = countx − t ;y = county − t ;r = sq r t ( x∗x + y∗y ) ;i f ( ( r <= (( f loat )d/2) ) | | ( va l == 1) )

a s s i gn ( countx + s , county + s , l , 1 , g r i d ) ;else

a s s i gn ( countx + s , county + s , l , phi (x , y , d ) + val ,g r i d ) ;

void pr in t ( f loat ∗a , FILE ∗t , int s )

// expor t s data from memory space to a f i l e

int x , y ;

94

f loat va l ;

for ( x = −s ; x <= s ; x++)

for ( y = −s ; y <= s ; y++)

va l = ∗( g r i d + (x + s ) + (y + s )∗ l ) ;f p r i n t f ( t , ”%f \ t ” , va l ) ;

f p r i n t f ( t , ”\n” ) ;

void p r i n t e f i e l d ( int s )

// c a l c u l a t e s the suare o f the magnitude o f the e l e c t r i cf i e l d from the array o f s t o r ed p o t e n t i a l

// and expor t s the data to a f i l e

f loat eex , eey ;int cx , cy ;s −= 1 ;for ( cx = 1 ; cx < 2∗ s ; cx++)

for ( cy = 1 ; cy < 2∗ s ; cy++)

eex = ( r e t r i e v e ( cx + 1 , cy , l , g r i d ) − r e t r i e v e ( cx − 1 ,cy , l , g r i d ) ) / 2 ;

eey = ( r e t r i e v e ( cx , cy + 1 , l , g r i d ) − r e t r i e v e ( cx , cy− 1 , l , g r i d ) ) / 2 ;

eex = eex∗ eex ;eey = eey∗ eey ;eex += eey ;f p r i n t f ( e f i e l d , ”%f \ t\ t ” , eex ) ;

f p r i n t f ( e f i e l d , ”\n” ) ;

f loat fy ( int x , int y , int s )

95

// determines the f o r c e in the y d i r e c t i o n v ia : F y = d(Eˆ2)/dy computing Eˆ2 from the

// va l u e s o f p o t e n t i a l s t o r ed in memory and then performing acentered d i f f e r e n c e method

// to determine the d e r i v a t i v e in the y−d i r e c t i o n

f loat temp , ea , eb ;

x += s ;y += s ;

temp = ( r e t r i e v e (x + 1 , y + 1 , l , g r i d ) − r e t r i e v e (x − 1 , y+ 1 , l , g r i d ) ) / 2 ;

ea = temp∗temp ;temp = ( r e t r i e v e (x , y + 2 , l , g r i d ) − r e t r i e v e (x , y , l ,

g r i d ) ) / 2 ;ea += temp∗temp ;

temp = ( r e t r i e v e (x + 1 , y − 1 , l , g r i d ) − r e t r i e v e (x − 1 , y− 1 , l , g r i d ) ) / 2 ;

eb = temp∗temp ;temp = ( r e t r i e v e (x , y , l , g r i d ) − r e t r i e v e (x , y − 2 , l ,

g r i d ) ) / 2 ;eb += temp∗temp ;

return ( ea − eb ) / 2 ;

f loat fx ( int x , int y , int s )

// determines the f o r c e in the x d i r e c t i o n v ia : F y = d(Eˆ2)/dy computing Eˆ2 from the

// va l u e s o f p o t e n t i a l s t o r ed in memory and then performing acentered d i f f e r e n c e method

// to determine the d e r i v a t i v e in the x−d i r e c t i o n

f loat temp , ea , eb ;

x += s ;

96

y += s ;

temp = ( r e t r i e v e (x + 2 , y , l , g r i d ) − r e t r i e v e (x , y , l ,g r i d ) ) / 2 ; // cen te r d i f f e r e n c e f o r e− f i e l d in xd i r e c t i o n at po in t x++

ea = temp∗temp ;temp = ( r e t r i e v e (x + 1 , y + 1 , l , g r i d ) − r e t r i e v e (x + 1 , y

− 1 , l , g r i d ) ) / 2 ; // cen te r d i f f e r e n c e f o r e− f i e l d iny d i r e c t i o n at po in t x++

ea += temp∗temp ;

temp = ( r e t r i e v e (x , y , l , g r i d ) − r e t r i e v e (x − 2 , y , l ,g r i d ) ) / 2 ; // cen ter d i f f e r e n c e f o r e− f i e l d in xd i r e c t i o n at po in t x−−

eb = temp∗temp ;temp = ( r e t r i e v e (x − 1 , y + 1 , l , g r i d ) − r e t r i e v e (x − 1 , y

− 1 , l , g r i d ) ) / 2 ; // cen te r d i f f e r e n c e f o r e− f i e l d inx d i r e c t i o n at po in t x−−

eb += temp∗temp ;

return ( ea − eb ) / 2 ; // centered d i f f e r e n c e o f e− f i e l d

void move( int s )

// t r a c k s the movement o f the c e l l v i a a f i n i t e time s t ep as// r ( t+1) = 0.5∗ a ( t )∗ dt ∗ dt + v ( t )∗ dt + r ( t )// v ( t+1) = a( t )∗ dt + v ( t )

int x = c e l l . x ;int y = c e l l . y ;f loat a , b ;f loat accx , accy ;a = fx (x , y , s ) ;b = fy (x , y , s ) ;

accx = dep coe f ∗a − drag∗ c e l l . vx ;accy = dep coe f ∗b − drag∗ c e l l . vy ;

a = 0.5∗ accx∗dt∗dt + c e l l . vx∗dt + tx ; //+ x ;tx = a ;

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c e l l . vx += accx∗dt ;

b = 0.5∗ accy∗dt∗dt + c e l l . vy∗dt + ty ; //+ x ; y ;ty = b ;c e l l . vy += accy∗dt ;

c e l l . x = ( int ) round ( a ) ;c e l l . y = ( int ) round (b) ;

f p r i n t f ( truth , ”%f \ t%f \ t%d\ t%d\n” , a , b , c e l l . x , c e l l . y ) ;

void dest roy (void )

// f r e e s used memory and c l o s e s f i l e s

f r e e ( g r i d ) ;f c l o s e ( out ) ;f c l o s e ( t ruth ) ;f c l o s e ( pos ) ;f c l o s e ( e f i e l d ) ;

int main ( int argc , char ∗argv [ ] )

int count ;int d , s , t ;f loat a , b ;int c = 0 ;int ys ;f loat yr = 0 ;f loat r a t e ;f loat high ;i f ( ( argc != 4)&&(argc !=6) )

f p r i n t f ( s tde r r , ”Correct usage i s : ’ dep diameter spac ingra t e ’ ( a l l in r e l a t i v e un i t s )\n” ) ;

f p r i n t f ( s tde r r , ” Al te rnate usage : ’ dep diameter spac ingra t e y−s t a r t y−r a t e ’\n” ) ;

e x i t (1 ) ;

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d = a to i ( argv [ 1 ] ) ;s = a t o i ( argv [ 2 ] ) ;r a t e = ato f ( argv [ 3 ] ) ;

f p r i n t f ( s tde r r , ”\nCel l DEP w i l l be based on the f o l l ow i ngparameters ( a l l numbers truncated to i n t e g e r s ) : \n” ) ;

f p r i n t f ( s tde r r , ”\nDiameter o f e l e c t r o d e s : %d\n” , d) ;f p r i n t f ( s tde r r , ” Spacing in between e l e c t r o d e s : %d\n” , s ) ;f p r i n t f ( s tde r r , ”Rate o f f low : %f \n” , r a t e ) ;

ys = 0 ;yr = 0 ;

i f ( argc == 6)

ys = a to i ( argv [ 4 ] ) ;yr = a to f ( argv [ 5 ] ) ;

i f ( ( d <= 0) | | ( s <= 0) | | ( r a t e < 0) )

f p r i n t f ( s tde r r , ”Diameter , Spacing , and Rate must be non−zero p o s i t i v e numbers\n” ) ;

return 0 ;i f (d > s )

f p r i n t f ( s tde r r , ”Diameter must be l e s s than or equal tospac ing \n” ) ;

return 0 ;

i f ( s % 2 == 0)

t = s / 2 ;s = 2∗( s+1) ;

else

t = s ;d ∗= 2 ;

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s = 2∗ s+1;

l = 2∗ s + 1 ;

i f ( ( ( ys < − l ) | | ( ys>l ) )&&(argc == 6) ) // | | ( yr < 0)

f p r i n t f ( s tde r r , ” | y−s t a r t | must be a p o s i t i v e i n t e g e rl e s s than 4∗ spac ing \n” ) ;

// f p r i n t f ( s tde r r , ”and y−ra t e must be p o s i t i v e .\n”) ;e x i t (2 ) ;

i f ( yr )f p r i n t f ( s tde r r , ”y−r a t e : %f \n” , yr ) ;

i n i t (d , s , r a t e ) ;

c e l l . y = ys ;c e l l . vy = yr ;

f p r i n t f ( s tde r r , ”\ n I n i t i a l p o s i t i o n : (%d , %d)\n” , c e l l . x ,c e l l . y ) ;

f p r i n t f ( s tde r r , ”\nEcce Signum !\n\n” ) ;f i l l ( s , d , t ) ;p r i n t ( gr id , out , s ) ;high = r e t r i e v e (0 , 0 , l , g r i d ) / 2 ;

tx = c e l l . x ;ty = c e l l . y ;

while ( ( c e l l . x <= ( s − 2) )&&( c e l l . x >= (2 − s ) )&&( c e l l . y <=( s − 2) )&&( c e l l . y >= (2 − s ) ) )

a = tx ; // c e l l . x ;b = ty ; // c e l l . y ;c++;f p r i n t f ( pos , ”%d\ t%d\ t%f \ t%f \ t%f \ t\n” , c e l l . x + s , c e l l . y

+ s , high , c e l l . vx , c e l l . vy ) ;

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f p r i n t f ( s tde r r , ”p = (%f \ t%f ) −> (%d , %d)\n” , a , b , ( int) round ( s i g ∗a ) , ( int ) round ( s i g ∗b) ) ;

move( s ) ;f p r i n t f ( s tde r r , ”p ’ = (%f \ t%f ) −> (%d , %d)\n\n” , tx , ty ,

( int ) round ( s i g ∗ tx ) , ( int ) round ( s i g ∗ ty ) ) ;i f ( ( a == tx ) && (b == ty ) )

break ;

f p r i n t f ( s tde r r , ”%f \ t%f \n” , tx , ty ) ;

f p r i n t f ( pos , ”%d\ t%d\ t%f \ t%f \ t%f \ t\n” , c e l l . x + s , c e l l . y +s , high , c e l l . vx , c e l l . vy ) ;

p r i n t e f i e l d ( s ) ;

des t roy ( ) ;return 0 ;

101

Appendix C

Ferrofluid Genetic Algorithm

#include <math . h>#include <con io . h>#include <s t d i o . h>#include <s t d l i b . h>#include <time . h>#include <s t r i n g . h>#include ”vmath . h”

#define rho 1236#define g 9.80665#define pi 3 .14159#define ch i 1 .172#define mu 0 1.256637062 e−6#define div 6e−3 // P i x e l S i z e#define i s t e p 1e−3#define s i z e 100 // Array s i z e#define rad 15e−2 // Radius o f loop in meters

#define s t 0 .0305 // sur f a ce t ens ion

double ∗hf , ∗ energy ; // Magnetic f i e l d and t o t a l energyarray

int ∗ g r id ; // Binary r ep r e s en t a t i on o f occupied p i x e l s

f loat cur = 0 . 0 1 ; // Current

vec to r (∗ f i e l d ) ( f loat x , f loat y , f loat z ) ;

102

int q = s izeof ( f loat ) ;

f loat abso lu t e ( f loat a )

return s q r t ( a∗a ) ;

double f (double z ) // For Newton ’ s method

double mu r = mu 0∗( ch i + 1) ;return z − mu r∗ ch i ∗ cur∗ cur∗pow( rad , 4) / (8∗ rho∗g∗pow( ( rad

∗ rad + z∗z ) , 3) ) ;

double fp (double z ) // Der i va t i v e f o r Newton ’ smethod

double mu r = mu 0∗( ch i + 1) ;return 1 + 6∗ z∗pow(mu 0 , 3)∗ ch i ∗ cur∗ cur∗pow( rad , 4) / (8∗

rho∗g∗mu r∗mu r∗pow( ( rad∗ rad + z∗z ) , 4) ) ;

double s o l v e r (void ) // Newton ’ s method f o rf i n d i n g h e i g h t in vary ing f i e l d

double old , new ;o ld = 0 ;new = 1e−6;while ( abso lu t e ( o ld − new) > div / 1000)

old = new ;new = old − ( f ( o ld ) / fp ( o ld ) ) ;

return new ;

f loat e l l i p e ( f loat k ) // E l l i p t i c i n t e g r a l (E type ) re turnsE( k )

f loat s tep = 0 ;f loat i n t e g r a l = 0 ;

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while ( s tep < pi /2)

i n t e g r a l += sq r t (1 − k∗k∗ s i n ( s tep )∗ s i n ( s tep ) ) ;s tep += i s t e p ;

i n t e g r a l ∗= i s t e p ;return i n t e g r a l ;

f loat e l l i p k ( f loat k ) // E l l i p t i c i n t e g r a l (K type ) re tu rnsK( k )

f loat s tep = 0 ;f loat i n t e g r a l = 0 ;while ( s tep < pi /2)

i n t e g r a l += 1 / sq r t (1 − k∗k∗ s i n ( s tep )∗ s i n ( s tep ) ) ;s tep += i s t e p ;

i n t e g r a l ∗= i s t e p ;return i n t e g r a l ;

// f i e l d f o r a loop current :

vec to r l f i e l d ( f loat x , f loat y , f loat z )

// Ca l cu l a t e s the magnetic f i e l d produced by a loop o f wireat a p o s i t i o n <x , y , z> r e l a t i v e to the cen te r o f the loop

vec to r temp ;f loat th ing ;f loat s tep = 0 ;f loat r , Q, alpha , beta , gamma, k ;f loat ee , ek ;f loat bottom ;

temp . x = temp . y = temp . z = 0 ;

104

r = sq r t ( x∗x + y∗y ) ;

alpha = r / rad ;beta = z / rad ;gamma = z / r ;Q = pow((1 + alpha ) , 2) + pow( beta , 2) ;k = sq r t (4∗ alpha / Q) ;th ing = cur / (2∗ rad ) ;

ee = e l l i p e (k ) ;ek = e l l i p k (k ) ;

bottom = Q − 4∗ alpha ;

temp . z = thing ∗ ( (1 − pow( alpha , 2) − pow( beta , 2) )∗ ee /(bottom ) + ek ) /( p i ∗ s q r t (Q) ) ;

temp . x = thing ∗gamma∗ ( (1 + pow( alpha , 2) + pow( beta , 2) ) ∗ ee/( bottom ) − ek ) /( p i ∗ s q r t (Q) ) ;

temp . y = (y / r )∗temp . x ;temp . x ∗= (x / r ) ;

return temp ;

// Constant f i e l d :

vec to r c f i e l d ( f loat x , f loat y , f loat z )

// Ca l cu l a t e s the magnetic f i e l d a t a po in t <x , y , z> due toa cons tant and uniform magnetic f i e l d

vec to r temp ;

i f ( ( x > (0 − s i z e ∗div / 4) )&&(x <= ( s i z e ∗div / 4) ) )

temp . y = 0 ;temp . x = 0 ;temp . z = cur / (2∗ rad ) ;

105

else

temp . x = 0 ;temp . y = 0 ;temp . z = 0 ;

return temp ;

// Var iab l e f i e l d

vec to r v f i e l d ( f loat x , f loat y , f loat z )

// Ca l cu l a t e s and re turns the magnetic f i e l d a t a po in t <x , y, z> due to a constant , but vary ing magnetic f i e l d

// The f i e l d decays wi th the cube o f the h e i g h t

vec to r temp ;

temp . x = 0 ;temp . y = 0 ;i f ( ( x > (0 − s i z e ∗div / 4) )&&(x <= ( s i z e ∗div / 4) ) )

temp . z = ( cur / 2) ∗ ( ( rad∗ rad ) / pow( rad∗ rad + z∗z , 1 . 5 ) ) ;else

temp . z = 0 ;

return temp ;

void pr in t ene r gy (char ∗a )// Exports the energy per array po in t to a f i l e

FILE ∗out ;out = fopen (a , ”w” ) ;for ( int y = 0 ; y < s i z e ; y++)

for ( int x = 0 ; x < s i z e ; x++)

106

f p r i n t f ( out , ”\ t%e” , (∗ ( energy + x + s i z e ∗y ) ) ) ; // (∗(g r i d + x + y∗ s i z e ) )∗

f p r i n t f ( out , ”\n” ) ;

f c l o s e ( out ) ;

void pr in t (char ∗a )// Pr in t s boo lean array o f occupat ion s t a t e s to a f i l e

FILE ∗out ;out = fopen (a , ”w” ) ;for ( int y = 0 ; y < s i z e ; y++)

for ( int x = 0 ; x < s i z e ; x++)f p r i n t f ( out , ”\ t%d” , ∗( g r i d + x + y∗ s i z e ) ) ;

f p r i n t f ( out , ”\n” ) ;

f c l o s e ( out ) ;

f loat g rav i ty ( f loat z )// Grav i t a t i ona l Energy : dUg = rho∗g∗ z∗dx∗dz

return z ∗( div ∗div ∗g∗ rho ) ;

f loat s u r f a c e t e n s i o n ( int i , int j )// Surface t ens ion : dUs = sigma∗(# of untouched s i d e s ) ∗( s i d e

l en g t h )

int count = 0 ;int a l l = 0 ;

i f ( i > 0)

count++;a l l += ∗( g r id + i − 1 + j ∗ s i z e ) ;

i f ( i < ( s i z e − 1) )

107

count++;a l l += ∗( g r id + i + 1 + j ∗ s i z e ) ;

i f ( j > 0)

count++;a l l += ∗( g r id + i + ( j − 1)∗ s i z e ) ;

i f ( j < ( s i z e − 1) )

count++;a l l += ∗( g r id + i + ( j + 1)∗ s i z e ) ;

return s t ∗div ∗( count − a l l ) ;

f loat t o t a l s u r f (void )

// Adds the t o t a l su r f a c e t ens ion energy f o r the en t i r eoccupat ion s t a t e s

f loat temp = 0 ;for ( int i = 0 ; i < s i z e ; i++)

for ( int j = 0 ; j < s i z e ; j++)

i f ( (∗ ( g r i d + i + j ∗ s i z e ) != ∗( g r id + ( i + 1) + j ∗s i z e ) )&&( i < ( s i z e − 1) ) )

temp += 0 . 5 ;i f ( (∗ ( g r i d + i + j ∗ s i z e ) != ∗( g r id + ( i − 1) + j ∗

s i z e ) )&&( i > 0) )temp += 0 . 5 ;

i f ( (∗ ( g r i d + i + j ∗ s i z e ) != ∗( g r id + i + ( j + 1)∗s i z e ) )&&(j < ( s i z e − 1) ) )

temp += 0 . 5 ;i f ( (∗ ( g r i d + i + j ∗ s i z e ) != ∗( g r id + i + ( j − 1)∗

s i z e ) )&&(j > 0) )temp += 0 . 5 ;

108

temp ∗= div ∗ s t ;

return temp ;

f loat t o t a l g r a v (void )

// Sums a l l g r a v i t a t i o n a l ene r g i e s f o r a l l occupied p i x e l s .

f loat temp = 0 ;for ( int i = 0 ; i < s i z e ; i++)

for ( int j = 0 ; j < s i z e ; j++)

i f (∗ ( g r i d + i + j ∗ s i z e ) )temp += rho∗g∗div ∗div ∗div ∗ j ;

return temp ;

f loat tota l mag (void )

// Sums a l l magnetic energy over the en t i r e occupat ion matrix

f loat a , temp = 0 ;for ( int i = 0 ; i < s i z e ; i++)

for ( int j = 0 ; j < s i z e ; j++)

a = 1 − ( ch i + 1) ∗ (∗ ( g r i d + i + j ∗ s i z e ) ) ;// a = 1 i f unocciped , −ch i i f occupieda ∗= 0.5∗mu 0∗ (∗ ( hf + i + j ∗ s i z e ) )∗div ∗div ;temp += a ;

return temp ;

void en e r g i z e (void )// Determines the energy per p i x e l : dU = dUm + dUs + dUg

109

for ( int i = 0 ; i < s i z e ; i++)for ( int j = 0 ; j < s i z e ; j ++)∗( energy + i + j ∗ s i z e ) = −0.5∗(( ch i + 1) )∗mu 0∗div ∗div

∗ (∗ ( hf + i + s i z e ∗ j ) ) + s u r f a c e t e n s i o n ( i , j ) +grav i ty ( j ∗div ) ;

int swapper (void )

// Determines the element which i s occupied and has theh i g h e s t d i f f e r e n t i a l energy and t ha t wi th the l owe s tenergy , which i s

// unoccupied , and swaps the va l u e s o f the occupat ion matrix

f loat tempp , tempc , en , high , low ;int ih , i l , jh , j l , ret , v , a , b ;

high = −10000;low = 10000;

tempp = tempc = to t a l g r a v ( ) + tota l mag ( ) + t o t a l s u r f ( ) ;// t o t a l energy in current c on f i g u r a t i on

for ( int i = 0 ; i < s i z e ; i++)for ( int j = 0 ; j < s i z e ; j++)

v = ∗( g r i d + i + j ∗ s i z e ) ;en = ∗( energy + i + j ∗ s i z e ) ;

i f ( v )

i f ( en > high )

ih = i ;jh = j ;high = en ;

else

i f ( en < low )

110

i l = i ;j l = j ;low = en ;

r e t = 0 ;

i f ( high >= low )∗( g r i d + ih + jh ∗ s i z e ) = 0 ;∗( g r i d + i l + j l ∗ s i z e ) = 1 ;tempc = to t a l g r a v ( ) + tota l mag ( ) + t o t a l s u r f ( ) ;i f ( tempc <= tempp)

en e r g i z e ( ) ;r e t = 1 ;

else∗( g r i d + ih + jh ∗ s i z e ) = 1 ;∗( g r i d + i l + j l ∗ s i z e ) = 0 ;r e t = 0 ;

return r e t ;

int swap (void )// Swaps h i g h e s t energy occupied p i x e l wi th l owe s t energy

unccupied p i x e l

f loat high , low , en ;int ih , i l , jh , j l , v , r e t ;

high = −1000;low = 1000 ;

r e t = 0 ;

for ( int i = 0 ; i < s i z e ; i++)

111

for ( int j = 0 ; j < s i z e ; j++)

v = ∗( g r i d + i + j ∗ s i z e ) ;en = ∗( energy + i + j ∗ s i z e ) ;

i f ( v )

i f ( en > high )

ih = i ;jh = j ;high = en ;

else

i f ( en < low )

i l = i ;j l = j ;low = en ;

i f ( high >= low )∗( g r i d + ih + jh ∗ s i z e ) = 0 ;∗( g r i d + i l + j l ∗ s i z e ) = 1 ;en e r g i z e ( ) ;r e t = 1 ;

return r e t ;

void i n i t ( f loat vp ) // I n i t i a l i z e s arrays , p i x e l ss t a r t by be ing f i l l e d in from the bottom , dUm = −(1/2)∗ ch i∗mu 0∗Hˆ2∗dx∗dz

g r id = ( int ∗) mal loc ( s izeof ( int )∗ s i z e ∗ s i z e ) ;hf = (double ∗) mal loc ( s izeof (double )∗ s i z e ∗ s i z e ) ;energy = (double ∗) mal loc ( s izeof (double )∗ s i z e ∗ s i z e ) ;

112

for ( int x = 0 ; x < s i z e ; x++)for ( int y = 0 ; y < s i z e ; y++)∗( g r id + x + s i z e ∗y ) = 0 ;i f ( y < vp∗ s i z e )∗( g r i d + x + s i z e ∗y ) = 1 ;

∗( hf + x + s i z e ∗y ) = pow(mag( (∗ f i e l d ) ( div ∗(x − 0 . 5∗ (s i z e − 1) ) , 0 , div ∗y ) ) , 2) ;

void random init ( f loat vp ) // I n i t i a l i z e arrays , randomp i x e l s t a r t

int count = s i z e ∗ s i z e ∗vp ;int num;

g r id = ( int ∗) mal loc ( s izeof ( int )∗ s i z e ∗ s i z e ) ;hf = (double ∗) mal loc ( s izeof (double )∗ s i z e ∗ s i z e ) ;energy = (double ∗) mal loc ( s izeof (double )∗ s i z e ∗ s i z e ) ;

for ( int x = 0 ; x < s i z e ; x++)for ( int y = 0 ; y < s i z e ; y++)∗( g r id + x + s i z e ∗y ) = 0 ;∗( hf + x + s i z e ∗y ) = pow(mag( f i e l d ( div ∗(x − 0 . 5∗ ( s i z e

− 1) ) , 0 , d iv ∗y ) ) , 2) ;

while ( count > 0)

num = ( s i z e ∗ s i z e − 1)∗ rand ( ) / RANDMAX;i f (∗ ( g r id + num) == 0)∗( g r i d + num) = 1 ;count−−;

void done (void ) // Destroys array po in t e r s

113

f r e e ( g r i d ) ;f r e e ( hf ) ;f r e e ( energy ) ;

void p r i n t f i e l d (void ) // Pr in t s magnetic f i e l d array tof i l e

FILE ∗out ;vec to r f ;int x , y ;double xv , zv ;

out = fopen ( ” h f i e l d . out” , ”w” ) ;for ( int y= 0 ; y < s i z e ; y++)

for ( int x = 0 ; x < s i z e ; x++)

f p r i n t f ( out , ”%e\ t ” , ∗( hf + x + y∗ s i z e ) ) ;

f p r i n t f ( out , ”\n” ) ;

f c l o s e ( out ) ;

vec to r max height (char c , f loat vp ) // Determines themaximum he i g h t by theory and computat iona l model

int b ig y = 0 ;int l i t t l e y = s i z e ;f loat theory , mu r , b , d e l t a ;vec to r temp ;

mu r = mu 0∗( ch i + 1) ;

for ( int y = 0 ; y < s i z e ; y++)

for ( int x = 0 ; x < s i z e ; x++)i f (∗ ( g r i d + x + y∗ s i z e ) )

b ig y = y ;else

114

i f ( y < l i t t l e y )l i t t l e y = y ;

l i t t l e y −−;

// f p r i n t f ( s tde r r , ” l i t t l e y = %d b i g y = %d\n” , l i t t l e y ,b i g y ) ;

de l t a = div ∗( b ig y − l i t t l e y ) ;

// f p r i n t f ( s tde r r , ”Maximum de l t a h e i g h t : %e\n” , d e l t a ) ;

switch ( c )

case ’ c ’ : case ’C ’ :

theory = ( ch i ∗mu r∗pow(mag( f i e l d (0 , 0 , 0 ) ) , 2) )/ (2∗ rho∗g ) ;

// f p r i n t f ( s tde r r , ”Max a n a l y t i c a l h e i g h t : %e\n” , theory ) ;

i f ( theory > s i z e ∗div )theory = 2∗ s i z e ∗div ∗(1 − vp ) ;

i f ( ( l i t t l e y == −1)&&(theory > de l t a ) )// f p r i n t f ( s tde r r , ”Can suppor t more

volume\n”) ;theory = de l t a ;

// e l s e// theory = 2∗( theory − vp∗ s i z e ∗ d i v ) ;// f p r i n t f ( s tde r r , ”Max d e l t a a n a l y t i c a l

h e i g h t ( cons tant f i e l d ) : %e Unitd i f f e r e n c e = %.2 f \n” , theory , ( d e l t a −theory ) / d i v ) ;

// f p r i n t f ( s tde r r , ”Ratio ( ana l y t i c / a l go ) : %f \n” , theory / d e l t a ) ;

// f p r i n t f ( s tde r r , ”Percent error : %.2 f%%\n” ,100∗ a b s o l u t e ( theory − d e l t a ) / theory ) ;

break ;case ’ v ’ : case ’V ’ :

115

theory = s o l v e r ( ) ;// f p r i n t f ( s tde r r , ”Max a n a l y t i c a l h e i g h t : %e\

n” , theory ) ;i f ( theory > s i z e ∗div )

theory = 2∗ s i z e ∗div ∗(1 − vp ) ;else

theory = 2∗( theory − vp∗ s i z e ∗div ) ;i f ( ( l i t t l e y == −1)&&(theory > de l t a ) )

// f p r i n t f ( s tde r r , ”Can suppor t morevolume\n”) ;

theory = de l t a ;

// f p r i n t f ( s tde r r , ”Max d e l t a a n a l y t i c a lh e i g h t ( v a r i a b l e f i e l d ) : %e Unitd i f f e r e n c e = %.2 f \n” , theory , ( d e l t a −theory ) / d i v ) ;

// f p r i n t f ( s tde r r , ”Ratio ( ana l y t i c / a l go ) : %f \n” , theory / d e l t a ) ;

break ;temp . x = theory ;temp . y = de l t a ;return temp ;

int main ( int argv , char ∗ argc [ ] )

int num;long count = 0 ;int t a f = 0 , end = 0 ;f loat vp ;FILE ∗ f ;t ime t s t a r t t ime , end time ;f loat d i f ;char cho i c e ;vec to r mh;

i f ( argv != 4)

116

f p r i n t f ( s tde r r , ”\nCorrect usage i s : %s cur vp v , c , l\n” , argc [ 0 ] ) ;

e x i t (1 ) ;

cho i c e = argc [ 3 ] [ 0 ] ;

cur = ato f ( argc [ 1 ] ) ;

switch ( cho i c e )

case ’ c ’ : case ’C ’ : f i e l d = c f i e l d ; break ;case ’ v ’ : case ’V ’ : f i e l d = v f i e l d ; break ;case ’ l ’ : case ’L ’ : f i e l d = l f i e l d ; break ;default :

f p r i n t f ( s tde r r , ”Not a va l i d cho i c e \n” ) ;e x i t (2 ) ;

srand ( time (NULL) ) ;

vp = ato f ( argc [ 2 ] ) ;

random init ( vp ) ;

p r i n t f i e l d ( ) ;

p r i n t ( ” s t a r t . out” ) ;time(& s t a r t t ime ) ;

/∗ f = fopen (” tva . out ” , ”w”) ;vp = 0 . 3 ; //0.2

/∗ f o r ( cur = 2000; cur < 10000; cur += 500)

random ini t ( vp ) ;ene r g i z e ( ) ;count = 0;wh i l e ( swapper ( ) )

count ++;mh = max height ( choice , vp ) ;

117

f p r i n t f ( f , ”%d\ t%f \ t%f \ t%f \n” , count , cur /(2∗ rad ) , mh. y, mh. x ) ;

f p r i n t f ( s t de r r , ” cur = %.0 f \n” , cur ) ;i f ( cur != 9500)

done () ;∗/

// f c l o s e ( f ) ;

while ( swapper ( ) )count++;

time(&end time ) ;d i f = d i f f t im e ( end time , s t a r t t ime ) / 16 ;f p r i n t f ( s tde r r , ”Time per p roce s s = %f \n” , d i f ) ;f p r i n t f ( s tde r r , ”Number o f g ene ra t i on s : %d\n” , count ) ;

p r i n t ( ” p i x e l s . out” ) ;p r i n t ene rgy ( ” energy . out” ) ;

done ( ) ;/∗max height ( choice , vp ) ; ∗/

118

Appendix D

Optical Ray Trace Program

#include <s t d i o . h>#include <math . h>

#define bump 5#define space 25#define s i z e 50#define de l t a 80#define theta 45 .0#define pi 3 .14159

double g r id [ s i z e ] [ s i z e ] ;f loat h , d i sp ;

struct vec f loat x , y , z ; ;

typedef struct vec vec to r ;

vec to r in , e1 , e2 , e3 , e4 ;

FILE ∗out ;

f loat he ight ( int dx , int dy )

return s q r t (bump∗bump − dx∗dx − dy∗dy ) ;

vec to r tangent ( int x , int y )

119

vec to r temp ;

temp . x = − ( g r i d [ x+1] [ y ] − g r id [ x − 1 ] [ y ] ) / 2 ;temp . y = − ( g r i d [ x ] [ y + 1 ] − g r id [ x ] [ y − 1 ] ) / 2 ;temp . z = 1 ;

return temp ;

void t r i a l (void )

for ( int x = 0 ; x < s i z e ; x ++)for ( int y = 0 ; y < s i z e ; y ++)

gr id [ x ] [ y ] = 0 ;

void i n i t i a l i z e (void )

for ( int x = space ; x < s i z e ; x += space )for ( int y = space ; y < s i z e ; y += space )

for ( int xc = − bump ; xc < bump ; xc++)for ( int yc = − bump ; yc < bump ; yc++)

i f ( s q r t ( xc∗xc + yc∗yc ) < bump)g r id [ x + xc ] [ y + yc ] = he ight ( xc , yc ) ;

elseg r id [ x + xc ] [ y + yc ] = 0 ;

void pr in t (void )

FILE ∗out2 ;out2 = fopen ( ” g r id . out” , ”w” ) ;vec to r temp ;

for ( int y = 0 ; y < s i z e ; y++)

for ( int x = 0 ; x < s i z e ; x++)f p r i n t f ( out2 , ”%f \ t ” , g r i d [ x ] [ y ] ) ;

f p r i n t f ( out2 , ”\n” ) ;

120

f c l o s e ( out2 ) ;

f loat mag( vec to r a )

return s q r t ( a . x∗a . x + a . y∗a . y + a . z∗a . z ) ;

vec to r d i v id e ( vec to r a , f loat b)

vec to r temp ;temp . x = a . x / b ;temp . y = a . y / b ;temp . z = a . z / b ;return temp ;

vec to r un i t ( vec to r a )

return d iv id e ( a , mag( a ) ) ;

void s o l v e ( vec to r a , vec to r b)

// n = <1, 0 , 1>// p0 = <disp , 0 , he i gh t>

f loat t ;f loat num;f loat d ;vec to r po int ;int p = 1 ;

d = disp + h ;

t = a . x + a . z ;

i f ( t != 0)

121

num = d − b . x − b . z ;t = num / t ;po int . x = a . x∗ t + b . x ;po int . y = a . y∗ t + b . y ;po int . z = a . z∗ t + b . z ;

else

p = 0 ;

i f ( po int . x > ( d i sp + ( ( f loat ) s i z e / 2)∗ cos ( theta ) ) )p = 0 ;

i f ( po int . x < ( d i sp − ( ( f loat ) s i z e / 2)∗ cos ( theta ) ) )p = 0 ;

i f ( po int . z > (h + ( ( f loat ) s i z e / 2)∗ s i n ( theta ) ) )p = 0 ;

i f ( po int . z < (h − ( ( f loat ) s i z e / 2)∗ s i n ( theta ) ) )p = 0 ;

i f ( po int . y < 0)p = 0 ;

i f ( po int . y > s i z e )p = 0 ;

i f (p)f p r i n t f ( out , ”%f %f %f \n” , po int . x , po int . y , po int . z ) ;

void r u n i t (void )

// out = in − 2∗( in . tan ) tan

vec to r out ;vec to r o f f s e t ;vec to r tang ;

for ( int y = 1 ; y < ( s i z e − 1) ; y++)for ( int x = 1 ; x < ( s i z e − 1) ; x++)

tang = uni t ( tangent (x , y ) ) ;

122

out . x = in . x − 2∗( in . x∗ tang . x )∗ tang . x ;out . y = in . y − 2∗( in . y∗ tang . y )∗ tang . y ;out . z = in . z − 2∗( in . z∗ tang . z )∗ tang . z ;out = uni t ( out ) ;

o f f s e t . x = x ;o f f s e t . y = y ;o f f s e t . z = gr id [ x ] [ y ] ;

s o l v e ( out , o f f s e t ) ;

void main (void )

t r i a l ( ) ;i n i t i a l i z e ( ) ;

out = fopen ( ” opt ix . out” , ”w” ) ;

h = ( f loat ) s i z e / (2∗ s q r t (2 ) ) ;d i sp = ( f loat ) s i z e − h ;

h += de l t a ;d i sp += de l t a ;

in . x = 1 ;in . y = 0 ;in . z = −1;

in = uni t ( in ) ;

r u n i t ( ) ;p r i n t ( ) ;f c l o s e ( out ) ;

123

Appendix E

Ferrofluid Control Program

#include <time . h>#include <s t d i o . h>#include <uni s td . h>#include <sys / i o . h>

#define MAXOR 765

void s e t box ( int data , int box )

int bank ;int bankbox ;

i f ( box>=1&&box<=8) bank=8; bankbox=box−1; i f ( box>=9&&box<=16) bank=16; bankbox=box−9; i f ( box>=17&&box<=24) bank=32; bankbox=box−17; i f ( box>=25&&box<=32) bank=64; bankbox=box−25; i f ( box>=33&&box<=40) bank=128; bankbox=box−33;

int base = 0x378 ;int c on t r o l = base + 2 ;

outb (1 , c on t r o l ) ;outb ( data , base ) ;outb (0 , c on t r o l ) ;outb (1 , c on t r o l ) ;outb ( bankbox+bank , base ) ;outb (3 , c on t r o l ) ;outb (1 , c on t r o l ) ;

124

int powt ( int num)

int count ;int temp = 1 ;for ( count = 0 ; count < num; count++)

temp ∗= 2 ;return temp ;

struct headchar bfType [ 2 ] ;int b fS i z e ;short bfReserved ;short bfReserved2 ;int b fO f f b i t s ;int b i S i z e ;int biWidth ;int biHeight ;short biPlanes ;short b i t s ;int biCompression ;int biS izeImage ;int biXPelsPerMeter ;int biYPelsPerMeter ;int biClrUsed ;int biClrImportant ;

h ;

void readhead (FILE ∗ f i l e , struct head ∗h)

f r e ad (&h−>bfType , 1 , 2 , f i l e ) ;f r e ad (&h−>bfS i ze , 1 , 4 , f i l e ) ;f r e ad (&h−>bfReserved , 1 , 2 , f i l e ) ;f r e ad (&h−>bfReserved2 , 1 , 2 , f i l e ) ;f r e ad (&h−>b fO f fb i t s , 1 , 4 , f i l e ) ;

f r e ad (&h−>b iS i z e , 1 , 4 , f i l e ) ;f r e ad (&h−>biWidth , 1 , 4 , f i l e ) ;f r e ad (&h−>biHeight , 1 , 4 , f i l e ) ;f r e ad (&h−>biPlanes , 1 , 2 , f i l e ) ;

125

f r e ad (&h−>b i t s , 1 , 2 , f i l e ) ;f r e ad (&h−>biCompression , 1 , 4 , f i l e ) ;f r e ad (&h−>biSizeImage , 1 , 4 , f i l e ) ;f r e ad (&h−>biXPelsPerMeter , 1 , 4 , f i l e ) ;f r e ad (&h−>biYPelsPerMeter , 1 , 4 , f i l e ) ;f r e ad (&h−>biClrUsed , 1 , 4 , f i l e ) ;f r e ad (&h−>biClrImportant , 1 , 4 , f i l e ) ;

void operate (FILE ∗ f )

int cx , cy ;unsigned char c ;int temp ;long index ;

int count , counter , a l t ;int dcount ;int data ;int box ;

data = 0 ;

dcount = −1;box = 1 ;

index = 0 ;

for ( cy = 0 ; cy < h . b iHeight ; cy++)

for ( cx = 0 ; cx < h . biWidth ; cx++)

temp = 0 ;f r ead (&c , 1 , 1 , f ) ;temp += c ;//temp = 0;f r e ad (&c , 1 , 1 , f ) ;temp += c ;//temp = 0;f r e ad (&c , 1 , 1 , f ) ;temp += c ;

126

a l t = h . b iHeight − cy ;

dcount = ( cx + h . b iHeight ∗cy + 1) ;

i f ( temp == 0)

i f ( ( dcount % 8) == 0)data += 128 ;

elsedata += powt ( ( dcount % 8) − 1) ;

i f ( ( ( dcount % 8) == 0) )

box = ( dcount / 8) ;s e t box ( data , box ) ;data = 0 ;

counter = h . biWidth % 4 ;for ( count = 0 ; count < counter ; count++)

f r ead (&c , 1 , 1 , f ) ;s e t box ( data , 13) ;data = 0 ;box++;

int main ( int argc , char ∗argv [ ] )

FILE ∗ f 1 ;

int base = 0x378 ;

ioperm ( base , 8 , 1) ;ioperm ( base + 2 , 8 , 1) ;

outb (0 , base ) ;outb (1 , base + 2) ;

i f ( argc != 2)

127

f p r i n t f ( s tde r r , ”Correct usage i s : \” con t r o l f i l e .bmp\”\n” ) ;

e x i t (1 ) ;

f 1 = fopen ( argv [ 1 ] , ” rb” ) ;i f ( f 1 != NULL)

readhead ( f1 , &h) ;f s e e k ( f1 , h . b fO f fb i t s , 0) ;

else

f p r i n t f ( s tde r r , ”Could not open f i l e \n” ) ;e x i t (2 ) ;

operate ( f 1 ) ;f c l o s e ( f 1 ) ;return 0 ;

128

Appendix F

Variable Channel

1. Clean standard microscope slide with acetone, isopropyl alcohol, deionized wa-

ter, nitrogen gas

2. Thermal evaporation of 150nm of aluminum

3. Spin Shipley 1813 45s@4000rpm

4. Pre-expose bake 1hr@90C

5. Expose 10s@16.4mWcm2

6. Develop in CD-30 approximately 45s

7. Aluminum etch in 8H3PO4 : 1HNO3 : 1H2O for approximately 3min

8. Remove resist with acetone, followed by IPA, DI, and N2 dry

9. Spin SU-8 100 10s@500rpm → 30s@3000rpm

10. Pre-expose bake 65o/15min → 95o/30min → Room Temperature

11. Expose 20s@16.4mWcm2

12. Post-expose bake 65o/1min → 95o/5min → Room Temperature

129

13. Develop in SU-8 Developer for 15min

14. Seal SU-8 channel using PDMS sheet and aquarium sealant

130

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