3/21/2013
OLD
DOMINION
UNIVERSITY
NANOFLUIDIC MOLECULAR SENSING
MAE 435 Mid-term Report | Laura Benson, Chris Hughes, and Zac Milne
i|Page
Contents
Section
Page Number
1. List of Figures
iii
2. Abstract
vi
3. Introduction
1
4. Nano Fluidics Background
1
a. Electrical Double Layer
1
b. Electroosmosis
3
c. Electrophoresis
4
5. Poisson-Boltzmann vs. Poisson-Nernst-Planck
5
6. Molecular Sensing (past and future)
6
7. Numerical analyses and software methodology
8
8. Bare SiO2 channel model, without stern layer assumption
9
9. The gated, charge regulated nanochannel
12
10. The gated, charge fixed nanochannel
17
11. Fabrication of the gated nanopore
20
a. Techniques
20
b. Walkthrough of the fabrication process
23
12. Dimensionless pore model design
27
13. Conclusion
30
14. Appendix
31
a.
MATLAB code
b. Figures
35
c. GANTT Chart
47
15. References
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31
48
List of Figures
1. Schematic of an EDL formed adjacent to a planar surface (Page 2)
2. Schematic of EOF in a slit channel bearing a uniform negative surface charge (Page 4)
3. Schematic of electrophoretic motion of a negatively charged particle (Page 5)
4. A Simple schematic of a coulter counter (Page 7)
5. 2-D representation of the gated nano-channel (Page 12)
6. COMSOL representation of 2 region model (Page 17)
7. ALD schematic (Page 22)
8. FIB conceptual representation (Page 22)
9. RF Sputtering conceptualization (Page 23)
10. Fabrication Process (Page 26)
11. Schematic Representation of a nanofluidic Coulter counter with an attached NFET (Page
35)
12. Velocity profile for various gate voltages; Circles represent approximate model (Page 36)
13. Velocity for various Stern Capacitances; Circles represent model without Stern
consideration (Page 36)
14. Velocity for various Stern Capacitances with a high salt concentration (Page 37)
15. Zeta Potential for various cases of Stern Capacitance; Medium Salt concentration (Page 37)
16. Experimental Data vs Model Prediction; Varied Salt (Page 38)
17. Experimental Data for Various Stern Capacitances; Varied Gate Voltage (Page 38)
18. Electrical Potential Distribution in COMSOL based Stern Layer model (Page 39)
19. Velocity Distribution in the Stern Layer COMSOL model (Page 39)
20. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various applied gate potential V g , CKCl  1 mM , and pH  8 (Page
40)
21. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various CKCl concentrations. Vg  5 V and pH  8 (Page 40)
22. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various solution pH. Vg  5 V and CKCl  1 mM (Page 41)
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23. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various softness degrees of the PE brush layer  at Vg  0 V
(Page 41)
24. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various softness degrees of the PE brush layer  at V g  5 V (Page
42)
25. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various thickness of the PE brush layer Rm at Vg  0 V (Page 42)
26. Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric
field Ez  20 kV/m for various thickness of the PE brush layer Rm at V g  5 V (Page 43)
27. A plot of zeta potential   
x 0
as a function of the applied gate voltage for various bulk
salt concentrations. The lower the CKCl and the higher the gate voltage the greater the
zeta potential. (Page 44)
28. A plot of zeta potential   
x 0
as a function of the applied gate voltage for various
densities of the PE chain grafted to the surface of the nanofluidic field effect transistor,
 PE . With the lower two values of  PE , the zeta potential changes signs. (Page 44)
29. A plot of zeta potential   
x 0
as a function of the applied gate voltage for various PE
layer thickness. The thicker the PE layer the lower the gate voltage at which the zeta
potential changes sign. (Page 45)
30. A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various applied gate potential V g . Interestingly, the flow
goes in the opposite direction for V g =20 V. (Page 45)
31. A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various densities of the PE chain grafted to the surface of
the nanofluidic field effect transistor,  PE . The higher the  PE the greater the EOF
velocity. (Page 46)
32. A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various degrees of softness of the PE layer, m . As m
decreases, so does the EOF velocity. (Page 46)
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33. Gantt Chart (Page 47)
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Abstract
The next generation of micro and nano-fluidic devices will require a deeper understanding of
fluid dynamics under the influence of electric fields. Currently, there is a very strong interest in
using the unique effects of nano-scale physical phenomena to make nano-pore based molecular
sensing devices. The work that this report summarizes consists of a study of nano-pores for
the purpose of DNA sequencing using several different models which take into account the
unique dynamics of poly-electrolyte and stern layers. In order to simulate the fluid dynamics,
mathematical models are developed and their accuracy verified by comparing numerical results
to published experimental data. Because the feasibility and accuracy of the mathematical
models are verified by experimental data, the study moves on to use the Computer Aided
Engineering package, COMSOL, to model a nano-pore under each unique condition: charge
regulated polyelectrolyte, charge fixed polyelectrolyte, and bare stern layer. Comparisons of
each model's molecular sensitivity, molecule velocity, and other important parameters are
drawn. Due to the cost of nanofabrication, a cursory investigation of the methods and cost
associated with manufacturing this technology is explored in the style of a literature review.
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Introduction
In 2009, the National Institute of Health (NIH) issued a challenge for researchers to
devise a way to sequence DNA for less than $1000 [1]. Making the equipment for less than
$1000 puts the ability to quickly and cheaply sequence DNA into every doctor’s office. This
ability is critical for doctor’s to take full advantage of genomic based medicine. This will allow
for doctors and pharmaceutical companies to personalize medicine for individuals and prevent
ineffective treatments from being pursued. From this challenge, nanopore based DNA sensors
have emerged as the technology most likely to meet this goal.
Our project makes no grand claims to have solved completely solved this problem. The
goal of the project is to completely and thoroughly study the underlying fluid mechanics that
makes this device possible, and to push the development of this technology one step further.
To do this, our first task has been to work with current researchers to develop three new
Nanofluidic Field Effect Transistor (NFET) models. These models are unique in that they each
increase the academic understanding of nanofluidic dynamics in nanochannels.
Next, our project plans to show that we can utilize the effects of NFETs to solve the
leading practical problems associated with Nanopore based DNA sequencing. Our plan is to
also further increase the academic understanding of this technology by proposing and
demonstrating a novel solution to one of these problems.
Nano Fluidics Background
Electrical Double Layer
Definition:
In micro and nanofluidics it is important to understand the electrical double layer (EDL).
This EDL plays an important role in understanding the potential and velocity profiles of flow
within a nanochannel. The EDL is formed on surfaces that come into contact with an ionic
aqueous solution. The surface could be a charged colloidal particle or a hard surface. With a
hard surface, a charged surface layer is created due to adsorption and ionization. When the
surface is immersed, counter-ions are attracted to this layer in order to neutralize it. Conversely,
co-ions are repelled. This forms two parallel layers of ions. The first layer is called the stern
layer, which is comprised of immobile ions. The ions are immobile due to a strong electrostatic
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force. The second layer is called the diffuse layer, which is comprised of mobile ions. These two
layers are called the electrical double layer. The ions in the EDL are mostly counter-ions[2, 3].
Governing Equations:
The Poisson Boltzmann equation, as given by [2]can be used to describe the electric
potential created from the charge within the diffuse layer.
∇2 𝑧𝐹𝜙
1
𝑧𝐹𝜙
=
sinh⁡(
)
𝑅𝑇
𝜆𝐷
𝑅𝑇
where 𝑧 = |𝑧𝑖 | where 𝑧𝑖 is the valence of the ith ionic species, 𝐹 is the Faraday constant, 𝜙 is
the electric potential within the fluid, 𝑅 is the universal gas constant, 𝑇 is the absolute
temperature of the electrolyte solution, and 𝜆𝐷 = √∑2
𝜀𝑜 𝜀𝑓 𝑅𝑇
𝑖=1 𝐹
2𝑧2𝐶
𝑖 𝑖𝑜
is the Debye length, which
characterizes the thickness of the EDL, where 𝜀𝑜 is the absolute permittivity of a vacuum, 𝜀𝑓 is
the relative permittivity of the fluid, and 𝐶𝑖𝑜 is the bulk concentration of the ith species.
The Debye length characterizes the thickness of the EDL.
The Poisson Boltzmann equation can be further manipulated to describe different
physical and chemical situations.
Figure 1: Schematic of an EDL formed adjacent to a planar surface as seen from [2]
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Electroosmosis
Definition:
Electroosmosis, or electroosmotic flow (EOF), is observed, as stated in [2], when an
external electric field is applied parallel to a stationary charged surface. The flow is induced by
the viscous electrolyte fluid being dragged by the counter-ions moving from the EDL on the
charged surface toward the oppositely charge electrode.
Governing Equations:
Electroosmosis is governed by the Navier-Stokes and the continuity equations as stated
by [2].
𝜕𝒖
𝜌 ( + 𝒖 ⋅ ∇𝒖) = −∇𝑝 + 𝜇∇2 𝒖 − 𝜀𝑜 𝜀𝑓 ∇2 𝜙𝑬
𝜕𝑡
∇⋅𝒖=0
where 𝜌 is the fluid density, 𝒖 is the fluid velocity, 𝑝 is the pressure, 𝜇 is the fluid dynamic
viscosity, and 𝑬 is the externally applied electric field. Also the term −𝜀𝑜 𝜀𝑓 ∇2 𝜙𝑬 is equal the
electrokinetic force acting on the fluid.
Further simplifications, as given in [2], such as there is no pressure gradient and the externally
applied electric field is much smaller than the one induced by the surface charge and the EOF is
fully developed, can lead to the following equation
𝜇
𝑑2 𝑢
𝑑2 𝜙
=
𝜀
𝜀
𝐸
𝑜 𝑓
𝑑𝑦 2
𝑑𝑦 2 𝑥
where 𝑢 is the fluid velocity in the x-direction and 𝐸𝑥 is the externally applied electric field in the
x-direction.
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Figure 2: Schematic of EOF in a slit channel bearing a uniform negative surface charge as seen
from [2]
Electrophoresis
Definition:
Electrophoresis is the movement, due to an externally applied electric field, of a charged
surface relative to a stationary fluid.
Governing Equations:
The equations describing electrophoresis, as given in [2], are the particle’s
electrophoretic velocity, steady fluid motion, electric potential, and ionic transport.
𝑼𝑝 = 𝜂𝑬
𝑛
2
−∇𝑝 + 𝜇∇ 𝒖 − 𝜀𝑜 𝜀𝑓 ∇𝜙 ∑ 𝐹𝑧𝑖 𝑐𝑖 = 0
𝑖−1
∇⋅𝒖=0
𝑛
−𝜀𝑜 𝜀𝑓 ∇2 𝜙 = ∑ 𝐹𝑧𝑖 𝑐𝑖
𝑖−1
∇ ⋅ (−𝐷𝑖 ∇ci − 𝑧𝑖
𝐷𝑖
𝐹𝑐 ∇𝜙 + 𝒖𝑐𝑖 ) = 0
𝑅𝑇 𝑖
where 𝑼𝑝 is the electrophoretic velocity of the particle, 𝜂 is the electrophoretic mobility of the
particle, 𝑐𝑖 is the molar concentration of the ith ionic species, and 𝐷𝑖 is the diffusivity of the ith
ionic species.
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Figure 3: Schematic of electrophoretic motion of a negatively charged particle as seen from [2]
Poisson Boltzmann vs. Poisson-Nernst-Planck
Poisson Boltzmann (PB)
The PB equation can be used to describe the EDL and electrokinetic flow. This equation
is only valid when the EDL is very thin, there are no overlapping EDL’s and when it is not
disturbed by the externally applied electric field[2].
The Boltzmann distribution is:
𝑐𝑖 = 𝐶0 exp⁡(−𝑧𝑖
𝐹𝜓
)
𝑅𝑇
where 𝑐𝑖 is the molar concentration of the ith ionic species, 𝐶0 is the bulk concentration of the
electrolyte solution, 𝑧𝑖 is the valence of the ith ionic species, and 𝜓 is the electric potential
arising from the surface charge.
And the Poisson equation is:
𝑛
2
−𝜀0 𝜀𝑓 ∇ 𝜓 = ∑ 𝐹𝑧𝑖 𝑐𝑖
𝑖=1
And the combined PB equation is:
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∇2
𝜀𝑜 𝜀𝑓 𝑅𝑇
where 𝜆𝐷 = √
2𝐹 2 𝐶0
𝐹𝜓
1
𝐹𝜓
= 2 sinh⁡( )
𝑅𝑇 𝜆𝐷
𝑅𝑇
is the Debye length, which characterizes the thickness of the EDL.
Because of the Boltzmann assumptions (that there is no overlapping EDL’s and that the
electrochemical potential is constant everywhere) the PB is only accurate in channels where the
EDL thickness is much smaller than the channel width. In order to accurately describe the flow
and potential distribution within a nanochannel, where the EDL’s are likely to overlap, the more
computationally expensive Poisson-Nernst-Planck equation is used [2, 4].
Poisson-Nernst-Planck (PNP)
When EDL’s overlap, it is necessary to use the PNP equations to analyze the
electrokinetic flow and ionic transport in a nanochannel. The Poisson equation is used to
describe the electrostatics and the Nernst-Planck equation for the ionic transport. The equations
as given in [2] are:
−𝜀0 𝜀𝑓 ∇2 𝜙 = 𝐹(𝑐1 𝑧1 + 𝑐2 𝑧2 )
𝜕𝑐𝑖
+ ∇𝑵𝑖 = 0
𝜕𝑡
where 𝐹 is the Faraday constant, 𝜙 is the electric potential within the fluid, 𝑅 is the universal
gas constant, 𝑇 is the absolute temperature of the electrolyte solution, 𝜀𝑜 is the absolute
permittivity of a vacuum, 𝜀𝑓 is the relative permittivity of the fluid, 𝑐1 and 𝑐2 are the molar
concentrations of cations and anions, 𝑧1 and 𝑧2 are the valences of cations and anions, 𝑵𝑖 =
𝐷
−𝐷𝑖 ∇ci − 𝑧𝑖 𝑅𝑇𝑖 𝐹𝑐𝑖 ∇𝜙 + 𝒖𝑐𝑖 is the ionic flux density of the ith ionic species where 𝐷𝑖 is the
diffusivity of the ith ionic species [2, 4].
Molecular Sensing (Past and Future)
Molecular sensing has been an important part of biomedical engineering for many years
now. One of the first devices ever used for a microscopic analysis is called a Coulter Counter.
First devised in the late 1940’s, this device detects the change in electrical resistance as a cell
passes through a micro-channel [5]. The user can then count the electrical resistance changes;
this number will then be proportional to the number of cells in the downstream (trans) side of
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the sample. This principle became known as the Coulter Principle [5], and has been utilized in
many different devices since the 1940’s.
Figure 4: A Simple schematic of a coulter counter[5]
In the mid 90’s, the first nanopore-based coulter counter was devised to [6] determine
the nucleotide sequence. Since its initial inception, the technology has evolved and changed in
significant ways. The original sensing devices were composed of 𝜶-hemolysin protein based
pores [6]. Now, due to the ever increasing abilities of nanotechnology manufacturers, nanopores built into solid-state devices are possible. This has become preferable, since it has
reduced the length of the pore considerably. The minimum length possible is preferred, since
smaller lengths can more easily discriminate between nucleotide bases in the DNA particle [7].
Despite the considerable development of this technology, two major technical hurdles
still exist. Due to the size of the electric field required for current discrimination and the
concentration of the field inside of the pore, the DNA particle translocates through the pore at
too great a speed [2]. In addition, the capture rate of the pore is too low for reasonable rates
of analysis. This means that too few DNA particles approach and pass through the pore within
a given amount of time. The capture rate is important for the design of a microfluidic system
able to analyze a DNA system within a reasonable amount of time.
Nanofluidic Field Effect Transistors have already been theoretically shown to significantly
slow the translocation of DNA through the pore [8]. This is possible because the hydrodynamic
forces against the particle nearly match the electrostatic force on the particle. To achieve these
hydrodynamic forces, the transistor gate must be on the surface of the pore. This gate
electrode adjusts the surface charge within the pore to create an electroosmotic flow in the
opposite direction of DNA translocation (Figure11).
Again using NFETs, we have proposed a method to increase the capture rate of the
pore. We propose to pattern another gate electrode on the upstream side of the membrane.
This new electrode will allow for us to control the electroosmotic flow perpendicular to the axis
7|Page
of the pore (Figure 11). We theorize that this will allow us to coax the DNA particles towards
the pore entrance, where the electrostatic forces will take the DNA through the pore [2].
In addition to using NFETs, DNA functionalized membranes have also been shown to
help retard DNA translocation speed [9]. Using charge regulated nucleotides; a very small layer
of polyelectrolyte can be attached to a hydrophilic dielectric such as SiO2. This ‘soft layer’ has
been shown to help increase the capture rate by relying on an effect called ion concentration
polarization. This effect creates a cloud of co-ions at the entrance to a charged pore. This
cloud of co-ions attracts DNA particles toward the pore [10].
Numerical Analyses and software methodology
Most problems in nanofluidics cannot be solved with analytical solutions. Because of the
size scale, many forces that are neglected at the macro-scale become important on the nanoscale. Thus, solving for a variable such as velocity also requires a solution of the electrical
potential distribution. To perform these calculations you must employ several different
methods, depending on the situation.
The simplest method, utilized for the Stern Layer model, is a zero-finding technique. For
our problem, must utilize the MATLAB (Mathworks, Natick, MA) program and the function fsolve
to compute a solution. The function fsolve uses three different algorithms to find the root of a
system of nonlinear equations [11]. The algorithms used are 'Trust region dogleg', 'Trust
region reflective', and 'Levenberg-marquardt'. The trust-region techniques are more robust
than Newton's method, since they can handle situations in which the Jacobian of a function is
singular [11]. To contrast, the simpler fzero function utilizes a combination of bisection
methods and interpolation and can only solve for the root of one equation, not a system of
equations.
Another method used in the project's numerical analysis utilizes the functions
bvp4c/bvp5c. These functions can solve multiple differential equations put into state-space
format. The only difference between bvp4c and bvp5c is their method of handling error
tolerances [12]. In general, bvp5c is best when small error tolerances are required. For the
purpose of this project, bvp4c is used in developing our Solid State model without the stern
layer assumption and bvp5c is used in our poly-electrolyte models.
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The most robust technique for handling nanofluidic numerical problems is COMSOL
(Burlington, MA) multiphysics. This finite element software allows for the simulation of
problems with coupled governing equations. The software solves numerical problems by
iteratively solving each problem averaged across each element. Therefore, one is able to create
models that are spatially representative of the actual structure.
To use COMSOL, one must first create a geometric representation of our problem. Once
complete, you then choose the governing equations in vector form and their corresponding
boundary conditions for a given problem. With this, you then input the constants and variables
into each equation's window so that the problem is completely defined. In developing the
mesh, you can create specific parameters for each region or boundary so that the automatic
mesh generator can appropriately mesh your problem. In order to minimize errors, you must
choose a particularly fine mesh for boundaries near where the potential or velocity changes
significantly.
Bare SiO2 Channel Model, Without Stern Layer Assumption
The first model that was analyzed for use in a NFET is the Bare SiO2 channel. In order
to learn the general effects of NFETs in a nanochannel, you must simplify the first model
significantly. One must consider the channel height to be infinite, and the fluid wall to be
planar. These assumptions allow us to utilize a Poisson-Boltzmann (PB) distribution of ions,
where the concentration of ions at any given point is proportional to the potential at that point.
The PB distribution of ions becomes invalid when the EDL of two or more opposite surfaces
becomes comparable to their characteristic length [13].
In order to simplify the calculations, most investigations into NFET operation have
ignored the Stern Layer, the first layer in an EDL. In this layer, the ions within the channel
react with the SiO2 dielectric to form charged ions. These ions alter the potential distribution
across the layer, reducing it to a significantly lower value. In this model, 𝜓0 is the potential at
the wall and 𝜓𝑑 is the potential at the interface between the stern layer and diffuse layer (Zeta
potential). The relationship between these two values and the stern layer's capacitance, from
[14], is:
𝜎0
= 𝜓0 − 𝜓𝑑 ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(1)
𝐶𝑠
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Where 𝐶𝑠 is the stern layer capacitance and 𝜎0 is the surface charge density on the
nanochannel wall [15].
To develop a relationship for the surface charge density, one must first realize that the
dielectric SiO2 has a charge regulated nature, meaning that the surface charge density is
strongly dependant on pH and concentration [15]. Therefore, the dielectric surface undergoes
the following protonation/deprotonation reactions.
𝑆𝑖𝑂𝐻2+ ↔ 𝑆𝑖𝑂𝐻 + 𝐻 + ⁡⁡𝑎𝑛𝑑⁡⁡𝑆𝑖𝑂𝐻 ↔ 𝑆𝑖𝑂− + 𝐻 + ⁡⁡⁡⁡⁡⁡⁡
Let 𝐾𝐴 and 𝐾𝐵 be the equilibrium constants, therefore:
𝐾𝐴 =
Γ𝑆𝑖𝑂− [𝐻 + ]
Γ𝑆𝑖𝑂𝐻 [𝐻 + ]
⁡⁡⁡⁡⁡𝑎𝑛𝑑⁡⁡𝐾𝐵 =
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(2)
Γ𝑆𝑖𝑂𝐻
Γ𝑆𝑖𝑂𝐻2+
Where Γ is the surface site density of each respective ion and [𝐻 + ] is the concentration of 𝐻 +
ions at the surface interface. The total number of molecules at the dielectric surface is a
constant and can be expressed with the equation,
𝑁𝑇𝑜𝑡𝑎𝑙 = Γ𝑆𝑖𝑂− + Γ𝑆𝑖𝑂𝐻 + Γ𝑆𝑖𝑂𝐻2+ ⁡⁡⁡⁡⁡⁡⁡⁡⁡(3)
From [15] we can express the surface charge density as
𝜎0 = −𝐹(Γ𝑆𝑖𝑂− − Γ𝑆𝑖𝑂𝐻2+ )⁡⁡⁡⁡⁡⁡⁡(4)
Where F is Faraday's constant. Combining equations 2, 3, and 4 one can now simplify the
charge regulated surface charge density as
[𝐻 + ]
𝐾
𝐹𝑁𝑇𝑜𝑡𝑎𝑙 ( 𝐾 − 𝐴+ )
[𝐻
]
𝐵
𝜎0 =
⁡⁡⁡⁡⁡⁡(5)
+
[𝐻 ]
𝐾
(1 + 𝐴+ + 𝐾 )
[𝐻 ]
𝐵
[𝐻 + ] can be found using a Poisson-Boltzmann distribution
𝐹𝜓0
[𝐻 + ] = 10−𝑝𝐻 𝑒 − 𝑅𝑇
𝐾𝐴 = 10−𝑝𝐾𝑎 ⁡; 𝐾𝐵 = 10−𝑝𝐾𝑏
Since pKa, pKb, and 𝑁𝑇𝑜𝑡𝑎𝑙 are material dependent and therefore known for SiO2[14, 16-18]
and one can assume a constant temperature, 𝜎0 = 𝑓(𝑝𝐻, 𝜓0 ).
The conservation of energy equation at the surface is written as
𝜀0 𝜀𝑑
𝑑𝜙
𝑑𝜓
− 𝜀0 𝜀𝑓
= 𝜎0 ⁡⁡⁡⁡⁡⁡⁡⁡𝑎𝑡⁡𝑥 = 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(6)
𝑑𝑥
𝑑𝑥
Where 𝜙 is the potential within the dielectric layer and 𝜓 is the potential in the liquid layer;
𝜀𝑓 ⁡&⁡𝜀𝑑 are the relative permittivities of the fluid and dielectric respectively. The first term in
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equation 6 is easily found by integrating equation 7 with the boundary conditions 8 and 9 to get
equation 10.
𝑑2 𝜙
= 0⁡⁡⁡⁡(7)
𝑑𝑥 2
𝜙 = 𝜓0 ⁡⁡𝑎𝑡⁡𝑥 = 0− ⁡⁡⁡⁡⁡(8)
𝜙 = 𝑉𝑔 ⁡⁡⁡⁡𝑎𝑡⁡⁡𝑥 = −𝛿⁡⁡⁡⁡⁡⁡⁡(9)
𝜀0 𝜀𝑑
𝜓0 − 𝑉𝑔
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(10)
𝛿
To find an expression for the second term in equation 6, one must integrate equation 11
subject to boundary conditions 12 and 13. From [19], the liquid potential distribution is
𝑁
𝑧𝑖 𝐹𝜓
𝑑2 𝜓
𝜌𝑒
1
=−
=−
(∑ 𝐹𝑧𝑖 𝐶𝑖0 𝑒 − 𝑅𝑇 )⁡⁡⁡⁡⁡⁡⁡⁡(11)
2
𝑑𝑥
𝜀0 𝜀𝑓
𝜀0 𝜀𝑓
𝑖=1
Where 𝜌𝑒 is the net space charge density, 𝑧𝑖 is the ion valence charge, and 𝐶𝑖0 is the bulk
concentration of each ionic species.
𝜓 = 𝜓𝑑 ⁡⁡⁡𝑎𝑡⁡𝑥 = 0+ ⁡⁡⁡⁡⁡(12)
𝜓=
𝑑𝜓
= 0⁡⁡𝑎𝑠⁡𝑥 → ∞⁡⁡⁡⁡⁡⁡⁡⁡⁡(13)
𝑑𝑥
Therefore, the second term in equation 6 can be expressed as
𝑁
𝑧𝑖 𝐹𝜓𝑑
𝑑𝜓
𝜀0 𝜀𝑓
⁡⁡(𝑎𝑡⁡𝑥 = 0+ ) = 𝑠𝑖𝑔𝑛(𝜓𝑑 )√2𝜀0 𝜀𝑓 𝑅𝑇 ∑ 𝐶𝑖0 (𝑒 − 𝑅𝑇 − 1) ⁡⁡⁡⁡⁡⁡⁡(14)
𝑑𝑥
𝑖=1
Later analyses of this model will compare 3 different variations of this model, multi-ion,
approximation, and analytical. The multi-ion model will include the ions 𝐾 + , 𝐶𝑙 − , 𝐻 + , 𝑎𝑛𝑑⁡𝑂𝐻 − ,
the approximation model will only include the salt ions, and the analytic model will be an
expression derived in [15]. To find each ions concentration, one must satisfy conditions for
electroneutrality [20, 21].
𝑝𝐻 ≤ 7
𝐾 + = 𝐶10 = 𝐶𝐾𝐶𝑙
𝐶𝑙 − = 𝐶20 = 𝐶𝐾𝐶𝐿 + 𝐻 + − 𝑂𝐻 −
𝐻 + = 𝐶30 = 103−𝑝𝐻
𝑂𝐻 − = 𝐶40 = 10𝑝𝐻−11
𝑝𝐻 > 7
+
𝐾 = 𝐶10 = 𝐶𝐾𝐶𝑙 − 𝐻 + + 𝑂𝐻 −
𝐶𝑙 − = 𝐶20 = 𝐶𝐾𝐶𝐿
11 | P a g e
𝐻 + = 𝐶30 = 103−𝑝𝐻
𝑂𝐻 − = 𝐶40 = 10𝑝𝐻−11
One can now solve for the zeta potential of the surface. Equations 1, 5, 6, 10, and 14 can be
combined to create two nonlinear equations with two unknowns (𝜓0 ⁡&⁡𝜓𝑑 ). One can now use a
numerical solver such as fsolve to find the zeta potential.
Now, to find the velocity distribution across the cross section one must simultaneously
numerically solve for the potential and velocity distribution using the governing equations 11
and 15, subject to the boundary conditions 12, 13, 16, and 17. Boundary condition 12 requires
that we have already solved for the zeta potential.
𝑑2 𝑢
𝜌𝑒 𝐸𝑧
=−
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(15)
2
𝑑𝑥
𝜇
Where 𝐸𝑧 and 𝜇 are the electric field and dynamic viscosity respectively.
𝑢 = 0⁡⁡⁡⁡𝑎𝑡⁡𝑥 = 0+ ⁡⁡⁡⁡⁡(16)
𝑑𝑢
= 0⁡⁡⁡⁡⁡𝑎𝑠⁡𝑥 → ∞⁡⁡⁡⁡⁡⁡⁡(17)
𝑑𝑥
The numerical solver bvp4c can be used to simultaneously solve each of these equations for
their appropriate boundary conditions. The important results from this analysis to NFET design
is the zeta potential and velocity distribution.
The Gated, Charge-Regulated Nano Channel
Figure 5: 2-D representation of the gated nano-channel
12 | P a g e
Intro, Background, and progress
The second part of the project is to develop an analytical model for a gated (Vg) and
charge regulated (CR) nano pore of a scale so diminutive that this work becomes unique not
only because of the CR and Vg, but also because of the difficulty in manufacturing it (this is
treated in a separate section). To help the reader proceed it is necessary to point out that the
terms polyelectrolyte, soft layer, and charge regulated region are used interchangeably and
synonymously.
Charge regulation of the soft layer by salt concentration or pH gradients is a more
realistic scenario for certain fabrication processes than approximating the space charge density
of the region. This aspect of the project has been studied by Zac Milne in strong collaboration
with Laura Benson.
Additionally, the ability to control the voltage at the dielectric face lends one the ability
to regulate the translocation velocity of the DNA through the pore. This is poignantly important
because current technology does not have a handle on this, and subsequently, the status-quo
of DNA sensing is that much time is spent preparing many samples in the hopes that one DNA
strand of the many is read properly by the sensing mechanisms.
This process can be alleviated by controlling the speed of the DNA through the pore.
Doing so requires manipulating the effects of ion electro osmosis. This is where the gate
electrode becomes very useful.
Figure 5 shows a schematic of the potential distribution in the pore and adjacent ionic
fluid. This potential counters the primary ‘cis’ electrode voltage which is used to induce the
initial motion of the DNA (Figure 11). Next, is a discussion about the equations which govern
the potential distribution due to the bulk ionic fluid, gate electrode, and soft layer charge
density.
Governing Equations, description, derivation
The governing equations come from Poisson’s equation for space charge conservation
[3]. Each region has a different charge density term depending on its physical layout. Figure 5
is useful to reference during these derivations so it is included again below.
13 | P a g e
The Dielectric layer has zero charge density and thus the second derivative of the potential
𝑑2 𝛹
𝑑𝑥 2
is zero. The Soft and bulk regions have the bulk space charge density (19) plus one more for
the soft layer (20), which is defined as 𝜌𝑓𝑖𝑥 .
Governing Equations
𝑑2 𝛹
𝑑𝑥 2
𝑑2 𝛹
𝑑𝑥 2
=
𝑑2 𝛹
𝑑𝑥 2
2𝑧𝑒𝐶𝑏𝑢𝑙𝑘 𝐴
𝑧𝑒𝛹
sinh ( ) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑑
𝜀𝑟 𝜀𝑜
𝑘𝑇
=
(18)
= 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡0 < 𝑥 < 𝑑
2𝑧𝑒𝐶𝑏𝑢𝑙𝑘 𝐴
𝑧𝑒𝛹
sinh ( ) −
𝜀𝑟 𝜀𝑜
𝑘𝑇
𝜌𝑓𝑖𝑥
𝜀𝑟 𝜀𝑜
< 𝑥 < 𝑚 (19)
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑚 < 𝑥 (20)
The boundary conditions take into account continuity between the interfaces of different
regions of both the potential and the electrical field. There is also the condition that the
potential and electrical field far away from the gate electrode are zero.
Boundary Conditions:
𝛹(0) = 𝑉𝑔
𝛹(𝑚− ) = 𝛹(𝑚+ )
𝜖𝑜 𝜀𝑟
𝑑𝛹
𝑑𝛹
= 𝜖𝑜 𝜀𝑑
|
|
𝑑𝑥 𝑥=𝑚+
𝑑𝑥 𝑥=𝑚−
𝛹(𝑑− ) = 𝛹(𝑑+ )
𝜖𝑜 𝜀𝑟
𝑑𝛹
𝑑𝛹
= 𝜖𝑜 𝜀𝑑
|
|
−
𝑑𝑥 𝑥=𝑑
𝑑𝑥 𝑥=𝑑+
𝛹(𝑥) = 0⁡𝑎𝑠⁡𝑥−→ ⁡∞
𝑑𝛹
−→ 0⁡𝑎𝑠⁡𝑥−→ ⁡∞
𝑑𝑥
14 | P a g e
Derivation of Soft Layer Charge Density as a Function of pH; Finding 𝝆𝒇𝒊𝒙
The charge density of the polyelectrolyte region is a function of the acid and base
association/ dissociation rates. Its derivation is shown below.
∴ 𝜌𝑓𝑖𝑥 = 𝐹([𝐵𝐻2+ ] − [𝐴− ])
𝐴𝐻 ↔ ⁡ 𝐴− + 𝐻 + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡, 𝐾𝐴 ⁡
𝐵𝐻2+ ↔ ⁡𝐵𝐻 + 𝐻 + ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡, 𝐾𝐵 ⁡⁡
𝐾𝐴 =
[𝐴− ][𝐻 + ]
[𝐴𝐻]
 [𝐴− ] =
𝐾𝐴 [𝐴𝐻]
[𝐻 + ]
𝑁𝐴 = [𝐴𝐻] + [𝐴− ] = [𝐴𝐻] +
𝐾𝐵 =
𝐾𝐴 [𝐴𝐻]
[𝐻 + ]
[𝐵𝐻][𝐻 + ]
[𝐵𝐻2+ ]
[𝐵𝐻2+ ] =
𝐾𝐵
𝐾
= [𝐴𝐻](1 + [𝐻𝐴+ ])
𝑁𝐵 = [𝐵𝐻2+ ] + [𝐵𝐻];⁡⁡⁡⁡⁡[𝐵𝐻] = ⁡
[𝐴𝐻] =
[𝐵𝐻][𝐻 + ]
𝐾𝐴 𝑁𝐴
[𝐻 + ]+𝐾𝐴
[𝐵𝐻2+ ] =
𝑁𝐵
𝑁𝐵
[𝐻 + ]
1+
𝐾𝐵
[𝐻+ ]
1+
𝐾𝐵
[𝐻 + ]
𝐾𝐵
𝑁 [𝐻 + ]
= 𝐾𝐵+𝐻+
𝐵
𝑵𝑩 [𝑯+ ]
𝑲𝑨 𝑵𝑨
∴ 𝝆𝒇𝒊𝒙 = 𝑭(
− +
)
+
𝑲𝑩 + [𝑯 ] [𝑯 ] + 𝑲𝑨
******************************
Derivation of Governing Equations
The derivation of the governing equations is included so that the reader may gain
insight into the physical laws behind the potential distribution in the Nano pore.
𝑑2 𝛹
𝜌𝑒𝑙 (𝑥)
=−
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑥 > 𝑚
2
𝑑𝑥
𝜀𝑟 𝜀𝑜
𝑑2 𝛹
𝜌𝑒𝑙 (𝑥) + 𝑍𝑒𝐶𝐴
=
−
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡0 < 𝑥 < 𝑚
𝑑𝑥 2
𝜀𝑟 𝜀𝑜
#⁡𝑠𝑝𝑒𝑐𝑖𝑒𝑠
𝜌𝑒𝑙 = 𝑍𝑒𝐶𝐴 = 𝑍𝑒𝐴 ∑𝑖=1
in the polyelectrolyte region.
𝑛𝑖
Following Boltzmann’s law for number density of the electrolyte ions
𝑛𝑖 = 𝑛𝑖∞ exp⁡(−
𝑧𝑖 𝑒𝛹(𝑥)
)
𝑘𝑇
Therefore
#⁡𝑠𝑝𝑒𝑐𝑖𝑒𝑠
𝜌𝑒𝑙 = 𝑍𝑒𝐶𝐴 = 𝑒𝐴
∑
𝑖=1
15 | P a g e
𝑧𝑖 𝑛𝑖∞ exp⁡(−
𝑧𝑖 𝑒𝛹(𝑥)
)
𝑘𝑇
Method of implementation in MATLAB:
The original equations are:
𝑑2 𝛹
= 0⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡0 < 𝑥 < 𝑑
𝑑𝑥 2
𝜌𝑓𝑖𝑥
𝑑2 𝛹 2𝑧𝑒𝐶𝑏𝑢𝑙𝑘 𝐴
𝑧𝑒𝛹
=
sinh (
)−
⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡0 < 𝑥 < 𝑚
2
𝑑𝑥
𝜀𝑟 𝜀𝑜
𝑘𝑇
𝜀𝑟 𝜀𝑜
𝑑2 𝛹 2𝑧𝑒𝐶𝑏𝑢𝑙𝑘 𝐴
𝑧𝑒𝛹
=
sinh (
) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝑚 < 𝑥
2
𝑑𝑥
𝜀𝑟 𝜀𝑜
𝑘𝑇
For a multi-ion model, all of the concentrations must be taken into account. They can
be simply added using a summation. The Formula below is the multi-ion equivalent Poisson
equation for the charge-regulated soft layer, and is the one used in MATLAB and COMSOL.
Recall that 𝜌𝑓𝑖𝑥 is the charge density in the polyelectrolyte region, derived above.
d 2
1

2
dx
 f 0
N
 Fz C
i 1
i
i0
exp(
 fix
zi F
)
RT
 f 0
The built in ODE solver “bvp5c” is implemented in MATLB to solve the governing
equations and boundary conditions for the potential distribution in the nano pore [22].
Progress:
Significant progress has been made in developing the model and implementing it in
MATLAB and COMSOL. We’ve obtained some interesting results and have been able to
compare between Laura and Zac’s to see the effects of considering the charge regulation of the
soft layer. However, there have been some snags that need to be unraveled. This will be
discussed next.
Limitations on the code and COMSOL model
Although the MATLAB code is promising, there is a large disparity in results with
implementing a two region model (in COMSOL) versus a three region model (in MATLAB). This
is because, to go from three regions to two we’ve derived the surface charge density at the
meeting of the dielectric face and the soft layer using the linearity of the potential in the
dielectric. This boundary condition (derived by Dr. Qian) is −𝜀𝑜 𝜀𝑑
16 | P a g e
(𝑉−𝑉𝑔 )
𝑑
which, as stated, is the
surface charge density. Using this in COMSOL we get drastically different results as the three
region model in MATLAB (and even three region in COMSOL). This is an issue that should be
resolved in for our final presentation.
COMSOL Model, explanation
Soft Layer (region 1)
Bulk ionic fluid (region 2)
The base two-region model (already meshed) used in COMSOL is shown in figure 2 a.
The mesh was made very dense at the interface of region one and two, where the soft layer
ends and the bulk layer begins. It has a maximum element size of .2e-9.
All of the important figures (for comparison) are obtained using this model and varying
specific parameters. The Charge regulated and fixed charge layer comparisons will be of great
interest to the academic world of nano fluidics, as will proving that the gate potential regulation
assists in counteracting the EOF through the channel due to the cis electrode.
The Gated, charge fixed Nanochannel
Governing Equations:
17 | P a g e
Defining region I as the dielectric channel wall, region II as the polyelectrolyte layer, and region
III as the bulk fluid, the following equations be derived from the Laplace and Poisson Boltzmann
equations and can be used to describe the electric potential in the three regions for this
particular model.
d 2
0
dx 2
In region I:
In region II:
d 2
1

2
dx
 f 0
In region III:
N
 Fz C
i
i 1
d 2V
1

2
dx
 f 0
exp(
i0
N
 Fz C
i 1
i
i0
 fix
zi F
)
RT
 f 0
exp(
zi FV
)
RT
Where 𝜙, 𝜓, and 𝑉 are the electric potentials inside the regions I, II, and III repectively, 𝜀𝑓 is
the relative permittivity of the electrolyte solution, 𝜀0 is the absolute permittivity of a vacuum, 𝐹
is the Faraday constant, 𝑅 is the universal gas constant, 𝑇 is the absolute temperature, 𝐶𝑖0 and
𝑧𝑖 are the bulk concentration and the valence of the ith ionic species respectively, and 𝜌𝑓𝑖𝑥 =
𝑒𝑍𝜎𝑃𝐸
)𝜒
𝑅𝑚
(
where 𝑒 is the elementary electric charge,⁡𝑍 is the valence of the dissociable groups
per polyelectrolyte (PE) chain, 𝜎𝑃𝐸 is the density of the PE chain grafted to the surface of the
nanofluidics field effect transistor, 𝑅𝑚 is the PE layer thickness, and 𝜒 is the dissociated degree
of functional groups in the PE layer.
The boundary conditions for the above equations are:
  Vg at x  
   at x  0
 0 f
d
d
  0 d
  w  0 at x  0
dx
dx
  V at 𝑥 = 𝑅𝑚
d dV

at 𝑥 = 𝑅𝑚
dx
dx
V  0 at x  
18 | P a g e
dV
 0 at x  
dx
and
where 𝑥 is measured from the polyelectrolyte layer with 𝛿 being the thickness of the silicon
dioxide dielectric layer, 𝜀𝑑 is the relative permittivity of the dielectric layer, and 𝑉𝑔 is the voltage
of the gate electrode.
The governing equations, assuming the simplifications of developed flow and the absence of a
pressure gradient in the z-direction, for the EOF velocity in the z-direction are given below.
E
d 2u
 z
In region II:
2
dx

N
 Fz C
i
i 1
E
d 2u
 z
In region III:
2
dx

i0
exp(
N
 Fz C
i 1
i
i0
zi F  u
)
RT

exp(
zi FV
)
RT
where 𝜇 is the dynamic viscosity of the electrolyte solution, 𝛾 is the hydrodynamic frictional
coefficient of the PE layer, and 𝐸𝑧 is the externally imposed electric field in the z-direction.
The boundary conditions for the above equations are:
𝑢1 = 0 at 𝑥 = 0
𝑢1 = 𝑢2 at 𝑥 = 𝑅𝑚
𝑑𝑢1
𝑑𝑥
=
𝑑𝑢2
𝑑𝑥
at 𝑥 = 𝑅𝑚
and
𝑑𝑢2
𝑑𝑥
= 0 at 𝑥 ⟶ ∞
All the above equations are as given in [3]. This model is very similar to the chargeregulated model. The only difference in the equations is the definition and calculation of 𝜌𝑓𝑖𝑥 .
This difference becomes apparent in the comparison of the model simulations. This model is
considered a two-ion model where 𝐶10 is the bulk concentration of 𝐾 + , 𝐶20 is the bulk
concentration of 𝐶𝑙 − , and 𝐶𝑘𝐶𝑙 is the background concentration of the 𝐾𝐶𝑙 electrolyte. Due to
electroneutrality, the following condition is applied to the model.
19 | P a g e
𝐶𝑘𝐶𝑙 = 𝐶10 = 𝐶20
The values used for each of the parameters are given in Table 2 in Appendix A. After
this model was created in COMSOL, several different plots were created and plotted as the
parameters were changed and different parameter sweeps were run. These graphs are shown
in Appendix A.
Fabrication of the Gated Nanopore
Nanopores can be created using several well tested techniques. Gated nanopores have
been in development to take advantage of several unique characteristics of the electrode aspect
of the nanopore [23]. For one, the applied voltage at the gate (Vg) can be used to control the
translocation velocity of the DNA. An added benefit is the direct sensing of molecules traveling
through the nanopore via a change in conductance while the molecule is traveling through (this
latter characteristic is not investigated this report).
This section reviews the techniques suggested for fabrication of our nano-channel.
Additionally, a detailed walkthrough of the fabrication process is presented.
Acronyms
FIB-Focused Ion Beam
RF (Radio Frequency)
ALD (atomic Layer Deposition)
RIE (Reactive Ion Etching)
XPS (X-Ray Photo-Electron Spectroscopy)
AFM (Atomic Force Microscopy)
ALD
The heart of the operation of creating the many layers which together comprise our
channel lies in the Atomic Layer Deposition technique (ALD). This process has been in
development since its inception in the 1960s by Russian scientists V. B. Aleskovskii and S.I.
20 | P a g e
Kot’sov at Leningrad Technological Institute and was developed further and patented by
Finnish scientist Tuomo Suntola [24].
ALD is the deposition of uniform layers of atomic or molecular material onto a target
substrate. Its process typically involves the use of two or more “precursors”. The first
precursor attaches to the substrate and the second prepares the first for further deposition.
Several materials have been studied and developed for myriad uses. Nevertheless, in this
study, the interest is in depositing a dielectric material.
The dielectric precursors have been chosen to be H2O and SiCl4 due in part to the
large amount of information available on the creation of ultrathin dielectric layers using
these materials, and also to the high dielectric constant of the dielectric formation : SiO2.
The process takes place in vacuum to ensure minimal contamination from remnant
gases; at elevated temperatures to increase the surface energy of the SiO2 / OH terminated
substrate, leading to more ordered and quicker bonding of the first precursor SiCl4. One
The SiCl4 bonds with the surface 2OH substrate, creating the HfCl2O2 and releasing 2HCl.
The next phase is to pulse the second precursor: H2O. The reaction of 2H2O with the
protruding SiCl2 forms 2HCl and Si(OH)2. The 2HCl is removed with inert gas and the
surface is terminated once again with OH, allowing for the process to be reiterated.
The realization is of a process limited thickness for each layer, since the reactions
are limited to stoichiometric constraints. Thus a uniform layer ~1-3 Ǻ forms during each
step, and repeated steps increase layers to the desired thickness. XPS and Ellipsometry,
combined with AFM are useful tools for determining the surface characteristics e.g.
roughness, thickness, composition (presence of contaminants) [25].
21 | P a g e
Figure 7: ALD schematic
FIB
The Focused ion beam method has been utilized towards two very different goals,
surface machining or sculpting, and milling. The latter method is of interest to this report since
the milling method is desired to cut a small-radius hole to realize the channel.
The Focused ion beam milling uses ionized Gallium+ ions at high currents to in effect
barricade and remove surface molecules. Continued exposure to the beam will mill a channel.
The beam can be focused through a series of filters and magnets to create a beam width of 20
nm practically [26].
Because of this limitation on beam width using the FIB, and the constraint that the
channel be no more than 10 nm in radius, the final deposition of the polyelectrolyte will serve
the dual duty not only of reducing the overall energy in the channel, but of thickening the walls
of it to meet the small-radius criteria.
Figure 8: FIB conceptual representation
RF Sputtering
22 | P a g e
RF sputtering is a process by which electrodes made from Pt or Cu are deposited
onto a substrate. This process uses an ionized noble gas in a vacuum chamber to bombard
a “target” material, in this case, Cu. The gas is ionized using a agnetron producing high
power RF signals. The Cu is anodized to direct the positive ions of the Noble gas, while the
substrate is cathodized to attract the Cu being discharged from the target. The thickness of
the resultant film is a function of target-substrate spacing, exposure time, and ion energy.
Some practical concerns with using this method are the temperature of the substrate as
exposure time increases. Cooling of the substrate-film system will induce non-equilibrium
stress at the interface due to differences in thermal coefficients.
A mask can be used over the substrate to ensure the desired geometry of film is
deposited. In our case, a mask is utilized to shape the electrode [27]. This technique
applied at the nano-scale has much room for study and promises control of electrode
fabrication at ever decreasing scales.
Figure 9: RF Sputtering conceptualization
Walkthrough of Fabrication Process
1: Growth and isolation of crystalline SiC wafer. Deposition of SiO2 and activation of OH
surface using ALD of H2O vapor
The first step in the fabrication process is to grow the SiC wafer. Since this technique
has been used in industry for many years it would suffice to order the crystalline structure from
a microelectronics manufacturer. It is important to have a wafer with no defects, that is, the
23 | P a g e
cubic structure must be continuous. A polytype of SiC, (β)3C-SiC, which has a zincblende FCC
structure, is the necessary material for this particular fabrication process.
Preparing the SiC for ALD is a two-step process. First, it must be cut to the desired
thickness (~20 nm) and mounted. A chemical etching process can be used for the coarse
dimensioning, finished off with laser etching to create as smooth a surface as possible.
The second step is to create OH groups. This can be done with an ALD of SiO2 and a
wash with H2O, at certain temperature and pressure conditions. The OH groups will be the first
link to the SiCl4 precursor.
2: Marker Creation.
It is important to have a physical reference on the substrate for alignment of the several
masks and beams that will be used later on in the pore’s manufacturing. Therefore, we will use
either chemical etching or FIB to introduce markers surrounding the area of interest on our
surface. When this and the previous steps are complete, we obtain (representatively) the
image shown in the first step of the schematic shown in Fig 10.
3: Mask Electrode
The next step is to mask out where the electrode is to go. A mask that can withstand
many hundreds of Celsius degrees is necessary because the RF sputtering process is highly
energetic and will heat up any material that comes into contact with the plasma.
4: RF Sputtering of Cu or Pt electrode.
After the mask is applied the electrode can be created using the RF Sputtering method.
It may be necessary to perform this task in several iterations and check the thickness of the
electrode using an AFM. The process can be terminated when the desired thickness is
obtained.
5: Wash using inert gas
At this, and after every successive step, it is necessary to wash the product with an inert
gas to remove any contaminants or free molecules that may later cause problems with the
depositions, i.e. dislocations or undesirable reactions. The mask is also removed in this step.
24 | P a g e
6: Dielectric Deposition
To avoid creating a “hill” of dielectric because of the height gradients due to the
electrode a mask must be deposited on the electrode. this can be done by using another mask
to make sure it deposits only on the electrode. Now that the electrode is protected, the
dielectric can be deposited using ALD. One useful aspect of ALD is the large amount of control
and understanding there is of the height of each layer. Thus, the height corresponds perfectly
linearly to the number of iterations of ALD that the manufacturer makes. The process is
terminated after the dielectric is as high or slightly higher than the electrode.
After this, the mask and dielectric covering the electrode are washed away using a
solution that etches the mask only. Now more dielectric is deposited to create a geometry that
both covers and straddles the electrode.
7: Next electrode
Using RF Sputtering again, another electrode is deposited on top of the dielectric. This
electrode will be used to help capture the DNA molecules by creating an electric field oriented
to push the molecules towards the pore entrance.
9,10: Reactive etching of both sides of wafer
To prepare for iso-etching of both sides of the wafer-electrode-dielectric assembly, a
mask is placed on both side, the bottom mask width depending on the thickness of the SiC
portion of the assembly, since the iso-etch will need to proceed deeper than on the top.
There are several fluids used to wet-etch an isometric shape, each of which must be selected
specifically for the material being etched. Because three materials are being etched, three
etching solutions must be used. Here is a list of wet-etching solutions: HF, H3PO4, H2SO4,
KOH, H2O2, HCl.
11: FIB pore creation using wide beam
To create the thoroughfare for DNA translocation an FIB will be used to penetrate the
etched portion of the assembly. FIBs have a practical beam width of 20 nm, but a slightly
wider beam may be used because of our ability to narrow the pore later on with the
polyelectrolyte deposition. ALD of the dielectric is the next step to coat the inside of the new
pore.
25 | P a g e
It is important to make all of the previously deposited films wide enough to withstand
the pressure created from the bombarding ions. A thorough wash is necessary after this step.
12: PEG Coating
The two precursors that will be used to ALD the polyelectrolyte layer are poly acrylic
acid (PAA) and polyacrylamide (PAAm). These have been chosen because of their ability to be
charge regulated and because of the volume of literature available on their application.
The ALD of the Polyelectrolyte marks the last step in the gated nano-pore device.
Figure 10: Fabrication Process
Practical concerns
There are many precautions that need to be taken during this process, many of which
are beyond our group’s current education. Researchers in nanotechnology spend their whole
lives working to minimize the occurrence of breakage, contamination, or failure of process.
Primarily, it is necessary to be trained in clean room procedure to obviate the possibility of
26 | P a g e
contamination. The facilities should be state-of-the-art with zero vibration and high vacuum for
the ALD, RF sputtering, and various surface imaging processes.
Additional concerns are of coefficient of expansion mismatches. Each layer will react
differently to heat and surface energy (strain) at the interface of two materials may develop
dislocations due to the mismatch.
It should be noted that the process described above should be considered a rough draft.
It could take several years of research to nail down the details enough to realize the entire
pore.
Dimensionless Pore Model Design
In order to determine the feasibility of applying NFETs to nanopore molecular sensors,
one must numerically solve our governing equations for the velocity distribution in the pore.
Since the pore size is comparable to the size of the electrical double layer, one must use a
Nernst-Plank distribution of ions instead of the Poisson Boltzmann distribution. In addition, to
maximize accuracy, one must solve the Nernst-Plank equation for 4 different ions (multi-ion).
This makes for a total of 6 partial differential equations that must be solved simultaneously.
Due to the computational load, you must utilize a supercomputer to find this solution.
To find the velocity distribution, one must use an axially symmetric, two-dimensional,
incompressible Navier-Stokes equation system. To find the velocity profile, the body forces
must be used in the NS equation system. In nanofluidic systems, this force can be extremely
small. To avoid numerical errors due to calculating small numbers and large numbers
together[13], you must use a dimensionless model.
For the model geometry, there are two reservoirs separated by a nanopore. The length
and radius of each reservoir is 200nm; the chosen length of the nanopore varies between
different analyses between 10nm and 30nm. For the purpose of molecular sensing, the smaller
the nanopore length the better the sensing ability [28]. However, this is eventually limited by
the structural integrity of the membrane and the abilities of fabrication processes. Due to the
limits of FIB, you must choose a pore radius of 10nm. The thickness of the soft layer varies
between different analyses between 1nm and 6nm. Fortunately, the fabrication process of soft
layers is much more versatile, and can allow for very small effective radii of the pore.
27 | P a g e
The 6 governing equations for the model are, Nernst-Plank without electroneutrality,
Electrostatics, and Incompressible Navier-Stokes.
Nernst-Plank:
∇ ∙ (−𝐷𝑖 ∇𝑐𝑖 − 𝑧𝑖 𝑢𝑚𝑖 𝐹𝑐𝑖 ∇𝜓) = −𝑼 ∙ ∇𝑐𝑖 ⁡⁡⁡⁡⁡⁡⁡(𝑖 = 1: 4)⁡⁡⁡⁡⁡(21)
Where 𝐷𝑖 is the diffusion coefficient, 𝑢𝑚𝑖 is the mobility, and 𝑼 is the velocity vector.
Electrostatics:
−∇ ∙ ε0 εr ∇ψ = ρe ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(22)
Incompressible Navier-Stokes:
𝑭 + ∇ ∙ [−𝑝𝑰 + 𝜇(∇𝑼 + (∇𝑼)𝑇 )] = 0⁡⁡;⁡∇ ∙ 𝑼 = 0⁡⁡⁡⁡⁡⁡⁡⁡(23)
Where the density is zero to remove the inertial term [2], 𝑝 is the pressure, and 𝑭 is the force
vector. The force vector is the body forces on the fluid. In nanofluidics, this relationship is
given by [2]:
𝑁
𝑁
𝑭 = −𝐹 ∑ 𝑧𝑖 𝑐𝑖 ∇ψ⁡(𝐵𝑢𝑙𝑘) = −𝐹 ∑ 𝑧𝑖 𝑐𝑖 ∇ψ − γ ∙ 𝐔⁡(PE⁡layer)
𝑖=1
𝑖=1
Where 𝛾 is the friction factor.
𝑁
𝑁
ρe = 𝐹 ∑ 𝑧𝑖 𝑐𝑖 ⁡⁡(𝐵𝑢𝑙𝑘⁡𝐹𝑙𝑢𝑖𝑑) = 𝐹 ∑ 𝑧𝑖 𝑐𝑖 + 𝜌𝑝𝑒 ⁡(𝑃𝐸⁡𝐿𝑎𝑦𝑒𝑟)⁡
𝑖=1
𝑖=1
In order to normalize these equations, one defines five different scales [2] as:
𝐶𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛⁡𝑆𝑐𝑎𝑙𝑒:⁡⁡⁡𝐶𝐾𝐶𝑙
𝐿𝑒𝑛𝑔𝑡ℎ⁡𝑆𝑐𝑎𝑙𝑒: 𝑅𝑡 = 1⁡𝑛𝑚
𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙⁡𝑆𝑐𝑎𝑙𝑒:
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦⁡𝑆𝑐𝑎𝑙𝑒:⁡𝑈0 =
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒⁡𝑆𝑐𝑎𝑙𝑒:
𝑅𝑇
𝐹
𝜀0 𝜀𝑓 𝑅 2 𝑇 2
𝜇𝑅𝑡 𝐹 2
𝜇𝑈0
𝑅𝑡
Using these scales, one can normalize (variables denoted by *) each constant and variable used
in our system of equations. One can now re-write our governing equations in dimensionless
form as:
∇∗ ∙ (𝑼∗ 𝑐𝑖∗ − 𝐷𝑖∗ ∇∗ 𝑐𝑖∗ − 𝑧𝑖 𝐷𝑖∗ 𝑐𝑖∗ ∇∗ ⁡𝜓 ∗ ) = 0⁡⁡(𝑖 = 1: 4)⁡(24)
28 | P a g e
𝑁
(𝜅𝑅𝑡 )2
−∇ 𝜓 =
∑ 𝑧𝑖 𝑐𝑖∗ ⁡⁡(𝐵𝑢𝑙𝑘) ⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡
2
∗2
∗
𝑖=1
𝑁
(𝜅𝑅𝑡 )2
∗
=
∑ 𝑧𝑖 𝑐𝑖∗ + 𝜌𝑝𝑒
⁡⁡(𝑃𝐸⁡𝐿𝑎𝑦𝑒𝑟)⁡(25)
2
𝑖=1
𝑁
−∇∗ 𝑝∗ + ∇∗2 𝑼∗ =
(𝜅𝑅𝑡 )2
(∑ 𝑧𝑖 𝑐𝑖∗ ) ∇∗ 𝜓 ∗ ⁡⁡(𝐵𝑢𝑙𝑘)
2
𝑖=1
𝑁
(𝜅𝑅𝑡 )2
=
(∑ 𝑧𝑖 𝑐𝑖∗ ) ∇∗ 𝜓 ∗ + γ∗ ∙ 𝐔 ∗ ⁡(𝑃𝐸⁡𝐿𝑎𝑦𝑒𝑟)⁡(26)
2
𝑖=1
∇∗ ∙ 𝑼∗ = 0⁡⁡⁡(27)
These equations require three different sets of boundary conditions. For the Nernst-Plank
system of equations, one can consider each ion concentration to satisfy the electroneutrality
condition at the ends of each reservoir. In addition, the membrane and walls must satisfy the
insulation condition where equation 24 is symmetric normal to the wall.
For the Navier-Stokes boundary condition, you can assume there is no pressure gradient
across the pore. In addition, one can assume the membrane and walls satisfy the no-slip
boundary condition. This is an accurate assumption, since the dielectric (SiO2) is hydrophilic.
For hydrophobic materials, the no-slip boundary condition isn't valid in nanofluidics [13].
In our electrostatics boundary conditions are where one applies external forces. For the
purpose of molecular sensing, there must be an applied electric field. This is realized at the
ends of each reservoir by setting one boundary to a voltage and the other to a ground. To
simulate the response with a NFET, one defines two different surface charges, the surface
charge at the channel walls due to the channel electrode and the surface charge on the
upstream (cis) side of the membrane. At each respective boundary, you can set the boundary
condition to be that surface charge. All other surfaces will be symmetric normal to the surface.
In order to maximize our post-processing ability with the solution, you must also define
several different projection coupling variables in COMSOL. Once integrated in the COMSOL
solver, these variables will give a cross-sectional average of each variable along the axis of the
system. With this data, you can see how the velocity varies as the DNA approaches the pore,
and is inside of the pore. To generate the mesh, you must utilize the free mesh generator with
a few constraints. Our first constraint ensures that there are a minimum of 10 elements across
the thickness of the soft layer due to the dynamic response to ionic concentrations within the
29 | P a g e
soft layer. The rest of our constraints ensure that there is a fine enough mesh for the
projection coupling variables to effectively capture the physical response in the pore.
To verify our model, you must use two approaches. You must replicate similar
conditions to those published in [2] in order to qualitatively check our results. In addition, you
must use the conservation of current and equation 28 to verify the results. Due to the
conservation of current, the calculated value of current at each reservoir end should be nearly
the same. For this model, the difference between the currents are <0.01%.
𝑁
𝐼 = ∫ (∑ 𝑧𝑖 𝑁𝑖∗ ) ∙ 𝐧𝑑𝑆 ∗ ⁡⁡⁡⁡⁡⁡⁡(28)
∗
𝑖=1
Where 𝑁 is the total ionic flux and S is the surface.
Conclusion
So far, our report has discussed the required background knowledge to design
nanofluidic systems. Before the project begins to tackle the problem of molecular sensing
devices, a great deal of time has been spent learning the operation of Nanofluidic Field Effect
Transistors and distinguishing between different fabrication methods. Using the NFET
technology, a novel solution has been proposed to the two greatest hurdles facing nanofluidic
molecular sensing devices, translocation speed and capture rate. In addition, our project has
proposed a method to fabricate this device using existing fabrication technology.
To validate the design, a dimensionless model has been built in COMSOL to simulate the
velocity profile in the pore. Using this model, the rest of our project will be spent simulating a
variety of conditions as time will allow. Our final report will present the results of this analysis,
as well as the results of our analyses of each NFET model. Using these analyses, the report will
also present a series of design recommendations to fabricate a working molecular sensing
device.
30 | P a g e
Appendices
Appendix A: MATLAB Code
Stern Layer Code
function y=multiionpotential
global XX
solinit=bvpinit(XX,[0 0 0 0]);
sol=bvp4c(@equation,@equationbc,solinit);
y = deval(sol,XX);
function dydx=equation(x,y)
global eps F FRT C10 C20 C30 C40 Ez dynamic
elecdens=F*(C10*exp(-y(1)*FRT)+C20*exp(-y(1)*FRT)-C30*exp(y(1)*FRT)C40*exp(y(1)*FRT));
dydx=[y(2);-elecdens/(eps);y(4);-elecdens*Ez/dynamic];
function res=equationbc(ya,yb)
global zeta
res=[ya(1)-zeta;yb(2);ya(3);yb(4)];
function y=multiionzeta(V)
global Ntotal KSiO KSiOH2 C10 C20 C30 C40
global F eps R T FRT Cs epsd detad Vg
%
phi0=V(1);
phid=V(2);
sum1=C20*(exp(-FRT*phid)-1)+C10*(exp(-FRT*phid)-1)+...
C30*(exp(FRT*phid)-1)+C40*(exp(FRT*phid)-1);
H0=C10*exp(-FRT*phi0);
sigma0=F*Ntotal*((H0/KSiOH2 - KSiO/H0)/(1+KSiO/H0+ H0/KSiOH2));
sigmad=-sign(phid)*sqrt(2*eps*R*T*sum1);
y(1)=sigma0-Cs*(phi0-phid);
y(2)=sigma0+sigmad-epsd*(phi0-Vg)/detad;
function y=approxzeta(V)
global Ntotal KSiO KSiOH2 C0 C10
global F eps R T FRT Cs epsd detad Vg
%
%
phi0=V(1);
phid=V(2);
sum1=C0*(exp(FRT*phid)-1)+C0*(exp(-FRT*phid)-1);
31 | P a g e
sigmad=-sign(phid)*sqrt(2*eps*R*T*sum1);
H0=C10*exp(-FRT*phi0);
sigma0=F*Ntotal*((H0/KSiOH2 - KSiO/H0)/(1+KSiO/H0+ H0/KSiOH2));
y(1)=sigma0-Cs*(phi0-phid);
y(2)=sigma0+sigmad-epsd*(phi0-Vg)/detad;
function Vd=LowPotential()
global Ntotal pKSiO pKSiOH2 pH C0
global F eps R T FRT Cs epsd detad Vg
kaba=sqrt(2*F^2*C0/(eps*R*T));
sum1=(Cs+eps*kaba)/Cs;
II=2*10^(pKSiOH2-2*pH);
Phi=10^(-pKSiO)-10^(pKSiOH2-2*pH);
Omega=10^(-pKSiO)+10^(-pH)+10^(pKSiOH2-2*pH);
tri=Phi*(10^(-pH)+II)/Omega^2;
Vd=(epsd*Vg-detad*sum1*F*Ntotal*Phi/Omega)/(epsd+eps*kaba*detad+...
detad*sum1*F^2*Ntotal/(R*T)*(II/Omega+tri))*...
(1+FRT*F*Ntotal/Cs*(II/Omega+tri))+F*Ntotal/Cs*Phi/Omega;
Explanation of Code for the charge regulated nanochannel
Here is a schematic of the structure of the code’s execution in MATLAB:
Main Code:
“ThetaCalc2Region.m”
Gather ODEs, Boundary Conditions, initial
Guesses for Potential and execute (solve) using
BVP5c. Create Graphs.
create ODEs
32 | P a g e
Ex (V/m),
Theta Initial Guesses
“ThetaInit”
Theta ODE code:
“odefun1”
Output: Theta(V),
Graphs
Theta Boundary Conditions
“bc1” Create boundary function
Create initial guesses
Soft Layer Code:
function [sol1]=ThetaCalc2Region(m,d)
%ThetaCalculator.m is the primary function which uses MATLABs built in code
%'bvp5c' to solve a 2nd order differential equation for the potential
%distribution versus distance from the surface layer of a soft particle
%immersed in an electrolyte solution.
global z1 z2 z3 z4 F R T Na Nb Ka Kb Vg pH ef eo ed Co d
N=Na+Nb;
options=bvpset('AbsTol',1e-11);
%Creating the domain
xinit1 = [linspace(0,m),m,linspace(m+m*.05,10e-8)];
%Setting up the initial conditions using bvpinit
solinit1=bvpinit(xinit1,@ThetaInit);
%Utilizing bvp5c to solve for the potential
sol1=bvp5c(@odefun1,@bc1,solinit1,options);
%Plotting to check for anomalies
plot(sol1.x,sol1.y(1,:))
end
%Setting up the ODEs for BVP5c
function dydx=odefun1(x,y,region)
global z1 z2 z3 z4 F R T Na Nb Ka Kb Vg pH ef eo ed Co
e=1.6e-19;
MATLAB and COMSOL variables used
FRT=F/(R*T);
Co (CKCl concentration)= varies;
C10=10^(-pH+3);C40=10^(-(14-pH)+3);
d (core layer thickness) = varies;
H=C10*exp(-z1*FRT*y(1));
m (polyelectrolyte layer thickness)=varies
rhofix=F*(Nb/(1+Kb/H)-Na/(1+H/Ka));
z1, z3 (electrolyte cation valence) = 1
if pH>=7
z2,z4 (electrolyte anion valence)=-1
C20=(Co-C10+C40);
F (Faraday Constant) = 96487
C30=Co;
R (Ideal Gas Constant) = 8.3145
else
C20=Co;
Na (acid concentration) = 600 Mol
C30=(Co+C10-C40);
Nb (base concentration) = 600 Mol
end
Ka (Acid rate) = (10^-3.5)*1000
rhoe=F*(z1*C10*exp(Kb(Base Rate)= (10^-8.5)*1000
z1*FRT*y(1))+z2*C20*exp(Vg (electrode potential) = Varies
z2*FRT*y(1))+z3*C30*exp(pH (electrolyte pH) = Varies
z3*FRT*y(1))+z4*C40*exp(-z4*FRT*y(1)));
ef (relative permittivity of KCl electrolyte) = 34
switch region
eo (Vacuum Permittivity) = 8.85 e-12
case 1
ed (Core permittivity ) = 3.9
dydx=[y(2) -(rhoe+rhofix)];
case 2
dydx=[y(2) -rhoe];
otherwise
33 | P a g e
error('regions not correct');
end
end
%Boundary Conditions for Potential ODEs
function res1=bc1(yl,yr)
global z Z F R T Na Nb Ka Kb Vg pH ef eo ed Co d
res1=[ yl(1,1)+d*yl(2,1)-Vg
yr(1,1)-yl(1,2)
yr(2,1)-yl(2,2)
yr(2,2)
];
Parameters Fixed Charge Model
end
𝛿
function yinit=ThetaInit(x,region)
global Vg
switch region
case 1
yinit=[1 -1];
case 2
yinit=[1 1];
otherwise
error('wrong');
end
end
34 | P a g e
𝑅𝑚
𝜀𝑓
𝜀0
𝜀𝑑
𝐹
𝑅
𝑇
𝑧1
𝑧2
𝑍
𝑒
𝜎𝑃𝐸
𝐶𝐾𝐶𝑙 = 𝐶10 = 𝐶20
𝑉𝑔
𝐸𝑧
𝜆
𝜒
𝜇
Value or Range
30 (nm)
3~15 (nm)
80
8.854× 10−12 (C/Vm)
3.9
96487 (C/mol)
8.31 (1/mol K)
300 (K)
1
-1
-1
1.602× 10−19
0.1× 1018 ∼0.6× 1018 (1/m2 )
1∼1000 (mM)
-20∼20 (V)
20 (kV/m)
0.5~10 × 10−9
0.5
0.001 (kg/sm)
Appendix B: Figures
Figure 11: Schematic Representation of a nanofluidic Coulter counter with an attached NFET [8]
35 | P a g e
Velocity vs X for pH = 4 and CKCl = 10 mM
400
300
200
U (m/s)
100
V = -3V
0
g
V = 0V
g
-100
V = 3V
g
V = 6V
-200
g
-300
-400
-500
0
2
4
6
8
10
X (nm)
12
14
16
18
20
Figure 12: Velocity profile for various gate voltages; Circles represent approximate model [29]
Velocity vs X for pH = 8 Vg = 5 V CKCl = 1 mM
2000
1800
1600
1400
U (m/s)
1200
1000
Cs = 0.3 F/(m2)
2
800
Cs = 1 F/(m )
2
Cs = 2 F/(m )
600
2
Cs = 2.9 F/(m )
400
200
0
0
5
10
15
X (nm)
20
25
30
Figure 13: Velocity for various Stern Capacitances; Circles represent model without Stern
consideration [29]
36 | P a g e
Velocity vs X for pH = 8 Vg = 5 V CKCl = 1 M
1200
1000
2
Cs = 0.3 F/(m )
2
Cs = 1 F/(m )
2
Cs = 2 F/(m )
800
2
U (m/s)
Cs = 2.9 F/(m )
600
400
200
0
0
0.5
1
1.5
2
X (nm)
2.5
3
3.5
4
Figure 14: Velocity for various Stern Capacitances with a high salt concentration [29]
Zeta vs Vg for pH = 10 CKCl = 100 mM
-40
2
Cs = 0.3 F/(m )
-60
2
Cs = 1 F/(m )
2
Cs = 2 F/(m )
 d (mV)
-80
2
Cs = 2.9 F/(m )
-100
-120
-140
-160
-180
0
5
10
Vg
15
20
Figure 15: Zeta Potential for various cases of Stern Capacitance; Medium Salt concentration
[29]
37 | P a g e
Zeta vs C0 for experimental data Gaudin and Fuerstenau
-60
 d (mV)
-80
-100
2
Cs = 0.3 (F/m )
2
= 1 (F/m)
2
= 2 (F/m)
2
= 2.9 (F/m)
-120
0
0.2
0.4
0.6
0.8
1
C0 (mM)
Figure 16: Experimental Data vs Model Prediction; Varied Salt [29]
Zeta vs Vg for experimental data Oh et al
100
2
Cs = 0.3 (F/m )
2
= 1 (F/m)
2
50
= 2 (F/m)
2
= 2.9 (F/m)
 d (mV)
0
-50
-100
-45
-30
-15
0
V (V)
15
30
45
g
Figure 17: Experimental Data for Various Stern Capacitances; Varied Gate Voltage [29]
38 | P a g e
Figure 18: Electrical Potential Distribution in COMSOL based Stern Layer model
Figure 19: Velocity Distribution in the Stern Layer COMSOL model
39 | P a g e
Figure 20
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various applied gate potential V g , CKCl  1 mM , and pH  8
V g (0, 5, 10, and 15 V)
Figure 21
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various CKCl concentrations. Vg  5 V and pH  8
CKCl (1, 10, 100, 1000 mM)
40 | P a g e
Figure 22
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various solution pH. Vg  5 V and CKCl  1 mM
pH (4, 6, 8, 10)
Figure 23
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various softness degrees of the PE brush layer  at Vg  0 V
 (0.5, 2, 5, 8 10 9 (m) )
41 | P a g e
Figure 24
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various softness degrees of the PE brush layer  at V g  5 V
 (0.5, 2, 5, 8 10 9 (m) )
EOF vs x for varying Rm (Fig 7c)
0.001
EOF(m/s)
0.0008
0.0006
3e-9 m
0.0004
6 e -9
9.00E-09
0.0002
1.50E-08
0
0
Figure 25
2E-08
4E-08
6E-08
8E-08 0.00000011.2E-07
x(m)
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various thickness of the PE brush layer Rm at Vg  0 V
Rm (3, 6, 9, 15 10 9 (m) )
42 | P a g e
EOF vs x for varying Rm; Vg=5v (Fig 7d)
0.001
0.0009
0.0008
EOF (m/s)
0.0007
0.0006
Rm=3e-9
0.0005
Rm=6e-9
0.0004
Rm=9e-9
0.0003
Rm=15e-9
0.0002
0.0001
0
0
Figure 26
2E-08
4E-08
6E-08
8E-08 0.0000001 1.2E-07
x(m)
Spatial variations of the electroosmotic flow (EOF) velocity driven by an applied electric field
Ez  20 kV/m for various thickness of the PE brush layer Rm at V g  5 V
Rm (3, 6, 9, 15 10 9 (m) )
43 | P a g e
V vs VG varying CKCL
0.15
0.1
0
-0.05
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
V (V)
0.05
-0.1
-0.15
-0.2
Vg (V)
CKCL=1 (mM)
CKCL=10 (mM)
CKCL=100 (mM)
CKCL=1000 (mM)
Figure 27: A plot of zeta potential   
x 0
as a function of the applied gate voltage for various
bulk salt concentrations. The lower the CKCl and the higher the gate voltage the greater the
zeta potential.
V vs Vg varying sigmaPE
0.15
0.1
0
-0.05
-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
16
18
20
V(V)
0.05
-0.1
-0.15
-0.2
Vg (V)
sigmaPE=0.1*10^18 (1/m^2)
sigmaPE=0.2*10^18 (1/m^2)
sigmaPE=0.4*10^18 (1/m^2)
sigmaPE=0.6*10^18 (1/m^2)
Figure 28: A plot of zeta potential   
x 0
as a function of the applied gate voltage for various
densities of the PE chain grafted to the surface of the nanofluidic field effect transistor,  PE .
With the lower two values of  PE , the zeta potential changes signs.
44 | P a g e
V vs Vg (varying Rm)
0.15
0.1
3nm
0
-0.05
-20
-17
-14
-11
-8
-5
-2
1
4
7
10
13
16
19
V (V)
0.05
5nm
10nm
-0.1
15nm
-0.15
-0.2
Figure 29: A plot of zeta potential   
Vg (V)
x 0
as a function of the applied gate voltage for various
PE layer thickness. The thicker the PE layer the lower the gate voltage at which the zeta
potential changes sign.
Figure 30: A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various applied gate potential V g . Interestingly, the flow goes in
the opposite direction for V g =20 V.
45 | P a g e
Figure 31: A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various densities of the PE chain grafted to the surface of the
nanofluidic field effect transistor,  PE . The higher the  PE the greater the EOF velocity.
Figure 32: A plot of the spatial variation of the electroosmotic flow velocity driven by an applied
electric field Ez  20 kV/m for various degrees of softness of the PE layer, m . As m
decreases, so does the EOF velocity.
46 | P a g e
Appendix C: Gantt Chart
Figure 32: Gantt Chart; We are currently about 67% complete, we have begun to fall
slightly behind due to time spent troubleshooting our dimensionless pore model
47 | P a g e
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
M. Kircher and J. Kelso, "High-Throughput DNA sequencing - concepts and limitations,"
BioEssays, vol. 32, pp. 524-536, 2010.
S. Qian and Y. Ai, Electrokinetic Particle Transport in Micro/Nanofluidics; Direct Numerical
Simulation Analysis. Boca Raton, FL 33487-2742: CRC Press, 2012.
H. Ohshima, "Theory of electrostatics and electrokinetics of soft particles," Science and
Technology of Advanced Materials, vol. 10, 2009.
Y. S. Choi and S. J. Kim, "Electrokinetic flow-induced currents in silica nanofluidic channels,"
Journal of Colloid Interface Science, vol. 333, pp. 672-678, 2009.
(2013). Coulter Counter.
J. J. Kasianowicz, E. Brandin, D. Branton, and D. W. Deamer, "Characterization of individual
polynucleotide molecules using a membrane channel," Proceedings of the National Academy of
Sciences of the United States of America, vol. 93, pp. 13770-13773, 1996.
J. O. Tengenfeldt, C. Prinz, H. Cao, R. L. Huang, R. H. Austin, S. Y. Chou, et al., "Micro- and
Nanofluidics for DNA analysis," Anal Bioanal Chem, vol. 378, pp. 1678-1692, 2004.
Y. Ai, J. Liu, B. Zhang, and S. Qian, "Field Effect Regulation of DNA Translocation through a
Nanopore," Analytical Chemistry, vol. 82, pp. 8217-8225, 2010.
L.-H. Yeh, M. Zhang, S. Qian, and J.-P. Hsu, "Regulating DNA translocation through functionalized
soft nanopores," Nanoscale, pp. 2685-2693, 2012.
L.-H. Yeh, M. Zhang, N. Hu, S. W. Joo, S. Qian, and J.-P. Hsu, "Electrokinetic Ion and Fluid
Transport in Nanopores Functionalized by Polyelectrolyte Brushes," Nanoscale, pp. 5169-5177,
2012.
fsolve. Available: http://www.mathworks.com/help/optim/ug/fsolve.html
bvp4c. Available: http://www.mathworks.com/help/matlab/ref/bvp4c.html
A. T. Conlisk, Essentials of Micro- and Nanofluidics; with Applications to the Biological and
Chemical Sciences. New York, NY: Cambridge University Press, 2013.
S. H. Behrens and D. G. Grier, "The Charge of Glass and Silica Surfaces," Journal of Chemical
Physics, pp. 6716-6721, 2001.
L.-H. Yeh, S. Zue, S. W. Joo, S. Qian, and J.-P. Hsu, "Field Effect Control of Surface Charge
Property and Electroosmotic Flow in Nanofluidics," Journal of Physical Chemistry C, vol. 116, pp.
4209-4216, 2012.
F. H. J. v. d. Heyden and D. S. C. Dekker, "Streaming currents in a single nanofluidic channel,"
Physical Review Letters, vol. 95, 2005.
M. B. Andersen, J. Frey, S. Pennathur, and H. Bruus, "Investigation of the solid/liquid interface of
coated silica nanochannels during capillary filling," Journal of Colloid Interface Science, vol. 353,
pp. 301-310, 2011.
M. Wang, Q. Kang, and E. Ben-Naim, "Modeling of electrokinetic transport in silica nanofluidic
channels," Analytica Chimica Acta, vol. 664, pp. 158-164, 2010.
N. Hu, Y. Ai, and S. Qian, "Field Effect Control of Electrokinetic Transport in Micro/Nanofluidics,"
Sensors and Actuators B: Chemical, vol. 161, pp. 1150-1167, 2011.
L.-H. Yeh, J. P. Hsu, S. Qian, and S. Tseng, "Counterion Condensation in pH-Regulated
Polyelectrolytes," Electrochemical Communications, vol. 19, pp. 97-100, 2012.
L.-H. Yeh, M. Zhang, N. Hu, S. W. Joo, S. Qian, and J. P. Hsu, "Controlling pH-Regulated
Bionanoparticles Translocation through Nanopores with Polyelectrolyte Brushes," Analytical
Chemistry, vol. 84, pp. 9615-9622, 2012.
48 | P a g e
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
L. F. Shampine, J. Kierzenka, and M. W. Reichelt, "Solving Boundary Value Problems for Ordinary
Differential Equations in MATLAB with bvp4c," 2000.
M. Taniguchi, M. Tsutsui, K. Yokota, and T. Kawai, "Fabrication of the gating nanopore device,"
Applied Physics Letters, vol. 95, p. 3, 2009.
T. Suntola, "International workshop on science and technology of thin films for the 21st
century," Thin Solid Films, pp. 84-89, 1992.
J. M. P. Alaboson, Q. H. Wang, J. D. Emery, A. L. Lipson, M. J. Bedzyk, J. W. Elam, et al., "Seeding
Atomic Layer Deposition of High-K Dielectrics on Epitaxial Graphene with Organic SelfAssembled Monolayers," American Chemical Society, vol. 5, pp. 5223-5232, 2011.
F. A. Lasagni and A. F. Lasagni, Fabrication and Characterization in the Micro-Nano Range, 2011.
S. Chandra, V. Bhatt, and R. Singh, "RF Sputtering: A viable tool for MEMS fabrication," Sadhana:
Academy Proceedings in Engineering Sciences, vol. 34, pp. 543-556, 2009.
S. Majd, E. C. Yusko, Y. N. Billeh, M. X. Macrae, J. Yang, and M. Mayer, "Applications of biological
pores in nanomedicine, sensing, and nanoelectronics," Current Opinion in Biotechnology, vol. 21,
pp. 439-476, 2010.
C. Hughes, L.-H. Yeh, and S. Qian, "Title," unpublished|.
49 | P a g e