Post on 06-Feb-2020
Numerical Analysis of Sessile Drop Flowfor Electrowetting-on-Dielectric devices
Mahmoud Kadoura
A thesis submitted in partial fulfilmentof the requirements for the degree of
BACHELOR OF APPLIED SCIENCE
Supervisor: A.N. Sinclair
Department of Mechanical and Industrial EngineeringUniversity of Toronto
March, 2008
Abstract
Microfluidics has been an area of active research for the past decade. This piece analyses the frictionless motion of sessile droplets in a specific design of an electrowetting-on-dielectric (EWOD) chip. The analysis is numerical and employed the commercial computational fluid dynamics package FLUENT. Focus areas included mesh and time step analysis as well as the effect of property change on fluid flow, with general correlation of some observations with the Young-Lippmann (Electrowetting) equation and the Navier-Stokes equations. The paper also includes a background study of the model-excluded phenomena currently only partially comprehended: contact angle saturation and contact angle hysteresis. Interpretations of the numerical simulations undertaken comprised, for droplet translation and coalescence, faster flow for lower viscosity and higher surface tension values. For droplet splitting, high dependencies on the droplet size, electrode-actuation symmetry, and most importantly the mesh type were observed. The elimination of the aforementioned two phenomena resulted in highly magnified velocities.
Keywords:
Microfluidics – Electrowetting – Contact angle saturation – Contact angle hysteresis
Acknowledgements
This paper would not have been possible to write without the help of Professor P. E.
Sullivan, my teaching assistant Mike Schertzer, and the support of friends and colleagues.
Thanks also to ANSYS and FLUENT for the provision of software licensing and updates.
Table of Contents
Page
List of Symbols……………………………………………………………………. iv
List of Figures…………………………………………………………………...... v
List of Tables……………………………………………………………………… vii
Chapter 1: Objectives and Methodology……………………………………….. 1
Chapter 2: Introduction and Motivation……………………………………….. 2
Chapter 3: Mathematical Modeling and Background………………………… 5
Section 3.1: Mathematical derivation of the electrowetting equation………...
Section 3.2: Notes on and variation from the Young-Lippmann equation……
Section 3.3: Contact angle saturation…………………………………………
Section 3.4: Contact angle hysteresis…………………………………………
5
7
9
13
Chapter 4: Computational Methodology and Model………………………….. 16
Section 4.1: Equations of fluid flow…………………………………………..
Section 4.2: Volume of Fluid method…………………………………………
Section 4.3: Microfluidic chip geometry……………………………………...
16
17
18
Chapter 5: Results and Discussion……………………………………………… 20
Section 5.1: Convergence study; Optimal mesh and time step……………….
Section 5.2: Droplet Translation……………………………………………… 5.2.1: Base Translation……………………………………………………………... 5.2.2: Effect of viscosity change……………………………………………………. 5.2.3: Effect of surface tension change……………………………………………...
20
25252627
Section 5.3: Droplet Mixing………………………………………………….. 5.3.1: Base mixing and effect of viscosity change…………………………………. 5.3.2: Effect of surface tension change……………………………………………...
Section 5.4: Droplet Splitting………………………………………………… 5.4.1: Base splitting………………………………………………………………… 5.4.2: Effect of contact angle symmetry and droplet size…………………………... 5.4.3: New mesh
Section 5.5: Summary and Conclusions………………………………………
292931
32323335
37
Chapter 6: Future Recommendations…………………………………………… 39
References…………………………………………………………………………..
Appendix A…………………………………………………………………………
Appendix B………………………………………………………………………….
I
III
IV
List of Symbols
Symbol Definition
γSG Solid-Gas interface surface tension
γLG Liquid-Gas interface surface tension
γSL Solid-Liquid interface surface tension
θ Contact Angle (subscript 'naught' for no voltage, ‘sat’ for
saturated, ‘A’ for advancing, and ‘R’ for receding)
capE Stored electrical energy of an infinite plate capacitor
ε0 Permittivity of free space or air
εr Permittivity of the insulator
d Dielectric thickness
ASL Solid-liquid contact (wetted) area
V (ch. 3) Applied potential difference
σ Capacitive charge density
V (ch. 4) Instantaneous velocity vector
p Instantaneous pressure gradient
R1, R2 Principal radii of curvature
µ Absolute viscosity (subscript ‘w’ for water)
List of Figures
Figure Caption
1 Sessile drop setup for electrowetting measurements
2 EWOD illustrations
3 Contact angle definition and surface energies at different interfaces
4 Electrowetting response of 0.1M KCl (pH=5.6) on 1.8μm thick Teflon 61% PDD (T2)
5 Illustration of contact line pinning and hysteresis
6 Effects of contact angle hysteresis in the EWOD device
7 Microfliudic Chip Geometry
8 Droplet mixing of two different mesh densities
9 Pressure Distribution (Pa) in y and x directions:Resultant droplet of mixing two 550-micron radius water droplets –many time steps –
10 Velocity Distribution (m/s) in y direction:Resultant droplet of mixing two 550-micron radius water droplets –many time steps –
11 Simple Translation of a 550micron-radius drop of water in air with contact angles of 117° (OFF, bottom) and 90° (ON, top)
12 Viscosity variation of a translating droplet
13 Surface tension variation of a translating droplet
14 Viscosity variation of merging droplets
15 Surface tension variation of merging droplets
16 Splitting of a 580micron-radius droplet with contact angles of 117° (OFF, middle), 90° (ON, bottom), and 89° (ON, top)
Figure Caption
17 Failed splitting of a 580micron-radius droplet with contact angles of117° (OFF, middle) and 90° (ON, bottom and top)
18 Tetrahedral mesh: Successful splitting of a 580micron-radius droplet with contact angles of 117° (OFF, middle), 90° (ON, bottom), and 89° (ON, top)
List of Tables
Table Title
1 Summary of interpretations of contact angle saturation
2 Some collected data on saturated contact angles
3 Summary of convergence study
4 Droplet split summary for a 100x20x20 hexahedral mesh
Chapter 1: Objectives and Methodology
The purpose of this thesis is to develop a comprehensive model for modeling droplet
behavior in a specific EWOD device. A computational fluid dynamics (CFD) software,
FLUENT, will be employed to numerically simulate sessile droplets' movement and
eventual mixing. The environment shall be in 3D and include as many parameters as
possible or needed. The simulations will include parameters and boundary conditions that
correlate with a PhD thesis on EWOD devices (Mike Schertzer). A lot of time will be
spent on arriving at a rudimentary, yet concise, model. With that a deeper look at
different parameters will lead to more insight of the droplet(s') behavior. The analysis
will include modifying initial and boundary conditions to assess the effect of a parameter
on the liquid motion.
Chapter 2: Introduction and Motivation
In the past few decades a trend is noticed of making devices smaller and smaller. Devices
such as Lab-On-A-Chip have been the focus of wide areas of research in digital
microfluidics and biology. These devices have a lot of advantages including the use of
tiny amounts of reagents for biological experiments, the repetitive nature for the study of
droplet behavior, process control of such droplets, development of microvalves [1] and
micropumps, and more [2].
The major difference between liquid behavior at the micro- and macroscale is the much
larger surface-to-volume ratio for the former, dictating the significance of capillary and
surface tension effects in predicting droplet motion [5]. Considering a discrete droplet on
a dielectric surface, electrowetting, or wetting as a result of applied electric field, is the
phenomenon observed when the droplet experiences a voltage difference and so a change
in the contact angle (angle between dielectric-liquid surface and the tangent to liquid-
ambient boundary)(Figure 1, details in next section). This change in contact angle is
believed to cause droplet motion in the direction of decreasing induced pressure gradient.
This is called Electrowetting-On-Dielectric (EWOD) (Figure 2, details in next section).
The voltage application causes an electric field to redistribute charges, resulting in a
hydrophobic surface to become hydrophilic [6]. However, the eventual motion of the
droplet is not yet physically well understood. The contact angle change may not be a
direct consequence of voltage application but is rather a result of another variable that
was in turn induced by voltage application, for instance.
Figure 1: Sessile drop setup for electrowetting measurements (Quinn et al. [4])
Figure 2: EWOD illustrations(top: Walker and Shapiro [3], bottom: Armani et. al [5])
ON
The limitations on droplet motion induced by electrowetting, and hence the challenges to
modeling of such a flow, include mainly contact angle saturation, three phase contact line
(TCL) pinning, and contact angle hysteresis [3]. Contact angle saturation is observed
when an extra applied voltage does not anymore induce a change in the contact angle.
TCL pinning, or sticking, is due to frictional forces at the surface (imagine a stationary
droplet on a mildly tilted surface). The difference in contact angles between the front and
rear ends of a drop is the result of such contact line pinning, and is known as contact
angle hysteresis.
The concept of 'electrowetting' began in 1993 by Berge, and was first applied by
Washizo in 1998 [10].
Chapter 3: Mathematical Modeling and Background
Mugele and Baret illustrate how different approaches result in a relation of contact angle
with applied voltage [2]. Theories included thermodynamic, energy minimization, and
electromechanical views. Initially the Bond number, measuring gravitational effects with
respect to surface tension, was shown to be less than unity and hence gravitational effects
were ignored [6] (Refer to Appendix A for details). Referring to figure 3, the following is
a simple derivation of the electrowetting equation [1]. The equation is strictly only
applicable to relatively low voltages where saturation effects are negligible:
Figure 3: Contact angle definition and surface energies at different interfaces. (A) No applied voltage and (B) with an applied voltage between the droplet and the substrate dielectric wall (Mohseni and
Dolatabadi [1])
Section 3.1: Mathematical Derivation of the Electrowetting Equation
A simple balance of surface forces in Figure 3 (B) above results in the famous Young relation:
cosSG LGSL (1)
where:γSG: Solid-Gas Interface surface tensionγLG: Liquid-Gas Interface surface tensionγSL: Solid-Liquid Interface surface tension (subscript 'naught' for no voltage)θ: Contact Angle (subscript 'naught' for no voltage)
The droplet-electrode setup is modeled as an infinite plate capacitor, and therefore the stored electrical energy is given by:
20
2VA
dE SL
rcap
(2)
where:(ε0 εr): dielectric constant of insulator (usually SiO2), or as shown the product of the permittivity of the ambient (ε0) and the relative permittivity of the insulator (εr)d: dielectric thicknessASL: solid-liquid contact areaV: applied potential difference.Note that the electric energy essentially modifies the surface energy and hence the contact angle.
The Lippmann relation states that the surface tension changes with voltage as follows:
VAddV
dSL
rSL 0 (3)
where σ is the charge density of the capacitive model. This is a simple differential equation with the solution as follows:
200 2
)( Vd
V rSLSL
(4)
Substituting in the Young relation yields the Young-Lippmann equation for electrowetting on dielectric:
200 2
cos)(cos Vd
VLG
r
(5)
The following shows a graph of the theoretical response with a chosen sample experimental response [4]:
Figure 4: Electrowetting response of 0.1M KCl (pH=5.6) on 1.8μm thick Teflon 61% PDD (T2). The solid line was plotted using the EWOD equation (Quinn et al.)
Section 3.2: Notes on and variations from the Young-Lippmann equation
A large change in contact angle for a given voltage can be achieved with increased
capacitance, and therefore electrodes must be coated with a dielectric layer of high
dielectric constant. A hydrophobic layer, typically Teflon®, is also coated on the
dielectric to allow for an initial un-wetted state (θ>90°).
At the TCL line, on the molecular level, the electric fields deviate from those causing
the simple force relation depicted in figure 3. The electric fields are in fact highly
divergent and non-uniform [7]. Hence at higher voltages, and even before saturation,
deviations from the theoretical line are observed (See last point).
Contact angle saturation occurs at a theoretically unpredictable point, after which the
equation would not hold anymore (contact angle remains constant for increasing
voltage).
It is argued that the physics behind the electrowetting equation is not accurate. It is
even suggested that the droplet displacement is independent of contact angle [8]. Also
suggested, a bit beyond the EWOD scope, is the fact that meniscus curvature change,
when a fluid column rises upward in a capillary tube, is not responsible for driving
the liquid upward [9]. This supports the theory in [8]. On the other hand, specifically
for EWOD, motion was proposed to be due to the electrostatic forces acting on the
droplet/surrounding interface (around the TCL). A pressure gradient arises from the
asymmetry, which induces motion [10]. Finally, it is widely believed that the
operation is by the local deformation of the liquid-gas interface which is caused by
'capacitively' charging the dielectric beneath the droplet. The motion then arises from
competing effects of 1) energy storage in the 'capacitor' (dielectric) and 2) surface
energy of the liquid-gas interface [3, 5].
Analysts have shown that anions of the droplet (OH-) create, by adsorption, an
electric double layer (dielectric + charges) that results in the deviation of
experimental records from the EWOD equation, even before saturation [4]. The
deviation was observed to be more pronounced for positive voltages than negative
ones. The authors also noticed that anion size had no effect on when the deviation
occurs, but the following dictated delayed (hence favored) deviation: increased molar
concentration, decreased pH, and decreased temperature. However, other
experimenters have shown that the deviation is independent of voltage polarity and/or
molar concentration [13, 15].
To summarize the dynamics of droplet motion on an electrode, one can start with low
voltage application. The result is that some electrical energy is stored in the dielectric,
and the electric fields hence redistribute themselves, especially around the TCL. To
satisfy conservation of energy, this electrical energy alters the surface (interfacial)
energies in a way that allows for a decrease in the contact angle. Consequently, and
particularly due to confinement of a droplet in EWOD devices (Figure 2), a pressure
gradient is generated between the front and rear ends of the droplet.
Intuitively, the droplet cannot move instantaneously upon voltage application. Friction
will dictate that there shall be a minimum voltage needed for droplet movement (refer to
hysteresis below). This is called the actuation voltage. Baviére et al. describe three phases
for motion [10]. First, only the front edge deforms (changes contact angle) but the drop
doesn’t move. Second, the rear edge starts to deform. Finally, the drop moves in steady
state, as opposed to the transient nature of phases one and two, at constant velocity. In
this phase the motion is due to front and rear contact angles being constant but different,
and this difference is what gives rise to the aforementioned pressure gradient.
The authors above also show how the actuation voltage is the same for droplets of
different viscosities, and is hence dependent on contact angles. However, intuitively, the
velocities reached at steady state would be higher for lower viscosities (again due to
hysteresis).
Section 3.3: Contact Angle Saturation
Contact angle saturation has been an area of active research for the past decade.
Depending on the background of experimenting scientists/engineers, many different
physical interpretations have been proposed. One suggestion was that air ionizes
profoundly at the three phase contact line (where electric fields diverge rapidly) and
hence at some point completely suppresses the decrease of the contact angle with
increased voltage [11]. Other unexplainable results were observed with [11]: at high
voltages and beyond saturation, pure water was unstable and satellite droplets emerged
(in agreement with wave behavior, for instance the distance between the emerged
droplets was analogous to wavelength). This segregation effect was suppressed by the
addition of salt to the water.
Another interpretation of saturation is in terms of the Maxwell Stress Tensor at the TCL
[12]. The argument is that, theoretically, the contact-angle reduction-resisting component
of the tensor (the outward normal one) increases with contact angle reduction. In other
words, as contact angle decreases (wetting) the work required to cause further wetting
increases. Eventually there will simply be inadequate work and the contact angle
saturates. An additional interpretation of the phenomenon is charge trapping (charges
bonding more closely to dielectric than droplet) [13, 16]. This lowers the electric field,
and the authors mathematically updated the electrowetting equation to include VT, the
voltage of trapped charge. A final electrical interpretation is that saturation is due to the
liquid being slightly resistive on a highly resistive dielectric [6]. The Lippmann
(electrowetting) equation is exact only for a perfectly conducting droplet. Knowing that,
the resistance of the droplet would increase with spreading (decreased contact angle),
until it is high enough to cause saturation. When the authors explain electrowetting, they
argue that at small contact angles the interfacial energies always 'beat' the electrical
energy, and the equation approximates well. However at high voltages, a force balance at
the TCL is insufficient, and the bulk fluid needs to be analyzed.
All the above interpretations were backed up by good or excellent experimental
correlations. It is also worth noting that higher AC frequencies delay contact angle
saturation [14] and speed up the travel by an order of magnitude [15].
Analysts Interpretation Focus Saturates by
1 Vallet et al. [11] Elec. field divergence @ TCL
Air ionization @ TCLSuppression of further surface
energy modification
2 Kang [12]Maxwell Stress Tensor
@ TCLComponent that resists contact angle decrease
Increase of component as
angle decreases
3Verheijen and
Prins [13] Charge TrappingFormation of double layer
of chargeOverly condensed
charge suppression
4 Shapiro et. al [6] Liquid finite resistanceTCL force balance
insufficientIncreasing
resistance with decreasing angle
Table 1: Summary of Interpretations of Contact Angle Saturation
Some authors fundamentally suggest otherwise. Mentioned above was the claim that
contact angle change and movement of the droplet are two completely independent
phenomena. Conclusively, contact angle saturation is a complicated phenomenon with
very many factors. For example, a question that arises from the last point in the previous
section is: How could saturation be dependent on molarity, pH, and other properties but
not the size of the anion in the solution? Table 2 below shows some collected values of
saturated contact angles for different parameters.
Nature Liquid Liquid size
Dielectric Sat. Contact Angle
Interpretation of Saturation
Ref
Experimental Water 1.6-1.8µL
1000Å SiO2 w/ 200Å Teflon
~ 80° N/A [15]
Theoretical Water Any 1µm SiO2 w/ 0.02µm Teflon
75° Liquid slight resistance
[6]
Experimental KCl or K2SO4
10µL 10µm Parylene coating
60° Charge Trapping
[13]
Experimental Electrolyte Solution N/A
(30, 15, 4)µm Parylene w/
100nm Teflon~ (60, 50, 65)°
**
Electric Double Layer (charge
trapping)
[16]
Experimental Decane 1µL 50µm Paraffin ~ 75° Charge Trapping
[17]
Experimental 10-2M Na2SO4
~ 3mm. dia.
50µm PTFE 27° Air ionization at TCL
[11]
Table 2: Some collected data on saturated contact angles(** Read from graph)
An examination of the above table can lead to some conclusions, when referring to
compatible data. I stress that the following observations are crude.
Noting that the first two rows describe the same dielectric, the theory in [6] was ‘proven’
by experiments in [15], at least to within 5°. This slight error suggests that the theory was
a bit conservative (underestimating θsat), which is a consequence of authors omitting
hysteresis. Turning to the next two rows, note that, for the sake of argument, KCl, K2SO4,
and an electrolyte solution behave chemically similarly. Also, both dielectrics are
polymers of thicknesses of the same order. Therefore, the values for saturated contact
angle can arguably be said to roughly ‘agree’ with a value of about 60°. Next, consider
the fifth row. The value of saturated contact angle agrees with the first two. However, we
have no grounds on which to base this similarity, since Decane is a quite different
substance from water, and a dielectric of very different material and much higher
thickness was employed. The last row of data is rather surprising, due to the relatively
low value of saturated contact angle. A possible explanation is the use of a droplet that is
larger than preceding experiments by orders of magnitude. The size effect can lead to
totally different physical reactions at the liquid boundary.
Despite in need of deeper analysis, one can state that the following are the most
influential parameters undermining contact angle saturation:
1. Dielectric thickness and material.
2. Droplet size and viscosity.
In the end, table 2 can suggest some values that experimenters can use as guidelines to
compare their obtained values with (if compatible with their materials):
Very thin (1µm) SiO2 and very small (1µL) water dropOR θsat ~ 75-80°.50µm thick Paraffin and very small (1µL) decane drop
Thin (5-30µm) Parylene and 10µL Chloride/Sulphate drop θsat ~ 60-65°.
Large drop of Sulphate on 50µm PTFE θsat ~ 25-30°.
Section 3.4: Contact Angle Hysteresis
As mentioned before, contact angle hysteresis can be viewed as the analogue of frictional
forces in contacting solids [3]. It is the direct consequence of contact line pinning, which
is the sticking of the contact line to the surface. Since the fluid as a whole can flow,
pinning leads to contact angles that are different at the leading and lagging edge of a
droplet. Namely, as per the figure below, θA > θ0, and θR < θ0:
Figure 5: Illustration of contact line pinning and hysteresis(Walker and Shapiro [3])
Figure 5 shows the case of a stationary particle on a slope. For the droplet to move along
a slope, it must deform at the leading edge and at the rear edge as shown [18]. Even if the
slope is big enough to cause motion by contribution of droplet weight (after begin
initially fixed), the motion will be suppressed and slow. Hysteresis is very similar for the
case of a moving particle on a horizontal surface (e.g. EWOD). The first deals with static
friction, while the latter with dynamic friction, both of which retard the droplet.
Consider an EWOD device, where particle confinement between an upper and a lower
electrode exists. Now picture a droplet moving on a very smooth electrode, where
hysteresis is negligible. The difference between the front and rear contact angles would
be large due to minimal resistance to motion. Noting that the pressure gradient is
proportional to the pressure difference, this results in a high pressure gradient to move the
droplet, and so droplet speed is relatively large. Now picture the same droplet on a not so
smooth electrode. Some of the pressure net force will be consumed to compensate for the
frictional losses. The difference between the front and rear contact angles, and hence the
driving pressure gradient, are therefore smaller, and the speed is relatively low:
Figure 6: Effects of contact angle hysteresis in the EWOD device(Droplet moving rightwards. Walker and Shapiro [3])
Contact angle hysteresis can also be viewed from an energy conservation point of view.
Consider for instance the drop on a tilted surface. The liquid-ambient boundary would
deform in such a way that allows it to reach a ‘new’ equilibrium to minimize interfacial
tension originally arising from tilting [19].
Alternatively, beyond the scope of the matter, contact angle hysteresis can also occur on
smooth surfaces, due to chemical heterogeneity [20, 21].
Referring back to figure 2, the combination of Teflon and the dielectric allows for easier
operation, i.e. operation at relatively low voltages, due to decreased hysteresis [15]. The
impregnation of a dielectric coating (Parylene for example) with oil also decreases
hysteresis [13].
The physics and chemistry behind electrowetting in general is not very well understood,
for instance there is dispute about what the liquid profile is near the contact line [14].
Only time and a lot of experimentation might lead to a standard understanding of the
phenomena.
Chapter 4: Computational Methodology and Model
Section 4.1: Equations of Fluid Flow
Quickly introduced here are the Navier-Stokes equations. Not specific to our project,
these generally govern incompressible, Newtonian fluid flow. Two fundamental
conservation principles are those of momentum and mass (energy excluded for brevity;
there is no heat transfer or work done in our droplet simulations) [22]:
Vpgdt
dV 2 (momentum) (6)
0 V (mass) (7)
where:
V is the velocity vector at time t: V = iu(x, y, z, t) + jv(x, y, z, t) + kw(x, y, z, t)
p is the pressure gradient at time t: z
p
y
p
x
pp
To include surface tension, γ, since it showed up in the electrowetting equation, the
relation with pressure gradient is used for a certain surface of principal radii of curvature
R1 and R2:
21
11
RRp (8)
For example, ignoring gravity for our situation, the momentum equation in the x direction
would be:
2
2
22
2
z
u
y
u
x
u
x
p
z
uw
y
uv
x
uu
t
u
This is a cumbersome, non-linear, second order partial differential equation.
Computational fluid dynamics methods rely on discretization, with given time steps to
advance with, to solve the above equations one step at a time until convergence or until
specified. The solution is not exact since it is not solved analytically but rather estimated
numerically.
Section 4.2: Volume of Fluid Method
A fundamental method of solving two-phase flow problems is the volume of fluid (VOF)
method. The algorithm essentially tracks the interface at specified advances in time.
First, and to accommodate for solvers’ discrete capabilities, meshing is essential. The
geometry needs to be input in a Finite Modeling package (Gambit for this project).
Meshing will create multiple finite volumes (or cells) constituting as a whole the entire
geometry. When run, the solver (Fluent) assigns a function f(x, y, z, t) called the fractional
function to each cell at each time step [23]. If a cell is empty (all air), it is assigned a
value of zero. If it is full of water the function is 1. A cell is understood to contain an
interface if the fractional function is between 0 and 1, and if there is at least one
neighboring cell with an assigned value of zero. Then, the cells assigned as containing an
interface will be fit by Piecewise Linear Interface (re)Construction (PLIC).
Clearly from how the method progresses, the tracking is highly dependent on the mesh
density. Higher densities imply more cells to perform the piecewise interpolation on,
resulting in higher accuracies and smoother profiles. The advantage of using the volume
of fluid method is that the boundary conditions for the upcoming time step can be known
by exploiting the fractional function. However, a disadvantage of this technique is that
fractional functions have to be assigned to every cell in the geometry (the solver can’t
know where, for example, the drops are). Moreover the cells assigned as containing an
interface have to be screened out. This results in (unnecessarily) high computational
effort, time, and computer memory usage.
Section 4.3: Microfluidic Chip Geometry
The chip design chosen for the subsequent analyses was that with two reservoirs and six
electrodes. Important dimensions included, first, 1mm by 1mm linear electrodes, and also
a chip thickness (distance from top to bottom cover) of 80 microns.
The surface types had to also be defined in the geometric package. The electrodes were
chosen as ‘walls’, allowing for contact angle specification when exported to Fluent. The
vertical, top, and bottom sides were chosen as ‘pressure inlets’, allowing for ambient
pressure specification. Note that the simulations DO NOT account for contact angle
saturation or hysteresis. Defining electrodes simply as walls will hence result in much
faster droplet speeds as per literature review in chapter 3. Next are plane views of the
design, with the origin at the bottom left corner of the bottom cover:
Figure 7: Microfliudic Chip Geometry(left: bottom cover; right: top cover)
Chapter 5: Results and Discussion
Section 5.1: Convergence Study; Optimal Mesh and Time Step
The simulations undertaken were specifically:
One drop moving: from one electrode to a neighboring one.
Two drops merging: from two different electrodes to a common one.
One drop splitting: from one electrode to two neighboring ones.
In order to arrive at accurate and trustworthy results, we needed to check that the
simulations were independent of the mesh density. Furthermore, the time step chosen
should also be such that a smaller step wouldn’t result in highly contrasting results. By
trying out different combinations of x-, y-, and z- mesh densities an optimal mesh was
arrived at. The same was applied for finding an optimal time step. It is important to note
that by optimal we mean the best tradeoff between runtime and acceptable accuracy. The
following shows the logic used in obtaining these two parameters:
Standard boundary conditions were chosen at contact angles of 117° for electrodes turned
off (hydrophobic), and 90° for ones turned on (relatively hydrophilic).
A mesh volume of (x, y, z) = (100, 100, 10) microns was first used. Mixing was chosen
for testing this mesh. The radius of each drop was 550 microns, in other words, the
maximum overlap with any neighboring electrode was 50 microns vertically. The initial
time step was 10-5 seconds. Each drop constituted about 700 volume elements. After
mixing occurred, the contact line did not converge on a circular one (that resembles a
relatively big drop); there were vertical lines on the sides.
When a mesh volume of (x, y, z) = (100, 20, 20) microns was then tested, where each
drop constituted about 350 cells, a circular profile was indeed arrived at. We conclude
that despite the overall decreased density, the significant y-density proved important in
having a fine resolution. The figure below summarizes the above (electrodes are not
shown, unfortunately, but it is easy to visualize them as the droplet diameter is very close
to the size of an electrode, and a drop rests initially on the centre of one):
100x20x20 microns
100x100x10 microns
(From left, t (s) = 10-4, 3x10-4, 10-3, 2x10-3, 3x10-3, 3.6x10-3, 3.7x10-3, 4x10-3, 5x10-3, 6x10-3, 7x10-3, 0.01)
Figure 8: Droplet mixing of two different mesh densities
To check that the solution was not comprehensive for the ‘weaker’ mesh, the final static
pressure and velocity profiles were obtained and compared. From the next figure, we
conclude that the 100x20x20 mesh is convergent since the velocity gradients are
negligible over the domain of the whole droplet at equilibrium. Also, the static pressure is
symmetric and uniform, implying a settled droplet. The weaker mesh shows velocity
variations as well as pressure inconsistencies, even after thousands of iterations:
100x100x10 mesh (y – vertical –) 100x20x20 (y)
100x100x10 (x – horizontal –) 100x20x20 (x)
Figure 9: Pressure Distribution (Pa) in y and x directions:Resultant droplet of mixing two 550-micron radius water droplets – many time steps –
(0.0025m: x-coordinate of the centre of the electrode over which droplets mixed; 0.006m: y-coordinate of the centre of the electrode. The y direction is the vertical of every interface plot in this paper)
100x100x10 100x20x20
Figure 10: Velocity Distribution (m/s) in y direction:Resultant droplet of mixing two 550-micron radius water droplets – many time steps –
Similar plots to the above were extracted after every simulation to ensure convergence
and physical comprehensiveness.
Next in the convergence study was testing a finer mesh in the x-direction. The division
was (x, y, z) = (50, 20, 40) microns. The first (movement) and second (mixing) type of
simulations progressed in a similar fashion to their equivalents in the previous 100x20x20
mesh. However, the third (splitting) type showed peculiar and unreal behavior of the
small, third daughter droplet (more on this later). The drop moved around arbitrarily
between the three employed electrodes and then eventually mixed with one of the main
daughter droplets at high speed! This is expectedly primarily due to the fact that the size
of the third drop was comparable to the z-direction mesh density (80/40 = 2 cells only),
which doesn’t achieve the goal of mesh density independence. The use of a finer x-
density, with no change in the first two types of simulation, implies that we can work
with the previous mesh without sacrificing much of the accuracy. This will be backed
with more proof below.
After that, and finally for mesh density testing, a fine mesh, taking a very long time to
run, was used. The all-fine mesh was 50x20x10 microns. The accuracy was very slightly
better for the above runs. However, it was worth forfeiting since the already chosen mesh
(100x20x20) made runs in less than one fifth of this mesh’s time
Lastly for our convergence study, an appropriate time step needed to be assigned. All the
above was done with a time step of 10-5 seconds. After experimenting, observed was that
a time step of 10-4.5 will not achieve convergence at all with any type of motion or mesh.
Also, a time step of 10-6 showed better results, yet again not accurate enough to
encourage the large amounts of computational time needed (days). As a result, the
original time step of 10-5 seconds was optimal and is used for every simulation.
To sum up, a mesh resolution of 100x20x20 microns was utilized with a time step of 10-5
seconds. For the splitting case, some accuracy was notably lost. The study allowed for
identifying which accuracy in meshing could be given up in order to attain lower running
times. By exploiting which meshing detail is most influential, even lower overall density
could prove more accurate. The following shows the rationale behind the above choice in
a compact form:
Mesh Density
(microns)Time Step(seconds)
Convergent? Accuracy Run Time(1GB RAM)
Importance of x-density
Importance of y-density
Importance of z-density
100, 100, 10 10-5 No Low > 1 hour
100, 20, 20 10-5 Yes Very Good ~ 1 hour
50, 20, 40 10-5 Yes Very Low < 1 hour
50, 20, 10 10-5 Yes High ~ 1 day
Any 10-4.5 No - -
50, 20, 10 10-6 Yes Very High days
Low High Very High
Table 3: Summary of Convergence Study
Section 5.2: Droplet Translation
5.2.1: Base Translation
Simple translation was first investigated. A droplet of radius 550 microns was placed on
an electrode (recall the crucial need of an overlapping droplet – here a maximum of 50
microns was introduced –). The boundary conditions were contact angles of 117° for all
‘walls’ – that is the electrodes, reservoirs, and covers –, one of 90° for the only activated
electrode, and atmospheric pressure for the chip’s edges. Appendix B shows all the
details relevant to the simulation, and can be compared easily for other simulation inputs.
Snapshots of the droplet’s advancement are shown:
(From left: t (s) = 10-4, 5x10-4, 10-3, 2x10-3, 3x10-3, 4x10-3, 5x10-3, etc…)
Figure 11: Simple Translation of a 550micron-radius drop of water in air with contact angles of 117°(OFF, bottom) and 90° (ON, top)
By utilizing a suitable mesh and time step, a smooth transition between interface profiles
was noticed. It is very close in transformation to what is observed in experiments using
comparable parameters. The next subsections will compare changes in the physical
parameters of water. In doing so, only viscosity was changed first, and then only surface
tension was changed. Values chosen for viscosity were the original (0.001003 kg/m-s),
and half and quarter that of the original. Those of surface tension (with air) were the
original (0.07275 N/m), and half and twice the original.
5.2.2: Effect of viscosity change
The following shows the simulations undertaken to investigate viscosity effects. As
expected, decreasing viscosity results in a faster motion. Recalling the Navier Stokes
equations, the viscosity appears in the ‘frictional’ term. Increased viscosity implies
increased forces that retard the motion. Interestingly, with quarter of the viscosity the
droplet ‘sloshed’ about before settling down. Since it was fastest, inertial forces
noticeably kept it moving even after it arrived over the destined electrode. A ‘blocking’
pressure gradient, due to the next (third, unwetted) electrode being turned off, is
established to ‘stop’ the drop. Also, velocity increase was not linear with viscosity
decrease, rather the increase in velocity was less sharp and more variable than linear (it
doesn’t double for a halved viscosity value, for instance).
µw
µw/2
µw/4
(Times as per base case)
Figure 12: Viscosity variation of a translating droplet
5.2.3: Effect of surface tension change
The above was also applied to observe effects of changing the surface tension of pure
water in air. The simulations predicted slower, more constrained flow with decreased
surface tension. In relation to the electrowetting equation (5), a decrease in γLG, with all
other parameters being constant, results in decreased velocity (analogous to the driving
potential). In relation to the Navier Stokes equations, lowered surface tension yields
lowered pressure gradients (equation 8 in previous chapter), which in turn results in
lowered velocities.
It makes sense that, if an interface is experiencing less ‘tension’ to act as a moving force,
it will progress slower. Also, velocity decrease was not linear with tension decrease, and
was, like before, rather less sharp and variable.
2γair-water
γair-water
γair-water/2
(Times as per base case)
Figure 13: Surface tension variation of a translating droplet
Section 5.3: Droplet Mixing
5.3.1: Base mixing and effect of viscosity change
In a very similar fashion to the base droplet translation simulation, a base mixing
simulation was obtained. It is the one already shown in the convergence study. Two
electrodes were assigned contact angles of 117° and a buffer electrode between them was
assigned one of 90°. Each hydrophobic electrode had the same size droplet and position
as the aforementioned. Viscosity change is also shown next.
The droplets mixed faster as expected with decreased viscosity. After mixing however,
the resultant drop velocity was not as large as the original droplets due to increased mass
for the same pressure gradient. Yet the instantaneous velocities of fluid particles within
the resulting drop were high due to fast motion and coalescence of original droplets.
Therefore the interface, after mixing and even almost settling, was observed to
experience wave-like shakes that were more prominent for lower viscosities. Because of
this phenomenon, lower viscosities required more time steps to converge during
simulations despite the shorter time to merge. Recall that friction is ignored and so the
shakes are magnified even more. What is shown in the figure, before droplet settlement,
are ‘sloshes’ occurring between mixing and settling down as opposed to those shakes that
are more subtle.
µw
µw/2
µw/4
(From left, t (s) = 10-4, 3x10-4, 10-3, 2x10-3, 3x10-3, 3.6x10-3, 3.7x10-3, 4x10-3, 5x10-3, 6x10-3, 7x10-3, 0.01. For µw/4: continues to the ‘shakes’ at 0.0111, 0.0121, and 0.0131 seconds – equilibrium for µw/4 not shown)
Figure 14: Viscosity variation of merging droplets
5.3.2: Effect of surface tension change
Below is a figure showing the effect of changing surface tension on mixing as done for
translation (times as above). Notice that decreased surface tension resulted in slower
mixing and in more abrupt changes in interface shifts.
2γair-water
γair-water
γair-water/2
Figure 15: Surface tension variation of merging droplets
Section 5.4: Droplet Splitting
5.4.1: Base Splitting
To avoid redundancy and because splitting proved troublesome, a change of boundary
conditions (contact angles) was investigated instead of physical parameter change like the
previous two sections. The problem with the simulation is that splitting always resulted in
a third, tiny droplet as well as the two expected daughter droplets. A lot of trials were run
to investigate how different meshes, time steps, and boundary conditions affect the
emergence of this third droplet. Unfortunately, all runs resulted with more than two
droplets, a case that doesn’t occur in experiment.
The reason behind the above is expectedly due to the exclusion of contact angle
saturation and hysteresis. The original drop splits much faster than in reality and hence
inertial forces are higher than the viscous forces – the latter being essentially what
prevents the formation of a third droplet. After building the base case, this phenomenon
was taken for granted and effects of boundary conditions were explored.
The droplet is originally 580 microns as opposed to 550 microns used previously (a
maximum overlap with a neighboring electrode of 80 microns as opposed to 50 microns).
Also, the electrodes to be turned on (above and below the drop) were assigned contact
angles of 90° and 89°. The differences will be explained in the next subsection.
(From left: time (s) = 5x10-5, 2x10-4, 5x10-4, 10-3, 1.5x10-3, 2x10-3, 3x10-3, 4x10-3, 5x10-3, 7x10-3,9x10-3, 0.0102, 0.0104, 0.011, 0.0115, 0.0125)
Figure 16: Splitting of a 580micron-radius droplet with contact angles of 117° (OFF, middle), 90° (ON, bottom), and 89° (ON, top)
5.4.2: Effect of contact angle symmetry and droplet size
At the early stages of simulating droplet splitting, symmetric boundary conditions were
implemented, i.e. the splitting electrodes between which the droplet initially sits are
assigned the exact same contact angle value. The result was that, even with smaller time
steps than 10-5, the simulation made only half the progress anticipated. The droplet never
split, and at some point in time the fluid was as if it were ‘stuck’.
A physical interpretation could be that the simulation was ‘too perfect’. The forces
pulling the drop apart were completely cancelling each other out when in opposite
directions, something that only happens in an ideal world. Contrast a rope that, if pulled
from one side with a force and from the exact opposite side with an exactly equal force,
will be immobilized at a certain time. The cubic cells employed in the mesh are, even
when very fine, not accurate enough to encapsulate the details of fluid flow just before
splitting. The droplet failing to split is illustrated in the next interface profile (times not
shown):
Figure 17: Failed splitting of a 580micron-radius droplet with contact angles of 117° (OFF, middle) and 90° (ON, bottom and top)
In addition to symmetry, droplet size can be limiting to splitting even with some definite
asymmetry! The reason why a radius of 580 microns was used is because a 550 micron
radius failed to split, even with contact angle difference for electrodes to be activated. A
maximum overlap of 50 microns was not enough to initiate fluid flow with our
hexahedral mesh.
In addition, strange observations included a ‘dominant’ electrode in the case of
intensified asymmetry (90° and 85°). The drop seemed like it will split, but then the more
hydrophilic electrode ‘sucked’ the whole fluid and splitting was not accomplished. Also,
an attempt to initiate splitting by introducing a higher pressure gradient (120°, not 117°,
and 90°) failed with the smaller drop in the symmetric case. Without further to mention,
the table next shows a summary of all the trials and observations carried out for different
sizes and boundary conditions:
Main Electrode
(°)
Top Electrode
(°)
Bottom Electrode
(°)
Droplet Radius
(microns)Flow? Split?
Third Drop?
117 90 90, 89, 88, 87 550 No - -
120 90 90, 89, 88, 87 550 No - -
117 90 90 580 Yes No -
117 90 89 580 Yes Yes Yes
117 90 85 580 Yes No -
Table 4: Droplet split summary for a 100x20x20 hexahedral mesh
5.4.3: New Mesh
At the end of the testing period, the above information regarding splitting motivated a
different approach. The symmetry mentioned above is also due to the hexahedral mesh
used in all of the simulations so far. For our simple block design, this meshing produced
perfectly symmetric cubes, and the argument made for failed splitting (no splitting or the
formation of a third droplet) holds. Therefore, a different type of mesh, namely
tetrahedral, was employed for a final set of runs. The density was the same (100x20x20),
but this mesh is twice as fine since it constitutes meshing pyramids instead of cubes.
There were 700 cells forming a droplet as opposed to the original 352 in a hexahedral
mesh. This increased accuracy allows for capturing flow details just before and just after
the splitting instant. As observed in figure 16, at that instant the fluid-air interface forms
very sharp, pointed profiles. A tetrahedral mesh will, due to its triangular base nature that
‘fits’ in corners and complicated geometric surfaces, be able to allocate for such forms of
profiles with higher accuracy. Presented next is a figure showing successful splitting of a
drop meshed tetrahedrally, where no third drop was noticed. A final note is that this mesh
resulted in a more realistic, relatively slower and smoother splitting. It worked for
symmetric cases as well, proving more physical comprehensiveness than with hexahedral
meshes.
Figure 18: Tetrahedral mesh: Successful splitting of a 580micron-radius droplet with contact angles of 117° (OFF, middle), 90° (ON, bottom), and 89° (ON, top)
Section 5.5: Summary and Conclusions
To put all the above in short form, a convergence study first showed that a mesh of (x, y,
z) = (100, 20, 20) microns yielded results that were very closely independent of the mesh.
For the splitting simulations however, a hexahedral mesh proved inadequate and the more
accurate, same-density tetrahedral mesh was essential. The study also showed that results
were nearly independent of a time step of 10-5 seconds (note, however, that the
tetrahedral mesh was not analyzed for independence of this time step. It nevertheless
yielded realistic splitting).
In investigating effects of parameter modification, lower viscosities experienced faster
motion and mixing, while lower surface tensions implied slower motion and mixing as
well as more sudden interface advancements when just mixing. The parameters were
modified while keeping all other properties constant. This complies with the
Electrowetting and Navier Stokes equations governing unsteady, incompressible,
Newtonian fluid flow in EWOD devices. Our awkward splitting case with a hexahedral
mesh proved interesting when investigating boundary condition amendment: for splitting
to occur, a minimum radius was needed, otherwise flow would simply not ‘commence’.
If it did, splitting did not occur.
All the simulations predicted rather large speeds and very smooth flow since contact
angle saturation and hysteresis were not accounted for in Fluent’s code. The
incorporation of these phenomenon, by numerical techniques or inclusion of additional
constants for instance, would help in achieving more realistic flow times, speeds,
pressure gradients, etc… [6] [10]. When that is possible, a better understanding of
parameter variation could be arrived at in details. Then, optimal designs of chips,
actuation voltages, dielectric properties, etc… could be tested and utilized for a
corresponding application.
Chapter 6: Future Recommendations
The following includes some tips for future analysts to consider when running
simulations similar to the ones provided in this project:
For every separate run, an optimal mesh and time step should be found by trial.
The above analysis used one optimal mesh for all runs due to time constraints.
For all runs, use a tetrahedral mesh instead of a hexahedral one. On average,
according to the tests run, this will yield 4-5 times more computation time given
all other parameters constant.
The maximum number of iterations for each time step was set to 20 here. For the
tetrahedral mesh however, the use of 30 would be more accurate, on the expense,
of course, of slower calculations and extensive computer memory use.
As shown in Appendix B, the under-relaxation factors were all set to 1 for
simplicity of the numerical method. However, it is actually crucial to find optimal
values, especially for pressure and momentum, to ensure correct convergence.
The study of these parameters was outside the scope of this project.
The pressure difference (indirectly, the contact angles) were always input into the
program, explicitly. A better understanding of the physics behind actuation could
be achieved via modeling dielectric material and thickness, and hence obtaining
contact angles implicitly. The same could be suggested for contact angle
saturation and hysteresis, which is at the moment an area of extensive study for
incorporating into numerical codes.
To understand more of the flow behavior, other influential factors could be tested,
such as the ambient medium and the inclusion of heat transfer and/or evaporation.
Since Fluent is a commercial package, it will almost always converge to a
solution. The use of educational packages instead, where an incorrect setup would
probably yield non-convergent solutions, is more comprehensive. This will allow
researchers to investigate why a solution did not converge.
References
[1] Kamran Mohseni and Ali Dolatabadi 2006 An electrowetting microvalve Numerical Simulation Ann. N. Y. Acad. Sci. 1077: 415-425
[2] Frieder Mugele and Jean-Christophe Baret 2005 Electrowetting: from basics to applications Institute of Physics Publishing 17 706-725, 732-734
[3] Shawn W. Walker and Benjamin Shapiro 2006 Modeling the Fluid Dynamics of Electrowetting on Dielectric (EWOD) Journal of Microelectromechanical systems 15 4 986-999
[4] Quinn A, Sedev R, and Ralston J 2003 Influence of electrical double layer in electrowetting J. Phys. Chem. B 107 1163-1169
[5] Armani M, Chaudhary S, Probst R, Walker S, and Shapiro B 2005 Control of microfluidic systems: two examples, results, and challenges International Journal of Robust and Nonlinear Control 15 16 785-803
[6] Shapiro B, Moon H, Garrell R L, and Kim C J 2003 Equilibrium behavior of sessile drops under surface tension, applied external fields, and material variations J. Appl. Phys. 93 9 5794-5810
[7] Welters W J J and Fokkink L G 1998 Fast electrically switchable capillary effectsLangmuir 14 1535
[8] Jones T B 2005 An electromechanical interpretation of electrowetting J. Micromech, Microeng. 15 1184-1187
[9] Hendriksson U and Eriksson J C 2004 Thermodynamics of capillary rise: why is the meniscus curved J. Chem. Educ. 81 150-155
[10] Baviér R, Boutet J, Fouillet Y 2007 Dynamics of droplet transport induced by electrowetting actuation (Research Paper) Microfluid Nanofluid
[11] Vallet M, Vallade M and Berge B 1999 Limiting phenomena for the spreading of water on polymer films by electrowetting Eur. Phys. J. B 11 583-586, 588
[12] Kwan Hyoung Kang 2002 How electrostatic fields change contact angle in electrowetting Langmuir 18 10318-10322
[13] Verheijen H J J and Prins M W J 1999 Reversible electrowetting and trapping of charge: model and experiments Langmuir 15 6616-6619
[14] Wang K L and Jones T B 2005 Saturation effects in dynamic electrowetting Applied physics letters 86 054104
[15] Cho S K, Moon H, Kim C J 2003 Creating, transporting, cutting, and merging of liquid droplets by electrowetting-based actuation for digital microfluidic circuits J. Microelectromech. Syst. 12 1 70-80
[16] Peykov V, Quinn A, and Ralston J 2000 Electrowetting: a model for contact-angle saturation Colloid Polym. Sci. 278 792
[17] Janocha B, Bauser H, Oehr C, Brunner H, and Göpel W 2000 Competetive electrowetting of polymer surfaces by water and decane Langmuir 19 3352-3353
[18] Gao L and McCarthy T J 2006 Contact angle hysteresis explained Langmuir 226234
[19] Hennig A at al. Contact Angle Hysteresis: Study by Dynamic - Cycling Contact Angle Measurements and Variable Angle Spectroscopic Ellipsometry on Polyimide Langmuir 20 6686
[20] Iwamatsu M 2006 Contact angle hysteresis of cylindrical drops on chemically heterogeneousstriped surfaces Journal of Colloid and Interface Science 297 772
[21] Ramos S M M, Charlaix E, Benyagoub A 2003 Contact angle hysteresis on nano-structured surfaces Surface Science 540 355
[22] Frank M. White Fluid Mechanics Fifth Edition, Ch. 4; pp. 226-243.
[23] Mousavi, Jafari, Yaghamaei, Vossoughi, Sakomaa (2006) Computer simulation of fluid motion in a porous bed using a volume of fluid method: Application in heap leaching. Minerals Engineering Vol. 10, no. 10; pp. 1078-1082.
Appendix A
The Bond Ratio
The Bond Ratio, measuring the effects of gravity relative to interfacial effects, can be
shown to be very small [6]:
The Bond ratio is defined as: B = R2ρg / γ
where R is the drop radius, ρ is the fluid density, g is gravitational acceleration, and γ is
the liquid-vapor surface tension.
Let’s take R = 100µm, then the bond ratio for a water drop (γLG = 7.3x10-2 kg/s2) would
be B = 0.0013, suggesting a 0.1% contribution of gravity relative to interfacial energy.
Appendix B
Detailed Entries into Fluent for Simple Translation
Solver: Unsteady; Implicit Body Force, Courant Number 0.25.
Multiphase: Volume of Fluid, 2 phases.
Materials: air at room temperature (ρ = 1.225 kg/m3, µ = 1.7894x10-5 kg/m-s); pure
liquid water at room temperature (ρ = 998kg/m3, µ = 0.001003 kg/m-s).
Phases: Primary phase: air; Secondary phase: water (with surface adhesion, γ = 0.07275
N/m).
Operating Conditions: Atmospheric pressure at zero gravity.
Boundary Conditions: All walls at 117°, except ON electrodes at 90° (splitting also used
120, 118, and 89, 88, 87, 86, 85); sides of 80 micron thickness at atmospheric pressure.
Solution Controls: PISO Pressure-Velocity coupling (more accurate than SIMPLE and is
needed for two-phase flows. Under-relaxation factors (pressure, density, body force, and
momentum) were 1 for simplicity.
Sequence Definition:
Time Step: 1e-5.
Number of Time Steps: 1000-2000.
Maximum number of iterations per time step (reached only by tetrahedral mesh): 20
Frame rate: 10 time steps per one frame, 5 for splitting.