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.