Simulation of Spreading Dynamics of a EWOD Droplet With Dynamic ...

Proceedings of the ASME 2009 2nd Micro/Nanoscale Heat & Mass Transfer International Conference MNHMT2009 December 18-21, 2009, Shanghai, China Proceedings of MNHMT2009

ASME 2009 2nd Micro/Nanoscale Heat & Mass Transfer International Conference December 18-21, 2009, Shanghai, China

MNHMT2009-18558 MNHMT2009-18558 Simulation of Spreading Dynamics of a EWOD Droplet with Dynamic Contact Angle and Contact Angle Hysteresis Fangjun Hong, Ping Cheng, Zhen Sun, Huiying Wu Ministry of Education Key Laboratory of Power Machinery and Engineering, School of Mechanical & Power Engineering Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 200240, P.R. China

dielectric layer is often covered by a thin hydrophobic layer.

ABSTRACT In this paper, the electrowetting dynamics of a droplet on a dielectric

surface

was

investigated

numerically

by

EWOD

overcomes

the

shortcoming

of

electrolytic

a

decomposition of water in traditional electrowetting, where the

mathematical model including dynamic contact angle and

electrode is in direct contact with the liquid, and broaden the

contact angle hysteresis. The fluid flow is described by laminar

applications of electrowetting. The recent growing interest of

N-S equation, the free surface of the droplet is modeled by the

EWOD comes from its promising applications, ranges from

Volume of Fluid (VOF) method, and the electrowetting force is

droplet-based microfluidic chips [3] to adjustable microlens

incorporated by exerting an electrical force on the cells at the

[4], optical and fluidic switches [5] as well as new kinds of

contact line. The Kilster’s model that can deal with both

electronic displays [6].

receding and advancing contact angle is adopted. Numerical

Generally, the relationship of the contact angle change

results indicate that there is overshooting and oscillation of

with applied voltage can be described by Young-Lippmann

contact radius in droplet spreading process before it ceases the

equation. However, many experimental studies [7-9] have

movement when the excitation voltage is high; while the

revealed that Young-Lippmann equation is only valid for a

overshooting is not observed for low voltage. The explanation

small to medium voltage. Beyond a certain voltage, which

for

special

obviously depends on the detailed experimental configurations,

characteristics of variation of contact radius with time were

the contact angle does not change anymore with the further

also conducted.

increasing of the voltage. This phenomenon is so called contact

the

contact

line

overshooting

and

some

angle saturation. Although some theories based on the INTRODUCTION

experimental observations, like the injection of charges in the

Electrowetting on dielectric (EWOD), first proposed by

dielectric layer at high voltage, provoking a screening effect

Berge [1] in the early 1990s, describes the phenomenon of

[8], or the ionization of air near the contact line inducing a loss

contact angle changing of a droplet on the surface of a

of electric energy [9], can partially explain the contact angle

dielectric layer separating the conductive liquid from the

saturation, the conclusive physical mechanism is not yet clearly

metallic electrode applied with electrical voltage. According to

established. More theoretical studies about static characteristics

the electromechanical point of the view, the reduction of

of electrowetting can be found in the review article [10].

1.

contact angle is due to the electrical force (Columbic force) at

From the application point of the view, a thorough

the tri-phase contact line [2]. To make the surface less friction

understanding of the droplet dynamics in electrowetting will be

and achieve larger contact angle variation capability, the

helpful in device design. Both DC and AC voltage can be used

1

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in EWOD. Under the DC excitation, the new equilibrium state,

kHz. It was found that at large spacing of the two plates, for

(a new contact angle and a new droplet final shape) is achieved

less viscous aqueous liquid, an obvious overshooting and

through a dynamic spreading process of the droplet under the

damped oscillatory behavior were observed. Recently, the

constant electrical wetting force at triple-phase contact line;

dynamics of millimeter sized liquid droplet on EWOD

while for AC excitation, the conditions are much more

configuration under the excitation of low frequency AC from ~

complicated because of two aspects of reasons. Firstly, in AC

10 Hz to ~100 Hz were investigated [14,15] Since at this

field, the electrical property of droplet depends on the

frequency range the droplet hydrodynamic response can follow

frequency [11], causing the different source of electrical force:

the periodical change of electrical force at the contact line, the

for low AC frequency(~1 MHz for normal electrolyte water

droplet vibrates. Theoretically, its primary droplet oscillation

solution), the droplet can be treated as a perfect conductor, and

frequency should be the twice of AC frequency [14], since the

is mainly under the electrical force of EWOD on tri-phase

electrical force is proportional to the square of voltage.

contact line; for high frequency (~10 MHz for normal

However, according to the experimental data of the variation of

electrolyte water solution), the droplet can be assumed to be

contact angle and contact radius with time from the reference

perfectly dielectric, and the droplet is mainly under the

[15], it seems that the droplet vibration frequency is not the

electrical force of dielectrophoresis (DEP), which is mainly

twice of excitation AC frequency, and they are not changing as

exerted on the interface of the droplet and ambient fluid; and in

a sinusoidal function, but with some normal and abnormal

a certain frequency range, the droplet is leaky dielectric, having

stops, which can be ascribed to the effect of contact angle

both conductor and dielectric properties, and both EWOD and

hysteresis.

DEP force will be existing. Therefore, for different AC

The droplet dynamics of EWOD is complicated because

frequency, the electrical force type and magnitude may be

the relative importance of viscosity, capillary wetting, and

different. It should be noted that when DEP force exits, it is not

surface tension is changing in the process, companying with the

in the field of EWOD anymore. Secondly, the periodical

variation of droplet profile. Therefore, theoretical analysis with

varying of voltage leads to the periodical change of the

the consideration of the detailed physical process is impossible.

electrical force. In the case of slow variation of the applied

Numerical method is an effective approach to study

voltage (low frequency ac), the contact angle and droplet shape

complicated fluid dynamics in electrowetting. Wang and Jones

can follow the momentary equilibrium values. If the AC

[13] developed a simplified hydrodynamic mode based on the

frequency exceeds the hydrodynamic response time of the

assumption of fully developed Poiseuille flow to study the

droplet (~100 Hz for millimeter sized droplet), the liquid

dynamic rising process of liquid between two parallel plates.

response depends only on the time-average of the applied

The contact line friction force depending on the wetting line

voltage and the behavior of the droplet is like that excited by

velocity was introduced to consider the dynamic wetting effect.

DC voltage.

The time averaged voltage is used to compute the electrical

Some experimental studies on dynamics aspect of

force. It was found that the line friction force dominates

electrowetting have been done. Decamps and Coninck [12]

viscous drag in all the tests in their study. It should be noted

investigated the spontaneous response of the contact angle of a

however that the adoption of the time averaged electrical force

3

5mm glycerol-water droplet on a Teflon surface with the

is not appropriate for some frequencies, ~100Hz, because the

thickness of 100um upon applying a 1kHz AC voltage with the

hydrodynamic response time of the liquid may be in the same

Vrms ranged from 400V to 800V. Since the AC frequency

order, and a time varying electrical force should be considered

exceeds the hydrodynamic response time of the droplet due to

in the numerical model. Some researches [16, 17] numerically

the high viscosity of the liquid and the relative high AC

studied the dynamic transport of the droplet (disk-like) between

frequency, the liquid droplet response likes it is excited by DC

the two plates in EWOD droplet-based microfluidic chips.

o

voltage. The contact angle decreased from 110 to its voltage

These models however do not consider the dynamics contact

dependent final value within less than 0.2s. Wang and Jones

angle and contact angle hysteresis, which theoretically play a

[13] studied the dynamic rising of several types of liquid with

key role in the capillary force driven flow.

different viscosity between vertically installed parallel plates

In this paper, we present a numerical model to study the

with electrodes powered by DC or AC voltage at 100 Hz to 5

electrowetting dynamics of a droplet in EWOD with the

2

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detailed consideration of the dynamic contact angle and contact

2.1 Electrical Force

angle hysteresis. The research is focused on the DC EWOD,

As stated in the introduction, in EWOD, the electrical

but it is also applicable for AC EWOD with the AC frequency

force is only exerted on the trip-phase contact line. With no

much higher than the hydrodynamic response time of the

voltage applied, the force balance in the horizontal direction at

droplet, which can be deemed as DC excitation as discussed in

the contact line according to Young-Laplace equation is:

γ lv cos θ 0 + γ sl = γ sv

the above text.

2.

(1)

where γ lv is the surface tension between the liquid and vapor,

MATHEMATICL SCHEME

θ 0 the equilibrium contact angle with no voltage applied, γ sl

Fig.1 shows the two typical kinds of testing setup of EWOD with the droplet in the ambient of the air. In Fig.1 (a), the grounding is realized by a metal wire inserted into the

the interfacial tension between the solid and the liquid, and γ sv

droplet, while in Fig.1 (b), the grounding line is buried inside

the interfacial tension between the solid and vapor.

the flat plate. Since the metal wire or lines is very small

When an electrical voltage is applied, an additional

compared to the electrode, they are neglected in the numerical

electrical force Fe will be imposed as shown in Fig. 2, and the

simulation, and therefore for both types of EWOD testing setup

force balance at the contact line at equilibrium state becomes:

Fe + γ sv = γ lv cos θ v + γ sl

the axis-symmetric computational domain as shown in Fig.2 is

(2)

adopted. The outlet boundary of the computational domain is

According to Young-Lippmann equation, the effect of

chosen to be large enough so that it will not have any effect on

electrical voltage on contact angle can be expressed as:

the computational results. The conductive substrate (electrode),

1 ε 0ε

dielectric and hydrophobic surface layers are not included in

2 d γ LG

the computational domain. The reason is that the present

where ε

numerical method does not simulate the electrical filed to

0

r

V = γ lv (cos θ v − cos θ 0 )

(3)

2

is the vacuum permittivity, ε

r

the relative

calculate electrical force, but utilize Young-Lippmann equation

permittivity of the dielectric material, d the thickness of the

to derive the electrical static force at the contact line as detailed

dielectric layer, θ v the equilibrium contact angle with the

in the following text. application of voltage. Combining Eq. (1) - Eq. (3), the electrostatic force at the grounding wire

contact line can be expressed as: droplet

droplet hydrophobic surface dielectric conductive substrate Activation voltage

grounding line

Fe =

hydrophobic surface dielectric conductive substrate

2 d γ lv

r

V

(4)

2

or

Activation voltage

(a)

1 ε 0ε

Fe = γ lv (cos θ v − cos θ 0 )

(b)

(5)

Although, Eq. (4) and Eq. (5) are derived according to the

Fig.1 two typical kinds of droplet EWOD testing setup

equilibrium state of the droplet, they are still valid when the contact line is moving, since the droplet is ideal conductor in electrowetting, and its shape will not affect the electrical force at the contact line. It should also be noted that Eq. (5) can also be used to calculate the electrical force when the contact line saturation occurs, with the experimental results on the relationship of contact angle θ v and applied voltage V being available.

2.2 Fluid Flow Fig.2 the computational domain

The water is Newtonian fluid and can be assumed to be

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incompressible considering the low speed of movement.

dynamic contact angle with contact line speed and other related

Because of low Reynolds number in the present study, it is also

parameters.

reasonable to assume the fluid flow is laminar. Based upon the

For the viscous and initial force dominated flow, such as

above assumptions, N-S equations, which describe the mass

droplet impinging, the effect of dynamic contact angle

and momentum conservation, can be formulated in Euler

normally can be neglected. For the present electrowetting case,

approach as:

however, the droplet movement is driven by the electrowetting

∇⋅V = 0

(6)

force, which is in the same order of the capillary force exerted

∂V 1 1 1 1 + ∇ ⋅ ( VV ) = − ∇p + ∇ ⋅ τ + g + Fb + Fe ρ ρ ρ ρ ∂t

(7)

by the wall. Therefore, the dynamic contact angle is extremely important and will play a determinant role.

is the velocity vector, t the time, ρ the fluid

The dynamic contact angle is a very complex problem,

density, p the scalar pressure, g the acceleration due to gravity,

and its mechanism is still open to question. The reference [18]

τ the viscous stress tensor and expressed as:

gives a detailed review on the research progress on the dynamic

where

V

⎛ 1 ⎡( ∇V ) + ( ∇V )T ⎤ ⎞ ⎦ ⎟⎠ ⎝2⎣

τ = 2μ ⎜

wetting line. As shown in Fig.3, the dynamic contact angle

(8)

differs from its static value and may refer to either an advancing or a receding contact angle, and their values

Fb, the body force derived from the continuum surface force

dependent on the contact line velocity. Because of the surface

(CSF) model, only exerted on the fluid cell at free surface to

roughness and chemically heterogeneous, even equilibrium

replace surface tension, and Fe the electrical force only exerted

contact angles may not be single-valued, but will depend on

on the fluid cell at the contact line.

whether the contact line has been advanced or recessed to stop,

To trace the free surface of droplet, Volume of Fluid

a phenomenon known as contact angle hysteresis.

(VOF) method is adopted. In this method, F is a scalar field, which defines the state of a particular cell in the computational domain:

⎧1 ⎪ F = ⎨0 < F < 1 ⎪0 ⎩

water partial air, partial water air

(9)

Because F moves with fluid (i.e., the total value of F for the droplet is constant), it satisfies the conservation equation: DF ∂F = + (V ⋅∇) F = 0 Dt ∂t

(10) Fig.3 Velocity dependent of the contact angle [19]

Piecewise-linear surface reconstruction is adopted to define the

Some empirical correlation of the dynamic contact angle

droplet profile where F=0.5.

with the velocity is available in the literature. However, most of models can only deal with the advancing contact angle. The

2.3 Dynamic Contact Angle and Contact Angle Hysteresis

present study adopted the dynamic contact angle model by

Non-slip condition is applied on the wall boundary. It is

Kistler [20], which can describe both the receding and

well known that there exists stress singularity at the contact line

advancing contact angle in the form as：

with the application of traditional non-slip condition. In order

−1 θ D = f Hoff [Ca + f Hoff (θ e )]

to relieve this stress singularity, mathematical models usually

（11）

introduce a slip or “cutoff” length rather arbitrarily. Another

where ， Ca = vc μ γ lv is the capillary number, with vc the

method to deal with the stress singularity is to add a contact

spreading velocity of the contact line, μ

line friction force, which is related to the contact line velocity.

the dynamic

viscosity of the liquid, and γ lv the surface tension of the liquid

However, the most reasonable and widely adopted method to

and vapor, θ e is the contact angle at zero contact line speed

deal with the stress singularity is to build the relationship of the

zero whose value depending whether the contact line is

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equilibrium contact angle of 112.7 o. This difference is due to

−1 advancing or receding as shown in Fig. 3, and f Hoff is the

the effect of contact angle hysteresis. Because, the contact line

inverse function of the Hoffman function, fHoff , defined as：

⎧

⎡

⎩

⎣

x ⎛ ⎞ ⎟ ⎝ 1 + 1.31x 0.99 ⎠

f Hoff ( x ) = arccos ⎨1 − 2 tanh ⎢5.16 ⎜

0.706

movement is stopped during the receding process in this case,

⎤⎫ （12） ⎥⎬ ⎦⎭

the receding static contact angle 107.7 should be used in Eq. (3), and then the computed contact angle is 77.1o, almost the same with that of measured from the final droplet profile. This finding may be useful for the explanation of why some

2.4 Numerical Solver

experimental result of static contact angles are above and the

The above numerical model was solved using commercial software

package,

CFD-ACE+

solver

(CFD

others are below the theoretically predicted value according to

Research

the equilibrium contact angle, because at some voltages the

Corporation). The electrical force at the contact line and the

contact line stop moving during the advancing process, while at

dynamic contact model was realized using user defined

other voltages the contact line stop moving during the receding

function (UDF). The computational grids close to the wall was

process.

specially refined with the minimum grid of 5um is used to

The phenomenon that contact radius during the spreading

capture the viscous force, and the effect of the dynamic contact

process is larger than the final contact radius at equilibrium

angle. The grid resolution was changed to achieve the

state is called contact line overshooting. The reason for the

grid-independent result.

overshooting may be explained as followings. Under the role of electrical force at the contact line, the droplet starts to spread and deform, the deformation of the droplet profile leads to the

3. RESULT AND DISCUSSION In this study, the water droplet with the volume of 7ul,

accumulation of pressure at the peripheral of the droplet

the surface tension of 0.0725N/m, the density of 980kg/m3 and

(restoring force), which prevents the further deformation of the droplet. When the maximum spreading achieves, the droplet

the kinetic viscosity of 0.89 × 10 m / s is assumed. The -6

2

starts to restore, if the applied voltage is high enough, the

dielectric material is assumed to be Parylene with thickness of

pressure accumulated during the deformation process will be

1 μ m and the hydrophobic layer is Teflon with the thickness of 0.25 μ m .

The

effective

relative

permittivity

of

larger enough to contradict the electrical force and contact

the

angle hysteresis (both preventing the contact line receding) to

Parylene-Teflon layer is assumed to be 3.0. The equilibrium

make contact line slide back.

o

contact angle before applying electrical voltage is 112.7 , and the receding and advancing contact angle at zero velocity is 107.7o and 117.7o, with the contact angel hysteresis of 10o. The applied voltage is in the range of 30V-70V and it is assumed the contact angle saturation does not occur. The initial droplet profile is determined according to the numerical simulation with no voltage applied. The present study considers the droplet dynamics after the application of electrical voltage. Fig. 4 shows the droplet profiles variation in the dynamic spreading process in the case of 60V. It can be seen at first the contact radius increases, till at about t=9ms, the contact radius achieve the maximum value, however the contact line does not stop moving, but starts receding (the contact radius start decreasing), at about t=21ms it begins advancing again, after an oscillatory motion, it finally stops moving at about t=30ms. The final static contact angle measured according to the droplet profile is about 77o. However, according to Eq. (3), at this

Fig. 4 variation of droplet profile with time in the case of 60 V

voltage, the final static contact angle is 81.8o, assuming the

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Fig.5 shows the contact radius varying with time for different applied voltages. It can be seen that for the voltage higher than 40V, the contact line overshooting and oscillation occurs, while for the low voltage of 30V, no contact line overshooting is observed. This is because for the low voltage, the accumulated pressure during the droplet spreading is not large enough to overcome the electrical force and contact angle hysteresis effect. As shown in Fig. 5, the contact line advancing speed at first stage decreases (the curve slope decrease), but Fig. 6 the experimental result on the variation of contact radius with time

after a certain time, it starts to increase again. This time point

during the droplet spreading under the effect of EWOD [15]

may be refer to turning point during the advancing of contact line. We also can see the turning point during the receding of

4.

contact line. These phenomena may be explained as followings.

CONCLUSION The numerical model incorporating the dynamic contact

After the application of voltage, the droplet start to spread,

angle and contact angle dynamics has been developed to study

because the electrical force of EWOD is exerted on the contact

electrowetting dynamics of the droplet. The contact line

line, therefore at the early stage, only the fluid near the contact

overshooting was found in the case of high voltage and was

line starts to move, while the remaining droplet keep almost

contributed to the accumulation of the pressure at the

stagnant, this leads to a drag force (viscous force), which slows

peripheral of the droplet during the contact line advancing,

down the contact line moving speed. At some time point, the

which finally contradicts the electrical force and contact angle

whole droplet starts to move, therefore, the viscous drag force

hysteresis effect and make the contact line slide back. It was

at the contact line decreases, causing the increasing of contact

also found that there are turning points in the curves of

line speed. The same explanation can be used to explain the

variation of contact radius with time, at which the contact line

turning point during the receding of the contact line. It should

speed changes from decreasing to increasing or from increasing

be pointed out that the curves for the variation of contact radius

to decreasing. The numerical results and numerical model of

with time from the present numerical simulation are very

this study are helpful to the design of EWOD based device.

similar to those of the experimental result by Sen and Kim [15] as shown in Fig. 6, where the turning points are also obvious.

ACKNOWLEDGEMENT

However, it is not appropriate to compare the two figures point

This research work was supported by the National Natural

by point because the numerical simulation conditions are

Science Foundation of China through Grant No. 50706026 and

different from those of the experiment.

Turning point   during the  advancing  of  contact line 

2.4 2.2 Contact Radius (mm)

70V

Grant No.50925624. REFERENCE 1.

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