Numerical study of the microdroplet actuation switching frequency in digital microfluidic biochips

RESEARCH PAPER

Numerical study of the microdroplet actuation switchingfrequency in digital microfluidic biochips

Ali Ahmadi • Kurt D. Devlin • Mina Hoorfar

Received: 29 June 2011 / Accepted: 15 August 2011 / Published online: 26 August 2011

� Springer-Verlag 2011

Abstract In this article, an electrohydrodynamic approach

is used to study the microdroplet actuation in contemporary

digital microfluidic biochips. The model is employed to

analyze the microdroplet motion, and investigate the

effects of the key parameters on the devices performance.

The modeling results are compared to the experimental

observations, and it is shown that the model provides an

accurate representation of digital microfluidic transport. An

extensive parametric variation is used to derive the maxi-

mum actuation switching frequency for ranges of the

microdroplet size, gap spacing between the top and bottom

plates and electrode pitch size. As a result, scalability of

the devices is investigated, and it is shown that the

microdroplet transfer rates change inversely with the sys-

tem size, and microdroplet average velocity is nearly the

same for different system scales. As a result of this study,

an adjustable force-based actuation switching frequency

implementation is proposed, and it is shown that faster

microdroplet motion is obtained by in situ adjusting of the

switching frequency. Finally, it has been observed that

fastest microdroplet motion, despite similar studies con-

ducted in the literature, is not achieved via actuating the

next electrode as soon as the microdroplet touches it.

Indeed, the switching frequency spectrum is dependent on

the physical and geometrical properties of the system.

Keywords Digital microfluidic � Microdroplet �Electrocapillary � Switching frequency

1 Introduction

Digital microfluidics is a new technology for generating

discrete microdroplet motion on arrays of electrodes (Pol-

lack et al. 2000, 2002; Lee et al. 2002; Moon et al. 2002;

Cho et al. 2003; Urbanski et al. 2006; Cooney et al. 2006;

Fair 2007; Fouillet et al. 2008; Brassard et al. 2008;

Abdelgawad and Wheeler 2008; Abdelgawad and Wheeler

2009; Fan et al. 2009). These devices are highly scalable

and reconfigurable, and allow for the efficient control

required for chemical and biological applications

(Srinivasan et al. 2004; Wheeler et al. 2005; Chang et al.

2006; Moon et al. 2006; Fair et al. 2007; Nichols and

Gardeniers 2007; Miller and Wheeler 2008; Sista et al.

2008; Jebrail and Wheeler 2009; Luk and Wheeler 2009;

Hua et al. 2010; Malic et al. 2010). As a result, these

devices are capable of performing high throughput analy-

sis, and their reconfigurability reduces both cost and weight

(Su et al. 2006; Fair 2007). A Schematic of a covered

digital microfluidic system is shown in Fig. 1.

Effective control of digital microfluidic devices can only

be achieved by having a deep understanding of the

microdroplet operational dynamics in the system. Without

an accurate prediction of the system kinetics, the system

will not operate at full throughput, and in certain cases, no

fluid transport will be achieved (Arzpeyma et al. 2008).

Therefore, a thorough understanding of the system kinetics

and determining the key parameters in the device perfor-

mance optimization are crucial tasks in the design, control,

and fabrication processes. Accurate models ultimately

allow us to find these key parameters and relate them to the

device throughput.

Geometrical and physical properties of the system and

the actuation switching frequency are the most important

key parameters influencing the throughput of the system,

A. Ahmadi � K. D. Devlin � M. Hoorfar (&)

School of Engineering, The University of British Columbia,

Vancouver, Canada

e-mail: [email protected]

123

Microfluid Nanofluid (2012) 12:295–305

DOI 10.1007/s10404-011-0872-8

and most of the recent studies have focused on investi-

gating and characterizing each effect individually

(Arzpeyma et al. 2008; Bhattacharjee and Najjaran 2010;

Bahadur and Garimella 2006). For instance, Fair (2010)

developed a hydrodynamic scaling model of droplet actu-

ation in a digital microfluidic systems. The proposed ana-

lytical model included the effects of the contact angle

hysteresis, drag from the filler fluid, drag from the solid

walls, and change in the actuation force while a droplet

traverses a neighboring electrode. It was determined that

reliable operation of digital microfluidic is possible as long

as the device is operated within the limits of the Lippmann-

Young equation. Bahadur and Garimella (2006) developed

a model based on the minimization of the system energy,

and predicted the overall performance of the device. They

investigated the origins and the fundamental physics

behind the microdroplet actuation, and used a parametric

variation to find the conditions which maximize the driving

force. In their analytical approach, the size of the micro-

droplet, however, is restricted to the electrode size.

Bhattacharjee and Najjaran (2010) modeled the digital

microfluidic systems for different microdroplet sizes,

actuation voltages, dielectric thicknesses, and electrode

sizes. However, in their study, electrode shapes were

assumed to have a simple square geometry. Arzpeyma

et al. (2008) developed a numerical model based on cou-

pling the hydrodynamic and electrostatic governing equa-

tions to investigate the optimum actuation condition. They

investigated the effects of the switching frequency on the

microdroplet motion and proposed a position-based

actuation algorithm for achieving higher microdroplet

velocities. They also recognized the limitations of their

model in terms of hysteresis effects. In a recent modeling

effort, SadAbadi et al. (2010) investigated the effects of

electrode switching frequency on the maximum possible

microdroplet velocity. They showed that the best time for

switching/actuating the next electrode is when the micro-

droplet leading edge contacts that electrode, and late or

early actuation will result in discontinuity in the micro-

droplet velocity.

In this study, a novel numerical electrohydrodynamic

approach is employed to model the microdroplet motion,

and investigate the effects of the key parameters on the

device performance. The friction factor (Ren et al. 2002)

and threshold voltage (Pollack et al. 2002) are the only

empirical parameters used in the developed model. The

numerical analysis employed here allows for modeling any

arbitrary system geometry and microdroplet shape. An

extensive parametric variation is used to derive the maxi-

mum actuation switching frequency (MASF) for wide

ranges of the microdroplet size, gap spacing, and electrode

pitch size. This study results in the development of a force-

based adjusting switching frequency algorithm based on

which optimum motion of the microdroplet is obtained.

The result of the new force-based switching frequency

algorithm will be compared to that of the position-based

algorithm. It will be shown that the latter provides slower

microdroplet motion compared to the force-based actuation

algorithm proposed in this article. In the following sec-

tions, the governing equations of microdroplet motion are

introduced, and the results obtained from the model and

proposed algorithm are presented and discussed.

2 Theory and methodology

In this section, microdroplet motion governing equations

and the boundary conditions used for solving these equa-

tions are explained. Details of the numerical scheme are

presented and discussed in Ahmadi et al. (2011).

2.1 Governing equations

To develop an accurate model for the microdroplet

dynamics, the driving and opposing forces must be quan-

tified. The governing transient equation for the microdro-

plet in the direction of the motion can be written as (Ren

et al. 2002)

mdvtransport

dt¼ Fdriving � Fwall � Ffiller � Ftpcl; ð1Þ

where m is the mass of the microdroplet, vtransport is the

microdroplet transport velocity, and Fdriving, Fwall, Ffiller,

top view

side view

AA

xz

xy

Fig. 1 Schematic of a covered digital microfluidic system is shown

296 Microfluid Nanofluid (2012) 12:295–305

123

and Ftpcl are the driving, wall, filler, and three-phase con-

tact line forces, respectively (Ahmadi et al. 2011; Ren et al.

2002; Bahadur and Garimella 2006; Ahmadi et al. 2009;

Buehrle et al. 2003; Baird et al. 2007; Arzpeyma et al.

2008).

The developed numerical scheme uses the finite volume

method (FVM) to solve the Maxwell equation (Buehrle

et al. 2003) in three material regions (dielectric layers,

filler, and microdroplet interior) (Ahmadi et al. 2011)

r � D ¼ r � ð�EÞ ¼ 0; ð2Þ

where D and E are the electric displacement and electric

field vector and e is the permittivity of the medium. The

electrostatic pressure, pel, can be obtained from the electric

field as

pel ¼�jEj2

2: ð3Þ

The driving (electromechanical) force acting on the

microdroplet interface can be calculated by integrating the

electrostatic pressure along the microdroplet–filler

interface as (Ahmadi et al. 2011)

Fdriving ¼Z

peldAinterface; ð4Þ

where dAinterface is the area element along the

microdroplet–filler interface. Calculating the wall and

filler forces, Fwall and Ffiller, involves the solution of

continuity and Navier–Stokes governing equations for the

hydrodynamic pressure. The continuity equation for

incompressible flow is

r � v ¼ 0; ð5Þ

where v is the velocity vector of the fluid particles. The

Navier–Stokes equation for the motion of the fluid is

q½otvþ ðv � rÞv� ¼ �rphyd þ lr2vþ qgþ qfE; ð6Þ

where phyd is the hydrodynamic pressure, q and l are the

fluid density and viscosity, g is the gravitational

acceleration [which can be ignored as the Bond number

is much smaller than 1 (Arzpeyma et al. 2008)], and qf is

the free charge density. The last term of Eq. 6 vanishes as

the Navier–Stokes equation is solved inside the conductive

liquid microdroplet and a dielectric filler fluid. While, both

the velocity and hydrodynamic pressure of the fluid are

unknown, it has been shown (Lomax et al. 2001) that the

continuity equation (5) and Navier–Stokes equation (6) can

be solved simultaneously using the numerical FVM. The

discretized regions of the digital microfluidic system are

shown in Fig. 2. After solving Eqs. 5, 6, the velocity vector

inside the microdroplet is then used to find the shear force

on the wall (Fwall) as

Fwall ¼Z

walls

s dA; ð7Þ

where s is the shear stress. Using the filler hydrodynamic

pressure (which is obtained from the solution of the Eqs. 5,

6) at the microdroplet interface, filler force, Ffiller can be

calculated as

Ffiller ¼Z

phyddAinterface: ð8Þ

The molecular-kinetic theory (Blake and Coninck 2002)

states that attachment or detachment of fluid particles is the

main source of energy dissipation at the moving three-

phase contact line. Although, dynamic of wetting can be

described by the microdroplet velocity and the dynamic

contact angle (Keshavarz-Motamed et al. 2010; Blake and

Coninck 2002), it was shown (Ren et al. 2002; Ahmadi

et al. 2009) that an additional force has to be added to the

dynamic equation of the microdroplet motion. Using the

molecular-kinetic theory, this three-phase contact line

force can be expressed as

Ftpcl ¼ 2Pnvtransport; ð9Þ

where n = 0.04 is the friction factor, P is the perimeter

length of the microdroplet. This linearly dependent friction

force is especially accurate at low and intermediate

velocities (Ren et al. 2002; Ahmadi et al. 2009).

2.2 Boundary conditions

2.2.1 Moving boundary

The boundary condition used for the microdroplet and filler

along the solid surfaces are the no-slip, no-penetration, and

zero pressure gradient condition. The microdroplet–filler

interface is moving with the microdroplet transport veloc-

ity, vtransport, and is implemented by defining fictitious

Cross-section A-AF=p Ael(i,j) (i,j)

(i,j)

Fig. 2 The discretized regions of the digital microfluidic system are

shown. Black circles show the center of each cell and the dashed linesshow the borders of each cell

Microfluid Nanofluid (2012) 12:295–305 297

123

velocities within solid cells adjacent to fluid cells (Buss-

mann et al. 1999). A method based on the fractional vol-

ume of fluid (VOF) method (Afkhami and Bussmann 2008;

Arzpeyma et al. 2008) is used for modeling the moving

boundary of the microdroplet. The main idea for the

implementation is that the microdroplet–filler interface is

moving with the transport velocity of the microdroplet.

Therefore, this volume fraction of the cells must satisfy the

advection equation

½otf þ vtransportoxf � ¼ 0: ð10Þ

The VOF method is based on the averaging phases at the

interface, in which the volume fraction, f, is advected with

the fluid flow. The volume fraction, f, for each cell is

defined as

f ¼ Vliq

Vcell

; ð11Þ

where Vliq and Vcell are the liquid volume and total cell

volume, respectively.

2.2.2 Microdroplet–filler interface

The relation between the electrostatic pressure, hydrody-

namic pressure, and microdroplet surface curvature can be

expressed as (Zeng and Korsmeyer 2004)

½½pel��n� ½½phyd��n ¼ cdfðr � nÞn; ð12Þ

where [[pel]] and [[phyd]] are the respective electrostatic

and hydrodynamic pressure changes across the droplet–

filler interface, respectively,n is the normal unit vector to

the interface, and cdf is the droplet–filler surface tension.

This discontinuity equation can be simplified by noting that

the surface force density of the electric origin must have no

shearing component. The pressure and surface tension

contributions are therefore normal to the interface (Kang

2002; Jones 2005). The curvature of the interface can then

be extracted from Eq. 2 by determining the electrostatic

pressure from the electric potential and field in the system

and by determining the hydrodynamic pressure from

Navier–Stokes and continuity equations inside the micro-

droplet and the filler.

2.2.3 Microdroplet contact angle

It is shown that the contact angle of a moving microdroplet

(dynamic contact angle) differs from its static value (static

contact angle) at equilibrium (Blake and Coninck 2002;

Keshavarz-Motamed et al. 2010). Using Frenkel–Eyring

activated rate theory of transport in liquids (Blake and

Coninck 2002; Keshavarz-Motamed et al. 2010), the static

contact angle, hS, and the dynamic contact angle, hD, can

be related to the microdroplet transport velocity as

cos hS � cos hD ¼vtransportn

cdf

: ð13Þ

2.2.4 Threshold condition

It has been shown before that there exists a threshold force

caused by pinning and hysteresis which prevents droplet

motion before sufficient applied voltage (Gao and

McCarthy 2006). However, implementing the threshold

condition to the proposed algorithm is not a trivial task.

Most of the recent modeling efforts suggest to subtract a

constant threshold force from the driving force (Ren et al.

2002; Kumari et al. 2008; Bahadur and Garimella 2006;

Ahmadi et al. 2009). However, since the threshold force

cannot be greater than the driving force, subtracting a

constant threshold force leads to inaccurate results.

Therefore, in this article the hysteresis condition is

implemented by considering an effective voltage as

Veff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2

app � V2tr

q; ð14Þ

where Veff and Vtr are the effective and the threshold

voltage, respectively.

3 Results and discussion

The results of the model are presented in this section. First,

to verify the accuracy of the model, the results obtained

from the model are compared to the experimental results.

After the verification, the scalability of digital microfluidic

devices, and the dependency of the MASF on the electrode

pitch size, gap spacing, and droplet radius are investigated.

Finally, an adjustable force-based switching frequency

method is proposed to allow for higher transport velocity.

3.1 Verification

It is required to evaluate the accuracy of the model in terms

of both displacement and velocity of the microdroplet.

Thus, the modeling results are compared and verified with

previous experimental observations. The experimental

setup introduced by Pollack et al. (2002) is used here. The

system consists of an 800 nm thick film of parylene C

which provides insulation over the control electrodes. Both

top and bottom plates have a 60-nm thick top-coating of

Teflon AF 1600. The filler fluid is 1 cSt silicon oil.

Dynamics of a 900 nl microdroplet of 0.1 M KCl solution

with a diameter of D = 1.9 mm moving between two

plates with a gap spacing of H = 0.3 mm and an electrode

pitch size of L = 1.5 mm is modeled.

The displacement and velocity obtained from the model

are shown in Fig. 3a,b, and compared to the experimental


123

observations. The modeling results are in excellent agree-

ment with the experimental data and the expected trend;

the rising edge of both the model and experimental results

for displacement (in Fig. 3a) are nearly the same; however,

the experimental results indicate that the microdroplet does

not complete the transport over the electrode. This

incomplete transport [as reported in Pollack et al. (2002)] is

contributed to insufficient voltage. Figure 3b shows a sharp

peak around the maximum driving force originated from

the sudden change in the rate of the increasing wetted area.

The difference between the experimental and modeling

values at the beginning of its motion is attributed to the

hysteresis effects in the system (Pollack et al. 2002).

Accurate models ultimately allow us to optimize the mi-

crodroplet motion over multiple electrodes. To maximize

the system throughput, the model must be capable of

predicting the actuating frequencies which deliver appro-

priately timed voltages to the underlying electrodes. It is

important to optimize the switching frequencies for the

desired microdroplet average velocity. The relationship

between these frequencies and the average velocity is

studied in this subsection.

Figure 4 shows the arrival time for the microdroplet

leading edge at various positions (i.e., displacements)

across the three-electrode structure. Results are shown for

four switching frequencies (2.5, 5, 10, and 15 Hz) and an

applied voltage of 26 V. The optimal case for transport

occurs with a switching frequency of approximately 13 Hz.

It is apparent from this figure that increasing frequency

leads to shorter arrival times at the end of the third elec-

trode: the 2.5 Hz case has an arrival time of 0.91 s; the

5 Hz case has an arrival time of 0.48 s; the 10 Hz case has

an arrival time of 0.24 s. However, as it can be seen in

Fig. 5, this trend does not apply to the 15 Hz case (which is

above the 13 Hz), as the microdroplet does not successfully

complete its transport over each individual electrode. Such

a phenomenon has been observed before (Pollack et al.

2002; Arzpeyma et al. 2008); there exists a maximum

actuation switching frequency (MASF) for the applied

voltages. This point is apparent from Fig. 6, which show

the maximum switching frequencies over a range of volt-

ages for model results. The results from the proposed

electrohydrodynamic model are compared to the experi-

mental observations (Pollack et al. 2002), and the results

obtained from the explicit electrostatic and hydrodynamic

modeling approach (Ahmadi et al. 2009) for oil and air

systems. It is clear that the model accurately calculates the

MASF over the range of the applied voltages reported in

the experimental results, and successfully predicts the

hysteresis conditions. The verified model can now be used

to obtain the optimum operational conditions to maximize

the device efficiency.

3.2 Effects of the microdroplet size

In this section, the effects of the microdroplet size on the

microdroplet electrohydrodynamics and the MASF is

studied. The microdroplet motion is modeled for its

transport over three electrodes (with an electrode pitch of

L = 1.5 mm and gap spacing of H = 0.3 mm). The

selected applied voltage is 35 V, which is between the

upper and lower values of the voltages (13 and 40 V) used

for the calculation of MASF (see Fig. 6). The numerical

MASF value calculated for this selected voltage (35 V) is

in good agreement with the experimental value. An arbi-

trary switching frequency of 15 Hz is used. Microdroplets

with three different diameters of 1.6, 2.25, and 3 mm are

modeled. The results of the model are presented in terms of

the microdroplet displacement and driving and opposing

forces during its motion. Figure 7 presents the behavior of

the microdroplet (with different sizes) under the same

actuation conditions. The leading edge position of the

microdroplet is shown as a function of time in Fig. 7a. The

difference in the initial positions is due to the fact that the

microdroplet center is aligned with the center of the initial

electrode. This is the most common initial position for a

(a) (b)Fig. 3 The a displacement and

b velocity are compared to the

experimental results (Pollack

et al. 2002). The applied voltage

is 26 V, and the results are

shown for a transition of the

microdroplet over one

electrode. Dynamics of a 900 nl

microdroplet of 0.1 M KCl

solution with a diameter of

D = 1.9 mm moving between

two plates with a gap spacing of

H = 0.3 mm and an electrode

pitch size of L = 1.5 mm is

modeled


123

stationary microdroplet as it self-centers itself with the

initial electrode after dispensing (Fair 2007). As it can be

seen in Fig. 7a, the smallest microdroplet reaches the last

electrode in the shortest time. However, since the same

switching frequency (i.e., 15 Hz) is used, the average

transport velocity is the same for all of the microdroplet

sizes. The driving force for each microdroplet is shown

Fig. 7b with respect to the leading edge position. The

driving forces acting on the larger microdroplets are bigger.

On the other hand, the opposing forces (i.e., shear, drag,

and three-phase contact line) increase due to the increased

microdroplet mass and surface area. Therefore, the mi-

crodroplets move slower as their sizes increase. The results

presented reveal the fact that the microdroplet size is

a key parameter in determining the microdroplet

electrohydrodynamics.

To investigate the relation between the microdroplet

size and the MASF, transport of three different microdro-

plet sizes over six electrodes (for the same system as

before) is studied. Figure 8 shows the time required for

completing the transport for three different microdroplet

sizes as a function of the switching frequency. For each

size, as the switching frequency increases the time required

decreases. However, the switching frequency cannot be

increased above a certain frequency which is called the

MASF. Interestingly, the MASF is strongly dependent on

the microdroplet size (see Fig. 8); smaller microdroplets

allow for higher MASF values.

3.3 Effects of system architecture

The system architecture plays a crucial role in microdroplet

dynamics. In essence, structural layers and geometric

parameters of the device have the most important effects on

the microdroplet motion (Ren et al. 2002). In this section,

the effects of the gap spacing and electrode pitch size on

the MASF are individually analyzed, and scalability of

digital microfluidic architectures is investigated. Figure 9

shows the MASF versus microdroplet diameter for three

different gap spacings (i.e., H = 0.6, 0.3, and 0.15 mm).

The electrode pitch size is kept constant as L = 1.5 mm.

The results from Fig. 9 confirm again that as the droplet

diameter increases, the MASF will decrease (given the

constant gap spacing). However, for the same microdroplet

diameter, as the gap spacing decreases, microdroplet

motion becomes slower due to the resulting increase in the

wall force. This is an interesting observation, as it shows

that for pre-fabricated top and bottom plates, faster trans-

port can be achieved simply by controlling the gap spacing

and the microdroplet volume. The effect of the electrode

Fig. 4 The arrival time for the modeled microdroplet leading edge is

shown for various positions (i.e., displacements) across the three-

electrode structure. The applied voltage is 26 V, and the results are

shown for four switching frequencies (2.5, 5, 10, and 15 Hz)

time = 0.4 s time = 0.8 s time = 1.2 stime = 0.0 s




Switching Frequency = 2.5 Hz



Switching Frequency =15.0 Hz

Fig. 5 The concept of

maximum actuation switching

frequency (MASF) is illustrated


123

pitch size on the MASF is shown in Fig. 10a. The gap

spacing is kept constant as H = 0.3 mm. The results show

that by decreasing the electrode pitch size the system

MASF increases. To determine the effect of the electrode

pitch size on the transport velocity, a similar plot is shown

in Fig. 10b. Interestingly, by decreasing the electrode pitch

size and keeping the gap spacing constant, for the same

diameter-electrode pitch size ratio, higher transport veloc-

ities can be achieved. This increase in the velocity can be

due to the decrease in the shear force because of the

reduced in the wetted area.

It has been shown before that digital microfluidic

structures are highly scalable (Pollack et al. 2002). The

scalable nature of these systems are studied here. Two

system designs are modeled: L = 1.5 mm and H = 0.6

mm; L = 0.75 mm and H = 0.3 mm. The size of the

second design is half of the size of the first design.

Figure 11 shows the MASF of these setups over a range of

microdroplet diameter-electrode pitch size ratios (D/L).

The MASF of the smaller design is larger than that of the

bigger design. However, the transport velocity of both

systems is the same (see Fig. 11b). This is a very important

observation which confirms the scalable nature of digital

microfluidic systems.

3.4 Adjustable force-based switching frequency

implementation

Constant switching frequency implementation is the sim-

plest actuation algorithm. However, by applying a constant

Applied Voltage(V)

MASF (Hz)

10 20 30 400

10

20

30

40

Experimental ResultsExplicit E/H ResutsCoupled EH Results

50 60 70

50

/

Oil Filler Air Filler

Fig. 6 The maximum switching frequencies for the proposed model

are shown for a range of voltages and compared to the explicit

electrostatic and hydrodynamic modeling (Ahmadi et al. 2009) and

(Pollack et al. 2002) experimental results for oil and air filler systems

Time (s)

Displacement (mm)

0.00 0.05 0.10 0.150

1

2

3

4

5

6D=1.60 mmD=2.25 mmD=3.00 mm

(a)

Displacement (mm)

Force ( N)

0.00 1.00 2.00 3.00 4.000

5

10

15

20

25

30

35

40D=1.60 mmD=2.25 mmD=3.00 mm

µ

(b)

Fig. 7 The effects of the microdroplet size on electrohydrodynamic properties are shown for three different diameters. a The microdroplet

leading edge position is shown as a function of time. b The driving force is shown as a function of the microdroplet leading edge position

Completion time(s)

Frequency (Hz)

0.20 0.25 0.30 0.35 0.40 0.45

10

15

20

25

30D=1.60 mmD=2.25 mmD=3.00 mm

Fig. 8 The time required to complete the transport over six

electrodes is shown as a function of the switching frequency for

microdroplets of three different diameters


123

frequency the device will not operate at its optimum

throughput and maximum transport velocity. In Fig. 12, the

maximum number of electrodes in which the microdroplet

can be transported (before it stops) is shown as a function

of the actuation switching frequency. Depending on the

constant switching frequency, the microdroplet will ulti-

mately stop after completing its transport over a limited

number of electrodes. Adjustable switching frequency

algorithms have been proposed to address this issue

(Arzpeyma et al. 2008; SadAbadi et al. 2010). Arzpeyma

et al. (2008) showed that optimum actuation is achieved

via actuating the next electrode as soon as the microdroplet

touches it. This has been referred to as an adjustable

position-based actuation algorithm. In this article, a new

electrode actuation algorithm is proposed in which actua-

tion is based on finding the maximum force acting on the

microdroplet (i.e., adjustable force-based actuation). As it

can be seen in Fig. 13, the proposed algorithm is based on a

feedback control submodule which monitors the micro-

droplet location. After sensing the microdroplet position,

the algorithm considers two potential actuation scenarios ,

as it is demonstrated in Fig. 14. After calculating the

driving force using the electrohydrodynamic model for

each scenario, the proposed algorithm chooses the opti-

mum actuation scheme which leads to higher velocity. This

performance of the proposed algorithm was examined with

respect to the other methods available (i.e., constant fre-

quency, adjustable position-based frequency). Figure 15

presents the results of this comparison. In essence, it shows

the time required to complete transport over six electrodes

as a function of the microdroplet leading edge position for

three different algorithms: constant frequency, adjustable

position-based frequency, and adjustable force-base fre-

quency. It is evident from the results that the adjustable

Droplet Diameter (mm)

MASF (Hz)

1.50 2.00 2.50 3.0010

15

20

25

30

35

H=0.6 mmH=0.3 mmH=0.1 5mm

Fig. 9 MASF is shown for three different gap spacings as a function

of microdroplet diameter

D/L

MASF (hz)

1.00 1.50 2.00 2.50 3.000

20

40

60

80

100

120 L=0.375 mmL=0.750 mmL=1.500 mmL=3.000 mm

(a)

D/L

Transport Velocity (mm/s)

1.00 1.50 2.00 2.50 3.0010

20

30

40

50

L=0.375 mmL=0.750 mmL=1.500 mmL=3.000 mm

(b)Fig. 10 The effect of the

electrode length on the MASF

and the microdroplet transport

velocity is shown. a The

increase in the MASF as the

length of the electrode decreases

is shown. b The increase in the

transport velocity due to the

decrease in the wetted surface

area is shown

D/L

MASF (hz)

1.00 1.20 1.40 1.60 1.80 2.000

20

40

60

80L=0.75 mm

L=1.50 mmH=0.60 mm

H=0.30 mm

(a)

D/L

Transport Velocity (mm/s)

1.00 1.20 1.40 1.60 1.80 2.000

20

40

60

L=0.75 mm

L=1.5 mmH=0.30 mm

H=0.60 mm

(b)Fig. 11 a The MASF of two

designs over a range of

microdroplet diameter-electrode

pitch size ratios (D/L) is shown.

b The microdroplet transport

velocity of two designs over a

range of microdroplet diameter-

electrode pitch size ratios

(D/L) is shown


123

force-based frequency results in faster microdroplet

motion. It should be noted that both adjustable position-

based actuation and adjustable force-based actuation will

lead to the same result for the case where the microdroplet

diameter is equal to the adjustable force-based electrode

length. However, as the ratio of the microdroplet diameter

and electrode size increases, the adjustable force-based

algorithm provides much faster motion compared to the

adjustable position-based actuation. The in situ control of

microdroplet motion in digital microfluidic systems is a

crucial task. Depending on the size of the system, the most

suitable switching frequency spectrum has to be found to

achieve the highest transport velocity.

4 Conclusion

In this study, the electrohydrodynamic method is imple-

mented numerically to investigate the effects of system

architecture and microdroplet size on the digital microflu-

idic system dynamics. The proposed methodology provides

a vital tool in the design and control processes, and can be

used for numerous applications including optimum routing

and achieving higher transport velocities. The modeling

results showed that for pre-fabricated top and bottom

plates, faster transport can be achieved simply by con-

trolling the gap spacing and the microdroplet volume. It

was also shown that depending on the constant switching

frequency, the microdroplet will ultimately stop after

completing its transport over a limited number of elec-

trodes. Therefore, a force-based adjustable switching fre-

quency was proposed and implemented. Compared to the

constant frequency and adjustable position-based algo-

rithms, the proposed adjustable forced-based algorithm

provides higher velocities.

Frequency (Hz)

Number of Electrodes Transported

22.00 24.00 26.00 28.00 30.000

5

10

15

Fig. 12 The maximum number of electrodes which the microdroplet

can be transported is shown as a function of the actuation switching

frequency (before it stops)

Fig. 13 The proposed algorithm is based on a feedback control

submodule which monitors the microdroplet location

Fig. 14 The algorithm considers two potential actuation scenarios.

After calculating the driving force using the electrohydrodynamic

model for each scenario, the proposed algorithm chooses the optimum

actuation scheme which leads to higher velocity

Displacement (mm)

Time (sec)

0.00 2.00 4.00 6.000.00

0.05

0.10

0.15

0.20Constant FrequencyAdjustable Force-based FrequencyAdjustable Position-based Frequency

Fig. 15 The time required to complete the transport over six

electrodes is shown as a function of the microdroplet leading edge

position for three different algorithms: constant frequency, adjustable

position-based frequency, and adjustable force-based frequency


123

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Numerical study of the microdroplet actuation switching frequency in digital microfluidic biochips

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