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In this paper, a general methodology for the dynamic study of electrostatically actuated droplets is presented. A simplified 1D transient model is developed to investigate the transient response of a droplet to an actuation voltage and to study the effect of geometrical and fluidthermal properties and electrical parameters on this behavior. First, the general approach for the dynamic droplet motion model is described. All forces acting on the droplet are introduced and presented in a simplified algebraic expression. For the retentive force, the empirically extracted correlations are used, and for the electrostatic actuation force, results from electrostatic finite element simulations are used. The dynamic model is applied to electrowetting induced droplet motion between parallel plates in the case of a single actuation electrode and for an array of electrodes. Using this methodology, the influence of the switching frequency and actuation voltage is studied. Furthermore, a linearized equivalent damped mass—spring model is presented to approximate the dynamic droplet motion. It is shown that the optimal switching frequency can be estimated by twice the natural frequency of the linearized damped mass—spring system.
Nomenclature  

Symbol  Description  Units 

Area  m^{2} 

Damping coefficient  Ns∙m^{−1} 

Electric displacement 
C∙m^{−2} 

Electric field  V∙m^{−1} 

Energy  J 

Force  N 

Contact line force  N 

Drag force  N 

Electrostatic force  N 

Friction Force  N 

Frequency  Hz 

Gap between control electrodes 
m 

Droplet height  m 

Spring constant  N∙m^{−1} 

Droplet mass  kg 

Radius  m 

Average droplet velocity  m∙s^{−1} 

Voltage  V 

Vertical droplet velocity  m∙s^{−1} 

Droplet volume  m^{2} 

Energy  J 

Power  W 

Width of an electrode  m 

Droplet position  m 



Tilt angle  rad 

Coefficient of contact line friction  Pa∙s 

Advancing contact angle  rad 

Receding contact angle  rad 

Fluid dynamic viscosity  N∙s∙m^{−2} 

Dynamic viscosity filler medium  N∙s∙m^{−2} 

Natural angular frequency  rad/s 
Electrowetting is a technique to manipulate fluids on a millimetre or micrometre scale by altering the wetting properties under the application of an electrical field [
Another application of digital microfluidics is the cooling of electronic systems. Researchers from Duke University [
In this paper, a generic simplified macroscopic model is described to predict the dynamics of droplet motion as a reaction to the application of a voltage at a single electrode or subsequently at an array of electrodes. Inputs for this model are the experimentallyderived material characterization properties and the calculated electrostatic actuation force. In literature, several dynamic models are reported [
A simplified onedimensional model is presented to describe the dynamic response of a single droplet, based on the forces acting on the droplet. The droplet is considered as a single mass
The driving force for the droplet motion is the electrostatic actuation force generated by the electrode. Important opposing forces acting on the droplet are the shear force between the droplet and channel
with
The differential Equation (1) describes the dynamic behavior of the droplet as a function of the driving force and the opposing forces. The formulation of the different forces acting on the droplet will be discussed more in detail in the next sections. In
The electrical modeling involves the computation of two coupled phenomena: the electric field distribution in the channel and the shape of the droplet. Because of their mutual influence, the two phenomena should be solved iteratively until a converged solution for net force and shape is achieved. The electrostatic actuation force
Because only the horizontal component of the electrostatic force contributes droplet motion and in the case where only the contribution of the electrostatic energy to the total energy is considered, the net actuation force is given by
First, the shape of the droplet is calculated using the software package Surface Evolver. This tool obtains the equilibrium shape of the droplet by minimizing the total potential energy of the system (gravitational, electrical and surface tension). Subsequently, the obtained shape is used in the finite element simulations using the software tool MSC.Marc to calculate the potential
Example of a grid used in a 2D electrostatic simulation (
The shear force
where
Assuming that a droplet is moving through the filler fluid as a rigid body, the viscous drag force
where,
In the case of air as a filler medium, the drag force is negligible compared to the dynamic contact line friction force and the viscous force.
The contact line friction force
where
Picture of the confined droplet between two Teflon coated surfaces during a tilt test (
Measurement data for the evolution of the advancing and receding contact angle (
In the dynamic case, these contact angles are a function of the velocity of the moving contact line
where
with
In the case of the actuation force being smaller than the threshold force, the static contact line friction force is equal to the actuation force and, as a result, the droplet will not move. The contact line force created by the droplet deformation will compensate for the actuation force on the droplet. Below the threshold force, the relation between the droplet deformation and the actuation force is given by Equation (8). For actuation forces higher than the threshold force, the droplet will move. For this dynamic case, the droplet motion can be described by Equation (1) using the description of the forces above:
The electrostatic actuation force is calculated by using finite element simulations. This force is calculated as a function of the applied voltage, the droplet volume, the electrode geometry (pitch and gap), the height of the channel or channel diameter, the dielectric constant and thickness of the insulation layer and the contact angle.
Overview of the dynamic model parameters and the reference values used in the test cases.
Category  Description  Symbol  Planar 

Geometrical  Droplet volume 

2.7 μL 
Channel height/diameter 

1 mm  
Electrode pitch  1 mm  
Electrode gap 

100 μm  
Drag coefficient 

30  
Insulation thickness 

1 μm  
Material properties  Droplet viscosity 

1.005 Pa∙s 
Droplet density 

1,000 kg/m^{3}  
Surface tension 

72 Mn/m  
Contact line friction (static)  [cos(

~8 μN  
Contact line friction coeff. 

0.08 Ns/m^{2}  
Contact angle 

110°  
Dielectric const. insulation 

3  
Application  Voltage 

45 V 
Start position 

−1 mm  
Switching frequency 

25 Hz 
The behavior of a moving droplet towards an activated electrode in the case of a sufficiently high actuation force
After linearization of the force profile, the droplet motion equation for a moving droplet in Equation (10), can be written as an equivalent equation of a damped springmass system:
Force profile as a function of the droplet position for an actuation voltage of 45 V: actual calculated profile (solid black line), difference between the actuation force and the static contact line friction (solid grey line) and the linearized force profile (dashed line).
Based on the nature of the motion equation, the droplet motion can exhibit oscillatory behavior. The linearized equation allows analysis of the droplet motion by calculation of an equivalent natural angular frequency
Force profile as a function of the droplet position for an actuation voltage of 45 V: actual calculated profile (solid black line), difference between the actuation force and the static contact line friction (solid grey line) and the linearized force profile (dashed line).
The value of the equivalent damping ratio
The validity of the dynamic model is limited to a certain range of the system parameters. For parameter values outside this range, the dynamic model can no longer be used to accurately predict the droplet motion. Firstly, the application of the dynamic model is only valid for sufficiently small droplet volumes. The force calculations are performed for a certain droplet shape, assuming that the effect of surface tension dominates the effect of the gravity. For large droplet volumes, the droplet shape is flattened, due to the increased importance of the gravity. The characteristic length to assess the relative impact of the surface tension and the gravity is the capillary length
In the case of a water droplet at 20 °C in air, the capillary length is around 2.7 mm. For droplets with dimensions smaller than the capillary length, surface tension dominates. For dimensions larger than the capillary length, gravity will have an impact on the droplet shape and the dynamic model cannot be used any more to predict the transient droplet motion.
Secondly, the dynamic model cannot be used for very high actuation voltages. The first reason is the effect of voltage saturation of the electrowetting effect. Above a certain threshold voltage, an increase in voltage will not result in a further contact angle decrease and consequently a higher actuation force acting on the droplet. The saturation limit of the voltage depends on the material properties of the solid surface and the liquid. The dynamic model should be used below this saturation limit. The second voltagerelated limitation is caused by the required rise time of the voltage signal. In the model, a step function of the voltage application is assumed. In reality, a finite rise time is required to reach the desired voltage. This rise time is not included in the dynamic model.
The electrostatic actuation force is known as a function of the droplet position
where
(
In the considered test case, the electrode pitch is 1 mm, the electrode gap is 100 µm, the height of the channel is 1 mm and the droplet volume is 2.7 µL. First, the dynamic droplet motion of an individual droplet is calculated for a single activated electrode. A constant voltage is assumed to be applied to an electrode close to the droplet. This will be first illustrated for voltages of 45 V and 75 V. This can be considered an infinitely long voltage pulse. Even if the voltage is kept switched on, the droplet will stop at a position where the actuation force is lower than the opposing forces and the droplet does not have enough inertia to move on. The position after actuation depends on the applied voltage and the opposing forces. In this case, the force at the starting point is 22 µN. This is higher than the threshold force of 8 µN and therefore the droplet will start moving. As long as the net actuation force is higher than the opposing forces, the droplet will accelerate. When the opposing forces are higher than the actuation force the droplet will slow down and eventually stop. Depending on the opposing forces and applied voltage, this can mean that the droplet could stop before reaching the centre of the powered electrode if the voltage is too low (in the case of 45 V), or on the other hand that the droplet will overshoot the centre (in the case of 75 V). When the droplet overshoots the centre of the electrode, the droplet is pulled back towards the centre by the reverse electrostatic force. As a result, the droplet will perform an oscillation around the centre until its inertia is too small to overcome the opposing forces. The result is a damped oscillation around the centre of the powered electrode. The point where the droplet stops is important to ensure actuation by the next electrode in an array of subsequent electrodes.
Results of the dynamic droplet position model for the parameters listed in
The purpose of the array of electrodes in the microfluidic system is to generate a continuous flow of discrete liquid droplets. To generate a continuous motion of the droplet through the channel, the droplet should be attracted from one electrode to the next by subsequently applying a voltage to the array of electrodes. This means that analysis for a single electrode can be repeated for all the electrodes in the array. What the droplet motion through the channel looks like, depends on the actuation voltage, the switching frequency, the starting position of the droplet and the opposing forces. The end position of the droplet motion for the first activated electrode is the starting position of the motion for the second actuated electrode. The droplet motion from one electrode to the next will be successful if the actuation force of the next electrode acting on the droplet is sufficiently high for the position the droplet is in at the moment. The voltage is switched off in the first electrode and switched on in the second. It is important to carefully control the switching frequency. If the frequency is set too high, the electrodes will switch before the droplet reaches the region of attraction of the next electrode and the droplet will stop. The safest option is to wait to switch until the droplet has stabilized. However, a successful droplet motion can be achieved by switching before this time at the right side of the oscillation. On the other hand, if the velocity is too low, the flow rate is suboptimal and the droplet will be waiting before being switched to the next electrode. In this case, the power consumption will be unnecessarily high. To develop an efficient control of the voltage and frequency of the droplet actuation, the transient droplet motion across multiple electrodes is studied more in detail.
(
From the previous graphs, it could be observed that the chosen frequency is safe to actuate the droplet over that particular array of electrodes. However, it is also clear that there is still room for improvement to increase the flow rate for the same voltage by increasing the frequency.
Maximum switching frequency as a function of the actuation voltage for the planar electrode configuration with air as filler liquid and with the system parameters listed in
In
where
In this article, a general methodology for the dynamic modeling of electroactuated droplets is presented to predict transient droplet motion. In this simplified approach, the droplet is dealt with as a single mass. The dynamic model predicts the macroscopic droplet motion, based on a force balance of all forces acting on the droplet. The opposing forces on the droplet are, as described, algebraic expressions as a function of the average droplet velocity. For the electrostatic actuation force, simulation results are used. Empirical models are derived from tilt tests of confined droplets to describe the contact line friction forces. Furthermore, a linearized equivalent damped mass—spring model has been derived to approximate the dynamic droplet motion. This linearized model can be used to develop control strategies for the switching of the electrodes.
Next, the methodology is applied to an array of electrodes. The voltage is switched subsequently from one electrode to the next. For each actuation voltage, a maximum switching frequency is found that the droplet will be able to follow. For higher frequencies, the droplet will not reach the next electrode before the voltage is switched. The maximal switching frequency can be approximated by twice the natural frequency of the linearized damped mass—spring system. Therefore, the linearized model can be used for the development of control strategies to optimize the switching of the electrodes. Using an optimized switching frequency, the dynamic model predicts droplet velocities in the range of 5 to 10 cm/s for a voltage range from 30 to 80 V.