# Modeling and Control of Electrowetting Induced Droplet Motion

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## Abstract

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Nomenclature | ||
---|---|---|

Symbol | Description | Units |

A | Area | m^{2} |

c | Damping coefficient | Ns∙m^{−1} |

D | Electric displacement | C∙m^{−2} |

E | Electric field | V∙m^{−1} |

E | Energy | J |

F | Force | N |

F_{CL} | Contact line force | N |

F_{D} | Drag force | N |

F_{el} | Electrostatic force | N |

F_{W} | Friction Force | N |

f | Frequency | Hz |

g_{e} | Gap between control electrodes | m |

H | Droplet height | m |

k | Spring constant | N∙m^{−1} |

M | Droplet mass | kg |

R | Radius | m |

U_{av} | Average droplet velocity | m∙s^{−1} |

V | Voltage | V |

v | Vertical droplet velocity | m∙s^{−1} |

Vol | Droplet volume | m^{2} |

W | Energy | J |

Ẇ | Power | W |

w_{e} | Width of an electrode | m |

x | Droplet position | m |

Greek Symbols | ||

α | Tilt angle | rad |

ζ | Coefficient of contact line friction | Pa∙s |

θ_{A} | Advancing contact angle | rad |

θ_{R} | Receding contact angle | rad |

μ | Fluid dynamic viscosity | N∙s∙m^{−2} |

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

ω_{0} | Natural angular frequency | rad/s |

## 1. Introduction

## 2. General Description of the Dynamic Droplet Model

_{av}, which is the volume-averaged velocity of the droplet. The internal motion inside the droplet and the consequent relative velocities are not considered here.

#### 2.1. Dynamic Model Formulation

_{w}, the contact line friction force F

_{CL}and the viscous drag F

_{D}due to the droplet moving through the filler liquid [27,28]. The contact line friction force includes the threshold effect that is observed for droplet movement. The droplet will deform under the electrostatic force and lead to contact angle hysteresis between the advancing angle θ

_{A}and the receding angle θ

_{R}[29]. In order to move the droplet, this electrostatic force needs to overcome a critical force due to the contact angle hysteresis. Since the droplet is considered as a single discrete mass moving through the channel, a one-dimensional force balance of the forces acting on the droplet, projected on the actuation direction, can be written as follows:

- M: mass of the droplet;
- U
_{av}: average velocity of the droplet; - F
_{el}: electrostatic driving force; - F
_{w}: shear force between the droplet and the channel; - F
_{CL}: contact-line friction force; - F
_{D}:drag force on filler liquid.

#### 2.1.1. Formulation for the Forces Acting on the Droplet

_{el}can be calculated as the negative gradient of the electrostatic energy. The total electrostatic energy U

_{el}in the system, with a volume Vol, is given by

**Figure 1.**Example of a grid used in a 2D electrostatic simulation (

**left**). Detail of the refined mesh around the three-phase line where high gradients in the electric field are expected (

**right**).

_{w}between the droplet and channel can be estimated when the velocity profile in the channel is known. A velocity profile can be assumed in the droplet with zero slip boundary conditions on top and bottom walls. Detailed simulations of the fluid flow inside the droplet show that a parabolic velocity profile is observed across the height of the channel if a no-slip boundary condition at the top and bottom wall is assumed [22]. Based on the average droplet velocity U

_{av}, the vertical velocity profile in the channel can be determined. The geometrical properties of the system are as follows: R

_{c}is the radius of the contact line circle and H is the gap between the top and bottom wall. The opposing forces can now be expressed as a function of the geometry of the droplet in the considered system. Assuming a parabolic velocity profile in the channel, the total shear force exerted by the top and bottom can be written as follows:

_{D}on the droplet can be estimated by:

_{D}is the drag coefficient and ρ

_{f}is the density of filler fluid and A

_{c}is the cross section area. The situation of a droplet moving through the filler fluid can be approximated by a cylinder in cross flow for the calculation of the drag coefficient. The drag coefficient C

_{D}depends on the Reynolds number of the flow. For a low R

_{e}number, which is the case in an electrowetting induced droplet flow, the drag coefficient C

_{D}is inversely proportional to the Reynolds number for the filler liquid and proportional to a constant a, depending on the geometry of the object. In the case of a sphere, a is 24. For a cylinder, a can be assumed to be 12. To estimate the drag coefficient on the droplet, a value between the sphere and cylinder case is chosen. As a result, the viscous drag force F

_{D}will scale linearly with the droplet velocity and can be written as follows:

_{CL}originates from intermolecular attraction forces near the contact line of the droplet on the solid surface. Many different approaches exist to account for this effect, ranging from experimental empirical correlations [28,30,31] to complex molecular-kinetics modeling [32]. An excellent review of recent theoretical, experimental and numerical progress in the description of moving contact line dynamics can be found in [33]. In the work presented in this paper, an approximation for the static and the dynamic contact line friction force is used, for which the coefficients are experimentally determined. A dedicated test fixture is designed and fabricated to perform tilt tests for a droplet confined between two parallel surfaces. During the tilting of the fixture, the advancing and receding contact angle and the positions of the droplet interface (in the case of droplet motion) and the tilt angle required to initiate droplet movement between two parallel Teflon coated surfaces were studied (Figure 2). The gravitational force for this tilt angle corresponds to the static contact line friction. This also corresponds to the maximal static contact angle hysteresis. For a higher force, the contact angle hysteresis will not increase any more and the droplet will start moving. From the experimental results, the maximum static contact angle hysteresis factor [cos(θ

_{R}) - cos θ

_{A}]

_{max}can be experimentally determined as a function of the droplet volume, the channel height and the surface. In Figure 3(a), an image of the droplet during the tilt test is shown, indicating the advancing and receding contact angles. Figure 3(b) shows the evolution of the contact angles as a function of the tilt angle in the regime where the gravitational force is smaller than the contact line friction force. The expression for this static contact line friction force is [34]:

_{A}and θ

_{R}are the advancing angle and the receding angle respectively and k is a constant depending on the shape of the droplet contact line.

**Figure 2.**Picture of the confined droplet between two Teflon coated surfaces during a tilt test (

**a**). Evolution of the advancing and receding contact angle as a function of the tilt angle (

**b**).

**Figure 3.**Measurement data for the evolution of the advancing and receding contact angle (

**a**) and the total contact line friction force (

**b**) as a function of the droplet velocity for a moving, confined droplet between two Teflon surfaces.

_{CL}. The left hand side of Figure 3 shows the evolution of the contact angles θ

_{A}and θ

_{R}as a function of the velocity. Since the contact line friction force depends on θ

_{A}and θ

_{R}, the friction force will also be a function of the velocity. Different algebraic expressions are presented in literature to approximate this dynamic behavior of the friction force. A commonly used expression based on the molecular interaction theory from [32] describes the dynamic part of the friction force as a power law function of the velocity:

_{c}the radius of the contact line and the droplet width w = 2R

_{c}. For an actuation force smaller than the threshold force, the actuation force is compensated for by static contact line friction force, due to the droplet deformation. When the actuation force is high enough to overcome the threshold value of the static contact line friction force the droplet will start moving. Even for an increasing actuation force, the static contact line friction force is assumed to remain equal to the threshold force.

#### 2.1.2. Summary for the Model Formulation

Category | Description | Symbol | Planar |
---|---|---|---|

Geometrical | Droplet volume | Vol | 2.7 μL |

Channel height/diameter | H/D | 1 mm | |

Electrode pitch | w_{e} + g_{e} | 1 mm | |

Electrode gap | g_{e} | 100 μm | |

Drag coefficient | C_{d} | 30 | |

Insulation thickness | t | 1 μm | |

Material properties | Droplet viscosity | μ_{d} | 1.005 Pa∙s |

Droplet density | ρ | 1,000 kg/m^{3} | |

Surface tension | γ | 72 Mn/m | |

Contact line friction (static) | [cos(
θ_{R}) - cos θ_{A}]_{max,st}_{.} | ~8 μN | |

Contact line friction coeff. | ζ | 0.08 Ns/m^{2} | |

Contact angle | θ | 110° | |

Dielectric const. insulation | ε_{r} | 3 | |

Application | Voltage | V | 45 V |

Start position | x_{0} | −1 mm | |

Switching frequency | f | 25 Hz |

#### 2.2. Model Linearization

_{el}> F

_{CL,static}can be described as a damped mass-spring system. This simplified model can be used to develop a control strategy for the switching of the electrodes. The dynamic contact line friction, the drag force and the viscous dissipation force are proportional to the droplet velocity and can be described with an equivalent viscous damping coefficient c

_{eq}(Ns/m). The net actuation force F

_{el}can be linearized in the region of the centre of the activated electrode and therefore be represented by an equivalent spring constant k

_{eq}(N/m). The equivalent damping coefficient can be obtained from linearization of the function of the net actuation force as a function of the droplet position. This linearization can be derived from the analytical formulation of the actuation force. Such analytical descriptions can be found in [26,27,31]. In this paper, the linearization is numerically derived from a response surface model (RSM) that is fitted to a large design of experiments (DOE) of electrostatic simulations performed for the parameters listed in Table 1. The equivalent damping coefficient is

**Figure 4.**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).

_{0}and an equivalent damping ratio ζ

_{eq}. In this way, the possible oscillatory behavior can be predicted and an optimal switching frequency for droplet motion over several subsequent electrodes can be obtained. The undamped angular frequency ω

_{0}, the damping ratio ζ

_{eq}and the damped natural frequency ω

_{d}are defined as:

**Figure 5.**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).

_{eq}determines the behavior of the droplet motion. For ζ

_{eq}> 1, the system is overdamped and the droplet moves towards the electrode centre without oscillations. For ζ

_{eq}= 1, the system is critically damped. For ζ

_{eq}< 1, the system is underdamped, and the system oscillates with a damped natural frequency ω

_{d}. In the case of the electrostatically actuated droplet, the natural frequency and damping ratio depend on the actuation voltage for a certain configuration (system and material parameters). A certain voltage corresponds to a critical damped droplet behavior ζ

_{eq}= 1. For lower voltages, the droplet motion will exhibit overdamped behavior and for higher voltages, oscillations will occur (Figure 5).

#### 2.3. Model Limitations

_{c}. The capillary length is defined as follows:

## 3. Droplet Motion and Control

#### 3.1. Model Solution Strategy

_{el}(x,t) for three consecutive power electrodes with frequency f, can be written as follows :

_{el,i}(x) is the actuation force as a function of the distance from the centre of activated electrode ‘i’. Using a similar method, the time dependent force can be described for any switching pattern, including multiple electrodes activated at the same time, using superposition of the different force profiles.

**Figure 6.**(

**a**) Electrostatic actuation force profile as a function of the droplet position at three subsequent electrodes. (

**b**) Schematic representation of the switching of the voltage in the actuation electrodes.

#### 3.2. Single Electrode Response

**Figure 7.**Results of the dynamic droplet position model for the parameters listed in Table 1 for the parallel plate test case for actuation voltages of 45 V and 75 V.

#### 3.3. Droplet Trajectory over an Array of Electrodes

**Figure 8.**(

**a**) Dynamic droplet response for the actuation over three subsequent electrodes for an actuation voltage of 40 V. (

**b**) Evolution of the actuation and opposing force on the droplet as a function of time.

#### 3.4. Influence of Switching Frequency

**Figure 9.**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 Table 1. The switching frequency is compared for the results of the dynamic model using the actual force profile and the results of the linearized damped spring-mass model (two times the natural frequency).

_{0}is the angular natural frequency of the damped mass-spring system in Equation (13). By comparing the switching frequency to the maximum frequency obtained from the simulations with the actual force profile, two differences can be observed. The first difference is that the damped mass-spring system predicts a continuous droplet motion from one electrode to the next for a lower voltage than the actual force profile. For voltages higher than 18 V, a droplet motion is realized. For this moving droplet, using the linearized model, a natural frequency and consequently a switching frequency can be found. In reality however, the droplet will not move far enough to be actuated by the next electrode. Only for a voltage of minimum 25 V can continuous droplet motion be achieved. The second deviation between the linearized model and the actual force profile is that the linearized model underpredicts the value of the switching frequency. This can be explained by the fact that the actuation force on the droplet already starts acting before the droplet reaches the centre of the previous electrode. As a result, the voltage can be switched slightly before this position is reached, leading to a slightly higher achievable switching frequency. Despite those two deviations, the linearized model provides a useful approximation of the optimal switching frequency within 10% of the results obtained by the dynamic model using the actual force profile. Therefore, the linearized model can be used as a fast tool to develop a control strategy for the activation of the electrodes in order to optimize the droplet flow.

## 4. Conclusions

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**MDPI and ACS Style**

Oprins, H.; Vandevelde, B.; Baelmans, M.
Modeling and Control of Electrowetting Induced Droplet Motion. *Micromachines* **2012**, *3*, 150-167.
https://doi.org/10.3390/mi3010150

**AMA Style**

Oprins H, Vandevelde B, Baelmans M.
Modeling and Control of Electrowetting Induced Droplet Motion. *Micromachines*. 2012; 3(1):150-167.
https://doi.org/10.3390/mi3010150

**Chicago/Turabian Style**

Oprins, Herman, Bart Vandevelde, and Martine Baelmans.
2012. "Modeling and Control of Electrowetting Induced Droplet Motion" *Micromachines* 3, no. 1: 150-167.
https://doi.org/10.3390/mi3010150