# A numerical technique to simulate display pixels based on electrowetting

## Abstract

We present a numerical simulation technique to calculate the deformation of interfaces between a conductive and non-conductive fluid as well as the motion of liquid–liquid–solid three-phase contact lines under the influence of externally applied electric fields in electrowetting configuration. The technique is based on the volume of fluid method as implemented in the OpenFOAM framework, using a phase fraction parameter to track the different phases. We solve the combined electrohydrodynamic problem by coupling the equations for electric effects—Gauss’s law and a charge transport equation—to the Navier–Stokes equations of fluid flow. Specifically, we use a multi-domain approach to solving for the electric field in the solid and liquid dielectric parts of the system. A Cox–Voinov boundary condition is introduced to describe the dynamic contact angle of moving contact lines. We present several benchmark problems with analytical solutions to validate the simulation model. Subsequently, the model is used to study the dynamics of an electrowetting-based display pixel. We demonstrate good qualitative agreement between simulation results of the opening and closing of a pixel with experimental tests of the identical reference geometry.

### Keywords

Electrowetting OpenFOAM Pixel Cox–Voinov Contact line### List of symbols

### Greek symbols

- \(\alpha\)
Phase fraction

- \(\beta\)
Ratio of viscosities

- \(\gamma\)
Interface tension (N/m)

- \(\delta\)
Weighting parameter (value fraction)

- \(\Delta n\)
Distance from cell centre to cell face (m)

- \(\varepsilon _0\)
Electric constant (F/m)

- \(\varepsilon _\mathrm{d}\)
Relative electric permittivity of dielectric

- \(\eta\)
Electrowetting number

- \(\theta\)
Contact angle \(({}^{\circ })\)

- \(\kappa\)
Interface curvature \((\hbox {m}^{-1})\)

- \(\mu\)
Fluid viscosity (Pa s)

- \(\rho\)
Density (\(\hbox {kg/m}^{3}\))

- \(\rho _\mathrm{E}\)
Electric charge density (\(\hbox {C/m}^{3}\))

- \(\sigma\)
Electric conductivity (S/m)

- \(\phi\)
Electric potential (V)

- \(\phi _\mathrm{m}\)
Mass flux \((\hbox {kg/m}^{2}/\hbox {s})\)

### Roman symbols

- \(Bo\)
Bond number

- \(C_\gamma\)
Interface compression coefficient

- \(Ca\)
Capillary number

- \(Ca_\mathrm{E}\)
Electrocapillary number

- \(d\)
Dielectric layer thickness (m)

- \(D\)
Aspect ratio

- \(\vec {E}\)
Electric field (V/m)

- \(\vec {F}\)
Force (N)

- \(\vec {g}\)
Gravitational acceleration \((\hbox {m/s}^{2})\)

- \(\vec {I}\)
Identity matrix

- \(\vec {n}\)
Unit normal

- \(p\)
Pressure \((\hbox {N/m}^{2})\)

- \(Q\)
Ratio of permittivities

- \(R\)
Ratio of conductivities

- \(R_\mathrm{d}\)
Radius (m)

- \(S_\mathrm{f}\)
Cell surface area \((\hbox {m}^{2})\)

- \(t\)
Time (s)

- \(\vec {\vec {T}}\)
Maxwell stress tensor \((\hbox {N/m}^{2})\)

- \(\vec {u}\)
Fluid velocity \((\hbox {m/s})\)

- \(\vec {u}_c\)
Compression velocity \((\hbox {m/s})\)

- \(\vec {x}\)
Position (m)

- \(y\)
Position (m)

## 1 Introduction

By applying an electric potential to the bottom electrode, while keeping the top electrode at ground voltage, the emerging electric field will interact with the oil–aqueous interface. The interface bends down due to the electrohydrodynamic force and creates a three-phase contact line. From that moment, electrowetting takes place and further moves the three-phase contact line (and thus the oil phase) to the edges of the pixel, revealing the pixel bottom. The advantage of this technique over conventional LCD displays is that the ambient light reflects on the surface of the display, instead of requiring an energy- and space-consuming backlight.

The creation of a design and testing of microfluidic electrohydrodynamic (EHD) devices requires a lot of time-consuming experimental work. For this reason, many researchers have used numerical simulations for their investigations. For instance, Ku et al. (2011) are among the first who attempted the simulation of different electrode patterns for use in electrowetting-based displays. Many studies have been devoted to electrohydrodynamics, for instance Tomar et al. (2007), Bjørklund (2009), López-Herrera et al. (2011) and Lima and d’Ávila (2013). While electrohydrodynamics is an important aspect of electrowetting, the effect of the three-phase contact line has not been studied in these works. For instance, while the Gerris EHD code due to López-Herrera et al. (2011) provides an accurate framework for EHD calculations alone, the explicit definition of the contact angle in three dimensions is not possible.

EWOD has been studied by others using different approaches. Arzpeyma et al. (2008) and Keshavarz-Motamed et al. (2010) study the actuation of a droplet onto an electrode using a volume of fluid (VOF) method. They implemented a two-way coupling scheme of the electric potential field and the VOF solver. The effect of electrowetting has been modelled by modifying the contact angle of the drop using the Young–Lippmann equation with local electric potential. Similarly, lattice-Boltzmann simulations have been performed by Aminfar and Mohammadpourfard (2009, 2012) in which both the electric field and the contact angle of moving and merging droplets have been resolved. In contrast to the VOF models mentioned earlier, these lattice-Boltzmann simulations derive the contact angle using adjusted surface tension coefficients that change with the applied electric potential. Arzpeyma et al. (2008) found a rather sharp transition between the equilibrium (zero voltage) contact angle and the contact angle on top of the electrode. Therefore, the approach of Dolatabadi et al. (2006) and Clime et al. (2010a, b) may be justified; their simulations of droplet movement are performed using a position-dependent contact angle, hereby omitting the need to solve for the electric field.

While the method used by Clime et al. (2010a, b) and Dolatabadi et al. (2006) has its physical justification for purely electrowetting cases (e.g. Buehrle et al. (2003), Mugele and Buehrle (2007), the display pixel case requires a combination of electrohydrodynamic and electrowetting modelling. As long as the electrolytic aqueous phase does not have a contact point with the dielectric layer, the problem is purely electrohydrodynamic since there is no three-phase contact line. When the three-phase contact line is formed, due to electrohydrodynamic retraction of the oil, the problem becomes of the electrowetting type. At this time, the electric field within the dielectric layer on which the droplet is deposited must be considered. Resolving the electric field in both the fluid phase and the solid phase is deemed essential for the purpose of simulating electrowetting pixels; when the oil has retracted, the two electrodes are separated by nothing than an electrolytic fluid, which short-circuits the system. A model for the display pixels hence requires accounting for the electric field distribution, as demonstrated in our earlier work (Manukyan et al. 2011) and explained in the modelling paper by Oh et al. (2012).

Hong et al. (2008), Drygiannakis et al. (2009), and Pooyan and Passandideh-Fard (2012) have simulated EWOD including the electric field strength in the dielectric layer and provide a thorough explanation of the numerical technique. Drygiannakis et al. (2009) intention is to investigate contact angle saturation due to dielectric breakdown with this model. However, it is not clear whether these methods also include pure electrohydrodynamic effects.

Multiphase flow solver: To distinguish between the electrolyte and the oil phase, a multiphase flow technique that can handle deformable interfaces including surface tension and topological transitions is preferred.

Dynamic contact angle model: The dynamic behaviour of the system should include a physically correct implementation of the dynamic contact angle model, for instance the Cox–Voinov model. This allows the simulation of a velocity-dependent contact angle.

- Electrostatic field solver: The solution of a Laplace equation (Gauss’s law) to determine the electric field is required, taking into account local variations of the electric permittivity.
Include electric field inside dielectric layer: Since the electric field is applied over both the spacing volume (including electrolyte and oil) and the solid dielectric layer, the electric field needs to be solved in a coupled fashion in both domains.

Capability of simulating perfect dielectric, perfect conducting and leaky dielectric compounds.

Ability of simulating arbitrary domain shapes

## 2 Model development

In this section, the governing equations are outlined. While the discussion on the fluid flow and electric field effects is based on our earlier work (Roghair et al. 2013), this work extends the model with appropriate Cox–Voinov (Cox 1986; Voinov 1976) contact angle boundary conditions and the incorporation of finite-size solid parts.

We have chosen to build our model using the OpenFOAM framework. OpenFOAM is an open-source software package capable of numerically solving a wide range of computational fluid dynamics (CFD)-related problems. Thanks to its open nature, we are able to incorporate the electrostatic field equations and the interaction of the electric field with the fluid–fluid interface into the existing framework. We use the source code of ‘interFoam’ as a base model (using version 2.1.1). While the basics of this model are described below, an extensive evaluation and verification of this multiphase flow solver is given in the work of Deshpande et al. (2012) and references therein.

### 2.1 Fluid flow

### 2.2 Electric equations

### 2.3 Dielectric solid layer

It is important to simulate the electric field in the dielectric solid layer. The fluid flow on top is solved dynamically, and the fluid–fluid configuration is used to perform an electrostatic computation resolving the electric field and charge distribution. The electric field strength in the solid layer is important as well, since it is affected by the fluid flow on top of it. Since no fluid flow is allowed inside the solid region, a distinction has to be made between two domains, which are to be coupled using their common electrical variable.

### 2.4 Dynamic contact angle

## 3 Validation

Various parts of the newly implemented model are validated through a series of cases, which have been inspired by the cases worked by López-Herrera et al. (2011). The validation cases all solve the entire set of equations. When only a specific aspect is verified (e.g. the electric equations), the other equations (e.g. the two-phase flow) are set up without a driving force.

### 3.1 Validation of electric equations

Exact solutions for the two-phase planar layer cases following López-Herrera et al. (2011)

Conductive–conductive | Dielectric–dielectric | Dielectric–conductive | |
---|---|---|---|

Property | \(R=\frac{\sigma _1}{\sigma _2}=0.25\) | \(Q=\frac{\varepsilon _1}{\varepsilon _2}=3\) | \(1/R=0,\qquad Q=3\) |

Fluid 1 | \(\phi _1^{\mathrm {exact}} = \frac{-2y+R}{1+R}\) | \(\phi _1^{\mathrm {exact}} = \frac{-2y+Q}{1+Q}\) | \(\phi _1^{\mathrm {exact}} = 1\) |

Fluid 2 | \(\phi _2^{\mathrm {exact}} = \frac{R(-2y+1)}{1+R}\) | \(\phi _2^{\mathrm {exact}} = \frac{Q(-2y+1)}{1+Q}\) | \(\phi _2^{\mathrm {exact}} = 1-2y\) |

### 3.2 Validation of multi-region implementation

Exact solution to the three-phase planar layer case with solid, fluid 1 and fluid 2 on the domain \(y \in \left[ -0.25, 0.5 \right]\) with the interfaces at \(y=0\), \(y=0.25\)

Solid | Fluid 2 | Fluid 1 | |
---|---|---|---|

Electric permittivity | \(\varepsilon _\mathrm{s} = 1\times 10^{-11}\) | \(\varepsilon _2 = 4\times 10^{-11}\) | \(\varepsilon _1 = 2\times 10^{-11}\) |

Solution | \(\phi ^{\mathrm {exact}} = \frac{y+0.25}{\varepsilon _\mathrm{s} \varepsilon _{\mathrm {tot}}}\) | \(\phi ^{\mathrm {exact}} = \frac{0.25}{\varepsilon _\mathrm{s} \varepsilon _{\mathrm {tot}}} + \frac{y}{\varepsilon _2 \varepsilon _{\mathrm {tot}}}\) | \(\phi ^{\mathrm {exact}} = \frac{0.25}{\varepsilon _\mathrm{s} \varepsilon _{\mathrm {tot}}} + \frac{0.25}{\varepsilon _2 \varepsilon _{\mathrm {tot}}} + \frac{y-0.25}{\varepsilon _1 \varepsilon _{\mathrm {tot}}}\) |

### 3.3 Gaussian charge bump

### 3.4 Charged cylinder case

Another case, verifying the implementation of the charge transport equation in combination with Gauss’s law, is the use of a conducting cylinder immersed in a perfect dielectric medium. When the cylinder is charged, the charges repel and travel to the edge of the cylinder. The charges generate an electric field in the dielectric medium, of which the magnitude is compared with analytical relations in Fig. 5b.

### 3.5 Rate of convergence

### 3.6 Validation of fully-coupled implementation

### 3.7 Contact angle validation

## 4 Electrowetting pixel simulations

The dynamic oil switching behaviour (e.g. response times, electric field distribution in the oil) has been investigated for pixel closing and pixel opening. In this work, we keep the geometry of the pixel fixed and discuss cases in which we vary as parameters the volume of oil and the applied voltage. Both electrodes extend to the entire length and width of the pixel, both in the simulations and in comparative experiments. Note that in practice, the electrodes would cover only part of the bottom and could even have a pattern to aid the quick withdrawal of oil (e.g. Ku et al. 2011). Alternative pixel designs, internal structures and tuning of the physical parameters lie outside of the scope of this study.

From a top-view perspective, the area enclosed by the pixel walls is rectangular, being twice as long as it is wide. The walls are \(4\,\upmu \hbox {m}\) high and are placed right on top of the bottom boundary of the fluid domain. While these walls are impermeable and serve as a physical boundary for the oil in a pixel, the domain boundaries above the walls up to the top of the domain (which may vary from 20 to 100 but is typically \(54\, \upmu \hbox {m}\) high) are open so that the aqueous fluid may freely flow in and out.

- 1.
The top region between the top electrode and the walls consists mainly of electrolytic fluid, which was modelled using 12 cells in height, using a mesh grading in the vertical direction, causing the cells on top of the walls being four times smaller in the vertical direction compared to the cells at the top of the pixel.

- 2.
The central region spanned by the pixel walls and the volume enclosed by them was modelled using 16 cells in height, giving a much higher resolution locally since this is where most of the dynamics takes place.

- 3.
The bottom region underneath the pixel (dielectric solid) is modelled by 10 cells in height, so that it properly captures the electric field in the solid layer.

Boundary conditions (BC) used for the different simulated variables

Phase fraction \(\alpha\) | Velocity \(\vec {u}\) | Pressure \(p\) | Potential \(\phi\) | Charge \(\rho _\mathrm{E}\) | Description | |
---|---|---|---|---|---|---|

Top electrode | \(\nabla \alpha =0\) | \(\vec {u} = 0\) | \(\nabla p = 0\) | \(\phi = 0\) V | \(\nabla \rho _\mathrm{E} = 0\) | Impermeable wall with ground potential |

Bottom electrode | N/a | N/a | N/a | \(\phi = \phi _\mathrm{v}\) V | N/a | Predefined potential, not adjacent to fluid region |

Fluid to dielectric | Contact angle BC | \(\vec {u} = 0\) | Fixed flux pressure | Mixed | \(\nabla \rho _\mathrm{E} = 0\) | Impermeable wall with coupled potential |

Fluid to wall | Contact angle BC | \(\vec {u} = 0\) | Fixed flux pressure | Mixed | \(\nabla \rho _\mathrm{E} = 0\) | Impermeable wall with coupled potential |

Wall to dielectric | N/a | N/a | N/a | Mixed | N/a | Solid–solid BC with coupled potential |

Outer domain BC | \(\nabla \alpha =0\) | \(\nabla \vec {u}=0\) | \(\nabla p = 0\) | \(\nabla \phi = 0\) | \(\nabla \rho _\mathrm{E} = 0\) | Coupling to neighbour pixels, free fluid in-outflow |

First, pixel closing dynamics is investigated, which is a process purely driven by capillarity. The volume of oil used inside a single pixel is varied, and the response times (from opened state to closed state) are evaluated. Subsequently, the pixel opening is studied, which is driven by an imposed voltage.

### 4.1 Pixel closing

The graphs are displayed on semilogarithmic scale to emphasize the different closing stages. While the initial part of the graphs shows a quick decrease in white area (the bulk stage), the lion’s share of the response time is depicted linear meaning an exponentially decaying visible white area.

### 4.2 Pixel opening

- 1.
Interface deflection. This stage is fully electrohydrodynamic (i.e. no electrowetting effects), as the three-phase contact line remains pinned at the top of the pixel walls. Within the pixel, oil separates the aqueous phase from the dielectric layer everywhere. Charges build up at the fluid–fluid interface and the Maxwell stress pulls the interface towards the bottom electrode. The oil film progressively thins during this stage throughout the central area of the pixel. This process is the same as for electrowetting-functionalized superhydrophobic surface (Manukyan et al. 2011) and electrically tunable optical apertures (Murade et al. 2011).

- 2.
Water–dielectric contact formation. The oil–water interface reaches the dielectric layer and the oil film breaks up (Staicu and Mugele 2006). This leads to the formation of a three-phase contact line—possibly including a molecularly thin residual oil layer that is not resolved with the present simulation technique. Charge accumulates at the oil cleared area, which restrict the electric field to the dielectric layer only.

- 3.
Expansion. The oil moves to the pixel sides forming two bodies at either side of the pixel. The electric field is strong on the oil side of the moving contact line which is the driving force for the retraction of the oil.

- 4.
Relaxation. The contact line of the bulk stops moving, but the oil in the small filaments along the pixel side corners is still flowing to the bulk. During the simulation, small amounts of oil have been left on the pixel bottom, fractions that are lower than the visualization threshold requires to draw an interface (recall that the VOF method used is a numerical technique that smears the interface), which gather in the cells in the pixel centre. This process is affected by deficiencies of the smeared interface. We are therefore unable to assess the quantitative accuracy of the behaviour of these satellite drops. Note, however, that the occurence of satellite drops is physical. They appear in the present experiments (see Figs. 13, 14, 15), and they were observed before (Murade et al. 2011; Staicu and Mugele 2006; Sun and Heikenfeld 2008). Using stability analysis in lubrication approximation of thin oil films between an electrode and a water film, it is possible to show that satellite drops form due to a linear instability (Staicu and Mugele 2006).

Comparing Figs. 13, 14 and 15, it is clear that the oil withdrawal behaviour is strongly dependent on the voltage applied; both the distribution of the oil and the absolute time of pixel opening vary substantially. A distinctive feature of the 25 V case (and to some extend for the 20 V case as well) is that the interface deflection takes place at two places instead of in the pixel centre. This results in an oil bulk at the pixel side walls in the final state, which in the simulation eventually lies on top of the walls.

Minor differences between experiments and simulations may occur for several reasons. First of all, the conductivity of the fluids may not be constant, especially if the pixels are actuated a lot. The simulation assumes insolubility of the two phases, but in reality a very small amount of solubility or dispersion of one fluid in the other may occur, although this has not been verified by experiments. Also, a closer investigation of the dielectric layers and pixel bottoms of other samples has shown that varying dielectric layer thickness, as well as differences in the pixel depth, may occur, which of course may influence the moment of touch-down and final shape of the interface. This is another reason why feature matching has been used to compare simulations and experiments from the moment of touch-down onward, since after the moment of touch-down the dynamics depend much less on variations in dielectric layer thickness or pixel bottom height.

Moreover, it should be noted that the final dynamics of break-up of the oil layer, when the interface is only 1 grid cell thick, cannot be captured by our CFD simulations. Yet, the overall agreement between our simulations and the experiments suggests that these effects have a minor influence on the global dynamics of pixel opening.

Experiments on EWOD devices require a dielectric layer to be present between the electrode and the fluid phases to prevent electrolysis. Yet, using the numerical model, it is possible to perform a simulation without a dielectric layer (i.e. using the bottom boundary of the fluid flow domain as electrode with a constant electric potential). The lack of a dielectric layer essentially increases the electrostatic force on the fluid–fluid surface as compared to a simulation (or experiment) that does include the dielectric layer. We have observed that a simulated pixel opening corresponds to a higher-voltage experiment that does include a dielectric layer. Hence, a reliable, quantitative result can only be obtained when taking the dielectric layer into account.

### 4.3 Mesh dependency

It was found that the coarse mesh produces a much faster touch-down stage (recall that numerical interface break-up happens faster with larger cells), but evolves into the similar shapes eventually. The base case and finer meshes produce very similar results, based on visual inspection of the simulation time steps. The touch-down event occurs on a very comparable time step (with only \(20\, \upmu \hbox {s}\) in between, a fraction of the total of 19 ms required for opening the pixel). The absence of symmetry is somewhat distracting and is caused by a non-symmetrical initial position of the interface. Eventually, the simulation reaches a symmetrical final state.

The white area is one of the most important characteristics of the pixels, during opening and closing behaviour, and is quantifiable using the visualization of the interface. However, the comparison between mesh sizes has been made in a qualitative way only, since the occurence of satellite droplets (due to thin film instability) occurs differently for different mesh sizes. In actual pixels, thin films/droplets are translucent, but in the simulations, they do block the pixel bottom. For this reason, different mesh sizes are not quantitatively comparable, and the comparison has been done in a qualitative way, similar to our comparison with experiments.

## 5 Discussion and conclusion

An electrohydrodynamic model for the simulation of electrowetting on dielectric devices has been implemented and described in this work. The current implementation is based on the OpenFOAM framework, using a volume of fluid method to account for the different fluid phases. Via a charge transport equation and Gauss’s law, electrostatic field calculations are performed based on a hydrodynamically resolved fluid–fluid interface, and the electric force acting on the fluid–fluid interface is taken into account. Furthermore, a multi-region approach has been employed to simulate the effect of a solid dielectric layer that is impermeable for the fluids and electric charges, but takes into account the electric field distribution.

Different aspects of the model have been verified using a number of synthetic benchmark cases, after which the model has been used to simulate the closing and opening behaviour of display pixels based on electrowetting. The pixels employ a transparent aqueous phase and an opaque oil phase which are actuated via an applied electric potential. Pixel closing has been studied by initializing an amount of oil on the side of the pixel, which spreads out over the pixel bottom. The effect of different oil volumes on the closing time and on the uncovered area has been described, along with a discussion of different stages of the closing behaviour.

Pixel opening behaviour has been simulated and compared to experiments. At three voltages, 15, 20 and 25 V, the simulations show a very good correspondence of the whole dynamics with measurements. Again, a number of different stages in opening behaviour have been described.

Several improvements of the algorithm can be considered. First of all, the VOF method as used in this work is not ideal, since the smeared interface may cause for instance spurious currents and (potentially artificial) satellite droplets. The impact of these aspects is small for the situations studied here. Further improvements could therefore be achieved by incorporation of geometric interface reconstruction (e.g. Maric et al. 2013). Additionally, a more accurate calculation of the interface curvature can be achieved by using height functions for curvature calculations (Afkhami and Bussmann 2009; Popinet 2009). These methods are at this point not incorporated in the public version of OpenFOAM.

## Notes

### Acknowledgments

This work is part of the VICI research programme ‘Switchable Superhydrophobic Surfaces’, which is financed by the Netherlands Organization for Scientific Research (NWO) and the Dutch Technology Foundation (STW).

### Conflict of interest

The authors declare that they have no conflict of interest.

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