Numerical Model Related to Impact Fluid / Solid Under the action of an Electric Field

The electrowetting is an area of significant interest. Many experimental studies have been conducted to find the relationship that binds the physical parameters of said phenomenon: the percentage of the white area inside the pixels of different sizes depending on the applied voltage etc. Our study is to develop a CFD model to validate the experimental results for the behavior of fluids (oil colored) within the reflector screens under the action of electric field.


Introduction
Electrowetting has become one of the most widely used for manipulating small amounts of liquids on surfaces under the influence of an electric field. This phenomenon is used in many industrial processes such as microfluidic "lab-on-a-chip", the devices (plans) of adjustable lenses to modify the convergence so in the industry of display screens. Our study is to develop a CFD (Computational Fluid Dynamics) method based on the VOF (Volume of Fluid) to follow the behavior of the liquid (in our case we choose the oil) inside a pixel under action of an electric field.

Mathematical Model
When a drop is placed on top of an electrode and the latter is then charged, the contact angle between   the liquid surface (the drop) and the solid surface (electrode) is reduced. This is called electrowetting. Lippmann equation [1] reflects the change in the interfacial tension between the solid  SL and the liquid as a function of the voltage V of the drop by the equation (1): Where, C is the specific capacity of the dielectric layer. From Young's equation (2) which describes the relationship between the contact angle  and the interfacial tension  SL between the interface solid-gas, solid-liquid and liquid-gas in the line of contact at the triple point (see Figure 1) This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

:
The dielectric constant of the dielectric layer.
t: The thickness of the dielectric layer.

Numerical Model
A fundamental method to solve two-phase flow is the volume of fluid method (VOF), which was developed by Hirt and Nichols (1981). VOF method is a fixed mesh with the interface between immiscible fluids is modeled by a characteristic function (called volume fraction) [2].

The model equations
The Equations of mass and momentum conservation (4) and (5) for each phase are given by [3]: Where V is the velocity vector, P is the pressure, and the FSF is the surface force vector, μ is the viscosity and density . The mixture density is calculated as follows: Where αk is the volume fraction of the liquid. Any other property of the mixture is calculated as follows: Where: α k =0 : The cell is empty. The interface between the two phases was followed by solving the continuity equation for the function of the volume fraction: The surface tension has been modeled as a regular variation of the capillary pressures through the interface.
Where n is the surface normal, n is the unit normal of the curvature. The surface normal n was evaluated in cells containing the interface and requires knowledge of the amount of volume of fluid present in the cell.
To calculate the electric potential in every area, the second order discretized form of the Laplace equation is solved at the beginning of each time step iteration [4].

Geometry and boundary conditions
We chose to study the problem in two dimensions. For this, the geometry is a square of dimension 1x1m 2 . We used a structured grid square of size 5 x 10 -4 using the preprocessor Gambit 2.2.30.For the boundary conditions we used two conditions: the input condition is defined as a pressure inlet and the side walls are declared type Wall. Where the electrode is fixed between the insulator (Teflon) and the substrate is a white rectangle 15nm thick. The liquids used in this study are dodecane and water. A drop of dodecane whose radius is 10  m is defined in the solver Ansys

Results and Discussion
VOF method available in the code Ansys Fluent calculation allowed us a better observation of the drop profile and from the equation of fluid dynamics. The movement of the drop is observed when a voltage of 15V is applied to the electrode of time ranging from t = 0 to t = 2.16 e -2 s (FIG. 4).
It is clear that, at different t, the drop of oil contracts until the contact angle at the triple point reaches its maximum value.