# An empirically validated model of the pressure within a droplet confined between plates at equilibrium for low Bond numbers

- First Online:

- Received:
- Revised:
- Accepted:

- 7 Citations
- 146 Downloads

## Abstract

An analytical model is presented that describes the equilibrium pressure within a confined droplet for small Bond numbers without prior knowledge of the interface shape. An explicit equation for the pressure was developed as a function of the gap height, surface tension, and contact angle. This equation was verified empirically. The shape of the interface was found based on the pressure predicted by both the proposed model and a model commonly used in electrowetting on dielectric (EWOD) investigations. These shapes were compared against experimentally observed interfaces for aspect ratios between 3.5 and 18. The pressures and shapes predicted by the proposed model were at least an order of magnitude more accurate than those predicted with a more commonly used model. At an aspect ratio of 3.5, the average error in the predicted shape was almost 4%, but decreased below the experimental error at an aspect ratio of 6. An aspect ratio of 15 is required for an EWOD device to split water droplets in air. The error in the model pressure and its predicted interface in this case were approximately 0.3%. The analytical pressure model proposed here can be used to increase the accuracy of models of practical EWOD devices. Better accuracy can be attained for small aspect ratios by iteratively calculating pressure using the model proposed here.

### List of symbols

- \( g \)
Acceleration due to gravity, \([g] \) = m/s

^{2}- \( \bar{h}\,,\,h \)
Half height of the gap, \( [ {\bar{h}}] \) = m

- \( N_{B} \)
Bond number

- \(\user2{n} \)
Outward facing normal vector

- \( \bar{p}\,,\,p \)
Pressure within the droplet, \([ {\bar{p}}] \) = Pa

- \( \bar{R} \)
Droplet radius in the horizontal plane, \([ {\bar{R}}] \) = m

- \( \bar{r}\,,\,r \)
Droplet radius in the vertical plane, \( \left[ {\bar{r}} \right] \) = m

- \( \bar{z}\,,\,z \)
Position on the vertical axis, \([ {\bar{z}}] \) = m

- \( \gamma \)
Surface tension between the droplet fluid and the surrounding medium, \([ \gamma] \) = N/m

- \( \eta \)
Change in \( \xi \) with respect with \( z \)

- \( \theta \)
Angle between tangent to interface and horizontal plane, \( \left[ \theta \right] \) = rad

- \( \left. \theta \right|_{ - h} \equiv \theta_{ - } \)
Interface angle at \( z = - h \) (also \( \pi \) minus the contact angle)

- \( \bar{\kappa }\,,\,\kappa \)
Total local curvature of the interface, \( \left[ {\bar{\kappa }} \right] \) = 1/m

- \( \bar{\xi }\,,\,\xi \)
Horizontal distance between the interface and the contact line, \( \left[ {\bar{\xi }} \right] \) = m

- \( \bar{\rho }\,,\,\rho \)
Circular-cylindrical coordinate, \( \left[ {\bar{\rho }} \right] \) = m

- \( \rho_{F} \)
Fluid density, \( \left[ {\rho_{F} } \right] \) = kg/m

^{3}- \( \varphi \)
Circular-cylindrical coordinate, \( \left[ \varphi \right] \) = rad

- \( \bar{\phi }\,,\,\phi \)
Level set function, \( \left[ {\bar{\phi }} \right] \) = m

- \( \overline{\nabla }\,,\,\nabla \)
Gradient operator, \( \left[ {\overline{\nabla }} \right] \) = 1/m