ECMath OT6: Optimization and Control of
Electrowetting on Dielectric for Digital
Microfluidics in Emerging Technologies This research is carried out in the framework of MATHEON supported by Einstein Foundation Berlin.


A number of emerging key technologies in microbiology, medical diagnostic devices, personal genomics, as well as nextgeneration lowenergy OLED displays and liquid lenses make use of a phenomenon known as electrowetting on dielectric (EWOD); see, e.g., [HZK${}^+$09, WMK${}^+$04, CMK03, GFK04]. Electrowetting involves the manipulation of small (microscopic) droplets on a dielectric surface by the actuation of the underlying current. In fact, droplets in a typical EWOD device are situated between two separated hydrophobic surfaces, one of which contains an array of controllable electrodes. The airliquidsolid contact angle can then by changed by varying the voltages on separate electrodes, which causes the droplets to move. Thus, the voltages are a natural choice for influencing (controlling) the motion of a droplet. For diagnostic devices it is indeed essential to create, move, split, and merge droplets, whereas EWODbased OLED displays typically only need to change the shape of a droplet, e.g., from round to flat, though splitting may also prove useful. Currently, there exist several very recent mathematical models for the simulation of EWOD, but little has been done in terms of control or optimization of EWOD devices. From the above applications, however, it is clear that for enhanced functionality of the underlying devices, proper control mechanisms are of paramount importance, also with respect to energy efficiency. In light of this, the proposed project will provide a mathematical, algorithmic, and computational framework for the optimization and control of EWOD.
Figure: A (sub)optimal control pattern for barycenter matching using a regularized
sharp interface model in finitehorizon model predictive control; see [AHN$^{+}$].
The project strives to provide a framework allowing engineers to design and test the viability of a device for subsequent fabrication, and to automate preexisting devices. We will pursue both sharp interface [WSN09] and phase field models [NSW14], for the movement of droplets in an EWOD device. Both models make use of a macroscopic description for contact line pinning, which is due to contact angle hysteresis as well as molecular adhesion at the solidliquidair interface, for a faithful representation of the droplets velocity and cover different aspects properly.
The sharp interface model (and its corresponding timediscrete model) [WSN09], based on a quasistatic HeleShaw flow, provides an accurate description of the droplet's location and the associated pressure boundary condition, but is problematic in case of topological changes. On the other hand, the phase field model [NSW14] couples a CahnHilliardNavierStokes system to a linear elliptic PDE for the electrostatic behavior and a parabolic equation for the electric charges. It is thermodynamically consistent and allows for topological changes, but at the expense of knowing the precise location of the droplet. Depending on the a priori fixed functionality of a device we will resort to the corresponding model.
The optimal control of either sharp interface model over all time intervals or of phase field model would result either in a sequence of controls or a boundary control that minimize/s a respective overall objective. However, due to the nontrivial dependencies on the moving interface variables in the sharp interface context, the proof of existence of an optimal control remains impossible without further restrictive assumptions or constraints, e.g., on the geometry. Moreover, the complexity of the phase field model poses severe challenges for a fast (realtime) numerical solution as needed for EWOD devices. For these reasons, instead of computing timediscrete or optimal controls we will pursue an idea from model predictive control (MPC). Although this control strategy removes a certain amount of difficulty, the remaining control problem in the case of the sharp interface model is a socalled mathematical program with equilibrium constraints (MPEC) in function space.
We aim to extend the bundlefree implicit programming solver [HS16] to each MPEC in MPC and to account for topological changes. This avoids smoothing schemes [AHN$^{+}$] which compromise the physics of the original model. We then investigate reducing the phase field model to one that is more amenable to a control or optimization setting via dimensional analysis. The original paper [NSW14] provides a dimensionality study which will aid in this work package. The results of the aforementioned work packages will dictate the type of MPC approach for the phase field model. The finitehorizon MPC approach will then evolve the timediscretized equations in time as done in the sharp interface setting. As a result, we obtain a series of PDEconstrained optimization problems with (almost) linear constraints.
[AHN$^{+}$]  H. Antil, M. Hintermüller, R.H. Nochetto, T.M. Surowiec, and D. Wegner. Finite horizon model predictive control of electrowetting on dielectric with pinning. Journal: Interfaces and Free Boundaries, 19 (2017), 130. 
[CMK03]  S. K. Cho, H. Moon, and C.J. Kim. Creating, transporting, cutting, and merging liquid droplets by electrowettingbased actuation for digital microfluidic circuits. Microelectromechanical Systems, Journal of Microelectromechanical Systems, 12(1):7080, Feb 2003.  
[HS16]  M. Hintermüller and T. Surowiec. A bundlefree implicit programming approach for a class of elliptic mpecs in function space. Math. Prog., 160(1):271305, 2016.  
[HZK$^{+}$09]  J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz. Electrofluidic displays using YoungLaplace transposition of brilliant pigment dispersions. Nat Photon, 3(5):292296, 05 2009.  
[GFK04]  J. Gong, S.K. Fan, and C.J. Kim. Portable digital microfluidics platform with active but disposable labonchip. In Micro Electro Mechanical Systems, 2004. 17th IEEE International Conference on. (MEMS), pages 355358, 2004.  
[NSW14]  R.H. Nochetto, A.J. Salgado, and S.W. Walker. A diffuse interface model for electrowetting with moving contact lines. Math. Models & Meth. in Appl. Sci., 24(1):67111, 2014.  
[WMK$^{+}$04]  A.R. Wheeler, H. Moon, C.J. Kim, J. A. Loo, and R. L. Garrell. Electrowetting based microfluidics for analysis of peptides and proteins by matrixassisted laser desorption/ionization mass spectrometry. Anal. Chem., 76(16):48334838, 08 2004.  
[WSN09]  S. W. Walker, B. Shapiro, and R. H. Nochetto. Electrowetting with contact line pinning: Computational modeling and comparisons with experiments. Physics of Fluids (1994present), 21(10), 2009. 