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.

 Project head Prof. Dr. Michael Hintermüller (1,2) Staff Dr. Soheil Hajian (1) Project period 1 June 2017 - 31 December 2018 Affiliations (1) Humboldt-Universität zu Berlin (2) Weierstrass Institute Berlin Cooperations (ECmath) C-SE1, OT8 External cooperations Prof. T. M. Surowiec (U Marburg) Prof. S. W. Walker (Louisiana State Univ.) Prof. H. Antil (GMU)

## Background of the project

A number of emerging key technologies in microbiology, medical diagnostic devices, personal genomics, as well as next-generation low-energy 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 air-liquid-solid 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 EWOD-based 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 finite-horizon model predictive control; see [AHN$^{+}$].

## Research program

The project strives to provide a framework allowing engineers to design and test the viability of a device for subsequent fabrication, and to automate pre-existing 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 solid-liquid-air interface, for a faithful representation of the droplets velocity and cover different aspects properly.

The sharp interface model (and its corresponding time-discrete model) [WSN09], based on a quasi-static Hele-Shaw 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 Cahn-Hilliard-Navier-Stokes system to a linear elliptic PDE for the electrostatic behavior and a parabolic equation for the electric charges. It is thermo-dynamically 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 non-trivial 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 (real-time) numerical solution as needed for EWOD devices. For these reasons, instead of computing time-discrete 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 so-called mathematical program with equilibrium constraints (MPEC) in function space.

We aim to extend the bundle-free 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 finite-horizon MPC approach will then evolve the time-discretized equations in time as done in the sharp interface setting. As a result, we obtain a series of PDE-constrained optimization problems with (almost) linear constraints.

## Publications

 [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), 1-30.

## References

 [CMK03] S. K. Cho, H. Moon, and C.-J. Kim. Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. Microelectromechanical Systems, Journal of Microelectromechanical Systems, 12(1):70-80, Feb 2003. [HS16] M. Hintermüller and T. Surowiec. A bundle-free implicit programming approach for a class of elliptic mpecs in function space. Math. Prog., 160(1):271-305, 2016. [HZK$^{+}$09] J. Heikenfeld, K. Zhou, E. Kreit, B. Raj, S. Yang, B. Sun, A. Milarcik, L. Clapp, and R. Schwartz. Electrofluidic displays using Young-Laplace transposition of brilliant pigment dispersions. Nat Photon, 3(5):292-296, 05 2009. [GFK04] J. Gong, S.-K. Fan, and C.J. Kim. Portable digital microfluidics platform with active but disposable lab-on-chip. In Micro Electro Mechanical Systems, 2004. 17th IEEE International Conference on. (MEMS), pages 355-358, 2004. [NSW14] R.H. Nochetto, A.J. Salgado, and S.W. Walker. A diffuse interface model for electrowetting with moving contact lines. Math. Models & Meth. in Appl. Sci., 24(1):67-111, 2014. [WMK$^{+}$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 matrix-assisted laser desorption/ionization mass spectrometry. Anal. Chem., 76(16):4833-4838, 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 (1994-present), 21(10), 2009.