Microscale 2D Particle Position Control: The Individual and Group Cases

Our goal is to design and build a printer-like system for assembling microparticles into engineered patterns. The assembly into desired patterns is based on feedback control that tracks the particles and makes corrections to the particle positions until the desired pattern is achieved (see “Summary” and “Xerography for Microelectronics” for more details on xerography and assembly for microelectronics). Micro- and nanoscale particle manipulation have received a lot of research interest due to their significant applications in microfabrication, biology, and medicine. Recent work [1] , [2] demonstrated a microchiplet control policy based on a one-step model predictive control approach. The 1D model used a capacitance-based model. However, the actuation mechanism was based on spiral-shaped electrodes that limited the number of simultaneously actuated electrodes. The electrodes were connected through wires to a digital-to-analog converter power source that set the electrodes’ electric potentials. The spiral-based experimental setup allows for radial chiplet motion only and does not scale to a large number of electrodes (for example, in the thousands) due to wiring challenges [3] . Colloids are at the core of many microassembly technologies. They are solution-processed assemblies of nano- to micrometer-sized particles, whose collective properties are controlled by both particle properties and the superstructure symmetry, orientation, phase, and dimension [4] (see “Colloids” for more details on colloids). A control scheme for individual and ensemble control of colloids is described in [5] . In particular, it is shown how inhomogeneous electric fields are used to manipulate individual and ensembles of colloidal particles (1 to 3 μ m in diameter) in water and sodium hydroxide solutions through electrophoresis (EP) and electro-osmosis. The authors use various electrode-to-particle-size ratios, various media in which the particles are immersed, and different mathematical models. The authors of [6] demonstrated location-selective particle deposition, where EP forces are the primary drive for particle (2- μ m polystyrene beads) manipulation. The control scheme was based on building large energy wells close to the desired location of the nanoparticles. Several works [7] , [8] describing the control of a stochastic colloidal assembly process that drives the system to the desired high-crystallinity state are based on a Markov decision process optimal control policy. The dynamical model is based on actuator-parametrized Langevin equations. In this work, individual particles are not directly manipulated. Hence, it is unclear how this approach can be used when assembling nonperiodic patterns, such as electrical circuits and heterogenous systems. Moreover, the particle size (≈3 μ m in diameter) is so small that the particles pose little disturbance to the electric field, which is completely shaped by actuation potentials. In addition, the long timescale for achieving the desired state would make the goal of high throughput challenging to achieve. Other self-assembly control approaches [9] , [10] , [11] would require significant modification to be used with our experimental system. Electro-osmosis solutions are a popular choice for particle control [12] , [13] . In such cases, both EP forces and fluid motions of electro-osmotic flows are used to drive particles. An electrode structure similar to our setup in [1] and [2] was used to study the effect of dielectrophoresis (DEP) on cancer cells [14] . Unlike our setup, particles are assumed to be small enough that the electric field is not disturbed by their presence. Accurate control of cells, quantum dots, and nanowires based on electro-osmosis is used in [15] and [16] . The authors use linear models of the electrode potentials, and the particles’ effect on the electric field distribution is negligible. In the work presented here, linearity in the electrode potentials does not hold since the driving forces are primarily DEP. In addition, the electric field is affected by the chiplet position. In [17] , the authors describe a DEP-based feedback control scheme for microsphere manipulation. The authors control the phase shifts of the voltages applied a micro-electrode array combined with closed-loop cascade control strategy based on real-time numerical optimization. For comparison, we formally derive a control policy for which we present empirical results that may be optimal as well. More importantly, our policy is easy to implement in real time since it does not require solving an optimization problem. This article describes a 2D control policy where the actuation is done using an electrostatic actuator array. The phototransistor-based array is optically addressed to enable dynamic control of the electrostatic energy potential and manipulate the position of small objects. The approach is sufficient to be applied to both nano- (<1 μ m) and micro-objects (hundreds of micrometers). The system uses dielectric fluids (for example, Isopar M) and supports both EP and DEP forces. We design policies for both individual particle control and the control of groups of particles. In both cases, we use an optimization-based approach to policy design. The difference comes from the types of dynamical constraints. In the individual particle control case, we use the dynamical model of motion for a particle as a constraint. In the case of controlling groups of particles, the dynamical constraint is given by a transport equation expressed in terms of particle density. This type of formulation is part of the broader class of optimal control problems, where the dynamical constraint is the Liouville partial differential equation (PDE). Controllability of the Liouville equation, together with optimal control of its moments for some special cases (for example, the linear case), are discussed in [18] . An analysis of problems of optimal control of ensembles governed by the Liouville equation is found in [19] , where the results apply to particular classes of problems (for example, the Liouville equation with an unbounded drift function with linear and bilinear control mechanisms) and classes of cost functionals. In [20] , the authors introduce a dynamic output feedback control of Liouville equations, which is applied to single-input, single-output discrete-time linear systems. Our formulation does not fit any of the problem setups enumerated in the preceding. The problem addressed in this article can be set in the larger context of optimal mass transport (OMT) theory that addresses the transport of mass from a source distribution to a target distribution, with minimum effort. A review of OMT-related problems and recent algorithms to solve such problems can be found in [21] and [22] . A particular formulation of the OMT problem is the density control problem, where a cost function expressed in terms of the velocity field is minimized, while the density is constrained by the transport equation, together with boundary conditions. Our problem can be put in this context by viewing the particles as a discretization of the mass that needs to be transported. One fundamental difference in our formulation, as compared to solutions proposed to solve the density control problem, is that the velocity field depends nonlinearly on the control variables, that is, the electrode potentials that shape the electric field. When using the electric potentials as optimization variables in an optimal control formulation, an analytic solution for the optimal control becomes illusive, leading to the need to employ a numerical approach.