Modeling, Simulation, and Optimization of ... - Semantic Scholar

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 25, NO. 2, FEBRUARY 2006

Modeling, Simulation, and Optimization of Electrowetting Jan Lienemann, Andreas Greiner, and Jan G. Korvink

Abstract—Electrowetting is an elegant method to realize the motion, dispensing, splitting, and mixing of single droplets in a microfluidic system without the need for any mechanical—and fault-prone—components. By only applying an electric voltage, the interfacial energy of the fluid–solid interface is altered and the contact line of the droplet is changed. However, since the droplet shape is usually heavily distorted, it is difficult to estimate the droplet shape during the process. Further, it is often necessary to know if a process, e.g., droplet splitting on a given geometry, is possible at all, and what can be done to increase the system’s reliability. It is thus important to use computer simulations to gain an understanding about the behavior of a droplet for a given electrode geometry and voltage curve. Special care must be exercised when considering surface-tension effects. We present computer simulations done with the Surface Evolver program and a template library combined with a graphical user interface (GUI) that facilitates standard tasks in the simulation of electrowetting arrays. Index Terms—Biochip microfluidics, droplet pumping, electrowetting arrays, simulation, surface tension.

Fig. 1. Four main operations of a microfluidic electrowetting array: droplet creation (1), droplet motion (2), droplet splitting (3), and droplet merging (4).

primed at the start. This paper focuses on an alternate fluiddisplacement mechanism: electrowetting. Electrowetting works without mechanical parts; the only moving mass is the fluid itself. It is technologically much easier as well: The manufacturing process requires only one step to pattern a metallic layer, whereas other micropumps require a number of lithographic steps and complex etching procedures.

A. Electrowetting Setup and Devices

I. I NTRODUCTION

M

ICROFLUIDICS is currently one of the fields of microsystem engineering with the largest market opportunities. Reproducible parallel-batch fabrication of large numbers of low-cost devices is ideal for the varied disposable devices dictated by contamination concerns in biology and medicine. The design of such devices will need to focus on exploiting device scaling while optimizing for reliability and lifetime. In the world of microsystems, where all dimensions are downscaled by several orders of magnitude, surface and edge effects become more important as the size shrinks. For example, a certain amount of water will form a droplet, the shape of which is barely influenced by gravity; further, the influence of electrostatic forces increases, while the effect of inertia decreases. The displacement of fluid volume is a fundamental design issue in microfluidic devices. A variety of micropumps [1] have been proposed that use movable mechanical parts like membranes for displacing fluid volumes or spotting droplets. They mostly operate with a continuous stream of fluid after being

Manuscript received February 15, 2005; revised April 28, 2005. This work is supported by the Commission of the European Communities under Contract G5RD-CT-2002-00744, Competitive and Sustainable Growth Program, Micrometer Scale Patterning of Protein and DNA chips, MICROPROTEIN, and by an operating Grant of the University of Freiburg, Germany. This paper was recommended by Associate Editor K. Chakrabarty. The authors are with the Institute for Microsystem Technology (IMTEK), Albert Ludwig University, Freiburg 79085, Germany. Digital Object Identifier 10.1109/TCAD.2005.855890

The underlying idea of electrowetting is to change the wetting properties of a liquid on a substrate. By applying an electric voltage, surfaces can be switched between a wetting and nonwetting state. If the substrate is only partially wetted, the liquid seeks to cover this part to minimize its energy. A phase boundary between liquid and the surrounding air would thus be shifted towards the wettable spot, and fluid motion can be observed. This spatial control of wetting is accomplished by applying voltage only on certain parts of the substrate—it is partitioned into an array of controllable spots by an assembly of electrodes. One possible application to biochips is in switching between flow channels. Channel-based biochips [2], [3] are typically configured at design time. In contrast, the use of an electrode array that controls wettability offers the possibility for reconfiguration of the “virtual” fluid channel at runtime [4], [5]. One can imagine the device to be like the field-programmable gate arrays used in microelectronics. Here, fluidic gates can allow the fluid meniscus to traverse a certain spot in the channel, inhibit the motion, or alter the fluid path [6]. It is even possible to omit preprocessed channels at all and form virtual channels by a suitable actuation of an assembly of electrodes. Since the effect mostly acts at phase boundaries, these devices usually operate with a quantized flow of singular droplets instead of a continuous flow. Fig. 1 shows an illustration of a possible electrowetting electrode array. By using the electrowetting effect, the droplets are moved from one electrode to another. Four main operations

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LIENEMANN et al.: MODELING, SIMULATION AND OPTIMIZATION OF ELECTROWETTING

need to be possible for the technology to be useful for fluid processing: 1) creation: to take a certain amount of liquid from a reservoir to form droplets of a given size; 2) transport: to move the droplet along a path to or from other functional components like detectors, catalytic converters, and supplies and waste outlets; 3) splitting: to split a droplet into smaller parts for parallel processing; 4) merging/mixing: to merge droplets and mix their contents. This can be achieved by diffusion aided by periodic actuation. Virtual reaction vials can be formed at a single spot of the array. Possible applications are arrayed bioassays and custom combinatorial synthesis of, e.g., deoxyribonucleic-acid (DNA) probes. But there are also other applications beyond the scope of biochips, e.g., for computer displays [7] or adaptive lenses [8], each with their own requirements. B. Device Design The shapes of droplets during this process can often become quite distorted and difficult to estimate. Computer simulations give an insight on the driving forces leading to the motions of a droplet. Calculated energy curves give hints to help the designer understand what happens energetically, and show optimization potential to increase the speed or reliability of the motion. They can also show if a process, like splitting, is possible at all for a given configuration, and which parameters need to be tuned to allow for a reliable operation. This enables the designer to experiment without the need to wait for possibly expensive prototypes; simulations hence speed up development cycles and allow for a shorter time to market that is crucial in such innovative fields as biomicroelectromechanical system (BioMEMS). Some design goals that a simulation could help achieve are the following. 1) Fluid-processing algorithms. An actuation scheme must be found to achieve each of the four operations listed above, and its parameters must be tuned. It is important to estimate the droplet geometry to design the size of electrodes and the amplitude of electric voltages. 2) Fluid process flow. Many single operations must be combined to form the complete process sequence. The start and end of one sequence must fit, and if parallel processing is wanted, the operations must not interfere. 3) Reliability. It is of utmost importance to determine the reliability and the limits of a certain design, and under which circumstances the success of an operation is not sensitive to parameters that are difficult to control. The design should be made such that independence from those is achieved. Further, the influence of tolerances should be quantified. 4) External constraints. Since the device will be connected to external equipment, other constraints might have to be considered, such as power consumption, processing time, external forces, etc.

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For the electrowetting setup as treated in this paper, most design goals are strongly influenced by geometrical quantities. The droplet shape and the device setup play an important role: electrical fields and substrate geometries influence the motion of the droplet. Also, the electrowetting actuation voltage curve V (t) determines the result. For droplet transport—also called pumping—some questions that will be asked are as follows. 1) Which droplet volumes can be transported with a given electrode setup? 2) How should the electrodes be shaped to allow transport with maximal speed at minimal actuation voltages for a large range of droplet volumes? 3) How accurate are the processes? 4) Is there a voltage limit where, e.g., complete wetting of a surface occurs? 5) If the liquid is transported in channels, how would it be possible to fill a larger chamber? Are there optimal “flange” shapes? 6) To understand the motion and to be able to optimize, it is important to know how the potential energy distribution that a droplet sees during the process is influenced by the setup. For droplet splitting, both droplet shapes and actuation parameters are of interest; an optical engineer would be interested to extract the geometry of a droplet to determine the focal length of a droplet lens; if fluorescent markers are inside the droplet, one would be interested in knowing the thickness variation of the droplet to calibrate the light output.

C. Computer-Simulation-Aided Design Experiments can answer many of those questions, and for the given setups are fairly easy to perform and hence quite satisfying. But there are some limitations involved. Due to the small size of some features (dielectric-layer thickness, electrode fine structure), facilities for the production and measurement of prototypes must be available. Especially if cost is an issue, experiments should be prepared using estimates of the results. Optimizations with a large number of evaluations might be easier narrowed down by computer simulation; otherwise, we have found it fairly costly to make quantitative and qualitative experimental comparisons when it comes to, e.g., finding optimal electrode shapes. Further, effects of changes can be estimated without interferences and contaminations in a simulation. Finally, one can also estimate the response of the system to inputs that are difficult to reliably apply in a laboratory setup, giving potential to perform thought experiments. This motivated the development of a modeling tool with which we could quickly perform what-if calculations. However, simulations also have their limits. First, they are always based on a model. The amount of detail, the number of physical effects involved, and the validity of assumptions and simplifications determine the accuracy of such a model. Material and geometrical data need to be obtained, and solver and discretization parameters must be chosen correctly. The possible

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resolution of details depends on the speed of the implementation. In conclusion, simulations should not replace experiments but complement and accompany them. In this paper, we present a tool that is very effective in helping to understand the process of electrowetting. A number of approaches are possible. One could implement a simulation coupling of at least the electrostatics and fluidic domain and provide a treatment of the droplet shape by, e.g., a levelset or volume-of-fluid method, which is the way to go if a design should be characterized before the production of prototypes. Another possibility is to view the droplet motion in its quasistatic limit, which is the approach we have used in our model. The goals of the simulation tool are to provide a fast methodology to compute the change of the shape of a droplet subject to electrowetting and to investigate the effect of this change. It is not meant as a full computational fluid-dynamics (CFD) tool as presented in [9]; the questions asked to a CFD tool are different from the ones we want to answer here. Our simulation is based on the energy equilibrium of surface tensions. Dynamic effects and the fluidic transport process within the moving droplet are excluded from the simulation, thus yielding results in the quasi-static limit of very slow motion and long time. This was motivated by: 1) the complete overshadowing of inertial forces by electrostatic forces (Re = 0.01) and 2) the conservative nature of quasi-static computations. In short, we clearly see whether a droplet can get “stuck” in a local minimum and hence block a fluidic path. While this kind of simulation makes no statement about the exact time response as CFD, it still provides general physical insight as shown in the electrode-finestructure optimization presented further on in this paper. There, the potential energy curves of certain device variations give strong hints on the performance, and the effect of modifications is much more visible and detectable than in other simulation and experimental approaches. This potential energy is the main quantity in the Surface Evolver, and thus easily accessible, allowing us to obtain a fairly accurate picture of, e.g., the energy saddle point that gives rise to droplet splitting. The modification of the geometry of, e.g., an electrode is a matter of a few-seconds work, and allows the quick performance of parameter-variation studies, the results of which aid in developing compact models of droplet operations. A further design decision is the representation of the model in a computer program. We use the Surface Evolver program for our simulation, which explicitly represents the fluid surface by a mesh consisting of triangles. The spatial constraints that the Surface Evolver makes available for use with nodes, edges, and faces are a useful modeling tool with which it is possible to simplify, and hence, speed up the computation of droplet motion. Due to the exclusion of internal fluid transport, the number of equations is strongly reduced; there is also no need for a boundary-element treatment of the droplet interior. This simplified model allows for a fast integration with a much lower CPU time compared to full CFD simulations. For example, the droplet splitting (Section VI-B) needed only a few minutes for solving. Further, there is no need to store a three-dimensional (3-D) grid, but only a two-dimensional (2-D) surface. This makes this approach well suited for optimization loops. However, these decisions also have some disadvantages that will be

TABLE I COMPARISON BETWEEN FULL CFD SIMULATION AND OUR QUASI-STATIC APPROACH (QS)

discussed later. Table I gives a short comparison of the two options.1 In the following section, we describe the basic physical effects and the typical setup of electrowetting devices, and we discusses the application of the Young–Lippmann equation to electrowetting on dielectrics for biochip applications as already formulated in the literature. We then present our simulation methodology along with a discussion of its limits. The methodology was integrated into a template library for the Surface Evolver program with a graphical user interface (GUI), addressing a number of specific problems of the design of electrowetting arrays. Finally, we present results of these simulations, which answer some of the questions posed above. II. W ETTING AND E LECTROWETTING Electrowetting is a method for altering the wetting properties of a surface. A voltage is applied, and an electrostatic field is built up. The energy stored in this field can be formulated as equivalent interfacial tension, and thus, related to the liquid– gas–substrate contact angle of the droplet. This leads to a deformation of the droplet shape, which can be used for the fluidic operations described above. In this section, we describe wetting in general and the influence of an applied electric potential. Moving droplets by an applied electric potential (without additional energy transducers like piezos or electrostatic actuators) can be achieved by two main effects: dielectrophoresis and electrocapillarity, and can be described under the framework of electrohydrodynamic forces [9]. The setup we will consider in this paper is an electrocapillarity approach called “electrowetting on dielectrics” (EWOD), which modulates the wetting properties of a substrate via electrostatic energy. A. Wetting on Surfaces An amount of a single-phase liquid L forms a spherical droplet if no external influences are present. This is the configuration with the smallest surface area for a given volume. The effect that leads to this minimization of surface area is the 1 A “+” in this table means that the method is appropriate or good, a “−” means that there might be some difficulties, and a “+/−” means a limited suitability.


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Fig. 2. Droplet on a substrate. Left: hydrophobic surface. Right: hydrophilic surface.

Fig. 3.

Schematic picture of the virtual displacement of the contact line.

surface tension, measured as energy per area [10], [11]. The reason for this property of the droplet surface is the differing environment of a liquid molecule. A molecule feels forces from neighboring molecules: Van der Waals forces for nonpolar molecules, the Keesom interaction (orientation effect) for polar molecules, and the Debye interaction (induction effect) for a polar and a nonpolar molecule. In the interior of the liquid, these forces equilibrate in the time average, while on the surface, only one half of the surrounding is contained in the liquid interior. After normalization to the surface area, the sum of these forces gives the Laplace pressure [10] ∆p = 2σH

Fig. 4. Typical setup of an electrowetting device. The contact angle θ is lowered if a voltage U is applied.

Fig. 5. Droplet changing its contact angle due to electrowetting.

Equilibrium, and thus, an energy minimum is reached when δF/δA = 0. This leads to the Young equation

(1) cos θ =

with ∆p as the pressure difference, σ as the surface tension, and H as the local mean curvature of the surface. If the droplet is not surrounded by vacuum but by another medium V (for vapor phase), the surface tension is replaced by the interfacial tension γLV , since now the second medium also causes a force on a liquid molecule. The energy of such a surface A can be calculated with the surface integral E=

γLV dA .

(2)

A

Now consider a droplet sitting on a surface S (Fig. 2). The droplet is in contact with two materials: The vapor phase and the substrate. Now, three different interfacial energies interact: γLV for the liquid–vapor interface, γSL for the substrate/liquid interface, and γSV for the substrate/vapor interface. The line where the three phases meet is called the contact line, and the angle of the liquid phase is called the contact angle θ. It can be calculated by considering the variation of the free energy F due to a virtual displacement of the contact line (Fig. 3) δF = γSL 2πrδx − γSV 2πrδx + γLV 2πrδx cos θ where r is the radius of the contact line.

(3)

γSV − γSL . γLV

(4)

B. Electrowetting The typical setup of an electrowetting device is shown in Fig. 4. The system consists of a dielectric layer of thickness d with metal electrodes below, while a droplet of conducting liquid (electrolyte) is situated on the upper exposed surface. It is essential that the dielectric layer is a good insulator with no pinholes, and that ions cannot easily be trapped inside the layer; this would inhibit the correct functioning of the device. In this particular setup, the droplet is in contact with a wire as shown in Fig. 4, further possibilities are discussed later. By applying an electric voltage U between the electrode and the droplet, charge is accumulated as in a capacitor. This decreases the interfacial tension between the droplet and the dielectric layer due to the stored electrostatic energy, leading to a change of the contact angle of the droplet (Fig. 5) [12]. The variation of free energy (3) then reads [13], [14] δF = γSL 2πrδx − γSV 2πrδx + γLV 2πrδx cos θ + δU −δWB (5) where U is the energy stored in the electric field in the dielectric layer and WB the work done by the voltage source to build up the potential between droplet and electrode.

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The energy stored in a capacitor with large area A, small plate distance d, and relative dielectric constant εr of the material in between for a voltage V is given as U=

1 1 εr ε0 A 2 CV 2 = V 2 2 d

(6)

where ε0 is the dielectric constant of vacuum. Now we assume that the droplet changes its area by δA = 2πrδx because of movement of the contact line. Then the energy of the electric field changes by δU 1 εr ε0 2 = V . δA 2 d

(7)

The additional energy is fed into the system by the voltage source, so that δWB εr ε0 2 = V . δA d

(8)

δU/δA and δWB /δA can be combined into an electrowetting term γEW = δWB /δA − δU/δA, whereupon (5) reads δF = γSL − γSV + γLV cos θ − γEW δA

(9)

with γEW =

1 εr ε0 2 V . 2 d

(10)

The Young equation (4) then becomes cos θ =

γSV − γSL + γLV

1 εr ε0 2 2 d V

.

(11)

This can be modeled as an equivalent interfacial tension of the liquid to the substrate, i.e., γSL (V ) = γSL (0) −

1 εr ε0 2 V 2 d

(12)

on those parts of the contact area where it overlaps with the respective electrode. By applying a voltage to an adjacent electrode pad, and provided that the contact interface overlaps this second electrode, a droplet seeks to increase its contact area on that pad at the cost of the area on the current pad. Therefore, a motion to the next electrode takes place. Subsequent application of this algorithm allows the transport of the droplet over a larger distance. By moving two droplets to the same spot, mixing can be achieved. Splitting requires more complicated actuation schemes, which can benefit from proper design tools. An analysis of droplet splitting was presented in [15] and [16]. C. Electrowetting Devices The setup described above requires the tracking of the droplet and the movement of the wire accordingly. Further, the wire also distorts the droplet shape, impeding use in optical applications. Therefore, some more advanced setups are used, as demonstrated in Fig. 6.

Fig. 6. Different actuation setups for electrowetting. (a) Droplet in contact with a wire; (b) two capacitive contacts; (c) confined droplet (electrode is complete upper substrate); (d) inverted setup; (e) liquid in a channel.

Fig. 6(a) shows the classical setup with one wire providing an ohmic contact and the capacitive coupling on the substrate. It is also possible to operate the droplet with two capacitive contacts [Fig. 6(b)]; the contact line must overlap with two electrodes, between which the voltage is applied. The droplet then wets both electrodes. In this case, only half of the applied voltage is available for each pad, since the electric field passes the dielectric layer twice. Another solution is to use a conductive plate instead of the wire, such that the droplet is confined between two substrates [Fig. 6(c)]. This also facilitates the splitting of droplets [16], since the Laplace pressure of the droplet surface is lower. Using a transparent conductor like indium tin oxide (ITO), optical monitoring is still possible. Especially for optical purposes, it is useful to invert the setup [Fig. 6(d)]: a nonconducting liquid is immersed into an electrolyte; the voltage is applied between the surrounding medium and the substrate. The main advantage is that the droplet shape is not distorted by an electrode, and setups with radial symmetry are easy to build, so that good adaptive lenses can be created. Finally, electrowetting can also be used to pump liquid in a channel; besides moving droplets in capillaries, one possible use is the priming of a microfluidic device, to avoid bubbles of air clogging the system. III. S IMULATION W ITH THE S URFACE E VOLVER An analytical solution to these equations is only possible for very simple cases; in general, numerical models are required. We implemented the electrowetting model with the Surface Evolver, a powerful program for the numerical modeling of minimal surfaces. The Surface Evolver by K. A. Brakke is an interactive program for the study of surface shapes arising from surfacetension effects and other energies. It “evolves” the surface to an energy minimum by a gradient descent or conjugate-gradients minimization. It is possible to introduce spatial constraints as


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well as global surface integral constraints like a fixed volume [17], [18]. By formulating appropriate energy terms, the effect of nonuniform surface tensions can be integrated. A. Numerical Representation In the Surface Evolver, the droplet is represented by its bounding facets, which are flat triangles defined by three vertices (points in the Euclidean R3 space) and three connecting edges. The basic operation for the evolution of the surface is the iteration step that moves the vertices along the energy gradient. The actual displacement is the product of the energy slope of the respective degree of freedom and a global scale factor, which can be specified by the user or optimized by the Surface Evolver. An additional quantity-correcting motion enforces global quantity constraints. For a facet with edges s0 and s1 , the facet energy due to surface tension γ can be calculated by E=

γ |s0 × s1 |. 2

(13)

It is straightforward to show that the gradient gi = ∂E/∂xi of the first edge s0 is then g s0 =

γ s1 × (s0 × s1 ) . 2 |s0 × s1 |

(14)

Summing up all gradient parts of the adjacent faces yields the total free-energy gradient of the vector motion [17]. B. Substrate–Liquid Interfaces The interface of the droplet to air is modeled by a triangle mesh as described. However, for the interface to the substrate, a mesh is inappropriate for a number of reasons. First, on those parts of the interface with constant interfacial tension, there is no gradient for the vertices sitting on the interface; this could lead to numerical problems and mesh degradation. Second, to model a varying interfacial tension as needed for electrowetting, the surface energy of the triangles would have to be updated whenever the triangle changes its position. Finally, it would be a waste of resources, since there is a very elegant way to solve this problem: instead of an explicit representation of the interface between droplet and substrate, the energy is added to the total energy by transforming the surface integral (2) into a line integral over the surface boundary [18]–[20]. This boundary is represented by the edges of the triangles at the contact line. With the Green–Gauss theorem, we have A

= γSLndA

g dl

(15)

∂A

with normal vector n and γSLn = ∇ × g . = kdA , where k is the unit Since on the bottom surface dA vector in the z direction, we require a g such that the third

Fig. 7. Droplet on a square electrode (left) and on an electrode with a jagged edge (right).

component of its rotation, fz = ∂gy /∂x − ∂gx /∂y, is equal to the interfacial tension γSL . Choosing gx = 0, we get   0 g =  x γSL (x , y)dx  . (16) 0 On the top (confined droplet), the sign is inverted. This approach can also handle spatially varying interfacial tensions. If only the interface on the bottom is replaced, the volume calculation is left to the Surface Evolver. For a confined droplet, the removed interface at the top must also be manually integrated in the volume calculation, as shown in [18], [19], and [21]. C. Electrowetting Electrowetting effects are modeled by using (12) and setting up γSL (x, y) such that, above the electrode, the second (electrowetting) term is switched on, and is left zero otherwise. Multiple electrodes with different voltages can be treated analogously. γSL (x, y) is then integrated according to (16) and written to the Surface Evolver script file. With the use of parameters, the voltage can be changed during runtime. Typically, the edges of the electrodes for this kind of electrowetting pump feature spikes reaching into the adjacent electrode. The reason for this arrangement is that the dynamics at the start of the droplet motion is essentially determined by the shape of the potential energy curve at the adjacent electrode, and thus, by the drag force on the contact line. For a flat electrode edge, the interfacial tension is likewise approximately flat with a transition at the pad boundary (Fig. 7). According to (2), this results in a flat potential curve as long as the contact line does not touch the actuated pad, and thus, a zero force. With the jagged pad edge, interdigital structures are possible that are also in touch with a droplet on the adjacent electrode. Thus, there exists an energy gradient, resulting in a driving force. The shape of these interdigital structures determines the drag force, and thus, the character of the initial motion. By optimizing its shape, it is possible to account for different droplet sizes and chemical contaminations on the substrate. Those contaminations can lead to a contact-angle hysteresis [10] and even inhibit the motion of the droplet. These shapes are not implemented in detail, because to resolve a jagged-electrode shape in all its complexity would require a very-fine mesh resolution of the contact line; further mesh degeneracy and instabilities were observed in numerical experiments. Instead, we assume that a spikes’ size is small

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Fig. 8. EDEW user interface. Left: simulation parameters; Top: Surface Evolver control window; Bottom: graphics window (provided by the Surface Evolver).

enough so that its effect can be averaged along the edge direction [22] y2 γ(x) = y1

γ(x, y )dy . (y2 − y1 )

(17)

IV. EDEW, A T OOL FOR E LECTROWETTING C HIP D ESIGN To implement a simulation, some experience in writing of Surface Evolver script files is required to specify the model along with constraints and surface energies. Writing new models can slow down the design process. However, ready-made solutions for standard problems could be used. We therefore provide a tool to simplify this process: a script template library is provided along with a user-friendly GUI for all relevant model parameters. For experienced users, direct interaction with the Surface Evolver still remains possible. The frontend is written in Java for portability reasons. Fig. 8 shows the main components of the program: the panel on the left allows entering parameters for the template library. Then, after starting the simulation, the control window (top right) opens, which allows interactive control of the simulation process. Each template set provides its own parameter and control panel. Currently, three models are implemented; extending the library is easily done by extending the Simulation Java class. Details of the available models and the Java class are provided in the user manuals [23], [24]. The first model (1DPath) provides a line of electrode pads both for confined and nonconfined droplets (Fig. 9). It allows one to test the basic operations of an electrowetting array like moving, dispensing, merging, and splitting. Since the topology of the droplet remains unchanged during the simulation, splitting and merging is detected by the designer using the graphical output.

Fig. 9.

1DPath model and the adjustable geometry parameters.

This model is also useful to explore the exact droplet shape, which is, e.g., valuable for the estimation of optical properties for microlens design [8], and how the shape behaves during the basic operations. This may also be critical for the optical detection of fluorescently marked biological molecules where refraction effects and signal variation due to the droplet depth must be compensated. Another very interesting question is evaporation [25], which is highly dependent on the local curvature; this can induce fluid flow inside the droplet. It also helps to solve practical questions, e.g., how long it takes until the droplet is evaporated, and how fast measurements must be done until the decrease of volume affects the results. The electrode edge structure is averaged as described above but still indicated in the graphical output for visualization purposes. To give the designer the possibility to optimize these interdigital structures, an extended version of the 1DPath model is provided. The SpikeShape model allows for selection from either a number of predefined spike shapes (sinusoidal, triangular, rectangular, rectangular with user-definable pulsewidth, see Fig. 10) or the definition of additional shapes.


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where d is the scale factor chosen by the Surface Evolver. By using x˙ ≈ (xn+1 − xn )/d, linearizing and reordering x˙ + Kx = fext .

(19)

Now let us have a look at a mass/damper/spring system subject to an external force fI + fD + fS = fext

Fig. 10. Variation of the interfacial tension γ(x) at the pad edge for different shapes.

For the latter, two steps are necessary: 1) find the function for γ EW (x) and normalize its support to the interval [0, 1] such that the normalized new function f (x) fulfills f (0) = 0 and f (1) = 1; x 2) according to (16), find the integral F (x) = 0 f (x )dx . These functions can also feature parameters, which can even be changed during runtime. It is possible to operate the model in a free-motion mode, where the droplet moves only according to the electrowetting forces, or in a constrained mode, which is the recommended mode for spike-shape optimization: Here, the centroid of the droplet is forced to a given position, resulting in a potential energy-over-centroid curve. Based on this curve, a dynamic model can be extracted that also allows the estimation of inertial effects. A third model simulates a liquid meniscus in a rectangular channel. The mesh consists only of the meniscus area; the liquid volume is modeled through surface integrals and constraints. For all four-channel walls, different properties and voltages can be specified. This model can be interesting if electrowetting serves for priming a fluidic structure by placing electrodes on one of the walls, e.g., for estimating the minimum voltage for wetting the complete channel wall. To avoid numerical problems, the voltages should be changed in small steps. V. L IMITS As already indicated above, the chosen approach results in a number of limitations. In this section, we discuss the consequences for the use of the presented model. 1) Inertia and Damping: The model evolves the droplet shape and position to a point of minimal potential energy. The trajectory of the degrees of freedoms is not necessarily the path a fully dynamic simulation would take. This also means that inertia and damping effects are not included in the model. However, under some assumptions, there is nevertheless a close relation to a more complete simulation due to the way the Surface Evolver calculates the motion of the mesh vertices. Their motion is proportional to the (negative) energy gradient, i.e., the resulting force f acting on the vertex, subject to constraints [17] xn+1 = xn + d ∗ f (x)

(18)

(20)

where fI = M x ¨ is the reaction force of the inertial mass M subject to acceleration, fD = C x˙ is the damping force of the system, fS = Kx is the reaction force of the stiffness K, and fext is the external force. x may also be vector-valued; M , C, and K then turn into matrices. The external force is balanced by the inertial, damping, and stiffness force. The work applied by the external force is converted into kinetic energy, potential energy, and dissipation by damping. At the beginning of the motion, energy mainly goes into the acceleration of the mass (kinetic energy), which can drive the system beyond the equilibrium point, where Kx = fext , leading to an oscillation. This is true if the ratio of the damping force over the inertia force is small enough. For a massless or strongly damped system where the damping force is much higher than the inertial force, fI ≈ 0 and the remaining ordinary differential equation (ODE) reads fD + fS = fext

(21)

C x˙ + Kx = fext .

(22)

or

With C = 1, this is the same formula as for the Surface Evolver evolution step except for the provision of constraints [17] and the timestep. The result is a damped motion—similar to what can be seen from movies of droplets moved by electrowetting [26], which is indicated by the scale effects discussed in the introduction. Damping was also found to be important in the context of droplet vibrations [27]. Still, this damping should not considered to be the real damping of the physical system, which is influenced by the fluid motion and other friction effects, but the equilibrium position after a long time is still the same. The main trait of such a damped system is the absence of overshooting effects that can push the system to a state that is not reachable in the quasistatic limit. One example is a droplet that is accelerated and moved to an electrode that is much larger than the droplet. In a full dynamic model, the droplet may end up further in the interior of the electrode. Droplet splitting is another example, where inertia may lead to an augmented droplet motion. The numerical experiment must therefore be carefully checked if it is necessary to include such effects—analogous to an RF switch consisting of two beams that are attracted to each other by electrostatic actuation: In one case, one would like to find a minimal voltage where the switch will close independent from squeeze film damping and from the applied voltage curve that may be distorted by parasitic line capacities; in another, one wants to find the maximum voltage one can

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Fig. 11. Finite-element-method (FEM) solution of the electrostatic energy near the contact line. The indicated values must be multiplied by 1014 J/m3 to obtain the energy density.

apply with a given curve such that no switching occurs. It is the first case where the main value of a quasi-static simulation lies: Even if due to a slow actuation, the inertia is not as high as expected, there is still the wanted effect, and the massless system gives a conservative design rule for these circumstances. 2) Energy Dissipation: There is no information on energy dissipation by damping. Therefore, the energy needed for a strongly damped process cannot be calculated by this simulation. We assume that the voltage source is capable of delivering all the energy needed to reach the equilibrium state. 3) Peripheral Electric Field: The electrostatic energy is calculated only below the droplet/substrate interface. Peripheral electric fields and the electric field in the air are not considered. However, the contribution of the air to the energy is small due to the fact that the field strength is smaller due to the lower dielectric constant (usually a factor of 2 or 3) and the longer length of the electric flux lines (the potential difference remains constant). Whereas the thickness of the dielectric layer is in the micrometer range or even below, the lengths of aerial flux lines are in millimeter dimensions. More errors could come from the region near the contact line, both from the contribution of the peripheral field inside the dielectric layer as well as in the surrounding air. We have performed a finite-element simulation to investigate the effect of this on the calculation of the modified interfacial tension. The simulation shows the region close to the contact line of a droplet. We assume a potential of 1 V at the droplet boundary and 0 V at the bottom electrode. The dielectric layer with a relative permittivity of 2 is 1 µm thick. The result (Fig. 11) shows that, except for a small region around the contact line, the electric-energy density is close to the assumed values of 0 away from the interface and 8.85 J/m3 just below it. Near the contact-line singularity, there is a small region where large values of the electrostatic energy are observed; nevertheless, this region is small compared to the remainder of the droplet. In conclusion, we observe a distortion of the electric field only at a region in the order of the size of the layer thickness, which is small compared to the droplet dimensions. Further, since only the energy difference of two systems (or the energy gradient) is important, we expect an influence

of these distortions only if the length of the contact-line part on the electrode or its curvature experience a large change. This happens, e.g., on that point where it intersects with the electrode boundary; still, the change of effective diameter is small for a small dielectric-layer thickness and a large droplet. The distortions inside and outside of the contact area also partly balance each other. 4) Charge Trapping: If the dielectric layer is penetrable by charged particles and the voltage is applied for a certain time, charges may be trapped inside [14]. This is often seen as one reason for the so-called contact-angle saturation, where the contact angle does not change any more if the voltage is increased above a certain limit; further, it impedes the reversibility of the interfacial tension change, leading to contact-angle hysteresis. This could be modeled by an additional voltage contribution, such that turning the voltage “off” means setting it to a finite value that models the trapped charges. 5) Charged Biomolecules: A further distortion of the process can come from large charged molecules—or molecules with a nonuniform charge distribution, which distort the Helmholtz layer of the droplet and modify the capacity of the droplet/electrode system. This could also cause contact-angle hysteresis, if the molecules remain attached to the substrate. This can be modeled by an additional “off” voltage as discussed above and by a modified layer thickness. However, these approaches need further experimental validation. 6) Topological Changes: Droplet splitting and merging is not fully implemented in the model, manual inspection remains necessary. This is due to the explicit surface representation; with a levelset or volume-of-fluid approach, this is only a minor issue. However, in these methods, the determination of the contact line and the surface reconstruction is more difficult, which is important for, e.g., optical applications.For droplet merging, on the other hand, it is easy to see from the graphical output whether the operation was successful and whether the droplets touch. Droplet splitting is more difficult to see, it occurs when the liquid bridge connecting the two parts collapses to a line, or even overlap and interpenetration occurs. Due to the implementation of the energy calculations, this singularity poses no numerical problems. VI. R ESULTS In this section, we show the results of a number of simulations performed with our model. All of them with the exception of the curved channel and the tube model can be performed with the EDEW tool; however the pinch-off simulation requires some manual input. A. Droplet Motion Fig. 12 shows the simulation of a nonconfined droplet moved by electrowetting with the material and operation data of Table II. There is no other external force to the droplet except for the change of interfacial energy. At the beginning of the motion [Fig. 12(a)], the change of the hydrophobic to hydrophilic behavior of the pad is clearly visible at the contact line on the actuated electrode. The droplet then


243

Fig. 13. Splitting of a droplet by electrowetting. The dark electrodes are actuated with a voltage of 25 V. Fig. 12. Simulation results for moving droplet: (a) after actuation of electrode; (b) moved to second pad, electrode actuated; and (c) relaxed after grounding electrode. TABLE II PARAMETERS FOR THE SIMULATION IN FIG. 12

TABLE III PARAMETERS FOR THE SIMULATION IN FIG. 13

natural hydrophobicity. If the parameters are well chosen, the droplet splits and two single droplets, each with half the volume, remain. We stop the simulation just before topological changes occur due to pinch-off, resulting in the shape shown in Fig. 13. The computation time for this simulation was about 3.5 min on an AMD Athlon 64 3000+ (1.8 GHz), the surface is discretized using about 1000 vertices. C. Rising Fluid in Tube This example shows a liquid column rising in a cylindrical tube due to capillary action. The capillary forces are balanced by gravity in the direction of the tube F c = Fg

moves without external influences, only because of the change in interfacial energy, to the next pad. After turning off the voltage, the droplet relaxes to its initial state. Another simulation where the droplet was not overlapping the adjacent electrode at the start shows no motion. This simulation can be used as a first validation against experiments. B. Droplet Splitting This simulation shows the successful splitting of a confined droplet. We repeat the experiment in [16] using the values in Table III. We place the droplet off center so that unbalanced splitting occurs as is sometimes seen in experiments. Another simulation with a centered droplet (not shown) resulted in an even partition. The procedure for splitting is as follows. 1) Spread the droplet over a number of electrodes (e.g., 3) by activating all of them. 2) Switch off electrodes in the center of the droplet. While the outer active electrodes still attract the droplet, the central inactive electrode repels the droplet due to its

(23)

2πrγ = πr2 &gh

(24)

2γ ⇒ h= r&g

(25)

where Fc and Fg are capillary and gravitational force, respectively, r is the tube radius, γ is the interfacial tension to the wall of the tube, & is the fluid density, g is the gravity constant, and h the height of the meniscus. The interfacial tension to the wall of the tube can be varied by electrowetting. Since an analytical solution is available, we can use this example as a verification for our approach. Fig. 14 shows a comparison between the analytical result and the Surface Evolver result (height average of meniscus vertices), yielding a very close match between the two. D. Pinch-Off in Confined Setup This simulation considers the case of a confined droplet losing volume, e.g., by evaporation. A failure of such a setup can occur because of two geometrical effects. The first danger is that the droplet becomes smaller than the electrode size. If it is then sitting in the interior of the electrode, with no overlap with an adjacent electrode, it is not possible any more to move the droplet away from this spot (see Figs. 6 and 7).

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still the meniscus stops at a certain point, and more voltage is needed for a further shift. However, for a straight channel as implemented in the EDEW model library, we observed that at a certain voltage we get a large increase in the proceeding of the contact line in the channel; its position increases further and further. Fig. 17 shows the different states of the meniscus for the system given in Table IV: Fig. 17(a) shows the equilibrium state for zero voltage. The other two graphs show the meniscus for a voltage of 86 V. This is not the equilibrium state; since complete wetting occurs for this value, the contact line proceeds further and further into the channel, until the finite resolution of the mesh leads to numerical instabilities. F. Optimization of Electrode Fine Structure Fig. 14. Height of a liquid column in a tube subject to electrowetting.

This problem can be easily tackled by making the electrodes smaller than the considered “worst case” droplet volume, such that even in that case transport remains possible. But since the confined setup only works properly as long as the droplet is in contact with both substrates, pinch-off must also be avoided at all circumstances. Assuming a contact angle θ at the substrate and a distance of h between top and bottom covers, we can calculate that the sufficient volume, where contact is always guaranteed, is V ≥ πh

3

1 1 − 1 − cos θ 3

.

(26)

If the contact angles on the substrates differ, the smaller of the two must be used, since a smaller contact angle decreases the height of the droplet and thus is the more critical part. Fortunately, there is a safety margin between the theoretical value and the actual pinch-off. As can be seen in Fig. 15, the shape of the evaporating droplet just before pinch-off is almost cylindrical near the hydrophobic part. This corresponds to a local energy minimum, which traps the surface in this shape. A further decrease in droplet volume finally results in the system leaving the local minimum. However, once the droplet has detached, recovery is impossible. This margin is clearly visible in Fig. 15, with minimal volume where the contact angles of both substrates are equal.

E. Channels When electrowetting is performed in channels, there is an additional constraint to the droplet motion: The surfaces of the channel walls heavily influence the droplet shape and thus the balance of surface-tension and interfacial energies. This becomes especially important if the channel changes its cross section or ends at a larger reservoir: The fluid might get stuck, because a large force is necessary to modify the surface. Fig. 16 shows a series of pictures of a liquid meniscus in such a channel with a varying cross section. The fluid itself is not discretized, but included by surface-integral transformations similar to (16). The voltage on the meniscus is increased from left to right, but

We calculated the free energy of a droplet being moved over actuated electrodes with different shapes of interdigital structures [22]. We studied the shapes shown in Fig. 10 for a structure length of 100 and 400 µm. The parameters of the model are shown in Table V. Initially, the droplet resides next to the pad to which the voltage is applied such that it does not touch the pad-edge structure of the actuated pad at all. We assume that only one pad is actuated at a time. We further assume that the motion happens on a much larger time scale than the fluidic relaxation of the droplet, i.e., the fluid shape follows the movement adiabatically. The droplet is then moved manually onto the pad. For every simulation step, the energy minimum for the droplet surface is calculated, with the constraint that the centroid of the droplet is fixed at a given location. The surface energy is evaluated and is plotted versus the centroid position. We compare the results to a geometric model, for which the following assumptions have been made: 1) The liquid–air interface does not contribute to the energy change, i.e., its area is approximately constant. 2) The base radius of the contact line does not change 3) The contact line always forms a circle (see Fig. 18). The potential energy change can then be calculated by evaluating rB 2 − ξ 2 γ(ξ + x )dξ 2 rB ∆E(xc ) = c

(27)

−rB

where rB is the radius of the contact line and xc is the position of the center of the contact line. The radius of the droplet base for a contact angle θ can be calculated with

3V . (28) rB = sin θ · 3 π(1 − cos θ)2 (2 + cos θ) The results of the Surface Evolver model are shown in Figs. 19–21. White circles indicate where the contact line arrives at the interdigital edge structure and where it arrives on the bulk pad. 1) Influence of the Spike Shape: The difference of the potential energy for the different shapes is clearly visible. The


245

Fig. 15. Minimal transportable volume of a droplet in a sandwich structure. Left: simulated minimal volume compared to sufficient transport condition for a plate distance of 100 µm and a constant contact angle of 110◦ on one plate. Right: development of the droplet shape with decreasing volume. TABLE V PARAMETERS FOR THE ELECTRODE-FINE-STRUCTURE OPTIMIZATION

Fig. 16. Liquid meniscus in a curved channel for different voltages.

Fig. 18. Schematic drawing of the geometric model.

Fig. 17. Meniscus in a rectangular channel. (a) Meniscus at low voltage. (b) and (c) Meniscus at higher voltage; a contact angle of 0◦ occurs. TABLE IV PARAMETERS FOR THE SIMULATION IN FIG. 17

rectangular shape shows a very steep energy descent from the beginning, which indicates rapid acceleration. The triangular shape shows a very shallow decrease and thus a vanishing energy gradient at the beginning, even the 10%spike shape performs better. However, the curve recovers very fast, and in the long run, the curves of b) and d) coincide.

Fig. 19.

Potential energy for different pad-edge shapes with length at 100 µm.

The sinusoidal shape lies in between. The energy gradient is larger than for the triangular shape at the beginning, but lower after half of the structure is passed.

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Fig. 20. Potential energy for different pad-edge shapes with length at 400 µm. The shape of the curves is similar to Fig. 19.

Fig. 22. Potential energy for different pad-edge shapes with a length 400 µm, calculated with the geometric model.

3) Comparison With Geometric Model: Fig. 22 shows the potential energy difference calculated with the geometric model. The curves are in excellent agreement with Fig. 20, showing the same features for the different shapes. For the droplet further on the pad, the curves were found to slightly diverge from the Surface Evolver curves; however, the curve shapes remain identical. VII. C ONCLUSION

Fig. 21. Potential energy for different spike lengths. The curves for sinusoidal shapes coincide after an initial energy difference; the curves for the (c) shapes show a clear shift to the right.

The spike shapes show a very low energy gradient. We also see that the energy curve is shifted to the right, because the structure does not cover half of the area as for the other examples, but only 10% and 25%, respectively. In contrast, the overall energy decrease is equal once the complete contact line of the droplet has passed the structure. A rectangular shape seems to be optimal with respect to the acceleration of the droplet; however, since the adjacent interdigital edge structures would touch at a pulse ratio of 50%, the fabrication of this ideal case is challenging and expensive. But since a smaller pulse ratio would impair the performance of the structure—as is visible for the c) shapes—either the sinusoidal shape or a mix of the triangular and the rectangular shape should be preferred. 2) Influence of the Spike Length: The curves for different spike lengths show good congruence for different sizes; the length does not affect the shape of the energy curve deformation, only its extent (Fig. 21). The contact line above the structure moves faster than the remaining part of the droplet, thus the effective structure length is smaller than the true length. The overall energy decrease is independent of the spike structure. Again, the c) shapes show a large shift towards positive x values. Since the initial energy gradient becomes lower the larger the spikes are, there is a tradeoff between a large size to reach small drops and a small size for a large gradient.

We have presented a modeling-and-simulation methodology for electrowetting effects, which enables the designer to calculate droplet shapes and provides insight into the energy configuration of electrowetting arrays, which is useful for the dimensioning and layout of biochips. A method for the calculation of the fine structure of the electrodes was presented and applied to the optimization of spike shapes for interdigital edge structures, which help to make the electrowetting process more reliable. The comparison with an analytic model confirms the resulting energy curves. All in all, the Surface Evolver simulation does much more than merely simulate the motion of electrowetted droplets, for it enables us to obtain a clear picture of the potential energy landscape for a specific electrode setup together with a moving droplet. In this way, we can go back and reshape the electrodes until the obtained energy landscape is of a configuration that allows controlled behavior of the “gadget” we are implementing with the electrode, be it a mover, splitter, or merger. These simulations were integrated into a user-friendly simulation tool based on the Surface Evolver code. A template library provides ready-made scripts, so that in most cases the simulation can be performed without the need for manual script input. The tool is available from [28]. R EFERENCES [1] D. J. Laser and J. G. Santiago, “A review of micropumps,” J. Micromech. Microeng., vol. 14, no. 6, pp. R35–R64, Jun. 2004. [2] D. R. Reyes, D. Iossifidis, P.-A. Auroux, and A. Manz, “Micro total analysis systems. 1. Introduction, theory, and technology,” Anal. Chem., vol. 74, no. 12, pp. 2623–2636, Jun. 2002. [3] P.-A. Auroux, D. Iossifidis, D. R. Reyes, and A. Manz, “Micro total analysis systems. 2. Analytical standard operations and applications,” Anal. Chem., vol. 74, no. 12, pp. 2637–2652, Jun. 2002. [4] M. G. Pollack, R. B. Fair, and A. D. Shenderov, “Electrowetting-based actuation of liquid droplets for microfluidic applications,” Appl. Phys. Lett., vol. 77, no. 11, pp. 1725–1726, Sep. 11, 2000.


[5] J. Ding, K. Chakrabarty, and R. B. Fair, “Scheduling of microfluidic operations for reconfigurable two-dimensional electrowetting arrays,” IEEE Trans. Circuits Syst., vol. 20, no. 12, pp. 1463–1468, Dec. 2002. [6] A. H. Tkaczyk, D. Huh, J. H. Bahng, Y. Chang, H.-H. Wei, K. Kurabayashi, J. B. Grotberg, C.-J. Kim, and S. Takayama, “Fluidic switching of high-speed air–liquid two-phase flows using electrowettingon-dielectric,” in Proc. 7th Int. Conf. Miniaturized Chemical and Biochemical Analysis Systems, Squaw Valley, CA, Oct. 5–9, 2003, pp. 461–464. [7] R. A. Hayes and B. J. Feenstra, “Video-speed electronic paper based on electrowetting,” Nature, vol. 425, no. 6956, pp. 383–385, Sep. 25, 2003. [8] B. Berge and J. Peseux, “Variable focal lens controlled by an external voltage: An application of electrowetting,” The Eur. Phys. J. E, vol. 3, no. 2, pp. 159–163, Oct. 2000. [9] J. Zeng, “Electrohydrodynamic modeling and simulation and its application to digital microfluidics,” in Proc. SPIE, Philadelphia, PA, L. A. Smith and D. Sob, Eds. Bellingham, WA: SPIE, 2004, vol. 5591, pp. 125–142. [10] P. G. de Gennes, “Wetting: Statics and dynamics,” Rev. Mod. Phys., vol. 57, no. 3, pp. 827–863, Jul. 1985. [11] J. Israelachvili, Intermolecular and Surface Forces, 2nd ed. New York: Academic, 1991. [12] M. G. Lippmann, “Relations entre les phenomenes electriques et capillaires,” Ann. Chim. Phys., vol. 5, no. 11, pp. 494–549, 1875. [13] M. Vallet, B. Berge, and L. Vovelle, “Electrowetting of water and aqueous solutions on poly(ethylene terephthalate) insulating films,” Polymer, vol. 37, no. 12, pp. 2465–2470, 1996. [14] H. J. J. Verheijen and M. W. J. Prins, “Reversible electrowetting and trapping of charge: Model and experiments,” Langmuir, vol. 15, no. 20, pp. 6616–6620, 1999. [15] S. K. Cho, H. Moon, J. Fowler, and C.-J. Kim, “Splitting a liquid droplet for electrowetting-based microfluidics,” in Proc. ASME Int. Mechanical Engineering Congr. Expo., New York, Nov. 11–16, 2001. IMECE2001/MEMS-23831. [16] S. K. Cho, H. Moon, and C.-J. Kim, “Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits,” J. Microelectromech. Syst., vol. 12, no. 1, pp. 70–80, Feb. 2003. [17] K. A. Brakke, “The surface evolver,” Exp. Math., vol. 1, no. 2, pp. 141– 165, 1992. [18] ——, Surface Evolver Manual, Version 2.20. Selinsgrove, PA: Susquehanna Univ., Aug. 2003. [19] J. Lienemann, “Modeling and simulation of the fluidic controlled selfassembly of micro parts,” Diplomarbeit, Inst. Microsyst. Technol., Albert Ludwig Univ., Freiburg, Germany, May 28, 2002. [20] J. Lienemann, A. Greiner, J. G. Korvink, X. Xiong, Y. Hanein, and K. F. Böhringer, “Modelling, simulation and experimentation of a promising new packaging technology—Parallel fluidic self-assembly of micro devices,” Sens. Update, vol. 13, no. 1, pp. 3–43, Mar. 2004. [21] J. Lienemann, A. Greiner, and J. G. Korvink, “Surface tension defects in micro-fluidic self-alignment,” in Symp. Design, Test, Integration and Packaging MEMS/MOEMS (DTIP), Cannes-Mandelieu, France, May 5–8, 2002, pp. 55–63. [22] ——, “Electrode shapes for electrowetting arrays,” in Proc. Nanotechnology Conf. (Nanotech), San Francisco, CA. Cambridge, MA: NSTI, Feb. 2003, vol. 1, pp. 94–97. [23] ——, EDEW Version 1.0, A Simulation Tool for Fluid Handling by Electrowetting Effects, Jan. 2004, Freiburg, Germany: IMTEK, Albert Ludwig Univ. [24] ——, EDEW Version 2.0, A Simulation and Optimization Tool for Fluid Handling by Electrowetting Effects, Apr. 2004, Freiburg, Germany: IMTEK, Albert Ludwig Univ. [25] L. R. van den Doel, L. J. van Vliet, K. T. Hjelt, M. J. Vellekoop, F. Gromball, J. G. Korvink, and I. T. Young, “Nanometer-scale height measurements in micromachined picoliter vials based on interference fringe analysis,” in Proc. 15th Int. Conf. Pattern Recognition, A. Sanfeliu, J. Villanueva, M. Vanrell, R. Alquezar, T. Huang, and J. Serr, Eds., Barcelona, Spain, Sep. 3–7, 2000, vol. 3, pp. 57–62. ser. Image, Speech, and Signal Processing.

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[26] Digital Microfluidics, Dept. Elect. Comput. Eng., Duke Univ. [Online]. Available: http://www.ee.duke.edu/research/microfluidics [27] A. Prosperetti, “Free oscillations of drops and bubbles: The initial-value problem,” J. Fluid Mech., vol. 100, no. 2, pp. 333–347, 1980. [28] Electrowetting, Laboratory for Simulation, Dept. Microsystems Eng. (IMTEK), Univ. Freiburg. [Online]. Available: http://www.imtek.de/ simulation/microprotein

Jan Lienemann obtained the Diploma degree in microsystem engineering from the Institute for Microsystem Technology, Albert Ludwig University, Freiburg, Germany, in 2002, where he is currently working towards the Ph.D. degree in the Microsystem Simulation Group. He is interested in research on the modeling and simulation of electrowetting and the model-order reduction of nonlinear systems.

Andreas Greiner received the Ph.D. degree from the University of Stuttgart, Germany, in 1992. He held several grants as a Research Fellow at the University of Bologna, the University of Lecce, and the Centre National de la Recherche Scientifique (CNRS) Research Center for Microoptoelectronics, Montpellier, France. He holds the position of a Senior Scientist at the Institute for Microsystem Technology (IMTEK), Albert Ludwig University, Freiburg, Germany, as member of the Group for Simulation. Dr. Greiner is a Group Leader of physical modeling and, in addition to conventional fluid dynamical simulation, he also explores schemes such as Lattice–Boltzmann techniques, dissipative particle dynamics (DPD), as well as advanced simulation methods for continuum systems.

Jan G. Korvink obtained the M.Sc. degree in computational mechanics from the University of Cape Town, South Africa, in 1987, and the Ph.D. degree in applied computer science from the Swiss Federal Institute of Technology (ETH) Zurich, Switzerland, in 1993. After his graduate studies, he joined the Physical Electronics Laboratory of ETH Zurich, where he established and led the Modeling Group. This was followed by a move to the Albert Ludwig University, Freiburg, Germany, where he holds the position of Chair in Microsystem Technology and runs the Laboratory for Microsystem Simulation. Currently, he is the Dean of the Faculty of Applied Science. He has written more than 130 journal and conference papers in the area of microsystem technology, and coedits the review journal Applied Micro and Nanosystems. His research interests cover the modeling and simulation and low-cost fabrication of microsystems.