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The Equilibrium Shape of Liquids Under Electric Fields

N. Fortner, B. Shapiro* * Corresponding Author: 3178 Martin Hall, Aerospace and Bio-Engineering,

University of Maryland, College Park, MD 20742. Acknowledgement: This work has been supported under DARPA grant number FCP0.0205GDB191, contact monitor Dr. Anantha Krishnan.

1. Abstract Electrowetting, the shape change and motion of liquids that is created by applied electric fields, is caused primarily by a competition between electro-static and surface tension energies. In order to design systems that exploit electro-wetting actuation, it is necessary to have efficient models that predict electro-wetting behavior. This paper presents a minimum energy optimization technique to find the equilibrium shape of liquid packets under applied electric actuation in micro-fluidic devices. We find the shape of liquid packets that can be described by one-dimensional curves, this includes: two-dimensional planar liquid shapes relevant in thin planar micro-fluidic devices and three-dimensional rotationally symmetric liquid shapes. The approach works for any two-dimensional or rotationally symmetric device geometry, it holds for liquids and solids with arbitrary electrical and surface tension properties, and it is much more computationally efficient then the alternative approach which is to solve a dynamic coupled two-phase Navier Stokes / electro-static simulation till equilibrium. As a demonstration of the approach two rotationally symmetric and one two-dimensional planar scenario are solved in the results section of the paper.

2. Introduction We predict the shape of liquids inside micro-fluidic devices under the influence of (a type of) electrical actuation. Micro-fluidic devices enable a vast array of chemical and biological tests, reactions, and manipulations in a miniaturized format; they have a low incremental cost per device once the fabrication process has been developed (for conventional fabrication processes [1, 2]) or a low total cost altogether (for soft lithography and other non-traditional fabrication techniques such as [3-11]); and they enable rapid test results with miniscule quantities of reagent. In micro devices, surface effects usually dominate bulk volume effects [12, 13]: this is because surface effects scale as length squared while volume effects scale as length cubed so the ratio of the two increases as length scale decreases. This leads to two recognized implications for micro-fluidic systems: first, surface tension effects are critical on the micro-scale, and second, pressure actuation becomes significantly less effective [13]. As a result, liquid handling methods that do not rely on pressure actuation have become popular for micro-fluidic systems. In particular, electrical actuation is common. Here we focus on electrical actuation of fluids that relies on a competition between surface tension and dielectric energy storage [14]. In particular, our approach is relevant to EWOD (Electro-Wetting-On-Dielectric) actuation [15-18]. EWOD has been used to move, split, mix, and join liquid droplets inside micro-fluidic devices [16, 18, 19]. This actuation is chiefly a

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surface driven effect (dielectric energy is stored in a thin dielectric layer under the liquid and so the amount of energy storage scales primarily with the liquid/solid area) but bulk electrical effects are also important and can limit the effectiveness of the devices [14]. For the EWOD systems [15-18], the competition between surface tension and dielectric energy storage is the dominant physical phenomena. Accounting for this competition between surface tension and dielectric energy exactly recovers the Lippmann-Young equation in the ideal case of a conducting sessile liquid droplet on top of a dielectric solid [14], and it predicts experimental data in the UCLA EWOD devices including contact angle saturation which is recovered by considering a slightly resistive liquid atop a dielectric solid [14-16]. In [14] we predicted the behavior of a perfectly spherical sessile droplet on top of a flat solid surface. Our goal here is to extend this result to non-spherical liquid shapes inside or on top of more complex device geometries. In particular we consider liquid shapes that can be described by one-dimensional curves, which includes two-dimensional planar liquid shapes, and three-dimensional rotationally symmetric shapes (as shown in Figure 5). The same optimization methodology can be used to find three-dimensional shapes that are not rotationally symmetric but this requires a more sophisticated optimization over two-dimensional surfaces and will be the subject of future research. Moreover, we wish to find liquid shapes for any kind of liquid and solid materials and with any type of electrical boundary conditions. We do this by searching for liquid shapes that minimize the total system energy. Viewing equilibrium liquid shapes as energy minima is a standard approach: it is applied in [20-23] to find the contact angle of sessile liquid drops, and it is used in [22, 24] to predict more general liquid shapes which are in contact with arbitrarily shaped solids and are under the influence of a variety of force fields not including electrical effects. In this paper, we consider the influence of electric fields. This means that our system energy consists of surface tension and electrical energies. As in [14], we go back to physical first principles and solve Maxwells equations to find the electric fields that give rise to the electrical energies. The solution domain of Maxwells equations includes the liquid shape and so the electric fields depend on the shape of the liquid. As a result, we are concerned with a coupled surface tension / electrical energy minimization problem: the electric field depends on the liquid shape and the equilibrium liquid shape depends on the surface tension and the electrical energies. Efficiently solving this coupled problem is the main contribution of this paper. We solve the problem by implementing an iterative electrical solve / gradient descent optimization algorithm. Our approach is an efficient way to find the equilibrium shape of liquids in non-trivial device geometries under electrical actuation. Compared to the alternate approach which would simulate a dynamic coupled 2-phase Navier Stokes / electro-static set of partial differential equations till equilibrium and would take many hours or days of computer time, our method is numerically simple and runs in tens of minutes on a laptop computer. The method can handle liquid and solid materials with any type of surface tension and electrical properties including non-ideal dielectric materials; it is capable of handling all the cases covered in [14], conducting liquid on insulating dielectric solid, insulating dielectric liquid on conducting solid, insulating dielectric liquid on insulating dielectric solid, and the slightly resistive liquid on dielectric solid case which predicts

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contact angle saturation in the UCLA EWOD devices; and our approach predicts the shape change of insulating dielectric liquids on dielectric solids which has been demonstrated experimentally only recently [25]. The paper is organized as follows. Section 3 outlines the energy minimization approach and quantifies both the surface tension and electrical energy in terms of the liquid shape. Section 4 presents the numerical algorithm that is used to find the minimum energy liquid shapes. This section is organized according to Figure 1 which gives an overview of the algorithm. Finally, Section 5 shows three sample results and provides a discussion of the algorithm behavior and performance.

3. Modeling Approach Our goal is to find the equilibrium shape of small packets of liquids inside solid geometries of arbitrary shape and under the influence of surface tension forces and applied electric fields. It is well recognized that (stable) equilibrium shapes are the ones that minimize the potential energy of the system [22, 24, 26] and so our approach here is to solve an optimization problem for the shape of the liquid/gas interfaces that minimizes the total system energy. This shape is constrained by the fixed solid geometry of the device and by the amount of liquid contained in the device. Micro-fluidic devices that use EWOD (ElectroWetting On Dielectric) for liquid droplet handling [15-18] exploit the balance between surface tension and dielectric energy to actuate liquid packets. Hence in dealing with these devices we are concerned with two main contributions to the system energy: the surface tension energy of the liquid/solid, liquid/gas, and solid/gas interfaces; and the dielectric energy stored inside the liquid and solid materials as well as inside the external charging source. The effect of gravity is negligible on small scales [14]. It is appropriate to focus on the electrical energy stored only in dielectric materials because non-dielectric materials do not store electrical energy – however, their shapes and resistive properties can alter the electric fields that are available to the dielectric materials for energy storage. This is why a slightly resistive liquid can create contact angle saturation in the EWOD devices: as the liquid spreads it progressively deprives the underlying dielectric solid of its electrical field and the dielectric energy that is required for continued actuation. Our first goal is to quantify the surface tension energy. As in [20, 21] , we describe the surface tension energy of an interface between two materials X and Y by E = σXY AXY where σXY is the material dependent surface tension coefficient and AXY is the area of the interface. The net surface tension energy for a liquid in contact with a solid and a gas can now be written as

( ) ( ) [ ]

LG LG LS LS SG SG

LG LG LS LS SG S SL

LG LG LS SG LS SG S

E A A AA A A AA A A

σ σ σσ σ σσ σ σ σ

= + += + + −= + − +

where, in the second line, we have rewritten the solid/gas area ASG as the total solid area AS minus the solid/liquid area ASL. Since the total solid area is constant, the term in square brackets

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on the third line is independent of the shape of the liquid and can be discarded from the optimization. Thus the relevant surface tension energy becomes (1) ( )ST LG LG LS SG LSE A Aσ σ σ= + − .

If there are multiple types of solid, for example if the device is made out of multiple materials or has a variety of hydrophobic and hydrophilic coatings, then equation (1) is modified to

(2) ( )ST LG LG LSp GSp LSppE A Aσ σ σ= + −∑

where the index p identifies the different solid materials. Our next task is to quantify the electrical energy. When a dielectric material is charged by an electric field, the dielectric energy per unit volume inside the material is dE = ½ ( )D Ei dv where

( , , )E x y z is the local electric field, ( , , )D x y z is the polarazibility vector field, and dv is a differential volume element [26]. For an ideal dielectric material, the polarazibility or induced dipole moment is linearly related to the electric field by the dielectric constant thus D Eε= . The dielectric energy of a material with dielectric constant εX and volume vX is now given by

(3) 21

2 ( , , )X

DE XvE E x y z dx dy dzε= ∫∫∫ .

Equation (3) holds for both solid and liquid dielectric materials. If there are multiple dielectric materials then the dielectric constant ε is treated as a constant within each material but is allowed to jump across different materials. The electric field, which appears inside the above volume integral, is given by a solution of Maxwells partial differential equations with appropriate boundary conditions. Attention is restricted to the case where changes in the applied voltage are slow compared to the millisecond electrical time constants [14] and, under this assumption, Maxwells equations reduce to the electrostatic Poisson equation

(4) ( ) 0ε φ∇ ∇ =i .

with the associated electric field E φ= −∇ . In addition to the energy of equation (3), it is also necessary to include the electrical energy that is stored in the external charging source because this energy is part of the total system energy and it can depend on the liquid shape [14]. The energy stored in the charging source is twice again the energy stored in the dielectric materials but with opposite sign (see Section II-8-2 in Feynman [26] for a complete discussion) thus the net electrical energy is (5) - 2 -ELEC DE CS DE DE DEE E E E E E= + = = .

The total energy is the sum of the surface tension and electrical energy. To minimize the number of model parameters that must be considered, we non-dimensionalize this total energy by the device length scale L, by the liquid/gas surface tension coefficient σLG, and by the applied voltage V [14]. This yields

(6) 2

12 ˆ

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆLG LS ELEC LG LS vE A A WE A A W dvφ= +Γ + = +Γ − ∇∫

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where the hat denotes non-dimensionalized quantities, φ is the electric potential that solves equation (4), ( )LS GS LGσ σ σΓ = − is the non-dimensional surface tension coefficient, and

2LGW V Lε σ= is the non-dimensional electrowetting number. Equation (4) is now viewed in

non-dimensional variables and has a unit length and voltage scale. Equilibrium liquid shapes minimize the total energy defined by equations (4) and (6). Since the electric field solution and liquid shape problems are coupled (the electric field depends on the liquid shape and the optimal shape depends on the total energy that depends on the electric field) we use an iterative procedure to find the minimal energy shape. Specifically, we guess at an initial shape, solve Poissons equation for the electric field, find the gradient of the total energy with respect to chosen shape parameters, change the liquid shape to decrease energy but keep liquid volume constant, and then repeat the procedure until convergence is achieved. The next section describes the numerical algorithm that implements this method.

4. Numerical Implementation An overview of the numerical shape optimization algorithm is shown in Figure 1. The algorithm starts with an initial liquid shape guess quantified by a discretized one-dimensional curve y. A MATLAB script computes the corresponding liquid volume and the liquid/gas and liquid/solid interface areas. The partial differential equation solver FEMLAB (http://www.comsol.com/) [27] is now used to solve the Poisson equation (4) for the liquid domain defined by y and with chosen liquid and solid boundary conditions. This allows the computation of the non-dimensional energy of equation (6) and gives the total system energy for the initial shape y. A set of k hat basis functions h1, h2, … ,hk is now used to perturb the nominal shape from y to y1 = y+h1, …, yk = y+hk. The volume and total system energy is recomputed by the MATLAB/FEMLAB sub-routine for each of these k perturbed shapes. The energy and volume gradients are now found by comparing the nominal volume and energy with the perturbed volumes and energies. A step along the energy gradient “E will decrease the energy of the system most effectively. However, this shape change will also change the volume. In order to keep the liquid volume constant, we project the energy gradient onto the constant volume sub-space (this is equivalent to taking a constrained Lagrange multiplier step). We then use the Armijo rule to find the maximum possible step size d* that still respects the energy and volume change gradient approximation, and we update the current shape from y to y - d* h* where h* is the projected shape change that best minimizes the energy while keeping the volume constant. The process repeats until an optimal liquid shape y* is found. At present, our liquid interface is described by a one-dimensional curve or curves. This means we can address two-dimensional liquid shapes, as is appropriate for liquids in thin planar micro-fluidic devices, or we can address three-dimensional shapes that have a rotational symmetry, this is useful for dealing with rotationally symmetric droplets or plugs of liquid. The minimal energy/ electrostatic-solve methodology presented in this paper extends to arbitrary three-dimensional scenarios, but the associated mathematics and algorithms required to find optimal two-dimensional interfaces are more sophisticated [22]. We are currently working on combining the Surface Evolver [22] (http://www.susqu.edu/facstaff/b/brakke/evolver/), which finds minimal

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energy surfaces without electric fields, with the partial differential equation solver FEMLAB. This will allow us to extend the algorithm of Figure 1 to three spatial dimensions.

Figure 1: An overview flow-chart of the shape optimization algorithm.

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The FEMLAB solver can handle cases where the dielectric parameter ε is not constant and instead depends on the local electric field. This allows us to treat materials with non-ideal dielectric properties. Due to reliability and power considerations, the electric fields that are typically applied in electro-wetting systems are below the electric fields that would lead to non-ideal dielectric effects such as dielectric breakdown. Consequently, we do not consider any non-ideal dielectric cases in our results section (Section 5). Below, in Sections 4.1 to 4.9, we describe the steps that are carried out within each part of the algorithm shown in Figure 1.

4.1. Defining the Liquid Interface Shape The shape of the liquid material is quantified by a discretized curve y = (x1,y1, x2,y2, …, xk,yk) in the xy plane. This curve can either represent the xy average of a two-dimensional interface that extends into the z direction or it can be rotated around the y axis to represent a three-dimensional solid of revolution. It is convenient to represent each point (xj,yj) of y by a single scalar number sj. This can be done a number of ways. For the simple solid C geometry shown schematically above and in Figure 2, sj can simply be the horizontal distance of the jth point away from the vertical middle wall of the C. In this case y = (x1,y1, x2,y2, …, xk,yk) = ( s1,0, s2,Dy, s3,2Dy, … , sk,(k-1)Dy ) so the sj will vary as the optimization progresses but the Dy = 1/(k-1) terms will remain constant (Dy is appropriately chosen for a unit height C). For a slightly more complex geometry, such as the rotationally symmetric drop of Figure 7, sj can be measured along rays that emanate from the point on the underlying solid which is directly below the outside of the wire. Thus the first ray is along the bottom surface: varying s1 allows the liquid to move on this solid; and the last ray is along the outside of the wire: varying sk allows the liquid to move up and down the wire. For more general geometries, we replace the rays by a sequence of curves that span the liquid geometry. Hence for the two-dimensional plug between a flat plate and a parabola, see Figure 9, where bottom surface is defined by y=0 and the top surface is defined by y=(pd) x2 +d, we let the half of the points on the right be defined by (xj,yj)=[ sj, (p (j-1)Dd) sj

2 +(j-1)Dd ] for j ranging from 1 to k/2 and Dd is chosen so that (k/2-1)Dd = d, and the other half of the points is defined in a similar way but with j ranging from k/2+1 to k. In all cases we represent the free curve (or curves) y by k scalar parameters sj, thus: y = y(s1,s2,s3,… , sk).

4.2. Computing the Liquid Volume and Interfacial Areas by MATLAB In order to evaluate the surface tension energy, and to enforce the constant liquid volume constraint, it is necessary to compute the liquid volume and interface areas for each shape y. In two spatial dimensions, the liquid volume is simply the area circumscribed by the solid geometry and the discretized curve y = (x1,y1, x2,y2, …, xk,yk). In the case of the C geometry when y = ( s1,0, s2,Dy, s3,2Dy, … , sk,(k-1)Dy ) we compute this area by a mid point integral discretization

( )1 1121

kj jj

V s s y−

+=≈ + ∆∑ . Similar formulas are derived for more complex geometries and more

complex parameterizations y(s1,s2,s3,… , sk). In three dimensions, the volume is given by the discretization of the volume of revolution integral 2V x dyπ= ∫ . Once again, we use a mid point discretization to compute this integral; the formulas are easily derived and are not shown here.

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In order to evaluate the surface tension energy of equation (2), we must find the liquid/gas and liquid/solid interface areas. In two dimensions, the liquid/gas area is equal to the arc length of y

and is given by ( ) ( )2 2121 11

1 ( / ) kLG j j j jj

A dx dy dy x x y yψ −

+ +== = + ≈ − + −∑∫ . Likewise, the

liquid/solid areas are given by arc lengths along solid parts of the geometry. In the C geometry, the bottom liquid/solid area is given simply by ALSb = x1 = s1 and the top area is given by ALSt = xk = sk. For three spatial dimensions, the liquid/gas area integral becomes an area of revolution, specifically ( )12 2

112 1 ( / ) 1 ( / )k

LG j jjA x dx dy dy x x x y yπ π−

+== + ≈ + + ∆ ∆ ∆∑∫ where

1j jx x x+∆ = − and 1j jy y y+∆ = − . The liquid/solid areas are also straightforward, in particular, for the rotated C geometry of Figure 8, the bottom and top areas are simply ALSb = p x1

2 = p s12

and ALSt = p xk2 = p sk

2. Formulas for other geometries are also easily derived. In all cases, the necessary volume and interfacial area quantities are computed by MATLAB sub-routines during the algorithm evaluation.

4.3. Solving Poissons Equation and Finding the Dielectric Energy by FEMLAB Given the shape y, we now need to find the stored dielectric energy. It is best to demonstrate the procedure with a simple and concrete example, and then to describe what changes and additions must be made to address more complex cases. Consider a two-dimensional liquid packet inside a solid C geometry of unit size, as shown in Figure 2. The floor of the C is held at ground, the top of the C has an applied unit voltage, and the side of the C is insulating. Liquid shape is defined by the discretized curve y. In this case there is only one material, so the non-dimensional version of the Poisson equation (4) is simply “2f(x,y)=0 and this equation must be solved over the liquid domain with the stated boundary conditions. The resulting electric field is shown in the middle of the figure. Dielectric energy per volume dEDE/dv is given by half the magnitude of the electric field squared. Integrating this quantity over the liquid volume gives the net non-dimensional dielectric energy EDE stored in the liquid.

Figure 2: The electric field and dielectric energy for a two-dimensional liquid inside a solid C geometry. The figure on the far left shows the applied boundary conditions and the liquid geometry defined by the curve y.

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The middle figure shows the resulting electric field E (arrows) and electric potential f (shading). The last figure shows the dielectric energy per unit volume dEDE/dv=|—f|2: black denotes low energy and white denotes a high energy. The integral of this quantity over the volume gives the net dielectric energy EDE.

In order to deal with three-dimensional, rotationally symmetric volumes, such as the rotationally symmetric drop of Figure 7 or the rotationally symmetric liquid plug of Figure 8, the procedure above is modified as follows. The Poisson equation in three-dimensions “2f(x,y,z)=0 is re-written in cylindrical coordinates with an assumed rotationally symmetry about the y axis

2 22 2

2 2

1( , , ) ( , )x y x yx x x zφ φ φφ α φ ∂ ∂ ∂

∇ = ∇ = + +∂ ∂ ∂

,

and with appropriate non-dimensional boundary conditions. At zero radius (so at x=0) there is now a symmetry boundary condition ˆ (0, ) / 0n y xφ φ∇ = ∂ ∂ =i . The electric field is still given by

E φ= −∇ which is now written as ˆ ˆx yE e ex yφ φ∂ ∂

= − −∂ ∂

where ˆxe and ˆye are unit vectors in the x

(radial) and y (vertical) directions respectively. Likewise, the dielectric energy per unit volume is

still given by half the magnitude of the electric field squared 221

2DEdE

dv x yφ φ⎡ ⎤⎛ ⎞∂ ∂⎛ ⎞= +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎝ ⎠⎣ ⎦

.

However, the integral over the volume must include an integration over the azimuthal angle a,

this yields 2 DE Axy

E x dx dyπ φ= ∇∫∫ where Axy is the liquid filled region in the a=0 plane.

To handle multiple liquid and solid dielectric materials, it is necessary to phrase and solve Poissons equations with different dielectric constants e within different material domains. Figure 7 shows such an example where the liquid has a small dielectric constant eL and the underlying solid has a large dielectric constant eS. The simplest way to handle such cases is to let the dielectric constant e be a piecewise constant function of space. Then Poissons equation becomes ( )( , , ) ( , , ) 0x y z x y zε φ∇ ∇ = . This naturally enforces the correct electric field jump conditions,

specifically n̂ε φ∇ i is constant, between any two interfaces where n̂ is the unit normal to the interface. For the example shown in Figure 7 it implies that eL ∑fL/ ∑z equals eS ∑fS/ ∑z across the liquid/solid interface. It is also possible to phrase more involved scenarios: such as the one discussed in [14] where the liquid is not a dielectric but has a small amount of resistance compared to the solid, and this causes a potential drop in the liquid which deprives the solid dielectric of some of its dielectric energy. We use FEMLAB (http://www.comsol.com/) [27] to solve the two- and three-dimensional Poisson equations stated above. FEMLAB models can be created using MATLAB commands and so it is possible to define the liquid and solid domains, state the required partial differential equations, the boundary conditions, and evaluate the electric energy integrals using MATLAB *.m scripts.

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4.4. Computing the Non-Dimensional Total System Energy To get the total system energy we combine the results of Section 4.2 and Section 4.3. Equation (6) states the total non-dimensional system energy as 1

2ˆ ˆˆ ˆ

LG LS DEE A A WE= +Γ − . Computation of the non-dimensional liquid/gas and liquid/solid areas is described in Section 4.2. Computation of the non-dimensional energy stored in the dielectric materials is described above in Section 4.3. The three contributions are weighted by the non-dimensional contact angle ( )LS GS LGσ σ σΓ = − and the non-dimensional electrowetting number 2

LGW V Lε σ= . The end result is the total system energy as a function of the liquid shape y.

4.5. Perturbing the Liquid Shape: The Hat Basis Functions In order to quantify changes in the liquid shape using a finite number of parameters, as is required in the optimization routine, a set of shape perturbation basis functions must be chosen. Choosing the shape perturbation basis functions largely determines the convergence properties of the optimization routine and is the most critical aspect of the method. We have experimented with using Fourier modes, Chebyshev polynomials, and hat functions, and we have found that the hat functions provide the best convergence results (and they are also the easiest to implement). As noted in Section 4.1, the liquid shape disretized curve is defined by k parameters s1,s2,…,sk. Hence the jth basis function hj will be a k dimensional vector hj = ( hj1, hj2, … , hjk ). A new shape will be defined by a new s vector [ 1] [ ] [ ]

1 1 2 2 ...n n nk ks s s s a a aη η η+ = + ∆ = + + + + . Our choice of

the hat basis functions is shown in Figure 3: it simply states that the jth basis function modifies only sj by an amount h, so hj = ( 0, 0, … , 0, h, 0, … 0 ) with the h in the jth location.

Figure 3: Schematic of the shape perturbation hat basis functions.

The hat function height h must be chosen judiciously. If the height h is too small then the nominal shape and the perturbed shape will differ by so little that the difference in the system energy will fall inside the numerical noise floor of the FEMLAB simulation. Conversely, if the height h is too big then the change in shape and shape gradients will create a change in energy that is not well approximated by a linear gradient expansion. Figure 4 shows how the energy difference changes as a function of h for a typical case. Cleary, there is a clear-cut linear range where the energy difference scales linearly with the hat height. The same figure also shows the FEMLAB simulation noise floor. A choice of h = 10-3 yields energy differences that are inside the linear range but are well outside the noise floor.

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Figure 4: Choosing the optimal hat basis function height h. The figure shows the difference between a nominal and perturbed shape energy as a function of perturbation hat height h and the number of shape discretization points k. As the height h is increased, the energy difference grows linearly until the change in shape enters the nonlinear regime (h > 10-2). Also shown is the FEMLAB simulation noise floor, this is computed by comparing the energy for the same shape but with two different meshes. The best choice of h is big enough to be well above the noise floor but small enough to still be inside the linear range. Here, a choice of h = 10-3 yields energy differences three orders of magnitude above the noise floor (so gradient estimation errors will be in the range of 0.1%) but it is safely inside the linear range.

4.6. Estimating the Volume and Energy Gradients Energy and volume gradients with respect to shape change parameters are estimated using a finite difference approach. Starting from the current initial shape y = y(s1,s2,…,sk) with liquid volume v and net system energy E, we create k perturbed shapes using the basis of Section 4.5 (7) y1 = y(s1+h,s2,…,sk), y2 = y(s1,s2+h,…,sk), … , yk = y(s1,s2,…,sk+h).

For each shape yj we find the corresponding volume vj and total system energy Ej via the approach of Sections 4.2, 4.3, and 4.4. Energy and volume gradients are then written as (8) ( )1 2, , , /kE E E E E E E h∇ = − − −…

(9) ( )1 2, , , /kv v v v v v v h∇ = − − −… .

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4.7. Finding the Optimal (Volume Conserving) Shape Change Direction Equation (8) tells us which shape change direction will yield the steepest descent in the net system energy. Specifically, if we change the shape from the nominal ( )1 2( ) , ,..., ks s s sψ ψ ψ= =

shape to the new shape *( )s sψ ψ δ= + ∆ with * /s E E∆ = −∇ ∇ then the net system energy will decrease the maximum amount per step size δ . However, in addition to decreasing the system energy we must also keep the liquid volume constant, and, since we have not yet used the volume gradient information of equation (9), there is no guarantee that the *s∆ stated above will conserve volume. To ensure that volume is conserved, we project the steepest energy descent direction

* /s E E∆ = −∇ ∇ onto the constant volume sub-space. Doing this projection is equivalent to

carrying out a constrained Lagrange multiplier optimization [28]. Let ˆ /v v v∇ = ∇ ∇ be the unit normal vector along the volume gradient “v. Within the linearization approximation, a shape change parameter vector s∆ will cause a zero change in volume only if ˆ 0s v∆ ∇ =i : in other words, if the shape change vector is orthogonal to the volume gradient and therefore lies in the constant volume sub-space. The part of the steepest energy descent vector *s∆ that lies outside the constant volume sub-space is given by ( )* * ˆ ˆs s v v⊥∆ = ∆ ∇ ∇i . Hence the part that lies inside the

sub-space is simply ( )* * * * * ˆ ˆs s s s s v v⊥∆ = ∆ −∆ = ∆ − ∆ ∇ ∇i and this is the steepest energy descent direction that respects the constant volume constraint. Substituting back and rewriting this in terms of the energy and volume gradients gives the optimal shape change direction as

(10) ( )* * * * * ˆ ˆ E E v vs s s s s v vE E v v⊥

⎡ ⎤⎛ ⎞∇ ∇ ∇ ∇∆ = ∆ −∆ = ∆ − ∆ ∇ ∇ = − −⎢ ⎥⎜ ⎟⎜ ⎟∇ ∇ ∇ ∇⎢ ⎥⎝ ⎠⎣ ⎦

i i .

4.8. Choosing the Maximal Shape Change Step Size for Fast Convergence At each iteration of the algorithm, we update the shape of the liquid/interface y by a step in the direction of the steepest energy descent, constant volume direction of equation (10). In order to guarantee fast convergence of the overall algorithm, it is advantageous to take the maximum possible step size d. We use the Armijo rule [29] to find this step size. The Armijo rule affords a practical compromise between the computational effort devoted to a line search along the optimal *s∆ direction and the computational effort dedicated to the remainder of the optimization algorithm. We implement this rule as follows: we start with a minimal step size dmin that corresponds to the FEMLAB noise floor; and we then progressively increase the step size by factors of five until the change in energy and volume predicted by the gradients, and the actual change in energy and volume computed by MATLAB and FEMLAB differ by more then 5%. The liquid shape in the next iteration of the algorithm is then give by ( )* *s sψ ψ δ= + ∆ where

* *sδ ∆ is the largest optimal shape change that is still accurately represented by a linear gradient approximation.

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We have also found that it is prudent to limit the maximum shape change to less than 10% of the size of the liquid geometry: since our geometries are of unit size we require that every point (xj,yj) of the discretized liquid/gas interface y move a distance less than 0.1 units during each iteration step. If this is not done, then the optimization can take steps that are so large that they force the interface into non-physical shapes and this leads to unrealistic local energy minima. Accordingly we choose the step size d*+ that is the minimum of the Armijo rule d* and the 10% maximum step size d+.

4.9. Algorithm Stopping Conditions In the above sections we have described the majority of the liquid shape optimization algorithm shown in Figure 1. During each iteration of the algorithm the liquid shape is updated by a computed energy gradient that is projected onto a constant volume subspace. This iterative procedure continues until one of two stopping conditions is met. First, the algorithm is terminated if the energy of the subsequent iteration is larger then the energy of the current iteration. Second, the algorithm is terminated if the calculated shape change is comparable to the FEMLAB/MATLAB noise floor. Both instances occur when the shape, or shape gradient, has approached the optimal, or zero value, to within the energy computation error.

4.10. Summary of the Optimization Algorithm A summary of the algorithm is shown in Figure 1. Starting at the top left and following the logic flow defined by the arrows: Section 4.1 uses k shape parameters s1,s2,s3,… , sk to define a liquid/gas interface shape y; Sections 4.2 and 4.3 describe MATLAB and FEMLAB routines that compute the interface areas, volume, and dielectric energy for the shape y; Section 4.4 combines the results of Sections 4.2 and 4.3 to find the total system energy for the shape y; the process is repeated for the k perturbed shapes y1, y2, … yk discussed in Section 4.5; the energy and volume gradients with respect to the shape parameters s1,s2,s3,… , sk are found by finite difference in Section 4.6; and the optimal, energy minimizing, shape change direction that keeps the liquid volume constant is found in Section 4.7; Section 4.8 defines the choice of the largest feasible step size which insures fast convergence of the overall algorithm; and Section 4.9 outlines the optimization termination conditions. Some results of this algorithm are presented below.

5. Results We have tested the shape optimization methodology for a number of physical geometries, three of which are presented below (see Figure 5), and have found that the optimization converges to physically meaningful shapes – under prudent algorithm settings the optimization does not seem to get caught in non-physical local optima. The optimisation runs for any two-dimensional planar or three-dimensional rotationally symmetric geometry: the three examples below are simply convenient choices to demonstrate the method. It has been verified that the optimization results are invariant both under changes in the FEMLAB mesh spacing and under changes in the number of points k that are chosen to discretize the liquid/gas interface curve y.

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Figure 5: The geometry and boundary conditions for the three cases discussed in Sections 5.1, 5.2, and 5.3. In each case: bold lines denote the solid geometry; medium thickness blue curves y denote a sample liquid shape; grey shading represents an applied equipotential surface; all other surfaces have insulating (no flux) boundary conditions; and the grided area is the relevant computational domain that is either rotated (for cases I and II) or extruded (for case III) to create physical three-dimensional volumes. For case I, the liquid and solid have different dielectric constants eL and eS respectively.

However, we note that algorithm convergence can be substantially slower for bigger k. This is due to an obvious and a subtle reason. The obvious reason is that we have to solve the Poisson equation k times to find the energy gradient of Section 4.6: since the Poisson FEMLAB solve is the most expensive part of the algorithm, increasing k proportionally increases the algorithm run time. A more subtle reason is the following: if we use a large number of points to discretize the liquid shape, then some of the points, especially points at the solid/liquid boundaries, can get stuck for a very long time. This is because a non-optimal small linear segment will accrue only a very small surface tension area penalty, and if, in addition, the small segment is in an area of low electric field then it will also have a small electric energy penalty. As a result, such a small linear segment can get stuck. A good work around is to first do a coarse optimization with a small k of five to twenty points, and then to interpolate the near optimal coarse curve with more points and carry out a fine scale optimization. Using this coarse/fine approach, convergence is usually achieved within twenty to a hundred iterations for the coarse part of the algorithm, and then just a few more iterations at the fine scale (typically less than ten) yields the final solution. In most cases, a fine optimization is not really needed since even the coarse optimization provides a sufficiently smooth curve.

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Algorithm solutions have also been verified against known cases. The algorithm correctly recovers the optimal spherical cap shape for a liquid droplet on a flat surface with no applied electric field. For the C solid shape of Figure 2 it can be shown, by the calculus of variations [30], that if there is no energy associated with the solid/liquid interface (G of equation (6) is zero) then the optimal liquid/gas shape y is a vertical line because this shape minimizes the electrical energy and also keeps the liquid/gas area to a minimum. It has been verified that the energy minimization algorithm recovers this optimal vertical interface. Finally, Figure 6 shows a comparison between the optimization and an experiment for an electrically actuated liquid drop.

Figure 6: The equilibrium shapes for an electrically actuated water droplet on top of a solid dielectric. The left panel shows an experiment from Hyejin Moon at UCLA: when a voltage is applied between the inserted electrode and an electrode underneath the dielectric layer, then the left spherical cap shape deforms into the shape on the right. The right panel shows the results of our energy minimization algorithm. In this case, the liquid is treated as a perfect conductor so there is no electric field or stored dielectric energy in the liquid.

5.1. Case I: Rotationally Symmetric Dielectric Liquid Drop on Dielectric Solid The first case we show is an extension of the one presented in Figure 6. In that example there is no electric field in the liquid and the electric field in the solid immediately underneath the liquid is uniform, this means it is possible to include the effect of the electric field inside the G liquid/ solid surface tension parameter of equation (6) and thus there is no need to numerically solve Poissons equations. We now consider the case where both the liquid and the solid can store dielectrical energy, as would be the case if we had a dielectric liquid like oil placed on top of a dielectric solid. This case shows that it is possible to electrically actuate an insulating liquid as has been demonstrated by recent experiments at UCLA [25]. The solid geometry and the imposed electrical boundary conditions are shown in the first panel of Figure 5. Figure 7 shows the optimal liquid shape as the non-dimensional electrowetting number W= eSV2/LsLG of equation (6) is increased. Both the liquid and the solid are coloured by the dielectric energy per unit volume. The small potential drop across the liquid deprives the solid of some of its electrical energy that would otherwise create an even more pronounced droplet actuation.

Figure 7: A three-dimensional rotationally symmetric drop of dielectric liquid on top of a dielectric solid with a ratio of dielectric constants of R = eL/eS = 1/10. The optimal shape of the liquid is shown as the voltage

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increases from zero voltage on the left to a high voltage on the right. Three parameters quantify the behavior: the non-dimensional surface tension coefficient G=(sLS-sGS)/sLG , the non -dimensional electrowetting number (of the solid) W = eSV2/LsLG, and the dielectric constants ratio R. The color denotes the electrical field strength: black for no field, white for a high field. The notch is the inserted electrode.

5.2. Case II: Rotationally Symmetric Liquid Plug between Conducting Plates Next we show a case where a dielectric liquid is confined between two conducting plates. This case shows that the shape dependent dielectric energy that is stored in an insulating liquid is sufficient to create electro-wetting actuation. The case is shown schematically in the second panel of Figure 5. There is a bottom hydrophobic horizontal electrode held at ground, a top hydrophobic horizontal electrode at a vertical height L that is held at a voltage V, and a plug of dielectric liquid between them. The liquid/gas interface is insulating. Our goal is to find the shape of the liquid as the voltage increases. Figure 8 shows a xy slice through the resulting liquid shapes. As in the other examples, all computations are done in non-dimensional variables: thus there is a unit distance between the electrodes and the voltage is quantified by the electrowetting number W=eLV2/LsLG.

Figure 8: The equilibrium shape of a three-dimensional rotationally symmetric liquid plug between conducting electrodes: an xy slice of the liquid is shown colored by its electrical energy per unit volume. As the voltage is increased from a zero voltage on the left to a high voltage on the right, the liquid approaches a vertical cylinder configuration that minimizes its electrical energy. (Note that the electrical equations are always non-dimensionalized to a unit applied voltage hence the color scale is the same for low and high voltages cases. It is the electrowetting number W in equation (6) that quantifies the increase in voltage.)

The liquid starts out with a concave shape because the parallel electrodes are hydrophobic. As the voltage is increased the liquid attempts to reach a vertical cylinder configuration that minimizes its electrical energy. The reason that the vertical liquid configuration is an electrical energy minimum is interesting. From equation (5), we know that the electrical energy EELEC is at a minimum when the dielectric energy EDE is at a maximum. The dielectric energy is the integral of (half of) the magnitude of the electric field squared. Thus the vertical shape must correspond to a, on average, maximal electric field inside the liquid. When the liquid shape is deformed to any non-vertical shape, then there are regions of constriction with higher electric fields and regions of expansion with lower electric fields (the total flux through any horizontal plane must remain constant). This effect is clearly visible in Figure 2 and means that regions of expansion

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have lower dielectric energy per unit volume. It turns out that the loss in electric field at these regions dominates the gain in electric field at regions of constriction and any change in the shape away from a vertical cylinder leads to a lower net dielectric energy and hence a higher total energy.

5.3. Case III: Two-Dimensional Liquid Plug in a Parabolic Constriction Here we consider a planar case of a two-dimensional liquid plug between a flat and a parabolic electrode. This would correspond to a planar micro fluidic device where the vertical gap is small compared to the horizontal length scale, the floor and ceiling are made out of an insulating material, and the device walls consist of two planar electrodes -- one with a straight shape and the other with a parabolic profile (see last panel of Figure 5). Both electrodes are hydrophobic and the liquid/gas interface is a zero-electric-flux insulating surface. Figure 9 shows the result. The liquid plug initially starts at the outside of the parabolic channel (since the channel is symmetric about x=0 the liquid could equally well have started at a symmetric equilibrium on the left side). Electrical energy is at a minimum when the dielectric energy EDE of equation (5) is at a maximum. This happens when the average electric field inside the liquid is at a maximum, and this in turn occurs when the liquid is located at the constriction because then the potential drop is across the smallest possible gap. Thus as the voltage is increased, the liquid plug is sucked into the constriction at the center.

Figure 9: A two-dimensional dielectric liquid plug between a parabolic electrode at voltage V and a straight electrode at ground. As the voltage is increased from left to right the liquid is pulled into the constriction. As previously, the liquid is colored by its dielectric energy per unit volume.

6. Conclusion We find the equilibrium shape of liquids actuated by electro-wetting, on top of, or inside, micro-fluidic devices of arbitrary shape. The equilibrium liquid shape is determined by minimizing the sum of surface tension and electrical energies. Although electro-wetting is a complex phenomena that includes many physical effects, it has been verified experimentally in prior work [14] that surface tension and electro-static energy are the two dominant energies. Our optimization approach is numerically efficient and evaluates in tens of minutes of computer run time. It holds for liquids and solid materials with arbitrary electrical and surface tension properties. In

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particular, the approach can handle all the electrical material property cases discussed in [14] including the slightly resistive liquid on insulating dielectric solid case that predicts contact angle saturation in the UCLA EWOD devices. This approach also predicts the ability to electrically actuate insulating liquids that has just been demonstrated experimentally [25]. In its current form, our optimization method is restricted to liquid shapes that can be characterized by one-dimensional curves: this includes rotationally symmetric and planar shapes as shown in Figure 5. Three-dimensional liquid shapes that do not have a rotational symmetry will require an optimization over two-dimensional liquid/gas interfaces and will be addressed in future research.

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