Capillary force actuation

Transcription

1 J. Micro-Nano Mech. 9 5:57 68 DOI.7/s RESEARCH PAPER Capillary force actuation Carl R. Knospe & Seyed Ali Nezamoddini Received: 3 June 9 / Accepted: 4 December 9 / Published online: 6 January # Springer-Verlag Abstract Microactuators based upon the alteration of the capillary pressure within a conducting liuid bridge via the application of electric potential are investigated. The physical principles of actuation are elucidated and the euations expressing bridge shape and force produced as a function of applied voltage are developed. It is shown that capillary force actuators are capable of achieving significant greater forces at lower voltage than electrostatic actuators. Charts for the design of actuators are presented. A comparison to electrostatic actuation provides a novel perspective on the mechanism of capillary force actuation. A simplified model of the actuator is also developed. Keywords Microactuators. Electrowetting. Capillary force Introduction A variety of actuation principles are available for microelectromechanical systems MEMS. These are typically classified into four families: electrostatic electromagnetic piezoelectric and thermal. Within each of these families a variety of technologies are present each technology having advantages and disadvantages with respect to force capability power reuirements bandwidth and ease of device manufacture [ 3]. The most commonly used microactuators belong to the electrostatic family including comb and parallel plate configurations among others. In general actuators in this C. R. Knospe * : S. A. Nezamoddini Department of Mechanical and Aerospace Engineering University of Virginia Charlottesville VA 93 USA knospe@virginia.edu family are simple to fabricate using photolithographic micromachining since they do not reuire materials or elements that are difficult to integrate into this process e.g. ferrous cores. Electrostatic actuators typically reuire voltages higher than desirable for integrated circuits often greater than 6 V. In many applications these actuators are large due to several factors including: limited actuation stroke necessitates significant mechanical amplification to achieve the reuired range of motion; limitations on device voltage result in large electrostatic surfaces to achieve the force production reuired for the application; and for comb drive in particular 3 manufacturing limitations results in relatively large electrostatic gaps lowering actuator effectiveness; and 4 side snap-over instability limits achievable force and stroke. In this paper the potential of employing electrically activated capillary forces for MEMS actuation is investigated. A careful physical analysis shows that such capillary force actuators CFA can produce significantly greater force than achieved by similarly sized electrostatic actuators when the voltage levels used are restricted to those commonly employed in integrated circuits. Furthermore the analysis demonstrates that large actuation strokes may be achieved without mechanical amplification. We consider in this paper an actuator consisting of a conducting liuid bridge between two electrodes each covered with a thin dielectric layer as shown in Fig.. Upon application of a voltage the apparent contact angle of the fluid upon each surface will change due to the phenomenon of electrowetting on dielectric EWOD. As will be shown the change in apparent contact angles will result in a reduction in the bridge s capillary pressure and therefore a change in the force acting upon both surfaces [4 5]. While electrowetting has been used for moving fluids laterally along a surface its potential for creating large

2 58 J. Micro-Nano Mech. 9 5:57 68 Liuid Bridge forces normal to a surface has previously been unrecognized. The difference between CFA and EWOD droplet transport is not superficial. EWOD droplet transport results from an imbalance in the lateral electrostatic forces acting at the contact line and occurs with negligible change in capillary pressure. In CFA lateral forces are balanced and normal force production is primarily due to changes in capillary pressure acting over the liuid-solid interface. Evaporation is a concern for the small liuid bridges in CFA. Many factors will influence the degree of evaporation that occurs including the volume and type of gas surrounding the bridge; the volume of the bridge; the initial vapor concentration; vapor pressure and surface tension of the liuid; the temperature; and the evaporation and condensation coefficients. A preliminary study has shown that evaporation losses will be negligible if the bridge is enclosed and materials are properly chosen [5]. Electrowetting on dielectric In static euilibrium without applied electric field a liuid s surface satisfies Laplace s euation constant mean curvature subject to the boundary condition at the contact line dictated by Young s euation s gs s ls s gl cos ¼ Electrode Dielectric Fig. Schematic of a capillary force actuator not to scale for purpose of illustration height of liuid bridge is greatly exaggerated ðþ which expresses the contact angle of the liuid θ in terms of the interfacial energies σ ij between the liuid solid and gas phases [6]. To simplify the discussion we consider the case of a perfectly conducting liuid bridge. From the perspective of actuator performance high liuid conductivity aids both actuator gain and bandwidth as a simple euivalent circuit model indicates [5]. With the wide range of liuids and dielectrics available liuid conductivity 5 to5 orders of magnitude greater than that of the dielectric layer may be achieved. Therefore such an assumption is not only helpful in analysis but also accurate for devices of interest. Electrowetting was first interpreted in the literature in terms of a reduction in the interfacial energy of the liuidsolid interface σ sl with applied voltage. The balance of the surface tensions at the contact line described by E. was thereby altered resulting in a lower contact angle. Lately another view of electrowetting has been advanced in which the phenomenon is viewed as a purely electromechanical effect. In this interpretation the applied electric field causes an additional electrostatic pressure on the liuid surface resulting in a change in contact angle [7 8]. Adding further support to the electromechanical interpretation it has recently been shown by Mugele et al. [9 ] that under an electric field the actual contact angle on the surface remains unchanged and only the apparent contact angle is altered. This change in apparent contact angle is captured by the Young-Lippmann euation: cosð v Þ ¼ cosð Þþ " d v d ðþ s gl t d where θ v is the apparent contact angle of the liuid ε d is the permittivity of the dielectric t d is the thickness of the dielectric layer and v d is the voltage across the layer. In many applications the dielectric layer may not have desirable wetting properties e.g. native contact angle contact angle hysteresis and a very thin topcoat typically fluoropolymer is used. Depending upon its thickness the topcoat may significantly affect the electric field. In this case effective values for ε d and t d must be used in E.. Here we outline the physics of the electromechanical origin of electrowetting. When voltage is applied the electric potential induces charge double layers at the dielectric / electrode interface and the dielectric / liuid interface with free charge in the conducting bridge and electrode and polarization charge in the dielectric [ 3]. The capacitance of this arrangement will be essentially eual to that of the dielectric layer itself. From an electromechanical viewpoint the presence of free charge in the liuid interacting with the electric field imparts electrohydrodynamic forces to the fluid [3]. In the perfectly conducting case free surface charge in the liuid bridge screen the electric field E from the bridge s interior and charge only appear at the liuid surface. The tangential component of the electric field is zero at the liuids surface [9]. The normal electric field at the liuid-gas interface hence the pressure will be negligible except very close to the contact line within a distance of a few times the dielectric thickness [4]. The transition of the profile tangent angle from the native contact angle θ to the apparent contact angle θ v occurs in this region. Excluding this region we may state that no electrical force acts upon the liuid-gas interface. As the contact line is approached on either the solidliuid or liuid-gas interface the charge density and field increase rapidly [4]. On the scale of the device considered

3 J. Micro-Nano Mech. 9 5: the three-phase contact line may be treated as onedimensional. In this spirit integration of Maxwell s stress tensor in the vicinity of the contact line yields a contact line force per unit lenth. The component tangential to the solid surface i.e. radially directed in the case of a liuid bridge is given by ˆF e r ¼ " d v d ð3þ t d A balance of this contact line force and the interfacial tensions acting upon the contact line then yields the Lippmann-Young euation E. [9 ]. Using Es. 3 may be alternatively written as ˆF e r ¼ s glðcosð v Þ cosð ÞÞ ð4þ Euation predicts that the effective contact angle will go to zero at a certain voltage and complete wetting of the surface will occur. However in practice the contact angle ceases to decrease with increased voltage a phenomenon known as contact angle saturation. At present six different mechanisms have been proposed as the cause of contact angle saturation: dielectric charge trapping [5]; air ionization in the vicinity of contact line [4]; 3 instability of the contact line and the ejection of satellite drops [4]; 4 reaching zero effective liuid-solid surface tension [6 7]; 5 electrical resistivity of the liuid [8]; and 6 local electrical breakdown of the dielectric [9 ]. Independent of the other mechanisms electrical breakdown is certain to place a fundamental limit on the contact angle that can be achieved and conseuently the force levels that may be generated by CFA. 3 Capillary bridges without electric field 3. Bridge shape To begin the analysis of the device shown in Fig. we first consider the determination of the force exerted upon the two parallel surfaces by a capillary bridge between them when an electric field is not present. It is assumed that the device size is such that gravity may be ignored and that the surfaces and their alignment are ideal. In this case the bridge shape will be axisymmetric and may be specified by its profile. Denote the spacing between the plates and the bridge volume by h and V respectively. Define the aspect ratio of the bridge as a ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p h 3 4 V ð5þ Delaunay [] determined that the profile of a capillary bridge extending between two parallel plates may be parameterized by the two radii at which the profile curve achieves a tangent line parallel to the central axis of the bridge. One of these radii r will be the radius of the neck or haunch that lies between the two plates. If the plates have the same contact angle as achieved with the CFA shown in Fig. r will be the radius at the mid point between the plates. The second characteristic radius r is not physically manifested for a stable bridge. Rather it occurs when the Delaunay profile curve is mathematically continued beyond the physical confines of the plates [6]. This is illustrated in Fig.. In the case of eual contact angles θ on both plates a stable bridge may be specified by providing two of the following values: bridge volume V radius of the contact line r c and spacing between the plates h. Note that the contact angle is the angle of profile tangent at the plate surface. As the bridge is axisymmetric and satisfies Laplace s euation the bridge profile will belong to the Plateau seuence of shapes [6]. This seuence consists of nodoids and unduloids along with the special cases of sphere cylinder and catenoid. Plateau [] introduced a non-dimensional capillary pressure to categorize the bridge shape: Fig. Important radii and profile tangent angle in parameterization of an unduloid profile z r p ¼ P c r ð6þ s gl where the capillary pressure P c is the difference between bridge interior and exterior pressures. As demonstrated by Kralchevsky and Nagayama [6] the bridge shapes within the Plateau seuence may be parameterized by this dimensionless pressure see Table. Since these are the only solutions to the axisymmetric Laplace euation the profile of any capillary bridge without electric field must be realized as a scaled version of a section of one of these profiles. The analytical expressions for bridges within the Plateau seuence are listed in Table which was previously provided in part in Kralchevsky and Nagayama [6]. However here all the expressions are presented in nondir ϕ In the case of a cylindrical bridge ah/d where D is the diameter of the cylinder. With aspect ratio fixed h may be employed as a scaling factor.

4 6 J. Micro-Nano Mech. 9 5:57 68 Table Classification of Plateau seuence of shapes parameterized by dimensionless pressure Bridges with Neck Bridges with Haunch p < p < p < ½ p ½ ½ < p < p < p Nodoid Catenoid Unduloid Cylinder Unduloid Sphere Nodoid mensional form. In Table is the nondimensional profile radius; ζ is the nondimensional bridge axial coordinate with ζ corresponding to nondimensional neck/haunch radius ; is the angle between the profile tangent and the radial coordinate axis see Fig. ; and υ is the nondimensional bridge volume between ζ and ζζ. The functions Eϕ and Fϕ in Table are the elliptic integrals Eðϕ; Z ϕ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ¼ Δ sin ðuþdu ¼ Z ϕ Fðϕ; Þ¼ Δ du ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ sin ðuþ Z sin ϕ Z sin ϕ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t t dt dt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð t Þð t Þ The values of the radii axial coordinate and volume may be related to their nondimensional counterparts via: r ¼ kr ; r ¼ kr ; r c ¼ kr c ; r ¼ kr; z ¼ kz; V ¼ k 3 ð7þ u where k is the scale factor. The nondimensional radii and are related to the dimensionless pressure p as: r ¼ jrj; r ¼ j rj ð8þ Given contact angle and nondimensional pressure dimensionless contact radius c may be calculated as: r c ¼ Pð ; pþ ð9þ where P p gives the nondimensional radius for given pair of nondimensional pressure and contact angle and is defined as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pðf; pþ ¼ r r þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin ðþ f sin ðþþ4r f r sin ðþ f ðþ where the positive / negative sign is determined by whether a bridge has haunch positive or neck negative. Scale factor k may be found by euating the profile height z at the contact line with one half of the actual bridge height i.e. spacing between plates: zðr c Þ ¼ kzðr c ; pþ ¼ h ðþ Solving for κ yields: k ¼ h½ zðpð ; pþ; pþš ðþ The volume of the bridge may be calculated as: V ¼ k 3 ur ð c ; p Þ ¼ k 3 uðpð ; pþ; pþ ð3þ Substitution for κ from E. yields an expression for the bridge volume in terms of dimensionless pressure contact angle and bridge height: V ¼ h3 uðpð ; pþ; pþ ½ zðpð ; pþ; pþš 3 ð4þ Given a bridge height volume and contact angle the dimensionless pressure may be determined by finding the root of E. 4. This euation may alternatively be written in terms of aspect ratio a: rffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p h a ¼ 3 p½zðpð ; pþ; pþš 3 ¼ 4 V uðpð ; pþ; pþ ð5þ which indicates that the value of the dimensionless pressure is dependent only on aspect ratio and contact angle; it is independent of scaling. Thus the profile s shape is governed only by these nondimensional parameters. Bridge shape without electric field may be determined using the following algorithm. First dimensionless pressure is determined by solving E. 4. The scale factor may then be calculated using E.. Last the bridge profile rz also r r r c is found from E. 7.

5 3. Force exerted 3.. Introduction Given the bridge shape the axial force without electric field may be determined in semi-analytic fashion via either of two approaches: summation of the capillary pressure and surface tension contributions the Stevin approach [6]; and differentiation of the total potential energy with respect to a virtual displacement of one plate. Both approaches will be introduced here; the later approach will be particularly useful when a simplified model is developed in the seuel. Table Euations specifying capillary bridge profile surface area and volume parameterized by dimensionless pressure Bridges with Neck Bridges with Haunch p < p < p < p < p < p < p Nodoid χ Catenoid Unduloid χ Cylinder Unduloid χ Sphere Nodoid χ Notation / / sin - / / sin Shape χ ζ / F E F E ζ Slope / cos / tan ϕ ζ ϕ d d ζ 4 / d s Area / ] [ 4 / E 4 E ζ ζ ζ υ ] [ d d Volume [ 3 β β F E 3 χ β ] [ 3 β F E 3 χ β J. Micro-Nano Mech. 9 5:

6 6 J. Micro-Nano Mech. 9 5: Stevin approach The force exerted by the bridge F c is expressed as the sum of contributions from the liuid s surface tension F σ and the capillary pressure F p : F c ¼ F s þ F p F s ¼ p r s gl F p ¼ p r P c Using E. 6 we have ð6þ ð7þ ð8þ F c ¼ pr s gl ð pþ ð9þ To find the force exerted the following algorithm may be performed. Given contact angle θ and bridge volume V the nondimensional pressure p and radius r may be found using Es. 4 and 7 respectively. The force may then be determined using E Virtual work approach The force exerted by the bridge may also be determined by differentiating the potential energy stored in the device with respect to a virtual displacement of one surface. With no electric field the energy stored is the sum of three surface energies each the product of the surface tension coefficient and the area of the corresponding two-phase surface: E ¼ s gs S gs þ s gl S gl þ s ls S ls ðþ where S ij is the area of the interface between the i and j phases. The sum of the gas-solid and liuid-solid areas is eual to the total area of solid and is therefore constant: S gs þ S ls ¼ S s ðþ Combining Es. and the total energy may be written as: E ¼ s gl S gl S ls cosð Þ þ sgs S s ðþ Since the last term is constant it will not contribute to the force exerted. The capillary force F c is conseuently: ds gl F c ¼ s gl dh þ s gl cosð Þ ds ls ð3þ dh To evaluate the variation of the areas S gl and S ls with bridge height h must be determined. The areas S gl and S ls may be found via S ls ¼ p r c ð4þ S gl ¼ Z h sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p rðzþ þ dr dz ðzþ dz ð5þ In this formulation use of the tangent slope dz dr ðrþ in the expression for S gl has been avoided as it would result in significant numerical difficulties for values of z near. Calculation of S gl using the euation reuires that the radius be expressed as a function of bridge height i.e. rz the inverse of the function zr introduced previously. Since analytical inversion of zr is not possible the inverse function is constructed numerically via polynomial interpolation of discrete sampled points. Force may be determined using the following algorithm. Using Es. 4 and 5 the areas are calculated for varying values of bridge height h and fixed contact angle θ. For the fixed value of θ a polynomial curve in terms of independent variable h is fitted to the numerical results. The derivatives needed for evaluating E. 3 are found by analytically differentiating with respect to h the polynomial representations of S gl and S ls. The authors have found that this algorithm yields force predictions that are eual to those found using the Stevin approach [5]. 4 Capillary bridges with electric field 4. Bridge shape Consider the diagram of a liuid bridge profile illlustrated in Fig. 3. The profile is shown for a bridge with an applied voltage solid lines and without dashed lines. For the bridge with applied voltage the profile tangent angle achieves the native contact angle θ at the plates surfaces cross sections C and C. The apparent contact angle θ v is the profile tangent angle near the surface cross sections B and B. The electrostatic pressure is only significant on the liuid-gas interface in a region within region B-C and region B -C in Fig. 3. For purposes of illustration these regions are depicted in the figure as a much larger fraction of C-C than would actually occur. Over the rest of the liuid-gas interface i.e. between B and B only the capillary pressure P c is significant. The balance between the capillary pressure and the surface tension dictates that the profile over region B-B is within the Plateau seuence of shapes the apparent contact angle serving as the boundary condition at B and B for this purpose. Given bridge volume height and apparent contact angle the bridge shape in region B-B and the capillary pressure within may be determined in the same manner as for a bridge without electric field. Since B-C is very small in comparison with A-C the volume in region B- B is essentially the entire volume of the bridge.

7 J. Micro-Nano Mech. 9 5: Force exerted Let us first considers the axial forces acting upon the box shown in Fig. 3 which contains one of the plates and half of the bridge bounded by cross section A at the neck of the bridge neck radius r A r profile tangent angle at A A 9. A balance of the forces in euilibrium reuires that the capillary force at the center of the bridge labeled F A c be eual to the total force exerted on the plate by the actuator F c. Hence the force exerted on the plate is determined by the bridge shape within region B-B and is dependent on bridge volume height and apparent contact angle. To calculate the force the Lippmann-Young E. is used first to determine the apparent contact angle. Then the force may be found using Es. 7 4 and 9. Thus the force with electrical potential may be determined in the same manner as the force without potential; the only difference being that the apparent contact angle is used rather than the native contact angle. 5 Design of actuators 5. Principles Design of an actuator for a particular application reuires the choice of bridge volume nominal bridge height bridge liuid dielectric layer composition and thickness and hydrophobic top-coat if any. These choices affect variables σ gl ε d t d h θ and V that enter into the actuator model developed in the preceding sections and through these variables determine the relationship between force and voltage. Adding to the complexity of design these choices will also influence the breakdown voltage contact angle hysteresis and contact angle saturation which affect force production but are not specifically included in the model presented earlier. Design charts are provided in this section for the actuator configuration shown in Fig.. These charts which encapsulate graphically the relations found in the previous section θ v C B A may be used to ease the selection process of material properties and configuration dimensions. Design employs two charts. The first of these charts relates force to the cosine of the apparent contact angle i.e.cosθ v see Fig. 4a. This relationship is provided by curves for various aspect ratios. For universality the force is presented on the ordinate as a surface energy F c /σ r h where F c and h are in units of μn and μm respectively euivalently N and m andσ r denotes the surface tension of the liuid/gas interface relative to that of water with its vapor ðs w ¼ :78 NmÞ i.e. s r ¼ s gl sw. Note that repulsive forces are shown as positive on the chart while attractive forces are negative. The second chart relates the change in the cosine of the apparent contact angle n ¼ cosð v Þ cosð Þ; to the voltage across the dielectric layer v d see Fig. 4b.This of course is simply a graph of the Lippmann-Young relation. However for ease of design this curve is given in the chart for various values of dielectric thickness parameter T: T ¼ft d in nanometersg s r ð6þ " r In terms of this parameter the Lippmann-Young euation may be written as: n ¼ " 9 s w T v d ð7þ The second design chart also contains curves of constant breakdown field strength parameter B: B ¼fe b in MVcmg " r s r ð8þ where e b is the dielectric breakdown field strength. The breakdown voltage may be expressed as the product of B and T with a correction factor for the mixed units used: v b ¼ t d e b ¼ : TB ð9þ The curves of constant breakdown parameter B in the chart are computed via substitution of Es. 9 into 7 to eliminate T yielding: n ¼ " 8 B s w v d ð3þ θ B C Fig. 3 Capillary bridge profile with solid lines and without dashed lines applied voltage relative height of section B-C is greatly exaggerated for purpose of illustration 5. Design example To illustrate the use of these design charts we consider an example. An actuator with a maximum attracting force of mn and stroke of μm is desired. Therefore the bridge height is chosen to be 7 μm so that the stroke needed is

8 64 J. Micro-Nano Mech. 9 5:57 68 less than /3 of the bridge height. We assume that the maximum force should be achieved over the entire range of actuator stroke from 7 μm bridge height to 5 μm. Itis desired that the device fit within an area of 5mm 5mm; we shall choose the bridge diameter at 7μm height as 3.5mm. The approximate aspect ratio of the bridge may be 7 mm computed as a ¼ 3:5 mm ¼ :. Suppose that the liuid and dielectric are chosen to be water and silicon dioxide respectively i.e. σ r ε r 3.9. Also suppose that the hydrophobic coating on top of the dielectric has a native contact angle θ and is sufficiently thin that it may be ignored in Lippmann-Young analysis. This supposition can only be verified after the initial design is complete. The force value in the first design chart may be computed as F mn ¼ s r h 7 mm 43Nm: where the negative sign indicates attraction. From Fig. 4a the contact angle cosine needed to generate the desired force is cosθ v.5 corresponding to θ v 6. The change in contact angle cosine may be computed as: n ¼ cosð v Þ cosð Þ ¼ :84 We will use a conservative estimate of the breakdown field strength of the silicon dioxide dielectric layer: e b ¼ 5 MV cm. Thus the chart parameter B may be computed as: B ¼fe b in MV cmg " r s r ¼ 5 3:9 ¼ 9:5 From Fig 4b if thickness parameter T is chosen to be less than approximately 4 dielectric breakdown will occur before the reuired change in contact angle is Fig. 4 Design charts for standard configuration: a capillary force in terms of contact angle cosine b change in contact angle cosine as a function of voltage Fc σ h r a. a.3 a.4 a cos θ v n b b v d

9 J. Micro-Nano Mech. 9 5: achieved. Choosing any value of parameter T greater than 4 will dictate the dielectric layer thickness via 6 and the voltage necessary to achieve the desired force. To obtain the lowest value of reuired voltage for this example T4 is chosen which corresponds to a desired dielectric thickness of: ft d in nanometersg ¼T " r ¼ 4 3:9 ¼ 5:6 nm s r From Fig. 4b the voltage across the dielectric layer is v d 7.5 volts. Since the device has dielectric layers on both ends of the bridge the reuired electrical potential for achieving mn is 5 volts. Let us compare this result to electrostatic actuation. Force production in a parallel plate electrostatic actuator is governed by the euation F ¼ " A h v ð3þ where A is the plate area h is the spacing between plates and and ε is permittivity of free space. Using h7μm A 5mm 5mm and v5 volts we find an electrostatic attractive force of 5 μn times smaller than CFA. To obtain a force of mn 665 volts must be applied to the electrostatic actuator. We note that while contact angle saturation will play a limiting role in CFA force production apparent contact angles of 6 have been previously obtained via electrowetting [3]. As Fig. 4a indicates higher levels of F/h are achieved with lower aspect ratio bridges. This has conseuences for CFA operation. Since the bridge has constant volume the aspect ratio will decrease as the plates move closer together. When sufficient voltage is applied and an attractive force is generated the aspect ratio will decrease the operating point will move to a lower aspect ratio curve on the chart and the force generated will be greater than that which would occur if there was no motion at all. This phenomenon will result in a snap-in instabily for CFA. parallel plate and comb configurations see Fig. 5 this perspective dictates that the force follows the formulae F v C ¼ " A h where C is the capacitance of gap between the electrodes A is the effective area h is the gap length and dx is the virtual displacement of one electrode. In this discussion the details of the charging of the capacitance under constant voltage will be omitted. In short a virtual displacement results in a change in the energy stored in the gap. This energy is eual to the sum of the mechanical work resulting from the attractive force and the electrical work done by the charging source when the potential difference is held constant during the displacement. The interested reader is referred to [4] for further insight. The variation in the capacitance with displacement C/ x is due to different mechanisms in the two actuator configurations []. For the parallel plate configuration the capacitance changes with a virtual displacement dx since the plate spacing h is increased. As a conseuence this mechanism reuires that the gap spacing must be larger than stroke. Unless the stroke needed is small μm the force achieved will be uite low as it depends inversely on the suare of the gap as E. 3 indicates. It should be noted that the working stroke is often limited to one third of the plate spacing as pull-in instability prevents greater utilization. With the necessary stroke specified the designer has two avenues to increase the achievable force of the parallel plate configuration: larger voltage and greater plate area. The breakdown field strength of the gap ultimately limits the first of these. But more importantly the voltage level desirable or practical for an integrated circuit device volts is usually much lower than that prescribed by electrical breakdown. Since the force achieved is proportional to voltage suared this implies that the actual energy densities in the capacitive gap euivalent to pressure are far less than the theoretical values provided in the literature 6 Electrostatic perspective 6. Electrostatic actuation The benefits of capillary force actuation may be understood from an electrostatic perspective. This is an alternative to the electrohydrodynamic perspective offered in the previous section. To begin this analysis the operational principles and limitations of electrostatic actuation are briefly reviewed. The force generated by an electrostatic actuator may be analyzed from the viewpoint of virtual work. In both the x a Fig. 5 Electrostatic actuator configurations. a parallel plate and b comb drive. The direction of motion x and the capacitive gap h are marked h h b x

10 66 J. Micro-Nano Mech. 9 5:57 68 which are based on breakdown field. Thus the full force capabilities of electrostatic actuators are rarely realized. For the comb configuration the change in capacitance with displacement is due to a change in the capacitive area A see Fig. 5b. As a result the spacing between comb teeth h may be made much smaller than the actuator stroke boosting force capacity. Indeed the primary advantage of the comb configuration is that it breaks the link between gap dimension and stroke allowing the stroke to be many times the capacitive gap. In practice however manufacturing reuirements typically limit the teeth spacing to several microns. Achievable stroke and force for the comb configuration is also limited by a lateral instability referred to as side snap-over. This instability results from the significant change in capacitance that occurs with lateral motion. The avenues by which a designer might improve comb force capability especially decreasing h all result in a greater tendency toward lateral instability. 6. Capillary force actuation We now turn our attention to the analysis of CFA via the principle of virtual work. For this we will ignore the energy associated with the liuid s surfaces and consider only the capacitive contribution. All terms are considered in the analysis in the subseuent section. The capacitive element in CFA is the solid dielectric layer on the electrode surface. The area of the dielectric element that stores energy and conseuently determines capacitance is the area wetted by the liuid bridge the liuid-solid interface area S ls. The difference in electrical potential occurs chiefly across this dielectric layer since the liuid is chosen to be significantly more conductive than the dielectric. The capacitance is given by: C ¼ " ds ls t d where the factor results from considering dielectric layers at both ends of the liuid bridge. As the volume of liuid is conserved a virtual displacement of one surface away from the other dx results in a change in wetted area ds ls and hence capacitance. Thus the liuid bridge may be viewed from a virtual work perspective as a deformable electrode acting to vary the capacitive area in the circuit. In a similar fashion as an electrostatic actuator the resulting force is given by F v ¼ " d t v d ð33þ For a simple analysis of this assume that the liuid bridge is cylindrical with area S ls and height h and so the volume is given by VS ls h. A virtual displacement of one plate away from the other then yields the volume conservation euation: V ¼ S ls h ¼ ðs ls þ ds ls Þðh þ dxþ which provides the variational relationship for the wetted area of each dielectric layer: ds ls dx ¼ S ls h The force between the plates is F CFA ¼ " ds ls t d h v d or in terms of applied voltage to the device ð34þ F CFA ¼ " ds ls 4 t d h v ð35þ Note that Es. 34 and 35 are approximations since the contributions of the liuid s surface energies are ignored in their development. A comparison of the force provided by the simple CFA model E. 35 and that of electrostatic actuation E. 3 indicates the advantages of capillary force actuation. Assuming the same plate spacing the ratio of forces is: F CFA F PLATE ¼ " d h " Sls A t d ð36þ We may assume that the wetted area S ls will be a significant fraction of the plate area for similarly sized actuators. While electrostatic actuators must have a gap that is several microns the thickness of the dielectric in CFA t d can be reliably manufactured to be much thinner on the order of to nm. As a result h/t d will be greater than for a μm gap and over for a μm gap. Furthermore most dielectrics will have ε d /ε greater than 3 with several having the ratio greater than 5. Therefore the force generated by CFA at a given low voltage will be considerably greater than that produced by a similarly sized electrostatic actuator. This analysis does not consider contact angle saturation or dielectric breakdown. As a result it cannot make a conclusion about the relative effectiveness of CFA when higher voltages are employed. 7 Simplified model The analysis of the previous section provides a means to develop a simplified model for capillary force actuation that avoids the elliptic integrals present in our earlier model. Simplified models permit direct analytical investigation of the relationship between actuation force and design parameters. Here we seek to expand upon the work presented

11 J. Micro-Nano Mech. 9 5: previously by considering both the surface energy and capacitive energy in the virtual work analysis. This is straightforward as we may simply superimpose the results of Es. 3 and 35 to gl F c ¼ s þ s gl cosð þ " d ls ð37þ 4 t Independent of contact angle we may state < ð38þ As a result the first and third terms of E. 37 are always negative i.e. attractive force between the plates. The second term is negative i.e. attractive only when cosθ > i.e. θ <9. While it is not explicit in E. 37 the first and second terms are functions of voltage as S gl and S ls are dependent on bridge shape. For low aspect ratio bridges the dependence of gl / h on bridge shape hence v is much stronger than that of S ls / h. In the case of a low aspect ratio cylindrical bridge a<. with contact angle near 9 i.e. θ v [6 ] one may use a cylindrical bridge approximation: S gl ¼ p rh S ls ¼ p r V ¼ p r h ð39þ ð4þ ð4þ where r is the radius of the bridge. It may be demonstrated ¼ p r ð4þ V ¼ pr r V fixed h ð43þ Since r h >> for low aspect ratio ls > Substitution into E. 37 then yields a useful approximation for the actuation force : ef c ¼ prs gl p r h or ef c ¼ prs gl þ r h cosð vþ s gl cosð Þ p r " d 4 ht d v ð44þ ð45þ In Fig. 6 the ratio of the force estimate E. 44 to that calculated using the actual liuid bridge profile Section 4. d F c ¼ bf c.f c is presented for a fixed volume of V ¼ 7:85 7 mm 3 euivalent to a cylindrical bridge of mm diameter and. mm height with varying aspect ratio. The approximate force model is very accurate for low aspect ratio bridges with errors less than %. 8 Conclusions The electrical manipulation of the capillary pressure within a conducting liuid bridge between dielectric-covered electrodes offers a novel means of generating large actuation forces in MEMS at low voltage. Electrowetting results in a change in apparent contact angle and the alteration of bridge shape and capillary pressure thereby generating force. An analytical comparison to electrostatic actuation indicates that capillary force actuation yield much larger forces at low voltage due to CFA dielectric layers that are much thinner than the air gaps needed by electrostatic actuation; and have higher dielectric permittivity than air. The force produced by capillary force actuation may be analyzed by considering the capillary force at the midpoint of a bridge without electric field where the contact angle is eual to the apparent contact angle dictated by the Young- Lippmann euation. However the force exerted upon the Fig. 6 Ratio of force estimate to actual force as a function of cos θ v and aspect ratio a bridge volume V ¼ 7:85 7 mm 3 a δf c cos θ v

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