Statics and dynamics of an underwater electrostatic curved electrode actuator with rough surfaces

Here, we present a model, design, static and dynamic testing, and analysis of an electrostatic curved electrode actuator in deionized water. The actuator is integrated within a microfluidic device designed for high throughput cell sorting. The actuator shifts the bifurcation point of a Y-shaped microfluidic channel to simultaneously increase the width of one channel while decreasing the width of another channel, thus changing the bias in hydrodynamic resistance between outlet channels. The actuator is modeled as a clamped-roller beam and the static displacement is calculated based on Rayleigh–Ritz energy methods. The model accounts for oxide growth and surface roughness that occurs during fabrication. We observe that modeling a rough contact surface improves the maximum displacement prediction to within less than 20% error from the experimental value. Additionally, the model predicts a release voltage within less than 8% error of the experimental value. We also present dynamic experiments to test the actuator displacement at frequencies from 1 to 4096 Hz and show that the actuator achieves large displacements ( > 8 µ m) at high frequencies ( > 100 Hz).


Introduction
Underwater microscale actuators and sensors are critical for active cell processing [1], drug delivery [2], and pathogen Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.detection [3,4], among others.However, actuation, in particular, is limited by the high viscosity of the immersing media, water, leading to Q-factors less than 1, or surface-tension effects at air/water interfaces [5,6].Of the set of known actuation modalities, electrostatic actuators are a compelling choice because of the larger achievable displacements and bandwidth frequencies available, and the ability to use low applied voltages (<10 V) [7].However, parallel-plate electrostatic actuators are limited by the well-known pull-in instability, where at a voltage greater than the pull-in voltage, VPI , there is an effective spring-softening nonlinearity that results in unstable pull-in of the actuator and the loss of the ability to modulate actuator displacement by voltage control [8].Due to the nonlinearity, the actuator can then be released by lowering the voltage below the release voltage, VR < VPI .This voltage to actuation relationship is demonstrated for a curved electrode actuator in figure 1, where the hysteresis loop is the area inside the curve of this cycle.Idealized electrostatic models assuming smooth, flat plate geometries predict that VR = 0, which does not agree with experimental observation.
Our group [7,9] and others [10][11][12] have investigated curved electrode actuators (figure 2), because of their ability to achieve large displacements (>4µm), high bandwidth frequencies (>1000 Hz), and stable displacements that are manipulable above the pull-in instability.In a curved electrode actuator, the applied voltage is purposefully set above VPI , and increasing the voltage beyond VPI results in the contact point a 2 'zipping' along the face of the curved electrode.This zipping action shortens the electrode sprung length, resulting in a spring hardening nonlinearity that counteracts the electrostatic spring softening nonlinearity, leading to a stable, thus tunable, displacement to voltage relationship [7,9,13].However, as with parallel plate electrostatic actuators, the smooth, flat plate assumption is incorrect and leads to poor predictions of the displacement profile and VR .Here, we have significantly expanded on our prior published modeling work [9] to incorporate surface roughness effects and to predict the displacement output in response to a decreasing voltage input.Hence, the model improves the predictions of VR .
Curved microbeams undergo a mechanical instability phenomenon to produce large displacements, while our work uses an electrostatic instability [14,15].Researchers have developed static and dynamic physics-based models to analyze bifurcation points that prevent high-speed operation [16,17].The mechanics of such beams in different geometries [18,19] is well-understood.The coupled physics of snap-through and pull-in in actuators to achieve O(10 −6 ) m displacement have been extensively studied and widely reported [14,[20][21][22][23].
A force-based model [24] was employed and simplified by utilizing the Galerkin and mode superposition methods to yield a set of coupled ordinary differential equations (ODEs).Here, for the curved electrode actuator, we use Raleigh-Ritz energy methods to calculate static displacement as we showed in Preetham et al [9].However, in this paper we demonstrate solution methods that do not require Taylor series approximations.
Our target application is for cell sorting using the electrostatic curved electrode actuator integrated within a microfluidic device.In particular, curved electrode actuators may be useful in cell sorting applications in which the encapsulation of cells in droplets for dielectrophoretic separation, as is done in flow cytometry [25], is not permissible.MEMS-based actuators have been successfully used for cell sorting [26][27][28] based on changing the bias in hydrodynamic resistance at a microfluidic bifurcation.In our previous work [29], we found a bias of over 7 µm is needed to sort particles at a bifurcation with over 90% enrichment, corresponding to a channel width ratio of w 2 /w 1 = 1.23.Additionally, high throughput sorting (>100 Hz) is a target metric.
This paper investigates the role of roughness from the microfabrication process and native oxide growth on a quasistatic model of the voltage to displacement relationship.We show that incorporating roughness into the model enables us to predict maximum actuator displacement within 20% of the experimental value and VPI and VR within 8% of the experimental values.Furthermore, we show that curved beam actuators achieve the displacements and bandwidths required for cell sorting applications in an underwater environment.Section 2 presents the design and modeling of the actuator and microfluidic system, section 3 shows the methods used to experimentally characterize actuator performance in DI water, and section 4 presents the modeling and experimental results.Finally, section 5 summarizes conclusions and future work.

Design
The actuator is integrated within a fusion bonded glass-silicon microfluidic chip with all channel dimensions between 2.5-200 µm, so the microfluidic networks can manipulate individual biological cells (figure 3(a)).The device has a single microfluidic inlet, a filter region to prevent debris from clogging critical channels, a sheath flow region to center particles [30], a bifurcation point shaped by the actuator architecture at which a particle will be driven to one of two particle outlet channels (Channel 1 or Channel 2) by the intra-channel pressure gradient, and one water outlet channel (Channel 3).The inlet channel, bifurcation point, and outlet channels are fabricated in a single plane.Fluids enter and leave this plane perpendicularly via standard microfluidic connectors interfaced with a custom jig.
The actuator is designed to actively modulate the hydrodynamic resistance bias in the microfluidic channels at the Y-shaped bifurcation by acting as a switch to move the bifurcation vertex to simultaneously increase the width of channel 1 and decrease the width of channel 2 (figures 3(b) and (c)), where w 1 = 30 µm and w 2 = 38 µm so that w 2 /w 1 = 1.27 in the OFF state.Thus, to sufficiently bias the hydrodynamic resistance for high throughput cell sorting such that the bias is flipped (w 1 /w 2 ⩾ 1.27), the target actuator displacement is over 8 µm at high frequencies over 100 Hz.Design parameters of the actuator used for modeling are included in table 1.Unlike the actuator reported by Preetham et al [9], we rely on natural oxide growth to insulate the curved and beam electrodes [31].

Actuator model
The actuator static modeling problem is to find the beam electrode displacement profile, w(x), as a function of the independent variable static voltage, V.The model presented here most closely resembles work presented in Preetham et al [9]; however, differences include that we solve the equations without using Taylor series expansions, incorporate surface roughness, and calculate the release voltage, VR , and corresponding reverse voltage displacement profile.The geometry of the problem is given in figure 2, where the actuator is modeled as a clamped-roller beam.For our model, we assume that the fabrication process results in a rough silicon surface that traps water between the two electrode faces in addition to growth of a thin oxide layer to prevent electrical shorting upon contact (figure 2(b)).Here, we solve for w(x) using the functional form of the partial differential equation that defines a beam, with appropriate boundary conditions, and that is subjected to an electrostatic force and a point force from the spring force transmitted through the reinforcing beam.We then use the Rayleigh-Ritz method to minimize the energy of the system for a trial function, with unknown parameters, that also satisfies the boundary conditions of the problem.Specifically, the deflection profile of the curved beam actuator, w(x), as a function of position x is the solution that minimizes the total energy, Π, in the system where U b is the bending potential energy and V el is the electrostatic energy.The bending potential energy is expressed as where L is the actuator length, E is the elastic modulus, I = 1 12 ht 3 is the bending moment of inertia, h is the beam height, t is the beam electrode thickness, and k r is the reinforcing beam stiffness.The electrostatic potential energy is expressed as where ϵ 0 is the permittivity of free space, V is the applied voltage, d 1 is the insulator thickness, d 2 is the electrode gap distance, d 3 is the contact roughness, ϵ 1 is the relative permittivity of the oxide growth layer, ϵ 2 is the relative permittivity of the deionized (DI) water, and 2 is the function that defines the electrode curve, where δ m is the maximal deflection.Equations ( 2) and (3) can be rewritten in terms of the non-dimensional distance variables

and
Here, we include the roughness variable, d 3 , which accounts for contact between 2 rough surfaces, in the model to predict the stable actuator displacement and account for a non-zero release voltage.Surface roughness, r RMS is quantified as the measured surface profile differences between a rough versus smooth surface relative to a mean line or plane [33][34][35] (figure 2(b)).To account for peaks and valleys, roughness r RMS , is quantified as the root-mean-square (RMS) value [34]: where Z(h) is the surface profile adjusted to be centered at the mean line and L R is the length of the analyzed segment (figure 2(b)).Due to fabrication processes, the electrodes may not be fabricated with a uniform surface roughness along the entire electrode height.Here, we calculated the total contact roughness, d 3 , using a different roughness metric for the top and bottom halves of the electrode height, r RMS and g, respectively.We observe a change in side wall profile as a function of depth, from a depth dependent performance of the DRIE process, which produces an extra gap distance, g, between surfaces that prevents contact (figure 2(b)).Contact roughness, d 3 , is an average of the total surface roughness for the top and bottom parts of the electrodes: = r RMS + g.Our analysis assumes that both sides of the contacting electrodes have the same roughness profile.

Definition of trial functions
The admissible trial function for the solution to minimize equation (1) must satisfy the boundary conditions for the given problem.We consider two configurations: (1) unzipped deflection, where V < VPI for rising voltage input or V < VR for falling voltage input and (2) zipped deflection, where V > VPI for rising voltage input or V > VR for falling voltage input.Details of the derivation are provided in the supplementary information.

Unzipped case.
We assume that the beam deflection is defined by Euler-Bernoulli bending of a beam subjected to both a uniform load, c 1 , and a point load representing the spring force k r w(L) applied at x = L.There is an imposed clamped boundary condition at x = 0 and roller boundary condition at x = L. Euler-Bernoulli bending assumes constant material properties and geometry along the length of the beam and small angle deflection.Deflection is the solution to the differential equation: with boundary conditions w(0) = 0, dw dx (0) = dw dx (L) = 0, and shear V(0) = c 1 L − k r w(L) and where δ(x) is the Dirac delta function.Applying the dimensionless variables from section 2.2, the differential equation becomes where EI , and the boundary conditions become W(0) = 0, dW dX (0) = dW dX (1) = 0, and shear (1).By application of the appropriate boundary conditions and solving for W(1), equation ( 7) becomes the trial function with unknown parameter C 1 : where a =  the beam problem is defined by a distributed load on the beam with a point force at the beam end, clamped boundary condition at X = A 2 , and roller boundary condition at X = 1.The equation to solve is: with boundary conditions Combined, the trial functions for the zipped configuration with unknown parameters C 1 , C 2 , A 1 , and A 2 are: where γ 1 , γ 2 , and γ 3 are coefficients defined in the supplementary information (equation (S13)).The solution to minimize Π in equation ( 1), has voltage bounds that define the unzipped ( V < VPI or V < VR ) and zipped ( V > VPI or V > VR ) configurations for rising and falling voltage inputs.We find VPI at which the configuration switches from unzipped to zipped by solving ∂Π ∂C1 = 0 and ∂ 2 Π ∂C 2 1 = 0 for unknown constants C 1 and V using the unzipped trial function (equation ( 8)).The function minimization becomes: min .
Next, we solve for VR at which the configuration switches from zipped to unzipped by solving for unknown variables = 0 using the zipped trial function (equation (10)).This problem is transformed to the function minimization problem: .

Solution algorithm.
There is no closed-form solution to compute VPI , VR , or the unknowns of the trial functions.Our solution leverages function optimization algorithms to solve for unknowns, using the Matlab function fmincon.Each scalar value, , is solved using the symbolic package in Matlab to compute partial derivatives and then numerically integrated to solve the integral.In going from the unzipped configuration ( V < VPI ) to the zipped configuration ( V > VPI ), we plug in an increasing voltage increment.In going reverse from the zipped configuration ( V > VR ) to the unzipped configuration ( V < VR ), we plug in a decreasing voltage.Details of the partial derivatives relevant to the solution are provided in the supplementary information.

Silicon patterning.
The actuator chips are fabricated from 100 mm silicon-on-glass (SOG) wafers following the fabrication steps shown in figure 4. The SOG wafers are constructed from anodic bonding p-type, < 100 >, low resistivity (0.01-0.02Ω-cm) Si to a 500 µm thick Borofloat 33 glass wafer followed by thinning and polishing the Si side to 45 µm thick (PlanOptik, Elsoff, Germany) (figure 4(a)).After cleaning wafers in Piranha solution (3:1 90% H 2 SO 4 :30% H 2 O 2 ), a 900 nm layer of SiO 2 is deposited using plasma enhanced chemical vapor deposition (PECVD) at 250 • C using N 2 O and SiH 4 He gases to serve as a Si etch mask (figure 4(b)).Next, Shipley 1813 positive photoresist (MicroChem) is patterned as an etch mask for the SiO 2 by spin-coating the photoresist onto the wafer, exposing the wafer to light through a patterned reticle (Abeam, Hayward, CA, USA) using projection lithography, and developing the pattern in MF-319 developing solution (MicroChem).A subsequent descum step removes residual photoresist on the wafer surface using O 2 .After lithography, the SiO 2 etch mask is etched using inductively coupled plasma-reactive ion etching with CHF 3 and Ar gases to pattern the SiO 2 , followed by photoresist removal using acetone (figure 4(c)).Next, the wafer is diced into individual chips.The chips are loaded into a deep reactive ion etcher (DRIE) to etch the Si channels down to the glass, which acts as an etch stop, using standard Bosch protocols with 7 s:13 s step cycles of C 4 F 8 :SF 6 (figure 4(d)).Prior to starting the etch, the Si chips sit on the loading platform within the DRIE chamber with a surface temperature of 0 • C to allow the samples to sufficiently cool to avoid overheating.Cleaning steps are implemented next to clean up residue on the chips as the following: an acetone and methanol soak with N 2 dry, O 2 plasma cleaning for 10 min in a plasma asher, Piranha clean (3:1 90% H 2 SO 4 :30% H 2 O 2 ), and Aqua regia clean (3:1 HCl:HNO 3 ).After etching, the thin members that compose the actuator are released from the glass substrate using a timed etch in 1:1 H 2 O: 49% HF v/v solution (figure 4(e)).Note that the methods for actuator fabrication are similar to those reported elsewhere [6,9], except we do not use a Au/Cr metal layer for the electrodes because the metal layers were found to be unnecessary [36].

Glass cap patterning.
A glass cap that is patterned to have a 5 µm recess and drilled holes for inlets, outlets, and electrical contact holes is aligned with and bonded to the Si using fusion bonding.The glass cap patterning process follows protocols similar to Lomasney et al [37].Glass microscope slides (76.2 mm × 25.4 mm × 1.1 mm) coated with 250 nm of chrome are purchased (Deposition Research Lab, St. Charles, MO, USA).Next, lithography is performed following the same protocols as the MEMS fabrication except using contact lithography with a transparency mask (FineLine Imaging, Colorado Springs, CO, USA) instead of projection lithography.The exposed chrome is etched using Nichrome etchant (Transene Company, Danvers, MA, USA).Next, a mixture of 5:1:3 H 2 O:70-75% HNO 3 :49% HF etches the glass slide to 5 µm deep features followed by a water rinse.The glass is then diced into individual chips on a dicing saw.Next, fluidic and electrical access holes are drilled using 0.8 mm and 1.8 mm diamond drill bits (His Glassworks, Asheville, NC, USA), glass chips are cleaned in acetone to remove the photoresist, and the chrome is etched off using Nichrome etchant.Next, the SOG chips and the glass cap are cleaned in a Piranha solution, a water bath, an RCA-1 solution (6:1:1 H 2 O:NH 4 OH:H 2 O 2 ), and a final water bath.SOG chips are carefully extracted from the solutions to avoid chipping of the edges.Finally, glass caps are manually aligned with the SOG chip under a stereomicroscope (Nikon SMZ800, Nikon, Tokyo, Japan); assembled chips are placed under 400 g weights consisting of a Macor ceramic plate weighted down by stainless steel plates.The chips are then fusion bonded in a furnace (Lindberg/Blue M Model BF51634PC-1, ThermoFisher Scientific, Waltham, MA, USA) at 615 • C for 8-10 h.The bonded chips (figure 4(f)) are removed after the furnace cools to 50 • C, which takes approximately 8 additional hours.A representative chip is shown in figure 4(g).

Surface roughness quantification.
To quantify surface roughness, a chip that has not yet been fusion bonded is prepared for imaging by coating the SOG chip in a 10 nm layer of Au/Pd using a Technics Hummer VI Sputtering System.Then, the actuator region is imaged using a Zeiss Ultra Plus scanning electron microscope (SEM) with 3 kV accelerating voltage (figure 5(a)).The sputter-coated actuator chip is diced near one of the outlet ports using a scribe tool and a crosssection is imaged (figure 5(b)).Using the cross-section SEM image, we apply the Matlab image processing toolbox Canny filter for edge detection to extract Z(h) and L R (figure 2(b)) to calculate r RMS = 259 nm (equation ( 5)).Additionally, the parameter g was measured as approximately 367 nm based on a step change along the height of the etched silicon, where the bottom part of the silicon was etched deeper into the wall than the top.

Experimental setup
3.2.1.Fluid flow.Following the methods of Tkachenko et al [38], we designed a jig for the chip-to-world fluid and electrical connections (figure S1).Fluid is flowed into the device using a syringe pump.First, devices are primed with methanol at 1 ml hr −1 , then the pump drives DI water mixed with 0.01% Pluronic F-127 at 25 µl hr −1 into the inlet for the rest of the experiment.

Characterization of actuator statics and dynamics.
The actuators are tested underwater to determine feasibility for the eventual realization as part of an active cell sorting device.The flow is stabilized for 10 min at 25 µl hr −1 and  then a voltage, V A , is applied to the electrodes to move the actuator using a custom LabVIEW code and function generator to apply the amplitude modulation signal (figure 6).The actuator displacement is measured at a region-of-interest with a 63X objective using a custom MATLAB image processing code.The image processing code provides a displacement trace in time to extract w(L, t) using image correlation [39].We supply a high frequency voltage signal to the electrodes to flip the electrical polarity and thus the water molecule orientation to mitigate charge shielding in DI water [1,6,9].The voltage signal is:  We perform quasi-static testing at a frequency of f = 10 −1 Hz and dynamic testing at higher frequencies f = 2 0 , 2 0.5 , . . ., 2 11.5 , 2 12 Hz.The recording frame rate is increased from 40 fps for low actuation frequencies up to 25 000 fps for high actuation frequencies to prevent aliasing by collecting data at a rate above the Nyquist rate (table S1).

Static model and quasi-static experiment results
The actuator achieved measurable displacements (figure 7), up to 7.4 µm (figure 8).In the channel design, that corresponds to a bias of w 1 /w 2 = 1.22, which is less than our target of w 1 /w 2 ⩾ 1.27 defined in section 2.1.The actuator tip displacement, w(L), increases with increasing voltage amplitude, V, starting gradually until reaching VPI .At VPI , the actuator jumps in displacement into the zipped configuration.Upon decreasing V gradually, the actuator tip deflection w(L) jumps down in displacement at VR (figure 8).We calculated the model error in predicting the actuator displacement for a rough surface (d 3 = 626 nm) and the model error for a smooth surface (d 3 = 0 nm) (table 2).The static model results for d 3 = 626 nm compared to the experimental results are within 20% error of w max (L), within 5% of VPI , and within 8% of VR .The static model results for d 3 = 0 nm compared to the experimental results are over 36% error from w max (L), over 15% from VPI , and 6.28 × 10 4 % from VR .The visual overlay of the smooth and rough modeling results are shown in figure S2.For the d 3 = 626 nm model, the modeling parameters are plotted to show in the unzipped configuration, the parameter C 1 is the same for rising and falling voltage inputs (figure

S3
).While in the zipped configuration, there are slight variations in the parameters for rising and falling voltage inputs (figure S4).

Dynamic experiment results
High frame rate recording enables tracking of the actuator tip at high frequencies (figure 7(c), videos S1-S3).The actuator achieves the design objectives of >8 µm displacement (can flip the bias to w 1 /w 2 > 1.27) at frequencies of >100 Hz for  V ⩽ 10 V. Up until the bandwidth frequency, the actuator fully realizes the steady-state displacement, creating a clipped displacement profile as the actuator is unable to further deflect past the maximum due to the stiffening spring (figure 9(a)).Additionally, with increasing V, the actuator demonstrates increasing bandwidth frequency due to the beam electrode acting as a stiffening spring [36].For example, for V = 8 V, the bandwidth frequency is f BW = 229 Hz and for V = 10 V, the bandwidth frequency is f BW = 574 Hz (figure 9(b)).Overall, the actuator achieves large displacement, high frequency actuation, demonstrating feasibility to be implemented within a microfluidic cell sorting device.

Conclusion
This paper demonstrates a static model of electrostatic curved electrode actuator physics that incorporates the surface roughness from fabrication processes.We also demonstrated quasistatic and dynamic experiments to characterize actuator performance.By incorporating roughness, the model results demonstrate that microfabrication surface irregularities are a benefit to enable a small hysteresis loop.From the model, VPI matches closely within 5% of the quasi-static experiments, while the release voltage is within 8% of the experimental value (table 2).In particular, by comparing the model results for the measured surface roughness to the results for a smooth surface, the prediction of VR is considerably improved.Furthermore, we demonstrated an actuator to achieve over 8 µm displacement to flip the channel bias to the target w 1 /w 2 ⩾ 1.27 at high frequencies (>100 Hz).For the quasi-static case, the actuator displacement was just under the target, flipping the channel bias to w 1 /w 2 = 1.22.However, the channel width design could be adjusted so the actuator would meet the target even at very low frequencies.Future work would adapt this system for a cell sorting application by incorporating realtime fluorescence detection.Additionally, here the actuator was tested in DI water for large displacement underwater actuation but for implementation as a cell sorter, more channels would be added to immerse cells in cell media for viability, while immersing the actuator in DI water.As the actuator operates in a fully closed microfluidic system to bias fluid flow within a microchannel, the actuator has the potential to be used in closed, liquid-filled environments compatible with biological or chemical samples.

Figure 1 .
Figure 1.Representative electrostatic curved electrode actuator displacement and hysteresis changes depending on surface roughness between contacting electrodes.The direction of the arrows correspond to a rising or a falling voltage input.

Figure 2 .
Figure 2. Schematic of curved-electrode sorter actuator.(a) Actuator in zipped configuration.(b) Inset of the cross-section shows an illustration of the surface roughness between the beam electrode and curved electrode surfaces with space such that water may not fully evacuate even when actuator enters the zipped configuration.

Figure 3 .
Figure 3. Schematics of (a) the overall actuator chip and inset (not drawn to scale) showing the actuator motion.Particle sorting is shown as the targeted design application with the (b) actuator in the OFF state and particles entering the wider channel, channel 2 and the (c) actuator in the ON state and particles entering the wider channel, channel 1.
details are included in the supplementary information.

2. 3 . 2 .
Zipped case.The admissible trial function for zipped deflection is divided into three different domains, defined based on the end positions and the beam contact points at a 1 and a 2 (figure2(a)).The domains have the boundaries: 0-A 1 , A 1 -A 2 , and A 1 -1, where A 1 = a1 L and A 2 = a2 L .This section provides the appropriate boundary conditions for each domain and the resultant admissible trial function.(i) Domain 0-A 1 .This domain defines the deflection between the beam base and first contact with the curved beam.The beam problem is defined by a distributed load with the clamped-clamped boundary conditions W(0) = 0, dW dX (0) = 0, W(A 1 ) = A 2 1 + D 2 , and dW dX (A 1 ) = 2A 1 .(ii) Domain A 1 -A 2 .Deflection in this domain is simply the shape of the curved beam: W(X) = X 2 + D 2 .(iii) Domain A 2 -1.Similar to the unzipped deflection profile,

Figure 4 .
Figure 4. Silicon device fabrication overview using glass as an etch stop.(a) Anodically bond Si to glass wafer.(b) Deposit SiO 2 using PECVD.(c) Pattern and etch SiO 2 through to Si.(d) DRIE exposed Si through to glass.(e) Wet etch using HF to release actuator.(f) Bond glass cap to Si. (g) Reflected light microscope image of the fusion bonded device showing recessed area around actuator which did not bond glass to Si, as desired, while the outside of the recessed area was successfully bonded.

Figure
Figure SEM images of (a) actuator and curved electrode and (b) rough side wall of Si-glass microfluidic channel used for roughness characterization in the indicated region of interest (ROI).

Figure 6 .
Figure 6.Experimental setup for quasi-static and dynamic testing.(a) Schematic of experimental setup.(b) Voltage input signal with a fundamental wave that is sinusoidal and the (c) corresponding actuator displacement profile.
kHz) (figure 6(b)).The sinusoidal AM input enables us to measure the output displacement for inputs at different amplitudes up to V within the same test (figure 6(c)).

Figure 7 .
Figure 7. Transmitted light microscope images of actuator in (a) OFF state and (b) ON state.(c) Actuator tip magnified 63X and moving at 1024 Hz to show the region of interest that was recorded at high frame rates.

Figure 8 .
Figure 8. Actuator model and quasi-static experimental displacement.(a) Beam deflection profile calculated from the model at select voltages.(b) Quasi-static displacement model and experimental results capturing hysteresis in tip deflection in response to a rising or falling voltage input.Error bars represent 1 standard deviation above and below for nine periods of displacement data from one device.

Figure 9 .
Figure 9. Actuator dynamic response to a sinusoidal AM signal input.(a) Time trace of actuator displacement with a normalized time axis for V = 10 V at select frequencies.(b) Actuator displacement over a range of frequencies at select V levels.The shaded region shows where the target displacement is achieved.

Table 1 .
[32]ator nominal and measured parameters.Note, insulator thickness is selected based on Goodman and Breece[31]and insulator permittivity is selected based on Robertson[32].Measured values are from the tested device.

Table 2 .
Comparison of model and experiment results.Percent error is calculated between the model and experiment for the d 3 = 0 nm and the d 3 = 626 nm cases.