pubs.acs.org/Langmuir © 2010 American Chemical Society
An Empirically Validated Analytical Model of Droplet Dynamics in Electrowetting on Dielectric Devices M. J. Schertzer, S. I. Gubarenko, R. Ben-Mrad, and P. E. Sullivan* Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8 Received September 15, 2010. Revised Manuscript Received October 21, 2010 Explicit analytical models that describe the capillary force on confined droplets actuated in electrowetting on dielectric devices and the reduction in that force by contact angle hysteresis as a function of the three-dimensional shape of the droplet interface are presented. These models are used to develop an analytical model for the transient position and velocity of the droplet. An order of magnitude analysis showed that droplet motion could be modeled using the driving capillary force opposed by contact angle hysteresis, wall shear, and contact line friction. Droplet dynamics were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The first four parameters describe the device geometry and fluid properties; the remaining parameters were determined experimentally. Images of the droplet during motion were used to determine the evolution of the shape, position, and velocity of the droplet with time. Comparisons between the measured and predicted results show that the proposed model provides good accuracy over a range of practical voltages and droplet aspect ratios.
Introduction The popularity of droplet based microfluidic devices as a means to improve throughput and reduce operating costs of biological protocols is growing.1-4 Several mechanisms have been introduced that manipulate droplets by chemical,1 thermal,1 acoustic,2 and electrical3 means. One promising platform for these devices uses electrowetting on dielectric (EWOD) to move droplets by applying asymmetric electric fields.3-6 Electrowetting on dielectric devices can extract, move, split, and mix droplets of fluid without using external pumps. EWOD devices have low power consumption, high reversibility, wide applicability to different fluids, and lower viscous losses than continuous flow devices.5-8 A comprehensive review of EWOD devices can be found in ref 7. Despite the demonstration of their practical capabilities, the translation of droplets through these devices and the flow patterns within the droplets are complex and not yet fully understood.9 It has been shown that both energy and force based analyses can be used to model the dynamic motion of droplets in EWOD devices. Analytical models of droplet dynamics give physical insight into the problem but do not generally provide transient information; conversely, numerical models provide more detail at high computational expense. The energy of a droplet in an EWOD device is the sum of interfacial energies between the liquid and the substrate, the liquid and the surrounding medium, and the substrate and the surrounding medium. (1) Brochard, F. Langmuir 1989, 5, 432–438. (2) Wixforth, A.; Strobl, C.; Gauer, C.; Toegl, A.; Scriba, J.; Guttenberg, Z. Anal Bioanal Chem. 2004, 379, 289–991. (3) Washizu, M. IEEE Trans. Ind. Appl. 1998, 34, 732–737. (4) Cho, S. K.; Moon, H.; Kim, C. J. J. Microelectromech. Syst. 2003, 12, 70–79. (5) Ren, H.; Fair, R.; Pollack, M.; Shaughnessy, E. Sens. Actuators, B 2002, 87, 201–206. (6) Chatterjee, D.; Hetayothin, B.; Wheeler, A. R.; King, D.; Garrell, R. L. Lab Chip 2006, 6, 199–206. (7) Berthier, J. Microdrops and Digital Microfluidics; William Andrew Publishing: Norwich, NY, 2008. (8) Walker, S. W.; Shapiro, B. J. Microelectromech. Syst. 2006, 15, 986–1000. (9) Mugele, F. Soft Matter 2009, 5, 3377–3384.
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Applying a voltage reduces the energy in the surface above the active electrode. If the voltage is applied asymmetrically, the total system energy is reduced as the droplet moves to cover the active electrode. The force on the droplet can be found by differentiating the system energy with respect to position.10-13 This method predicts that the force on the droplet is independent of confinement distance and positive for the entire motion if the dielectric layer covering the active electrode is thicker than that covering the ground.11 In an EWOD device, the application of an asymmetric electric field on a droplet creates an asymmetric deformation in the droplet interface. This creates a nonuniform surface tension force around the periphery of the droplet which drives it toward the active electrode.4,5,8,14-20 Since the change in the interface shape is over 100 times faster than the motion of the bulk fluid,8 transient effects that occur during the deformation of the interface can be neglected. This approach has been used to create analytical models for the average velocity of the droplet,1,5 numerical models for droplet motion and splitting,8,18-20 and a criteria for splitting droplets in EWOD devices.4 Although capillary force drives the droplets in these investigations, the transient shape of the droplet interface is rarely observed. Instead, the deformation of the interface is correlated to the applied voltage, and the assumed shape is used to model the capillary force.4,5,8,14,16,19 (10) Jones, T. B. Langmuir 2002, 18, 4437–4443. (11) Bahadur, V.; Garimella, S. V. J. Micromech. Microeng. 2006, 16, 1494– 1503. (12) Baird, E.; Young, P.; Mohseni, K. Microfluid. Nanofluid. 2007, 3, 635–644. (13) Kumari, N.; Bahadur, V.; Garimella, S. V. J. Micromech. Microeng. 2008, 18, 085018. (14) Kang, K. H. Langmuir 2002, 18, 10318–10322. (15) Bavier, R.; Boutet, J.; Fouillet, Y. Microfluid. Nanofluid. 2008, 4, 287–294. (16) Pollack, M. Ph.D. Thesis, Duke University, 1999. (17) Lee, J.; Moon, H.; Fowler, J.; Schoellhammer, T.; Kim, C. J. Sens. Actuators, A 2002, 95, 259–268. (18) Lu, H. W.; Glasner, K.; Bertozzi, A. L.; Kim, C. J. J. Fluid Mech. 2007, 590, 411–435. (19) Walker, S. W.; Shapiro, B.; Nochetto, R. H. Phys. Fluids 2009, 21, 102103. (20) Kuo, J. S.; Spicar-Mihalic, P.; Rodriguez, I.; Chiu, D. T. Langmuir 2003, 19, 250–255.
Published on Web 11/16/2010
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Figure 1. Sketch of the experimental facility used in this investigation with enlarged view of the EWOD device (not to scale).
It has been shown that using an assumed interface shape for a confined droplet can lead to significant errors in the equilibrium case.21 One issue with modeling droplet motion in EWOD devices is that the motion of the droplet increases the apparent contact angles on the leading edge of the droplet and reduces them on the trailing edge.8 This effect is referred to as contact angle hysteresis and a qualitative model can be found in ref 7. Quantitative models for this effect have been incorporated in numerical investigations by defining the hysteresis force as a fraction of the driving pressure defined by a hysteresis coefficient.8,19 Due to the complexity of incorporating dynamic contact angle information into these models, constant values that approximate the average effect of the capillary forces are often employed.18 This paper presents an analytical model for droplet dynamics in EWOD devices. The model predicts the transient position and velocity of the droplet based on the surface tension force found through observation of the droplet interface. This is in accordance with the electro-quasistatic assumption that the interface shape is dependent on applied voltage.22 Explicit analytical models for the driving capillary force and the effect of contact angle hysteresis are developed as functions of the three-dimensional curvature of the droplet. These models give rise to a semiempirical model for droplet dynamics which is experimentally validated for effects of applied voltage and aspect ratio. The proposed model provides transient position and velocity information at little computational expense, and the form of the equations provides insight into droplet motion in EWOD devices.
Experimental Section Measurements of the transient position and contact angle were made in the facility shown in Figure 1. The facility consisted of an EWOD device, the electrical control system, and the optical measurement system. The EWOD device consisted of two silica glass slides. The upper and lower slides were approximately 25 75 mm2 and 75 75 mm2, respectively. Both slides were cleaned using an organic solvent before fabrication. The lower slide was coated with a conductive layer of chromium and gold that was patterned into an array of square electrodes separated by 60 μm. Electrode sizes of 1.0, 1.5, and 2.0 mm were used in this investigation. The lower slide was then coated with a dielectric Parylene layer and a hydrophobic Teflon layer. The upper slide had an ITO layer which was coated with Teflon. The thicknesses of the chromium, gold, ITO, Parylene, and Teflon layers were approximately 10 nm, 100 nm, 150 nm, 2 μm, and 50 nm, respectively. The Parylene and Teflon layers above the electrical bond pads were manually scratched away to provide electrical contact. The gap between the substrates was achieved using eight pieces of double sided 3M Scotch tape. This gap was measured optically (21) Schertzer, M. J.; Gubarenko, S. I.; Ben Mrad, R.; Sullivan, P. E. Exp. Fluids 2009, 48, 851–862. (22) Haus, H. A.; Melcher, J. R. Electromagnetic Fields and Energy; Prentice-Hall: Upper Saddle River, NJ, 1989.
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in each case, and the nominal distance was 715 μm. Droplets of deionized water were deposited on the bottom slide using a VWR VE2 Signature pipettor before adding the upper slide. The uncertainty in the pipet measurements is reported by the manufacturer as 1%. In all cases, the droplet diameter was equal to the diagonal of the electrode (1.2-4.6 μL). Selective application of the electric field was achieved with a control system consisting of a National Instruments PXI 8195 controller, a PXI 2529 matrix-switching device, an Agilent 33120A signal generator, and a custom amplifier similar to that used in ref 23. Output channels were connected to bond pads for each addressable location on the EWOD device using custom fixtures (Figure 1). Electrical connections were automated using Labview Real Time 8.2. The frequency was fixed at 10 kHz, and the voltage varied between 90 and 120 Vrms. The applied voltage is not expected to affect the temperature of the droplet, as voltages were generally applied for less than 300 ms and the lack of current means that there is no resistive heating in the system. Droplet motion was not repeatable for voltages below 90 Vrms, and dielectric breakdown would occur frequently at voltages greater than 120 Vrms. The optical measurement system consisted of a Leica MZ16F fluorescence stereomicroscope and a Canadian Photonics Laboratory MS5K black and white camera with a CCD array of 1280 1020 pixels. The optical measurement system was mounted on an X-Y stage to focus the image and adjust the horizontal field of view. The EWOD device was mounted on a Delrin block connected to a Melles Griot manual jack that was used to adjust the vertical field of view. The device and the microscope were aligned using custom fixtures. All images were backlit using a Schott DCR III light source. Bitmap images of the droplet in transit were analyzed in MatLab. Calibration was performed using an in-plane calibration target. The resolution for the position measurements was between 3.4 and 3.6 μm/pixel. Since images were recorded at 200 fps, the resolution of the velocity data is on the order of 1.4 mm/s, approximately an order of magnitude below the average droplet velocities seen here. A typical droplet image is shown in Figure 2. The initial grayscale image was vertically divided into 10 equal sections. The grayscale threshold in each section was found using the Otsu method,24 and the initial image was converted into 10 binary images. The curves from the region of interest in each binary image were used to reconstruct the shape of the interface (Figure 2). After the droplet interface was found, the center of the droplet was assumed to be the midpoint between the average positions of the two interfaces (Figure 2). The contact angles of the moving droplet were found by solving the differentiation of a second degree polynomial fit of the interface at the contact points. This method is similar to that used in ref 21. Velocity data were found by differentiating the position of the droplet with time.
Analytical Model The proposed model predicts the transient position and velocity of a droplet in an EWOD device. The droplet has a radius R, it sits atop electrodes with a side length L, and it is confined between parallel plates separated by a distance of 2h (Figure 3). Dimensionless variables in the model were found using scales for length (R), force (γR), pressure (γ/R), time (T0 = R(2πhFF/γ)1/2), and velocity (u0 = (γ/2πhFF)1/2), where γ is the surface tension between the fluid and the surrounding medium and FF is fluid density. When the same symbols are used to denote corresponding dimensional and nondimensional quantities, a dash “-” indicates a dimensional value. Although the three-dimensional shape of the droplet is accounted for in the model, the droplet motion is assumed to be one-dimensional since the droplet is symmetric about the plane Oxz (Figure 3a,b). (23) Schertzer, M. J.; Ben Mrad, R.; Sullivan, P. E. Sens. Actuators, B 2010, 145, 340–347. (24) Otsu, N. IEEE Trans. Syst., Man, Cybern. 1979, SMC-9, 62–66.
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Figure 2. Image of a droplet moving left to right over a 2 mm electrode actuated at 115 V. The droplet interface (solid line) and the center line (dashed line) are found using the image processing toolbox in MatLab.
Figure 3. View of the confined droplet from (a) the top and (b) the side, and (c) a meridional section of the droplet and interface angles θ-(j), θþ(j) for the top and bottom contact lines, respectively. (d) Top view image of the droplet as it translates between electrodes with relevant parameters.
This motion is characterized by x. The droplet is driven by the capillary force that arises from the asymmetric deformation of the droplet interface; this force is opposed by contact angle hysteresis, viscous friction, and contact line friction. An expression for capillary force as a function of the droplet curvature is required. In cylindrical coordinates {F,j,z}, an arbitrarily shaped droplet can be described by F = 1 þ y(j,z), where y(j,z) characterizes the droplet interface and |y(j,z)| , 1; |j| e π; |z| e h. The angle j defines a meridional section of the droplet containing the vertical axis z (Figure 3a,c,d). The shape of the interface is described using the interface angle θ(j,z) around the perimeter of the droplet. Interface angles θ-(j) and θþ(j) can be used to describe the contact angles along the top (z = -h) and bottom (z = h) contact lines, respectively. The contact angle for the top contact line is given by π - θ-. It was shown in ref 21 that the droplet interface can be assumed to be semicircular with a shape y = s(j,z) determined by h and θ-(j), θþ(j) (Appendix, eqs A.1-A.3). The pressure inside a 19232 DOI: 10.1021/la103702t
stationary undeformed droplet was also found in ref 21. A more general form of the pressure p(j) that allows for asymmetry in j is pðjÞ ¼ χ - ðjÞ=2h þ HðjÞ=8, χ - ðjÞ cos θ - ðjÞ - cos θ þ ðjÞ
ð1Þ
where H(j) accounts for the deviation in the droplet shape from a sphere (Appendix, eqs A.4 and A.5). Since h , 1, the first term in eq 1 may be used as a good approximation of the pressure within the droplet. By integrating eq 1 along the droplet interface, the horizontal capillary force (Fx) becomes Z 1 π ð0Þ ð1Þ ð0Þ ð2Þ Fx ¼ Fx þ hFx , Fx ¼ ΘðjÞ cos j dj 2 0 where ΘðjÞ G½θ - ðjÞ - G½θ þ ðjÞ,
GðθÞ 2θ - sin 2θ
ð3Þ
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Figure 5. Side view of the confined droplet illustrating the model of contact-angle hysteresis.
Figure 4. Analytical results for coarse and accurate estimations of the horizontal component of capillary force; θ-(j) = 74° and θ2þ = 106°. (1) Here F (0) x is a first approximation of Fx and hF x can be added to increase accuracy (Appendix, eq A.6). For practical values of θ-(j) and θ2þ θþ(π), Fx monotonically decreases with θ1þ θþ(0), but F (0) x does not (Figure 4). This suggests that Fx is more accurate than F (0) x , but the two predictions are in good agreement in the operating regime of EWOD devices. This also suggests that the capillary force has a weak dependence on the gap distance (h), but that dependence can be reasonably neglected in EWOD investigations. Now that expressions for the driving capillary force have been found (eqs 2 and 3), the interface angles at the contact lines (θ-(j), θþ(j)) must be specified. Dynamic contact angles above active and inactive electrodes depend on equilibrium contact angles (θv, θ0) (Figure 3d), the angular position around the periphery (j), and contact angle hysteresis5,7,8,19 (Figure 5). Hysteresis reduces the driving capillary force and is thought to be related to contact line pinning which results from molecular adhesion at the contact line.19 Intuitively, the hysteresis angle will not be constant around the periphery of the droplet. It is expected that it will be at a minimum when the tangent to the contact line is parallel to the velocity and a maximum when it is perpendicular. In this investigation, contact angle hysteresis is defined for |j| e π by the function
ψh ðjÞ ¼ Rh ð1 - 2jjj=πÞ;
ψh ð ( π=2Þ ¼ 0,
ψh ð0Þ ¼ Rh ,
ψh ð ( πÞ ¼ - Rh
ð4Þ
where Rh is the amplitude of hysteresis. It is assumed that deformation of the interface only occurs on sections of the contact line directly above the active electrode. As such, the contact angle on the lower contact line will be a piecewise function dependent on the droplet position (Figure 3d). Based on the static contact angles (θ0, θv) and the contact angle hysteresis model eq 4, interface angles θ-(j), θþ(j) for both contact lines can be specified as functions of |j| e π θ - ðjÞ ¼ π - θ0 - ψh ðjÞ, ( θ0 , beyond active electrode θ þ ðjÞ ¼ ψh ðjÞ þ θv , on active electrode Langmuir 2010, 26(24), 19230–19238
ð5Þ
The methodology for experimental identification of θ0, θv, and Rh is given in the Appendix by√eqs A.11 and A.12. If the droplet radius is L/ 2, then the contact line for |j| e π/4 is initially above the active electrode. This portion of the contact line remains above the electrode until the leading edge of the droplet is tangent to the far edge of the √ electrode. This occurs at x = x/, where x/ = 3l//2 - 1, l/ = 2. As such, the capillary force and corresponding droplet dynamics will be different for the segments [0,x/] and [x/,l/]. Using eqs 2-5, the total horizontal capillary force can be found in the form Fx ¼ Fc þ Ffh ,
Fc Y0 G1 ðxÞ, Y0
Ffh Rh ½Y1 þ Y3 G3 ðxÞ
pffiffiffi 2½Gðθ0 Þ - Gðθv Þ=4
Y1 a1 cos 2θ0 þ a2 cos 2θv - 8=π,
ð6Þ ð7Þ
Y3 cos 2θv - cos 2θ0 ð8Þ
where Fc and Ffh are the driving capillary force and contact angle hysteresis effect, respectively, and constants a1, a2 and nonlinear functions G1(x), G3(x) are given in the Appendix by eqs A.7-A.9. This suggests that the dimensionless horizontal capillary force is dependent on the contact angles (θ0, θv), the hysteresis angle (Rh), and the droplet coordinate (x). The driving capillary force is always positive, the force due to the hysteresis effect is always negative, and the total horizontal force becomes negative at approximately 1.57 mm for a hysteresis angle of 3° and electrode size of 2 mm (Figure 6). This suggests that contact angle hysteresis plays an important role in droplet dynamics. Now that the horizontal capillary force has been modeled, its magnitude can be used to see what forces play an important role in droplet motion in EWOD devices. Additional forces that oppose droplet motion include contact line friction (F fl), and viscous shear forces in the fluid from the wall (F fw) and the surrounding medium (F fa). Models for these frictional forces were taken from literature.7,11,15 2
jF fw j ¼ 6πμF uR =h,
jF fl j ¼ 4πRμ l u,
jF fa j ¼ 2KC FA h Ru2
ð9Þ
Here μhF and μhl are friction coefficients, uh is the droplet velocity, KC is the drag coefficient for a cylinder, and FA is the density of air. Values of μhF, μhl, and KC were 0.001 N 3 s/m2,7,15 0.04 N 3 s/m2,11 and 1.15,11 respectively. A comparison of these forces is given in the Table 1 for h = 353.1 μm, R = 1.38 mm, θ0 = 105.9°, θv = 92.9°, and uh = 1 cm/s. From this comparison, viscous shear from the surrounding medium can be neglected if the surrounding medium is air. DOI: 10.1021/la103702t
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and velocity of the droplet are determined by integrating eq 11 on the segments [0,t/] and [t/,tmax]. Time instants t/ and tmax are found from the conditions x(t/) = x/ and u(tmax) = 0. For the interval [0,t/], G1(x) = 1 and G3(x) = 0 and eq 11 can be analytically solved as a linear system such that xðtÞ ¼ at=μ þ a½expð - μtÞ - 1=μ2 , uðtÞ ¼ a½1 - expð - μtÞ=μ
ð12Þ
where a Y0þRhY1. The value of t/ is the root of equation x(t) x/ = 0, and the corresponding velocity u(t/) = u/ can also be determined. These parameters can be used as initial conditions to integrate the nonlinear system eq 11 numerically at [t/,tmax]. One of the advantages of an analytical model is that the equation gives physical insight into the behavior of the system. When looking at the form of eq 12, it is apparent that the time constant of the droplet motion is given by τ = 1/μ and that the steady state velocity is given by uss = a(θ0,θv,Rh)/μ.
Results and Discussion Figure 6. Analytical results for driving capillary force (F c), the opposing force due to contact angle hysteresis (F fh), and total capillary force (F x) as a function of droplet position for a contact angle hysteresis angle of 3°. Table 1. Comparison of Driving and Opposing forces driving force
opposing forces
|F c|
|F fh|
|F fw|
|F fl|
|F fa|
30.9 μN
22.0 μN
1.0 μN
6.9 μN
0.00013 μN
In the proposed model, the droplet is considered to be a material point11 with the same mass, driving forces, and opposing forces as the droplet. Similar assumptions were also made in refs 5 and 13. Although it is assumed that the droplet moves through the device as a rigid body, the three-dimensional interface shape is used to determine the capillary forces acting on the droplet. This method may not provide the detail of a model obtained from the Navier-Stokes equations,8 but it does provide a framework for a transient analytical model of droplet dynamics. These models are important tools, and the form of the equations can provide physical insight into the problem. After assuming that the droplet moves as a rigid body, its motion can be described by dx=dt ¼ u,
mF du=dt ¼ F c þ F fh þ F fw þ F fl
ð10Þ
where mF = 2πFFhR2 is the mass of the droplet. This suggests that the motion of the droplet is dependent on {h,R,γ,FF; θ0,θv,Rh; μhF, μhl}. This list can be simplified by introducing a single coefficient ( μh) to incorporate the effects of viscous losses and contact line friction. The first four parameters are defined by the device geometry and fluid properties; the remaining parameters can be determined experimentally. The normalized equations of motion are given by x_ ¼ u,
u_ ¼ - μu þ Y0 G1 ðxÞ þ Rh ½Y1 þ Y3 G3 ðxÞ
ð11Þ
where a dot above a variable indicates a derivative relative to nondimensional time t and parameter μ is a dimensionless friction coefficient (eq A.10) that accounts for losses due to viscous and contact line friction. An estimate of the friction coefficient (μˆ ) can be found from experimental data. The algorithm used to find μˆ is given in the Appendix (eqs A.13-A.18). The transient position 19234 DOI: 10.1021/la103702t
The proposed model assumes that the droplet translates between electrodes in an EWOD device as a rigid body. The change in the droplet length over time for an electrode pitch of 1.5 mm and an applied voltage of 115 Vrms is presented in Figure 7. As the droplet accelerates, its length increases by approximately 6%. More than half of this increase occurs in the first 5 ms as a result of the initial application of the electric field. The droplet length is reduced by approximately 2% at the end of the motion. This appears to be an inertial effect as the compression occurs after the leading interface has stopped moving. The ratio of inertial forces to surface tension forces is given by the Weber number (We = FV2r/γ). An appropriate length scale when examining the compression of the droplet in this case is the radius of curvature in the horizontal plane. Weber numbers for the current application range between 1.5 10-3 and 0.05. Although this shows that surface tension forces are much larger than inertial forces, it is not unreasonable to suggest that inertial effects may be responsible for the slight compression seen in Figure 7. A similar result was presented in ref 25 where the motion of the leading and lagging interfaces was different. The minimal deformation of the droplet supports the rigid body assumption. For the purpose of the experimental results, the droplet motion begins after the initial deformation of the interface and ends when the droplet regains its original length. The evolution of the contact angles (θ1-, θ2-, θ1þ, θ2þ) with time for each measured position is shown in Figure 8. Contact angles for the trailing interface (θ2-, θ2þ) remain relatively constant ((2°), while those on the leading interface (θ1-, θ1þ) remain constant for the first 80% of the motion before approaching undeformed values. Values for θ0, θv, Rh, and μ, were 103.7°, 67.9°, 8.1°, and 2.4, respectively (Appendix, eqs A.11-A.18). These values were used in the model to predict droplet dynamics. The observed position and velocity of a droplet traveling between 1.5 mm electrodes actuated at 115 Vrms are compared with model predictions in Figure 9. The maximum position and the time at which it is achieved are underpredicted by 2.3% and 0.5%, respectively. The average error in position was approximately 20 μm. This error is 5.1% of the transient position, or 1.5% of the total displacement. The observed and predicted average velocities agree to within 0.03 cm/s, or 2.6%. The maximum velocity is underpredicted by approximately 10%. Higher errors (25) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Lab Chip 2002, 2, 96–101.
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Figure 7. Experimentally observed change in the length of a droplet for a 1.5 mm electrode actuated at 115 V. Droplet motion begins at the left dashed line and ends at the right dashed line.
Figure 8. Experimentally observed change in the transient contact angles for the contact lines on the ground electrode for the (a) trailing and (b) leading interfaces as well as those above the actuation electrodes above the (c) trailing and (d) leading interfaces.
in the instantaneous velocity data are expected, as they are attained by differentiating the experimentally observed position with respect to time. The average transient error in this case was approximately 0.05 cm/s. The average absolute transient error was 0.17 cm/s, which is similar to the experimental resolution (0.14 cm/s). Underprediction of the velocity during deceleration (t > 90 ms) is due to droplet deformation. Although the relative error in velocity here is large, it does not translate into significant errors in the transient position (0.8-2.4%) because the droplet displacement in this region is small. Model predictions were also experimentally verified over a range of voltages (90-120 Vrms) for the same aspect ratio. Here, the maximum and minimum deformations of the droplet were 7.5% and 2.5%, respectively (Figure 10). Interestingly, the change in the droplet radius in these cases suggests that there are two types of motion over the range of applied voltages. For voltages between 100 and 120 Vrms, the change in the droplet length with Langmuir 2010, 26(24), 19230–19238
Figure 9. Comparison between the observed (square) and predicted (line) (a) droplet position and (b) droplet velocity with time. DOI: 10.1021/la103702t
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Figure 10. Experimentally observed maximum (closed) and minimum (open) droplet radius as a function of applied voltage for electrode sizes of 1.0 mm (tilted square), 1.5 mm (square), and 2.0 mm (triangle).
Figure 12. Error in the (a) average absolute transient position (average population dashed line) and (b) average (closed square) and absolute average (open square) velocity as a function of applied voltage for an electrode size of 1.5 mm.
Figure 11. Experimentally observed evolution of (a) hysteresis angle and (b) friction coefficient with applied voltage for an electrode size of 1.5 mm.
time was similar to that presented in Figure 7, but at 90 and 95 Vrms the length was consistent throughout the motion. At lower voltages, capillary forces may not dominate over wall effects, which are not accounted for in the proposed model. The evolution of the hysteresis angle (Rh) and friction coefficient ( μ) with applied voltage are shown in Figure 11. The hysteresis angle increased exponentially with applied voltage. This was an expected result, as an increase in voltage corresponds to an increase in droplet velocity. The friction coefficient decreased by approximately an order of magnitude when the voltage was greater than 100 Vrms. This further suggests that a transition in droplet dynamics related to wall effects occurred between 95 and 100 Vrms. The average error in the maximum position for 90 and 95 Vrms was 12.7%, but this value decreased to 2.3% at larger voltages. The average maximum time was accurate to within 13% when the applied voltage was 100 Vrms, and 3% in all other cases. The average transient error in these cases was approximately 40 μm, or 3.5% of the maximum displacement (Figure 12a). 19236 DOI: 10.1021/la103702t
The velocity profiles for voltages between 90 and 120 Vrms were similar to that presented in Figure 9, with a rapid acceleration followed by a steady state and a rapid deceleration. The error in the average velocity over all cases was 8.1%. The average transient (0.004 cm/s) and absolute (0.17 cm/s) errors again show that the velocity data are evenly distributed about the predicted values with errors comparable to the experimental resolution (Figure 12b). For small voltages, the average error in the steady state velocity was 11%. For voltages greater than or equal to 100 Vrms, this value fell to 3.8%. This shows that the proposed model can be used to predict droplet dynamics in EWOD devices over a range of applied voltages. Experimental results were also compared against model predictions for electrode sizes of 1.0 and 2.0 mm. The maximum displacement in the 1.0 and 2.0 mm cases was predicted to within 7.5% and 1.2%, respectively, and the droplet travel time was predicted to within 2.5% in both cases (Figure 13a,b). The error in the average position for the 1.0 and 2.0 mm cases was 6.3% and 13.6%, respectively. The average error in the 2.0 mm case is higher because of errors on the order of 45% that occur within the first 15 ms of the motion. Since the droplet radius is larger here, the surface tension forces that preserve the droplet radius are lower and the droplet stretches more during acceleration at the beginning of the motion (Figure 10). The average error in this case after the first 15 ms of motion was 3.7%. The error in the predicted average velocity for the 1.0 and 2.0 mm electrodes was 5.0% and 3.0% (Figure 13c,d), and the predicted steady state velocity for the 1.0 mm case was accurate to within 2.6%. Interestingly, in the Langmuir 2010, 26(24), 19230–19238
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Figure 13. Comparison between the observed (square) and predicted (line) droplet position with time for electrode sizes of (a) 1.0 and (b) 2.0 mm and velocity for electrode sizes of (c) 1.0 and (d) 2.0 mm for an applied voltage 115 Vrms.
2.0 mm case, the droplet began to decelerate before reaching steady state. This change in the shape of the velocity profile was captured by the proposed model. The average error in the droplet velocity during acceleration for the 2.0 mm case was 3.7%. The velocity was again underpredicted during deceleration. The change in the shape of the velocity curve in the 2.0 mm case may be due to a change in the friction coefficient. The average value of μ for the 1.0 mm case and the 1.5 mm cases with applied voltages greater than 95 Vrms was approximately 4; for the 2.0 mm case, μ was 0.3. This highlights the importance of the surface quality in EWOD devices. One of the advantages of an analytical model is that the form of the equation can provide information about the physics of the problem. In this case, the time constant for the 2.0 mm case was greater than 20 ms (T0/μ), and the predicted maximum velocity was approximately 5.5 cm/s (au0/μ). If the droplet was moving at the maximum velocity for the majority of the translation, the transit time would be approximately 36 ms. Since this time is of the same order of magnitude as the time constant, the model predicts that the acceleration of the droplet will take the majority of the transit time. As such, an average velocity model would not be appropriate in this case.
Conclusions The experimental data show that the analytical model developed here can reliably predict droplet dynamics in EWOD devices for a variety of voltages and aspect ratios. The model predicts the transient position and velocity at low computational expense, and the equations developed provide physical insight into the problem. An order of magnitude analysis of the forces present showed that opposing forces from contact angle hysteresis, viscous shear at the wall, and contact line friction should be included with driving capillary force in the dynamic model. The transient position and velocity of the droplet were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The model also provides an analytical argument for neglecting the effect of gap distance on the capillary force in practical EWOD devices. Profile images of droplets in transit were used to determine the droplet position and the shape of the droplet interface. The gap Langmuir 2010, 26(24), 19230–19238
distance in these devices remained approximately constant at 715 μm, but the electrode pitch was varied between 1 and 2 mm. Actuation voltages ranged between 90 and 120 Vrms at a frequency of 10 kHz. The hysteresis angles in this investigation ranged from approximately 2° to 10°, and the friction coefficient decreased by an order of magnitude between 95 and 100 Vrms. In the base case, the maximum position and travel time of the droplet were predicted to within 2.5% and 0.5%. The average error in position was found to be 5.1% of the transient position, or 1.5% of the total displacement. The average velocity was predicted to within 2.6%, and the predicted steady state velocity was accurate to 3.7%. The average error in the transient velocity was approximately 0.04 cm/s, and the average error in the absolute transient velocity was comparable to the experimental resolution. The droplet velocity was underpredicted during droplet deceleration. This is likely due to the compression of the droplet during deceleration, which pushes the droplet center of mass further ahead than it would if the droplet remained cylindrical. For voltages below 100 Vrms, droplet velocities were generally overpredicted. An examination of the deformation of the droplet during translation and the evolution of the friction coefficient with applied voltage suggests that wall effects may retard the droplet motion in these cases. The errors in the predicted position and velocity were similar to the base case for voltages greater than 100 Vrms. Errors were greater at lower voltages, where wall effects may play a greater role in droplet dynamics. For electrode pitches of 1.0 and 2.0 mm, errors in the predicted position were similar to the base case as was the velocity prediction for the 1.0 mm case. In the 2.0 mm case, the droplet began to decelerate before reaching steady state. The model predicted this behavior, and the average error in velocity during the droplet acceleration was less than 4%. Acknowledgment. The support of the Canadian Foundation for Innovation, Ontario Graduate Scholarships (OGS), Mathematics of Information Technology and Complex Systems (MITACS), and National Sciences and Engineering Research Council (NSERC) of Canada through Discovery grants and PGS D scholarships is greatly appreciated. We would also like to acknowledge the assistance of the ECTI Open Research Facility in preparing the DOI: 10.1021/la103702t
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Schertzer et al.
devices described here, and Engineering Services Inc. (ESI) for their collaboration on this work.
Appendix A semispherical droplet y = s(j, z) at the plane {z, y} is defined by its center (z0, y0) and radius r as functions of j sðj, zÞ ¼ y0 ðjÞ þ rðjÞ ¼ 2h=χ - ðjÞ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 ðjÞ - ½z - z0 ðjÞ2
ðA.1Þ
Method for Experimental Identification of Contact Angles θ0, θv and Hysteresis rh. Contact angles (θ1-,θ2-,θ1þ,θ2þ) depend on θ0, θv and hysteresis angle Rh from eqs 4 and 5
χ þ ðjÞ cos θ - ðjÞ þ cos θ þ ðjÞ
ðA.3Þ
Function H(j) describes the pressure (eq 1) inside the droplet HðjÞ fπ - G½θ - ðjÞg=cos θ - ðjÞ þ fπ - G½θ þ ðjÞg=cos θ þ ðjÞ GðθÞ 2θ - sin 2θ
ðA.4Þ ðA.5Þ
1 ¼ 8
Z
π
f0 ðjÞ cos j dj,
0
f0 ðjÞ ΘðjÞ HðjÞ=χ - ðjÞ ðA.6Þ
G1 ðxÞ
8 < 1,
h i : 1 - 2ð1 - X 2 Þ 1=2 ,
x_ ¼ u, xð0Þ ¼ 0 u_ ¼ - μu þ a, uð0Þ ¼ 0
G3 ðxÞ
0 e x e x/ x/ e x e l /
;
A Δ~ xN , x 1 :::Δ~
ðN1Þ
u1 :::^ aΔt - Δ~ uN ðA.14Þ U a^ Δt - Δ~
ðN1Þ
Δ~ xi x~i - x~i - 1 ,
Δ~ ui u~i - u~i - 1
^ h Y1 θ^0 , θ^v a^ Y0 θ^0 , θ^v þ R
X 1 þ x/ - x
ðA.9Þ The nondimensional friction parameter μ is used in the droplet dynamic model (eq11) R μF 3kF pffiffiffi μF , h h sffiffiffiffiffiffiffiffi 1 2π 2kF pffiffiffi μl , kF γFF h
19238 DOI: 10.1021/la103702t
ðA.13Þ
ðA.15Þ
ðA.16Þ
An OLS solution for this equation is given by μˆ = (ATA)-1ATU, which can be reduced to the form
0, 0exex/ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 pffiffiffi G1 ðxÞ - 1 þ x/ - x þ 1 - X arccos X , x/ exel/ π 2
μ μF þ μl ,
ðA.12Þ
The overdetermined linear equation for μ takes the form Aμ = U, where vectors A, U are
ðA.7Þ
ðA.8Þ (
^h θ^v ¼ θ~1 þ - R
^ h, θˆ 0 ,θˆ v where θ1-, θ2- are contact angles, not interface angles. R are used in eqs 7, 8, and 11. Identification Algorithm for Friction Coefficient μ. Linear regression with ordinary least squares (OLS) can be used to identify ( μ). Suppose that data for droplet positions (~ xi) and velocities (~ ui) on [0,t/] are given at discrete instants of time ti = iΔt (i = 1, ..., N), where Δt = t//N, N g 2. Then identification xi,~ ui (i = 1, ..., N )}. algorithm uses discrete arrays {ti,~ Over the segment [0,t/], the droplet dynamic model eq 11 becomes the linear system
Constants a1, a2 and functions G1(x), G3(x) are calculate the horizontal capillary force in eqs 6-8 pffiffiffi pffiffiffi a1 24 þ 4 2 - π 2 =4π 2:006, pffiffiffi pffiffiffi a2 8 - 4 2 þ π 2 =4π 0:540
ðA.11Þ
^ h ¼ ðθ~1 - - θ^2 Þ=2, R
^h, θ^0 ¼ θ^2 þ R
Function Fx(1) characterizes the correction of capillary force in (eq 2) Fxð1Þ
θ2 - ¼ θ0 - Rh
θ^2 ¼ ðθ~2 þ þ θ~2 - Þ=2,
ðA.2Þ
χ - ðjÞ cos θ - ðjÞ - cos θ þ ðjÞ,
θ 2 þ ¼ θ 0 - Rh
θ1 - ¼ θ0 þ Rh ,
Estimations for θ0, θv, and Rh can be found using data for θ~1þ, θ~1-, θ~2þ, and θ~2-:
z0 ðjÞ ¼ hχ þ ðjÞ=χ - ðjÞ,
y0 ðjÞ ¼ - 2h sin θ - ðjÞ=χ - ðjÞ
θ 1 þ ¼ θ v þ Rh ,
μl
ðA.10Þ
μ^ ¼ ð^ ax~/ Δt - βN Þ=RN
ðA.17Þ
where RN
N X i¼1
ðΔ~ xi Þ2 ,
βN
N X
Δ~ xi Δ~ ui ;
x~/ ¼ x~N
ðA.18Þ
i¼1
6 0 and RN > 0, so eq A.17 is always For a moving droplet, Δx~i ¼ correct. From eq A.10, μ combines both μhF and μhl. To find the latter coefficients separately, experiments for the same fluid must be performed at different gap distances (2h1, 2h2) or droplet radii (R1, R2). Supporting Information Available: Sketch of the top view of the droplet showing the intersecting portion of the lower contact line in various positions. This material is available free of charge via the Internet at http://pubs.acs.org.
Langmuir 2010, 26(24), 19230–19238
