pubs.acs.org/Langmuir © 2010 American Chemical Society
Charge-Coupled Transient Model for Electrowetting Luis Casta~ner,* Vito Di Virgilio, and Sandra Bermejo MNT Group, Department of Electronic Engineering, E.T.S.E. Telecomunicaci o, Universitat Polit ecnica de Catalunya, Jordi Girona 1, Barcelona 08034, Spain Received July 12, 2010. Revised Manuscript Received August 16, 2010 Electrowetting is widely used as a means to increase the wettability of droplets on a substrate covered by a dielectric. Although static or quasi-static models of the triple-line movement already exist, little research has been published on transient modeling coupled to the charge transient. This work describes a model of two differential equations coupling the charging to the movement taking into account friction. The model results are validated by comparison to published experimental results. The model focuses on applications, and hence the time to respond, the power consumption, and the energy and its breakdown into components are calculated. Moreover, the use of a generalized voltage source allows us to model successfully the results of a “corona charge” experiment as a means to increase wettability without contact between the electrode and the liquid sample. Finally, the model is extended to an ideal “charge-driven mode” electrowetting proposal resulting in better controllability of the speed and transient time between two contact angle values with applications to lab-on-a chip or displays.
Introduction Miniaturization helps the development of new components in microelectronics and other application areas such as in microfluidics, where electrowetting1,2 is becoming an important method of controlling droplets below the microliter range for lab-on-achip devices,3-5 displays,6,7 or adjustable lenses.8 Understanding electrowetting has been the subject of much research, developing theoretical models for steady-state and dynamic operating conditions, performing experiments on different surfaces, analyzing the effects of charge trapping,9 studying the contact angle saturation,10 and developing applications. The purpose of this work is to build a theoretical model for the transient dynamics of the triple line (TPL), coupling the electric charge dynamics with the triple-line movement. As will be shown below, this allows us to model the contact angle change and the triple-line velocity after a voltage step is applied and also to compute the energy budget and its breakdown into components during the transient. The model is valid either for conventional geometry in which the electrode contacts the liquid or for a contactless approach, also described in this article, which is based on the corona charge. Most of the work analyzing the triple-line dynamics assumes quasi-static conditions in which the TPL speed is constant either because the bottom plate moves at a constant speed, such as in the experiments by Blake,11 Berge,12 and Vallet, Berge, and Vovelle,13or *Corresponding author. E-mail:
[email protected]. (1) Quilliet, C.; Berge, B. Curr. Opin. Colloid Interface Sci. 2001, 6, 34. (2) Mugele, F.; Baret, J. C. J. Phys.: Condens. Matter 2005, 17, R705–R.774. (3) Cho, S. K.; Moon, H.; Kim, C. J. J. Microelectromech. Syst. 2003, 12, 70. (4) Satoh, W.; Loughran, M.; Suzuki, H. J. Appl. Phys. 2004, 96, 835. (5) Pollack, M. G.; Shenderov, A. D.; Fair, R. B. Lab Chip 2002, 2, 96. (6) Beni, G.; Tenan, M. A. J. Appl. Phys. 1981, 52, 6011–6015. (7) Hayes, R. A.; Feenstra, B. J. Nature 2003, 425, 383–385. (8) Berge, B.; Peseux, J. J. Eur. Phys. J. E 2000, 3, 159. (9) Quinn, A.; Sedev, R.; Ralston, J. J. Phys. Chem. B 2003, 107, 1163–1169. (10) Quinn, A.; Sedev, R.; Ralston, J. J. Phys. Chem. B 2005, 109, 6268–6275. (11) Blake, T. D.; Clarke, A.; Stattersfield, E. H. Langmuir 2000, 16, 2928–2935. (12) Berge, B. C. R. Seances Acad. Sci., Ser. B 1993, 317, 157–163. (13) Vallet, M.; Berge, B.; Vovelle, L. Polymer 1996, 37, 2465–2470. (14) Ren, H.; Fair, R. B.; Pollack, M. G.; Shaughnessy, E. J. Sens. Actuators B 2002, 87, 201–206.
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because the droplet moves at a constant speed in a capillary, driven by pressure.6,14 Experiments conducted by Decamps15 recorded the instantaneous value of the contact angle after depositing a drop of glycerol from a syringe until equilibrium was reached. The resulting transients were then fitted to a quasi-static model based on the molecular kinetic (MK) theory.16 Transient measurements of the triple-line velocity by measuring the current transient have been used in the past.17 The oscillation of a drop in ac electrowetting for a half-region spherical drop and the unsteady spreading of drops in air have also been described.18,19 The complexity of dynamic wetting includes the presence of a very thin precursor film propagating ahead of the nominal contact line20,21 with applications to micro- and nanodrug delivery. In this work, we formulate a set of coupled differential equations to fully model the transient of the triple-line movement after an electrostatic potential is applied. These equations couple the electrical charge dynamics with the triple-line movement, beyond the quasi-static approach and the conventional voltage source driving and have been extended to a generalized power source. As mentioned above, besides the contact electrowetting, we also explore in this article a contactless electrowetting method in which charging occurs via “corona charging”, thus avoiding direct contact between the electrode and the liquid. A coronacharging procedure was used earlier to assist electrostatically the experiments performed by Blake11 and to analyze the effect of the electrohydrodynamic thrust in the movement of particles suspended in the liquid.22 In this article, we take full advantage of this contactless method to induce contact angle change. Taking into account that corona charge devices have been successfully (15) (16) (17) (18) (19) (20) (21) (22)
Decamps, C.; De Conink, J. Langmuir 2000, 16, 10150–10153. Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421–423. Verheijen, H. J. J.; Prins, M. W. Rev. Sci. Instrum. 1999, 70, 3668–3673. Oh, J. M.; Ko, S. H.; Kang, K. H. Langmuir 2008, 24, 8379–8386. Oh, J. M.; Ko, S. H.; Kang, K. H. Phys. Fluids 2010, 22, 032002. de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–861. Yuan, Q.; Zhao, Y. P. Phys. Rev. Lett. 2010, 104, 246101. Arifin, D.; Yeo, L.; Friend, J. Biomicrofluidics 2006, 1, 01410.
Published on Web 09/21/2010
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In Figure 1, the main surface tensions are shown. γLV is the liquid-vapor surface tension, γSV is the solid-vapor surface tension, γSL is the solid-liquid surface tension, and γF is the friction component written in the same units of surface tension (N/m) and defined below. The contact angle at an arbitrary value of time is θ. The substrate is covered by a thin layer of dielectric (EWOD)28 of thickness d. Under equilibrium conditions, the Young-Dupre29 equation can be written as ð1Þ γSV ¼ γLV cos θ þ γSL þ γF Figure 1. (a) Schematic view of the geometry considered. (b) Equivalent circuit.
miniaturized,23 added value can be found in those lab-on-a-chip applications where the lack of electrode-to-sample contact avoids electrolytic reactions and reduces nonspecific adsorption in systems involving biological entities.22,24,25 In this article, we first develop the theoretical model equations and describe the main features. We then analyze the energy budget breakdown during the transient. This provides better insight into the dynamics. The model is validated, on the one hand, by comparing the results with published experimental measurements of the contact angle dynamics from the literature15,17 and, on the other hand, by experimental measurements made by us using the coronacharging technique. The model is then extended to cover the excitation of electrowetting induced by an ideal charge source, and the results of this model, compared with the conventional voltage source, shows that the transient can be speeded up or slowed down by the proper choice of the parameters of the charge source.
Charge-Coupled Transient Model We will consider the simplest geometry, shown in Figure 1, where a droplet lies on top of a dielectric-covered substrate in the open air at atmospheric pressure. Once equilibrium is reached, a dc voltage is applied and the transient that develops thereafter is modeled. Our model involves the movement of the nominal contact line without taking into account the precursor film monolayer. We will consider that the movement works against friction and that viscous effects can be neglected. The model below is hence applicable when the fluid motion has negligible effects. It has been found that friction accounts for the major part of the dissipation in submillimeter geometries.14 It is also well known that it can be assumed that friction gives rise to a force per unit length proportional to the velocity by a friction coefficient ζ6,11 and that this coefficient is independent of the applied voltage.15 Despite viscous effects being neglected in this work, they can easily be accommodated in the model and hence it will also be useful for other geometries such as displays.6,7,26 We also consider that the gravitational effects are negligible27 because an error smaller than 1% was calculated for a 10 μL drop. We do not consider the effects of charge trapping in the dielectric layer, which introduces a shift in the steady-state Lippmann-Young equation (eq 27).
We consider that the liquid is electrically conductive and that, prior to the application of any electric potential, equilibrium is reached and a contact angle of θ0 is established. We also consider that in t = 0þ a voltage is suddenly applied, triggering a transient in which the contact angle changes from θ0 to a new equilibrium value θF. This transient involves not only the triple-line movement and the contact angle change but also a dynamic change in the electric charge from the source. We hence face a transient in which the contact angle, the charge, and the triple-line velocity are functions of time. The effect of a small change in the electric energy per unit area, UA, stored in the solid-liquid capacitance involves an infinitesimal change in the solid-liquid surface tension as follows,30 dγSL ¼ - dUA
thereby indicating that an electric energy increase reduces the value of the solid-liquid surface tension. This may be a limitation of the accuracy of the model shown below, which is based on the Lippmann equation and the parallel plate capacitor approximation, in view of the other model developed31,32 showing charge accumulation at the triple line based on the Schwarz-Christoffel transformation. We take into acount that dUA ¼ CA V dV ¼ qA dV
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ð3Þ
where CA is the capacitance per unit area, CA = ε0εr/d, d is the dielectric thickness, ε0 is the vacuum permittivity, εr is the relative permittivity, V is the liquid-solid potential, and qA is the charge per unit area defined as follows, q ð4Þ qA ¼ A where q is the total charge and A is the area. Using eq 3 in eq 2 and taking into account that dγSL ¼
dγSL dt dt
ð5Þ
after substituting eq 3, - qA dV ¼
dγSL dt dt
ð6Þ
and - qA
(23) Chua, B.; Wexler, A. S.; Tien, N. C.; Niemeier, D. A.; Holmen, B. A. J. Microelectromech. Syst. 2008, 17, 115–123. (24) Malaquin, L.; Vieu, C.; Genevieve, M.; Tauran, Y.; Carcenac, F.; Porciel, M. L.; Leberre, V.; Trevisiol, E. Microelectron. Eng. 2004, 73-74, 887–892. (25) Li, H.; Zheng, Y.; Akin, D.; Bashir, R. J. Electromech. Syst. 2005, 14, 103– 111. (26) Lao, Y.; Sun, B.; Zhou, K.; Heikenfeld, J. J. Display Technol. 2008, 4, 120– 122. (27) Verheijen, H. J. J.; Prins, M.W. J. Langmuir 1999, 15, 6616–6620.
ð2Þ
dV dγ dt ¼ SL dt dt dt
ð7Þ
(28) Moon, H.; Cho, S. K.; Garrrell, R. L.; Jin, C-; Kim, C. J. J. Appl. Phys 2002, 92, 4080. (29) Probstein, R. F. Physicochemical Hydrodynamics, 2nd ed.; Wiley: Hoboken, NJ, 2003. (30) Zeng, J.; Korsmeyer, T. Lab Chip 2004, 4, 265–277. (31) Kang, K. H. Langmuir 2002, 18, 10318–10322. (32) Kang, K. H.; Kang, I. S.; Lee, C. M. Langmuir 2003, 19, 5407–5412.
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and finally we have - qA
dV dγ ¼ SL dt dt
ð8Þ
Taking the time derivative of eq 1 and considering that γLV and γSV do not depend on the applied voltage33 and hence are independent of time, it follows that dγSV dθ dγ dγ ¼ - γLV sin θ þ SL þ F ¼ 0 dt dt dt dt
ð9Þ
For the friction term, as mentioned above, the frictional force is proportional to the velocity, so the friction force per unit length, γF, is given by γF ¼ ςv
ð10Þ
where ζ has units of Ns/m accordingly. As can be seen, eq 10 does not include static friction, which is assumed to be negligible because of the fact that we are neglecting gravity effects and hence the normal force, which in this case is the weight of the drop, and is neglected here. Because the velocity is a function of time, eq 9, by taking into account eq 10, becomes 2
- γLV sin θ
dθ dv dV þ ς - qA ¼ 0 dt dt dt
ð11Þ
and by taking into account that qA
dV qA dqA ¼ dt CA dt
ð12Þ
dθ dv qA dqA þς ¼ 0 dt dt CA dt
ð13Þ
eq 11 becomes - γLV sin θ
The triple-line velocity, v, depends on the contact angle; assuming that the volume of the drop is constant and that the gravitational effects can be neglected, the shape of the drop can be considered to be a spherical cap.15 Although the spherical cap geometry can be distorted,18,19 it can still be valid for small droplet sizes or large surface tension. The radius r, shown in Figure 1, is a function of the contact angle and is given by sin θ
r ¼ R
ð2 - 3 cos θþcos3 θÞ1=3
ð14Þ
ð15Þ
where vol is the drop volume. Then, v ¼
ΩðθÞ ¼
dr dθ ¼ - RΩðθÞ dt dt ð1 - cos θÞ2 ð2 - 3 cos θþcos3 θÞ4=3
(33) Digilov, R. Langmuir 2000, 16, 6719–6723.
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Equation 18 is the first differential equation of our model for the contact angle under dynamic conditions, where the excitation is the last term. Let us now consider a model for qA. The charge qA can be modeled with the help of Figure 1b, where we model a generalized voltage source composed of a voltage source, VS, a series resistor R (which includes the internal resistance of the source and eventually that of the charging path and that of the droplet itself), and the liquid-solid capacitance, C(θ),which is a function of the contact angle. It can be writen that VS ¼ R
dq q þ dt C
ð16Þ
ð17Þ
ð19Þ
In ref 17, this equation in the particular case of R = 0 was used to measure the contact angle by monitoring the current supplied to the drop by the source. Although the value of the resistor R will generally be small, it plays a crucial role in the charging dynamics and in the energy loss. By including R, the model also allows us to conclude that VS and R are the elements of a generalized Thevenin equivalent circuit, thereby covering other charging options such as the corona-charging technique illustrated below. The area, A, is a function of the contact angle, A ¼ πr2 ¼ πR2
sin2 θ ð2 - 3 cos θ þ cos3 θÞ2=3
Using eq 20 in eq 19, we have dqA VS qA dA dθ 1 þ ¼ dt AR A dθ dt CA R
ð20Þ
ð21Þ
Equation 21 is the second differential equation of our model. Equations 18 and 21 finally become ! d2 θ 1 dΩ dθ 2 γLV sin θ 2qA 2 dθ ¼ þ dt2 ΩðθÞ dθ dt ςRΩðθÞ CA ςr dt qA qA þ - VS ð22Þ CA ςRΩðθÞRA CA dqA VS 2RΩðθÞ dθ 1 þ ¼ - qA r dt ACA R dt AR
with 3 vol 1=3 R ¼ π
Using eqs 14-17, eq 13 becomes d2 θ dΩðθÞ dθ 2 dθ qA dqA - γLV sin θ - ςRΩðθÞ 2 - ςR ¼ 0 dt dθ dt dt CA dt ð18Þ
ð23Þ
Equations 22 and 23 are two coupled differential equations of three state variables: the contact angle θ, the time derivative of the contact angle dθ/dt, and the charge per unit area qA. These equations model the charge-coupled transient electrowetting dynamics and can be easily written in a suitable way for a numerical solver. Besides, the triple-line velocity is given by eq 16, Figure 2 shows an example of the results of the model, solved using MATLAB, for given values of the parameters involved as shown in the Figure caption. The initial values of the three state variables are θ(0) = θ0, the time derivative of the contact angle (dθ/dt)t=0 = 0, and the charge per unit area qA(0) = 0. As can be seen, the model predicts that for small values of the internal resistance of the source (e.g., R = 1Ω) the charge builds up much faster than the contact angle starts to change because the Langmuir 2010, 26(20), 16178–16185
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Taking into account eqs 2 and 3, Z Z 1 qA dqA þ B - qA dV þ B ¼ γSL ¼ CA
ð25Þ
where B is an integration constant that is easily calculated by taking into account that at t = 0, qA = 0, and γSL = γSL(0), hence B = γSL(0), γSL ¼ γSL ð0Þ -
qA 2 2CA
ð26Þ
The value of γSL(0) can be found using eq 1 in equilibrium (γF = 0), in the absence of any charge and knowing the value of the contact angle in t = 0: θ0 and the values of γSV and γLV. Equation 24 can now be written as Z Z Z qA 2 dA ¼ ðγLV cos θ þ γSL ð0Þ - γSV Þ dA þ γF dA A 2CA A A ð27Þ Figure 2. Plots of the three state variables as a function of time: contact angle θ (up), triple-line velocity v (middle), and charge per unit area qA (bottom). Each line corresponds to a different value of R (10 Ω (-), 1 108 Ω (---), 1 109 Ω (- 3 -), and 1 1010 Ω ( 3 3 3 ). VS = 500 V, ζ = 4 Ns/m2, vol = 20 10-9 L, d = 30.92 μm, θ0 = 110°, εr = 2.62, γLV = 72.8 10-3 N/m, γSL(0) = 44.7 10-3 N/m, and γSV = 19.8 10-3 N/m.
time constant of the RC circuit is much smaller than the transient time of the contact angle change. It also can be seen that increasing the value of the internal source resistance to really large and impractical values (>108 Ω) reduces the speed of the triple line because the charging of the capacitance is slower. The model allows us to compute the conmutation times between the two contact angle values θ0 and θF . The fall time, tf, is defined as the time required for the contact angle to go from 90% of (θ0 - θF) to 10% of (θ0 - θF) in the spreading phase, and the rise time, tr, is the time required to go from 10% of (θ0 θF) to 90% of (θ0 - θF) in the recovery phase. In the example explored in Figure 2, the fall time was 0.528 s and the recovery time was 0.2448 s. A better understanding of the dynamics is provided in the next section, where the main energy components are analyzed. Because an ideal current source is equivalent to a voltage source with an infinite value of the internal resistance, the use of a current source is a straightforward extension of the model. Moreover, the geometry of a corona-charging scheme cannot be properly modeled with an ideal voltage source, and it is better modeled with our generic VS, R-equivalent circuit as will also be shown below. For ac electrowetting, the value of VS should be the rms value of the applied voltage.
where the left-hand side is the mechanical work done by the electrostatic force and the two terms of the right-hand side are the two components of this work: the conservative component, ECONS (first term), and the dissipative component, EF (second term), arising from friction. This is easily demonstrated by performing a full transient cycle of the voltage source from 0 to VS and back to 0 again. The conservative term is zero after integrating in this closed trajectory whereas the dissipative term is not. The energy budget breakdown is then the following: the energy supplied to the droplet by the source at a given time t, ES is equal to the sum of the energy stored in the capacitor EC (electrical conservative part) plus the energy dissipated in the resistor (electrical dissipative part) ER plus the conservative component ECONS and the dissipative friction component EF ES ¼ EC þ ER þ E CONS þ EF where
Z ES ¼
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A
qA 2 A 2CA
ER ¼
1 R
Z VS -
qA CA
ð29Þ
ð30Þ 2 dt
ð31Þ
Z ECONS ¼ A
ðγLV cos θ þ γSL ð0Þ - γSV Þ dA
ð32Þ
Z
The solid-liquid-gas system receives all the energy from the electric power supply, which is used to do (a) electrical work (charging the capacitance and dissipating power in the resistor) and (b) mechanical work (dissipative work to overcome friction and nondissipative work changing the contact angle and the shape of the droplet). Taking into account that the units in eq 1 are J/m2 (energy per unit area), the energy components of the transient can be calculated by integrating equation1: Z Z Z Z γSV dA ¼ γLV cos θ dA þ γF dA þ γSL dA ð24Þ A
dq dt ¼ VS qA A dt
EC ¼
Energy Budget Breakdown
A
VS
ð28Þ
A
EF ¼ A
γF dA
ð33Þ
and, dA ¼
dA dr dθ dθ dt ¼ - 2πrRΩðθÞ dt dr dθ dt dt
ð34Þ
Figure 3 shows the energy budget breakdown into components for the off-on transient that develops after a voltage VS is applied at t = 0þ (drop spreading). In this example, we have used the specific parameter values shown in the Figure caption and the values shown in Figure 2 for the rest. DOI: 10.1021/la102777m
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Figure 3. Components of the energy for VS = 500 V, ζ = 4 Ns/m2, and R = 10 Ω. Total energy delivered by the source (-), energy stored in the capacitor ( 3 3 3 ), energy dissipated by the resistor (---), energy dissipated by friction (- 3 -), and the conservative energy term (-) .
As can be seen, at the very beginning of the transient, the source energy goes almost completely into resistor losses because of the very large current spike at t = 0þ. (For VS = 500 V and R = 10, the current at t = 0þ is 50 A provided the source has no current limitations.) Soon after, the capacitance is fully charged and the source energy splits in half between resistor losses and capacitor storage because the contact angle has not yet begun to change. When the friction starts to be overcome, the area of the base of the drop increases and hence the source has to deliver more current to fill up the increasing capacitance to the same voltage. This is seen in Figure 3 for t > 0.01 s. Thereafter, the energy goes into capacitance storage and to the conservative component with little further increase in resistor losses. Figure 4 shows the on-off transient (recovery), which we trigger by setting VS = 0 after equilibrium has been reached at the end of the off-on transient (θ = θF) just explained. This is a continuation of the results shown in Figure 3. As can be seen, the energy stored in the capacitor goes to resistor dissipation at the beginning of the off-on transient and the conservative component is dissipated by friction at the end of the transient. Table 1 shows quantitatively the energy component values at the end of the off-on transient and that at the end of the on-off transient. Neglecting heat conduction to the substrate, convection to the air, and radiation, the temperature increase can be estimated to be 1.64 10-4 K for a 20 μL water drop at 25 °C for the data values considered in Table 1. The results in Figures 3 and 4 validate the consistency of the model and show how the different components evolve during the transient. It is also clear that the friction in the off-on transient accounts for roughly 9% of the total energy from the source and rises to 43.2% at the end of the full off-on-off transient. This is consistent with the results shown in ref 14. Similarly, the energy dissipated by the resistor is 6.7% at the end of the off-on transient and rises to 56.7% at the end of the recovery phase.
Model Validation To validate the model, we have done two different things. On the one hand, we have performed comparisons of the results of our model with published experimental results taken from the literature, and on the other hand, we have carried out a 16182 DOI: 10.1021/la102777m
Figure 4. Breakdown of energy components in the on-off transient. Energy stored in the capacitor ( 3 3 3 ) energy dissipated by the resistor (---), energy dissipated by friction (- 3 -), and conservative energy term (-) . The parameter values are the same as in Figure 3. Table 1. Breakdown of the Energy Components for VS = 500 V, R = 10, and ζ = 4 Ns/m2 end of off-on end of on-off transient (Figure 3) transient (Figure 4) (drop spreading) (recovery) energy from the source, ES (10-6 J) energy stored in the capacitor, EC (10-6 J) energy dissipated in the resistor, ER (10-6 J) energy dissipated by friction, EF (10-6 J) conservative component ECONS (10-6 J)
13.574
13.574
6.79
0
0.914
7.704
1.22
5.87
4.65
0
corona-charging experiment on our own. In this section, let us first describe the results of the literature comparisons. There is little experimental data on the non-quasi-static transient dynamics of the triple line. We have selected two published experimental results. The first comparison concerns the experimental data shown in ref 15 where plots are shown of the contact angle values as a function of time for several applied voltages. Experiments were made on top of a 100-μm-thick PTFE plate using a glycerol droplet of 5 mm3 volume. In Figure 5, a comparison of the results of our model with the experimental data points estimated from Figure 5 in ref 15 for 700 V is shown. From the 700 V plot in Figure 5 of ref 15 we estimated an initial contact angle of θ0 = 110° and a final contact angle of θF = 82°. From these two values and using eq 1 with γF = 0, γLV was estimated to be 0.1135 N/m. A value of γSV = 0.019 N/m was used for Teflon. From these values, we used a trial and error procedure to fit the model to the experimental values. This was quite simple because the effect of the source resitance changes the slope of the contact angle close to the origin of time whereas the friction coefficient values are the governing parameter for the rest of the transient. The fit was for ζ = 9 Ns/m2 and R = 100 Ω . As can be seen, the agreement is quite good. The second comparison concerns the results published in ref 17, where a plot of the TPL velocity as a function of time is shown for a water droplet on top of a 10-μm-thick parylene-Teflon film. The comparison with our model is shown in Figure 6. In Figure 5 of ref 17, the TPL speed values are plotted for an applied voltage of -225 V for a 10 μL drop. The value of θ0 = 120° Langmuir 2010, 26(20), 16178–16185
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Figure 7. (a) Corona-charging setup. (b) Thevenin equivalent circuit.
Figure 5. Comparison of the model results with data estimated from Figure 5 in ref 15 for VS = 700 V, 5 mm3 volume. All other parameters estimated from the initial and final contact angle values are reported. ζ was fitted to ζ = 9 Ns/m2, R = 100 Ω, and εr = 2.7.
Figure 6. Results of the model for the same geometry as in ref 17 and data points estimated from Figure 5, also in ref 17. ζ = 0.15 Ns/m2, εr = 2.7, d = 10 μm, and 10 μL volume of water.
was estimated from Figure 3b in ref 17. The value of γSV = 0.019 N/m was also taken from ref 17. The same trial and error procedure was used to fit the model as described above for Figure 5. The value of the friction coefficient in our model was fitted to ζ = 0.15 Ns/m2, R = 100 Ω, and a time shift in the data of 1 ms was required to fit the model to the experimental data. The experimental points used in Figures 5 and 6 above correspond to different materials (silicon PTFE in the case of Figure 5 and a stack of aluminum, parylene, and a highly fluorinated layer in Figure 6). We attribute the different values obtained for the friction coefficient to the different surface finishes and roughness. These comparisons indicate that the model developed is able to reproduce experimental data reported by other researchers in the field.
Application to a Corona-Charging Electrowetting Experiment We have performed a corona-charging experiment as an example of the applicability of the model to a generalized power source. As shown in this section, the results of this technique demonstrates that a contactless procedure changes the wettability properties of a surface by avoiding physical contact between the electrodes and the liquid sample. Although it has been observed in conventional contact electrowetting that at high voltages air ionization in the vicinity of the triple line induces charge leakage (34) Vallet, M.; Vallade, M.; Berge, B. Eur. Phys. J. B. 1999, 11, 583–591.
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and hence limits the contact angle change,34 air ionization has not been used deliberately as a means to increase wettability. The increase in wettability observed after this corona charging contrasts with the decrease in wettability observed after electron irradiation.35 Figure 7 shows a schematic view of the geometry used. The charging path of the liquid-solid capacitance is subject to the air breakdown and ionization pattern, and hence a Thevenin equivalent circuit composed of a voltage source Vth and a series resistance Rth can model the effective energy source. These two values, Vth and Rth, take the place of VS and R, respectively, in the circuit shown in Figure 1b. Therefore, the values of VS and R in our model are Thevenin equivalent values rather than physically real values. The experiment consisted of the application of the corona charge to a 20 μL droplet of DI water on top of a silicon substrate covered by a 30-μm-thick PDMS layer. We used a Sylgard 184 kit, a mixture ratio of 10:1, spinning for 90 s at 1000 rpm, and curing at 70 °C for 30 min. The corona charge instrument is composed of five PTFE needles that support a high-tension wire and an Ultravolt power source. The surface charge was measured using a Trek model 541 electrostatic voltmeter. The contact angle was measured using KSV Cam 200 equipment with a Basler A602F camera with 1 telecentric optics. The distance between the corona charge probe and the drop surface was 6 mm. In Figure 8, a comparison between the experimental observations and the model is shown. The model was adjusted using the values of the parameters as follows: ζ = 2 Ns/m2, Vth = 390, and Rth = 100Ω. The charge density in our model, qA = CAxVth = 2.9864 C/m2, compares very well with the measured charge of 3.05 C/m2 using the electrostatic voltmeter.
Extension to Charge-Driven-Mode Electrowetting The above modeling of the conventional voltage-driven electrowetting has revealed speed limitations in the droplet boundary movement. This is basically due to the fact that the energy transfer to useful movement is hampered by resistor losses at the very beginning of the transient, and those losses cannot be recovered. Trying to shorten the transient time by allowing very large currents at t = 0þ soon becomes impractical and does not provide any advantage because the resistor losses increase similarly. It can be envisioned that, instead of using a voltage source, a charge source can be advantageous. This can be easily accomplished by using a current source having a nominal value of Ion gated for a given length of time ton, as schematically shown in Figure 9. This is exactly the same as feeding the electrowetting system with a charge of up to Ion ton. Because it is now possible to (35) Aronov, D.; Molotskii; Rosenman, M. G. Phys. Rev. B 2007, 76, 035437.
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Figure 8. Comparison of the corona charge electrowetting experiment with the model adjusted to ζ = 2 Ns/m2, Vth = 390 V, and Rth = 100 Ω. The charge density in the model is 2.9864 C/m2.
design and implement close to ideal current sources36 and there are instruments commercially available that can withstand an output impedance of up to 1 1014 Ω,37 dissipative resistor losses can be much reduced. Potentially, the use of a charge source, in the absence of resistive losses, could lead to faster transients. However, slower transients are also achievable (e.g., to avoid contact angle overshoot) by slowing down the charge supply by increasing the value of ton. This is shown below. The governing equations have to be changed to accommodate a charge source of energy and become Z q 1 t ¼ Ion dt ð35Þ qA ¼ A A 0 and Z
t
ð36Þ
Ion dt 0
Equation 22 changes to d2 θ 1 dΩ dθ 2 ¼ dt2 ΩðθÞ dθ dt γ sin θ 2πr þ - LV ςRΩðθÞ A3 CA ς
2 !
Z
1 ðIon CA ςRΩðθÞA2
Ion dt Z
t
Ion dtÞ
dθ dt ð37Þ
0
and now the two differential equations that solve the transient problem are eqs 36 and 37. All definitions for the components of energy in eqs 29-33 are still valid here with the exception of the resistor energy, which is (36) Ghovanloo, M.; Najafi, K. IEEE Trans. Biomed. Eng. 2005, 52, 97–105. (37) www.keithley.com/products/dcac/currentsource.
16184 DOI: 10.1021/la102777m
charge-driven case with Ion ton = 2.745 10-8 C. θ0 = 110°, θF = 0.2443 rad, total energy delivered by the source (upper solid line), energy stored in the capacitor ( 3 3 3 ), energy dissipated by friction (- 3 -), and conservative energy component (-). (Inset) Plot of the contact angle for the charge-driven case (solid line) compared to the voltage-driven case (dashed line) for VS = 500 V, R = 10 Ω. The rest of the parameter values are the same as in Figure 2.
now zero, and the source energy, which now becomes Z 1 t Ion 2 t dt ES ¼ CA 0 A
Figure 9. Schematic of charge-mode-driven electrowetting.
dqA Ion 1 dA ¼ dt A A2 dt
Figure 10. Components of the energy as a function of time for a
ð38Þ
A better understanding of the significant changes that appear in the triple-line dynamics can be appraised by analyzing the results shown in Figure 10 where the energy components for an off-on transient have been plotted as a function of time. To make the comparison easier, we have selected an amount of total charge such that the final equilibrium angle θF = 0.2443 rad is the same as in the example shown in Figure 3. We have also adjusted the parameters of the charge source (Ion = 1.16 10-7 A, ton = 0.227 s) so as to have the same value of the total energy delivered by the source in the two cases, namely, 13.574 10-6 J as in Table 1. As we can see in the main axis in Figure 10, there is no dissipation at the beginning of the transient and it does not start until the capacitance has not been significantly charged. The rate of charging depends only on source parameters Ion and ton. We see that there is even an overshoot of the energy transferred to the capacitance having a maximum just at t = ton. This is due to the fact that for t > ton on, no further charge is available from the source and as the triple line moves, the capacitance increases but, because the total charge is the same, the charge per unit area qA has to decrease, as does the energy stored in the capacitor (eq 35). The rest of the energy from the source goes to the conservative component (e.g., movement) and to friction. This totally changes the dynamics of the movement compared to that of conventional voltage-driven electrowetting, as can be seen in the inset of Figure 10 where the contact angle transient is compared between the charge-driven case and the voltage-driven case. As can be seen for the same energy supplied by the source, the transient is considerably faster in the charge-driven case. If we measure the time that the contact angle lasts to get a value of [θF þ 0.1(θ0 - θF)], which we call the conmutation time, then the charge-driven case is about 2.8 times faster than the equivalent voltage-driven case. An example of the controllability of the movement that can be achieved using charge control is illustrated in Figure 11, where the triple-line velocity is plotted for Ion ton = 2.745 10-8 C. Each plot corresponds to a different value of ton while keeping the Langmuir 2010, 26(20), 16178–16185
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Figure 11. Triple-line velocity as a function of time for four values of ton such that Ion ton = 2.642 10-8 C. ton = 0.01 s (-), ton = 0.02 s (---), ton = 0.1 s (- 3 -), ton = 0.5 s ( 3 3 3 ), and ζ = 4 Ns/m2. (Inset) Energy from the source as a function of the conmutation time. The rest of the parameter values are the same as in Figure 2.
product Ion ton constant. By comparing the results shown in Figure 11 with the results shown in Figure 2, it can be seen that the maximum velocity can be increased by a factor of 7 for ton = 0.01 s compared to the voltage-driven case with R = 10. Using shorter values of the ton time can shorten the transient. This is accompanied by an increase in the consumed energy from the source, as can be seen in the inset of Figure 11. Conversely, increasing the value of ton can also slow the transient down without modifying the end contact angle. Thus, charge mode driving provides good controllability for electrowetting devices with Ion and ton values within practical and achievable limits.
Conclusions A charge-coupled model of the movement of the triple line in electrowetting has been developed by taking into account friction
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Article
and resistor dissipation as losses in energy transfer. It is shown that increasing the value of the series resistor can slow the dynamics of the transient. The model also allows us to study the breakdown into the components of energy transfer between the source and the system: capacitance storage, resistor losses, conservative component, and friction losses. However, the current model does not include a model for the saturation of the contact angle, and as far as our results are concerned, it does not show any overshoot in the spreading of the base area in contrast with some experimental observations38 because no inertial term has been taken into account. The model has shown that it is able to reproduce experimental results published in the literature15,17 and also can be extended to model the corona-charging method that is experimentally developed in this work. The use of a generalized voltage source, which is the Thevenin equivalent, allows us to predict the contact angle change accurately after corona charging. As an extension of the model described above, the use of an ideal charge source to drive droplets is shown to change the transient dynamics by adding controllability. This reminds us of the findings of our past work in controlling MEMS devices,39,40 also by using a charge source. In fact, the transient can be speeded up or slowed down by the proper selection of the parameters of the charge source withour modifying the end contact angle. Speeding up the transient involves a moderate increase in the total energy requested from the source. Acknowledgment. This work has been supported by MICINN project number TEC-2007-67081. V.D.V. gratefully acknowledges an FPI grant (BES-2008-007481) that was also provided by the MICINN. (38) Paneru, M.; Priest, C.; Sedev, R.; Ralston, J. J. Phys. Chem. C 2010, 114, 8383–8388. (39) Casta~ner, L.; Senturia, S. D. IEEE J. Microelectromech. Syst. 1999, 8, 290. (40) Nadal-Guardia, R.; Dehe, A.; Aigner, R.; Casta~ner, L. J. Microelectromech. Syst. 2002, 11, 255–263.
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