Electrowetting Dynamics Facilitated by Pulsing - American Chemical

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Electrowetting Dynamics Facilitated by Pulsing M. Marinescu,*,† M. Urbakh,*,‡ T. Barnea,† A. R. Kucernak,† and A. A. Kornyshev*,† Department of Chemistry, Faculty of Natural Sciences, Imperial College London, SW7 2AZ London, U.K., and School of Chemistry, Faculty of Exact Sciences, Tel AViV UniVersity, Tel AViV 69978, Israel ReceiVed: June 8, 2010; ReVised Manuscript ReceiVed: September 23, 2010

Electrovariable optical components such as variable-focus lenses, microfluidic systems, and electronic displays exploit the electrowetting effect, where the shape of a liquid/fluid interface changes in response to an applied electric field. Low-voltage electrowetting devices cannot make use of an insulating polymeric film coating and are susceptible to contact angle hysteresis. When bare solid electrodes are used, pinning forces originate from surface roughness and heterogeneity, making hysteresis in these devices practically unavoidable. Recent experimental studies of an electrowetting system based on an interface between two immiscible electrolytic solutions (ITIES) have demonstrated that hysteresis can be eliminated by applying short voltage pulses in addition to the steady-state bias. The applied microsecond pulses activate the droplet, facilitating the depinning of the contact line and its motion. Here, a theoretical model describing the effect of electric pulses on droplet dynamics is presented. Key factors in determining the electrowetting response under pulsing are the driving force due to the applied voltage, the static friction force resulting from energetic and morphological nonuniformities of the surface, the viscous-like energy dissipation, and the inertia of the moving liquid. The significance of the present description of hysteresis-free, pulse-assisted electrowetting is not confined to the development of opto-fluidic devices using this technique. In a wider context, it provides a new framework for the theoretical analysis of wetting dynamics, pinning, and friction in systems with liquid droplets on nonideally smooth, solid substrates. 1. Introduction The term electrowetting refers to a change in the shape of a liquid/fluid interface in contact with an electrode as a result of an applied voltage. The change occurs to minimize the sum of the capillary and electrostatic energy terms. Established optofluidic technologies1 use this phenomenon in variable-focus lenses,2–4 microfluidic systems,5 and electronic displays.6 Most current systems are based on the interface between a conducting and a nonconducting liquid. In order to obtain a significant electrowetting response in such systems, it is necessary to apply voltages that trigger undesired side effects, such as corrosion of the electrode surface and electrolyte decomposition. To overcome these side effects, the electrode surface is protected by an insulating layer; this causes additional voltage losses and increases the operating voltage. The corresponding increase in power demand may be undesirable for portable applications. There is significant interest in finding alternative systems with lower operating voltages. A new class of electrowetting systems based on the interface between two immiscible electrolytic solutions (ITIES) has been proposed and investigated theoretically7 and has shown to be feasible experimentally.8 Significant electrowetting responses are obtained at operating voltages below ∼1 V, the threshold for electrochemical reactions.9 Ions remain in their respective solvent when the free energy of transfer across the liquid/liquid interface is large compared to the applied voltage. In systems where the droplet is not an electrolyte, the potential drop is distributed across the entire droplet. In ITIES systems, the * To whom correspondence should be addressed. E-mail: Monica. [email protected], [email protected], and A.Kornyshev@ imperial.ac.uk. † Imperial College London. ‡ Tel Aviv University.

voltage drop is concentrated in a narrow region at each interface, of approximately a Debye length thickness, where the ions screen the potential completely. As a result, electric fields of up to 107 V cm-1 form across the liquid/liquid interface, even for relatively low applied voltages (∼0.5 V). The magnitude of the localized field depends on the concentration of electrolytes. Screening of the electric field near the three-phase contact line (CL) allows ITIES systems to benefit from the removal of field singularities in the vicinity of the CL, thereby significantly reducing the risk of chemical reactions. Moreover, in the absence of field distortions and when the droplet is small enough for buoyancy effects to be negligible (when the droplet size is smaller than the capillary length), the shape of the droplet on a macroscopically smooth surface is an ideal truncated sphere, as shown by both theoretical calculations10 and experimental images.8 This shape is desirable for electro-variable lens applications and is maintained on a real surface as long as the correlation length of roughness or inhomogeneity is much shorter than the droplet size. A comprehensive model describing the equilibrium state of ITIES systems has been developed.11 The two back-to-back double layers at the liquid/liquid interface are described using the Verwey-Niessen theory. The electrical double layers at the electrode/surrounding electrolytic solution and the electrode/ droplet interfaces are described by using the nonlinear Gouy-Chapman theory of a diffuse double layer in series with an inner layer. The equilibrium shape of the droplet depends on system parameters such as dielectric constants, electrolyte concentrations, and surface tensions. A major obstacle to the operation of microfluidic low-voltage devices based on bare electrodes is the contact-angle hysteresis caused by random pinning forces originating from surface roughness and heterogeneity of solid electrodes. Vibrational

10.1021/jp1052634  2010 American Chemical Society Published on Web 12/06/2010

Electrowetting Dynamics Facilitated by Pulsing energy may be used to overcome random pinning and reduce hysteresis in wetting systems12 or to move droplets on inclined surfaces.13 Such energy is supplied by mechanical shaking, which is both difficult to perform in a reproducible, wellcontrolled manner and of little use in small-scale devices. A more feasible method is “electric shaking”, provided by voltage pulsing14 or sinusoidal voltage modulation.15 The latter reduces hysteresis for high-voltage electrowetting; for ITIES-based lowvoltage electrowetting, the pulse-assisted technique8 was effective. The method has been successfully applied to a submillimeter droplet of nitrobenzene with dissolved tetrabutylammonium-tetraphenylborate (TBA+-TPB-) electrolyte, on a sputtered gold electrode, in an aqueous solution of lithium chloride. Pulses of -2 V were applied for 50 µs in addition to a negative stationary bias. As a result, the droplet was depinned and brought to a significantly more contracted geometry. The procedure was then reversed, and 50 µs pulses of 2 V on top of a 0 V bias were applied, bringing the droplet back to its initial geometry. The short duration of the pulse prevented significant Faradaic processes from taking place. The dynamics and range of contact-angle variation of the droplet were found to be significantly different from those in the absence of the pulse. Inspired by these findings, we formulate here a model to describe the effects of pulsing on the droplet dynamics and the resulting stick-slip motion. Although the effects of surface roughness on the statics of wetting have been studied intensively, the understanding of its impact on the dynamics is limited.16 We therefore propose a phenomenological model in the spirit of the work done by the group of de Gennes.17 We assume that wetting dynamics is determined by the competition between two acting forces, a driving force caused by an applied voltage and a resisting force corresponding to the friction on the rough metal surface. The applied pulses activate the droplet, facilitating the depinning of the three-phase CL toward its final position. The pulse duration can be very short, to minimize energy costs and prevent electrochemical reactions. In the section Results and Discussion, we show that the theoretical predictions of this model correctly describe many features of the droplet response, as measured experimentally.8 The model proposed here is not just a one-experiment theory. Understanding the mechanism through which the pulsing technique significantly reduces the unfavorable effects of surface roughness may prove relevant to future applications. Furthermore, it validates the pulsing technique as a tool for facilitating electrowetting, as well as a new electroanalytical method for studying wetting dynamics, a fundamental issue in physics and physical chemistry. 2. The Model and Basic Equations 2.1. Droplet Geometry. The ideal truncated sphere shape of a macroscopic droplet lying on a macroscopically smooth yet possibly microscopically rough or inhomogeneous surface is unambiguously described by the radius of the wetting spot, Figure 1. If R denotes the radius of the circular three-phase CL, the equations describing the droplet dynamics can be formulated in terms of a single time-dependent function R(t). This quantity fully determines the instantaneous contact angle. 2.2. Balance of Forces. At any applied voltage V, there are two forces acting on the CL: (i) a driving force Fd that tends to bring the droplet to the new equilibrium contact angle corresponding to V and (ii) a friction force Ff that opposes the motion caused by Fd. If the CL forms an ideal circle, the friction force

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Figure 1. The system: droplet on a plane electrode, immersed in a surrounding liquid; droplet contact angle R, radius of the contact line R, applied potential V.

may be considered homogeneous and isotropic over the whole surface, in the absence of macroscopic patterning of the solid surface. The friction force includes a velocity-independent contribution from the chemical and geometrical roughness of the electrode surface and a velocity-dependent term that accounts for viscous dissipation associated with the rheological flow:

Ff ) -F0 sgn(R˙) - ηR˙

(1)

The limiting value of the friction force for vanishing velocity is approximated by the static friction F0;17,18 the CL starts moving if Fd exceeds F0. The surface roughness of the electrodes in the system is of the order of 5 nm, much smaller than the size of the droplet (∼0.5 mm). In such cases, the effect of roughness on the droplet motion can be expressed as a static friction F0 independent of position. If the surface is ideally flat and uniform, F0 ) 0, and no hysteresis is present. We are unaware of a general quantitative relation between roughness and friction; there are, however, experimental techniques allowing for the direct measurement of F0.19 The existence of a dependence of F0 on the radius R is not straightforward in the case of buoyancy-free electrowetting. The assumption of a radius-independent F0 is often used,20,21 and the present model employs it as a first approximation, which is easy to refine, if needed. The driving force acts to bring the droplet to an equilibrium position under the applied voltage and takes the form of a restoring force. It is calculated in terms of the free energy functional for the three-phase system Fd ) -∂G/∂R and decreases as the droplet approaches the equilibrium shape Re(V) for the applied bias V on an ideally smooth surface. Under the assumption that Fd is linear with displacement, the driving force is

Fd(V) ) k(V)(Re(V) - R)

(2)

where R is the instantaneous radius of the CL circle, and k(V) ) ∂2G/∂R2 is the effective stiffness taken at R ) Re(V). Equation 2 resembles Hooke’s law and is verified by the quasi-parabolic shape of the calculated free energy of the system G(R) around the equilibrium radius, Figure 2. The equilibrium model of electrowetting with ITIES and our description of the dynamics of electrowetting under pulsing are linked through the effective stiffness under the driving force, k(V).

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Marinescu et al. of viscosity, are completely balanced by the static friction. As soon as the CL radius enters the interval [Re(Vp), Re(Vb)], the driving forces during bias and pulse are of opposing directions. Each further pulse activates the CL to move further, but the sliding distance is shorter each time. For zero acceleration and velocity, the CL radius is

Rpin(b,p) ) Re(b,p) -

Figure 2. The calculated free energy of the system can be approximated by a parabola around its minimum. Calculations are based on the energy functional described by Monroe et al.11 for R0 ) 1°, cd ) 0.1 M, cs ) 0.5 M, and the indicated values of applied voltage V. Values of k ≈ 0.1 N m-1 are obtained.

The free energy G(R) was derived for the ideal case of an ion-impermeable ITIES in a system formed by a droplet on an atomically smooth, inert electrode.11 There the contributions of surface-tension, electrostatic, and entropic terms are included within the mean-field approximation. This model provides a fair description of the driving force on a droplet, as long as the distribution of ions follows instantaneously its changing geometry. This is true at times longer than the time needed to recharge the double layers via ion migration through the bulk of the droplet. In the case of a bias voltage, this restriction is met; for short pulses, it is not. Under the influence of a short pulse, the effective magnitude of the applied potential Veff, as reflected in the surface charge at interfaces, is significantly lower than the potentiostat value; a method to approximate Veff is presented in Appendix A. 2.3. Equation of Droplet Motion. The equation governing the motion of the CL under applied potential is similar to previous models of mechanical shaking17 and reads

mR¨ + ηR˙ ) k(b,p)(Re(b,p)) - R) - F0 sgn(R˙)

(3)

where indices b and p denote states under applied bias Vb and pulse Vp, respectively. In eq 3, m is the accelerated/moving mass in the system, and η is the damping coefficient determined by the viscosity at the solid/liquid interface. An applied pulse Vp of the same sign as the bias voltage affects the droplet motion by enhancing the driving force. The higher Vp, the larger the driving force: k(V) increases with V, as does the difference between the initial set position and the new equilibrium one Re(Vp). By drawing an analogy with the damped harmonic oscillator, the three fundamental parameters that characterize this system are: k/m (the natural frequency), η/2mk (the damping ratio), and F0/m. Models describing the wetting dynamics of droplets on a solid in the presence of friction2,3 do not usually include an inertial term. However, much like the response of a solid/solid system to mechanical shaking,17 the mass-dependent term gives a significant contribution to the droplet motion under short voltage pulses. Any droplet displacement is negligible during the short pulse. The CL however may acquire a finite velocity because of the increased driving force. After the pulse is turned off, the CL continues its motion due to inertia. The CL stops when the driving force, proportional to the distance from the equilibrium position under the bias, and the inertial force, decreasing because

F0 sgn(R˙) F0 ) Re(b,p) ( k(b,p) k(b,p)

(4)

where ‘+’ corresponds to a contracting motion and ‘-’ to spreading; Rpin are the limiting radii for infinitely slow contraction or spreading. Under any applied potential, the CL radius has a stationary value within [Re - F0/k, Re + F0/k], an interval of metastable positions around the equilibrium radius on a friction-less surface. The value of the CL radius at any given time during and after each pulse is given by the solution of eq 3. Equation 4 illustrates that, for infinitely slow motion of the droplet, the contact-angle hysteresis caused by surface roughness is given by the static friction force Rpin+ - Rpin- ) 2F0/k, as suggested by Tadmor.22 2.4. Methods of Solution. Equation 3 presents an initial value problem that must be solved iteratively for every pulse in the sequence. While an analytical solution is possible, in principle, the expressions involved become prohibitively cumbersome. Presented here are the numerical results obtained in MATLAB,23 by using an explicit Runge-Kutta formula based on the Dormand-Prince method.24 The motion of the droplet can be calculated for any sequence of applied pulses, if the driving forces due to the applied bias and pulse are known. A closed-form analytical solution to the equation of motion is derived in Appendix B for a set of simplifying assumptions. Although limited, this solution helps rationalize the droplet dynamics and correctly predicts its main qualitative features. The stick-slip motion under successive pulsing, as well as the asymptotically decreasing change in contact angle with each subsequent pulse, are retrieved. Analytical solutions are illustrated in Figures 6 and 7. 2.5. Model Parameters. To apply the mathematical model developed here, typical values for the system parameters must first be specified; some can be experimentally obtained for each setup, while others must be estimated or considered as fitting parameters. Viscosity and Damping. At 25 °C, the bulk viscosity of water is ηbulk,w ) 9 × 10-4 kg m-1 s-1 and that of nitrobenzene is ηbulk,n ) 2 × 10-3 kg m-1 s-1. The relevant viscosity involved in the CL motion is an effective property of a liquid film during slip and can be different from the bulk value by up to two orders of magnitude.25 The damping coefficient η in eq 3 is related to viscosity through η ) ηsurfr, where r is a characteristic length in the system, such as the droplet radius r ) 1 mm. With these values, η ≈ 10-6-10-3 kg s-1. MoWing Mass. The parameter m characterizes the fraction p of the liquid mass of the droplet and of the surrounding, which rearranges around the three-phase CL, either pFnVd or p(Fn Fw)Vd. Vd denotes the volume of the droplet, and Fn, Fw the densities of the droplet (nitrobenzene) and surrounding liquid (water), respectively. An approach that reflects the droplet motion at the microscopic level is required for correctness. For the two liquids used in the experimental setup (25 °C, 1 atm, Fw ) 997.0 kg m-3, Fn ) 1199 kg m-3, Vd ≈ 10-10 m3) the two expressions give similar results. For p ) 0.1 one obtains m ≈ 10-9-10-8 kg.

Electrowetting Dynamics Facilitated by Pulsing

Figure 3. Disappearance of contact-angle hysteresis under pulsing and strong dependence of the evolution of the droplet geometry on the static friction force. Curves calculated for parameter values: m ) 1.2 × 10-10 kg, η ) 0.3 × 10-5 kg s-1, R0 ) 1°, Vpzc ) 0.3 V, cd ) 0.1 mol dm-3, cs ) 0.5 mol dm-3. Following the steps of the experimental study, values on the x-axis mark the following events: -1, droplet brought to a halt by positive pulsing on top of a 0 V bias; 0, bias of -0.65 V applied; 1-10, pulses of -1 V applied; 11, bias of 0 V applied; 12-21, pulses of 1 V applied.

Static Friction Force. Values of the static friction force should be comparable with the lateral friction force measured for sessile liquid droplets of hexadecane on Teflon and OTA treated mica,19 for which F0 ≈ 10-5 N. In the present model, the static friction term is considered independent of the CL radius. This assumption is not necessarily justified, since the friction between a spreading or contracting liquid and a solid surface is caused by the molecular interactions at the contact area; for a droplet close to dewetting, the static friction term is expected to approach zero. Assuming that most of the resistance during spreading and contracting is met within a narrow region around the CL, the relation F0 ) βR holds, where β is a proportionality constant. The equation of motion in eq 3 appears unchanged for (k f k + β sgn R˙) and (F0 f βRe(b,p)), and the dynamics of the droplet motion can be mapped completely onto those of a system in which the static friction is independent of R. Point of Zero Charge. Values of Vpzc between 0 and 0.3 V are measured on a single-crystal gold surface for KPF6 solution, depending on the crystal phase and the concentration of the solution.26 A positive shift in the apparent Vpzc is observed for concentrations increasing from 0.001 mol dm-3 to 0.030 mol dm-3. In the experimental setup, the electrode surface is polycrystalline, and the solution concentrations are at least one order of magnitude larger. A value of 0.3 V was chosen to illustrate the effects of a nonzero Vpzc. 3. Results and Discussion 3.1. Evolution of the Droplet Shape under Pulsing. The change in the contact angle for a typical system under a cycle of bias voltage and pulsing is illustrated in Figure 3. The main features of the system dynamics discussed in the previous section are visible. As a bias voltage is applied, the droplet contracts because of the electrowetting effect. The CL is pinned by a nonzero friction force, but successive applications of a pulse of the same sign as the bias voltage can provide enough energy to depin it. Key features of pulse-assisted electrowetting are reproduced, such as the stepwise contraction and spreading of the droplet, the asymptotic change in the contact angle, and the disappearance of hysteresis. The hysteresis-free behavior observed experimentally8 can be seen as a result of the preparation of the system prior to a

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Figure 4. Pulsing affects the ultimate contact angle. Calculated contactangle values after 10 (-1 V) pulses for the indicated values of the friction force, in comparison to the frictionless case (thick line). Each time the CL reaches zero velocity, if the resulting angle lies inside the gray area of metastable droplet geometries, it remains pinned. The area of metastability is shown for the smallest friction force, F0 ) 1 × 10-5 N; for larger F0, the area widens.

cycle of bias voltage and pulsing. In experiments8 the droplet reaches its initial position (Step -1) as a result of five negative and five positive pulses on top of a 0 V bias. The final position (Step 21) is reached after a complete potential cycle of bias voltage (from 0 to -0.65 V and back to 0 V), with ten negative pulses on top of the -0.65 V bias (Steps 1-10), and ten positive pulses on top of the 0 V bias (Steps 12-21). After a complete pulsing cycle, the contact angle is independent of the initial droplet geometry. The contact angle in Steps -1 and 21 is, thus, identical, and hysteresis appears to be eliminated. Although the initial and final contact angles are identical, the CL motion as depicted in Figure 3 is qualitatively different in the contracting and spreading modes. The electrowetting behavior is symmetric around Vpzc rather than around 0 V; for the effects of the positive and negative pulses to be comparable, their magnitudes must be equidistant from Vpzc. As this was not the case in the experimental setup, asymmetry of the curve is expected. The driving force during spreading is significantly lower than during contraction; negative pulses act in the same direction as the bias, whereas positive pulses decrease the effective voltage value during the pulse. Furthermore, the effective stiffness k decreases as the applied voltage gets closer to Vpzc. 3.2. Effect of the Friction Force. Figure 3 shows that, as the friction force increases, the CL exhibits a smaller displacement with pulsing. The static term of the friction force acts to create an interval of metastable positions around the ideal equilibrium value, the one reached on a friction-less electrode. After a set of negative pulses, the final position lies inevitably in this interval, irrespective of the value of the friction force or of the bias, as shown in Figure 4. The exact position is not an intrinsic characteristic of the system, like in the case of a frictionless one; it depends on the amplitude and duration of each pulse. For larger values of the friction force, the droplet geometry changes less compared to its initial, hysteresis free configuration. In the limit of very large friction forces, the CL does not move. In the case of very low friction forces, the net effect of pulsing becomes zero; all energy accumulated during a pulse is dissipated through the oscillatory motion, and the droplet eventually reaches the same equilibrium position as that before the pulse. For a set of different bias voltages, the net change in contact angle brought by electric pulsing is plotted in Figure 5 against

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Marinescu et al. of the pulse on the motion of ions in the two liquids. Modeling the elastic coefficient k as both voltage- and radius-dependent could improve the absolute accuracy of the theoretical predictions, especially for low applied bias voltages, where the G(R) is not ideally parabolic. A possible dependence of the static friction term F0 on the CL radius should be investigated, as it could improve the accuracy of the theoretical predictions. These aspects can be analyzed once more in-depth experimental studies validate the theory. 4. Conclusion

Figure 5. Dramatic effect of friction on the net change of the contact angle after a pulsing cycle. The net change in contact angle is shown as the difference between the contact angle reached after 10 pulses of -1 V and the initial equilibrium geometry for the indicated values of the bias.

Figure 6. The exact numerical solution of the model can reproduce experimental data for reasonable parameter values. Parameters: -0.65/0 V biases, -1/1 V pulses, F0 ) 1.65 × 10-5 N, m ) 1.2 × 10-10 kg, η ) 0.3 × 10-5 kg s-1, R0 ) 1°, Vpzc ) 0.3 V, cd ) 0.1 mol dm-3, cs ) 0.5 mol dm-3. Labeling of the x-axis as in Figure 3.

the value of the static friction. The value for the initial contact angle of the droplet was not a free parameter of the model. It was calculated as the value to be reached after a set of five negative pulses on top of a 0 V bias, as in experiments. In a system with a larger friction force or for low bias voltages, the interval of contact angles reached during the pulsing cycle decreases, as expected. The dependence is approximately linear. In the limit of a frictionless electrode, the net change in CL radius is given by the difference between the equilibrium contact angle under the respective bias and the Young Laplace contact angle. 3.3. Comparison with Experimental Data. In Figure 6, we display theoretical curves obtained from the exact numerical solution of the model, as well as from the simplified analytical solution presented in the Appendix B, in comparison to the experimental data.8 The exact solution can reproduce the data quantitatively relatively well, for the expected parameter values. The analytical solution only captures the basic qualitative character of the experimental curve, for reasons detailed in Appendix B. The minor quantitative differences between the exact theoretical results and experimental data may be due to the onset of secondary processes affecting the surface of the electrode that are not taken into account in the model presented, such as voltage-induced surface reconstruction.27 From a theoretical stand point, the presented model could benefit from a more detailed description of the nonlinear effects

The effects of electric pulsing on the spreading and contracting motion of a droplet at a rough electrode in electrowetting with ITIES are rationalized. The phenomenological theory of electrowetting dynamics presented here successfully describes the experimentally observed stick-slip motion of the droplet and the apparent removal of contact-angle hysteresis by pulsing. Although the theoretical model reproduces the experimentally measured curves, achieving an exact fit cannot be a target, as long as the values of most model parameters are not independently measured or more precisely evaluated. Among such parameters are (i) the roughness of the electrode surface characterized by scanning probe techniques, (ii) the liquid/liquid surface tension, (iii) the Young-Laplace contact angle, and (iv) the electrode/liquid potentials of zero charge. The role of the latter in a system with three competing interfaces is, in itself, conceptually challenging. Most parameters are, in principle, measurable, and diminishing their number is highly desirable for a rigorous fit. We hope that the present analysis will stimulate interest in such measurements. Once supported by a more complete set of experimental data, both for electrowetting dynamics in different systems and for the system parameters listed above, the presented framework can form the basis for detailed studies of the friction force at a liquid/electrode interface and its dependence on applied voltage, of the effective potential of zero charge of a three-phase system, and, more generally, for a better understanding of nonequilibrium aspects of wetting on solid surfaces. Appendix A: Details of the Pulse Effect The experimentally applied pulses8 were 50 µs long, much shorter than the time needed for the extra charges to travel between the electrical double layers inside the droplet. The times for ionic migration to charge the double layers can be estimated according to td ) LLd/D where Ld ≈ 10-7 cm is the Debye length, D ≈ 10-5 cm2/s is the diffusion constant, and L is the distance over which the ions must migrate; Ln ≈ 0.1 cm in the droplet, Lw ≈ 1 cm between the working and counter electrodes in water. The corresponding migration times are tdn ) 10-3 s and tdw ) 10-2 s in the nitrobenzene droplet and aqueous surrounding, respectively. The double layers, thus, do not have time to rearrange to equilibrium during the applied pulse of τp ≈ 10-5 s, and the voltage drop extends throughout the droplet and surrounding, not only across the interfaces. Initially, all the extra potential of the pulse, Vp - Vb, is unscreened and drops over the whole system (∼1 cm), causing a weak electric field with a negligible impact on the droplet geometry. Under the influence of this field, all four types of ions start moving toward the respective interfaces, thus contributing to the screening of the extra voltage, Vp - Vb. The higher the amplitude and the longer the duration of the pulse, the larger the amount of charge that gathers at the interfaces during the pulse.

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After a change in the applied voltage, the value of the electric field in the bulk decreases with time as a result of screening. The electric-field variation leads to a decrease in the intensity of charge currents that reach each interface. For an accurate description of the effect of pulsing on charge distribution, one should consider the nonlinear electrokinetics of the ion motion through the solution. A much simplified case is considered here, by estimating the ion velocity to

V)

zeD Vp - Vb kT Lw

(A1)

where ze is the ion charge, and D is the diffusion constant for ions. The extra charge density δQ accumulated at each liquid/ electrode interface during the pulse can thus be estimated as

δQ )

(ze)2nD Vp - Vb τp kT Lw

(

1 δQ 1 + ) δQ K Kdl Kil

Rpin,b ≡ R(0) ) Re,b +

)

(A3)

where Kdl and Kil are the diffuse- and inner-layer integral capacitances, respectively. For the short pulses under consideration, the diffuse layer is not allowed to rearrange under Vp. It is reasonable to assume that under a short pulse Kdl and Kil11 each maintain the value reached during the bias. For the set of parameter values listed above, the total charges present at the end of the pulse are Qse ) 160 µC cm-2 at the surrounding electrolyte/electrode interface and Qde ) 74.3 µC cm-2 at the droplet/electrode interface. As expected, Qse > Qde, because of the higher dielectric constant and electrolyte concentration of the aqueous solution. The extra charges δQs and δQd accumulated under the pulse and the corresponding extra screened potential are each approximately a quarter of their value under Vb. The value of δQ given by eq A2 is an upper estimate, because the flux of charge toward the interfaces in fact decreases with time as a result of the gradual screening of the applied voltage. This effect is not included in the above estimations. All in all, changes in charge density at the three interfaces lead to an instantaneous driving force different from that in the absence of the pulse. In all calculations in this paper, the driving force at the end of the pulse is assumed to be equal to that acting on the CL under a bias voltage of Veff ) Vb + δU.

F0 kb

R˙(0) ) 0

(B1)

for infinitely slow droplet motion. As mentioned, the position in eq B1 is the zero-velocity limit of the CL radius. This position is usually overshot because of nonzero inertia and velocity, and the CL radius reaches a stationary position within the interval of metastable geometries given by [Re,b - F0/kb, Re,b + F0/kb]. Below, Rpin,b is considered as the initial radius of the CL, assumption which has no impact on the main predictions of the approximate solution. During the first pulse, 0 < t < τp, the equation of motion reads

mR¨ + ηR˙ ) kp(Re,p - R) - F0 sgn(R˙)

(A2)

where n is the concentration of the ion species in the electrolyte. It is assumed for simplicity that all ions have the same hydrodynamic radii in both media. For typical values of the system parameters: Vp ) 1 V, Vb ) 0.65 V, Vpzc ) 0.3 V, D ) 10-5 cm2 s-1, Lw ) 1 cm, τp ) 50 µs, T ) 295 K, ns ) 0.5 mol dm-3, nd ) 0.1 mol dm-3, and z ) 1, one obtains δQs ) 32.8 µC cm-2 and δQd ) 6.57 µC cm-2 for the surrounding/electrode and the droplet/electrode interfaces, respectively. Any charge δQ accumulated at an electrolyte/electrode interface leads to an extra potential drop δU

δU )

As a result, the CL radius decreases, R˙ < 0, and the droplet is pinned at

(B2)

Because the pulse is very short, the condition R˙ < 0 remains valid throughout its duration; the solution is

R(t) ) Rpin,p - (A1e-λ1t + A2e-λ2t)

(B3)

( 

(B4)

with

λ1,2 )

η 1( 2m

1-

4kpm η2

)

Rpin,p is analogous to Rpin,b in eq B1, and coefficients A1,2 are derived from the initial conditions in eq B1

( (

4kpm 1 A1,2 ) (Rpin,p - Rpin,b) 1 - 1 2 η2

) ) -1/2

(B5)

The closed-form solution becomes

R(t) ) Rpin,p + (Rpin,b - Rpin,p)f(t)

(B6)

with f(t) )

(

1 12

1



1-

) ( e-λ1t +

4kpm η2

1 1+ 2

1



1-

)

e-λ2t

4kpm η2

(B7)

The position of the CL at the end of the pulse R(τp) and its velocity R˙(τp) represent the initial conditions for the motion during the bias interval. At later times, t > τp, when only Vb is applied, the equation of motion is

mR¨ + ηR˙ ) kb(Re,b - R) - F0 sgn(R˙)

(B8)

Appendix B: A Simplified Analytical Solution A step-by-step analytical description of the droplet contraction under pulsed voltage is presented below. It is assumed that, at t ) 0,the droplet is brought from a zero-volt bias to a Vb * 0.

and the CL continues to contract in a decelerated motion. For the initial contracting motion R˙ < 0, the solution can be written as

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R(t) ) Rpin,b - (B1e-ω1(t-τ) + B2e-ω2(t-τ))

(B9)

( 

(B10)

with

η 1( 2m

ω1,2 )

1-

4kbm η2

)

The initial conditions for eq B9 ensure that the CL displacement and velocity are continuous at time τp, when the pulse voltage is switched off and only the bias voltage is applied:

R(τp) ) Rpin,p + (Rpin,b - Rpin,p)f(τp) R˙(τp) ) (Rpin,b - Rpin,p)f˙(τp)

(B11)

From eqs B9 and B11 it follows that

{

(

B1,2 ) (Rpin,p - Rpin,b) (f(τ) - 1) 1 -

)

ω 1 + ω2 2f˙(τ) ω1 - ω2 ω1 - ω2

}

(B12)

R(t) ) Rpin,b + (Rpin,b - Rpin,p)g(t)

(B13)

with

[

(

)

]

ω1 + ω2 2f˙(τp) 1 (f(τp) - 1) 1 e-ω1(t-τp) 2 ω1 - ω2 ω1 - ω2 ω1 + ω2 2f˙(τp) 1 + (f(τp) - 1) 1 + + e-ω2(t-τp) 2 ω1 - ω2 ω1 - ω2 (B14)

[

(

)

]

When the CL velocity reaches zero, the direction of the motion could change under the effect of the driving force, and the CL would start expanding. As soon as the velocity changes sign however, so does the friction term in the expression for Rpin,b. For realistic values of the static friction,19 an analysis of the equation of motion shows that, if the CL velocity reaches zero while within the interval [Re,b - F0/kb, Re,b + F0/kb], the droplet is trapped. This interpretation is supported by experimental data,8 where no oscillatory motion is visible. Let t* > τp denote the time at which the droplet stops after the first pulse at maintained bias and Rf,1 ) R(t*) its corresponding CL radius. An equation for t* is obtained from the condition R˙(t*) ) 0:

e(ω1-ω2)(t*-τp) ) -

Although the system is oscillatory in nature for the chosen values of parameters m, k, η, the droplet is effectively pinned at Rf,1. Were the CL to change its direction of motion, the following equations would be fulfilled immediately before and after t*:

ηR˙ ) k(b,p)(Re(b,p) - R) + F0 - mR¨ ηR˙ ) k(b,p)(Re(b,p) - R) - F0 - mR¨

and the solution becomes

g(t) )

Figure 7. Time dependence of the CL motion. Although typical values of the system parameters allow for oscillations, F0 ) 1.65 × 10-5 N, m ) 1.2 × 10-10 kg, η ) 0.3 × 10-5 kg s-1, R0 ) 1°, Vpzc ) 0.3 V, cd ) 0.1 M, cs ) 0.5 M, τp ) 5 × 10-5 s, the CL is effectively pinned at R(t ) t*) when the velocity first reaches zero.

(

Then, eq B13 can be rewritten as a definition for Rf,1:

Rf,1 ) Rpin,b + (Rpin,b - Rpin,p)g(t*)

(from eq 3 rewritten for the viscous term). The CL cannot continue its motion, since they cannot both be fulfilled. During the second pulse, the time dependence of R(t) is similar to that during the first pulse. By taking into account the conditions at the beginning of the second pulse, R ) R(t*) and R˙(t*) ) 0, one obtains

R(t) ) Rpin,p + (Rf,1 - Rpin,p)f(t) ) Rpin,p + (Rpin,b - Rpin,p)(1 + g(t*))f(t)

fi+1(t) f (1 + g(t*))fi(t),

The motion of the droplet CL during a one pulse at constant bias as described by eqs B6 and B13 is shown in Figure 7.

˙fi+1(t) f (1 + g(t*))f˙i(t) (B19)

under the assumption that the duration of motion after each applied pulse is independent of the pulse number. Then, the CL final position after the second pulse is

Rf,2 ) Rf,1 + (Rpin,b - Rpin,p)g(t*)(p(t*) - 1) ) Rpin,b + (Rpin,b - Rpin,p)p(t*)g(t*)

(B20) with

[ (

)

]

ω1 + ω2 2f˙(τp) 1 f(τp) 1 e-ω1(t-τp) 2 ω1 - ω2 ω1 - ω2 ω1 + ω2 2f˙(τp) 1 e-ω2(t-τp) + f(τp) 1 + + 2 ω1 - ω2 ω1 - ω2 (B21)

p(t) )

(B16)

(B18)

By comparing the expression in eq B18 to the displacement after the first pulse in eq B6, the following substitution can be inferred for each subsequent pulse i

)

B1ω1 B1ω1 1 ⇒ t* ) ln + τp B2ω2 ω1 - ω2 B2ω2 (B15)

˙ t f t*, - R < 0 ˙ t f t*, + R > 0 (B17)

[ (

)

]

Electrowetting Dynamics Facilitated by Pulsing

J. Phys. Chem. C, Vol. 114, No. 51, 2010 22565

It can be deduced that the position reached after n pulses is n

Rf,n ) Rpin,b + (Rpin,b - Rpin,p)g(t*)

∑ p(t*)i-1 i)1

(B22) and that the displacement of the CL caused by pulse i is

∆i ) (Rpin,b - Rpin,p)g(t*)p(t*)i-1

(B23)

The final radius of the CL after an infinite number of pulses is obtained by taking the limit n f ∞ in the geometric progression of eq B22

Rf,∞ ) Rpin,b + (Rpin,b - Rpin,p)

g(t*) 1 - p(t*)

(B24)

because it can be shown that 0 < p(t*) < 1 for relevant physical system parameters. An analogous set of expressions can be derived for the spreading motion of a droplet under the effect of positive pulses on top of a bias. In this case, the expressions obtained in eqs B6, B13, and B22-B24 remain valid, with the modified notations

Rpin,b ≡ R(0) ) Re,b -

F0 F0 Rpin,p ) Re,p kb kp

and eq B24 provides values for the droplet geometry during a complete bias and pulse cycle, as displayed in Figure 6. In this simplified recurrent scheme, two key approximations have been adopted. The time-interval between two consecutive pulses is long enough to allow the droplet to come to rest before the next pulse is applied. This approximation is easy to control and warrant experimentally. The closed-form solution comes at the price of another approximation: t* in eq B24 remains the same after each pulse. The exact numerical solution of eq 3 shows that t* decreases exponentially for each subsequent pulse. This is the reason for the difference between the two theoretical curves in Figure 6, visible from the second pulse on. The closedform solution however catches many of the important physical features that govern droplet motion in pulse-assisted electrowetting and does so in a more transparent fashion than numerical solutions can. Equation B24 predicts that the final geometry of the droplet after pulsing is between the equilibrium position under pulse and that under the bias voltage. The exact position is given by the interplay between the system parameters.

Acknowledgment. The authors thank Charles Monroe and Alice Sleightholme for important discussions. The work emerged from the project “Electrified liquid interfaces: structure, dynamics, functioning” supported by the Leverhulme Trust (UK), F/07058/P. T.B. acknowledges the fellowship of ETH, Zu¨rich that allowed him to spend a year at Imperial College, London. M.U. acknowledges the support by the Israel Science Foundation, Grant No. 1109/09. References and Notes (1) Shamai, R.; Andelman, D.; Berge, B.; Hayes, R. Soft Matter 2008, 4, 38–45. (2) Berge, B.; Peseux, J. Eur. Phys. J. E 2000, 3, 159–163. (3) Hendriks, B. H. W.; Kuiper, S.; Van As, M. A. J.; Renders, C. A.; Tukker, T. W. Opt. ReV. 2005, 12, 255–259. (4) Dong, L.; Agarwal, A. K.; Beebe, D. J.; Jiang, H. Nature 2006, 442, 551–554. (5) Cho, S. K.; Moon, H. BioChip J. 2008, 2, 79–96. (6) Hayes, R. A.; Feenstra, B. J. Nature 2003, 425, 383–385. (7) Monroe, C. W.; Daikhin, L. I.; Urbakh, M.; Kornyshev, A. A. Phys. ReV. Lett. 2006, 97, 136102. (8) Kornyshev, A. A.; Kucernak, A. R.; Marinescu, M.; Monroe, C. W.; Sleightholme, A. E. S.; Urbakh, M. J. Phys. Chem. B 2010, 114, 1488514890. (9) Girault, H. H. J.; Schiffrin, D. J. Electroanal. Chem. 1989, 15, 1– 141. (10) Monroe, C. W.; Urbakh, M.; Kornyshev, A. A. J. Phys. Condens. Mat. 2007, 19, 375113. (11) Monroe, C. W.; Urbakh, M.; Kornyshev, A. A. J. Electrochem. Soc. 2009, 156, P21–P28. (12) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077– 2080. (13) Daniel, S.; Chaudhury, M. K.; de Gennes, P. G. Langmuir 2005, 21, 4240–4248. (14) Monroe, C.; Kornyshev, A. A.; Kucernak, A.; Sleightholme, A.; Urbakh, M. US Patent US 2008/0283414, 2008. (15) Li, F.; Mugele, F. Appl. Phys. Lett. 2008, 92, 244108. (16) Mugele, F. Soft Matter 2009, 5, 3377–3384. (17) Buguin, A.; Brochard, F.; de Gennes, P. G. Eur. Phys. J. E 2006, 19, 31–36. (18) Mueser, M. H.; Urbakh, M.; Robbins, M. O. AdV. Chem. Phys. 2003, 126, 187–272. (19) Tadmor, R.; Bahadur, P.; Leh, A.; N’guessan, H. E.; Jaini, R.; Dang, L. Phys. ReV. Lett. 2009, 103, 266101. (20) Bowden, F. P.; Tabor, D. The friction and lubrication of solids; Oxford University Press: Oxford, 2001; p 424. (21) Gao, J. P.; Luedtke, W. D.; Gourdon, D.; Ruths, M.; Israelachvili, J. N.; Landman, U. J. Phys. Chem. B 2004, 108, 3410–3425. (22) Tadmor, R. Surf. Sci. 2008, 602, L108–L111. (23) MATLAB, version 7.10.0.499(R2010a); The MathWorks Inc.: Natick, MA, 2010. (24) Dormand, J. R.; Prince, P. J. J. Comput. Appl. Math. 1980, 6, 19– 26. (25) Klein, J. Phys. ReV. Lett. 2007, 98, 056101. (26) Hamelin, A.; Stoicoviciu, L. J. Electroanal. Chem. 1987, 236, 267– 281. (27) Kornyshev, A. A.; Vilfan, I. Electrochim. Acta 1995, 40, 109–127. (28) Bazant, M. Z.; Kilic, M. S.; Storey, B. D.; Ajdari, A. New J. Phys. 2009, 11, 075016.

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