Article pubs.acs.org/Langmuir
Fast Electrically Driven Capillary Rise Using Overdrive Voltage Sung Jin Hong,§,† Jiwoo Hong,§,‡ Hee Won Seo,† Sang Joon Lee,‡ and Sang Kug Chung*,† †
Department of Mechanical Engineering, Myongji University, Yongin, Gyeonggido 17058, South Korea Department of Mechanical Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja-dong, Pohang 790-784, South Korea
‡
ABSTRACT: Enhancement of response speed (or reduction of response time) is crucial for the commercialization of devices based on electrowetting (EW), such as liquid lenses and reflective displays, and presents one of the main challenges in EW research studies. We demonstrate here that an overdrive EW actuation gives rise to a faster rise of a liquid column between parallel electrodes, compared to a DC EW actuation. Here, DC actuation is actually a simple applied step function, and overdrive is an applied step followed by reduction to a lower voltage. Transient behaviors and response time (i.e., the time required to reach the equilibrium height) of the rising liquid column are explored under different DC and overdrive EW actuations. When the liquid column rises up to a target height by means of an overdrive EW, the response time is reduced to as low as 1/6 of the response time using DC EW. We develop a theoretical model to simulate the EW-driven capillary rise by combining the kinetic equation of capillary flow (i.e., Lucas−Washburn equation) and the dynamic contact angle model considering contact line friction, contact angle hysteresis, contact angle saturation, and the EW effect. This theoretical model accurately predicts the outcome to within a ± 5% error in regard to the rising behaviors of the liquid column with a low viscosity, under both DC EW and overdrive actuation conditions, except for the early stage (140 V). The deviations between the experimental results and the theoretical prediction for low voltages are attributed to the CAH. The CAH arises from the chemical and topographical heterogeneity of the solid surface. It manifests as pinning force and consequently hinders the displacement of liquid on the solid surface.29−31,40 However, the deviations at high voltages (>140 V) are originated from the CAS. To explore the effect of the applied voltage on the dynamic behavior of the capillary rise between the parallel plates, the temporal evolution of the height of the liquid rise is measured under different DC actuations (Figure 4). The liquid columns rose monotonically up to the equilibrium height for applied DC voltages in the range of 80−120 V. This response is called an overdamped response. However, the liquid column rose to the maximum height and then fell to the equilibrium height at DC voltage of 140 V. This response is called an underdamped response. This system can be generally simplified into a dynamic model of the second-order mass-damper-spring 13720
DOI: 10.1021/acs.langmuir.5b02921 Langmuir 2015, 31, 13718−13724
Article
Langmuir Table 1. Time to Reach the Target Height As Function of Applied Voltage time to reach the target height (ms) applied voltage (V)
target height of 2.23 mm (i.e., equilibrium height corresponding to DC voltage of 80 V)
target height of 3.76 mm (i.e., equilibrium height corresponding to DC voltage of 100 V)
target height of 5.84 mm (i.e., equilibrium height corresponding to DC voltage of 120 V)
80 100 120 140
160 31.7 26.0 23.2
− 63.5 34.4 27.7
− − 74.5 35.9
Figure 5. Temporal evolution of the height of liquid column under different overdrive EW actuations at the target voltages of (a) 80 V and (b) 100 V. The bottom illustrates the waveform of applied voltage.
4.5 times and 5.8 times faster than the response time for the DC EW actuation condition at the same target voltage. Similarly, the overdrive method was employed at the target voltage (100 V) at two different overdrive conditions (120 V for 34 ms and 140 V for 27 ms). A comparison of response times of the heights of water columns under the DC and overdrive EW actuation conditions is summarized in Figure 6. In DC or low-frequency AC EW actuations, the dielectric layers covering electrodes prevent the passage of the Ohmic current and the entire voltage drop occurs in them.32−34 Accordingly, the power consumption (P) of EW actuations can be approximately estimated by the following relation: P ≈ (1/ 2)(εoεd/sδ)V2t. Here, εo is the vacuum dielectric permittivity, εd is the dielectric constant, V is half of the time-averaged applied voltage Va, δ is the thickness of the insulator, and t is the time duration of applied voltage. We summarized a comparison of the power consumption under the DC and overdrive EW actuation conditions in Table 2. As illustrated in Table 2, there is no difference between the power consumptions under these actuation conditions. This may be because that an overdrive voltage is applied during a short duration of time. A theoretical model for the prediction of the EW-driven capillary rise is required in order to find the optimal conditions of the overdrive signal when the physical properties of the liquid (e.g., density, viscosity, and surface tension) and device size are varied. Additionally, it may guide us in the right direction for explaining EW-driven rise dynamics of liquid columns. For this, we modified Chen and Hsieh’s model18 considering CAH and CAS. Chen and Hsieh developed a theoretical model to simulate the EW-driven capillary rise by
Figure 6. Comparison of the response time of the height of the water column under DC and overdrive EW actuations (OA): (I) 100 V → 80 V, (II) 120 V → 80 V, (III) 140 V → 80 V, (IV) 120 V → 100 V, and (V) 140 V → 100 V. The experimental error is typically within ±5% of the indicated value. A table provides the numeric values used in bar chart.
combining the kinetic equation of capillary flow (i.e., the Lucas−Washburn equation),19,20 and the dynamic contact angle model considering the CLF21 and the EW effect.22 Herein, we briefly explain their model but additional details can be found in Chen and Hsieh’s paper.18 On the basis of the force balance between the inertial force, surface forceconsidering 13721
DOI: 10.1021/acs.langmuir.5b02921 Langmuir 2015, 31, 13718−13724
Article
Langmuir ⎛ ⎞ cos θ − cos θo η = L⎜ ⎟ cos θs − cos θo ⎝ (cos θs − cos θo) ⎠
Table 2. Comparison of the Power Consumption of the Height of the Water Column under DC and Overdrive EW Actuations (OA): (I) 100 V → 80 V, (II) 120 V → 80 V, (III) 140 V → 80 V, (IV) 120 V→ 100 V, and (V) 140 V → 100 V EW actuation
P (mW)
EW actuation
P (mW)
80 V OA (I) OA (II) OA (III)
1.1 1.2 1.3 1.4
100 V OA (IV) OA (V)
1.8 1.9 2.0
where θs is the saturation angle, and L is the Langevin function L(X) = coth(3X)−1/3X. Note that only this law can estimate the tendency of CAS by introducing a proper mathematical function but cannot explain the origin of the saturation phenomenon. To predict temporal evolution of the height-of-rise of the water column under different DC EW actuations and overdrive EW actuations, we used the values ρ = 998 kg/m3, γ = 7.0 × 10−2 N/m, d = 1 × 10−3 m, μ = 1 × 10−3 Pa·s, θo = 108° ± 2°, εd = 3.4, εo = 8.854 × 10−12 F/m, δ = 2.0 μm, g = 9.81 m/s2, θs = 77° ± 3°. The cpin can be determined from the CAH, which is given by cpin= γ (θr − θa)/2. Here, θr and θa are 103° ± 3° and 113° ± 3°, respectively. The numerical analysis is performed by using values of these properties and adjusting value of ζ. This theoretical model accurately predicts the outcome to within a ± 5% error in regard to the rising behaviors of the liquid column with a low viscosity, under different DC EW actuation conditions, except for the early stage (
