1. Introduction
A sessile droplet on a flat dielectric-coated electrode wets the substrate more when a voltage is applied between the droplet and the electrode. The so-called electrowetting-on-dielectric (EWOD) phenomenon has been rapidly explored in recent years in both fundamental research and industrial applications. Spreading of droplets can be well controlled using the EWOD effect, making it an ideal tool to study contact-line dynamics including wetting, dewetting (Hong et al. Reference Hong, Kim, Kang, Kim and Lee2014; Vo & Tran Reference Vo and Tran2018), contact angle hysteresis (Nelson, Sen & Kim Reference Nelson, Sen and Kim2011; Sawane et al. Reference Sawane, Datar, Ogale and Banpurkar2015), droplet and soft-surfaces interactions (Dey et al. Reference Dey, Van Gorcum, Mugele and Snoeijer2019), as well as coalescence-induced jumping droplets (Boreyko & Chen Reference Boreyko and Chen2009; Farokhirad, Morris & Lee Reference Farokhirad, Morris and Lee2015; Vahabi et al. Reference Vahabi, Wang, Mabry and Kota2018). Moreover, EWOD has been emerging as a powerful technique in various industrial applications such as droplet manipulation (Fair et al. Reference Fair, Pollack, Woo, Pamula, Hong, Zhang and Venkatraman2001; Fair Reference Fair2007; Pollack, Fair & Shenderov Reference Pollack, Fair and Shenderov2000), optical imaging systems (Berge & Peseux Reference Berge and Peseux2000; Kuiper & Hendriks Reference Kuiper and Hendriks2004; Hao et al. Reference Hao, Liu, Chen, He, Li, Li and Wang2014; Lee, Park & Chung Reference Lee, Park and Chung2019), liquid deposition (Baret & Brinkmann Reference Baret and Brinkmann2006; Leïchlé, Tanguy & Nicu Reference Leïchlé, Tanguy and Nicu2007) and energy harvesting systems (Moon et al. Reference Moon, Jeong, Lee and Pak2013; Xu et al. Reference Xu2020). An understanding of the droplet–substrate interactions under the electrowetting effect also helps in the design and optimisation of advanced surfaces such as anti-icing (Mishchenko et al. Reference Mishchenko, Hatton, Bahadur, Taylor, Krupenkin and Aizenberg2010) and self-cleaning surfaces (Blossey Reference Blossey2003).
Among applications utilising the electrowetting effect, induction of droplet detachment from a solid surface (Lee et al. Reference Lee, Hong, Kang, Kang and Lee2014; Vo & Tran Reference Vo and Tran2019; Wang et al. Reference Wang, Xu, Wang, Gu, Hu, Lyu and Yao2020; Weng et al. Reference Weng, Wang, Gu, Li, Wang and Yao2021; Xiao & Wu Reference Xiao and Wu2021) is not only a topic of fundamental interests, but also a versatile tool in enabling manipulation of droplets in three-dimensional settings (Hong et al. Reference Hong, Kim, Won, Kim and Lee2015; He et al. Reference He, Zhang, Yang, Zhang and Deng2021). The principle of this technique relies on conversion of electrical energy to droplet surface energy by overstretching the droplet when the electrowetting effect is activated. Subsequently, when the electrowetting effect is deactivated, the excess surface energy of the overstretched droplet converts to kinetic energy inducing droplet retraction and detachment (Vo & Tran Reference Vo and Tran2019). The critical condition of droplet detachment using electrowetting actuation is that the excess surface energy of the overstretched droplet overcomes the sum of viscous dissipation and elastic energy of the contact line during droplet retraction (Vo & Tran Reference Vo and Tran2019).
In practice, in addition to the applied voltage used to control the strength of the electrowetting effect, the activating duration of the voltage 
$T_{p}$ (see figure 1a, top panel) is also an important parameter determining jumping behaviours of the actuated droplets. Two major approaches to modulate 
$T_{p}$ to induce droplet jumping by electrowetting are actuation from equilibrated state (AES) and actuation from maximum deformation state (AMS). In the AES approach (figure 1a, middle panel), the activating duration 
$T_{p}$ is held until the droplet reaches its new equilibrium. In other words, 
$T_{p}$ is set as 
$\geq \tau _{e}$, where 
$\tau _{e}$ is the time required for the droplet to reach the new equilibrium after the electrowetting effect is activated (Cavalli et al. Reference Cavalli, Preston, Tio, Martin, Miljkovic, Wang, Blanchette and Bush2016; Vo & Tran Reference Vo and Tran2019; Wang et al. Reference Wang, Xu, Wang, Gu, Hu, Lyu and Yao2020). As 
$\tau _{e}$ is well defined using the system's parameters (Vo, Su & Tran Reference Vo, Su and Tran2018), the AES approach is simple and well controlled. However, the ability to induce droplet jumping using the AES method is limited by contact angle saturation, i.e. the electrowetting effect saturates at high applied voltage (Mugele & Baret Reference Mugele and Baret2005) resulting in a saturation of the excess surface energy supporting droplet detachment. In contrast, in the AMS approach (figure 1a, bottom panel), 
$T_{p}$ is set equal to 
$\tau _{m}$, i.e. the time for the droplet to reach maximum deformation state after the electrowetting effect is activated (Lee, Lee & Kang Reference Lee, Lee and Kang2012; Lee et al. Reference Lee, Hong, Kang, Kang and Lee2014). As the excess surface energy at the maximum deformation state is higher than that at the equilibrated state, the AMS method is a more effective approach compared to that using AES. For instance, for the same actuating system, the critical voltage for droplet detachment to occur from AMS is lower than that from AES (Lee et al. Reference Lee, Hong, Kang, Kang and Lee2014; Wang et al. Reference Wang, van den Ende, Pit, Lagraauw, Wijnperlé and Mugele2017). As a result, AMS is a more favourable method in practical applications (Hong & Lee Reference Hong and Lee2015; Hong et al. Reference Hong, Kim, Won, Kim and Lee2015). Nevertheless, there is yet to be a parametric study on the dynamical behaviours of droplet detachment using AMS. Moreover, the critical conditions for droplet detachment using AMS is not yet established. This tremendously limits the applicability of the AMS method.
Figure 1. (a) Top panel, schematic illustrating modulation of the applied voltage. The duration of the voltage pulse is 
$T_p$ and the time reference (
$t = 0$) is set at the end of the pulse. Middle panel, schematic illustrating actuation from equilibrated state (AES). The voltage is turned on and maintained until the droplet reaches its equilibrated state; subsequently the voltage is turned off, causing droplet retraction. Bottom panel, schematic illustrating actuation from maximum deformation state (AMS). The voltage is turned on, causing a droplet to spread to its maximum deformation state, and is then turned off exactly at this moment. (b) Plot showing the time to reach maximum deformation, 
$\tau _{m}$ versus 
$U$. (c) Plot showing 
$\tau _{m}$ versus the underdamped characteristic spreading time 
$\tau = {\rm \pi}(\rho r_0^3/\eta \sigma )^{1/2}$. The solid line represents the relation 
$\tau _{m} = \tau$. (d) Snapshots showing the behavioural change of a water droplet with radius 0.5 mm using AMS when 
$U$ is increased. The scale bars represent 0.5 mm. (e) Plot showing the dependence of 
$h_{m} - r_0$ on 
$U$ for droplets jumping from AMS (blue circles) and AES (red squares). Here, the droplet has radius 
$r_0 = 0.5$ mm and viscosity 
$\mu = 1.0\ {\rm mPa}\ {\rm s}$; 
$U_{s} = 120$ V; the experiment was done in 2 cSt silicone oil.
In this paper, we systematically investigate detachment behaviours of droplets under electrowetting actuation using the AMS method by varying three control parameters: applied voltage 
$U$, droplet viscosity 
$\mu$ and droplet radius 
$r_0$. We first confirm that the AMS method is a more effective approach to induce droplet detachment compared to that using AES shown by both a lower critical detachment voltage and higher maximum jumping height. We then construct a comprehensive phase diagram separating detachable and non-detachable behaviours of droplets under AMS varying all the three control parameters. The dependence of the detach velocity and detach time on the control parameters are also examined in detail. Finally, we theoretically develop and experimentally verify a model describing the critical condition for droplet detachment using the AMS approach.
2. Experimental method
2.1. Experimental set-up and materials
To induce the electrowetting effect, we use a substrate made from an indium-tin-oxide (ITO) glass slide spin-coated with a layer of fluoropolymer (Teflon-1601, DuPont) (Vo & Tran Reference Vo and Tran2021a). We set the thickness of the Teflon layer to 
$d = 2.5\ {\mathrm {\mu }} {\rm m}$ to ensure electrical insulation for the ITO electrode. To apply a voltage between a droplet deposited on the substrate and the ITO electrode, we use an 
$18\ \mathrm {\mu }{\rm m}$ diameter tungsten wire dipped into the droplet bulk and connect it to the positive terminal of a direct current (DC) power supply via a normally open solid state relay (SSR) (see figure 1a). The negative terminal of the power supply is connected to the ITO electrode. We use a function generator to close the SSR, thereby applying a voltage 
$U$ in the range 
$60\ {\rm V} \leq U \leq 200\ {\rm V}$ between the electrodes for a controlled duration 
$T_{p}$ (figure 1a) to induce the electrowetting effect and droplet jumping. For our electrowetting substrates, the voltage causing contact angle saturation is 
$U_{s} \approx 110 \pm 10$ V (table 1).
Table 1. Measured values of viscosity 
$\mu$, density 
$\rho$ of glycerin solutions, and interfacial tensions 
$\sigma$ between glycerin solutions and 2 cSt silicone oil. The CAS voltage 
$U_{s}$ is determined experimentally by examining the saturation of the equilibrated contact angle 
$\theta _{e}$ when varying 
$U$ (Vo & Tran Reference Vo and Tran2019).
We use aqueous glycerin solutions consisting of glycerol, DI water and 0.125 M sodium chloride to generate droplets. The electrical conductivity of the working liquid is measure experimentally at 
${\approx }8.8 \times 10^{-4}\ {\rm S}\ {\rm m}^{-1}$. In our analysis and similar to other studies of electrowetting (Mugele & Baret Reference Mugele and Baret2005; Baret & Brinkmann Reference Baret and Brinkmann2006), we consider liquid droplets as perfectly conductive and neglect the minute effect of varying liquid permittivity. The viscosity 
$\mu$ of glycerin solutions is varied from 
$1.0\ {\rm mPa}\ {\rm s}$ to 
$68.7\ {\rm mPa}\ {\rm s}$ by adjusting the glycerol concentration (table 1). The droplet radius 
$r_0$ is also varied between 
$80 {\mathrm {\mu }}{\rm m}$ and 
$1.5\ {\rm mm}$. We immerse the substrate in a pool of silicone oil; the oil's temperature is kept at 
$20 \pm 0.5\,^{\circ }{\rm C}$ to maintain consistent experimental conditions. The use of silicone oil as the outer phase in our experiment is not only to reduce contact angle hysteresis, but also to increase initial contact angle of liquid droplets on the substrates (Baret & Brinkmann Reference Baret and Brinkmann2006; Hong & Lee Reference Hong and Lee2015). The contact angle of glycerin solution droplets deposited on the substrate in the silicone oils is 
$\theta _{0} = 160^{\circ }$ in the absence of the electrowetting effect. For simplicity, the viscosity of the oil is kept fixed at 
$\mu _{o} = 1.8~{\rm mPa}~{\rm s}$. Other properties of the working liquids, including density 
$\rho$ and interfacial tension 
$\sigma$, are measured experimentally and given in table 1.
To capture the behaviours of droplets under electrowetting actuation, we use a high-speed camera (Photron, SAX2), typically running at 5000 frames per second (FPS). The recorded images are processed using MATLAB to extract the contact radius 
$r$ and dynamic contact angle 
$\theta _{t}$ during actuation, as well as the jumping height 
$h$ of droplets after actuation. The measurements of 
$r$ and 
$\theta _{t}$, as well as the uncertainty analysis, follow the same experimental procedure described in our previous study (Vo & Tran Reference Vo and Tran2019). The uncertainty of the contact angle measurements is estimated within 
$2.5^\circ$. For each set of the control parameters (
$r_0, \mu, U$), the experiment is repeated three times.
2.2. Electrowetting actuation
To induce jumping of droplets using electrowetting actuation, we note that the activating duration 
$T_{p}$ plays a crucial role. Experimental studies have pointed to the time to reach maximum deformation 
$\tau _{m}$ as the most optimal time duration for jumping droplets, i.e. causing highest jumping (Wang et al. Reference Wang, van den Ende, Pit, Lagraauw, Wijnperlé and Mugele2017). We also note that the dynamics of droplets actuated by the electrowetting effect is either underdamped or overdamped, i.e. the electrowetting-induced driving force is opposed dominantly by either the droplet inertia or contact-line friction, respectively (Vo et al. Reference Vo, Su and Tran2018). Each one of these behaviours are characterised by a distinct spreading time scale. As it was shown that electrowetting-induced droplet detachment from solid substrates is only possible for spreading in the underdamped regime, we set the activating duration 
$T_{p}$ equal to the underdamped characteristic spreading time 
$\tau = {\rm \pi}(\rho r_0^3/\eta \sigma )^{1/2}$ (Vo et al. Reference Vo, Su and Tran2018; Wang et al. Reference Wang, Xu, Wang, Gu, Hu, Lyu and Yao2020; Xiao & Wu Reference Xiao and Wu2021):
\begin{equation} T_{p} = {\rm \pi}\left( \frac{\rho r_0^3}{\eta \sigma}\right)^{1/2}, \end{equation}
where the electrowetting number 
$\eta$, also known as the electrical capillary number (Fallah & Fattahi Reference Fallah and Fattahi2022; Hassan et al. Reference Hassan, Khalil, Khan, Moon, Cho and Byun2023), is defined as 
$\eta = \epsilon \epsilon _0 U^2/(2 d \sigma )$ for 
$U \leq U_{s}$ and 
$\eta = \epsilon \epsilon _0 U_{s}^2/(2 d \sigma )$ for 
$U > U_{s}$. Here, 
$\epsilon$ and 
$d$ respectively are the dielectric constant and thickness of the Teflon coating, 
$\epsilon _0$ the permittivity of free space, 
$\rho$ the droplet density, 
$\sigma$ the droplet–oil interfacial tension and 
$U_{s} \approx 110 \pm 10$ V the threshold voltage above which contact angle saturation (CAS) occurs. The threshold voltage 
$U_{s}$ is determined experimentally by examining the dependence of the equilibrated contact angle 
$\theta _{e}$ on the applied voltage 
$U$ (Vo & Tran Reference Vo and Tran2021b). Slight fluctuation of 
$U_{s}$ is within 
${\pm }10$ V and occurs when varying the liquid's viscosity (see table 1). We note that (2.1) implies that 
$T_{p}$ also saturates when 
$U > U_{s}$ due to the contact angle saturation effect, consistent with the experimental data shown in figure 1(b). In figure 1(c), we show an excellent agreement between the measured values of the time to reach maximum deformation and the characteristic spreading time 
$\tau$ of droplets actuation in the underdamped regime. This strongly suggests that 
$\tau$ can be used to describe the time to reach maximum deformation of droplets under electrowetting actuation. As a result, the activating duration 
$T_{p}$ in our experiment is determined by (2.1), i.e. dependent only on the experimental parameters and free of uncertainty from the experimental values of 
$\tau _{m}$.
3. Results and discussions
3.1. Droplet jumping by EWOD actuation from maximum deformation state
In figure 1(d), we show several series of snapshots of actuated droplets to illustrate their spreading and jumping dynamics. From the top panel to the bottom one, the applied voltage 
$U$ is increased from 80 V to 140 V, while the droplet radius and viscosity are fixed at 
$r_0 = 0.5\ {\rm mm}$ and 
$\mu = 1.0\ {\rm mPa}\ {\rm s}$, respectively. Generally, we observe that as soon as the electrowetting effect is activated on a droplet, it causes droplet spreading with an initial contact-line velocity 
$v_{I}$ and generates on the droplet's surface capillary waves propagating from the contact line towards the apex. At the end of the activating duration (
$t = 0$), the droplet immediately recoils and subsequently jumps off from the substrate if the applied voltage is sufficiently high. For instance, in the experiment shown in figure 1(d), droplet detachment from the solid substrate occurs for 
$U \geq 90$ V. We also observe that at high applied voltages, e.g. 
$U = 140$ V (figure 1d, last panel), a small satellite droplet is ejected from the actuated droplet. This is due to the effect of the strong capillary waves on the droplet's surface generated by the electrowetting effect (Vo & Tran Reference Vo and Tran2021b).
In the cases where an actuated droplet detaches from the substrate, the detach time 
$T_{d}$, measured from the time reference 
$t = 0$ to the moment the droplet detaches, reduces with the applied voltage. For instance, 
$T_{d}$ drops from 
$6.3$ ms to 
$5.8$ ms when 
$U$ increases from 120 V to 140 V (figure 1d). Moreover, the maximum jumping height 
$h_{m}$ defined as the maximum height of the droplet's centre of mass increases with 
$U$. For instance, 
$h_{m} - r_0$ significantly increases from 
$0.38$ mm to 
$1.55$ mm when 
$U$ increases from 100 V to 140 V (figure 1d). Comparing the maximum jumping height obtained in our experiment with that obtained in the case where jumping is induced by AES (figure 1e), we observe that for the same voltage, 
$h_{m}$ obtained from AMS (blue circles) is consistently higher than that from AES (red squares). Moreover, the maximum jumping height obtained from AMS is not limited by contact angle saturation, in contrast to that from AES.
In figure 2, we show the phase diagrams of droplet behaviours obtained by varying three of the control parameters 
$r_0$, 
$\mu$ and 
$U$. These observed behaviours, which are illustrated in figure 1(d), include non-detachable, detachable without splitting and detachable with splitting. The critical voltage at the transition for detachment is higher for smaller droplet size (figure 2a) or higher viscosity (figure 2b). We also observe that the detachment of droplet is limited for 
$\mu < 17.6\ {\rm mPa}\ {\rm s}$ as the transition from the underdamped regime to overdamped regime occurs at 
$17.6\ {\rm mPa}\ {\rm s}$ (figure 2b) (Vo et al. Reference Vo, Su and Tran2018). We highlight that the AMS method works for 
$r_{0}$ as small as 
$80\ \mathrm {\mu }{\rm m}$, a limit that was not possible using AES for the same substrate (Vo & Tran Reference Vo and Tran2019).
Figure 2. (a) Phase diagram showing different behaviours of 
$\mu = 1.0\ {\rm mPa}\ {\rm s}$ droplets under AMS with varying 
$U$ and 
$r_0$. (b) Phase diagram showing different behaviours of 
$r_0 = 0.5$ mm droplets under AMS with varying 
$U$ and 
$\mu$. In both diagrams, droplet behaviours are categorised into three major regimes: non-detachable (black squares), detachable without splitting (red circles) and detachable with splitting (blue triangles). The dash lines are used to guide the eyes along the boundary between the detachable and the non-detachable regimes. The dash-dotted lines indicate the average contact angle saturation (CAS) threshold (see table 1 for specific values of 
$U_{s}$ at different 
$\mu$).
3.2. Detach velocity and detach time
In the case where a droplet detaches from the solid substrate, the detach velocity 
$u_{d}$, defined as the droplet's centre-of-mass velocity at the moment it completely separates from the substrate, and the detach time 
$T_{d}$, i.e. the duration from the reference time 
$t= 0$ to the separating moment, are the most critical parameters representing the detaching dynamics. Understanding of these characteristic parameters help optimise design and operations of droplet-actuating systems using the electrowetting effect. In this section, we discuss the dependencies of 
$u_{d}$ and 
$T_{d}$ on the applied voltage 
$U$, viscosity 
$\mu$ and droplet radius 
$r_{0}$.
We first focus on the detach velocity 
$u_{d}$. In figure 3(a), we show the dependence of 
$u_{d}$ on the applied voltage 
$U$ for 
$r_{0}$ varying from 0.08 mm to 1.25 mm and 
$\mu$ fixed at 
$1~{\rm mPa}~{\rm s}$. Generally, we observe that 
$u_{d}$ increases with 
$U$ as a result of higher electrical energy applied to the system. However, the dependence of 
$u_{d}$ on 
$U$ becomes more irregular at high applied voltage, i.e. 
$U > 130$ V. For example, for 
$r_0 = 0.08$ mm, 
$u_{d}$ varies little between 
$U = 140$ V and 
$U = 150$ V. We attribute such irregular dependence of 
$u_{d}$ on 
$U$ to hydrodynamical and electrical instabilities of the system at high voltage. When the applied voltage increases, the electrical force applied to the contact line becomes larger, causing more abrupt and forceful deformation to the droplet, e.g. stronger capillary waves and even splitting of the droplet (figure 1d). Such violent behaviours induce nonlinear effects that reduce the energy transfer efficiency, from electrical to kinetic energy of the detached droplet. Moreover, electrical leakage may be possible through the dielectric layer at high applied voltage without breaking it down during the experiment (Moon et al. Reference Moon, Cho, Garrell and Kim2002). This also reduces the efficiency of the EWOD effect in generating higher 
$u_{d}$ for jumping droplets. Therefore, reducing irregularities at high voltage may require increasing viscosities of the liquids or using insulators with higher dielectric strength. Next, in figure 3(b), we plot the dependence of 
$u_{d}$ on 
$U$ for 
$\mu$ varying from 
$1\ {\rm mPa}\ {\rm s}$ to 
$8.2~{\rm mPa}~{\rm s}$ and 
$r_{0}$ fixed at 
$0.5$ mm. We observe that for 
$U<130$ V, 
$u_{d}$ linearly increases with 
$U$, whereas for 
$U\ge 130$ V, 
$u_{d}$ approaches a plateau. The increasing rate of 
$u_{d}$ with 
$U$ also reduces with higher 
$\mu$ due to larger viscous dissipation.
Figure 3. (a,b) Plots showing detach velocity 
$u_{d}$ versus 
$U$ for (a) different droplet radii 
$r_0$ and (b) different droplet viscosity 
$\mu$. (c,d) Plots showing detach time 
$T_{d}$ versus 
$U$ for (c) different droplet radii 
$r_0$ and(d) different droplet viscosity 
$\mu$. (e) Plot showing detach time 
$T_{d}$ versus 
$\tau$ for various values of 
$r_0$, 
$U$ and 
$\mu$. Inset is a zoomed-in plot showing data in the dashed box. The solid line indicates the best fit to the experimental data using the linear relation 
$T_{d} = k\tau$, where 
$k = 1.18 \pm 0.07$. The shaded areas indicate the average contact angle saturation (CAS) threshold (see table 1 for specific values of 
$U_{s}$ for different 
$\mu$).
We now examine the detach time 
$T_{d}$. In figures 3(c) and 3(d), we respectively show 
$T_{d}$ versus 
$U$ for various droplet radii and droplet viscosities. We note that for 
$r_0 = 1.25$ mm and 
$r_0 = 1.5$ mm, the voltage is limited at 110 V and 100 V, respectively, due to frequent electrical breakdowns of the dielectric layer separating the electrode and the liquids. Here, electrical breakdowns increase with the actuation time, as well as the contact area between the droplet and the substrate during actuation. Both factors increase with larger droplet radius.
Typically, 
$T_{d}$ decreases with increasing 
$U$ as long as the voltage is within the contact angle saturation (CAS) limit, i.e. 
$U_{s} = 110\ {\rm V}$ in our case. The detach time 
$T_{d}$ eventually reaches a plateau when the applied voltage exceeds 
$U_{s}$. Here, we note that 
$T_{d}$ for AMS is defined in the same way as the droplet's retracting time, i.e. from the moment the electrowetting effect is turned off at the droplet's maximum deformation to the moment the droplet detaches from the surface. In addition, the retracting time of the droplet only depends on its maximum spreading diameter, which is limited by CAS. As a result, we infer that 
$T_{d}$ also saturates when 
$U \geq U_{s}$, consistent with our experimental results shown in figures 3(c) and 3(d).
The detach time 
$T_{d}$, which is measured from the moment the voltage is released to the detaching moment, is defined in the same way as the retracting time of the droplets from maximum deformation. As a result, we hypothesise that 
$T_{d}$ is closely related to the time scale characterising the spreading (or retracting) dynamics of droplets. As the spreading droplets in our study are underdamped, we examine the relation between 
$T_{d}$ and the underdamped characteristic spreading time scale 
$\tau$ and show a plot of 
$T_{d}$ versus 
$\tau$ in figure 3(e). The plot consists of data obtained by varying 
$r_0$, 
$\mu$ and 
$U$ in their explored ranges. We observe that the dependence of 
$T_{d}$ on 
$\tau$ can be approximately described using a linear relation 
$T_{d} \approx k \tau$, where 
$k = 1.18 \pm 0.07$, suggesting that 
$T_{d}$ is reasonably characterised by 
$\tau$ in the explored ranges of the control parameters. Furthermore, for each dataset obtained using fixed 
$\mu$ and 
$r_0$, we note that figures 3(c) and 3(d) indicate that 
$T_d$ decreases with 
$U$ for 
$U< U_{s}$ and plateaus for 
$U>U_{s}$. This implies smaller data deviation from the linear fit in figure 3(e) at higher voltages. In other words, the linear fit better describes the relation between 
$T_{d}$ and 
$\tau$ in the high-voltage regime.
3.3. Critical conditions for jumping droplets by EWOD actuation from maximum deformation state
We now search for the critical condition for droplet detachment by AMS. We first note that the droplet actuation dynamics in our study is kept in the underdamped regime, as it is the requirement to enable detachment (Vo & Tran Reference Vo and Tran2019). This requirement also ensures that the excess surface energy at the maximum deformation state is not entirely dissipated by the viscous effect during retraction. Subsequently, we follow the energy balance approach used to determine the jumping conditions for droplets actuated by AES (Vo & Tran Reference Vo and Tran2019) to formulate the critical condition for droplets to detach from the solid substrate by AMS as
\begin{equation} \Delta E_{s} \geq E_{v} + E_{cl}, \end{equation}
where 
$\Delta E_{s} = E_{s}^{m} - E_{s}^{d}$ is the difference between the surface energy at maximum deformation 
$E_{s}^{m}$ and the surface energy at the detach moment 
$E_{s}^{d}$; 
$E_{v}$ is the viscous dissipation and 
$E_{cl}$ contact line elasticity energy during retraction. We note that the kinetic energy of a droplet vanishes at its maximum deformation state. The gravitational potential energy is negligible compared with the surface energy as the Bond number 
${Bo} = (\rho - \rho _{o}) g r_0^2 \sigma ^{-1}$ is small (
$2.14 \times 10^{-4} \leq {Bo} \leq 1.23 \times 10^{-1}$). Here, 
$\rho _{o} = 873 \ {\rm kg}\ {\rm m}^{{-3}}$ is the density of silicone oil and 
$g = 9.81\ {\rm m}\ {\rm s}^{{-2}}$ is the gravitational acceleration.
We first focus on the surface energy difference 
$\Delta E_{s}$ and note that it can be written as 
$\Delta E_{s} = E_{s}^{m} - E_{s}^{e} +E_{s}^{e} - E_{s}^{d}$, where 
$E_{s}^{e}$ is the surface energy at equilibrated state after applying the electrowetting effect without turning it off. On one hand, we note that the surface energy difference 
$E_{s}^{e} - E_{s}^{d}$ between the equilibrated state and detachment was already formulated (Vo & Tran Reference Vo and Tran2019): 
$E_{s}^{e} - E_{s}^{d} = [2(1 + \cos \theta _{e})^{-1} - 4 (r_0/r_{e})^2 - \cos \theta _0] \sigma {\rm \pi}r_{e}^2$, where 
$\theta _{e}$, 
$r_{e}$ are respectively the contact angle and contact radius of the droplet at the equilibrated state; 
$\theta _0$ is the contact angle of the droplet at the initial state. Moreover, we argue that the surface energy difference 
$E_{s}^{m} - E_{s}^{e}$ comes from the capillary wave generated at the beginning of droplet actuation; this wave would not occur if the voltage 
$U$ were to ramp up slowly to keep the spreading quasi-static. As a result, the surface energy difference 
$E_{s}^{m} - E_{s}^{e}$ is calculated by the energy carried by the capillary wave 
$E_{s}^{m} - E_{s}^{e} = 0.5 \rho a^2 \omega ^2 l S$, where 
$a \sim r_0 {We}^{1/2} \sin \theta _0 (T_{p} \omega )^{2/3}/[2{\rm \pi} (1 - \xi ^2)^{1/2}]$ is the wave amplitude at time 
$t = T_{p}$; 
$\omega \sim (\sigma /\rho r_0^3)^{1/2}$ is the angular frequency of the wave; 
$l \sim r_0$ is the wavelength; 
$\xi = \lambda /(\sigma \rho r_0)^{1/2}$ is the decaying ratio of the capillary waves; 
$\lambda$ is the contact line friction coefficient; 
$S = 2(1 + \cos \theta _e)^{-1} {\rm \pi}r_e^2$ is the area of a hypothetical spherical cap having contact angle 
$\theta _{e}$ and base radius 
$r_{e}$; and 
${We}$ is the contact line Weber number, defined to directly relate to the applied voltage 
$U$ by the relation 
${We} = [(\epsilon \epsilon _0/2 d \sigma )^{1/2} (U - U_{c}) + 1]^2$. Here, 
$U_{c}$ is the critical voltage for capillary wave generation on the droplet's surface (Vo & Tran Reference Vo and Tran2021b). In our experiment, 
$U_{c}$ varies from 
$70$ V to 
$110$ V depending on the viscosity and radius of the actuated droplet. We therefore obtain the expression of the surface energy difference:
\begin{equation} \Delta E_{s} = \left[\frac{2}{1 + \cos \theta_e} - 4 \left(\frac{r_0}{r_e}\right)^2 - \cos \theta_0 + {We} \frac{ (T_{p} \omega)^{4/3}}{4{\rm \pi}^2(1 - \xi^2)} \frac{\sin^2 \theta_0}{1 + \cos \theta_{e}} \right] \sigma {\rm \pi}r^2_{e}. \end{equation} We note that the damping ratio 
$\xi$ is essentially similar to the Ohnesorge number Oh 
$=\mu (\sigma \rho r_0)^{-1/2}$ in which the contact line friction coefficient 
$\lambda$ is used instead of the liquid's viscosity 
$\mu$ to represent dissipation. In our analysis, 
$\lambda$ is obtained empirically using the relation 
$\lambda = C (\mu \mu _{o})^{1/2}$, where 
$C = 32.9$ is a fitting parameter (Vo et al. Reference Vo, Su and Tran2018; Vo & Tran Reference Vo and Tran2019). By using 
$\xi$ instead of Oh, the dissipation in the liquid bulk is effectively neglected (Carlson, Bellani & Amberg Reference Carlson, Bellani and Amberg2012a,Reference Carlson, Bellani and Ambergb; Vo & Tran Reference Vo and Tran2019). Indeed, the ratio 
$\xi /\mathrm {Oh}$, calculated for all of our experiments, varies from 
$5.47$ to 
$45.3$, strongly suggesting that the dissipation at the contact line is dominant. As a result, we ignore bulk dissipation in our estimation of 
$E_{s}^{m} - E_{s}^{e}$ to arrive at (3.2). Also, by only considering dissipation at the contact line, we obtain an expression for 
$E_{v}$ (Vo & Tran Reference Vo and Tran2019):
\begin{equation} E_{v} \sim \lambda \frac{r_{m}}{T_{d}} {\rm \pi}r^2_{m} \approx \lambda \frac{r_{e}}{T_{d}} {\rm \pi}r^2_{e}. \end{equation} The contact line elasticity energy 
$E_{cl}$ is the surface energy accumulation due to pinning and subsequent stretching of the liquid–oil interface at the vicinity of the contact line (Joanny & de Gennes Reference Joanny and de Gennes1984; Vo & Tran Reference Vo and Tran2019) and is determined by a similar approach to that of Vo & Tran (Reference Vo and Tran2019):
\begin{equation} E_{cl} \sim \kappa \sigma {\rm \pi}r^2_{m} \approx \kappa \sigma {\rm \pi}r^2_{e}. \end{equation}
Here, 
$\kappa = {\rm \pi}\sin ^2 \theta _{r}/ \ln (r_0/\gamma )$, where 
$\theta _{r}$ is the receding contact angle and 
$\gamma$ is the defect's size. In our experiment, which was conducted using the same set-up and materials as Vo & Tran (Reference Vo and Tran2019), 
$\theta _{r} \approx 121^{\circ }$ and the average defect's size is 
$\gamma \approx 260$ nm. The parameter 
$\kappa$ only changes minutely within the explored droplet radius: 
$\kappa = 0.27\pm 0.03$ for 
$0.08\ {\rm mm} \leq r_0 \leq 1.25\ {\rm mm}$.
Substituting (3.2), (3.3), (3.4) into (3.1), the condition at the transition between non-detachable and detachable behaviours is
\begin{equation} \frac{2}{1 + \cos \theta_{e}} - 4\left(\frac{r_{\rm 0}}{r_{e}}\right)^2 - \cos \theta_0 + {We} \frac{(T_{p} \omega)^{4/3}}{4{\rm \pi}^2(1 - \xi^2)} \frac{\sin^2 \theta_0}{1 + \cos \theta_{e}} = \frac{r_{e} \lambda}{T_{d} \sigma } + \kappa. \end{equation} As shown in the previous sections, both the activation time 
$T_{p}$ and the detach time 
$T_{d}$ are well approximated by the underdamped characteristic spreading time 
$\tau$. We therefore simplify (3.5) as
\begin{equation} \alpha + \varPsi (\theta_{e}, \xi) {We} = {\rm \pi}^{{-}1} \beta \eta^{1/2} \xi + \kappa, \end{equation}
where 
$\beta = r_{e}/r_{0} = (1-\cos \theta _{e}^2)^{1/2} [4 (1-\cos \theta _{e})^{-2} (2 + \cos \theta _{e})^{-1}]^{1/3}$, 
$\alpha = 2 (1 + \cos \theta _{e})^{-1} - 4 \beta ^{-2} - \cos \theta _0$, and 
$\varPsi (\theta _{e}, \xi ) = 0.25\sin ^2 \theta _0 (1 + \cos \theta _{e})^{-1} ({\rm \pi} \eta )^{-2/3} (1 - \xi ^2)^{-1}$. We note that 
$\theta _{e}$ depends on the electrowetting number 
$\eta$ following the Young–Lippmann equation 
$\cos \theta _{e} - \cos \theta _{0} = \eta$ (Mugele & Baret Reference Mugele and Baret2005).
In figure 4, we show a plot of 
$\alpha + \varPsi \,{We}$ versus 
${\rm \pi} ^{-1} \beta \eta ^{1/2} \xi$ of all the data obtained for the non-detachable and detachable behaviours shown in figure 2. Here, the composite terms 
$\alpha + \varPsi \,{We}$ and 
${\rm \pi} ^{-1} \beta \eta ^{1/2} \xi$ respectively represent the driving energy and the energy cost for detachment. The shaded area indicates the transitional extent caused by variation in 
$\kappa$ (
$\kappa = 0.27\pm 0.03$) resulting from variations in both the defect's size 
$\gamma$ and the radius 
$r_0$. We note that all the terms in (3.6) are calculated using the system parameters except the contact line friction coefficient, which is determined empirically using 
$\lambda = C (\mu \mu _{o})^{1/2}$ with 
$C = 32.9$ (Vo et al. Reference Vo, Su and Tran2018; Vo & Tran Reference Vo and Tran2019), and the receding contact angle 
${\theta _{r} \approx 121^{\circ }}$ determined independently from our previous work (Vo & Tran Reference Vo and Tran2019). The excellent agreement between the experimental data and the formulated jumping condition (3.6) thus highlights our analysis as a predictive tool for utilizing modulated electrowetting in droplet actuation and detachment from surfaces.
Figure 4. Plot showing 
$\alpha + \varPsi \,{We}$ versus 
${\rm \pi} ^{-1} \beta \eta ^{1/2} \xi$ using the data of the non-detachable and detachable behaviours shown in figure 2. The shaded area indicates the transitional extent due to variation in 
$\kappa$ (
$\kappa = 0.27\pm 0.03$).
4. Conclusions
We aim to systematically investigate droplet detachment induced by EWOD using actuation from the maximum deformation state (AMS). By varying droplet radius, viscosity and applied voltage, we demonstrate a significant expansion of the detachable regime for droplets actuated by using the electrowetting effect with AMS and provide a comprehensive phase diagram of the detachment behaviours. The applied voltage for AMS, no longer bounded by contact angle saturation limit, enables detachment of droplets as small as 
$80\ \mathrm {\mu }{\rm m}$ in radius. This introduces a powerful tool for applications requiring actuation and detachment of droplets from solid surfaces, in particular, those dealing with small droplets or highly viscous liquids. We then provide a detailed characterisation of detach velocity and detach time of actuated droplets. Finally, we develop a theoretical prediction for the critical condition causing droplet detachment. The theoretical prediction is consistent with our experimental data for liquid viscosity ranging from 1 mPa to 68.7 mPa, droplet size from 0.08 mm to 1.5 mm and applied voltage from 60 V to 200 V.
We note that our study is limited within the droplet-in-oil setting and further studies may be required to explore the limit of our analysis in the droplet-in-air setting, which is typically known for stronger hysteresis effect and electrical instability at the three-phase contact line. Nevertheless, our study may serve as a strong basis for wider use of electrowetting in applications requiring precise actuation of droplets such as tissue engineering, digital microfluidics and three-dimensional (3-D) printing. Our results also provide key insights to mechanistic understanding of related phenomena such as coalescence-induced jumping of droplets (Boreyko & Chen Reference Boreyko and Chen2009; Liu et al. Reference Liu, Ghigliotti, Feng and Chen2014; Farokhirad et al. Reference Farokhirad, Morris and Lee2015) or droplet bouncing on solid substrates (Sanjay, Chantelot & Lohse Reference Sanjay, Chantelot and Lohse2023).
Funding
This study is supported by Nanyang Technological University, the Republic of Singapore's Ministry of Education (MOE, grant number MOE2018-T2-2-113), and the RIE2020 Industry Alignment Fund–Industry Collaboration Projects (IAF–ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner, HP Inc.
Declaration of interests
The authors report no conflict of interest.
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