# Droplet jumping by modulated electrowetting

Quoc Vo1<EMAIL_ADDRESS>Tuan Tran2<EMAIL_ADDRESS>1School of Medicine,
University of Pittsburgh, 4200 Fifth Ave, Pittsburgh, PA 15260, USA. 2School
of Mechanical & Aerospace Engineering, Nanyang Technological University, 50
Nanyang Avenue, 639798, Singapore.

###### Abstract

We investigate jumping of sessile droplets from a solid surface in ambient oil
using modulated electrowetting actuation. We focus on the case in which the
electrowetting effect is activated to cause droplet spreading and then
deactivated exactly at the moment the droplet reaches its maximum deformation.
By systematically varying the control parameters such as the droplet radius,
liquid viscosity, and applied voltage, we provide detailed characterisation of
the resulting behaviours including a comprehensive phase diagram separating
detachment from non-detachment behaviours, as well as how the detach velocity
and detach time, i.e, duration leading to detachment, depend on the control
parameters. We then construct a theoretical model predicting the detachment
condition using energy conservation principles. We finally validate our
theoretical analysis by experimental data obtained in the explored ranges of
the control parameters.

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##  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., 2014; Vo & Tran, 2018), contact angle
hysteresis (Nelson et al., 2011; Sawane et al., 2015), droplet and soft-
surfaces interactions (Dey et al., 2019), as well as coalescence-induced
jumping droplets (Vahabi et al., 2018; Boreyko & Chen, 2009; Farokhirad et
al., 2015). Moreover, EWOD has been emerging as a powerful technique in
various industrial applications such as droplet manipulation (Fair et al.,
2001; Fair, 2007; Pollack et al., 2000), optical imaging systems (Berge &
Peseux, 2000; Hao et al., 2014; Kuiper & Hendriks, 2004; Lee et al., 2019),
liquid deposition (Baret & Brinkmann, 2006; Leïchlé et al., 2007), and energy
harvesting systems (Moon et al., 2013; Xu et al., 2020). Understanding of
droplet-substrate interactions under electrowetting effect also helps design
and optimisation of advanced surfaces such as anti-icing (Mishchenko et al.,
2010) and self-cleaning surfaces (Blossey, 2003).

Among applications utilising the electrowetting effect, induction of droplet
detachment from a solid surface (Vo & Tran, 2019; Lee et al., 2014; Wang et
al., 2020; Weng et al., 2021; Xiao & Wu, 2021) is not only a topic of
fundamental interests, but also a versatile tool in enabling manipulation of
droplets in three-dimensional settings (He et al., 2021; Hong et al., 2015).
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,
2019). 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, 2019).

In practice, beside the applied voltage used to control the strength of the
electrowetting effect, the activating duration of the voltage $T_{\rm p}$ (see
Fig. 1a, top panel) is also an important parameter determining jumping
behaviours of the actuated droplets. Two major approaches to modulate $T_{\rm
p}$ to induce droplet jumping by electrowetting are actuation from
equilibriated state (AES) and actuation from maximum deformation state (AMS).
In the AES approach (Fig. 1a, middle panel), the activating duration $T_{\rm
p}$ is held until the droplet reaches its new equilibrium. In other words,
$T_{\rm p}$ is set $\geq\uptau_{\rm e}$, where $\uptau_{\rm e}$ is the time
required for droplet to reach the new equilibrium after the electrowetting
effect is activated (Cavalli et al., 2016; Vo & Tran, 2019; Wang et al.,
2020). As $\uptau_{\rm e}$ is well-defined using system’s parameters (Vo et
al., 2018), 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, 2005) resulting in a saturation of the excess surface energy
supporting droplet detachment. In contrast, in the AMS approach (Fig. 1a,
bottom panel), $T_{\rm p}$ is set equal to $\uptau_{\rm m}$, i.e., the time
for droplet to reach maximum deformation state after the electrowetting effect
is activated (Lee et al., 2012, 2014). As the excess surface energy at the
maximum deformation state is higher than that at the equilibriated state, AMS
is a more effective approach compared to 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., 2014; Wang et al., 2017). As a
result, AMS is a more favourable method in practical applications (Hong et
al., 2015; Hong & Lee, 2015). Nevertheless, there is yet 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.

In this paper, we systematically investigate detachment behaviours of droplets
under electrowetting actuation using AMS method by varying three control
parameters: applied voltage $U$, droplet’s viscosity $\mu$, and droplet radius
$r_{0}$. We first confirm that AMS is a more effective approach to induce
droplet detachment compared to AES shown by both lower critical detachment
voltage and higher maximum jumping height. We then construct a comprehensive
phase diagram separating detachable and nondetachable behaviours of droplets
under AMS varying all the three control parameters. The dependence of detach
velocity and detach time on the control parameters are also examined in
details. Finally, we theoretically develop and experimentally verify a model
describing the critical condition for droplet detachment using AMS approach.

##  2 Experimental method

###  2.1 Experimental setup 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, 2021a). We set the thickness of the Teflon
layer at $d=2.5\,{\rm\upmu m}$ 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 $18\,\upmu$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 Fig. 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 $T_{\rm p}$ (Fig. 1a) to induce the electrowetting effect
and droplet jumping. For our electrowetting substrates, the voltage causing
contact angle saturation is $U_{\rm s}\approx 110\,\pm 10$V (Tab. 1).

%wt glycerol |  $\mu$ (mPa$\cdot$s) |  $\rho$ (${\rm kg.m}^{-3}$) |  $\sigma$ (${\rm mN.m}^{-1}$) |  $U_{\rm s}$ (V)
---|---|---|---|---
0 | 1.0 | 1000 | $37.2\pm 0.5$ | $\approx 120$
20 | 2.2 | 1053 | $32.4\pm 1.0$ | $\approx 115$
41.5 | 4.8 | 1106 | $30.4\pm 0.9$ | $\approx 110$
55 | 8.2 | 1141 | $29.7\pm 0.5$ | $\approx 110$
67 | 17.6 | 1174 | $26.9\pm 0.4$ | $\approx 105$
74 | 32.7 | 1192 | $25.9\pm 0.5$ | $\approx 100$
80 | 68.7 | 1210 | $26.8\pm 0.4$ | $\approx 105$
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 $U_{\rm s}$ is determined experimentally by
examining the saturation of the equilibrated contact angle $\theta_{\rm e}$
when varying $U$ (Vo & Tran, 2019).

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 $\approx 8.8\times 10^{-4}\,{\rm
S\,m^{-1}}$. In our analysis and similar to other studies of electrowetting
(Mugele & Baret, 2005; Baret & Brinkmann, 2006), we consider liquid droplets
perfectly conductive and neglect the minute effect of varying liquid
permittivity. The viscosity $\mu$ of glycerin solutions is varied from
$1.0\,{\rm mPas}$ to $68.7\,{\rm mPas}$ by adjusting the glycerol
concentration (Tab. 1). The droplet radius $r_{0}$ is also varied between
$80\,{\rm\mu m}$ 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, 2006; Hong & Lee, 2015). The contact angle of
glycerin solution droplets deposited on the substrate in the silicone oils is
$\theta_{\rm 0}=160^{\circ}$ in the absence of the electrowetting effect. For
simplicity, the viscosity of the oil is kept fixed at $\mu_{\rm o}=1.8\,{\rm
mPas}$. Other properties of the working liquids, including density $\rho$ and
interfacial tension $\sigma$, are measured experimentally and given in Tab. 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 $\theta_{\rm t}$ during actuation, as
well as the jumping height $h$ of droplets after actuation. The measurements
of $r$ and $\theta_{\rm t}$, as well as the uncertainty analysis, follow the
same experimental procedure described in our previous study (Vo & Tran, 2019).
The uncertainty of 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.

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, $\uptau_{\rm m}$ versus $U$. (c) Plot showing
$\uptau_{\rm m}$ versus the underdamped characteristic spreading time
$\uptau=\pi(\rho r_{0}^{3}/\eta\sigma)^{1/2}$. The solid line represents the
relation $\uptau_{\rm m}=\uptau$. (d) Snapshots showing the behavioural change
of a water droplet with radius $0.5\,$mm using AMS when $U$ is increased. The
scalebars represent $0.5\,$mm. (e) Plot showing the dependence of $h_{\rm
m}-r_{0}$ 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
$\mu=1.0\,$mPa s; $U_{\rm s}=120\,$V; the experiment was done in 2 cSt
silicone oil.

###  2.2 Electrowetting actuation

To induce jumping of droplets using electrowetting actuation, we note that the
activating duration $T_{\rm p}$ plays a crucial role. Experimental studies
have pointed to the time to reach maximum deformation $\uptau_{\rm m}$ as the
most optimal time duration for jumping droplets, i.e., causing highest jumping
(Wang et al., 2017). 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., 2018). Each
one of these behaviours are characterised by a distinct spreading timescales.
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 $T_{\rm p}$ equal to the underdamped characteristic
spreading time $\uptau=\pi(\rho r_{0}^{3}/\eta\sigma)^{1/2}$ (Vo et al., 2018;
Wang et al., 2020; Xiao & Wu, 2021):

$T_{\rm p}=\pi\left(\frac{\rho r_{0}^{3}}{\eta\sigma}\right)^{1/2},$ (1)

where the electrowetting number $\eta$, also known as the electrical capillary
number (Hassan et al., 2023; Fallah & Fattahi, 2022), is defined as
$\eta=\epsilon\epsilon_{0}U^{2}/(2d\sigma)$ for $U\leq U_{\rm s}$ and
$\eta=\epsilon\epsilon_{0}U_{\rm s}^{2}/(2d\sigma)$ for $U>U_{\rm s}$. 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 $U_{\rm
s}\approx 110\pm 10\,$V the threshold voltage above which contact angle
saturation (CAS) occurs. The threshold voltage $U_{\rm s}$ is determined
experimentally by examining the dependence of equilibrated contact angle
$\theta_{\rm e}$ on the applied voltage $U$ (Vo & Tran, 2021b). Slight
fluctuation of $U_{\rm s}$ is within $\pm 10\,$V and occurs when varying the
liquid’s viscosity (see Tab. 1). We note that Eq. 1 implies that $T_{\rm p}$
also saturates when $U>U_{\rm s}$ due to the contact angle saturation effect,
consistent with the experimental data shown in Fig. 1b. In Fig. 1c, we show an
excellent agreement between the measured values of the time to reach maximum
deformation and the characteristic spreading time $\uptau$ of droplets
actuation in the underdamped regime. This strongly suggests that $\uptau$ can
be used to describe the time to reach maximum deformation of droplets under
electrowetting actuation. As a result, the activating duration $T_{\rm p}$ in
our experiment is determined by Eq. 1, i.e., dependent only on the
experimental parameters and free of uncertainty from the experimental values
of $\uptau_{\rm m}$.

##  3 Results and Discussions

###  3.1 Droplet jumping by EWOD actuation from maximum deformation state

In Fig. 1d, 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 $r_{0}=0.5\,$mm,
$\mu=1.0\,$mPa 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 $v_{\rm I}$ and generates on the
droplet’s surface capillary waves propagating from the contact line toward 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 Fig. 1d,
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 (Fig. 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, 2021b).

In the cases that an actuated droplet detaches from the substrate, the detach
time $T_{\rm d}$, measured from the time reference $t=0$ to the moment the
droplet detaches, reduces with the applied voltage. For instance, $T_{\rm d}$
drops from $6.3\,$ms to $5.8\,$ms when $U$ increases from 120 V to 140 V (Fig.
1d). Moreover, the maximum jumping height $h_{\rm m}$ defined as the maximum
height of the droplet’s center of mass increases with $U$. For instance,
$h_{\rm m}-r_{0}$ significantly increases from $0.38\,$mm to $1.55\,$mm when
$U$ increases from 100 V to 140 V (Fig. 1d). Comparing the maximum jumping
height obtained in our experiment with the one obtained in the case jumping is
induced by AES (Fig. 1e), we observe that for the same voltage, $h_{\rm m}$
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 Fig. 2, we show the phase diagrams of droplet behaviours obtained by
varying three the control parameters $r_{0}$, $\mu$, and $U$. These observed
behaviours, which are illustrated in Fig. 1d, include nondetachable,
detachable without splitting, and detachable with splitting. The critical
voltage at the transition for detachment is higher for smaller droplet size
(Fig. 2a) or higher viscosity (Fig. 2b). We also observe that the detachment
of droplet is limited for $\mu<17.6\,{\rm mPa\,s}$ as the transition from the
underdamped regime to overdamped regime occurs at $17.6\,{\rm mPa\,s}$ (Fig.
2b) (Vo et al., 2018). We highlight that AMS method works for $r_{\rm 0}$ as
small as $80\,{\rm\mu m}$, a limit that was not possible using AES for the
same substrate (Vo & Tran, 2019).

Figure 2: (a) Phase diagram showing different behaviours of $\mu=1.0$ mPa 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: nondetachable (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
nondetachable regimes. The dashed-dotted lines indicates the average contact
angle saturation (CAS) threshold (see Tab. 1 for specific values of $U_{\rm
s}$ at different $\mu$).

###  3.2 Detach velocity and detach time

In the case that a droplet detaches from the solid substrate, the detach
velocity $u_{\rm d}$, defined as the droplet’s center-of-mass velocity at the
moment it completely separates from the substrate, and the detach time $T_{\rm
d}$, 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 electrowetting effect. In this
section, we discuss the dependences of $u_{\rm d}$ and $T_{\rm d}$ on the
applied voltage $U$, viscosity $\mu$, and droplet radius $r_{\rm 0}$.

We first focus on the detach velocity $u_{\rm d}$. In Fig. 3a, we show the
dependence of $u_{\rm d}$ on the applied voltage $U$ for $r_{\rm 0}$ varying
from 0.08 mm to 1.25 mm and $\mu$ fixed at $1\,{\rm mPa\,s}$. Generally, we
observe that $u_{\rm d}$ increases with $U$ as a result of higher electrical
energy applied to the system. However, the dependence of $u_{\rm d}$ on $U$
becomes more irregular at high applied voltage, i.e., $U>130\,$V. For example,
for $r_{0}=0.08\,$mm, $u_{\rm d}$ varies little between $U=140\,$V and
$U=150\,$V. We attribute such irregular dependence of $u_{\rm d}$ 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
(Fig. 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., 2002). This also reduces the efficiency of the EWOD effect in
generating higher $u_{\rm d}$ 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 Fig. 3b,
we plot the dependence of $u_{\rm d}$ on $U$ for $\mu$ varying from $1\,{\rm
mPa\,s}$ to $8.2\,{\rm mPa\,s}$ and $r_{\rm 0}$ fixed at $0.5\,$mm. We observe
that for $U<130\,$V, $u_{\rm d}$ linearly increases with $U$, whereas for
$U\geq 130\,$V, $u_{\rm d}$ approaches a plateau. The increasing rate of
$u_{\rm d}$ with $U$ also reduces with higher $\mu$ due to larger viscous
dissipation.

We now examine the detach time $T_{\rm d}$. In Fig. 3c and d, we respectively
show $T_{\rm d}$ vs. $U$ 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, $T_{\rm d}$ decreases with increasing $U$ as long as the voltage is
within the contact angle saturation (CAS) limit, i.e., $U_{\rm s}=$110 V in
our case. The detach time $T_{\rm d}$ eventually reaches a plateau when the
applied voltage exceeds $U_{\rm s}$. Here, we note that $T_{\rm d}$ for AMS is
defined 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 $T_{\rm d}$ also saturates
when $U\geq U_{\rm s}$, consistent with our experimental results shown in Fig.
3c and d.

The detach time $T_{\rm d}$, which is measured from the moment the voltage is
released to the detaching moment, is defined the same way as the retracting
time of the droplets from maximum deformation. As a result, we hypothesise
that $T_{\rm d}$ is closely related to the timescale characterising the
spreading (or retracting) dynamics of droplets. As the spreading droplets in
our study are underdamped, we examine the relation between $T_{\rm d}$ and the
underdamped characteristic spreading timescale $\uptau$ and show a plot of
$T_{\rm d}$ vs $\uptau$ in Fig. 3e. The plot consists of data obtained by
varying $r_{0}$, $\mu$, and $U$ in their explored ranges. We observe that the
dependence of $T_{\rm d}$ on $\uptau$ can be approximately described using a
linear relation $T_{\rm d}\approx k\uptau$, where $k=1.18\pm 0.07$, suggesting
that $T_{\rm d}$ is reasonably characterised by $\uptau$ in the explored
ranges of the control parameters. Furthermore, for each dataset obtained using
fixed $\mu$ and $r_{0}$, we note that Fig. 3c and Fig. 3d indicate that
$T_{d}$ decreases with $U$ for $U<U_{\rm s}$ and plateaus for $U>U_{\rm s}$.
This implies smaller data deviation from the linear fit in Fig. 3e at higher
voltages. In other words, the linear fit better describes the relation between
$T_{\rm d}$ and $\tau$ in the high-voltage regime.

Figure 3:  (a, b) Plots showing detach velocity $u_{\rm d}$ versus $U$ for (a)
different droplet radii $r_{0}$ and (b) different droplet viscosity $\mu$. (c,
d) Plots showing detach time $T_{\rm d}$ versus $U$ for (c) different droplet
radii $r_{0}$ and (d) different droplet viscosity $\mu$. (e) Plot showing
detach time $T_{\rm d}$ versus $\uptau$ for various values of $r_{0}$, $U$,
and $\mu$. Inset: 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
$T_{\rm d}=k\uptau$, where $k=1.18\pm 0.07$. The shaded areas indicates the
average contact angle saturation (CAS) threshold (see Tab. 1 for specific
values of $U_{\rm s}$ for different $\mu$).

###  3.3 Critical conditions for jumping droplets by EWOD actuation from
maximum deformation state

We now seek 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,
2019). 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,
2019) to formulate the critical condition for droplets to detach from the
solid substrate by AMS as

$\Delta E_{\rm s}\geq E_{\rm v}+E_{\rm cl},$ (2)

where $\Delta E_{\rm s}=E_{\rm s}^{\rm m}-E_{\rm s}^{\rm d}$ is the difference
between the surface energy at maximum deformation $E_{\rm s}^{\rm m}$ and the
surface energy at the detach moment $E_{\rm s}^{\rm d}$; $E_{\rm v}$ is the
viscous dissipation and $E_{\rm cl}$ 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 to the surface energy as the Bond number ${\rm Bo}=(\rho-\rho_{\rm
o})gr_{0}^{2}\sigma^{-1}$ is small ($2.14\times 10^{-4}\leq\rm{Bo}\leq
1.23\times 10^{-1}$). Here, $\rho_{\rm o}=873\,{\rm kg\,m^{-3}}$ is the
density of silicone oil and $g=9.81\,{\rm m\,s^{-2}}$ is the gravitational
acceleration.

We first focus on the surface energy difference $\Delta E_{\rm s}$ and note
that it can be written as $\Delta E_{\rm s}=E_{\rm s}^{\rm m}-E_{\rm s}^{\rm
e}+E_{\rm s}^{\rm e}-E_{\rm s}^{\rm d}$, where $E_{\rm s}^{\rm e}$ is the
surface energy at equilibriated state after applying the electrowetting effect
without turning it off. On one hand, we note that the surface energy
difference $E_{\rm s}^{\rm e}-E_{\rm s}^{\rm d}$ between the equilibrated
state and detachment was already formulated (Vo & Tran, 2019): $E_{\rm s}^{\rm
e}-E_{\rm s}^{\rm d}=[2(1+\cos\theta_{\rm e})^{-1}-4(r_{0}/r_{\rm
e})^{2}-\cos\theta_{0}]\sigma\pi r_{\rm e}^{2}$, where $\theta_{\rm e}$,
$r_{\rm e}$ are respectively the contact angle and contact radius of the
droplet at the equilibriated state; $\theta_{0}$ is the contact angle of the
droplet at the initial state. Moreover, we argue that the surface energy
difference $E_{\rm s}^{\rm m}-E_{\rm s}^{\rm e}$ 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 $E_{\rm s}^{\rm m}-E_{\rm s}^{\rm e}$
is calculated by the energy carried by the capillary wave $E_{\rm s}^{\rm
m}-E_{\rm s}^{\rm e}=0.5\rho a^{2}\omega^{2}lS$, where $a\sim r_{0}{\rm
We}^{1/2}\sin\theta_{0}(T_{\rm p}\omega)^{2/3}/[2\pi(1-\xi^{2})^{1/2}]$ is the
wave amplitude at time $t=T_{\rm p}$; $\omega\sim(\sigma/\rho
r_{0}^{3})^{1/2}$ 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;
$S=2(1+\cos\theta_{e})^{-1}\pi r_{e}^{2}$ is the area of a hypothetical
spherical cap having contact angle $\theta_{\rm e}$ and base radius $r_{\rm
e}$; and ${\rm We}$ is the contact line Weber number, defined to directly
relate to the applied voltage $U$ by the relation ${\rm
We}=[(\epsilon\epsilon_{0}/2d\sigma)^{1/2}(U-U_{\rm c})+1]^{2}$. Here, $U_{\rm
c}$ is the critical voltage for capillary wave generation on the droplet’s
surface (Vo & Tran, 2021b). In our experiment, $U_{\rm c}$ 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:

$\Delta E_{\rm
s}=\left[\frac{2}{1+\cos\theta_{e}}-4\left(\frac{r_{0}}{r_{e}}\right)^{2}-\cos\theta_{0}+{\rm
We}\frac{(T_{\rm
p}\omega)^{4/3}}{4\pi^{2}(1-\xi^{2})}\frac{\sin^{2}\theta_{0}}{1+\cos\theta_{\rm
e}}\right]\sigma\pi r^{2}_{\rm e}.$ (3)

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 $\lambda=C(\mu\mu_{\rm o})^{1/2}$, where $C=32.9$ is a
fitting parameter (Vo et al., 2018; Vo & Tran, 2019). By using $\xi$ instead
of Oh, the dissipation in the liquid bulk is effectively neglected (Vo & Tran,
2019; Carlson et al., 2012a, b). Indeed, the ratio $\xi/{\rm Oh}$, 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 $E_{\rm s}^{\rm m}-E_{\rm s}^{\rm e}$ to
arrive at Eq. 3. Also, by only considering dissipation at the contact line, we
obtain an expression for $E_{\rm v}$ (Vo & Tran, 2019):

$E_{\rm v}\sim\lambda\frac{r_{\rm m}}{T_{\rm d}}\pi r^{2}_{\rm
m}\approx\lambda\frac{r_{\rm e}}{T_{\rm d}}\pi r^{2}_{\rm e}.$ (4)

The contact line elasticity energy $E_{\rm cl}$ 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, 1984; Vo &
Tran, 2019) and is determined by a similar approach with Vo & Tran (2019):

$E_{\rm cl}\sim\kappa\sigma\pi r^{2}_{\rm m}\approx\kappa\sigma\pi r^{2}_{\rm
e}.$ (5)

Here, $\kappa=\pi\sin^{2}\theta_{\rm r}/{\rm ln}(r_{0}/\gamma)$, where
$\theta_{\rm r}$ is the receding contact angle, and $\gamma$ is the defect’s
size. In our experiment, which was conducted using the same setup and
materials with Vo & Tran (2019), $\theta_{\rm r}\approx 121^{\circ}$ 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
$0.08\,{\rm mm}\leq r_{0}\leq 1.25\,{\rm mm}$.

Subtituting Eqs. 3, 4, 5 into Eq. 2, the condition at the transition between
nondetachable and detachable behaviours is

$\frac{2}{1+\cos\theta_{\rm e}}-4\left(\frac{r_{\rm 0}}{r_{\rm
e}}\right)^{2}-\cos\theta_{0}+{\rm We}\frac{(T_{\rm
p}\omega)^{4/3}}{4\pi^{2}(1-\xi^{2})}\frac{\sin^{2}\theta_{0}}{1+\cos\theta_{\rm
e}}=\frac{r_{\rm e}\lambda}{T_{\rm d}\sigma}+\kappa.$ (6)

As shown in the previous sections, both the activation time $T_{\rm p}$ and
the detach time $T_{\rm d}$ are well approximated by the underdamped
characteristic spreading time $\uptau$. We therefore simplify Eq. 6 as

$\alpha+\Psi(\theta_{\rm e},\xi){\rm We}=\pi^{-1}\beta\eta^{1/2}\xi+\kappa,$
(7)

where $\beta=r_{\rm e}/r_{\rm 0}=(1-\cos\theta_{\rm
e}^{2})^{1/2}[4(1-\cos\theta_{\rm e})^{-2}(2+\cos\theta_{\rm e})^{-1}]^{1/3}$,
$\alpha=2(1+\cos\theta_{\rm e})^{-1}-4\beta^{-2}-\cos\theta_{0}$, and
$\Psi(\theta_{\rm e},\xi)=0.25\,\sin^{2}\theta_{0}(1+\cos\theta_{\rm
e})^{-1}(\pi\eta)^{-2/3}(1-\xi^{2})^{-1}$. We note that $\theta_{\rm e}$
depends on the electrowetting number $\eta$ following the Young-Lippmann’s
equation $\cos\theta_{\rm e}-\cos\theta_{\rm 0}=\eta$ (Mugele & Baret, 2005).

Figure 4:  Plot showing $\alpha+\Psi{\rm We}$ versus
$\pi^{-1}\beta\eta^{1/2}\xi$ using the data of the nondetachable and
detachable behaviours shown in Fig. 2. The shaded area indicates the
transitional extent due to variation in $\kappa$ ($\kappa=0.27\pm 0.03$).

In Fig. 4, we show a plot of $\alpha+\Psi{\rm We}$ versus
$\pi^{-1}\beta\eta^{1/2}\xi$ of all the data obtained for the nondetachable
and detachable behaviours shown in Fig. 2. Here, the composite terms
$\alpha+\Psi{\rm We}$ and $\pi^{-1}\beta\eta^{1/2}\xi$ respectively represent
the driving energy and the energy cost for detachment. The shaded area
indicates the transitional extent caused by variation in $\kappa$
($\kappa=0.27\pm 0.03$) resulted from variations in both the defect’s size
$\gamma$ and the radius $r_{0}$. We note that all the terms in Eq. 7 are
calculated using the system parameters except the contact line friction
coefficient, which is determined empirically using $\lambda=C(\mu\mu_{\rm
o})^{1/2}$ with $C=32.9$ (Vo et al., 2018; Vo & Tran, 2019), and the receding
contact angle $\theta_{\rm r}\approx 121^{\circ}$ determined independently
from our previous work (Vo & Tran, 2019). The excellent agreement between the
experimental data and the formulated jumping condition (Eq. 7) thus highlights
our analysis as a predictive tool for utilizing modulated electrowetting in
droplet actuation and detachment from surfaces.

##  4 Conclusions

We aim to systematically investigate droplet detachment induced by EWOD using
actuation from 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 $\upmu$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 developed 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 3D printing. Our results also
provide key insights to mechanistic understanding of related phenomena such as
coalescence-induced jumping of droplets (Boreyko & Chen, 2009; Farokhirad et
al., 2015; Liu et al., 2014) or droplet bouncing on solid substrates (Sanjay
et al., 2023).

## Acknowledgments

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.

## References

  * Baret & Brinkmann (2006) Baret, Jean-Christophe & Brinkmann, Martin 2006 Wettability Control of Droplet Deposition and Detachment. Physical Review Letters 96 (14), 146106.
  * Berge & Peseux (2000) Berge, B & Peseux, J 2000 Variable focal lens controlled by an external voltage: An application of electrowetting. The European Physical Journal E 3 (3), 159–163.
  * Blossey (2003) Blossey, Ralf 2003 Self-cleaning surfaces — virtual realities. Nature Materials 2 (5), 301–306.
  * Boreyko & Chen (2009) Boreyko, Jonathan B. & Chen, Chuan Hua 2009 Self-propelled dropwise condensate on superhydrophobic surfaces. Physical Review Letters 103 (18), 2–5.
  * Carlson et al. (2012a) Carlson, A., Bellani, G. & Amberg, G. 2012a Contact line dissipation in short-time dynamic wetting. EPL (Europhysics Letters) 97 (4), 44004\.
  * Carlson et al. (2012b) Carlson, Andreas, Bellani, Gabriele & Amberg, Gustav 2012b Universality in dynamic wetting dominated by contact-line friction. Physical Review E 85 (4), 045302, arXiv: 1111.1214.
  * Cavalli et al. (2016) Cavalli, Andrea, Preston, Daniel J., Tio, Evelyn, Martin, David W., Miljkovic, Nenad, Wang, Evelyn N., Blanchette, Francois & Bush, John W M 2016 Electrically induced drop detachment and ejection. Physics of Fluids 28 (2), 022101.
  * Dey et al. (2019) Dey, Ranabir, Van Gorcum, Mathijs, Mugele, Frieder & Snoeijer, Jacco H. 2019 Soft electrowetting. Soft Matter 15 (32), 6469–6475.
  * Fair et al. (2001) Fair, R.B., Pollack, M.G., Woo, R, Pamula, V.K., Hong, R, Zhang, T & Venkatraman, J 2001 A micro-watt metal-insulator-solution-transport (MIST) device for scalable digital bio-microfluidic systems. International Electron Devices Meeting. Technical Digest, , 00 (C), 16.4.1–16.4.4.
  * Fair (2007) Fair, R. B. 2007 Digital microfluidics: is a true lab-on-a-chip possible? Microfluidics and Nanofluidics 3 (3), 245–281.
  * Fallah & Fattahi (2022) Fallah, Keivan & Fattahi, Ehsan 2022 Splitting of droplet with different sizes inside a symmetric T-junction microchannel using an electric field. Scientific Reports 12 (1), 1–12.
  * Farokhirad et al. (2015) Farokhirad, Samaneh, Morris, Jeffrey F. & Lee, Taehun 2015 Coalescence-induced jumping of droplet: Inertia and viscosity effects. Physics of Fluids 27 (10), 102102.
  * Hao et al. (2014) Hao, Chonglei, Liu, Yahua, Chen, Xuemei, He, Yuncheng, Li, Qiusheng, Li, K Y & Wang, Zuankai 2014 Electrowetting on liquid-infused film (EWOLF): complete reversibility and controlled droplet oscillation suppression for fast optical imaging. Scientific Reports 4 (1), 6846.
  * Hassan et al. (2023) Hassan, Rizwan Ul, Khalil, Shaheer Mohiuddin, Khan, Saeed Ahmed, Moon, Joonkyeong, Cho, Dae Hyun & Byun, Doyoung 2023 Electric field and viscous fluid polarity effects on capillary-driven flow dynamics between parallel plates. Heliyon 9 (6), e16395.
  * He et al. (2021) He, Xiaodong, Zhang, Jianfeng, Yang, Bo, Zhang, Xiaoping & Deng, Youquan 2021 Droplet three-dimension manipulation in parallel liquid-infused membrane plates configuration. Sensors and Actuators, B: Chemical 330, 129344.
  * Hong et al. (2014) Hong, Jiwoo, Kim, Young Kwon, Kang, Kwan Hyoung, Kim, Joonwon & Lee, Sang Joon 2014 Spreading dynamics and oil film entrapment of sessile drops submerged in oil driven by DC electrowetting. Sensors and Actuators, B: Chemical 196, 292–297.
  * Hong et al. (2015) Hong, Jiwoo, Kim, Young Kwon, Won, Dong-Joon, Kim, Joonwon & Lee, Sang Joon 2015 Three-dimensional digital microfluidic manipulation of droplets in oil medium. Scientific Reports 5 (1), 10685.
  * Hong & Lee (2015) Hong, Jiwoo & Lee, Sang Joon 2015 Detaching droplets in immiscible fluids from a solid substrate with the help of electrowetting. Lab on a chip 15 (3), 900–907.
  * Joanny & de Gennes (1984) Joanny, J. F. & de Gennes, P. G. 1984 A model for contact angle hysteresis. The Journal of Chemical Physics 81 (1), 552–562.
  * Lee et al. (2012) Lee, Seung Jun, Lee, Sanghyun & Kang, Kwan Hyoung 2012 Droplet jumping by electrowetting and its application to the three-dimensional digital microfluidics. Applied Physics Letters 100 (8), 081604.
  * Kuiper & Hendriks (2004) Kuiper, S. & Hendriks, B. H W 2004 Variable-focus liquid lens for miniature cameras. Applied Physics Letters 85 (7), 1128–1130.
  * Lee et al. (2019) Lee, Jeongmin, Park, Yuna & Chung, Sang Kug 2019 Multifunctional liquid lens for variable focus and aperture. Sensors and Actuators, A: Physical 287, 177–184.
  * Lee et al. (2014) Lee, Seung Jun, Hong, Jiwoo, Kang, Kwan Hyoung, Kang, In Seok & Lee, Sang Joon 2014 Electrowetting-induced droplet detachment from hydrophobic surfaces. Langmuir 30 (7), 1805–1811.
  * Leïchlé et al. (2007) Leïchlé, T., Tanguy, L. & Nicu, L. 2007 Electrowetting-assisted drop deposition for controlled spotting. Applied Physics Letters 91 (22), 224102.
  * Liu et al. (2014) Liu, Fangjie, Ghigliotti, Giovanni, Feng, James J. & Chen, Chuan Hua 2014 Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces. Journal of Fluid Mechanics 752, 39–65.
  * Mishchenko et al. (2010) Mishchenko, Lidiya, Hatton, Benjamin, Bahadur, Vaibhav, Taylor, J. Ashley, Krupenkin, Tom & Aizenberg, Joanna 2010 Design of Ice-free Nanostructured Surfaces Based on Repulsion of Impacting Water Droplets. ACS Nano 4 (12), 7699–7707.
  * Moon et al. (2002) Moon, Hyejin, Cho, Sung Kwon, Garrell, Robin L & Kim, Chang-Jin “CJ” 2002 Low voltage electrowetting-on-dielectric. Journal of Applied Physics 92 (7), 4080–4087.
  * Moon et al. (2013) Moon, Jong Kyun, Jeong, Jaeki, Lee, Dongyun & Pak, Hyuk Kyu 2013 Electrical power generation by mechanically modulating electrical double layers. Nature Communications 4 (1), 1486–1487.
  * Mugele & Baret (2005) Mugele, Frieder & Baret, Jean-Christophe 2005 Electrowetting: from basics to applications. Journal of Physics: Condensed Matter 17 (28), R705–R774.
  * Nelson et al. (2011) Nelson, Wyatt C., Sen, Prosenjit & Kim, Chang Jin C.J. 2011 Dynamic contact angles and hysteresis under electrowetting-on-dielectric. Langmuir 27 (16), 10319–10326.
  * Pollack et al. (2000) Pollack, Michael G., Fair, Richard B. & Shenderov, Alexander D. 2000 Electrowetting-based actuation of liquid droplets for microfluidic applications. Applied Physics Letters 77 (11), 1725–1726.
  * Sanjay et al. (2023) Sanjay, Vatsal, Chantelot, Pierre & Lohse, Detlef 2023 When does an impacting drop stop bouncing? Journal of Fluid Mechanics 958, 1–20.
  * Sawane et al. (2015) Sawane, Yogesh B., Datar, Suwarna, Ogale, Satishchandra B. & Banpurkar, Arun G. 2015 Hysteretic DC electrowetting by field-induced nano-structurations on polystyrene films. Soft Matter 11 (13), 2655–2664.
  * Vahabi et al. (2018) Vahabi, Hamed, Wang, Wei, Mabry, Joseph M. & Kota, Arun K. 2018 Coalescence-induced jumping of droplets on superomniphobic surfaces with macrotexture. Science Advances 4 (11), 1–8.
  * Vo et al. (2018) Vo, Quoc, Su, Haibin & Tran, Tuan 2018 Universal Transient Dynamics of Electrowetting Droplets. Scientific Reports 8 (1), 836.
  * Vo & Tran (2018) Vo, Quoc & Tran, Tuan 2018 Contact line friction of electrowetting actuated viscous droplets. Physical Review E 97 (6), 063101.
  * Vo & Tran (2019) Vo, Quoc & Tran, Tuan 2019 Critical Conditions for Jumping Droplets. Physical Review Letters 123 (2), 24502\.
  * Vo & Tran (2021a) Vo, Quoc & Tran, Tuan 2021a Droplet ejection by electrowetting actuation. Applied Physics Letters 118 (16), 161603.
  * Vo & Tran (2021b) Vo, Quoc & Tran, Tuan 2021b Dynamics of droplets under electrowetting effect with voltages exceeding the contact angle saturation threshold. Journal of Fluid Mechanics 925, A19.
  * Wang et al. (2020) Wang, Qinggong, Xu, Meng, Wang, Chao, Gu, Junping, Hu, Nan, Lyu, Junfu & Yao, Wei 2020 Actuation of a Nonconductive Droplet in an Aqueous Fluid by Reversed Electrowetting Effect. Langmuir 36 (28), 8152–8164.
  * Wang et al. (2017) Wang, Zhantao, van den Ende, Dirk, Pit, Arjen, Lagraauw, Rudy, Wijnperlé, Daniël & Mugele, Frieder 2017 Jumping drops on hydrophobic surfaces, controlling energy transfer by timed electric actuation. Soft Matter 13 (28), 4856–4863.
  * Weng et al. (2021) Weng, Ning, Wang, Qinggong, Gu, Junping, Li, Jindong, Wang, Chao & Yao, Wei 2021 The dynamics of droplet detachment in reversed electrowetting (REW). Colloids and Surfaces A: Physicochemical and Engineering Aspects 616 (104), 126303.
  * Xiao & Wu (2021) Xiao, Ke & Wu, Chen Xu 2021 Curvature effect of electrowetting-induced droplet detachment. Journal of Applied Physics 129 (23), 234701.
  * Xu et al. (2020) Xu, Wanghuai, Zheng, Huanxi, Liu, Yuan, Zhou, Xiaofeng, Zhang, Chao, Song, Yuxin, Deng, Xu, Leung, Michael, Yang, Zhengbao, Xu, Ronald X., Wang, Zhong Lin, Zeng, Xiao Cheng & Wang, Zuankai 2020 A droplet-based electricity generator with high instantaneous power density. Nature 578 (7795), 392-396.