# Spreading dynamics of droplets impacting on oscillating hydrophobic
substrates

Aditya Potnis1  Abhishek Saha1<EMAIL_ADDRESS>1Department of Mechanical
and Aerospace Engineering, University of California San Diego, La Jolla, CA
92093, USA

###### Abstract

Droplet impact on oscillating substrates is important for both natural and
industrial processes. Recognizing the importance of the dynamics that arise
from the interplay between droplet transport and substrate motion, in this
work, we present an experimental investigation of the spreading of a droplet
impacting a sinusoidally oscillating hydrophobic substrate. We particularly
focus on the maximum spread of droplets as a function of various parameters of
substrate oscillation. We first quantify the maximum spreading diameter
attained by the droplets as a function of frequency, amplitude of vibration,
and phase at the impact for various impact velocities. We highlight that there
can be two stages of spreading. Stage-I, which is observed at all impact
conditions, is controlled by the droplet inertia and affected by the substrate
oscillation. For certain conditions, a Stage-II spreading is also observed,
which occurs during the retraction process of Stage-I due to additional
energies imparted by the substrate oscillation. Subsequently, we derive
scaling analyses to predict the maximum spreading diameters and the time for
this maximum spread for both Stage-I and Stage-II. Furthermore, we identify
the necessary condition for Stage-II spreading to be greater than the Stage-I.
The results will enable optimization of the parameters in applications where
substrate oscillation is used to control the droplet spread and, thus, heat
and mass transfer between the droplet and the substrate.

###### keywords:

##  1 Introduction

Droplet impacts on stationary and non-stationary surfaces are frequently
witnessed in our daily lives. For instance, raindrops falling onto leaves
(Gart et al., 2015; Park et al., 2020) and onto beating wings of birds (Zhang
et al., 2019) and insects entail fluid-structure interaction wherein movement
of the surfaces greatly affects the outcomes of such impacts and thus governs
numerous phenomenon such as the spread of pathogens, environmental aerosol
dispersion, and repulsion of liquid from natural surfaces. Understanding this
droplet-surface interaction is necessary to improve agricultural practices and
to design superior bio-inspired water-repellent media. In addition, droplet
impact on vibrating surfaces is a critical aspect of numerous industrial
processes. Controlling the deposition of functional droplets can be useful in
sprays for thermal coating (Tropea & Roisman, 2000; Saha et al., 2009). The
impact of droplets on substrates is also critical for applications such as
inkjet printing and in additive manufacturing (Martin et al., 2008; Tang et
al., 2019b; Lohse, 2022), where the post-impact spreading of droplet
determines the quality of the final product. The spreading dynamics of
impacted droplets also control the efficacy of spray cooling (or heating) of
surfaces where the heat transfer is proportional to the contact area and
contact time between the droplet and the substrate (Breitenbach et al., 2018).
Furthermore, the spreading is an important parameter in applications where the
impact promotes chemical reactions during surface treatments or mass transfers
during cleaning processes. Owing to its broad applications, droplet impact on
stationary and non-stationary substrates is of scientific interest. Hence, a
large volume of studies investigated the mechanistic description of the
dynamics, and various post-impact outcomes, which will be reviewed next.

Over the past century, extensive work has been carried out to investigate the
impact of different liquids on stationary substrates with a variety of surface
properties and morphologies. Early studies published by Worthington (1877a, b)
explored the impact of water and mercury on a static glass surface. Since
then, numerous studies have explored the various aspects of droplet impact on
stationary media, which were periodically summarized in reviews by Yarin
(2006); Khojasteh et al. (2016); Josserand & Thoroddsen (2016). These studies
have established that the outcome of impact and post-impact dynamics are
governed by the properties of both liquid and substrate. The phenomena
observed during impact are the results of the balance between forces involving
impact inertia, capillary force (surface tension), viscous dissipation, and
gravitational force. The relative strength of these forces can be quantified
using non-dimensional numbers, such as Weber number ($We$), Reynolds number
($Re$), and Froude number ($Fr$).

$We=\frac{\rho V_{0}^{2}D_{0}}{\gamma},\quad Re=\frac{\rho
V_{0}D_{0}}{\mu},\quad Fr=\frac{V_{0}^{2}}{gD_{0}}$ (1)

where, $\rho$ is the liquid density, $V_{0}$ is the impact velocity of
droplet, $D_{0}$ is the initial droplet diameter, $\gamma$ is the air-liquid
surface tension, $\mu$ is the dynamic viscosity of the liquid, $g$ is the
gravitational acceleration. Previous studies have found distinct
characteristics of impact differentiated by their governing mechanisms and
classified as either viscous regime (low $We$ and $Re$) or inertia-capillary
regime (high $We$ and $Re$). Rioboo et al. (2001) presented the different
qualitative outcomes of droplet impact on solid surfaces characterized, namely
as prompt splash, corona splash, rebound, partial rebound, deposition, and
receding breakup. Splashing generally occurs when the inertia of the droplet
overcomes surface tension during high $We$ impacts (Xu et al., 2005; Liu et
al., 2010; Hao & Green, 2017; Mandre & Brenner, 2012; Khabakhpasheva &
Korobkin, 2020). For low $We$ impacts, the primary focus of a group of studies
was dynamics of droplet deformation and retraction during impacts on various
impact surfaces (Tang et al., 2019b; Bird et al., 2009; Clanet et al., 2004).
The outcome of droplet impact is generally quantified in the form of geometric
parameters, such as spreading factor ($D^{*}=D/D_{0}$), defined by the ratio
of instantaneous droplet diameter ($D$) to initial diameter and the
instantaneous height of the deformed droplet. Some studies (Antonini et al.,
2013; Bartolo et al., 2005) also analyzed key timescales, including contact
time, rebounding time, etc. The time evolution of the spreading factor was
investigated in several investigations, which highlighted the maximum spread
factor ($D^{*}_{max}$). Experimentally it was shown that the evolution of
$D^{*}$, and $D^{*}_{max}$ depend on $We$, $Re$ and substrate properties
including, wettability (substrate contact angle), surface roughness, etc
(Lagubeau et al., 2012; Eggers et al., 2010; Ukiwe & Kwok, 2005). The
experimental observations were complimented by several theoretical and
analytical studies (Pasandideh-Fard et al., 1996; Du et al., 2021; Chandra &
Avedisian, 1991; Clanet et al., 2004; Fedorchenko et al., 2005). These
approaches, often based on the assumption of simplified geometry of deformed
droplets, assess the role of viscous loss along the boundary layer to derive
either scaling laws or estimated droplet diameter (Bennett & Poulikakos, 1993;
Pasandideh-Fard et al., 1996; Attané et al., 2007). To obtain more detailed
insights and to analyze localized dynamics, a large number of studies used
numerical simulations (Šikalo et al., 2005; Raman et al., 2016a; Tropea &
Roisman, 2000; Rioboo et al., 2002; Eggers et al., 2010; Renardy et al., 2003;
Wildeman et al., 2016).

The impact of a droplet on moving surfaces differs from the static surface due
to the modification of the relative velocity between the droplet and the
substrate and the associated change in the interfacial dynamics. Lee & Kim
(2004) experimentally studied the impact of low-viscosity droplets on a moving
substrate and explored the influence of various trajectories of vertical
motion of a surface on the post-impact characteristics of spreading and
rebound. They showed that the observed modification in the dynamics could not
be solely attributed to the change of relative droplet velocity and that
different trajectories of substrate motion with similar relative velocity can
cause various degrees of deviation in the dynamics. Weisensee et al. (2017)
carried out experiments with vertically oscillating rigid as well as elastic
surfaces to determine the effect of surface motion on the rebound
characteristics of impacting droplets. They provided evidence that the contact
time is a strong function of the timescale of oscillation and demonstrated
that the phase of surface oscillation at the impact plays a critical role.
This was subsequently confirmed by Kim et al. (2018), who worked with flexible
superhydrophobic surfaces with varying natural frequencies. Recent
experimental and theoretical work by Upadhyay et al. (2021) on flexible
superhydrophobic surfaces showed that a spring-mass system model could
estimate the contact time of droplets before the rebound. Moradi et al. (2020)
used axisymmetric Lattice Boltzmann simulations and confirmed that, for low
adhesion surfaces, the amplitude, frequency of oscillation, and the phase at
impact dictate the spreading and rebound characteristics. Similar results were
also reported for impact on a superhydrophobic surface using a discrete
particle method based on many-body dissipative particle dynamics (MDPD) (Lin
et al., 2022) and a coupled level-set and volume of fluid (CLSVOF) method (Li
et al., 2022). Along with reaffirming that oscillation parameters are critical
to post-impact droplet dynamics, these studies have also reported that energy
dissipation plays a key role in determining droplet dimensions, such as the
maximum spreading diameter and height during the spreading process. For high-
energy destabilizing impacts, Khabakhpasheva & Korobkin (2020) used asymptotic
analysis to derive a model and conjectured that splashing is possible for
impacts on a rigid vibrating surface but not on elastically supported
substrates. While most of these studies employed substrates oscillating
parallel to the direction of the impact, some studies investigated the effects
of substrate motion perpendicular to the direction of the impact. In general,
it is shown that such motions also alter the post-impact behavior and that by
controlling the parameters of oscillation, droplet rebound can be suppressed
(Raman et al., 2016b; Raman, 2019).

The above short review of literature has shown that droplet impact dynamics
are markedly different for impacts on non-stationary substrates. Although
these works have investigated the impact on oscillating surfaces, a vast
majority have utilized superhydrophobic surfaces, wherein the droplet is prone
to rebound upon impact due to its low affinity to the substrate. Additionally,
there is a lack of systematic experimental studies to explore the effect of a
wide range of oscillation parameters on the maximum spread of impacting
droplets. Our work endeavors to experimentally show that the dynamics of
droplet impact can be manipulated by using a vertically oscillating rigid
hydrophobic substrate. We investigate the spreading behavior in the
‘deposition’ regime of impact as described in Rioboo et al. (2001). We will
provide evidence that the reported post-impact normalized maximum droplet
diameter ($D^{*}_{max}$) during spreading and the time taken to achieve this
quantity ($t_{max}$) is strongly influenced by parameters of surface
oscillations namely, amplitude ($A$) and frequency ($f$) of oscillation and
phase at impact ($\phi$). Finally, we will provide scaling analyses to
theoretically assess the effects of these oscillation parameters on the
$t_{max}$ and $D^{*}_{max}$.

Next, we will describe the experimental setup. After that, we will present the
experimental results of the impact on static and oscillating substrates,
although our primary focus will be on the latter. Subsequently, we will
present the scaling analyses for theoretical estimations of the maximum
spreading of droplets. Finally, we will conclude with a summary of the work.

##  2 Experimental setup

In our experiments, a single droplet of de-ionized (DI) water
($\rho=998~{}kg\textrm{-}m^{-3}$, $\gamma=72~{}mN\textrm{-}m^{-1}$, and
$\mu=0.89~{}mPa\textrm{-}s$) is generated at the tip of a vertically
positioned needle (outer diameter $0.25~{}mm$) by pushing liquid through it
using a syringe pump. When the gravitational force overcomes the surface
tension, the droplet detaches, yielding an almost constant initial diameter
$\text{D}_{0}$ ($\approx 2~{}mm$). Upon detaching from the needle, the droplet
takes a near-spherical shape with minimal deformation. The droplet accelerates
as it falls downwards and eventually lands on the substrate with an impact
velocity $\text{V}_{0}$. By changing the free-fall distance, $\text{V}_{0}$
was varied to yield approximate Reynolds number ($Re$, defined in Eq. 1) and
Weber number ($We$, defined in Eq. 1) in the ranges of $1890<Re<3765$ and
$18<We<77$, respectively. Our substrate is a mirrored glass surface mounted on
a speaker unit (STAPEZTM FP-SPK-M glass-composite-diaphragm woofer). The
substrate is coated with a commercially available hydrophobic coating (RainX),
which results in a static contact angle ($\theta_{eq}$) of $\sim 90^{\circ}$
with DI water. A schematic of our experimental setup is shown in Figure 1.

Figure 1: Schematic of the experimental setup for droplet impact on a
vibrating hydrophobic substrate.

A sinusoidal signal from the function generator is used to drive the speaker
to provide a controlled substrate oscillation. This results in a vertical
oscillation in the substrate in the form $y_{s}=A\ sin(2\pi ft+\phi)$, where
$y_{s}$ is the position of surface measured from the static position, $t$ is
time, $A$ is the amplitude of oscillation, $f$ is the frequency, and $\phi$ is
the phase at impact. Here, $t=0$ refers to the instance when the droplet
impacts the substrate. For this work, we assign $y_{s}$ to be positive in the
upward direction.

A high-speed camera (Phantom V7.3) coupled with a Nikon $50~{}mm$ lens, 2x
teleconverter, and an extension bellow is used to record impact dynamics. The
camera and lens are mounted at a slight incline to the substrate to ensure an
unrestricted view of the droplet even during high amplitude oscillations
(shown in Fig. 2). Images are recorded at $14,760$ frames per second with a
$512\times 384~{}pixel^{2}$ resolution yielding a spatial resolution of
approximately $12.9~{}\mu m/pixel$. A high-intensity diffused LED array is
used as a backlight. Each experimental condition was repeated at least three
times to ensure the repeatability of the result. The ensemble average of the
desired quantity is considered, and the standard deviation is recorded to
estimate the error. The error associated with the desired phase at impact
($\phi$) is $\pm 0.028\pi$ or $\pm 5^{o}$.

A laser-based sensor, placed slightly above the impacted surface, was used to
detect the proximity of the droplet from the substrate. The signal from the
sensor triggers the high-speed camera and the function generator through an
external delay generator. The phase of the oscillating substrate at the time
of impact (i.e., the phase at impact, $\phi$) is controlled by adding and
modulating a delay between the signal from the sensor and the function
generator. The frequency and amplitude of the oscillation in the substrate
were directly controlled through the function generator (direct digital
synthesis, DDS, signal generator from Koolertron), whose output was amplified
using an amplifier (Lepai
$\textsuperscript{\textregistered}~{}\text{LP-}220\text{TI}$) before sending
it to the speaker to oscillate the substrate. Before the experiments, the
relation between the input voltage to the speaker and ensued amplitude of the
oscillation was obtained through a detailed calibration process.

High-speed images obtained during the experiments are processed using a custom
MATLAB code to extract the quantitative information on droplet spreading. In
our study, the instantaneous diameter of the deformed droplet ($D(t)$) is
defined as the maximum horizontal extent of the droplet as seen in high-speed
images. It is worth noting that this diameter, $D(t)$, is different from the
contact diameter (diameter of the contact line), especially during the initial
period of the deformation.

Figure 2: high-speed snapshots showing stages of droplet impact for $We=27$,
$Re=2300$ with (a) static substrate (b) f = $100\text{Hz}$, A = $0.25mm$ and
$\phi=3\pi/4~{}rad$ (c) f = $400\text{Hz}$, A = $0.125mm$ and
$\phi=\pi/4~{}rad$. Multiple peaks in the time evolution of droplet spread are
observed for high frequency cases due to the effect of subsequent oscillations
as elaborated in Sec. 3.2.

##  3 Experimental results

This section will illustrate the experimental findings of our investigation.
Our interest lies primarily in the post-impact spreading behavior,
particularly the normalized maximal spreading of droplets. Henceforth, in this
exposition, we will highlight the observations till the instant of the maximum
spread of droplets, with limited scrutiny of the receding phase. After
establishing a baseline for our study by outlining the findings of the impact
on a static surface, we will present the dynamics for cases with oscillating
substrates, which is the focus of this study. The influence of phase at impact
($\phi$), the effect of frequency of oscillation ($f$), along with the effect
of oscillation amplitude ($A$) will be addressed systematically.

###  3.1 Impact on static substrates

The speaker was not actuated for these experiments, ensuring the substrate
remained static during the impact. The droplet free falls till it contacts the
substrate at $t=0$, with the velocity at impact $\text{V}_{0}$. A series of
high-speed images for impact on a static surface is shown in Figure. 2a. The
process begins with the spherical droplet making contact with the stationary
substrate. Promptly after the instant of impact ($t<0.1~{}ms$), a small air
bubble forms near the contact point (also observed for oscillating substrates
as shown in Fig. 2) due to the entrapment of air between the droplet and
surface. This has been observed in previous studies (Chandra & Avedisian,
1991; Pasandideh-Fard et al., 1996; Bouwhuis et al., 2012; Tang et al., 2019a)
of droplet impact on both solid and liquid substrates. This bubble formation
is caused by non-uniform pressure distribution in the interfacial gas layer
trapped between the droplet and impacted interfaces and the eventual collapse
of this gas layer at a location away from its center. The droplet starts
deforming after the impact (Fig. 2a), its bottom surface is flattened, and the
droplet spreads outwards. This initial stage of spreading (upto $t\approx
0.3~{}ms$ as illustrated in Figure 2a) is referred to as the ‘kinematic phase’
(Rioboo et al., 2002) where the impact inertia dominates over capillary and
viscous effects in controlling the spreading dynamics.

Figure 3: (a) Temporal evolution of post-impact normalized droplet diameter
($\text{D}/\text{D}_{0}$) for static impact for $We=18,27,45$ $\&$ $77$ – – –
denotes the normalized time $t_{max}/\tau$ for maximum spreading. (b)
Normalized maximum spread for static impact ($\text{D}_{max,s}^{*}$) and
normalized time for the maximum spread, with respect to ‘crashing time’
($t_{max,s}/\tau$) as functions of $We$ which display a power law, and linear
fitting respectively.

As the deformed portion of the droplet spreads past the initial droplet
diameter $D_{0}$, a lamella is formed, and it rapidly spreads radially, while
the upper portion of the droplet remains undeformed, resembling a truncated
sphere as seen at $t\approx 0.3~{}ms$ in Figure 2a. It is observed that this
undeformed part of the droplet continues to move downward with the velocity
(measured at the tip of the droplet) equal to the impact velocity
($\text{V}_{0}$), an observation also reported in earlier studies (Lagubeau et
al., 2012). This is the ‘spreading phase’ (Rioboo et al., 2002) where surface
tension and viscosity begin to affect the spread. At this stage, capillary
waves are seen to rise up through the droplet surface, and they travel
upwards, eventually reaching the top of the droplet ($\approx 1.3~{}ms$ in
Figure 2a), thereby completely deforming it. More details of these waves can
be found in studies by Pasandideh-Fard et al. (1996) and Renardy et al.
(2003).

As the top of the droplet reaches its lowest point, it no longer looks like a
spherical cap but resembles a pancake ($\approx 2.2~{}ms$ in Figure 2a). For
inertia-driven impact, the time taken for the droplet to reach this stage is
defined as the droplet ‘crashing time’ ($\tau=\text{D}_{0}/\text{V}_{0}$) and
is the characteristic inertial timescale for such phenomena. At this stage,
the droplet has expended most of its kinetic energy. The ‘spreading phase’
continues till the droplet completely deforms and all its kinetic energy is
traded for an increase in surface energy, with some energy lost to viscous
dissipation. The spreading results in thinning of the lamella and a thick rim
is formed on the droplet periphery, creating an almost toroidal geometry
(labeled as ‘maximum spread’ at $\approx 2.9~{}ms$ in Figure 2a). For lower
$\text{V}_{0}$ (and $We$) and thus lesser spread, the droplet forms a geometry
with a less pronounced rim described as pancake-like rather than toroidal (not
shown here). The time history of the instantaneous diameter of the droplet for
four different $We$ is shown in Fig. 3a. The normalized maximum diameter
($\text{D}_{max,s}^{*}=\text{D}_{max,s}/\text{D}_{0}$), achieved by the
droplets, and the normalized time ($\text{t}_{max,s}/\tau$) taken to achieve
the maximum spread are shown in Figure 3b. In general, we observe that
$\text{t}_{max,s}$ is greater than $\tau$ and the normalized time follows a
linear relation ($\text{t}_{max,s}/\tau=0.0109We+1.0346$) with $We$. On the
other hand, normalized maximum diameter ($\text{D}_{max,s}^{*}$) displays a
power-law dependence on $We$, with an exponent of approximately $1/4$, a
behavior also observed in previous studies (Clanet et al., 2004). These values
of $\text{D}_{max,s}^{*}$ and $\text{t}_{max,s}$ for static impact are
recorded as the ‘baseline’ to juxtapose with oscillating substrate cases and
thus, will be used to normalize corresponding length and time scales.

After achieving maximum spread, the system transitions to the ‘relaxation
phase’ (Rioboo et al., 2002), and the droplet recedes due to surface tension
in an effort to minimize its surface energy. The droplet settles into a damped
oscillation of its diameter till viscous losses eventually render it
stationary, and it achieves an equilibrium position during the
‘wetting/equilibrium phase.’ Sometimes, a Rayleigh jet (shown in Fig. 2) is
observed due to excess kinetic energy at the end of retraction. More details
on the ‘relaxation’ and ‘wetting/equilibrium phase’ can be found in literature
(Yamamoto et al., 2016; Richard et al., 2002; Antonini et al., 2013).

###  3.2 Impact on oscillating substrates

We will now present the experimental results for droplet impact on oscillating
substrates. For these experiments, the substrate was actuated with a
sinusoidal wave, and the phase at the impact ($\phi$) was controlled using the
laser-triggered delay generator as detailed in Sec. 2. The post-impact stages
of droplet spread on an oscillating substrate qualitatively resemble that on a
static one, as shown in Fig. 2b and c. However, the spreading time and the
maximum spread change due to the continuous movement of the substrate. The
dynamics of post-impact spreading for impact on oscillating substrate, thus,
depends on the motions of both the droplet and the substrate. Their combined
effect is quantified by the relative droplet velocity, defined as
$V_{rel}=V_{drop}+V_{s}$, which changes with time. Here, $V_{drop}$ is the
instantaneous downward velocity of the droplet. It is to be noted that based
on the phase at impact ($\phi$), the substrate velocity ($V_{s}$) at impact
can be upward ($0<\phi<\pi/2~{}\&~{}3\pi/2<\phi<2\pi$) or downward
($\pi/2<\phi<3\pi/2$), thereby increasing or decreasing the relative impact
velocity ($V_{rel}$), respectively. Let’s elaborate on this using the case
where the substrate oscillated with amplitude, $A=0.25~{}mm$, frequency,
$f=100\text{Hz}$, and phase at impact was $\phi=3\pi/4$. The temporal
evolution for this impact is shown in Fig. 4a. The top panel of the figure
illustrates the instantaneous droplet diameter during the spreading processes.
The solid line represents the impact on the oscillating substrate, and the
dotted line refers to the impact on the static substrate at the same $We$. The
morphology of the deformed droplets is shown for some key instances as insets.
The lower panel displays the instantaneous locations of the substrate during
the spreading process. Figure 4a shows that, at the time of impact, the
substrate moves downwards, away from the droplet with velocity
$\text{V}_{s}\approx-0.11~{}m/s$. Furthermore, after the impact,
$\text{V}_{s}$ remains negative as the substrate continues with a downward
motion. Naturally, the droplet experiences a lower value of $V_{rel}$
throughout the duration of spreading. The decreased $V_{rel}$ induces reduced
vertical compression, and hence a reduced spreading
($D_{max}^{*}=D_{max}/D_{0}=2.04$), compared to the spreading for impact on
the static substrate ($D_{max,s}^{*}=2.22$) for the same impact velocity or
$We$ as seen in Fig. 4a. The change in $V_{rel}$ also affects the time taken
by the droplet to reach maximum spreading ($t_{max}=3.3~{}ms$ and
$t_{max,s}=3.0~{}ms$ for impacts on oscillating and static substrates,
respectively). The observed dynamics quantitatively change with $\phi$ because
the instantaneous $V_{s}$ and hence, $V_{rel}$ follow different temporal
histories leading to either inhibited or assisted spreading of the droplet.
The modification in spreading diameter and time for spreading, due to
substrate oscillation is illustrated in Fig. 5, where we compared
$D^{*}_{max}/D^{*}_{max,s}$ and $t^{*}_{max}/t^{*}_{max,s}$ vs. $\phi$ for
different $A$ and $f$ for a fixed $We(\approx 27$) and $Re(\approx 2300$).

Figure 4:  Temporal evolution of post-impact normalized droplet diameter (
$\text{D}/\text{D}_{0}$) with snapshots illustrating droplet profiles (Fig. 2)
for $We=27$, $Re=2300$ and (a) f = $100\text{Hz}$, A = $0.25mm$ and
$\phi=3\pi/4~{}rad$ (b) f = $400\text{Hz}$, A = $0.125mm$ and
$\phi=\pi/4~{}rad$; The bottom plots show the evolution of substrate motion
($y_{s}/A=sin(2\pi ft+\phi)$) for both cases. – – – shows the spreading for
impact on a static surface.  Figure 5: (a,b) Normalized maximum spread,
$\textit{D}^{*}_{max}/\textit{D}^{*}_{max,s}$ as a function of phase, $\phi$
at impact for various frequencies for $We\approx 27$, and $Re\approx 2300$.
(a) $A=0.125~{}mm$ (b) $A=0.25~{}mm$. Here the error bars represent the extent
of standard deviation about the mean value. (c,d) Normalized time to maximum
spread, $\textit{t}_{max}/\textit{t}_{max,s}$ as a function of phase, $\phi$
at impact for various frequencies. (c) $A=0.125~{}mm$ (d) $A=0.25~{}mm$. The
significantly higher values of $\textit{t}_{max}/\textit{t}_{max,s}$ seen in
both figures are a consequence of Stage-II spreading. Here the error bars
represent the extent of standard deviation about the mean value.

From Fig. 5a and b it is evident that the oscillations rendered spreading
difficult compared to the impact on the static substrate
($D_{max}^{*}/D_{max,s}^{*}\ <1$), for $0\lessapprox\phi\lessapprox\pi$. On
the other hand, the oscillations enhanced the spreading
($D_{max}^{*}/D_{max,s}^{*}>1$) for $\pi\lessapprox\phi\lessapprox 2\pi$. At
lower amplitude ($A=0.125~{}mm$), for the lowest frequency ($f=100\text{Hz}$),
the minimum $D_{max}^{*}$ occurs close to $\phi=\pi$ and for higher $f$s the
minimum $D_{max}^{*}$ condition shifts towards lower $\phi$ values ($\pi/2$
for $f=220\text{Hz}$ and $f=300\text{Hz}$) as shown in Fig. 5a. On the other
hand, the maximum $D_{max}^{*}$ occurs close to $\phi=2\pi$ for
$f=100\text{Hz}$ and with increase in $f$, it shifts to lower $\phi$ ($3/2\pi$
for $220\text{Hz}$). Fig. 5a also shows that the value of the maximum
$D_{max}^{*}$ across various $\phi$ monotonically increases with $f$. However,
the value of the minimum $D_{max}^{*}$ first decreases with an increase in
frequency until $f\lessapprox 150~{}\text{Hz}$, then increases for higher
frequencies. These trends are qualitatively in close agreement with previous
numerical studies by Moradi et al. (2020); Lin et al. (2022); Li et al.
(2022). However, we also observe a modified behavior for higher frequencies
($f\geq 250\text{Hz}$) in that the enhancement in spreading
($D_{max}^{*}/D_{max,s}^{*}>1$ ) sustained for an increasingly large range of
$\phi$ values. For example, we observed $D_{max}^{*}/D_{max,s}^{*}>1$ across
all $\phi$s at $A=0.125~{}mm$ and $f=300~{}\text{Hz}$ (Fig. 5a).

At low frequencies, in the duration of droplet spreading, the substrate
completes only a partial cycle of oscillation, and depending on the direction
of the motion, the spreading is assisted or inhibited. We generally observe a
single local maximum in the instantaneous droplet diameter. This is identified
as Stage I spreading in Fig. 4a, which shows the dynamics for $100\text{Hz}$.
On the other hand, at higher frequencies, the substrate undergoes multiple
cycles of oscillations during the spreading process (Fig. 4). Here, the
initial spread is governed by the first cycle of oscillation where the $D^{*}$
profile reaches the first maxima, Stage-I. The droplet, subsequently,
initiates a ’relaxation phase’ as $D^{*}$ starts reducing. But, before the
retraction phase can occur, the substrate starts a downward acceleration,
causing the rim of the droplet to increase in height as the substrate rapidly
moves downwards. This is evident in snapshots for $400\text{Hz}$ shown in
Figs. 2b and 4b. The heightened rim subsequently collapses as the substrate
starts an upward movement during the next cycle of oscillation (Fig. 4b). This
yields an increase in instantaneous $D^{*}$, creating a second local maximum.
This is termed as Stage-II spreading as shown in Fig. 4b. For some $\phi$s,
the overall maximum spreading is observed during the Stage-II, as shown in
Fig. 4b. Since the maximum spread is achieved at a later cycle of oscillation,
the $\textit{t}_{max}/\textit{t}_{max,s}$ for the Stage-II maximum spread is
significantly higher, as illustrated in Figure. 5c and d.

The spreading dynamics ($D_{max}^{*}/D_{max,s}$) remain qualitatively similar
at various phases across the different amplitudes. More specifically, we
observe the maximum and minimum spread ($D_{max}^{*}$) to occur at the same
$\phi$ for different amplitudes (Fig. 5a vs. b). However, the minimum and
maximum values of $D_{max}^{*}$ are reduced and amplified with an increase and
decrease in $A$, respectively. Clearly, variations in amplitude do not alter
the time scales associated with substrate oscillation but affect the magnitude
of the effect or $D_{max}^{*}/D_{max,s}^{*}$. Comparison of
$D_{max}^{*}/D_{max,s}^{*}$ for a larger range of $A$ are shown in the
supplementary materials. We have also performed experiments with three
different $We$ (and $Re$), which results in qualitatively similar behavior.
The $D^{*}_{max}$ values from these experiments can be found in the
supplementary materials.

##  4 Scaling analysis

###  4.1 Stage-I spreading

####  4.1.1 Time for maximum spread

As the Stage-I spreading is kinematically controlled for an inertia-driven
impact, the characteristic timescale governing the spreading dynamics of a
droplet impacting on a substrate is the crashing time ($\tau$). Physically, it
is the time taken by the tip of the droplet to reach the substrate in the
absence of any form of deceleration, and its relation with droplet diameter
can be expressed as

$D_{0}=\int_{0}^{\tau}V_{rel}(t)dt.$ (2)

Imposing $V_{rel}=V_{0}$, for impact on static substrates (i.e., no substrate
motion), the crashing time can be shown to be
$\tau_{s}=\text{D}_{0}/\text{V}_{0}$. For a wide range of impact conditions
($2<We<900$), Clanet et al. (2004) proved that $\tau_{s}$ is indeed the
relevant timescale. The time for the droplet to achieve maximum spreading
($t_{max}$), however, is different than the crashing time ($\tau$). For impact
on static substrates, it has been shown that, although the droplet achieves a
significant part of its total deformation and spreading at $t\leq\tau_{s}$,
the droplet still possesses a small amount of kinetic energy which decays
almost asymptotically (Roisman et al., 2002). This remnant energy causes
further deformation and spreading, albeit at a much weaker rate compared to
$t\leq\tau_{s}$. Viscous loss and capillarity dominate at this stage
($t>\tau_{s}$), eventually restricting and stopping the spreading process. One
can impose an assumption of $t_{max,s}=\tau_{s}$, as it was done for several
studies. However, this assumption will lead to an under-prediction of
$t_{max}$ compared to experimental measurements. This introduces significant
errors in evaluating viscous losses while the modeling of droplet spread. The
accurate estimation of $t_{max,s}$ from simple scaling is challenging, which
is also recognized by other studies (Du et al., 2021). To get a realistic
estimate of $t_{max,s}$, we plotted the ratio of ${{t}_{max,s}}/{\tau_{s}}$
for impact on solid substrates as function of $We$ in Fig. 3b and evaluated a
linear relation with $We$, ${{t}_{max,s}}/{\tau_{s}}=\ 0.0109We+1.0346$. As
the overall $We$ does not change due to the substrate oscillation, we will
assume that this relationship between crashing time, time for the maximum
spread, and $We$ holds for impact on oscillating substrates and hence,

$\frac{{t}_{max}}{\tau}=\ 0.0109\ We\ +\ 1.0346.$ (3)

The crashing time for impact on oscillating substrates ($\tau$) is affected by
the motion of the substrate and hence, is different from its counterpart of
impact on static substrates ($\tau_{s}$). For impact on oscillating
substrates, $\tau$ is a function of impact velocity ($V_{0}$), and frequency
($f$), amplitude ($A$), phase ($\phi$) of the oscillation, and can be
evaluated by using $V_{rel}=V_{0}+2A\pi f\cos{(2\pi ft+\phi)}$ in Eq. 2. Here,
it is assumed that the downward velocity ($V_{0}$) of the droplet with respect
to a lab-fixed reference is constant for the crashing period, i.e.,
$0<t\leq\tau$. In supplementary material, we compared the trajectory of the
tip of the impacting droplet and the instantaneous droplet spread, which shows
that the droplet descends with nearly a constant velocity until $t=\tau$,
justifying the assumption. Once the crashing time ($\tau$) is calculated
theoretically (Eq. 2), we use the correlation in Eq. 3 to evaluate the
theoretical time taken for maximum spread during droplet impact on oscillating
substrates.

Figure 6a shows the theoretical (solid lines) and experimental (symbols)
normalized $t_{max}$ for impacts on oscillating substrates for various $\phi$
and $f$ at $We=27$, $Re=2300$, $A=0.25~{}mm$. For most conditions, the
theoretical values show good agreement with the experimental data in that it
captures both qualitative and quantitative changes with $\phi$ and $f$. Large
discrepancies were observed for the conditions (e.g., $f=250\text{Hz}$) for
which the maximum spreading was obtained during the subsequent oscillations of
the substrate (Stage-II spreading). The analyses of crashing time that led to
Eq. 2 account for the inertia controlled Stage-I spreading, but discount the
additional spreading which occurs during the retraction stage of the droplet
by the action of subsequent oscillations (Stage-II).

In Fig. 6b, we compare the theoretical vs. experimental $t_{max}$ for all
conditions studied in our experiments covering a $We$, $A$, $f$, and $\phi$.
Again, we observe good agreement between the experiment and theory for all the
conditions with Stage-I spreading with an error range of $\pm 10\%$. As
expected, the conditions affected by the Stage-II spreading (marked with open
symbols) show discrepancies due to the above-mentioned reason. We will derive
scaling for Stage-II spreading later in Section 4.2.

Figure 6: (a) Comparison of experimental and theoretically predicted values
for normalized maximum spreading time, $\textit{t}_{max}/\textit{t}_{max,s}$,
as a function of phase at impact, $\phi$, for $We\ =\ 27$, and $.25~{}mm$ for
various frequencies. (b) Comparison of experimental and theoretically
predicted the maximum spreading time, $\textit{t}_{max}$. The filled symbols
represent Stage-I spreading and open symbols are for Stage-II as seen from
experiments. The plots show that theoretical values for $\textit{t}_{max}$
show a good match for all data showing Stage-I spreading. Figure 7:
Illustration of mass-conservation during droplet impact; $\text{D}_{0}$ &
$\text{V}_{0}$ are initial droplet diameter and velocity, $\text{D}_{max}$ &
$h$ are dimensions of spreading droplet, $\text{V}_{s}$ is the velocity of
substrate, and $\text{V}_{rel}$ is the relative velocity.

####  4.1.2 Maximum spreading diameter

Recognizing that we have an good prediction of $t_{max}$ for impact on
oscillating substrates, we next proceed to analyze the energy balance to
obtain a theoretical expression of normalized maximum spreading diameter
($D^{*}_{max}$). As mentioned before, for impacts with relatively high $We$
and ${Re}$ the spreading is inertia driven. The hydrophobic nature of the
substrate ($\theta_{eq}=90^{\circ}$) allows the droplet to spread with the
effects of viscosity being confined mostly to the boundary layer close to the
substrate. Thus, the dynamics of spreading is an outcome of the balance
between the initial kinetic energy (KE), which in parts, is converted to the
surface energy (SE) of the droplet, and is lost to viscous dissipation (W). It
is generally assumed that the droplet has negligible kinetic energy at maximum
spreading. Furthermore, for impacts on oscillating substrates, the total
energy in the system is altered by the presence of an oscillating boundary,
which injects and withdraws energy from the system as the substrate moves
upwards and downwards, respectively. The energy balance between the states of
the droplet before impact and at the maximum spread, then, can be expressed as

$KE_{0}+SE_{0}+E_{s}=\ SE_{f}+W.$ (4)

Here, the kinetic and surface energies of the droplet before the impact are
$\text{KE}_{0}=(1/12)\pi\rho D^{3}_{0}V_{0}^{2}$ and $\text{SE}_{0}=\gamma\pi
D_{0}^{2}$, respectively. To simplify the analysis, it was assumed that the
geometry of the deformed droplet at the end of the spreading resembles that of
a shallow cylinder (or pancake) of diameter, $D_{max}$ and ‘splat height’ $h$
(shown in Fig. 7). Although the droplet takes a far more complicated shape
with rims during the impact, Eggers et al. (2010) showed that rim formation
occurs during the retraction phase. This justifies the assumption of a
cylindrical geometry at the instant of maximum spread before the retraction
begins. With this assumption, the final surface energy can be expressed as
$\text{SE}_{f}=\gamma\pi\left[D_{max}^{2}/4+D_{max}h\right]$. From mass
conservation, the height of the deformed droplet can be shown to be $h=(2/3)\
(D_{0}^{3}/D^{2}_{max})$. The overall viscous dissipation is estimated by
$W=\int^{t_{max}}_{0}\int_{\Omega}\Phi d\Omega dt\approx\Phi\Omega t_{max}$.
Here, $\Phi$ is the viscous dissipation rate per unit volume and can be
approximated as $\Phi\sim\mu\left({\text{V}_{0}}/{\delta}\right)^{2}$. The
boundary layer thickness is obtained by the relation
$\delta\sim{D_{0}}/\sqrt{Re}$ (White & Majdalani, 2006). The volume of the
boundary layer at the bottom of the cylindrical droplet, where the viscous
loss is significant, is given by $\Omega={\pi D_{max}^{2}\delta}/{4}$. Since
the expressions for evaluating $\Phi$ and $\delta$ are approximations, we
introduce $\alpha$ as the scaling factor to accurately assess the total
viscous loss in the boundary layer,

$W=\alpha\frac{\rho\,V_{0}^{3}}{\sqrt{Re}}\pi D_{max}^{2}t_{max}.$ (5)

The complexity of quantitative assessment of viscous dissipation in droplets
during impact on substrates is well-known. A scaling factor to account for the
discrepancies has been proposed previously by Eggers et al. (2010). The
process of evaluation for $\alpha$, in our study will be discussed later.

To account for the additional energy supplied by the moving substrate, we
introduce $E_{s}$, defined as the total energy imparted by the surface to the
droplet until maximum spread is achieved ($t=t_{max}$). Assuming the whole
droplet is moving with respect to the substrate, we can express
$E_{s}=(\rho\pi/8)\int^{t_{max}}_{0}(d\rotatebox[origin={c}]{180.0}{$A$}/dt)V_{s}^{2}(t^{\prime})dt^{\prime}$.
Here, we assume that the contact length of the droplet with the surface scales
with the droplet length-scale, i.e., the initial diameter $D_{0}$. Thus, the
rate of change of the volume of liquid affected by the surface movement can be
expressed as $(d\rotatebox[origin={c}]{180.0}{$A$}/dt)\approx D_{0}^{2}V_{s}$.
These leads to

$E_{s}\
\approx\rho\,\frac{\pi\,D_{0}^{2}}{8}\,\left(\,\int^{t_{max}}_{0}V_{s}^{3}(t^{\prime})\,dt^{\prime}\,\right).$
(6)

Here the energy transfer from the substrate to the droplet is assumed to be
lossless. We note that the above expression for $E_{s}$ accounts for the
‘directionality’ of the energy transfer. $E_{s}$ is positive and negative when
energy is added to and withdrawn from the droplet, respectively. By
substituting the expressions for various forms of energies in Eq. 4 and by
normalizing the lengths by $D_{0}$, and time by $\tau$, we find

$\frac{We}{12}\ +\ 1\ +\ \frac{E_{s}}{\gamma\ \pi\ D_{0}^{2}}\ =\
\frac{(D^{*}_{max})^{2}}{4}\ +\ D^{*}_{max}\,h^{*}_{max}\ +\ \alpha\
\frac{We}{\sqrt{Re}}\ (D^{*}_{max})^{2}\ \frac{t_{max}}{\tau}.$ (7)

Theoretical estimation of $\alpha$, the scaling factor arising from the
boundary layer analyses, is highly sensitive to the assumptions and
simplifications. Since an accurate calculation of the scaling factor is
difficult from theory, we use experimental data. Here, we utilize the data
from the experiments with impacts on static substrates ($E_{s}=0$). By
substituting the measured $D^{*}_{max}$ and $\tau$ in Eq. 7 for various impact
conditions, we can solve for $\alpha$. As shown in Fig. 8, the best-fit
$\alpha$ depends on the impact conditions as $\alpha=2.829We^{-0.55}$. A
similar approach was also successfully used by Eggers et al. (2010) for their
analyses of impacts on static substrates.

Figure 8: Empirical fit for $\alpha$ expressed in 7 obtained from experimental
values for static impact for $We\ =\ 18,\ 27,\ 45\ \&\ 77$

Equation 7 can be solved to evaluate $D^{*}_{max}$ for impacts on oscillating
substrates with various amplitudes, frequencies, and phases. Figure 9a
compares the experimental (symbols) and theoretical (lines) $D^{*}_{max}$ for
$We=27$, $Re=2300$, and a fixed value of amplitude ($A=0.25mm$) over a range
of frequency ($f$) and phase ($\phi$). We notice that the model predicts the
decrease and increase in $D^{*}_{max}$ as frequency and phase change with
reasonable accuracy. Furthermore, Fig. 9b compares $D^{*}_{max}$ measured in
all the experiments with the corresponding theoretical predictions. Overall,
good qualitative and quantitative agreements have been observed across all the
conditions.

We can make a few additional observations from the analyses and the results of
$D^{*}_{max}$. By using representative values, one can show that $E_{s}$ is
relatively insignificant compared to the other terms ($KE_{0}$, $SE_{0}$,
$SE_{f}$ and $W$) in energy balance equation (Eq. 4). This implies that the
modification in a maximum spread of the droplet for impact on the oscillating
substrate does not directly come from the additional energy introduced by the
substrate. The oscillations also modify the droplet’s relative velocity,
affecting the history of the deformation process and, thus, spreading time,
$t_{max}$. This, in turn, affects the viscous dissipation, $W$, and hence
$D_{max}$. Thus, it is essential that $t_{max}$ is modeled accurately.
Furthermore, an order of magnitude analysis of the other terms in Eq. 7 shows
that about $40-60\%$ of total initial energy ($KE_{0}+SE_{0}$) is lost through
viscous dissipation ($W$). This large viscous dissipation was also reported
for droplet impact on static substrates by Wildeman et al. (2016) and Li et
al. (2022).

Figure 9: (a) Comparison of experimental and theoretically predicted values
for $D^{*}_{max}/D^{*}_{max,s}$ for $We=27$, $Re=2300$, $A=0.25~{}mm$ as a
function of $\phi$. (b) Comparison of experimental and theoretically predicted
absolute values for $D^{*}_{max}$ at the instant of maximum spreading for all
experimental data. The filled icons represent $Stage~{}1$ spreading, and
unfilled icons are for Stage II as seen from experiments.  Figure 10: Temporal
evolution of post-impact normalized droplet diameter ($\text{D}/\text{D}_{0}$)
illustrating the important time instants for Stage-I and Stage-II spreading
for $We=27$, $Re=2300$, f = $400\text{Hz}$, A = $0.125mm$ and
$\phi=\pi/4~{}rad$. The top plot shows the evolution of substrate motion
($y_{s}/A=sin(2\pi ft+\phi)$). The illustration showcases the mechanism of
Stage II spreading.

###  4.2 Stage-II spreading

As discussed earlier and seen in Figures 4b, and 5, an impacting droplet on an
oscillating substrate may attain its maximum spread during the retraction
process due to the subsequent cycle of oscillations. If the condition is
conducive, the oscillation of the substrate may assist the droplet in
attaining a local instantaneous diameter greater than Stage-I spreading, which
occurs immediately after the impact (discussed and analyzed in the previous
section). We will now discuss the physical mechanism for Stage-II spreading
and the conditions required for $D_{max}$ to occur during this period.

Physically, at the point of maximum spread (Stage I), the droplet loses its
kinetic energy and contains excess surface energy in the form of a deformed
shape. In the absence of substrate motion, the droplet will undergo a
relaxation phase, where the diameter will reduce, and the height will
increase. It will eventually form a jet-like central liquid column (Fig. 2a).
The time for this jet formation ($t_{jet}$) is primarily controlled by the
capillary process and, in our experiments (shown in supplementary material),
is found to be

$t_{jet}\approx\frac{1}{2}\ t_{cap},$ (8)

where $t_{cap}=\sqrt{{\rho D_{0}^{3}}/{\gamma}}$ is the capillary time.
Yamamoto et al. (2016) also reported similar findings for their study with
droplet impact on the stationary superhydrophobic substrate. The relaxation of
the droplet can be approximated as a capillary process, and the instantaneous
diameter during the relaxation ($D_{2}$) can be expressed as

$D_{2}\approx D_{max,I}\cos(2\pi\Delta t/t_{cap}^{\prime}),$ (9)

where the $\Delta t$ is the time elapsed after maximum spread occurred at
$t=t_{max,I}$, and $D_{max,I}$ is the diameter at the maximum spread at Stage
I. Assuming that the droplet keeps its cylindrical shape during the relaxation
period, the reduction in the surface area ($\Delta SE_{rlx}=SE_{f}-SE_{rlx}$)
can be expressed as

$\Delta
SE_{rlx}=\gamma\pi\left(\frac{1}{4}(D_{max,I}^{2}-D_{2}^{2})+\frac{2}{3}D_{0}^{3}\left(\frac{1}{D_{max,I}}-\frac{1}{D_{2}}\right)\right).$
(10)

To achieve the maximum spread during Stage-II, the substrate oscillation must
provide additional energy to overcome the reduction in surface area during the
relaxation process, thereby causing a greater spread. From observation, it was
noted that this is achieved when the phase of the substrate oscillation at the
droplet impact is such that the substrate initiates the downward motion of its
sinusoidal trajectory during the relaxation phase of the droplet. This sudden
downward movement of the substrate momentarily pulls the adjacent (bottom)
part of the droplet while the top part undergoes the relaxation process. This
process lasts until the top of the droplet feels the downward motion of the
substrate, and the whole droplet starts moving downwards. Although short, this
process causes a momentary increase in droplet height ($\delta h$) during the
relaxation process. Recognizing that the disturbance from the bottom to the
top of the droplet is transported through capillary waves along the droplet
surface, the increase in droplet height can be estimated as $\delta h\approx
V_{s}t^{\prime}_{cap}$. Here, $t^{\prime}_{cap}=\sqrt{\rho
D_{max,I}^{3}/\gamma}$ the capillary time for the deformed droplet at the end
of Stage-I spreading. Once the substrate reaches its lowest position and
subsequently reverses the direction of its motion ($t=T_{trough}$), the
elongated droplet collapses, thereby causing the droplet to undergo Stage-II
spreading. The process, with the help of instantaneous droplet diameter and
substrate motion, is schematically shown in Fig. 10. Since the spreading
happens due to the vertical elongation of the droplet, which collapses, the
increase in surface area during Stage-II spreading ($\Delta SE_{II}$) can be
approximated as the additional potential energy stored in the elongated
droplet height, i.e.,

$\Delta SE_{II}=mg\delta h=\frac{\pi}{6}\rho D_{0}^{3}gV_{s}t^{\prime}_{cap}.$
(11)

Thus, the surface area and the diameter of the droplet after the Stage-II
spreading can be given by,

$SE_{II}=(SE_{f}-\Delta SE_{rlx})+\Delta
SE_{II}=\gamma\pi\left(\frac{D_{max,II}^{2}}{4}+\frac{2}{3}\frac{D_{0}^{3}}{D_{max,II}}\right),$
(12)

where $(SE_{f}-\Delta SE_{rlx})$ is the surface area of the droplet before the
Stage-II spreading has initiated. Clearly, for $D_{max}$ to occur during
Stage-II spreading ($D_{max,II}>D_{max,I}$) the required condition is

$\Delta SE_{II}>\Delta SE_{rlx}$ (13)

Furthermore, the aforementioned oscillation-assisted elongation of the droplet
and subsequent Stage-II spreading requires the downward motion of the
substrate to initiate during the relaxation period (bounded by $\tau$ and
$t_{jet}$) of the droplet. Hence the corresponding required condition is

$\tau<T_{crest}<t_{jet}.$ (14)

Here $T_{crest}$ is the time when the substrate initiates its downward motion,
identified by the crest of the sinusoidal trajectory of the substrate motion
(shown in Fig. 10). Equations 13 and 14, together, compose the necessary and
sufficient condition for maximum spreading to occur in the Stage-II. To test
this scaling analysis for Stage-II, in Fig. 11, we plotted the experimentally
measured difference between the maximum spreading diameter of Stage-I and
Stage-II as a function of $\Delta SE_{II}/\Delta SE_{rlx}$ for all
experiments. We observe that all the data with $D_{max,II}>D_{max,I}$
(identified by green symbols) are located in a range of $\Delta SE_{II}>\Delta
SE_{rlx}$. Similarly, the (red) data points which do not show the Stage-II
spreading ($D_{max,2}<D_{max,I}$), are mostly lying in the regime of $\Delta
SE_{II}<\Delta SE_{rlx}$. We also notice a few outliers, which show
$D_{max,II}<D_{max,I}$, even though they satisfy the theoretical condition for
Stage-II spreading ($\Delta SE_{II}>\Delta SE_{rlx}$). It is to be noted for
these points, the observed difference between $D_{max,I}$ and $D_{max,II}$ are
rather small ($<5\%$), and most lie within the experimental uncertainty.
Overall, we can infer that the experimental observation closely supports the
necessary conditions derived for Stage-II spreading.

Figure 11: Illustration of energy criteria for Stage-II spread. Experimentally
observed percentage change in droplet spread diameter from Stage-I and Stage-
II as a function of $\Delta SE_{II}/\Delta SE_{rlx}$ for all cases. The dotted
line $\Delta SE_{II}/\Delta SE_{rlx}=1$ marks the threshold for the occurrence
of maximum spread at Stage-II, differentiated by the green colored symbols.

Since Stage-II spreading occurs at the instance when the downward motion of
the substrate ends and the upward motion begins, the maximum spreading time
can be approximated by $t_{max,II}\approx T_{trough}$. Finally, by using Eqs.
10 \- 12, one can evaluate the theoretical approximation of $D_{max,II}$, the
maximum diameter obtained in Stage-II. In Fig. 12 a and b, we compare the
experimentally measured $t_{max}$ and $D^{*}_{max}$ with the theoretically
obtained values for all experiments, including both Stage-I and Stage-II
spreading. To evaluate the theoretical values, we used scaling for Stage-II if
the experimental condition satisfies Eq. 13 and 14. The comparison confirms
that the presented scaling analyses can satisfactorily predict the spreading
time and maximum spreading diameter.

Figure 12: Comparison of experimental and theoretically predicted values for
$\textit{t}_{max}$ and $D^{*}_{max}$ at instant of maximum spreading taking
theoretical estimate of Stage-II into consideration.

Before we conclude this section, we will discuss two additional observations,
which can be made from Eq. 14, one of the necessary conditions for Stage-II
spreading. From definitions, we know that $\tau\sim D_{0}/V_{0}$,
$t_{jet}\sim\sqrt{{\rho D_{0}^{3}}/{\gamma}}$, and thus,
$t_{jet}/\tau\sim\sqrt{We}$. Since Eq. 14 inherently implies $t_{jet}>\tau$,
Stage-II is expected to occur only for high impact inertia ($We>1$)
conditions. Furthermore, by using the properties of a sinusoidal trajectory
(See Fig. 10), we can express $T_{crest}=(5\pi/4-\phi)/2\pi f$. Substituting
this in Eq. 14 and recalling that $\phi>0$, we find $f>5/8t_{jet}$. This
suggests that a minimum frequency is required for substrate oscillation to
assist Stage-II spreading. In experiments, indeed, Stage-II spreading can be
observed only for $f\geq 250Hz$.

##  5 Conclusion

In summary, we presented a detailed experimental study highlighting the
effects of an oscillating hydrophobic substrate on the spreading process of an
impacting droplet. We showed that the maximum droplet diameter and the time
taken are greatly affected by parameters of substrate oscillation, such as the
frequency, amplitude, and phase at impact. The maximum spread can occur in two
stages. The oscillation may promote or inhibit spreading in Stage-I, which is
primarily controlled by impact inertia. The scaling analyses showed that the
instantaneous motion of the substrate alters the relative velocity between the
droplet and the substrate, thereby changing the timescale of spreading. The
energy budget confirms this change in spreading time will greatly affect the
viscous dissipation; thus, the droplet attains a different spreading diameter.
The phase of the oscillation at the impact greatly affects this process.

While inertia-controlled Stage-I spreading is observed for all conditions, a
secondary spreading process (Stage-II) can be observed for certain impact
conditions in which the droplet attains the maximum diameter at the latter
stage. The Stage-II spreading is controlled purely by substrate oscillations.
For a higher frequency of oscillation, if the phase at the impact is such that
the substrate initiates its downward motion after the Stage-I spreading, the
droplet undergoes sudden elongation. This allows the droplet to attain greater
potential energy, and when it collapses, the droplet spreads to a larger
diameter. Based on the scaling argument, the necessary conditions for the
Stage-II spreading were identified, which showed good agreement with
experimental data.

We end this exposition by discussing the relevance and limitations of the
finding in the context of applications. In many industrial processes, droplet
impact involves heat and mass transfer between the droplet and the substrate.
Thus, it is of interest to modulate the spreading diameter and time. Our study
provides a guideline to design the oscillation parameters for applications
where substrate oscillations can be used for such modulation. It should be
noted that the present study has been conducted in a frequency range of less
than 1kHz, for which we expect the proposed scaling to hold. However, very
different behavior of liquid droplets is observed for high (MHz-range)
frequencies, where the current scaling may not be applicable.

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