# Water drop impact on thin viscous oil layers

Surjit Bharatsingh†, Piyush Sahu†, Gaurav Salwan  Dileep Mampallil
<EMAIL_ADDRESS>Indian Institute of Science Education &
Research Tirupati, Yerpedu P. O. PIN 517619, Tirupati, AP, INDIA

(August 28, 2024)

###### Abstract

We conducted experimental investigations into the short-term and long-term
impact dynamics of pure water drops on silicone oil-coated surfaces. Our
observations revealed distinct phases: initially, rapid maximal spreading
followed by retraction, subsequent oscillations characterized by a relatively
stable contact angle, and a final phase involved the gradual spreading of the
drop on the underlying solid surface due to the rupturing of the oil layer.
The maximal spreading radius follows $We^{1/4}$ scaling, where $We$ is the
Weber number, independent of the viscosity of the underlying oil layer. By
introducing fluorescent tracer particles in the oil, we noted that the drop
expanded over the oil layer during the initial spreading without displacing
oil. Subsequently, as the oscillations dampen, the drop ruptures the oil
layer, initiating a dewetting process and spreads over the solid surface,
leaving a tiny oil droplet under the water drop. Upon completion of this
dewetting process, the onset time for the drop spreading nonlinearly increased
with the oil layer thickness as $\sim d_{o}^{2}$, which we attribute to the
dewetting dynamics. Our findings unveil intriguing dynamics of water drops
impacting on oil-coated surfaces.

Keywords: Drop impact, oil film
† Equal contribution

##  I Introduction

Fundamental understanding of drop impact [1, 2, 3, 4, 5] on various surfaces
such as pre-wetted [4], superhydrophobic [6], granular [7, 8] and liquid [1,
9] surfaces and subsequent spreading [10] has implications in various fields
such as inkjet printing [11], drop-impact printing [12], cooling systems [13],
spraying of pesticides and medicines [14], to list a few. These applications
also involve the impact of drops of complex fluids [15, 16], such as blood
drops, where the resulting patterns have implications in forensics [17, 16].
Thus, understanding the fundamental dynamics of drop impact is important in
many processes.

The dynamics of a liquid drop after impacting a surface depends upon the drop
size, impact velocities, properties of the liquid, and the physical nature of
surface, such as wettability and roughness [5, 2, 3]. The impact dynamics
demonstrate spreading, rebounding, splashing, and break-up depending upon the
liquid and surface proprieties and impact energies [2, 3]. Upon impact, the
inertia expands the liquid on the surface, which is balanced by viscous and
surface tension forces [18, 19]. Drop impact on hydrophobic and
superhydrophobic surfaces [20, 6] shows bouncing of the drop with a short
contact time that depends upon the inertia, surface tension, and liquid-
surface interactions [21, 22]. Relative temperature difference between the
impacting liquid and the solid can also lead to non-sticking effects [23]. The
ambient medium (usually air) also influences drop impact dynamics on solid
surfaces. The local increase in pressure between the impacting face of the
drop and substrate surface deforms the impacting face, eventually entrapping a
thin layer of air upon impact [24]. Drop impact on granular medium [25, 26,
27] can demonstrate the effects of this air entrapment as patterns on the
granular layer and also microbubble generation [8].

Impact drops on soft and viscoelastic surfaces [28, 29], and even immiscible
liquids [9], have also attracted scientific interest. The spreading and
retraction of drops on soft surfaces strongly depend upon the viscoelastic
properties of the surface [29, 30]. On such surfaces, the contact angle
hysteresis can dissipate much of the impact energy, especially during
retraction.

In the case of drop impact on liquid layers, the competition between the
inertial, capillary, and viscous effects determines the dynamics after impact.
An example is the water drop impact on oil layers [31] and oil-infused
surfaces [32, 33, 34]. These studies show that impact energy and oil viscosity
determines the dynamics of both the drop and the oil [31, 32]. Further, Lee et
al. showed that the viscosity of the oil film does not have significant effect
on the maximal spreading radius of impacted water drops [32]. However, at
lower oil viscosity, the bouncing of the drop is enhanced [33]. The crown
formation of the impacted drop also vivid at relatively higher impact
velocities [31]. Previous studies indicate that the interaction between water
drops impacting oil layers, or, more broadly, the impact of drops on liquid
layers, is a complex process that requires further exploration.

In this research, we experimentally study water drop impacting a highly
viscous silicone oil layer coated on various substrates and demonstrate
several previously unreported dynamics. When the drop impacts, it rapidly
spreads without disrupting the oil layer. However, during the subsequent
retraction phase, the contact line drags the oil layer inward and thus the
drop dissipates the impact energy. Additionally, the drop exhibits damping
oscillations and slowly penetrates the oil layer. We find that the time the
drop takes for this penetration increases with the oil layer thickness and
impact velocity, contrary to the expected rupturing the oil layer immediately
upon impact. We systematically investigate the different dynamics exhibited by
the impacted drop and draw comparisons with established scaling laws.

##  II Experiments

We spin-coated silicone oil on glass slides sequentially cleaned with
isopropyl alcohol and water followed by air drying. The spin coating of
silicone oil was performed at different rpm ranging 500 to 5000. Different rpm
resulted in different thickness of the oil, depending upon the oil viscosity
($\mu_{o}$), such as 370 and 10000 cSt (Sigma Aldrich). The oil thickness was
obtained by comparing the weight of the glass slide in a microbalance
(Quintix224-10IN) before and after the uniform coating with the oil. The
thickness was obtained as $d_{o}=\delta m\/\rho_{o}A$, where $\delta m$ is the
oil mass, $\rho_{o}$ is the density of oil, and $A$ is the area of the glass
slide. An average thickness was obtained from five different samples. The
relationship between the spin rpm and the oil layer thickness for different
oils are shown in Supplementary Fig.S1. We compared the measured oil
thicknesses to the ones obtained using the Meyerhofer model [35], which gives
the spin-coated thickness as $(3\mu_{o}/4\rho_{o}\omega^{2}T)^{1/2}$, where,
$\omega$ is the angular velocity of spinning and $T$ is the duration of the
spinning. The calculated values were higher than the measured ones up to two
times, especially at lower rpm. We used the experimental values of the
thickness for further analyses.

We coated oil on different substrates such as, glass, indium tin oxide (ITO),
and polydimethylsiloxane (PDMS). On these bare surfaces (without oil coat),
water drops have contact angles of 22 $\pm$ 2∘, 60 $\pm$ 2∘. and 100 $\pm$ 2∘,
respectively.

Water drops of volume 6.5 $\mu$l (with 5% deviation) having radius $R$ = 1.1
mm was dropped from a Teflon tubing connected to a syringe pump. The impact
height varied from 0 to 0.95 m. The corresponding impact velocity varied
between 0 and 4.3 m/s. Upon impact, the dynamics of the drop was imaged using
a high-speed camera (Phantom MICRO LAB110) at frame rates up to 5000. The
images were analyzed using ImageJ software and homemade Matlab code to obtain
the contact angle and base radius as a function of time.

Figure 1: Impact dynamics of water drops dispensed from three different
heights (3, 10, 30 cm) onto silicon oil layer of thickness 7 $\mu$m. The oil
viscosity was 370 cSt. From the video recorded at 2000 fps, the images
corresponding to different regimes are shown. The oscillation phase continues
to several hundreds of milliseconds. During this process the drop edge
penetrates into the oil layer. This aspect is characterized by the spreading
of the drop as it meets the underlying hydrophilic surface (glass).

## Results

To begin, we will provide an overview of the observations regarding the
behavior of liquid drops impacting on oil-coated glass slides as shown in
Fig.1. The dynamics of the drop’s base radius and contact angle are quantified
in Fig. 2(a) and 2(b), respectively. Upon impact, the drop rapidly spreads to
reach its maximum base diameter ($D_{max}$) in approximately 4 milliseconds.
As the impact velocity increases, the time required to achieve maximum
spreading becomes shorter (Fig.1). Just before the drop retracts, the contact
angle decreases to a minimum value in proportion to the impact velocity.
Subsequently, the drop enters a phase of retraction followed by damping
oscillations while maintaining a relatively constant average contact angle of
around 100 degrees in a quasi-equilibrium state. Notably, after about a
second, the drop undergoes a slow spreading phase, indicated by an increase in
the base radius and a decrease in the contact angle. We will delve into these
features of the curves in Fig 2 in the subsequent discussions.

Figure 2:  The base radius and contact angle of drops placed softly and
impacted from two different heights (3 and 10 cm) onto silicon oil layer of
thickness 7 $\mu$m coated on glass. The oil viscosity was 370 cSt. Upon
maximum spread, the contact angle decreases to a minimum and the contact line
is dragged on the oil layer during the retraction. With increasing impact
height, the energy loss during the dragging phase increases resulting in a
weak oscillation phase. The drop spreads on the surface at $t=t_{R}$, when the
contact line meets the substrate, manifested by a decrease in the contact
angle.  Figure 3:  The maximum spreading parameter $\beta$ as a function of
Weber number. The error bars are the standard deviation from at least five
measurements. The inset illustrates a drop spreading on the oil layer with
various parameters marked.  Figure 4:  Images showing the impact dynamics of a
drop on an oil layer (viscosity: 370 cSt) containing suspended fluorescent
particles (diameter 2 $\mu$m). (i)-(ii): From impact to maximum spreading no
particles are displaced from their original position, implying the integrity
of the oil layer during the drop spreading. (ii)-(iii): Only, a few particles
are dragged radially inward during the retraction phase. The yellow dashed
circle represents the edge of the water drop. (iv)-(v): From the beginning and
end of the rupturing of the oil layer, particles are displaced as the drop
penetrated the oil layer. The red dashed circle in (v) represents the oil
droplet trapped at the center.

The maximum spreading diameter: Upon impact, in the absence of splash, the
maximum spreading diameter [18, 36, 37, 38, 3, 39, 40] is an important
parameter in several applications involving drop impact. The maximum spreading
is characterized by a factor, $\beta$, which is the ratio of the diameter upon
maximum spreading ($D_{max}$) to that at the initial spherical drop ($D=2R$).
The value of $\beta$ results from the balance of kinetic energy, surface
energy, and viscous dissipation. Considering these aspects, numerous
relationships were derived connecting $\beta$ to dimensionless numbers such
as, Webber number, ${We}=\rho_{w}Du_{i}^{2}/\sigma_{w}$ and Reynolds number,
${Re}=\rho_{w}Du_{i}/\mu_{w}$, where, $\rho_{w}$ is the density of the
impacting liquid (water), $u_{i}$ is its impact velocity, $\sigma_{w}$ is the
surface tension, and $\mu_{w}$ is the viscosity of the impacting drop. In the
viscous regime where spreading occurs by a balance between the kinetic energy
and the viscous dissipation, $\beta\sim{Re}^{1/5}$ was reported [37, 3]. When
viscous effects are small, a comparison of the initial kinetic energy with the
capillary energy has predicted $\beta\sim{We}^{1/2}$ at asymptotic regimes of
large Webber numbers. Clanet et al. derived another scaling law,
$\beta\sim{We}^{1/4}$, considering the volume conservation between the initial
spherical and the final pancake shapes [18]. They considered that the final
thickness of the pancake drop is proportional to the capillary length
$a_{c}=\sqrt{\sigma_{w}/(\rho_{w}g^{\prime})}$, where $g^{\prime}\sim u_{i}/D$
is an increased acceleration experienced by the drop during the spreading.

In our experiments, we observe the scaling $\beta\sim{We}^{1/4}$. Although the
scaling ${We}^{1/4}$ was first demonstrated for low-viscosity drop impact on
strongly water repellent surfaces [18], impact on various surfaces [38, 3]
including oil-infused surfaces [32] have shown this scaling. Impact of viscous
drops such as water-glycerol (6 & 51 cSt) and blood drops on steel surfaces
have also shown ${We}^{1/4}$ scaling [40]. Various other works reported
different scaling laws as ${We}^{1/2}$, ${We}^{1/4}$, ${Re}^{1/4}$,
${Re}^{1/5}$ or complex relationships suggesting a universal scaling [40, 42,
3]. Since correlating the scaling nature with the experimental parameters is
difficult, recently, machine learning has been employed to explore the maximal
spreading dynamics [43].

The scaling law being independent of the viscosity of the oil layer can be
understood by comparing the viscous dissipation, $\int\mu(\partial u/\partial
z)^{2}dV$, inside the water drop and the underlying oil layer (Here, $dV$ is
the volume element; also see inset of Fig.3). During spreading, the ratio of
total viscous dissipation in oil to that in the water can be written as

$\frac{E_{o}}{E_{w}}=\frac{\mu_{o}(u_{o}/d_{o})^{2}\,d_{o}\pi
R_{p}^{2}}{\mu_{w}((u-u_{o})/d_{w})^{2}\,d_{w}\pi R_{p}^{2}},$ (1)

where $R_{p}$ is the radius of the pancake drop, $d_{o}$ is the oil layer
thickness, $d_{w}$ is the pancake drop thickness, $u$ is the fluid velocity at
the top of the drop, and $u_{o}$ is the fluid velocity at the water-oil
interface (see inset of Fig.3). At the interface between the water and oil,
the viscous stress must balance. Thus, we can write,
$\mu_{w}(u-u_{o})/d_{w}\approx\mu_{o}u_{o}/d_{o}$. Therefore, we can rewrite
Eq.1 approximately as

$\frac{\mu_{w}}{\mu_{o}}\frac{d_{o}}{d_{w}}\ll 1.$ (2)

Eq.2 implies that the dissipation of the impact energy inside the oil is
negligible compared to that in the spreading water drop [41]. Thus, the
viscous properties of the underlying oil layer do not play a crucial role in
the spreading dynamics. Thus, we can expect the same scaling laws regardless
of the oil layer viscosity.

To understand how the initial fast spreading of the water drop displace the
oil, we dispersed red fluorescent polystyrene particles of diameter 2 $\mu$m
in the oil (viscosity: 370 cSt). We observed the impact from below the glass
slide (Fig.4). Upon impact, the drop spreads without creating any noticeable
movement of the particles indicating that the spreading is mostly peripheral.
Also the side view of the drop demonstrated an advancing angle more than 90∘.
After the maximum spread, during the retraction, drop drags the oil radially
inward manifested by the corresponding movement of the fluorescent particles.
It demonstrates that the drop dissipates its energy in the oil mainly during
the retraction phase.

Other effects may also influence the dynamics of impacted drop spreading. For
instance, air film entrapment between the impacting drop and the oil layer
[24, 44, 45] might cause a differences in the spreading dynamics on the low
and high viscous oil surfaces. Since the surface of the relatively low viscous
oil deforms faster, a wide air film can be trapped there, which might be
minimal with a high viscous oil layer. While the specific mechanisms altering
the scaling behavior remain unclear, our measurements suggest that changes in
the viscosity of the impacting surface strongly affect the dynamics of
spreading and thus the scaling behaviour.

Figure 5:  (a) An example of fit of Eq.4 on experimental data of contact line
oscillations of a drop impacted from 1 cm height. (b) The fitting parameters,
$\gamma$ and $\alpha$, as a function of the Weber number. The red lines fitted
on $\gamma$ and $\alpha$ are having slops of 0.13 $\pm$ 0.015 and -0.15 $\pm$
0.013, respectively.

Drop oscillations: Upon retraction, the drop executes damping oscillations.
The oscillatory motion of the contact line can be seen in Fig.2. A simple
analogy with the spring-mass system can capture the damped oscillation
dynamics of the drop [46]. Therefore, we can write,

$m\ddot{y}+c\dot{y}+ky=0,$ (3)

where, $m$ is the mass, $y$ is the displacement, $c$ is the damping
coefficient and $k$ is the spring constant. The general solution to Eq.3 is

$y(t)=Ae^{-\alpha\,t}\sin(\gamma t+\phi),$ (4)

where $\alpha=c/2m$, $\gamma=\sqrt{\omega^{2}-\alpha^{2}}$, $\omega^{2}=k/m$,
and $A=y(0)$.

In Fig.3(a), we plotted the frequency $\gamma$ and the damping factor
$\alpha$, obtained by fitting Eq.4 to the oscillating regions of the curves,
as a function of ${We}$, i.e, with increasing impact velocity. We see that
$\alpha\sim{We}^{-0.15}$ and $\gamma\sim{We}^{0.15}$, where the same power
with opposite signs is expected as $\gamma=\sqrt{\omega^{2}-\alpha^{2}}$.

An increased degree of damping with increasing ${We}$ is expected. It is
important to note that the retraction phase is responsible for dissipating
most of the energy. When the impact velocity is higher, the droplet spreads
over a larger area, and the dissipation increases proportionally. Thus, with
increasing ${We}$, the available energy for oscillations becomes less and the
initial amplitude of oscillation decreases. Therefore, the effective decay
rate $\alpha$ also decreases with ${We}$.

The damping factor is related to the ratio of viscous force to inertia force
or the inverse of the Reynolds number. Therefore, in Supplementary Fig.S2, we
have plotted the values of $\alpha$ as a function of ${Re}$, which indicates a
relationship $\alpha\sim{Re}^{-1/4}$.

A typical damping time can be estimated as $t_{damp}=\rho_{w}R^{2}/\mu_{w}$,
which gives a value of about 1 s with $R\sim 1$ mm. Using the viscosity of oil
($\mu_{o}$), we estimate a damping time of 1 ms, which is much shorter than
the experimentally observed value of about 0.3 s. Thus, we assume that most of
the damping occurs within the water drop, also in accordance with Eq.2.

Figure 6:  (a) About 0.5 s after impact, water ruptures the oil layer (marked
by the arrow) and initiate the dewetting of the oil layer. Finally, under the
spreading water drop, an oil droplet is formed (see inside the dashed circle
on panel of 1.0 s and Supplementary video 1). The bright circular periphery of
the drop is due to light reflection. (b-c) The onset time of drop edge
spreading $t_{R}$ depends on ${We}$ and $d_{o}$. (b) $t_{R}$ demonstrates an
increase with the Weber number, with power approximately as ${We}^{1/4}$.
($d_{o}$ = 7 $\mu$m). (c) We note that $t_{R}$ shows an increasing trend with
the thickness of the oil layer. The red solid line has a slope of 2.0.  Figure
7:  After impact, the drop spreads across the solid surface beneath the oil
layer after a duration $t_{R}$ (marked by the arrows). The base radius $R_{b}$
follows a scaling relation of $t^{0.25}$. The impact heights are shown in the
panel. The schematic in the inset illustrates the initial (dashed line) and
final (solid line) shapes of the drop as it spreads.

Drop penetration into the oil layer and spreading: Throughout the damping
oscillations, both the base radius and the corresponding contact angle
maintained a consistent average value for a duration of $t_{R}$ as evident in
Fig.2. For example, the contact angle, constant around approximately 100∘,
decreases to approximately 35∘ at around $t_{R}$. The corresponding increase
in the base radius is also apparent. We observed similar spreading also with
PDMS and ITO surfaces coated with oil. When an oil layer (7 $\mu$m thick) was
applied to PDMS and ITO surfaces, after a duration $t_{R}$, the final contact
angles were measured to be 100 $\pm$ 2∘ and 65 $\pm$ 2∘, respectively. These
values closely align with the contact angles observed on the corresponding
bare surfaces, namely 100 $\pm$ 2∘ for PDMS and 60 $\pm$ 2∘ for ITO. These
findings strongly suggest that the change in contact angle occurs as the
drop’s contact line penetrates the oil layer and interacts with the underlying
substrate. This is expected, as the oil density (971 kg/m3) is slightly lower
than that of water.

On glass, the final contact angle did not match the contact angle of water on
the bare substrate. For example, while the drop displayed a contact angle of
22∘ on bare glass, it exhibited a slightly larger angle (approximately 35∘)
after spreading with an oil layer. This difference might be attributed to the
potential presence of a thin oil film trapped beneath the spreading drop [47,
48].

We further looked into the details of the water penetration into the oil
layer. The bottom view of the drop shows that, before the water drop spreads
over the underlying substrate, the water ruptures the oil layer and the
dewetting of the oil layer follows. The rupturing and dewetting process starts
only a few hundred microseconds after the impact. This short delay might be
due to the entrapped air layer between the drop and the oil as observed in
experiments involving water drops being engulfed by the oil with some initial
delay due to air entrapment [49].

The dewetting process is shown the images in Fig. 6(a) and Supplementary video
1. The entrapped air and the dewetting process delay penetration of the drop
edge into the oil. This overall delay in the drop edge penetration, denoted by
$t_{R}$, amounts to about one second on 370 cSt oil and over 50 seconds on
10000 cSt oil layers (Fig. 6(b)).

The rupturing and dewetting of the oil is initiated somewhere under the bulk
region of the drop (Fig. 6(a) and Supplementary video 1). Under the drop, as
the dewetting advances a hole is opened up in the oil layer and the water wets
the substrate. Assuming a thin oil layer at the dewetting region, the initial
dewetting velocity can be estimated as [50],

$v_{dw}=\frac{k\,\sigma_{ow}}{\mu_{o}}\theta_{E}^{3},$ (5)

where $\sigma_{ow}$ is the interfacial tension between oil and water,
$\theta_{E}^{3}$ is the equilibrium contact angle of oil on glass surface, and
$k$ is a numerical coefficient that ranges between 10-2 and 10-3 and related
to the molecular properties of the liquid [50]. With $\sigma_{ow}$ = 39 mN/m
[51], $\theta_{E}$ = 87∘, and $k$ = 10-2, we calculate a velocity of 3.7 mm/s
for the case $\mu_{o}$ = 370 cSt. This value is very close to the measured
dewetting velocity of 5 $\pm$ 2 mm/s for oil viscosity 370 cSt. The measured
value was about 0.03 mm/s for oil viscosity 10000 cSt.

As shown in Fig. 6(a), eventually, the dewetted oil converges to form a small
droplet at the central region under the water drop (Supplementary video 1).
This oil droplet formation is driven by the relatively strong interfacial
tension of oil/water interface (39 mN/m) [51]. Despite the oil having a
slightly lower density, the oil droplet adheres to the substrate without
ascending within the water drop.

We observed that only when the dewetted oil converges under the water drop,
the edge of the water drop starts to spread outward, i.e., at time $t_{R}$.
Thus, neglecting the delay (a few hundred microseconds) imparted by the air
entrapment, the value of $t_{R}$ is mainly the time for completion of the
dewetting process, which can be estimated as

$t_{R}\sim\frac{R_{b}}{v_{dw}},$ (6)

where $R_{b}$ is the equilibrium base radius of the water drop. With $R_{b}$ =
1.6 mm, we obtains $t_{R}$ as 0.43 s and 11 s for oils of viscosity 370 and
10000 cSt, respectively. The $t_{R}$ values for the highly viscous oil can be
even larger, as the values of $k$ may be smaller than 10-2. Therefore, the
estimated values of $t_{R}$ align closely with the observed ones in our
experiments.

We further study the influence of the impact parameters on the delay time
$t_{R}$. The value of $t_{R}$ rises with the ${We}$, as demonstrated in Fig.
6(b). This observation may seem counter-intuitive, as one might anticipate
that a greater impact velocity would expedite the drop’s penetration into the
oil layer, resulting in an earlier encounter with the underlying substrate.
However, the opposite trend suggests that the impacting drop does not promptly
displace the oil layer, as was also evident from Fig. 4. We observed that
$t_{R}$ follows a relationship of $t_{R}\sim{We}^{0.25\pm 0.05}$ for oil with
viscosity 370 cSt, and $t_{R}\sim{We}^{0.4\pm 0.1}$ for oil with viscosity
10000 cSt, as shown in Fig. 6(b).

Our measurements show that $t_{R}$ depends upon the oil thickness as well.
Remarkably, as illustrated in Fig. 6(c), it follows a power-law relationship,
denoted by $t_{R}\sim d_{o}^{1.9\pm 0.2}$. This exponent, exceeding 1.0,
implies that the drop’s edge isn’t sinking freely. The observed exponent can
be related with the dewetting process. When a hole appears on a viscous thin
layer of thickness $e$, the dewetting proceeds until its thickness increases
to a value $e_{c}=2\kappa^{-1}sin(\theta_{E}/2)$, where $\kappa^{-1}$ is the
capillary length [50]. Thus, as the layer thickness increases during the
dewetting, the velocity takes the form [50],

$v_{dw}\sim e_{c}^{2}-e^{2},$ (7)

with $e(t)\leq e_{c}$. Thus, $v_{dw}$ decreases as $e_{c}^{2}-e^{2}$ with
time. An estimated value of $t_{R}$ can be obtained from the average velocity
as $t_{R}\sim R_{b}/\left\langle e_{c}^{2}-e^{2}\right\rangle$. It aligns with
the observation that $t_{R}\sim d_{o}^{2}$. The exact expression connecting
the temporal variation of $v_{dw}$ and the observed scaling with initial oil
layer thickness ($d_{o}$) and Weber number may require further theoretical
investigations.

Followed by the dewetting, the drop spreads over the substrate surface. It
shows an increase in the base radius (see region after about 1 s in Fig.
2(a)). We plotted this region of the curves in Fig. 7. Our findings reveal a
scaling behavior of $R_{b}\sim t^{0.25\pm 0.07}$. Typically, the initial
spreading dynamics of drops conform to a power law. For instance, in the
inertial-capillary regime, a scaling law of $R_{b}\sim t^{0.5}$ has been
documented [52, 53]. However, in the viscous-capillary regime, various
exponents have been reported [52, 54]. As the viscous drop spreads, the
exponent diminishes over time, eventually reaching 1/10 on wetting surfaces
[55, 52]. Despite our drop being a low-viscosity liquid (water), the
resistance posed by the viscous oil affects the movement of the contact line.
Moreover, the spreading regime does not align with early times, as the drop
had already partially spread on the oil layer. Consequently, we anticipate a
smaller exponent, indicative of the late-time dynamics of viscous droplets, in
our case.

## Discussion

Our observation that the impacted drop initially does not deform the oil layer
contrast with the findings from other studies on water drop impact on oil [31,
32, 33]. For instance, Che et al. [31] examined water drop impact on silicone
oil layers with viscosities ranging from 5 to 50 cSt. They noted that the
impacted drop displaces the oil layer, but this effect diminishes with
increasing oil viscosity, consistent with our findings. Our measurements,
conducted at much higher viscosity (370 and 10000 cSt), reveal that the
impacted drop does not disturb the oil layer during its spreading phase.

While the drops spread on high-viscosity oil almost akin to a solid surface,
the retraction phase involves significant viscous dissipation. This could be
attributed to the dragging of the oil along with the retracting contact line.
Conversely, in experiments with very low viscosity, such as our trials with 20
cSt silicone oil, the retracted water drop retained considerable kinetic
energy, resulting in the formation of a vertically elongated jet shape (see
Supplementary Fig. S3). At large impact energies, this jet subsequently broke
up to form a small drop at the tip. This observation aligns with previous
findings indicating that the bouncing of the impacted drop occurs when the oil
viscosity is low [33].

## Summary

In our experimental study, we investigated the impact of water drops on thin
layers of viscous oil applied to solid surfaces. We observed a series of
dynamic phenomena, including maximal spreading and retraction, oscillations,
delayed penetration into the oil layer, and spreading on the solid surface. We
show that the viscosity of the underlying oil layer does not affect the
scaling relationship for the maximum spreading radius ($\beta\sim{We}^{1/4}$).

Following spreading and retraction, the drop undergoes oscillations,
maintaining a constant average base radius and contact angle for a duration
$t_{R}$ (a few seconds). During this time the water slowly rupture the oil
layer and dewet the oil under the drop, eventually converging as a small oil
droplet under the spreading water drop. Intriguingly, we observed a nonlinear
relationship between $t_{R}$ and the thickness of the oil layer ($t_{R}\propto
d_{o}^{2}$), which we relate to the dewetting dynamics of oil layer under the
water drop. Similarly, $t_{R}$ increases with the Weber number approximately
as ${We}^{1/4}$, indicating a complex interplay between the dewetting process,
oil layer thickness, and impact energy. In summary, we unveil previously
unreported short-term and long-term dynamics of drops impacting on oil layers.
Our findings may inspire new experimental and theoretical investigations in
related areas.

##  III Acknowledgment

DM acknowledges IISER Tirupati intramural funds and Science and Engineering
Research Board (India) grant CRG/2020/003117.

##  IV Competing financial interests

The authors declare no competing financial interests.

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