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# Miscibility and wettability: how interfacial tension influences droplet
impact onto thin wall films

Ronan Bernard1<EMAIL_ADDRESS>David Baumgartner2  Günter
Brenn2  Carole Planchette2  Bernhard Weigand1  Grazia Lamanna1  1Institute of
Aerospace Thermodynamics, University of Stuttgart, 70569 Stuttgart, Germany
2Institute of Fluid Mechanics and Heat Transfer, Graz University of
Technology, 8010 Graz, Austria

###### Abstract

The influence of miscibility and liquid wettability during droplet impact onto
thin wall films is investigated experimentally. Despite similar liquid
properties and impact conditions, differences in the splashing limit, the
crown extension and the duration of the ascending phase are observed. These
differences are related to the interfacial tension of the droplet/wall-film
liquid pairs, which is linked to their miscibility and wettability. More
precisely, by calculating the crown surface energy, we show that the energy
stored in the interface between droplet and wall-film (if any) is not
negligible and leads to smaller crown extensions and the need of more kinetic
energy to initiate splashing. Similarly, by calculating a modified capillary
time taking into account all surface and interfacial tensions, we show that
the interfacial tension acts as a non-negligible recoiling force, which
reduces the duration of the ascending phase. The dynamics of this ascending
phase is well captured for different wall-film thicknesses if accounting for
the variations of the liquid masses in movement. Overall, droplet/wall-film
interactions can be seen as inertio-capillary systems where the interfacial
tension between droplet and wall film plays a significant role in the storage
of energy and in the crown kinetics during the impact process. Besides, this
analysis highlights that viscous losses have already a significant effect
during the crown extension phase, by dissipating almost half of the initial
energies for droplet impact onto thin wall films, and most likely by
influencing the capillary time scale through damping.

###### keywords:

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##  1 Introduction

Droplet impact processes can be observed in many natural events and technical
applications, such as rain, combustion chamber with lubricating film, spray
coating, cosmetic and pharmaceutical production. In the last decade, the
interest in binary systems (i.e. with different liquids for droplet and
impacted liquid substrate) has increased. Besides their technical relevance,
they offer the possibility to get a deeper understanding of the interaction
between droplet and impacted liquid. By investigating this interaction, the
question of the influence of miscibility and wettability between the droplet
and the impacted liquid substrate arises.

Few studies on miscibility or wettability can be found for various droplet
impact configurations, but barely for droplet impact onto wall-films. For
droplet impact onto deep pools, the effect of wettability (in terms of
spreading parameter) has been studied numerically with water droplets
impacting into an oil bath (Wang et al., 2020). The authors analysed the
horizontal and vertical penetration into the oil cavity and found no
significant effects of the wettability compared to those of the Weber number
and the viscosity ratio for this bulky impact configuration where the droplet
spreading is rather small and thus, does not favor interfacial forces in the
process. Additional studies on deep pool considered different miscible and
immiscible pairs with pure water, ethanol and silicone oil impacting onto
water (Hasegawa & Nara, 2019). However, since the liquid properties were very
different, the effects of miscibility alone could not be quantified. Only a
qualitative observation of a Worthington jet during the recoiling phase could
clearly be attributed to the difference in miscibility. For droplet impact
onto a continuous jet, the effects of wettability and miscibility were studied
independently of the liquid properties by Baumgartner et al. (2020). This
study was carried out jointly with the present one in order to investigate the
importance of the impact configuration in similar physical droplet impact
processes with the same liquids. Droplet impact onto a continuous jet exhibits
generally lower droplet spreading. Furthermore, the absence of a solid wall
and the bulkier impact configuration leads to an impact kinetics fixed by the
encapsulated drop only, and most of the effects of wettability and miscibility
were observed in the recoiling phase or on the phenomenology of the impact
(e.g. whether the droplet merges, is encapsulated or spreads around the jet).
In the case of droplet impact onto thin wall films, which is the focus of this
work, the liquid structure formed is a lamella expanding from the impact
point, the so-called crown, as shown in the images of Fig. 1. Under certain
impact conditions, the expanding crown can destabilize at the rim, leading to
ejection of secondary droplets, referred to as crown-type splashing, which
needs particular attention in the above mentioned applications. While the
features of droplet impact onto thin wall films with similar liquids, miscible
per definition, have been quite extensively studied (see e.g. the review
article of Liang & Mudawar (2016)), studies on binary droplet/wall-film
systems are rarely found in the literature. Originally studied in the
pioneering work of Worthington & Cole (1897) with water and milk, most of the
studies on binary systems are with miscible liquid pairs (Geppert et al.,
2017; Bernard et al., 2018; Geppert, 2019; Geppert et al., 2016; Bernard et
al., 2017; Kittel et al., 2018b; Thoroddsen et al., 2006) and focus on
splashing limit, crown dynamic or on the viscosities. In contrast, binary
systems with exclusively immiscible droplet/wall-film systems have been
investigated mainly with respect to the characteristics of the secondary
droplets (Shaikh et al., 2018) and to the repartition of the droplet and wall-
film liquids in the crown depending on the viscosity ratio (Kittel et al.,
2018a).

Only few studies report directly or indirectly an effect of miscibility and
wettability for droplet impacts onto wall-film. Aljedaani et al. (2018)
focused on the occurrence of holes in the crown, similar to some extend to
those observed in the right-hand column of Fig. 1. The holes formed on the
crown wall grow till they join themselves and form a net-like structure which
finally disintegrates into secondary droplets (see the video in the
supplementary material of the impact process corresponding to the experiments
in Fig. 1). This particular type of splashing event was first observed for
water/glycerol droplet impacting onto very thin wall-films of ethanol
(Thoroddsen et al., 2006) and is a unique feature of binary droplet/wall-film
systems. The hole formation was attributed to Marangoni driven flows initiated
by small secondary droplets impacting onto the inside part of the crown wall
(Thoroddsen et al., 2006; Aljedaani et al., 2018; Kittel, 2019). This
phenomena was also observed with very similar surface tensions for droplet and
wall-film (hyspin and hexadecane liquids) and was thus attributed more
generally to inhomogeneities in viscosities and/or surface tensions in the
crown wall (Geppert et al., 2016). Although Aljedaani et al. (2018) noticed
only a weak reduction in the growth rate of the holes for increasing droplet
viscosity, a large viscosity ratio between droplet and wall-film influences
the liquid repartition in the crown. For droplets with much higher viscosity
than the wall-film, a two-stage crown was observed with the wall-film liquid
at the upper part of the crown. Similar crown morphologies with high viscosity
ratios have been observed for immiscible droplet/wall-film systems by Kittel
et al. (2018a), and can be explained by the strong differences in the time
scale of the ejecta sheets coming from the droplet and the wall-film (Marcotte
et al., 2019). By influencing the liquid repartition inside the crown, the
viscosity ratio also influences the occurrence of the holes in the crown wall.
In the present study instead, the droplet and wall-film viscosities are quite
similar. Hence, only one stage crowns are observed, where the droplet is
expected to cover uniformly the wall-film liquid in the crown wall. Besides
the miscible configurations leading to the formation of holes, Aljedaani et
al. (2018) studied the impact of immiscible liquid pairs. They did not report
any major differences in the crown dynamics (looking especially at the crown
angle), but the hole formation vanishes despite differences in surface tension
between droplet and wall-film. They observed that the patch formed by the
immiscible small droplets sitting on the crown wall do not grow. Holes in the
crown were indeed observed only for miscible liquid pairs till now in the
literature (Geppert, 2019; Kittel, 2019; Thoroddsen et al., 2006), although
Aljedaani et al. (2018) suggested that the hole formation could be influenced
by liquid wettability of immiscible pairs. The hole formation process is not
investigated in the present study, but it is encountered while studying
miscibility and wettability effects with a miscible droplet/wall-film system
involving glycerol, water and ethanol.

The wetting behaviour between droplet and wall film was studied by Che & Matar
(2018) with combinations of glycerol/water droplet impacting onto silicone oil
and vice versa. While a Worthington jet was formed at the end of the recoiling
phase for the water droplet onto the oil, it was not the case for the reversed
combination. The shape of the crowns were also slightly different because of
the different droplet spreading speed in the wall-film liquid. The authors
state that immiscible combinations share similar features with that on
miscible films without drastic differences in the impact process, which is a
qualitative observation though.

Only Chen et al. (2017) studied explicitly the effect of miscibility on crown
formation and splashing for droplet impact onto very thin films (maximum film
height of $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$, i.e. 0.017 times the
droplet diameter). They reported a significant role of the interfacial tension
(assuming a value of zero for miscible pair, Sec. 2) in the receding phase, as
for Che & Matar (2018) for the wettability. They did not mention any influence
for the extension phase, which is crucial for the ejection of secondary
droplets, but did observe a small shift in the splashing limit. They reported
that, in case of miscible liquids, a larger Weber number (based on droplet
properties) and film thickness are required to form a crown and to eject
secondary droplets, however without distinction between crown-type splashing
and prompt splash (i.e. formed as the droplet hits the surface, within the
first $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ (Thoroddsen et al., 2011)).
They explained this phenomena by the attenuation of the kinematic
discontinuity due to the absence of interface between droplet and wall-film.
To the contrary, Banks et al. (2013) noted that the immiscible liquid pair
FC-72 on water required a far higher Weber number than the other miscible
pairs studied for crown formation, despite having lower viscosity and surface
tension, which should rather promote splashing. They could not find a clear
explanation for this, attributing the observations to either the miscibility
effects or some indirect effects of viscosities on the impact morphology. The
effects of the interfacial tension found on the onset of splashing in the
present study could explain the observation of Banks et al. (2013).

The present review highlights that the question of the influence of
miscibility and liquid wettability for droplet impact onto wall films remains
open. To what extent can miscibility and wettability influence droplet impact
onto thin wall films, especially during the extension phase?

Since droplet impact onto wall-films leads to an extreme spreading of liquids,
for example from 2 to 5 times the droplet diameter for the rim displacement
(Bernard et al., 2020), the interface between droplet and wall-film liquids
increases drastically during the extension process. Thus, the interfacial
tension acting on this interface might have a significant influence on the
crown dynamics. Does the interfacial tension, linked to miscibility and
wettability, influence the impact process, and if yes, how? To address these
questions, three droplet/wall-film liquid pairs have been chosen to vary the
miscibility and wettability behaviour, as presented in Sec. 2, leading to a
variation of their interfacial tension. The differences observed between these
three droplet/wall-film pairs in terms of impact outcome, crown morphology and
crown kinetics are reported in Sec. 3. From these observations, the role of
the interfacial tension during crown extension is discussed in Sec. 4. The
conclusions are summarized in Sec. 5.

$\text{$\tau_{in}${}}=-0.53$ $\text{$\tau_{in}${}}=1.28$
$\text{$\tau_{in}${}}=2.25$ $\text{$\tau_{in}${}}=4.50$
$\text{$\tau_{in}${}}=6.75$ $\text{$\tau_{in}${}}=9.00$
GW/Hexa$\text{$\sigma_{d/f}^{*}${}}\approx{}0.35$$g$$4$ mm GW/SO
$\text{$\sigma_{d/f}^{*}${}}\approx{}0.28$ GW/EtOH
$\text{$\sigma_{d/f}^{*}${}}\approx{}0$

Figure 1: High-speed images of droplet impact onto wall films for three
droplet/wall-film liquid pairs at similar impact conditions (see the last row
of Table 2 for $\text{$\mathit{We}_{d}$}=363\pm 5$, and
$\text{$\delta${}}=\text{$h_{f}${}}/\text{$D_{d}${}}=0.122\pm 0.011$) at
different time instants from the impact
$\text{$\tau_{in}${}}=t\text{$V_{d}${}}/\text{$D_{d}${}}$. Only the relative
importance of the interfacial tension
$\sigma_{d/f}^{*}$=$\sigma_{d/f}$/($\sigma_{d}$+$\sigma_{f}$+$\sigma_{d/f}$)
varies significantly between the liquid pairs, linked to a variation of
wettability and miscibility.

##  2 Experimental setup and conditions

Three different droplet/wall-film liquid pairs listed in Table 1 are used to
vary miscibility and liquid wettability. All the binary droplet/wall-film
systems are similar, only their respective interfacial tension $\sigma_{d/f}$
is changing significantly so that it corresponds to immiscible liquids with
partial and total wetting conditions as well as to miscible liquids. The
experiments are recorded by high-speed shadowgraphy.

###  2.1 Experimental setup

The experimental setup is composed of a dropper, an impact area, and a high-
speed imaging system. A schematic is provided in Appendix A. The dropper
consists of a needle (Gauge 21), connected through flexible tubes (PTFE) to a
liquid reservoir. The liquid is set in motion by a peristaltic pump with a
flow rate kept low enough to let a single droplet fall by gravity. The height
from the wall-film surface to the needle tip is varied to obtain different
droplet impact velocities $V_{d}$. The impact area is a shallow pool in which
the liquid is progressively added with a micro-pipette, bounded by a metallic
ring of $0.6\text{\,}\mathrm{mm}$ height and $60\text{\,}\mathrm{mm}$ diameter
fixed on a sapphire plate. Sapphire is used for its high refractive index
compared to the liquids used. This allows a reliable use of a confocal
chromatic sensor (Micro-Epsilon, IFS2405-3) to measure the film thickness
$h_{f}$ with an accuracy of about 1% for the values investigated in this work
(principle described in Lel et al. (2008)).

The imaging system is designed for high-speed shadowgraphy, recording
simultaneously two orthogonal perspectives to track asymmetric features such
as holes in the crown. Typical high-speed images taken with a high-speed
digital video camera (Photron FASTCAM SA1.1) and LED lights operated in
backlit mode can be seen in Fig. 1 (only one perspective is shown). The
spatial and temporal resolutions of the high-speed imaging were
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{px}}^{-1}$ and
$0.05\text{\,}\mathrm{ms}$, respectively combined with an exposure time of
1/92 ms.

The high-speed images are post-processed with an in-house MATLAB routine that
extracts the primary geometrical parameters of the crown (rim radius $R_{R}$,
base radius $R_{B}$ and height $H_{R}$). The high-speed images are first
processed by subtracting the background image (first image before the droplet
enters the frame). This step, combined with an increase of contrast (grey-
scale spread over the full dynamic range), improves significantly the
observation of the liquid structures. Thus, the processed high-speed images,
as shown in Fig. 1 are preferred to the raw images (as in the Appendix B for
example) to illustrate the impact process. Finally, the images are binarized
in order to detect the four edges of the crown like the four orange dots in
Fig. 2. The distance between the detected crown edges enables the evaluation
of the crown geometrical parameters: crown rim radius $R_{R}$ and crown base
radius $R_{B}$ for the horizontal distances, and the crown rim height $H_{R}$
in the vertical. To detect these edges, the image is analysed pixel line by
pixel line. At the bottom, a search for the maximum radius is carried out
within the last five lines belonging to the crown. The detection of the crown
rim is more difficult, especially when the rim has corrugations or fingers.
For each pixel line at the crown top, the presence of fingers (alternation of
black and white pixels) is checked for. The height of the crown $H_{R}$ is set
at the finger base, meaning that the corrugations due to the fingers are not
taken into account. The values of the geometrical parameters are averaged
between the two perspectives of the experimental setup to reduce the
measurement uncertainty. The typical errors associated with the measurement of
the crown geometrical parameters are between 1.5% and 4.3% for the radii, and
between 1.6 and 5.2% for the crown height (Geppert, 2019). The droplet
diameter $D_{d}$ and impact velocity $V_{d}$ are also measured from the high-
speed images shortly before impact ($\sim$ $1.5\text{\,}\mathrm{ms}$). By
adding up the systematic and random uncertainties, which maximises the error,
the overall uncertainty for the droplet diameter remains smaller than 2.8%,
and below 3% for the impact velocity (Geppert, 2019). A detailed description
of the setup (which has been used with the same operating conditions), the
post-processing routine and the associated measurement errors can be found in
Geppert et al. (2016, 2017); Geppert (2019).

###  2.2 Variation of droplet/wall-film liquids

The liquids of the droplet/wall-film pairs have been chosen to vary the
wettability and miscibility behaviours, keeping other liquid properties almost
unchanged. The liquid properties are summarized in Table 1, together with the
abbreviations used for each liquid.

The droplet liquid is systematically composed of a mixture of glycerol and
water (abbreviation GW), which is coloured with Indigotin 85 (E 132, BASF,
Germany) similar to the liquids used by Baumgartner et al. (2020). The only
parameter varying significantly is the interfacial tension with the wall-film
liquids $\sigma_{d/f}$ (see Baumgartner et al. (2020) for more details on
their determination), the subscript $d$ standing for droplet and $f$ for wall-
film. The interfacial tension is associated with the miscibility behaviour
($\sigma_{d/f}=0$ for miscible liquids) and the wettability behaviour through
the spreading parameter $S$. The spreading parameter is defined as
$\text{$S${}}=\text{$\sigma_{d}$}-\text{$\sigma_{f}$}-\text{$\sigma_{d/f}$}$,
which takes negative values for partial wetting (P.W.) and positive values for
full wetting (F.W.) (De Gennes et al., 2013)). For the wettability,
n-hexadecane (abbreviation Hexa, third column in Table 1) provides partial
wetting with aqueous solutions as many alkanes. This is confirmed by the
estimation of $S$ for GW/Hexa which is negative with approximately
$-9\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$, and the observation of
n-hexadecane lenses on top of GW (Baumgartner et al., 2020). To the contrary,
silicone oil (abbreviation S.O., fourth column in Table 1) provides full
wetting with aqueous solutions (Ross & Becher, 1992; De Gennes et al., 2013).
The spreading parameter $S$ is positive and equal to approximately 15. The
full wetting behaviour is confirmed by the observation of an oil drop
spreading on the surface of a liquid bath of GW (Baumgartner et al., 2020).

The last wall-film liquid is a mixture of ethanol, water and glycerol
(abbreviation EtOH, fifth column in Table 1), which has been tuned to approach
the two other wall-film liquid properties. The mixture of ethanol is miscible
with the droplet liquid GW. In the case of miscible liquids, the interfacial
tension is assumed to be zero because no interface, in a strict sense, exists
between the liquids. Note that for miscible liquids with composition
gradients, $\sigma_{d/f}$ may transiently differ from zero before diffusion
takes place Truzzolillo & Cipelletti (2017), which may be the case for droplet
impact processes of a few milliseconds. Nevertheless, the measurements of
interfacial tensions for water and glycerol mixtures show values of
$1\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$ or lower (Petitjeans &
Maxworthy, 1996), and around $2\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$
for water and alcohols like butanol (Enders & Kahl, 2008). These values remain
within the experimental uncertainties, thus, the interfacial tension can be
approximated to be zero for GW/EtOH.

| Droplet | Wall-film
---|---|---
Abbreviation | GW | Hexa | S.O. | EtOH
Composition | G50 W50 | Hexa100 | SO100 | EtOH55 G30 W15
$\rho\,\left($\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$\right)$ | $1116\pm 2$ | $767\pm 10$ | $908\pm 5$ | $936\pm 10$
$\mu\,\left($\mathrm{mPa}\text{\,}\mathrm{s}$\right)$ | $4.97\pm 0.10$ | $3.50\pm 0.30$ | $5.10\pm 0.05$ | $4.58\pm 0.40$
$\nu\,\left(${\mathrm{mm}}^{2}\text{\,}{\mathrm{s}}^{-1}$\right)$ | $4.45\pm 0.10$ | $4.57\pm 0.45$ | $5.62\pm 0.09$ | $4.90\pm 0.48$
$\sigma\left($\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$\right)$ | $68\pm 2$ | $26.5\pm 1$ | $19.5\pm 0.5$ | $25.7\pm 0.7$
$\text{$\sigma_{d/f}${}}\left($\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$\right)$ | / | $50\pm 2$ | $34\pm 1$ | 0
$\text{$S${}}\left($\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$\right)$ | / |  $\approx-9$ (P.W.) |  $\approx 15$ (F.W.) | miscible
$\text{$\sigma_{d/f}^{*}${}}(-)$ | / | $\approx 0.35$ | $\approx 0.28$ | $\approx{}0$
Table 1: Liquid properties of droplet and wall-film liquids. In the line
Composition, the numbers indicate the mass percentage of G: glycerol, W:
water, Hexa: n-hexadecane, S.O.: silicone oil, EtOH: ethanol. The spreading
parameter $S$ is associated with the wettability behaviour: partial wetting
(P.W.) for $\text{$S${}}<0$ or full wetting (F.W.) for $\text{$S${}}>0$. The
non-dimensionalised interfacial tension $\sigma_{d/f}^{*}$ quantifies the
importance of $\sigma_{d/f}$ in the droplet/wall-film system.

In order to quantify the importance of the interfacial tension in the
droplet/wall-film system, the non-dimensionalised interfacial tension
$\sigma_{d/f}^{*}$=$\sigma_{d/f}$/($\sigma_{d}$+$\sigma_{f}$+$\sigma_{d/f}$)
is introduced. It corresponds to the ratio of the interfacial tension to the
sum of the surface tensions of droplet and wall film. Thus, it is zero for
miscible liquids as GW/EtOH, and increases with increasing importance of the
interfacial tension in the droplet/wall-film system as for GW/SO and GW/Hexa
in the last row of Table 1. Note that in some cases, it could be interesting
to have a unified parameter combining the importance of the interfacial
tension with the spreading parameter, e.g. as $S^{*}$=$\sigma_{d/f}$/$S$ in
the case where the spreading parameter is different from zero. This parameter
could help to understand the kinetics of the receding where the wettability
has a strong influence for example in the formation of a Worthington jet (Che
& Matar, 2018), whose magnitude could be related to $S^{*}$. In the present
study, we rather focus on the crown extension phase where the value of the
interfacial tension seems to play a significant role independently on the
wetting behaviour of the liquid pair, probably because of the large Weber
numbers involved (see Sec. 2.3).

###  2.3 Experimental range

The single droplets of the glycerol/water mixture exhibit diameters of $D_{d}$
= $2.20\pm 0.07\text{\,}\mathrm{mm}$ for the full database. The impact
velocity is varied from
$V_{d}$=$2.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}3.7\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
with four different fall heights. For each impact velocity investigated, all
three liquid pairs are used as wall-film. The dimensionless wall-film
thickness $\delta$=$h_{f}$/$D_{d}$ is kept quasi-constant at two investigated
ranges, i.e. at $\delta$=$0.122\pm 0.011\text{\,}\mathrm{,}$ and at
$\delta$=$0.259\pm 0.008\text{\,}$ In order to describe accurately the impact
process, the initial state of the droplet/wall-film system needs to be
quantified. Therefore, the range of Weber and Reynolds numbers based on
droplet properties, as
$\text{$\mathit{We}_{d}${}}=\text{$\rho_{d}$}\text{$V_{d}${}}^{2}\text{$D_{d}${}}/\text{$\sigma_{d}$}$
and
$\text{$\mathit{Re}_{d}$}=\rho\text{$V_{d}${}}\text{$D_{d}${}}/\text{$\mu_{d}$}$
respectively, are given in the two first columns of Table 2. Note that the
range indicated entails the experimental conditions of all three liquids pairs
of GW/Hexa, GW/SO and GW/EtOH. However, these numbers are not representative
of the full binary droplet/wall-film systems since the liquid properties of
droplet and wall-film differ. A proper way of taking both droplet and wall-
film liquid properties into account in these non-dimensional numbers is still
under discussion in literature (Geppert et al., 2014; Kittel et al., 2017).
Recently, averaged liquid properties (subscript avg) as
$\text{$\rho_{avg}${}}=\left(\text{$\rho_{d}${}+$\rho_{f}${}}\right)/2$,
$\text{$\sigma_{avg}${}}=\left(\text{$\sigma_{d}${}+$\sigma_{f}${}}\right)/2$
and $\text{$\nu_{avg}${}}=\sqrt{\text{$\nu_{d}$$\nu_{f}$}}$ have shown good
ability to capture the crown rim dynamic for binary droplet/wall-film systems
(Bernard et al., 2020). Note that this approach was developed for miscible
droplet/wall-film liquid pairs made of silicone oils. Hence, it does not
consider wettability and miscibility effects which are considered separately
in the present study (e.g. in terms of spreading parameter $S$ and
dimensionless interfacial tension $\sigma_{d/f}^{*}$ summarized in Table 1).
These averaged properties can be used in the dimensionless numbers to form
$\mathit{We}_{avg}$ and $\mathit{Re}_{avg}$ summarized in Table 2 for each
reference value of $\mathit{We}_{d}$ and all liquid pairs combined.
Furthermore, they can be combined in a single impact parameter
$\text{$K${}}=\text{$\mathit{We}$}^{0.5}\text{$\mathit{Re}$}^{0.25}$ (also
summarized in the last column of Table 2), which, considered together with
$\delta$, is fully representative of the droplet wall-film system for the
onset of splashing (Cossali et al., 1997). These non-dimensional impact
parameters between the three liquid pairs indicate very similar droplet/wall-
film systems for each set of experiments. The variation does not exceed $\pm
8\%$ in general at a given $\mathit{We}_{d}$, while the non-dimensionalised
interfacial tension $\sigma_{d/f}^{*}$ increases by more than 25% between
GW/SO and GW/Hexa, and is zero for GW/EtOH. Hence, the major varying parameter
between the liquid pairs investigated is $\sigma_{d/f}^{*}$, which, combined
to the surface tensions of droplet and wall-films defined the miscibility and
wettability of the liquid pairs.

$\delta$ | $\mathit{We}_{d}$ | $\mathit{Re}_{d}$ | $\mathit{We}_{avg}$ | $\mathit{Re}_{avg}$ | $K_{avg}$
---|---|---|---|---|---
$0.122\pm 0.011$ | $218\pm 2$ | $1212\pm 10$ | $286\pm 18$ | $1152\pm 76$ | $98\pm 2$
$269\pm 2$ | $1341\pm 8$ | $354\pm 30$ | $1274\pm 66$ | $112\pm 3$
$318\pm 3$ | $1448\pm 10$ | $417\pm 35$ | $1375\pm 71$ | $124\pm 4$
$363\pm 5$ | $1547\pm 16$ | $477\pm 42$ | $1469\pm 70$ | $135\pm 4$
$0.259\pm 0.008$ | $217\pm 1$ | $1205\pm 2$ | $284\pm 21$ | $1144\pm 65$ | $98\pm 2$
$268\pm 2$ | $1336\pm 7$ | $353\pm 28$ | $1272\pm 70$ | $112\pm 3$
$320\pm 2$ | $1453\pm 7$ | $420\pm 29$ | $1383\pm 85$ | $125\pm 2$
$363\pm 7$ | $1546\pm 24$ | $481\pm 40$ | $1482\pm 76$ | $136\pm 4$
Table 2: The investigation range of typical non-dimensional parameters. The
subscript avg refers to averaged liquid properties between droplet and wall-
film.

##  3 Experimental observations

The temporal evolutions of the three liquid pairs are shown in Fig. 1. Despite
similar impact conditions (see Table 2, row $\text{$\delta${}}=0.130\pm 0.001$
with $\text{$\mathit{We}_{d}$}=363\pm 5$), differences between the liquid
pairs can be observed in the impact outcome, the crown extension and the crown
kinetics.

###  3.1 Shift in the splashing limit

The first differences that can be observed qualitatively between the three
droplet/wall-film pairs in Fig. 1 are their impact outcomes. The full temporal
evolution of the impact process for each liquid pair side-by-side can be
observed in the video of the supplementary material. For GW/Hexa, the crown
rim stays relatively stable, undulations become significant only in the
receding phase (the maximum crown height is reached at about
$\tau_{in,H_{max}}$ = $t_{H_{max}}$ $V_{d}$/$D_{d}$ = 5.40) and fingers can
barely be observed at the end of the impact process. Thus, the outcome of
GW/Hexa can be categorised as a transition case (i.e. fingers formed) close to
deposition (i.e. no fingers). In the case of GW/SO, undulations on the rim can
already be observed in the ascending phase (up to $\tau_{in,H_{max}}$ = 6.53),
and at the end of the extension process, long fingers with droplets about to
detach are formed. The long fingers only disintegrate into secondary droplets
at the end of the impact process, leaving the droplets formed in the vicinity
of the impact location. Thus, the outcome of GW/SO can again be categorised as
a transition case, but close to splashing (i.e. secondary droplets are ejected
far beyond the crown dimensions). Note that the tiny droplets observed already
on the picture at $\tau_{in}$ = 2.25 are associated with prompt splash, i.e.
they are not considered as part of the crown-type outcome. The prompt
splashing consists in the fragmentation of a thin and fast liquid lamella
called ejecta sheet (Thoroddsen, 2002) formed at the base of the impacting
droplet. The distinction between prompt and crown-type splashing relies first
on their different time scales, since prompt splashing occurs quasi-
immediately after the droplet has impacted the liquid surface (within the
first $100\text{\,}\mathrm{\SIUnitSymbolMicro s}$, Liang & Mudawar (2016)).
Second, the ejecta sheet and the crown wall arise from different dynamics and
can be observed dinstinctly (Zhang et al., 2012). Third, prompt and crown-type
splashing can be observed independently of each other in a splashing regime
map (Deegan et al., 2007). Last, prompt splashing leads to the formation of
much smaller secondary droplets than crown-type splashing (Cossali et al.,
1997; Motzkus et al., 2009). Hence, there is most likely no or small
interdependence between the prompt splashing observed and the crown-type
splashing studied in the present experiments. In the video of the impact
experiments similar to those of Fig. 1 provided in the supplementary material,
one can see that prompt splashing occurs in all three cases as the droplets
penetrate the wall-film, although the tiny droplets produced are ejected
beyond the crown size and thus observable later only for GW/SO and GW/EtOH.
Furthermore, the video shows clearly that the crown wall arises far beyond the
location where these tiny droplets are form, in the vicinity of the
droplet/wall-film neck region. For the last liquid pair GW/EtOH, fingers are
already formed in the ascending phase (e.g. at $\tau_{in}$ = 4.50), and some
droplets are ejected from the fingers. Towards the end of the crown extension,
holes are formed on the crown wall, as already reported in literature for
droplet impacts with aqueous and ethanol solutions (Thoroddsen et al., 2006;
Aljedaani et al., 2018). As explained in the introduction, mixture gradients
in the crown wall and/or fine droplets that cannot be observed with the
current setup impacting on the crown wall could cause these holes. Despite the
formation of holes in the crown (the first occurrence could be back-tracked at
$\tau_{in}$ = 6.30), it can be observed that a few secondary droplets (the
bigger ones) are ejected from the fingers. Thus, the outcome of GW/EtOH can be
categorised as a splashing case, but close to transition since only few
droplets are ejected. Hence, a small shift in the splashing limit is observed
between the three cases despite similar liquid properties and impact
conditions: the miscible pair GW/EtOH starts to splash, the full wetting pair
GW/SO is close to splash and the partially wetting pair GW/Hexa is closer to
deposition. Note that the smallest viscosity (despite the small differences
between the pairs) is the one of n-hexadecane. Since less viscous losses are
expected, splashing with less kinetic energy should be observed (see the
review article of Liang & Mudawar (2016)). Yet it is not the case, hence,
small viscosity variations are not expected to modify the trend observed
between the liquid pairs. This shift in the fragmentation limit, classically
derived based on empirical observation of the outcome, is very small. Thus, it
is hard to evaluate it in terms of critical Weber or Reynolds numbers.
Furthermore, there is a smooth transition from deposition to splashing
(Cossali et al., 1997), which lead to the introduction of the transition
outcome (Geppert et al., 2016) as described above to refine the splashing
criteria. Qualitatively though, one could modify the averaged Weber number
$\mathit{We}_{avg}$ defined in Sec.2 taking into account all surface and
interfacial tensions with
$\left(\text{$\sigma_{d}${}}+\text{$\sigma_{f}${}}+\text{$\sigma_{d/f}${}}\right)/3$
instead of $\left(\text{$\sigma_{d}${}}+\text{$\sigma_{f}${}}\right)/2$. This
would lead to a characteristic Weber number of the impact multiplied by 0.98
for GW/Hexa, 1.08 for GW/SO and 1.5 for GW/EtOH, highlighting that GW/EtOH
would cross the splashing limit earlier than GW/SO and finally than GW/Hexa.
This difficulty to assert the splashing limit for such a narrow difference
highlights the need to consider another parameter to quantify the differences
observed between the liquid pairs.

###  3.2 Shift in crown extension

Besides the small shift in the fragmentation observed in Fig. 1, the liquid
pairs show differences in the size of the crown.

$\sigma_{d}$, $\rho_{d}$, $\mu_{d}$$\sigma_{f}$, $\rho_{f}$,
$\mu_{f}$$\sigma_{d/f}$$\bullet$$\bullet$$\bullet$$\bullet$$H_{R}$2$R_{R}$2$R_{B}$

Figure 2: Schematic of a meridional section of a crown with droplet (green)
and wall-film (blue) liquids, illustrating the full coverage of the droplet
liquid on the inner part of the crown. The interface between them is marked by
a red line. Note that the crown thickness is not representative of the
reality. The crown edges (orange dots) extracted from the post-processing are
used to define the crown geometrical parameters: rim radius $R_{R}$, rim
height $H_{R}$ and base radius $R_{B}$.

To quantify this observation, the crown surface is calculated from the high-
speed images by extracting the geometrical parameters of the crown as shown in
the schematic of Fig. 2. The crown rim height $H_{R}$, the rim radius $R_{R}$,
and the base radius $R_{B}$ are extracted from the crown edges marked with
orange circles. The crown surface, expressed in Eq. 1, is calculated by
summing the surface of the disc at the crown base, and the surface of the
crown wall counted twice for the inner and outer parts. The crown wall is
approximated by a conical frustum formed between the base and rim radii. This
assumes having a straight line between the base and rim radii, although
sometimes the crown wall is slightly bended inward. This bending and its
influence on the determination of the crown surface is discussed in Appendix
B. However, the difference in crown morphologies (between the crown edges) is
taken indirectly into account via the calculation of $\Sigma_{c}$. Besides,
the surface area at the rim due to the crown thickness is neglected.

$\text{$\Sigma_{c}${}}=\upi\text{$R_{B}${}}^{2}+2\pi\left(\text{$R_{B}${}}+\text{$R_{R}${}}\right)\sqrt{\text{$H_{R}${}}^{2}+\left(\text{$R_{R}${}}-\text{$R_{B}${}}\right)^{2}}$
(1)

The propagation of uncertainty from the measurement of $H_{R}$, $R_{R}$ and
$R_{B}$ given in Sec. 2.1 to $\Sigma_{c}$ can be calculated based on the
formula of Gauss-Laplace as
$\Delta_{\text{$\Sigma_{c}${}}}=\sqrt{\sum_{i}\left[\left(\partial\text{$\Sigma_{c}${}}/\partial
x_{i}\right)\;\Delta x_{i}\right]^{2}}$, where $x_{i}$ are the different
variables of $\Sigma_{c}$ (here the geometrical parameters) and $\Delta x_{i}$
their respective absolute uncertainty. The relative measurement uncertainty
$\Delta_{\text{$\Sigma_{c}${}}}/\text{$\Sigma_{c}$}$ is dependent upon the
crown morphology, i.e. $H_{R}$/$R_{R}$ and $R_{B}$/$R_{R}$. Note that these
ratios vary during the impact process, and between different impact
conditions. A typical crown configuration though in the present experimental
conditions is $\text{$H_{R}${}}/\text{$R_{R}${}}\approx{}0.5$ and
$\text{$R_{B}${}}/\text{$R_{R}${}}\approx{}1$. In this case, the relative
error $\Delta_{\text{$\Sigma_{c}${}}}/\text{$\Sigma_{c}${}}$ is equal to 5.7%.
In the present database, the ratio $\text{$H_{R}${}}/\text{$R_{R}${}}$ varies
from 0.05 to 0.65, and $\text{$R_{B}${}}/\text{$R_{R}${}}$ from 0.65 to 1.30.
By considering these extreme values, propagation of uncertainties up to 8.27%
can be found, tendentially for high $R_{B}$ and/or small $H_{R}$.

The temporal evolution of $\Sigma_{c}$ is shown in Fig. 3(a) for the same
experiments as in Fig. 1 that have similar impact conditions. Except at the
early stage of impact where kinetic energy dominates, the curves separate
progressively. As observed qualitatively in the high-speed images, the crown
surface of GW/Hexa is smaller than the one of GW/SO, which is smaller than the
one of GW/EtOH. Note that this ranking of crown surface extensions correlates
with the values of $\sigma_{d/f}^{*}$ reported in Table 1: The larger is
$\sigma_{d/f}^{*}$ (tendentially for immiscible liquid pairs with partial
wetting), the smaller becomes the crown surface. A similar ranking in the
spreading between miscible and immiscible droplet/wall-films during the crown
extension phase has also been measured by Chen et al. (2017), without being
explained. The formation of holes in the crown of GW/EtOH (grey symbols in
Fig. 3) does not influence this trend since the values of $\Sigma_{c}$ for
GW/EtOH are systematically above, but adds noise which prevents the exact
determination of the maximum crown surface $\Sigma_{c,max}$ (filled symbols in
Fig. 3(a,b)). $\Sigma_{c,max}$ for GW/EtOH is lying between the value where
the first hole appears (filled grey symbols and grey dashed line), and the
maximum value calculated before the destruction of the crown (filled red
symbols an red dashed line). Furthermore, the determination of the crown
parameters during extension remains repeatable despite stochastic holes
(Geppert et al., 2017).

$0$$2$$4$$6$$8$$0$$100$$200$$300$$t_{H_{max}}$$t$
$($\mathrm{ms}$)$$\Sigma_{c}$
(${\mathrm{mm}}^{2}$)$\text{$\mathit{We}_{d}$}=363\pm 5$
$200$$250$$300$$350$$400$$0$$100$$200$$300$$\mathit{We}_{d}$
(-)$\Sigma_{c,max}$ (${\mathrm{mm}}^{2}$)$\text{$\delta${}}\approx{}0.12$
$200$$250$$300$$350$$400$$0$$2$$4$$6$(c)$\mathit{We}_{d}$ (-)$t_{H_{max}}$
$\left($\mathrm{ms}$\right)$$\text{$\delta${}}\approx{}0.12$GW/HexaGW/SOGW/EtOH

Figure 3: (a) Temporal evolution of the crown surface $\Sigma_{c}$ for the
same experiments as in Fig. 1. The filled symbols with yellow edge correspond
to $\Sigma_{c,max}$. The dashed lines correspond to $t_{H_{max}}$, the time at
which $H_{R,max}$ is reached. (b) Maximum crown surface $\Sigma_{c,max}$ at
different Weber numbers $\mathit{We}_{d}$. (c) Time duration $t_{H_{max}}$ of
the crown ascending phase (marked with dashed lines in (a)) at different Weber
numbers $\mathit{We}_{d}$. The dimensionless wall-film thickness is kept
constant for the data of this figure at $\text{$\delta$}=0.122\pm 0.011$. The
grey symbols correspond to the measurements where holes in the crown are
observed (only for GW/EtOH).

The values of $\Sigma_{c,max}$ are reported for different Weber numbers in
Fig. 3(b). It is clear that the trend observed between the liquid pairs
observed in (a) remains the same: at a given $\mathit{We}_{d}$, the values of
$\Sigma_{c,max}$ are systematically smaller for GW/Hexa than for GW/SO, which
are smaller than GW/EtOH. For a given liquid pair, $\Sigma_{c,max}$ increases
with growing $\mathit{We}_{d}$, i.e. with increasing droplet kinetic energy.
This corroborates the larger rim expansions observed at higher droplet kinetic
energy for droplet impact onto wall-films (Bernard et al., 2020), or for other
impact configurations, e.g. onto dry surfaces (Huang & Chen, 2018), with
another droplet (Roisman et al., 2012), or with a continuous jet (Baumgartner
et al., 2020).

###  3.3 Shift in duration of crown ascending phase

It is interesting to note that the duration of the crown ascending phase (i.e.
the time at which $H_{R,max}$ is reached, marked by dashed lines in Fig. 3(a))
also exhibits a systematic ranking between the liquid pairs, although in
smaller amplitude than the differences in crown surfaces. The ascending
duration of GW/Hexa is smaller than the one of GW/SO, which may be smaller
than the one of GW/EtOH (the grey dashed line corresponds to the time at which
the first hole could be back-tracked). Here again, this ranking correlates
with the values of $\sigma_{d/f}^{*}$ reported in Table 1. The higher is
$\sigma_{d/f}^{*}$, the shorter becomes the ascending duration. Note that a
similar observation could not be made with $t_{\Sigma_{c,max}}$ (the time at
which the maximum crown surface is reached). In fact, the exact time of crown
surface receding is unclear because it has, in some cases, a plateau at its
maximum value. This effect is known as the stabilization phase of the crown at
the end of the surface extension (Zhang et al., 2019). This happens when the
radial extension of the crown is longer than the axial one (Bernard et al.,
2018). In this case, the decrease of $H_{R}$ is compensated by a continuous
increase of $R_{R}$ and/or $R_{B}$. This effect shifts $t_{\Sigma_{c,max}}$
compared to $t_{H_{max}}$ (as in Fig. 3(a) for GW/Hexa between the blue filled
symbol and the blue dashed line) and leads to scattered data. Hence,
$t_{H_{max}}$ is preferred to $t_{\Sigma_{c,max}}$. The durations of the crown
ascending phase $t_{H_{max}}$ are given in Fig. 3(c) for different
$\mathit{We}_{d}$. A trend between the liquid pairs is observed, similar to
the maximum crown surface $\Sigma_{c,max}$: the ascending phase of GW/Hexa is
slightly shorter than the one of GW/SO, which is (not systematically) slightly
smaller than GW/EtOH. These durations are slightly increasing with growing
$\mathit{We}_{d}$, as already observed for example with droplet impact onto
dry surfaces (Huang & Chen, 2018), but not for droplet head-on collisions or
impacts on a continuous jet where the kinetics is fixed by the encapsulated
drop only (Baumgartner et al., 2020).

In summary, slight differences between the liquid pairs were observed for the
onset of splashing, but more prominently, differences in maximum crown
surface, and in the duration of the crown ascending phase despite similar
impact conditions and liquid properties. Thus, these differences can be
attributed to their wettability and miscibility behaviours.

##  4 Importance of the interfacial tension

The miscibility and wettability behaviours responsible for the differences
observed in Sec. 3 are linked to the interfacial tension $\sigma_{d/f}^{*}$
characteristic of each liquid droplet/wall-film pair (see Table 1). During
droplet impact onto thin wall-films, the initial kinetic energy of the droplet
is partially converted into surface energy and partially dissipated due to
impact losses (initial droplet deformation) and viscous losses during the
extension phase. This energy transfer leads to an important crown extension
and a significant role of the surface forces. Although the role of surface
tension is quite clear, the importance of interfacial tension for droplet
impact onto wall-films during extension remains poorly documented and
understood. In the following, we investigate the role of the interfacial
tension, starting with energetical considerations followed by kinetic ones.

###  4.1 Energy storage

In order to estimate if the differences in the crown extension observed in
Sec. 3.2 can be explained by the differences in interfacial tension
$\sigma_{d/f}$, the temporal evolution of the crown surface energy
$E_{\sigma,c}$ is calculated. Each portion of the crown (i.e. base disc and
conical frustum) is multiplied by the corresponding surface/interfacial
tension (i.e. $\sigma_{d}$, $\sigma_{f}$ and/or $\sigma_{d/f}$). The droplet
liquid is assumed to completely cover the wall-film liquid on the inside of
the crown, as represented in Fig. 2. This full coverage by the droplet liquid
on the inner crown wall has been observed numerically for aqueous
droplet/wall-film systems Zhang et al. (2019), and experimentally for dyed
water droplet impacting on silicone oil films (Shaikh et al., 2018), for
$Re_{f}=\rho_{f}\text{$V_{d}${}$h_{f}$}/\mu_{f}$ below 400 (Kittel et al.,
2018a). In the current work, the maximum value of $\mathit{Re}_{f}$ is 196.
Hence, it can reasonably be assumed that the droplet liquid completely covers
the inner part of the crown, which leads to the following expression for the
crown surface energy:

$\text{$E_{\sigma,c}${}}=\left(\sigma_{d}+\sigma_{d/f}\right)\upi\text{$R_{B}${}}^{2}+\left(\sigma_{d}+\sigma_{d/f}+\sigma_{f}\right)\upi\left(\text{$R_{B}${}}+\text{$R_{R}${}}\right)\sqrt{\text{$H_{R}${}}^{2}+\left(\text{$R_{R}${}}-\text{$R_{B}${}}\right)^{2}}$
(2)

The propagation of uncertainty for $E_{\sigma,c}$ can be calculated similar to
that of the crown surface $\Sigma_{c}$, by considering additionally the
measurement uncertainty associated with the surface and interfacial tensions
given in Table 1. This provides a propagation of error for the reference
geometrical configuration ($\text{$H_{R}${}}/\text{$R_{R}${}}\approx{}0.5$ and
$\text{$R_{B}${}}/\text{$R_{R}${}}\approx{}1$) of 6.1% for GW/Hexa, 7.2% for
GW/SO and 7.6% for GW/EtOH. By considering the extreme values of the crown
geometrical configuration as presented for $\Delta_{\text{$\Sigma_{c}${}}}$,
the maximum propagation of uncertainty to $E_{\sigma,c}$ becomes 9.6%
(obtained for GW/EtOH).

The temporal evolution of $E_{\sigma,c}$ is shown in Fig. 4(a) for the same
cases as in Fig. 3(a). In contrast to $\Sigma_{c}$, the maxima are now
comparable (note that at a constant Weber number, $E_{k,d,0}$ is constant for
the three liquid pairs). Indeed, the overall surface energy of GW/Hexa, which
has the smallest crown surface, is relatively increased due to the high value
of $\sigma_{d/f}$ for this liquid pair. Likewise, the overall surface energy
of GW/EtOH, which has the biggest crown surface, is relatively decreased since
the interfacial tension is zero (miscible liquid pair). In between lies the
case of GW/SO. For all $\mathit{We}_{d}$ investigated in our study, all
$E_{\sigma,c,max}$ normalized with the droplet kinetic energy
$\text{$E_{k,d,0}${}}=\rho_{d}\upi\text{$D_{d}${}}^{3}\text{$V_{d}${}}^{2}/12$
are systematically coming together. Hence, the interfacial tension appears as
a good candidate to explain the differences in maximum crown extensions. This
means that, for immiscible liquid pairs, the interface between droplet and
wall film stores a non-negligible amount of energy during the extension phase,
proportionally to $\sigma_{d/f}^{*}$. This stored energy neither participates
in the crown extension, nor in the splashing process. Hence, liquid pairs with
smaller interfacial tension (the limit being zero for miscible cases) expand
more, and splash at lower droplet kinetic energy as observed in Fig. 1. For
impact processes with smaller extensions (e.g. at very small
$\mathit{We}_{d}$, or onto another impacted substrates), the interfacial
energy could be less significant and its effect may not be noticeable.

$0$$0.2$$0.4$$0.6$$0.8$$0$$0.2$$0.4$$0.6$(a)$\tau_{cap,H_{max}}$$\tau_{cap}$
(-)$E_{\sigma,c}$/$E_{k,d,0}$ $(-)$$\text{$\mathit{We}_{d}$}=363\pm 5$
$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$$0.8$$1$(b)$\mathit{We}_{d}$
(-)$E_{\sigma,c,max}$/$E_{k,d,0}$ $(-)$$\text{$\delta${}}\approx{}0.12$
$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$(c)$\mathit{We}_{d}$
(-)$\tau_{cap,H_{max}}$ (-)$\text{$\delta${}}\approx{}0.12$GW/HexaGW/SOGW/EtOH

Figure 4: (a) Temporal evolution of the crown surface energy $E_{\sigma,c}$
normalized with the initial droplet kinetic energy $E_{k,d,0}$ in function of
$\tau_{cap}$ (see Eq. 3) for the same experiments as in Fig. 1 and Fig. 3(a).
The filled symbols with yellow edge correspond to
$E_{\sigma,c,max}$/$E_{k,d,0}$. The dashed lines correspond to
$\tau_{cap,H_{max}}$, the time at which $H_{R,max}$ is reached normalized with
$t_{cap}$. (b) Maximum crown surface energy $E_{\sigma,c,max}$ normalized with
the initial droplet kinetic energy $E_{k,d,0}$ at different Weber numbers
$\mathit{We}_{d}$. (c) Capillary time duration $\tau_{cap,H_{max}}$ of the
crown ascending phase (highlighted with dashed lines in (a)) at different
Weber numbers $\mathit{We}_{d}$. The capillary time scale $t_{cap}$ used has a
typical spring constant of $\sigma_{d}$+$\sigma_{d/f}$+$\sigma_{f}$ and a
characteristic mass
$\text{$M_{d}${}}+\text{$M_{f}${}}=\left(\text{$\rho_{d}$}+\text{$\rho_{f}$}\right)\pi\text{$D_{d}${}}^{3}/6$.
The dimensionless wall-film thickness is kept constant for the data of this
figure at $\text{$\delta$}=0.122\pm 0.011$. The grey symbols correspond to the
measurements where holes in the crown are observed (only for GW/EtOH).

The ratio of $E_{\sigma,c,max}$ with $E_{k,d,0}$ in Fig. 4(b) indicates how
much the incoming kinetic energy has been transferred into surface energy. On
average, this ratio is lying around 50% of the initial kinetic energy in our
experimental range. This amount corroborates numerical simulations at similar
$\mathit{We}_{d}$ and wall-film thickness of aqueous droplet/wall-film systems
(Zhang et al., 2019). Similar values have also been found in numerical
simulations of droplet impact onto dry surface with slip condition (Wildeman
et al., 2016), i.e. with negligible surface friction which can be assimilated
to a liquid interface. At the end of crown extension, the quantity
$1-\text{$E_{\sigma,c,max}${}}/\text{$E_{k,d,0}${}}$ is representative of the
remaining energy and losses before the onset of retraction. Some kinetic
energy might remain due to internal flow motions and the complex shape of the
crown. Furthermore, the motion of a liquid mass in the vertical direction of a
liquid mass leads to a conversion of kinetic energy into gravitational energy.
Numerical studies of water droplet impact onto wall-films at $\delta$=0.1 and
$\mathit{We}_{d}$=250 (Zhang et al., 2019) show that at the end of the
expansion, the remaining kinetic energy is about 8% of $E_{k,d,0}$, and the
gravitational energy of 2%. Combined with a surface energy of about 50%, this
corresponds to an energy loss of about 40% of the droplet kinetic energy
during the extension phase. This corroborates the study on dry surface with
slip condition at the wall (Wildeman et al., 2016), where they found 10%
remaining energy (no gravitation energy since the spreading is horizontal),
and 40% of $E_{k,d,0}$ converted into heat. Hence, in the present
droplet/wall-film experiments, the remaining kinetic and gravitational
energies at the end of crown extension are expected to be about 10% of
$E_{k,d,0}$. Note that the gravitational energy is thus minor for
$\delta$=0.1, but for higher wall-film thickness like $\delta$=0.3 it can
represent up to 15% of the crown surface energy (Cossali et al., 2004).

A full energy balance of the impact process (always assuming a zero kinetic
energy at maximum crown surface) should also consider the initial surface
energies of droplet and wall-film. The initial surface energy of the droplet
can be calculated with
$\text{$E_{\sigma,d,0}$}=\upi\text{$D_{d}${}}^{2}\sigma_{d}$. For the wall-
film, the initial surface energy can be approximated with the free surface of
the wall-film covered by the crown base at maximum extension, i.e.
$\text{$E_{\sigma,f,0}$}\approx\upi\text{$R_{B,max}${}}^{2}\sigma_{f}$ (see
e.g. similar approaches for droplet impact onto dry surfaces by Vaikuntanathan
& Sivakumar (2016)). This surface corresponds to the free surface of the wall-
film covered by the crown base at maximum extension and is therefore an impact
dependent variable that can be estimated a posteriori in the present database.
The values of
$E_{\sigma,c,max}$/($E_{k,d,0}$+$E_{\sigma,d,0}$+$E_{\sigma,f,0}$) taking all
initial energies into account would be diminished from 5 to 10% compared to
$E_{\sigma,c,max}$/$E_{k,d,0}$ alone reported in Fig. 4(b). Since this shift
roughly corresponds to the value of the remaining kinetic and gravitational
energies of approximately 10% of $E_{k,d,0}$ at the end of the crown extension
discussed previously, this means that $E_{\sigma,c,max}$/$E_{k,d,0}$ gives an
estimation of the losses at the end of the crown extension, of about 50% of
all initial energies for these impact conditions.

Looking at the influence of the droplet kinetic energy, the maximum crown
surface energy normalized with the droplet kinetic energy slightly decreases
with increasing $\mathit{We}_{d}$. Hence, the percentage of total losses is
increasing with growing $\mathit{We}_{d}$, which can be explained by higher
deformation losses of the impacting droplet (Gao & Li, 2015) and a more
pronounced influence of the boundary layer because of a higher shear due to
the higher velocity gradient, and a deeper penetration of the droplet bringing
the flow closer to the wall (Lamanna et al., 2019).

The extension of the experimental database to $\text{$\delta${}}\approx{}0.26$
given in Appendix B shows a similar data reduction of the normalized crown
surface energies between the different liquid pairs at a given Weber number.
However, the values are now higher compared to
$\text{$\delta${}}\approx{}0.12$, lying around 0.6 and above. This
corroborates the influence of shear during the extension process since it is
expected to be reduced for higher wall-film thickness because of a decreased
influence of the solid wall. However, a more refined quantitative analysis is
not possible here since the gravitational energies and remaining kinetic
energy at the end of crown extension might be larger for wall-film thickness
above 0.2 (Zhang et al., 2019).

To conclude, the interface (if any) between the droplet and wall-film stores a
non-negligible amount of energy during the extension. Taking into account the
interfacial energy generalizes the estimation of viscous losses made for
miscible liquid pairs to immiscible ones with different interfacial tensions.
With about 50% loss of the initial surface and kinetic energies for $\delta$
$\approx{}$ 0.1, the viscous effects prevent the crown extension as much as
the capillary forces for thin wall-films. Hence, the droplet/wall-film system
should be considered as a inertio-capillary-viscous system.

###  4.2 Recoiling force

Droplet impact processes can be considered as oscillating mass-spring systems
(Planchette et al., 2017; Okumura et al., 2003), where surface tension is the
recoiling force opposing the deformation induced by the impacting droplet.
Hence, the droplet and the wall-film could be seen as springs in parallel
whose spring constants are given by the surface and interfacial tensions.
Following this reasoning, the system during impact could be represented by an
equivalent spring constant equal to the sum of each, i.e.
$\sigma_{d}+\sigma_{f}+\sigma_{d/f}$. Considering the crown as an inertio-
capillary system, the capillary time scales of the crown ascending phase (i.e.
increase of $H_{R}$) can be compared for the three liquid pairs to see whether
the trend between the liquid pairs observed in Fig. 3(c) is related to the
differences in interfacial tensions. Thus, we propose to evaluate the
capillary time based on the equivalent spring constant as:

$\text{$\tau_{cap}${}}=\frac{t}{\text{$t_{cap}${}}}=\frac{t}{\sqrt{\frac{\text{$M_{d}${}}+\text{$M_{f}${}}}{\sigma_{d}+\sigma_{d/f}+\sigma_{f}}}}$
(3)

The characteristic mass of the droplet/wall-film system entering $t_{cap}$ is
most likely a combination of droplet and wall-film masses as
($M_{d}$+$M_{f}$). While the characteristic mass of the droplet can be easily
calculated before impact as
$\text{$M_{d}${}}=\text{$\rho_{d}$}\pi\text{$D_{d}${}}^{3}/6$, the
determination of wall-film mass $M_{f}$ is more difficult. By assuming an
equal volume of droplet and wall-film liquid participating to the oscillation
process, it comes
$\text{$M_{d}${}}+\text{$M_{f}${}}=\left(\text{$\rho_{d}$}+\text{$\rho_{f}$}\right)\pi\text{$D_{d}${}}^{3}/6$.
This rough estimation has the advantage of being determined with pre-impact
parameters. In Fig. 4(a), the time is non-dimensionalized with $t_{cap}$ to
form $\tau_{cap}$. Compared to $t_{H_{max}}$ in Fig. 3(a), we observe that
$\tau_{cap,H_{max}}$ (the non-dimensional time at which $H_{R,max}$ is
reached) are coming together. Indeed, the differences of $\sigma_{d/f}$ in
$t_{cap}$ compensate the differences in extension duration for each liquid
pair, similar to the compensating effect for $E_{\sigma,c,max}$ in Fig. 4(b).
Hence, the interfacial tension is responsible for the different durations of
ascending phases. This means that, for immiscible liquid pairs, the interface
between droplet and wall-film acts as a non-negligible force preventing the
extension, proportionally to $\sigma_{d/f}^{*}$. This scaling leads to a
unified temporal evolution for the three liquid pairs of the normalized crown
surface energy with the capillary time during the ascending phase at a given
impact condition. The role of interfacial tension is observed for all impact
conditions investigated in our study as shown in Fig. 4(c), all data points
coming together at a given $\mathit{We}_{d}$. $\tau_{cap,H_{max}}$ is lying
around 0.4, similar to what has been also observed for much lower Weber
numbers (below 10) for the droplet oscillation alone impacting a shallow and
deep pool (Tang et al., 2019).

The values of $\tau_{cap,H_{max}}$ are slightly increasing with increasing
$\mathit{We}_{d}$. This indicates that $\tau_{cap}$ can not properly capture
the process for all impact conditions, but only highlights the influence of
the interfacial tension during the extension. The consideration of the
inertial time $\tau_{in,H_{max}}$=$t_{H_{max}}$$V_{d}$/$D_{d}$ would lead to
an even stronger dependency on the Weber number since $\mathit{We}_{d}$
increases with $V_{d}$. Furthermore, it would not bring the experiments
together at a given $\mathit{We}_{d}$ since no surface tensions are
considered. A viscous time as $\text{$D_{d}${}}^{2}/\nu$ would also let the
ranking of $t_{H_{max}}$ and the trend with $\mathit{We}_{d}$ unchanged given
the similar kinematic viscosities involved for all impact conditions (see
Table 1). The extension of the experimental database to
$\text{$\delta${}}\approx{}0.26$ represented in Fig. 5 shows a similar data
reduction of the ascending phase durations between the different liquid pairs
at a given Weber number. However, the values are now higher compared to
$\text{$\delta${}}\approx{}0.12$, lying around 0.6. Furthermore, the trend
with increasing Weber number becomes weaker compared to
$\text{$\delta${}}\approx{}0.12$, exhibiting almost constant values.

$200$$250$$300$$350$$400$$0$$2$$4$$6$$8$(a)$\mathit{We}_{d}$ (-)$t_{H_{max}}$
$\left($\mathrm{ms}$\right)$$\text{$\delta${}}\approx{}0.26$

$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$$0.8$$1$(b)$\mathit{We}_{d}$
(-)$\tau_{cap,H_{max}}$ (-)$\text{$\delta${}}\approx{}0.26$GW/HexaGW/SOGW/EtOH

Figure 5: (a) Time duration $t_{H_{max}}$ of the crown ascending phase at
different Weber numbers $\mathit{We}_{d}$. (b) Capillary time duration
$\tau_{cap,H_{max}}$ of the crown ascending phase at different Weber numbers
$\mathit{We}_{d}$. The dimensionless wall-film thickness is kept constant at
$\delta$=$0.259\pm 0.008\text{\,}\mathrm{f}$or the data of this figure.

These discrepancies between different Weber numbers and with higher wall-film
thicknesses suggest a variation of the characteristic mass used in $t_{cap}$
between these different experiments, since the variations of crown surfaces
and with it, of the recoiling forces are already taken into account. Indeed,
the kinetic energy of the droplet influences the droplet penetration inside
the film. This could lead to an increased interacting wall-film mass, which is
not captured with the current capillary scaling assuming an equal amount of
droplet and wall-film liquids. Similarly, larger wall-film thicknesses
increase the available amount of wall-film liquid that could enter the crown.
Hence, the estimation of the interacting mass of wall-film needs to be refined
to get a deeper insight into the oscillating behaviour. Therefore, the maximal
interacting wall-film mass $M_{f}$ can be estimated by considering the mass
contained in a cylinder of radius $R_{B}$ at $t_{H_{max}}$ and of height equal
to the difference of the wall-film height $h_{f}$ with the residual film
thickness $h_{res}$, as
$\text{$M_{f}${}}=\pi\text{$R_{B}${}}\left(\text{$t_{H_{max}}${}}\right)^{2}\left(\text{$h_{f}${}}-\text{$h_{res}${}}\right)$.
The residual thickness corresponds to the thickness of the wall-film which is
not set in motion by the droplet impact. It can be approximated with
$\text{$h_{res}${}}\approx(0.098\text{$\delta${}}^{4.0413}+0.79)\text{$\mathit{Re}$}^{-2/5}$
(van Hinsberg et al., 2010). Hence, the residual thickness increases with
increasing initial wall-film thickness, and decreases with increasing Reynolds
number due to a deeper penetration of the droplet in the wall-film. In the
present database, the Reynolds number can be approximated with averaged liquid
properties as introduced in the section on the experimental range,
corresponding to $\mathit{Re}_{avg}$. For the value of
$\text{$R_{B}${}}\left(\text{$t_{H_{max}}${}}\right)$, it has now the drawback
that it should be extracted post-impact. For predicting purposes however, one
could use available theoretical models predicting $R_{B}$ in the literature,
e.g. the one of Roisman et al. (2008) predicting the maximum value of $R_{B}$
during the impact process in function of $\mathit{We}_{d}$, $\delta$ and the
Froude number $\mathit{Fr}_{d}$. The experimental data scaled with the
modified capillary time scale $\tau_{cap,H_{max},m}$ based on the maximum mass
of the wall-film are given in Fig. 6 for $\text{$\delta${}}\approx{}0.12$ (a)
and $\text{$\delta${}}\approx{}0.26$ (b). It can be seen that the values are
now comparable for both wall-film thicknesses. Hence, the modified scaling
captures the oscillating dynamics for different wall-film thickness, due to
the increase of the characteristic mass. This correction also enables to
reduce the dependency with $\mathit{We}_{d}$ for
$\text{$\delta${}}\approx{}0.1$. Yet, a slight slowdown of the crown dynamics
with increasing $\mathit{We}_{d}$ can still be observed. Hence, additional
effects might be responsible for this remaining trend.

Considering a classical oscillating mass-spring system, the damping can
increase the inviscid oscillating period $t_{cap}$ as
$\text{$t_{cap,\nu}${}}=2\pi/\left(\text{$\omega_{cap}$}\sqrt{1-\text{$\eta$}^{2}}\right)$,
where $\omega_{cap}$ is the inviscid pulsation and $\eta$ the damping ratio.
Applied to oscillating droplets during an impact, the damping ratio can be
expressed as
$\text{$\eta${}}\approx{}32\nu\text{$t_{cap}${}}/(2\pi\text{$D_{d}${}}^{2})$
by approximating a damping coefficient of $32\nu/\text{$D_{d}${}}^{2}$ (Tang
et al., 2019). By replacing $t_{cap}$ in $\eta$, the damping ratio can be re-
expressed with the impact parameters of the present study in terms of a
modified Ohnesorge number as $\text{$\eta${}}=16\pi\text{$\mathit{Oh}$}/9$,
with
$\text{$\mathit{Oh}$}=\nu{}\sqrt{(\text{$\rho_{d}${}}+\text{$\rho_{f}${}})/\left(2\text{$D_{d}${}}(\text{$\sigma_{d}${}}+\text{$\sigma_{f}${}}+\text{$\sigma_{d/f}${}})\right)}$.
The maximum value of $\mathit{Oh}$ in the present database (considering
averaged kinematic viscosity as $\sqrt{\text{$\nu_{d}$$\nu_{f}$}}$ suitable
for binary droplet/wall-film system (Bernard et al., 2020)) is less than
0.015, leading to $\text{$\eta${}}=0.084$, and an increase of $t_{cap}$ of
less than 0.5%. Hence, the damping due to viscosity alone is negligible. Note
however that this damping does not take shear into account during the
extension, which has been shown to play a significant role for the crown
extension dynamics at the base (Geppert et al., 2020) and at the rim (Bernard
et al., 2020). This viscous losses due to shear might increase the damping,
and with it the oscillation period when shear increases. Since shear losses
are expected to increase with increasing $\mathit{We}_{d}$ as highlighted by
the increasing trend of $1-\text{$E_{\sigma,c,max}${}}/\text{$E_{k,d,0}${}}$
in Fig. 4(b), this could explain the increasing trend of $\tau_{cap,H_{max}}$
with growing $\mathit{We}_{d}$. This also corroborates that this trend is
weaker for $\delta$=0.2 where shear is smaller because the influence of the
solid wall is reduced.

Another mechanism that could lead to the increase of $\tau_{cap,H_{max}}$ with
growing $\mathit{We}_{d}$ is the amplitude of the oscillation. Since
droplet/wall-film systems are characterised by large extensions, the
isochronism of small oscillations might be altered. In this case, the
oscillation period increases with increasing amplitude (Fulcher & Davis, 1976)
which corresponds in our case to the Weber number.

$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$$0.8$$1$(a)$\mathit{We}_{d}$
(-)$\tau_{cap,H_{max},m}$ (-)$\text{$\delta${}}\approx{}0.12$

$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$$0.8$$1$(b)$\mathit{We}_{d}$
(-)$\tau_{cap,H_{max},m}$
(-)$\text{$\delta${}}\approx{}0.26$GW/HexaGW/SOGW/EtOH

Figure 6: (a) Time duration $t_{H_{max}}$ of the crown ascending phase at
different Weber numbers $\mathit{We}_{d}$. (b) Capillary time duration
$\tau_{cap,H_{max}}$ of the crown ascending phase at different Weber numbers
$\mathit{We}_{d}$. The dimensionless wall-film thickness is kept constant at
$\delta$=$0.259\pm 0.008\text{\,}\mathrm{f}$or the entire database.

To conclude, the interface (if any) between the droplet and wall-film brings a
supplementary recoiling force during the extension. Similar to a mass-spring
oscillating system, the interfacial tension influences the oscillation
behaviour of immiscible droplet/wall-film systems. Furthermore, the impact
configuration (especially the wall-film thickness and the droplet kinetic
energy) changes the characteristic mass of the system by modifying the mass of
wall-film interacting in the crown, and the influence of shear, which might
influence the damping significantly.

##  5 Conclusions

The influence of miscibility and wettability in terms of spreading parameter
is investigated for droplet impact onto thin wall-films: immiscible with
partial wetting, with full wetting or miscible, respectively. At similar
liquid properties, this corresponds to a variation of the interfacial tension
between the droplet and the wall-film. First, shifts in the splashing limit as
well as in the crown extensions are observed. Lower interfacial tension
promotes splashing and larger crown extensions. These discrepancies vanish by
taking into account the energy stored in the interface (if any) between the
droplet and the wall film. An energy balance at the end of crown extension
taking into account all surface and interfacial tensions also highlights that
almost half of the initial kinetic energy is lost for thin wall-films. Second,
a shift in the duration of the crown ascending phase is observed. Smaller
interfacial tensions result in longer duration. These differences vanish by
scaling time with a capillary time scale taking into account both surface and
interfacial tensions, indicating an inertio-capillary driven system. The
dynamics is well captured by the capillary time for different wall-film
thicknesses if accounting for the variations of the liquid masses in movement.
Shear at the crown base during extension might also prolong the crown
ascending phase by influencing the damping rate. To conclude, the interfacial
tension leads to a non-negligible energy storage and recoiling force for
droplet impact onto thin wall films where the extension is important. The
interfacial tension, which is linked to miscibility and wettability, needs to
be taken into account to understand the impact dynamics and to model it
accurately.

The authors gratefully acknowledge the financial support by the Deutsche
Forschungsgemeinschaft (DFG) within the international research training group
GRK2160/1 Droplet Interaction Technologies (DROPIT) in the frame of the Summer
School 2018.
Declaration of Interests. The authors report no conflict of interest.

## References

  * Aljedaani et al. (2018) Aljedaani, A. B., Wang, C., Jetly, A. & Thoroddsen, S. T. 2018 Experiments on the breakup of drop-impact crowns by Marangoni holes. J. Fluid Mech. 844, 162–186.
  * Banks et al. (2013) Banks, D., Ajawara, C., Sanchez, R., Surti, H. & Aguilar, G. 2013 Effects of drop and film viscosity on drop impacts onto thin films. Atomization and Sprays 23(6), 525–540.
  * Baumgartner et al. (2020) Baumgartner, D., Bernard, R., Weigand, B., Lamanna, G., Brenn, G. & Planchette, C. 2020 Influence of liquid miscibility and wettability on the structures produced by drop-jet collisions. Journal of fluid mechanics 885, A23.
  * Bernard et al. (2017) Bernard, R., Foltyn, P., Geppert, A., Lamanna, G. & Weigand, B. 2017 Generalized analysis of the deposition/splashing limit for one- and two-component droplet impacts upon thin films.
  * Bernard et al. (2018) Bernard, R., Geppert, A., Vaikuntanathan, V., Lamanna, G. & Weigand, B. 2018 On the scaling of crown rim diameter during droplet impact on thin wall-films. ICLASS 2018 .
  * Bernard et al. (2020) Bernard, R., Vaikuntanathan, V., Weigand, B. & Lamanna, G. 2020 On the crown rim expansion kinematics during droplet impact on wall-films. Experimental Thermal and Fluid Science 118, 110168.
  * Che & Matar (2018) Che, Zhizhao & Matar, Omar K. 2018 Impact of droplets on immiscible liquid films. Soft Matter 14 (9), 1540–1551.
  * Chen et al. (2017) Chen, N., Chen, H. & Amirfazli, A. 2017 Drop impact onto a thin film: Miscibility effect. Physics of Fluids 29 (9), 092106.
  * Cossali et al. (1997) Cossali, G. E., Coghe, A. & Marengo, M. 1997 The impact of a single drop on a wetted solid surface. Experiments in Fluids 22 (6), 463–472.
  * Cossali et al. (2004) Cossali, G. E., Marengo, M., Coghe, A. & Zhdanov, S. 2004 The role of time in single drop splash on thin film. Experiments in Fluids 36 (6), 888–900.
  * De Gennes et al. (2013) De Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2013 Capillarity and wetting phenomena: drops, bubbles, pearls, waves. Springer Science & Business Media.
  * Deegan et al. (2007) Deegan, R D, Brunet, P & Eggers, J 2007 Complexities of splashing. Nonlinearity 21 (1), C1–C11.
  * Enders & Kahl (2008) Enders, S. & Kahl, H. 2008 Interfacial properties of water+ alcohol mixtures. Fluid Phase Equilibria 263 (2), 160–167.
  * Fulcher & Davis (1976) Fulcher, L. P. & Davis, B. F. 1976 Theoretical and experimental study of the motion of the simple pendulum. American Journal of Physics 44 (1), 51–55.
  * Gao & Li (2015) Gao, X. & Li, R. 2015 Impact of a single drop on a flowing liquid film. Phys. Rev. E 92, 053005.
  * Geppert et al. (2020) Geppert, A., Bernard, R., Weigand, B. & Lamanna, G. 2020 Analytical Model for Crown Spreading During Drop Impact onto Wetted Walls: Effect of Liquids Viscosity on Momentum Transfer, pp. 177–190. Springer International Publishing.
  * Geppert et al. (2016) Geppert, A., Chatzianagnostou, D., Mesiter, C., Gomaa, H., Lamanna, G. & Weigand, B. 2016 Classification of impact morphology and splashing/deposition limit for n-hexadecane. Atomization and Sprays 26, 983–1007.
  * Geppert et al. (2014) Geppert, A, Gomaa, H, Meister, C, Lamanna, G & Weigand, B 2014 Droplet wall-film-interaction: Impact morphology and splashing/deposition boundary of hyspin/n-hexadecane two-component system. In Proceedings of the 26th Annual Conference on Liquid Atomization and Spray Systems, ILASS-Americas.
  * Geppert et al. (2017) Geppert, A., Terzis, A., Lamanna, G., Marengo, M. & Weigand, B. 2017 A benchmark study for the crown-type splashing dynamics of one and two-component droplet wall–film interactions. Exp. in Fluids 58, 172(1–27).
  * Geppert (2019) Geppert, A. K. 2019 Experimental investigation of droplet wall-film interaction of binary systems. PhD thesis.
  * Hasegawa & Nara (2019) Hasegawa, K. & Nara, T. 2019 Energy conservation during single droplet impact on deep liquid pool and jet formation. AIP Advances 9 (8), 085218.
  * van Hinsberg et al. (2010) van Hinsberg, Nils Paul, Budakli, Mete, Göhler, Sebastian, Berberović, Edin, Roisman, Ilia V., Gambaryan-Roisman, Tatiana, Tropea, Cameron & Stephan, Peter 2010 Dynamics of the cavity and the surface film for impingements of single drops on liquid films of various thicknesses. Journal of Colloid and Interface Science 350 (1), 336–343.
  * Huang & Chen (2018) Huang, Hai-Meng & Chen, Xiao-Peng 2018 Energetic analysis of drop’s maximum spreading on solid surface with low impact speed. Physics of Fluids 30 (2), 022106.
  * Kittel (2019) Kittel, H. M. 2019 Drop impact onto a wall wetted by a thin film of another liquid. PhD thesis, Technische Universität Darmstadt.
  * Kittel et al. (2017) Kittel, Hannah M, Roisman, Ilia V & Tropea, Cameron 2017 Splashing of a very viscous liquid drop impacting onto a solid wall wetted by another liquid. In Proceedings ILASS–Europe 2017\. 28th Conference on Liquid Atomization and Spray Systems. Universitat Politècnica València.
  * Kittel et al. (2018a) Kittel, H. M., Roisman, I. V. & Tropea, C. 2018a Content of secondary droplets formed by drop impact onto a solid wall wetted by another liquid. ICLASS 2018 .
  * Kittel et al. (2018b) Kittel, H. M., Roisman, I. V. & Tropea, C. 2018b Splash of a drop impacting onto a solid substrate wetted by a thin film of another liquid. Phys. Rev. Fluids 3, 073601(1–17).
  * Lamanna et al. (2019) Lamanna, G, Geppert, A & Weigand, B 2019 On the effect of a thin liquid film on the crown propagation in drop impact studies.
  * Lel et al. (2008) Lel, V.V., Kellermann, A., Dietze, G., Kneer, R. & Pavlenko, A.N. 2008 Investigations of the Marangoni effect on the regular structures in heated wavy liquid films. Experiments in Fluids 44 (2), 341–354.
  * Liang & Mudawar (2016) Liang, G. & Mudawar, I. 2016 Review of mass and momentum interactions during drop impact on a liquid film. Int. J. Heat Mass Transfer 101, 577–599.
  * Marcotte et al. (2019) Marcotte, Florence, Michon, Guy-Jean, Séon, Thomas & Josserand, Christophe 2019 Ejecta, corolla, and splashes from drop impacts on viscous fluids. Physical Review Letters 122 (1).
  * Motzkus et al. (2009) Motzkus, C., Gensdarmes, F. & Géhin, E. 2009 Parameter study of microdroplet formation by impact of millimetre-size droplets onto a liquid film. Journal of Aerosol Science 40 (8), 680–692.
  * Okumura et al. (2003) Okumura, K., Chevy, F., Richard, D., Quéré, D. & Clanet, C. 2003 Water spring: A model for bouncing drops. EPL 62 (2), 237–243.
  * Petitjeans & Maxworthy (1996) Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. part 1. experiments. Journal of Fluid Mechanics 326, 37–56.
  * Planchette et al. (2017) Planchette, C., Hinterbichler, H., Liu, M., Bothe, D. & Brenn, G. 2017 Colliding drops as coalescing and fragmenting liquid springs. Journal of fluid mechanics 814, 277–300.
  * Roisman et al. (2008) Roisman, I. V., van Hinsberg, N. P. & Tropea, C. 2008 Propagation of a kinematic instability in a liquid layer: Capillary and gravity effects. Phys. Rev. E 77, 046305.
  * Roisman et al. (2012) Roisman, I. V., Planchette, C., Lorenceau, E. & Brenn, G. 2012 Binary collisions of drops of immiscible liquids. J. Fluid Mech. 690, 512–535.
  * Ross & Becher (1992) Ross, Sydney & Becher, Paul 1992 The history of the spreading coefficient. Journal of Colloid and Interface Science 149 (2), 575–579.
  * Shaikh et al. (2018) Shaikh, S., Toyofuku, G., Hoang, R. & Marston, J.O. 2018 Immiscible impact dynamics of droplets onto millimetric films. Experiments in Fluids 59 (1), 7.
  * Tang et al. (2019) Tang, X., Saha, A., Sun, C. & Law, C. K. 2019 Spreading and oscillation dynamics of drop impacting liquid film. Journal of Fluid Mechanics 881, 859–871.
  * Thoroddsen (2002) Thoroddsen, S. T. 2002 The ejecta sheet generated by the impact of a drop. Journal of Fluid Mechanics 451, 373–381.
  * Thoroddsen et al. (2006) Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2006 Crown breakup by Marangoni instability. J. Fluid Mech. 557, 63–72.
  * Thoroddsen et al. (2011) Thoroddsen, S. T., Thoraval, M.-J., Takehara, K. & Etoh, T. G. 2011 Droplet splashing by a slingshot mechanism. Physical Review Letters 106 (3).
  * Truzzolillo & Cipelletti (2017) Truzzolillo, D. & Cipelletti, L. 2017 Off-equilibrium surface tension in miscible fluids. Soft matter 13 (1), 13–21.
  * Vaikuntanathan & Sivakumar (2016) Vaikuntanathan, V. & Sivakumar, D. 2016 Maximum spreading of liquid drops impacting on groove-textured surfaces: effect of surface texture. Langmuir 32 (10), 2399–2409.
  * Wang et al. (2020) Wang, Bo, Wang, Chenyu, Yu, Yude & Chen, Xiaodong 2020 Spreading and penetration of a micro-sized water droplet impacting onto oil layers. Physics of Fluids 32 (1), 012003\.
  * Wildeman et al. (2016) Wildeman, S., Visser, C. W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. Journal of fluid mechanics 805, 636–655.
  * Worthington & Cole (1897) Worthington, A. M. & Cole, R. S. 1897 Impact with a liquid surface, studied by the aid of instantaneous photography. Philosoph. Trans. Royal Soc. London Ser. A 189, 137–148.
  * Zhang et al. (2012) Zhang, L. V., Toole, J., Fezzaa, K. & Deegan, R. D. 2012 Splashing from drop impact into a deep pool: multiplicity of jets and the failure of conventional scaling. Journal of Fluid Mechanics 703, 402–413.
  * Zhang et al. (2019) Zhang, Y., Liu, P., Qu, Q. & Hu, T. 2019 Energy conversion during the crown evolution of the drop impact upon films. International Journal of Multiphase Flow 115, 40–61.

##  Appendix A Schematic of the experimental setup

High-SpeedCameraTriggerFilm thicknessmeasurement device (CCI)Impact
areaDropperLED Figure 7: Schematic of the experimental setup allowing a two-
perspective high-speed shadowgraphy of the impact area by an arrangement of
lenses and mirrors. Reprinted from Bernard et al. (2020) with permission from
Elsevier Inc. (2020).

##  Appendix B Crown surface energies at $\text{$\delta${}}\approx 0.26$

The maximum crown surfaces $\Sigma_{c,max}$ and normalized crow surface
energies $E_{\sigma,c,max}$/$E_{k,d,0}$ are given in Fig. 8 (a) and (b),
respectively, in function of $\mathit{We}_{d}$ for the three different liquid
pairs at $\text{$\delta${}}\approx{}0.26$. The trends observed are in general
similar to those of $\text{$\delta${}}\approx{}0.12$: the points of the
different liquid pairs come together if the interfacial energy between droplet
and wall film is taken into account, the crown surface $\Sigma_{c}$ tends to
increase with growing $\mathit{We}_{d}$, and $E_{\sigma,c,max}$/$E_{k,d,0}$
decreases with growing $\mathit{We}_{d}$. Considering the crown surface
energies (Fig. 8(b)), the values are higher than those of
$\text{$\delta${}}\approx{}0.12$ as explained in Sec. 4.1.

$200$$250$$300$$350$$400$$0$$100$$200$$300$(a)$\mathit{We}_{d}$
(-)$\Sigma_{c,max}$ (${\mathrm{mm}}^{2}$)$\text{$\delta${}}\approx{}0.26$

$200$$250$$300$$350$$400$$0$$0.2$$0.4$$0.6$$0.8$$1$(b)$\mathit{We}_{d}$
(-)$E_{\sigma,c,max}$/$E_{k,d,0}$
$(-)$$\text{$\delta${}}\approx{}0.26$GW/HexaGW/SOGW/EtOH

Figure 8: (a) Maximum crown surface $\Sigma_{c,max}$ at different Weber
numbers $\mathit{We}_{d}$. (b) Maximum crown surface energy $E_{\sigma,c,max}$
normalized with the initial droplet kinetic energy $E_{k,d,0}$ at different
Weber numbers $\mathit{We}_{d}$.

However, some discrepancies are observed for small $\mathit{We}_{d}$, where
the crown surfaces are much higher than expected. This is particularly
pronounced for GW/Hexa, whose trend with decreasing $\mathit{We}_{d}$ is
reversed, leading to crown surfaces even higher than those of GW/SO. Note that
a similar trend is also observed for GW/Hexa at
$\text{$\delta${}}\approx{}0.12$ for the lowest $\mathit{We}_{d}$, but in much
smaller amplitude. This sudden increase is peculiar since it corresponds to
higher values of normalized crown surface energies approaching 1, which would
indicate extremely low losses. The possible reasons for these discrepancies
are listed hereafter.
First, it is possible that these experimental conditions lead to a much better
energy transfer from the incoming droplet kinetic energy to surface energy.
Low Weber/Reynolds numbers and high wall-film thickness promote a smooth
energy transfer, corresponding to a clear deposition case (or full deposition
case), as it is confirmed by the unperturbed rim of Fig. 9. Furthermore, no
energy is lost to form the corrugations on the crown rim, and there is no mass
losses. Since the interfacial tension of GW/Hexa is higher, this effect could
happen at higher $\mathit{We}_{d}$ and stronger than for the other liquid
pairs, as it has been observed for the splashing limit and explained in Sec.
3.1.
Second, the particularly strong increase of GW/Hexa could be due to
contaminations that would decrease the value of the interfacial tension, and
thus increase the crown extension. However, this effect alone is unlikely
since it has been observed at several impact conditions, and also for all
liquid pairs (GW/SO and GW/EtOH) although in smaller amplitude.
Last, part of this increasing trend could be due to measurement errors of the
crown surface due to its concavity for these cases, especially for GW/Hexa as
illustrated in Fig. 9. Similar crown contractions have already been observed
in the literature for n-hexadecane droplet/wall-film experiments (Geppert et
al., 2017), but they were rather located at the top of the crown. In the
present cases, this curvature of the crown wall could be due to the
interfacial tension acting on it, since it is the most varying parameter
between the three liquid pairs.

$a$$b$$z$$r$$\bullet$$\bullet$$\bullet$$\bullet$$H_{R}$2$R_{R}$2$R_{B}$

Figure 9: Top: Typical high-speed image (without background substraction) of
GW/Hexa at $\mathit{We}_{d}$=320 and $\text{$\delta${}}\approx{}0.259$ (see
full experimental condition in Table 2). The scale bar measures
$2.00\text{\,}\mathrm{mm}$. Bottom: schematic of a cylindrical crown with
curved wall.

It could also result from the gradient of the crown thickness and/or liquid
properties inside the crown wall. To better understand these effects, the
spatial distribution of the liquids need to be resolved which goes beyond the
scope of this study, but could be for example further investigated with Direct
Numerical Simulations. Coming back to the evaluation of the crown surface,
this crown wall bended inward can lead to an overestimation of the real crown
surface. Indeed, the crown surface is calculated by assuming a straight line
between the rim and base radii (the crown wall being considered as a conical
frustum, as in Fig. 2). In order to get a rough idea of this overestimation,
one can compare the surface of a simplified cylindrical crown with the same
crown where the side wall is curved inward as illustrated in Fig. 9 (bottom).
In this simplified case, the envelop of the crown wall is assumed to follow an
ellipse centered at ($R_{R}$,$H_{R}$/2) and a vertical major axis $a$ equal to
$H_{R}$/2. The minor axis $b$ can be varied to see the influence of the crown
wall curvature. In this configuration, the crown radius $r$ is dependent on
the height $z$ as follows :

$r=\text{$R_{R}${}}-b\sqrt{1-\frac{\left(\text{$H_{R}${}}/2-z\right)^{2}}{a^{2}}}$

The crown surface can be calculated as :

$\text{$\Sigma_{c}${}}=\int_{0}^{\text{$H_{R}${}}}2\pi r(z)dz$

The solution of this integration for $a=\text{$H_{R}${}}/2$, normalized with
the cylindrical surface of reference $2\pi\text{$R_{R}${}}\text{$H_{R}${}}$ is
provided in function of the crown aspect ratio $H_{R}$/$R_{R}$ in Fig. 10 for
different values of the minor axis $b$. For $b=0$, the coefficient is equal to
one, the surfaces are equivalents. With increasing curvature, the ratio
decreases, and it decreases also with increasing aspect ratio. The highest
aspect ratio observed in the present database is $H_{R}$/$R_{R}$=0.65. The
highest value of b can be estimated from the high-speed images. In Fig. 2, the
value of $b$ is approximately 0.13$H_{R}$, which is a particularly pronounced
case. Hence, in the worst case scenario we consider the configuration of
$b=0.15\text{$H_{R}${}}$ with $\text{$H_{R}${}}/\text{$R_{R}${}}=0.65$, which
leads to an underestimation of almost 8%.

$0$$0.2$$0.4$$0.6$$0.9$$0.92$$0.94$$0.96$$0.98$$1$$\frac{\text{$H_{R}${}}}{\text{$R_{R}${}}}$
(-)$\frac{\text{$\Sigma_{c}${}}_{,\text{curved}}}{2\pi\text{$H_{R}${}$R_{R}${}}}$
(-)$b=0$$b=0.05\text{$H_{R}${}}$$b=0.10\text{$H_{R}${}}$$b=0.15\text{$H_{R}${}}$
Figure 10: Normalized crown surface in the case where the crown wall is bended
inward in function of the crown aspect ratio $H_{R}$/$R_{R}$, for different
values of the minor axis $b$ of the ellipse as shown in the schematic of Fig.
9 (bottom). This quantifies the underestimation of the measured crown surface
$\Sigma_{c}$ when the crown wall is assumed to be a straight line between base
and rim radii.