writekey=false,name=

Also at: ]School of Engineering and Applied Sciences, Wyss Institute for
Biologically Inspired Engineering, Harvard University, Cambridge, MA 02138
Also at: ]Department of Mechanical and Aerospace Engineering, University of
Florida, Gainesville, FL 32611 Also at: ]Intel Corporation, Hillsboro, OR
97124 Also at: ]Department of Mechanical Engineering, Virginia Polytechnic
Institute and State University, Blacksburg, VA 24060 Also at: ]ANSYS
Incorporated, Lebanon, NH 03766

# Increased solidification delays fragmentation
and suppresses rebound of impacting drops

Varun Kulkarni  [  Suhas Tamvada  [  Nikhil Shirdade  Navid Saneie  [
Venkata Yashasvi Lolla  [  Vijayprithiv Batheyrameshbapu  [  Sushant Anand
Corresponding Author<EMAIL_ADDRESS>Department of Mechanical and Industrial
Engineering
University of Illinois, Chicago, IL 60607

###### Abstract

The splat formed after drop impact on supercooled solid surfaces sticks to it.
On the contrary, a sublimating supercooled surface such as dry ice inhibits
pinning and therefore efficiently rebounds drops made of a variety of liquids.
While rebound is expected at lower impact velocities on dry ice, at higher
impact velocities the drop fragments leaving behind a trail of smaller
droplets. However, it is not known whether rebound can be entirely suppressed
or fragmentation be controlled on such surfaces and if it depends on the
extent of solidification inside the drop. In this work, we report on the role
played by solidification within drops in modifying the outcomes of their
impact on the supercooled ultra-low adhesive surface of sublimating dry ice.
We show that the solidification thickness depends on the impact velocity and
is the primary driver in suppression of rebound and also promotes a delay in
fragmentation. Our findings imply that sublimating supercooled surfaces can
present a broad spectrum of outcomes from complete bouncing to no-rebound
which are not seen in drop impacts on supercooled superhydrophobic surfaces.
We attribute this to thermo-elastocapillarity which considers bending of the
solidified layer and is used to demarcate regime boundaries and determine the
coefficient of restitution during rebound.

This is not the latest version of the paper and the latest one can be found
at:
$\bullet$ https://doi.org/10.1103/PhysRevFluids.9.053604 or,
$\bullet$ ResearchGate©, click here for pdf.

##  I Introduction

Liquid drop impact on supercooled surfaces maintained below the melting
temperature of the solidified liquid ($T_{m}$) is decisive to several
industrial applications and everyday life in icy climates [1, 2, 3]. Depending
upon liquid’s thermal and physical properties, substrate’s wettability and
impact conditions, a fascinating spectrum of post-impact behavior is observed
which ranges from complete bouncing [4] in the absence of solidification to
pinning-mediated adhesion of solidifying drops [5, 2]. In the latter, adhesion
of solidified material on surfaces, exemplified by ice accumulated on roads
[6], wind turbine blades [3], and aircraft wings[7] is quite undesirable,
disrupting everyday activities, industrial operations sometimes even
imperiling human safety. Conversely, in applications like thermal spraying,
splat quenching and additive manufacturing [2, 1, 8] adhesion may be used
advantageously by solidifying molten metals by impaction on an underlying
substrate resulting in the formation of solid film of desired functional
attributes. Whether the goal is to engineer surfaces and coatings for reduced
ice adhesion or to create solid thin films with specific functional
properties, achieving these objectives is crucially dependent on comprehending
the outcomes of drop impact and the role of solidification during such impacts
[9, 10].

Fundamentally, the drop impact outcomes are dictated by liquid density
$\rho_{l}$, surface tension, $\sigma_{l}$, and kinematic/geometric factors
including impact velocity ($V_{0}$) and initial drop diameter ($D_{0}$) and
succinctly described by the Weber number, We which represents the ratio of the
inertial ($\rho_{l}D_{0}^{2}V_{0}^{2}$) and surface tension force
($\sigma_{l}D_{0}$). Another factor besides these liquid properties which
plays an important role is the nature of substrate as communicated by its
wettability or roughness. Therefore, the liquid drop We and the substrate
properties, together can aptly describe both complete bounce (on highly non-
wetting surfaces) - with/without fragmentation and, pinning-mediated adhesion
[4].

Figure 1:  Drop impact on a typical non-wetting surface, (i) overlayed images
of the release of a drop of diameter, $D_{0}$, from a height, $H_{0}$ and,
moment of its impact with a velocity, $V_{0}$ forming a splat of maximum
diameter, $D_{max}$, (ii) rebound of a drop to height, $H_{1}$ (iii)
fragmentation of splat of molten tin (reprinted with permission, Aziz and
Chandra [Aziz2000]), (iv) corona splash (reprinted with permission, Xu [11])
(v) prompt splash (reprinted with permission, Almohammadi and Amirfazli [12]).
Dotted boxes in (iii) - (v) point to schematic side views of fragmentation and
splashing. Scale bars from (i) - (v) represent a length of 1 mm.

Physically, in simple terms we can understand rebound by considering a drop
released from a height of $H_{0}$ (as shown in Fig. 1(i), if its initial
kinetic energy, $KE_{0}\sim\rho_{l}D_{0}^{3}V_{0}^{2}$ exceeds its surface
energy at maximum spread, $SE_{max}\sim\sigma_{l}D_{max}^{2}$ where, $D_{max}$
is the maximum horizontal splat diameter), i.e. $\Delta E=KE_{0}-SE_{max}>0$.
The rebound height $H_{1}$ (as shown in Fig. 1 (ii)) then can be simply
estimated from $\Delta E$ by equating it to $\sqrt{mgH_{1}}$, where $m$ is the
mass of the drop. From this, it may be tempting to infer that an increase in
$KE_{0}$ with a comparatively lower increase in $SE_{max}$ leads to greater
rebound (or, higher value of $H_{1}$), but, in practice, this increase is
bounded. This is because, an increase in impact velocity, $V_{0}$ also
increases the horizontal spreading velocity $V_{s}$ (as shown in Fig. 1 (i),
whose spread is curtailed by surface tension. Consequently, liquid accumulates
at the edge of the spreading drop forming a rim bounding a thinner central
sheet of the denser fluid (drop), accelerating horizontally from 0 to
$V_{s}=3/8V_{0}$ into the lighter fluid (air) after the drop contacts the
substrate [13]. Such spreading induces Rayleigh-Taylor instability [4], with
waves manifesting as fingers at the periphery of the rim that eventually
disintegrate into smaller drops as shown in Fig. 1(iii) and the accompanying
schematic. It is noteworthy that such splat fragmentation is differentiated
from the frequently reported drop splashing [14, 15, 16], which refers to the
appearance of a lamella (liquid sheet) lifted upwards supported
aerodynamically by a lift force[14] (shown schematically in the dashed box
associated with Figs. 1(iv), (v). Usually, the expelled lamella in these cases
either forms a bowl-like structure known as corona splash and shown in Fig.
1(iv) or an inclined sheet (this known as prompt splash as shown in Fig. 1(v)
ultimately disintegrating into small droplets.

A key factor in the dynamics described above are dissipative losses which slow
the spread and recoil of the drop after impact. In isothermal impacts, it is
commonly seen that the dissipation becomes significant when, (i) dynamic
viscosity of the liquid, $\mu_{l}$ is high and/or, (ii) there is an increase
in viscous stresses ($\sim\mu_{l}V_{0}h_{pan}$) due to reducing splat
thickness, $h_{pan}$. The relative significance of dissipation due to
viscosity can be assessed by two dimensionless groups: (i) Reynolds number,
$Re$ which is the ratio of kinetic energy (inertia), $\rho_{l}V_{0}^{2}$ with
the energy lost due to viscous dissipation, $\mu_{l}V_{0}D_{0}$ per unit
volume resulting in, $\rho_{l}V_{0}D_{0}/\mu_{l}$ and, (ii) capillary number,
$Ca$ given by the ratio of viscous dissipation to the surface energy,
$\mu_{l}V_{0}/\sigma_{l}$. For instance, in liquids such as water,
$Re\gtrapprox\mathcal{O}(10^{2})$ and $Ca\lessapprox\mathcal{O}(10^{-3})$
allowing us to ignore viscous effects, whereas for glycerol,
$Re\lessapprox\mathcal{O}(1)$ and $Ca\gtrapprox\mathcal{O}(10)$ which
indicates a non-negligible role of viscosity in drop dynamics.

Contrary to traditional, isothermal drop impacts on surfaces at room
temperature described heretofore, impact of liquids (at room temperature) on
supercooled surfaces presents non-isothermal conditions. Dissipation in such
cases occurs thermally, through solidification, often represented in terms of
the dimensionless Stefan number, $Ste=c_{p,s}\Delta T/\mathcal{L}$ and Peclet
number, $Pe=V_{0}D_{0}/\alpha_{s}$ where, $c_{p,s}$ (in kJ$\cdot$kg-1K-1) is
the specific heat capacity of the solidified layer, $\mathcal{L}$ is the
latent heat of solidification (in kJ$\cdot$kg-1) and $\alpha_{s}$ (in
m2$\cdot$s-1) is the thermal diffusivity of the solidified layer,
respectively. Solidification of the liquid drop here commences immediately
upon contact with the surface and progresses gradually as the drop spreads
enhancing pinning to the surface. Adhesion of such nature [5] influences the
final splat morphology leading to intriguing outcomes such as self-peeling [2]
and fracture [17]. Consequently, conditions leading to the arrest of drop
spread are critical to the eventual shape assumed by these splats. In this
regard, two main approaches have been adopted, the first only considers
hydrodynamics at the drop scale and the second considers the dynamics at the
contact line exclusively. In the first approach, at the drop scale level,
solidification is either seen as reducing kinetic energy post impact [8] or
cooling of the liquid is seen to augment dissipation which now consists of
both the viscous and thermal boundary layer [18]. The second approach focuses
its attention to the contact line [19, 20] and argues that spreading of
solidifying drops is arrested when at the contact line, (i) the contact angle
of the spreading drop equals the angle of the freezing front (ii) a critical
volume at the contact line is solidified or, (iii) the liquid at the contact
line reaches a critical temperature determined by the effect of kinetic
undercooling [21]. Since the dynamics of drop spread are essential in
realizing the eventual fate of the deforming drop, solidification can play a
vital role in these impact scenarios.

The above survey highlights the role of adhesion and solidification in drop
impact on supercooled surfaces and since they act in concert with each other,
isolating their individual roles in the ensuing mechanics is often difficult.
For example, because drop pinning is omnipresent on engineered surfaces in
non-isothermal impacts, the extent to which the solidification within a drop
alone controls post-impact parameters such as maximal spreading, rebound
height, contact time, rebound and splat-fragmentation is not clearly
understood. The choice of test liquids and surfaces therefore becomes
extremely critical. Even though water is one of the most preferred liquids for
impact studies because of its large latent heat of fusion, surface tension and
high supercooling, the effects of solidification within it during impact on
supercooled non-wetting surfaces are small. In contrast, low surface tension
liquids like alkanes have low supercooling, low dynamic viscosity and low heat
of fusion making them prime candidates to study effects of solidification on
drop impact. However, engineering surfaces that are completely non-wetting to
them at all impact velocities is still challenging. One of the very few solid
materials that successfully meets such stringent requirements is dry ice (di),
a supercooled (CO2 gas) material that sublimates at $T_{di}$ = −78.9°C. Dry
ice’s combination of being a molecular and sublimating solid eliminates any
pinning between the drop and surface [22, 23] by providing near contactless
levitation similar to Leidenfrost drops [24] making it an ultra-low adhesive
surface even for low surface tension liquids. Hence, we choose this material
to isolate and show how solidification within a drop alone affects each of the
above-mentioned facets of drop impact. While rebound and fragmentation of
drops on dry ice has been known [22, 23], we show that in the absence of
pinning the extent of solidification controls both these outcomes delineating
them from a region of no bounce when the splat merely spreads and rests on the
surface. Since, the the layer closest to dry ice is solidified our examination
concentrates on fragmentation rather than splashing which precludes any
aerodynamic lifting of the lamella. Further, we also show that the rebound
height in cases where rebound is seen is determined by the amount of
solidification inside an impacting drop. In the course of our investigation,
we also derive and use the dependence of solidification thickness on impact
velocity which has not been tackled so far and provides a facile method to
analyze such phenomena. Our efforts use a combination of laboratory
experiments and theoretical arguments to explain the underlying physics behind
our observations.

Following this introduction, which constitutes Section I of the paper, we
organize the remaining text along the lines described here. We begin with a
description of experimental details of the materials tested and experimental
conditions in Section II following which we describe our experimental
observations on the different drop impact morphologies in Section III.
Thereafter, in Section IV we provide the theoretical foundation for the
dependence of solidification thickness on impact velocity extending Stefan’s
analysis of gradual solidification of a liquid gently placed over a cold
surface. Next, in Section V we describe the two observed regimes:
fragmentation and rebound accompanied by scaling arguments to determine the
criterion when these may be observed. In Section VI we focus on the drop
spread and derive the scaling for the maximum spreading of drop on impact.
Finally, in Section VII we discuss last of our results which details the
effect of solidification on rebound height for different liquids. Our paper
closes with Section VIII, which contains in brief, a summary of our findings,
potential applications, open questions, and suggestions for future work.

##  II Methods, materials, liquid properties, impact conditions and
dimensionless groups

###  II.1 Setup and surface topography

Figure 2:  (i) Sketch of experimental setup showing impact of drop of on dry
ice (not drawn to scale). Listed below are the tested liquids. (ii) Topography
of dry ice surface as measured using an optical profilometer giving an average
value of roughness which equals 10 $\mu$m.

Our setup (shown in Fig. 2(i)) consists of a needle attached to a glass
syringe which is connected to a syringe pump and operated such that a single
drop is ejected at a given time. Drops of diameters varying between 1.8$-$2.8
mm are tested and correspond to needles of gauge sizes 14, 16, 18, and 20
procured from Norsdon®. Five different test liquids, namely, water and long
chain alkanes, decane, tetradecane, pentadecane and hexadecane are chosen as
test liquids which shall be denoted by the color coding displayed below Fig.
2(i). The capillary length, $\ell_{cap}=\sqrt{\sigma_{l}/\rho_{l}g}$ where $g$
is the acceleration due to gravity ($\approx 9.81$ m$\cdot$s-2) for all these
liquids is greater than or nearly equal to the drop diameters generated,
ensuring that role of gravity is negligible in the drop deformation and impact
dynamics. After their release from the needle the drops were allowed to impact
on a dry ice slab at -78.9∘C. Room temperature/ambient conditions correspond
to a temperature of 25∘C (or 298 K) and atmospheric pressure, 1 atm.

To observe and record the drop impact behavior we used Photron®FASTCAM Mini AX
camera at 4000 frames per second (fps) with a pixel a resolution of 1024
$\times$ 1024 pixels and a shutter exposure time of 5 $\mu$s. The selected
frame rate gave us a temporal resolution of about 0.25 ms which was lesser
than the impact time scale, $D_{0}/V_{0}$ of 1 ms. To spatially resolve the
impact dynamics we used a high magnification lens (InfiniProbe®TS$-$160) with
a focal length between infinity and 18 mm which produced a magnification of
0$-$16$\times$ such that 1 pixel $\approx$ 15 $\mu$m. The background lighting
used to illuminate our setup consisted of an LED (Nila-Zaila®) light source
whose intensity of diffused and homogenized using an acrylic diffuser plates
placed between the light source and the impacting drop. Drops of size, $D_{0}$
(in m) and density, $\rho_{l}$ (in kg$\cdot$m-3) were released from heights,
$H_{0}$ between 2 and 12 cm to vary their impact velocity ($V_{0}$) 0.3 and
1.5 m$\cdot$s-1 on nitrogen-purged frost-free dry ice surface. These
experiment conditions correspond to Weber number, $We$ ranging from 12 to 120.
The videos obtained were analyzed using the open source software IMAGE J [25].

Lastly, the dry ice topography and roughness was experimentally measured using
the Bruker-Nano Contour GT-K Optical Profilometer at the Nanotechnology Core
Facility(NCF) at University of Illinois at Chicago(UIC). The samples were
acquired with a 5 $\mu$m scanning step in both x and y perpendicular
directions in the plane as shown in Fig. 2(ii) at a scanning velocity of
approximately 2 mm$\cdot$s-1 in both directions. The three-dimensional
topography revealed by these surface scans is shown in Fig. 2(ii) in the form
of surface heights data acquired by the microprofilometer with a region of
interest (ROI) within 1 mm2. The non-uniformity of the dry ice surface is
clearly seen with a height varying between -5 $\mu$m and 5 $\mu$m. The
experiments were repeated 5 times at different locations on the dry ice
surface with a maximum standard deviation of 0.01 $\mu$m in the maximum and
minimum height measured. Note that for a typical sublimation rate of 1$\%$
mass per hour, initial slab dimensions of $0.3\;\textrm{m}\;\times
0.15\;\textrm{m}\;\times 0.06\;\textrm{m}$, dry ice density of 1600
kg$\cdot$m-3 and experimentation time of 10 s the decrease in height of the
sample is approximately 0.1 $\mu$m. This is $<10\%$ of the measured roughness,
$R_{a}$ $\approx$10 $\mu$m and therefore we can conclude that sublimation does
not affect our profilometry measurements.

###  II.2 Liquid thermal, physical properties, impact conditions and
dimensionless groups

For our experiments, we chose water and four alkanes: decane ($T_{m}\approx$
$-30$°C), tetradecane ($T_{m}\approx$ 5°C), pentadecane ($T_{m}\approx$ 10°C)
and hexadecane ($T_{m}\approx$ 18°C). These alkanes were chosen because except
their $T_{m}$, their other thermal-fluid properties are all nearly the same so
any difference in their post-impact behavior can be directly attributed solely
to their thermal properties.

Table 1:  Thermal, fluid, interfacial properties, kinematic quantities and dimensionless groups |  |  |  Water |  Decane |  Tetradecane |  Pentadecane |  Hexadecane
---|---|---|---|---|---|---|---
Thermal Properties |  Specific Heat [kJ$\cdot$kg-1K-1] |  Liquid, $c_{p,l}$ |  4.18 |  2.21 |  2.19 |  2.20 |  2.22
|  Solid, $c_{p,s}$ |  1.70 |  2.20 |  1.90 |  2.00 |  2.20
Thermal Conductivity [W$\cdot$m-1K-1] |  Liquid, $k_{l}$ |  0.60 |  0.14 |  0.14 |  0.14 |  0.14
|  Solid, $k_{s}$ |  2.20 |  0.17 |  0.14 |  0.15 |  0.22
Thermal Diffusivity [m2$\cdot$s-1] |  Liquid, $\alpha_{l}$ [$\times$ 10-7] |  1.43 |  0.87 |  0.83 |  0.82 |  0.81
|  Solid, $\alpha_{s}$ [$\times$ 10-7] |  11.44 |  0.85 |  0.81 |  0.83 |  1.11
Latent Heat [kJ$\cdot$kg-1] |  Liquid $\rightleftharpoons$ Solid, $\mathcal{L}$ |  334 |  194 |  227 |  207 |  236
Fluid Properties |  Surface tension [N$\cdot$m-1] |  Liquid, $\sigma_{l}$ [$\times$ 10-3] |  72 |  23.83 |  26.56 |  27.07 |  27.47
Dynamic viscosity [Pa$\cdot$s-1] |  Liquid, $\mu_{l}$ [$\times$ 10-3] |  0.89 |  1.26 |  2.33 |  3.10 |  3.30
Density [kg$\cdot$m-3] |  Liquid, $\rho_{l}$ |  998 |  728 |  764 |  769 |  770
|  Solid, $\rho_{s}$ |  920 |  840 |  880 |  884 |  886
Impact Conditions |  Initial drop diameter [m] |  $D_{0}$ [$\times 10^{-3}$] |  2.80 |  1.82 |  1.88 |  1.98 |  1.98
Impact velocity [m$\cdot$s-1] |  $V_{0}$ |  0.24$-$1.29 |  0.25$-$1.44 |  0.20$-$1.45 |  0.27$-$1.22 |  0.19$-$1.30
Dimensionless Groups |  Stefan number |  Ste |  0.40 |  0.56 |  0.70 |  0.86 |  0.91
Peclet number |  Pe [$\times$104] |  0.53$-$2.84 |  0.52$-$2.98 |  0.48$-$3.54 |  0.65$-$2.90 |  0.59$-$3.12
Weber number |  We [$\times$10] |  0.23$-$6.60 |  0.34$-$11.35 |  0.22$-$11.77 |  0.46$-$8.85 |  0.21$-$9.21

The thermal, phase change and fluid properties for the liquid and solid phases
of water and the four different alkanes tested are detailed in this section.
These are used in this work and listed in Table 1 as: Thermal, fluid,
kinematic and interfacial properties and dimensionless numbers. The elastic
(Young’s) modulus, $Y$ for solid alkanes is taken to be $2\times 10^{8}$ Pa as
alkanes fall under the category of paraffin wax while that for ice is
$9.33\times 10^{9}$ Pa [17]. To calculate the thermal diffusivity for the
solid ($\alpha_{s}$) and liquid ($\alpha_{l}$) phases we use the formula,
$\alpha_{s}=k_{s}/\rho_{s}c_{p,s}$ and $\alpha_{l}=k_{l}/\rho_{l}c_{p,l}$.
Also, the properties of the liquids in their solidified form are considered at
their melting temperature, $T_{m}$ and their surface tension in their liquid
form, $\sigma_{l}$ is considered to be within the same order of magnitude as
that in solid form, $\sigma_{s}$. For water, $\sigma_{s}\approx 90$
mN$\cdot$m-1 and considering alkanes in their solid form to be similar to
solid paraffin wax it amounts to approximately a value of 35 mN$\cdot$m-1
about 1.3 times their value in the liquid form. Upon contact with dry ice
substrate, the liquid immediately solidifies and therefore the interfacial
tension between the solidified layer and dry ice is dropped from further
analysis. The roles of other interfacial tensions that may arise are described
in the relevant sections for regime transitions, Sections V.1 and V.2. The
thermal conductivity of dry ice is, $k_{di}=0.011$ W$\cdot$m-1K-1. All values
presented here are taken from literature [26, 27].

A decrease in temperature when the drop touches the dry ice surface can lead
to an increase in dynamic viscosity resulting in enhanced viscous dissipation,
offering a plausible reason for the arrest of movement of the spreading drops.
For this to be true, the viscous effects need to penetrate a larger thickness,
typically the height of the pancake ($h_{pan}$) formed after drop impact,
which would take longer compared to solidification of a thin micro-layer
($\delta$) of the drop in the vicinity of dry ice. To emphasize the role of
solidification over viscous effects, we can compare the time scales of
spreading, $D_{0}/V_{0}$, viscous penetration,
$\rho_{l}(D_{0}/4)^{2}/\mu_{l}$, and thermal penetration
($\delta^{2}/\alpha_{s}$), using the length scales, $D_{0}/4$ (example, Fig 3
(iii), $t$ = 4 ms) and $\delta$ for viscous and thermal penetration
respectively, at which arrest of spread may be expected. For typical
conditions representative of our experiments, $D_{0}=2$ mm, $V_{0}=1$
m$\cdot$s-1, $\alpha_{s}=10^{-7}$ m2$\cdot$s-1, $\rho_{l}=770$ kg$\cdot$m-3,
$\mu_{l}=$1 mPa$\cdot$s (from Table 1) and
$\delta\approx\mathcal{O}\left(10\right)\mu\textrm{m}$, we calculate the
aforementioned times scales (approximately) as, 2, 125 and 1 ms respectively.
This implies that only spreading and thermal penetration time scales are
comparable, and hence viscous effects can be neglected.

Similarly, for temperature dependent properties such as density, viscosity and
surface tension to influence drop spreading, the thickness of the thermal
boundary layer in time, $D_{0}/V_{0}$ corresponding to the drop
impact/spreading time scale should be the order of the thickness of flattened
drop. For our impact scenarios this value of the thermal boundary layer is
$\mathcal{O}\left(10\right)\mu\textrm{m}$ m which means that the temperature
decrease in the liquid drop is only restricted to a thin layer close to the
sublimating dry ice surface.

Hence, our choice of test liquids ensured that the role of temperature-
dependent liquid properties or the role of viscosity was negligible. At room
temperature, the non-dimensional Ohnesorge number,
$Oh=\mu_{l}/\sqrt{\rho_{l}D_{0}\sigma_{l}}$ for all liquids tested was
relatively low assuming values between 0.0025 and 0.008.

##  III Experimental Observations

Using the experimental methodology and liquids described above we
systematically investigate two main aims: (i) splat morphology after impact
and its dependence on the impact We and, (ii) coefficient of restitution as
measured by the rebound height. Our experimental findings addressing the first
aim are displayed as a sequence of images in Fig. 3(i) - (iv)(also see SM See
Supplemental Material for information related to the supplementary movies.
Movie 1, Movie 2). At low impact velocities all liquids bounce, similar to
findings reported in literature [22, 23, 24]. Here, to compare our results
with these earlier studies we used water as the test liquid and impact it with
dry ice surface at $We\approx 20$, an example of which is shown in Fig. 3(i).
We observe splat morphology and rebound behavior, identical to previous
works[22, 23, 24]. To examine the effects of solidification, we experiment
with hexadecane which should exhibit higher solidification as it has higher
Ste (see Table 1). For a hexadecane drop impacting at $We\approx 24$ shown in
Fig. 3(ii) when solidification was substantial we continue to observe bouncing
like water but with the major difference that the bottomost layer of drop
looked partially solidified. Usually in supercooled surfaces solidification
induced pinning restrict bounce off so this result on dry ice surface is
unique especially since it proves that bouncing can be observed if adhesion is
minimized or eliminated entirely.

Figure 3: Different regimes in drop impact (i) bouncing with negligible
solidification for water at We$\approx$ 20 (ii) bouncing with discernible
solidification for hexadecane, We$\approx$ 24 (iii) fragmentation for
We$\approx$ 80 (iv) no-bounce for We$\approx$ 84\. Raw data corresponding to
transition regimes boundaries between bouncing/fragmentation and bouncing/no-
bounce for different liquids as a function of (v) $D_{max}/D_{0}$ and, (vi)
$We$.

To test whether bouncing continues or ceases we gradual increasing the higher
impact velocities. At higher $We$ (or, impact velocities), two distinct
outcomes emerge for almost the same $We$ (see SM [1] Movie 3), the first being
drop fragmentation (see Fig. 3(iii) for decane, $We\approx 80$) and the other
wherein bouncing is suppressed altogether (Fig. 3(iv) for hexadecane,
$We\approx 84$). This suggests that impact outcomes are dominantly dictated by
solidification, which for low Ste liquid like decane and water are shown to
result in fragmentation producing a drop smaller in diameter which continues
to bounce. In contrast, for liquids like hexadecane, pentandecane with high
Ste or increased solidification, bouncing is suppressed entirely when the
thicknesses of the solidified layer increases.

Lastly, we collate the raw data gathered from our experiments with all five
liquids in terms of $We$ and $D_{max}/D_{0}$ and plot them as shown in Fig.
3(v) and (vi). Light green, light purple and light yellow color codes are used
for demarcating regimes of bounce, fragmentation and no-bounce. It is to be
noted that higher values of $We$ and $D_{max}/D_{0}$ typically imply higher
impact velocity, $V_{0}$.

In our second objective, we focus on the effect of solidification on the
bounce height. All 4 alkanes and water are tested to understand this behavior
in detail. A sample result from our experiments at $We\approx 20$ is shown in
Fig. 4(i) (also see SM [1] Movie 2). In this figure we see that from left to
right, with increasing solidification, the rebound height decreases. Our
results were further tested at different We to observe whether for a
particular liquid, an increase We leads to a decrease in rebound height. Fig.
4(ii) depicts the raw data relating to two $We\approx$ 20 and 45 and we note
that increasing $We$ leads to a decrease in rebound height for all liquids.
Our measurements indicate that solidification not only depends on the liquid
thermal and phase change properties but also impact velocity or We.

Figure 4: (i) Experimental images showing decrease in rebound height, $H_{1}$
with increasing solidification as indicated by the Ste number. (ii) $H_{1}$ at
two different We $\approx$ 20 (half filled semicircles) and 45 (filled
semicircles), demonstrating two specific influences, first, the decrease in
$H_{1}$ for all liquids with an increase in We and, second, a decrease in
$H_{1}$ at given We for liquids with higher Ste or with increased
solidification.

Both of above observations are studied in detail and rationalized
theoretically in the next section (Sec. IV) where effect of solidification
with We is quantified. We then use this information in deriving criteria for
regime transitions in Sec. V followed by a understanding how drop spread is
affected by solidification in Sec. VI which an ancillary goal in understanding
the drop dynamics of this nature better and conclude with Sec.VII where our
theoretical framework is used to explain our observation on rebound height.

##  IV Dependence of solidified liquid layer, $\delta$ on impact Weber number,
$We$

As suggested during the description of our experiments partial solidification
is important in understanding the mechanics behind our observations. In this
section we quantify it by calculating the thickness of the solidified portion
of a liquid ($\delta$) when it comes in contact with a supercooled dry ice
surface. An important consideration in this regard is that the surface of the
dry ice is not smooth (see Section II) and the contact made by the drop is not
perfect. Moreover, increasing $We$ can greatly affect this area of contact
too, thereby increasing the extent of heat transfer. Additionally, the
properties at the surface can differ significantly from that in the bulk.
Together, these effects lead to development of non-negligible thermal contact
resistance at this contact area developing a temperature distribution as shown
in Fig. 5(i). Two extreme limits of this at low and high We with the
difference in contact area are also sketched in Figs. 5(ii)(a) and (iii)(a)
with the resulting temperature distribution in Figs. 5(ii)(b) and (iii)(b). In
the following we meticulously analyze cases high We when contact resistance is
negligible and then the case for arbitrary We with finite contact resistance.

###  IV.1 Thickness of solidified liquid layer, $\delta$ at $We=0$

We first consider the energy balance, commonly known as Stefan’s condition [1,
29, 30] at the solidification interface [29] and written as follows,

$\rho_{s}\mathcal{L}\dfrac{d\delta}{dt}=k_{s}{\dfrac{\partial T}{\partial
z}}\biggr{|}_{s}-k_{l}{\dfrac{\partial T}{\partial z}}\biggr{|}_{l}$ (1)

Here, $\rho_{s}$ is the density of the solidified liquid, $\mathcal{L}$, is
the latent heat of solidification of the liquid and $\delta$ is the thickness
of the solidified layer. Eq. 1 is representative of the fact that the latent
heat for solidification $\rho_{s}\mathcal{L}\frac{d\delta}{dt}$ is provided by
the difference of conductive heat transfer in the liquid
$-k_{l}{\frac{\partial T}{\partial z}}\bigr{|}_{l}$ and solid
$-k_{s}{\frac{\partial T}{\partial z}}\bigr{|}_{s}$ phases across the
solidification front (see Fig. 5(i)-(iii)). $d\delta/dt$ is the velocity of
the interface while it is solidifying, $k_{s}$ is the thermal conductivity of
the solidified liquid and $k_{l}$ is the liquid thermal conductivity. In Eq.
(1), $-k_{l}{\frac{\partial T}{\partial z}}\bigr{|}_{l}$ is much smaller
compared to $-k_{s}{\frac{\partial T}{\partial z}}\bigr{|}_{s}$ as the
thickness of the solidified layer, $\mathcal{O}(10^{-6})$ is much smaller than
the diameter of the drop, $\mathcal{O}(10^{-3})$ and therefore the gradient,
${\frac{\partial T}{\partial z}}\bigr{|}_{s}$ is larger compared to
${\frac{\partial T}{\partial z}}\bigr{|}_{l}$. Here, $\Delta
T_{s}=T_{m}-T_{di}$ (for solid phase) is $\mathcal{O}\left(10\right)$ and
$\Delta T_{l}=T_{m}-T_{amb}$ (for liquid phase) is also
$\mathcal{O}\left(10\right)$. Going forward, since $\Delta T_{l}$ will not be
used, the subscript $l$ will be dropped and $\Delta T$ would mean $\Delta
T_{s}$.

Figure 5: (i) General scenario of a drop in contact with a surface with non-
negligible contact resistance, $R_{c}$ and the ensuing temperature
distribution at the dry ice/liquid interface with a contact temperature of
$T_{c}$. Portion of the solidified drop in contact with the supercooled
surface is marked with a solidification thickness of $\delta$. Two limiting
cases of contact resistance, $R_{c}$ when, (ii) $R_{c}\approx 0$ when $We$ is
high, sketched in (a) with corresponding temperature profile in liquid and
solid phases shown in (b) depicting $T_{c}\approx T_{di}$. Note that red line
within the zone of the solidified drop is a linear approximation to the
unsteady temperature distribution which in the limit of small $\delta$ is an
accurate representation of the temperature gradient at quasi steady or low
Ste. (iii) $R_{c}>>1$ when $We$ is low is sketched in (a) with corresponding
temperature profile in liquid and solid phases shown in (b) depicting
$T_{c}\approx T_{m}$ (iv) Section of the drop in contact with the dry ice
texture at one location (v) Exploded view showing depth of capillary
penetration, $h_{cap}$ and dynamic pressure head due to drop impact, $h_{dyn}$
used to estimate contact resistance at a given We.

Furthermore, we use the approximation $dT/dz$ as $\Delta T/\delta$ which is
true for a quasi-steady approximation when $Ste<1$ and implies that the
temperature distribution within the solid layer is linear as shown in Fig.
5(ii)(b) between the limits $T_{m}$ and $T_{di}$. We shall see in Section
IV.1.1 that this approximation is fairly accurate by considering the unsteady
nature of the problem. Finally, using the foregoing simplifications we can
express $\delta$ as,

$\rho_{s}\mathcal{L}\dfrac{d\delta}{dt}=k_{s}\dfrac{\Delta T}{\delta}$ (2)

Integrating Eq. 2 we derive the expression for $\delta$ using the definition
of non-dimensional $Ste$, $\left(=\dfrac{c_{p,s}\Delta T}{\mathcal{L}}\right)$
and the initial condition, $\delta\left(0\right)=0$ as given below,

$\delta=\sqrt{2Ste\;\alpha_{s}t}=2\lambda\sqrt{\alpha_{s}t},\textrm{ where,
}\lambda=\sqrt{Ste/2}$ (3)

$Ste$ used here indicate fast solidification where large ($>1$) and small
values ($<1$) correspond to slow solidification. Also, Eq. 3 is very useful to
determine the scaling for the solidification time scale by expressing $t$ as
$t_{sol}$ and rearranging Eq. 3.

$t_{sol}\sim\dfrac{\delta^{2}}{\alpha_{s}Ste}=\dfrac{\rho_{s}\delta^{2}\mathcal{L}}{k_{s}\Delta
T}$ (4)

We shall use this expression as a time scale in the next section but before
that we completely non-dimensionalize Eq. 3 by evaluating Eq. 2 at
$t_{con}\approx 5D_{0}/V_{0}$ [8], which denotes the time for which the drop
is in contact with the dry ice surface. By defining,
$\overline{\delta}=\delta/D_{0}$ we reduce Eq. 3 to,

$\overline{\delta}=\sqrt{\dfrac{10Ste}{Pe}}$ (5)

The value of $\lambda$ in Eq. 3 assumes a quasi steady approximation ($Ste<1$)
of the governing energy equation which gives rise to a linear temperature
distribution within the solidified layer[17]. For our case, Ste though less
than 1 is not significantly low to completely eliminate the time dependent
term in the governing equation. We explore this in the next section and see if
the quasi-steady (linear) approximation is actually reasonably justified for
us.

####  IV.1.1 Determining the prefactor, $\lambda$

To calculate $\lambda$ corresponding to the exact non-linear temperature
distribution in the solid and liquid layer we would need to solve the one
dimensional heat equation [1] for both the phases with Stefan condition as one
of the boundary conditions. This is expressed as below for the liquid (l) and
solid (s) layers,

$\dfrac{\partial T_{l,s}}{\partial
t}=\alpha_{l,s}\dfrac{\partial^{2}T_{l,s}}{\partial z^{2}}$ (6)

At $z>\delta$, Eq. 6 is solved for the liquid phase and for $z<\delta$ it is
solved for the solid layer with the matching Stefan condition at the
interface, $z=\delta$. A general solution [1, 29] to Eq. 6 can be written as,

$T_{l,s}=A_{l,s}+B_{l,s}\;\textrm{erf}\left(\dfrac{z}{2\sqrt{\alpha_{l,s}t}}\right)$
(7)

The four constants $A_{l},A_{s},B_{l}$ and $B_{s}$ can be determined from the
boundary conditions for the solid and liquid phases,

$\displaystyle T_{l}$
$\displaystyle=T_{amb}\quad\quad\quad\;\textrm{at}\;\;\;z\to\infty,\;\;t\geq
0\;\;\;\;\;\;\textrm{(liquid layer)}$ (7a) $\displaystyle T_{l}$
$\displaystyle=T_{m}\quad\quad\quad\quad\textrm{at}\;\;\;z\to\delta,\;\;\;\;t>0\;\;\;\;\;\;\textrm{(liquid
layer)}$ (7b) $\displaystyle T_{s}$
$\displaystyle=T_{di}\quad\quad\quad\quad\textrm{at}\;\;\;z=0,\;\;\;\;\;t\geq
0\;\;\;\;\;\;\textrm{(solid layer)}$ (7c) $\displaystyle T_{s}$
$\displaystyle=T_{m}\quad\quad\quad\quad\textrm{at}\;\;\;z\to\delta,\;\;\;\;t>0\;\;\;\;\;\;\textrm{(solid
layer)}$ (7d)

The temperature distribution obtained in the solid and liquid layers using the
boundary conditions above are therefore given by,

$T_{s}=T_{di}+\dfrac{T_{m}-T_{di}}{\textrm{erf}\left(\lambda\right)}\;\textrm{erf}\left(\dfrac{z}{2\sqrt{\alpha_{s}t}}\right)$
(8)
$T_{l}=T_{amb}-\dfrac{T_{amb}-T_{m}}{\textrm{erfc}\left(\lambda\sqrt{\alpha_{s}/\alpha_{l}}\right)}\;\textrm{erf}\left(\dfrac{z}{2\sqrt{\alpha_{l}t}}\right)$
(9)

In determining the temperature distributions in Eqs.8, 9 we have used the
relations,
$\textrm{erf}\left(q\right)=\int\displaylimits_{0}^{q}e^{-p^{2}}\,\mathrm{d}p$
and $\textrm{erfc}\left(q\right)=1-\textrm{erf}\left(q\right)$ where,
$\textrm{erf}\left(q\right)$ and $\textrm{erfc}\left(q\right)$ are the error
and complementary error function, respectively. Furthermore,
$\textrm{erf}\left(0\right)=0$ and $\textrm{erf}\left(\infty\right)=1$. We
also substitute for $\delta$ as $2\lambda\sqrt{\alpha_{s}t}$ in the
expressions for the two temperature distributions above however it introduces
another unknown $\lambda$. This can be evaluated using the Stefan boundary
condition stated in Eq. 1 amounting to the transcendental Eq. 10 which can be
solved for $\lambda$.

$\sqrt{\pi}\lambda\;e^{\lambda^{2}}\textrm{erf}(\lambda)=Ste$ (10)

The roots of Eq.10 for different $Ste$ lead to
$\lambda\approx\mathcal{O}(10^{-1})$ [29]. For small $\lambda$ at small $Ste$,
$e^{\lambda^{2}}\textrm{erf}(\lambda)\approx 2\lambda/\sqrt{\pi}$ leading to
$\lambda=\sqrt{Ste/2}$ (derived in the next section). In this linear
approximation, we see for our chosen values of Ste (refer Table 1) the values
for $\lambda$ lie between $0.5-0.67$ and are in close agreement those obtained
from the exact solution of Eq. 10 which lie in the range, $0.45-0.6$.

###  IV.2 Thickness of solidified liquid layer, $\delta$ at any $We$

The expression for dimensionless solidified layer thickness,
$\overline{\delta}$ ($=\delta/D_{0}$) at time, $t_{con}\approx 5D_{0}/V_{0}$
as given by Eq. 5 predicts increasing values of $\delta/D_{0}$ at lower
initial impact velocities, $V_{0}$ or $Pe$ (and by extension $We$). However,
we expect the solidified layer thickness to decrease with lower $We$ as lesser
contact is established at lower $We$ amounting to decreased heat transfer (see
schematic Fig. 5(iii)). To obtain this variation we recognize that contact
resistance, $R_{c}$ can inhibit growth of the dimensionless solidified layer,
$\overline{\delta}$. Here, we derive the relation for $\overline{\delta}$ with
$We$ which includes the effect of contact resistance, $R_{c}$ to be used in
theoretically determining the regime transition boundaries in Section V and
COR calculations used in Section VII pertaining exclusively to the regime
where the drops bounce back. We divide our discussion into two subsections:
first of which deals with determining $\delta=$ f(t, Rc), the second, on
establishing the dependence of Rc with impact velocity($V_{0}$) or impact We.

####  IV.2.1 Expression for $\delta$ as a function of contact resistance and
time

When a liquid drop touches the surface of dry ice heat is lost from the liquid
drop to the composite dry ice/CO2 thereby solidifying the drop. The CO2 gas
pockets lead to increase in contact resistance which becomes significant at
lower impact $We$ (see Fig. 5 (iii)). The transfer of heat occurs in three
layers - (i) contact layer comprising of the composite dry ice/CO2 surface,
(ii) solidified liquid layer and, (iii) the liquid layer (drop) as shown in
Fig. 5(i). Initially, the heat is transferred to the contact surface from the
liquid drop which is at a higher (ambient) temperature. Thereafter, the liquid
drop solidifies and creates another layer through which the heat is
transferred via conduction to the remaining liquid drop. The heat transfer in
these three cases may be written mathematically as follows [32],

  1. (i)

Solidification of liquid drop by releasing latent heat,
$\rho_{s}\mathcal{L}\dfrac{d\delta}{dt}$.

  2. (ii)

Conduction through solidified layer to liquid drop,
$k_{s}\dfrac{T_{amb}-T_{c}}{\delta}$. Here, $k_{s}$ is the thermal
conductivity of the solidified liquid, $T_{amb}$ is the initial drop
temperature and $T_{c}$ is the contact temperature of the interface when the
drop meets dry ice. The linear temperature distribution within the solidified
layer due to low $Ste$ justifies use of this simple expression (see section
IV.1.1 for further details).

  3. (iii)

Contact surface to solidified portion of the liquid drop,
$\dfrac{T_{c}-T_{di}}{R_{c}}$

Equating (i), (ii) and (iii) above, we obtain,

$\rho_{s}\mathcal{L}\frac{d\delta}{dt}=k_{s}\frac{T_{m}-T_{c}}{\delta}=\frac{T_{c}-T_{di}}{R_{c}}$
(11)

The above three equations contain two unknowns, $\delta$ and $T_{c}$ which we
can solve, bearing in mind that $R_{c}$ is known and shall be determined in
section IV.2.2. Considering last two of these expressions, (ii) and (iii), the
contact temperature $T_{c}$ can be obtained in terms of $\delta$ as,

$T_{c}=\frac{T_{m}k_{s}R_{c}+T_{di}\delta}{k_{s}R_{c}+\delta}$ (12)

Exploring Eq. 12 in the limits $R_{c}\approx 0$ and $R_{c}>>1$ we obtain
$T_{c}$ as $T_{di}$ and $T_{m}$ respectively. The temperature profile
corresponding to these are sketched in Fig. 5(ii)(b) and (iii)(b). To solve
for $\delta$ we consider the first two cases, (i) solidification of liquid
drop and, (ii) conduction through solidified layer in Eq. 11 and after a
little algebra arrive at a non-dimensional form of the differential equation
governing the evolution of $\overline{\delta}$ in time by choosing,
$\overline{\delta}=\delta/D_{0}$ and
$\tau=t/\left(D_{0}^{2}/\alpha_{s}Ste\right)$ where,
$D_{0}^{2}/\alpha_{s}Ste=\tau_{sol}$ as defined in Eq. 4 in Section IV.1.

$\frac{d\overline{\delta}}{d\tau}=\frac{1}{(Bi^{cr}_{s})^{-1}+\overline{\delta}}$
(13)

The Biot number of the solidified layer, $Bi^{cr}_{s}$ in Eq. 13 is equivalent
to the dimensionless contact resistance defined as, $D_{0}/k_{s}R_{c}$. Large
values of $Bi^{cr}_{s}$ denote low contact resistance as usually assumed (see
Fig. 5(ii)(a) and (b)) while finite, lower values signify increasing role of
contact resistance (see Fig. 5(iii)(a) and (b)). With this physical
description in mind we solve, Eq. 13 subject to the initial condition,
$\overline{\delta}\left(0\right)=0$ which represents the state when the
solidified layer has not formed just when the drop touches the dry ice
surface. The solution (neglecting the root which yields a negative value for
$\overline{\delta}$) to Eq. 13 is therefore given by,

$\overline{\delta}=\sqrt{(Bi^{cr}_{s})^{-2}+2\tau}-(Bi^{cr}_{s})^{-1}$ (14)

In eqn 14 the dependence of $Bi^{cr}_{s}$ on impact velocity or $We$ (in non-
dimensional terms) remains to determined, which we shall pursue in the next
section.

####  IV.2.2 Dimensionless contact resistance, $Bi^{cr}_{s}$ as a function of
impact $We$

One of the crucial considerations for estimating the contact resistance
($R_{c}$) is the area of the drop that is in contact with the composite dry
ice/CO2 gas surface. As one would expect, higher the impact velocity, more
would be the area of contact compared to the case when the contact is purely
due to gentle deposition. A simple way to account for this increase in contact
area is by considering the ratio of depth the liquid penetrates due to
impalement and total depth which includes the depth that is achieved upon
gentle deposition [33]. The depth to which a liquid penetrates a texture
($h_{cap}$) can be calculated as, $h_{cap}=\sigma_{l}/\rho_{l}gR_{a}$ which is
obtained by equating the Laplace pressure of meniscus of radius of curvature,
$R_{a}$ and the hydrostatic pressure of column of height, $h_{cap}$ (see Fig.
5(iv) and (v)). Whereas, the depth associated with impalement of the
asperities is given by the Bernoulli dynamic pressure head,
$h_{dyn}=V_{0}^{2}/2g$. The ratio, $f_{s}$ therefore assumes the form,

$f_{s}=\dfrac{V_{0}^{2}/2g}{\sigma_{l}/\rho_{l}gR_{a}+V_{0}^{2}/2g}$ (15)

On rearranging the above and defining, ${\overline{R}}_{a}=R_{a}/D_{0}$ we
get,

$f_{s}=\dfrac{We{\overline{R}}_{a}}{2+We{\overline{R}}_{a}}$ (16)

Figure 6: (i) Variation of dimensional solidified thickness $\delta$ with $We$
(in bouncing regime) as given by eqn 18. The dotted lines show the value of
$\delta$ when contact resistance is negligible corresponding to
$Bi^{cr}_{s}\to\infty$ (ii) For any other $Bi^{cr}_{s}$ corresponding to a
finite $We$, $\delta$ can be simply written as,
$2\lambda\lambda_{c}\sqrt{\alpha_{s}t}$, where, $\lambda=\sqrt{Ste/2}$ and
$\lambda_{c}=f\left(We\right)=0.2We^{2/5}$ or $\sim V_{0}^{4/5}$.
$\lambda_{c}$ assumes values from 0.2 to 1 which specifically for lower $We$
and $Bi^{cr}_{s}$ can be averaged to 0.5 and approximately 1 at higher $We$ or
$Bi^{cr}_{s}$ as used in order of magnitude analysis for the scaling arguments
to determine the regime boundaries (Fig. 8 (ii) in Section V).

In terms of heat transfer it means that the conduction across an interface
with a roughness (depth) $R_{a}$, effective reference flat area,
$f_{s}A_{\textrm{{ref}}}$ (adjusted for the true contact area by the factor,
$f_{s}$) and thermal conductivity of $k_{di}$ should be the same as that of an
interface with contact resistance $R_{c}$ and area, $A_{\textrm{{ref}}}$. For
a temperature difference, $\Delta T$ across the interface such an energy
balance implies, $k_{di}f_{s}A_{\textrm{{ref}}}\Delta
T/R_{a}=R_{c}^{-1}A_{\textrm{{ref}}}\Delta T$ which yields,
$f_{s}=R_{a}/R_{c}k_{di}$. Hence, $f_{s}$ can now be expressed as,
${\overline{R}}_{a}Bi^{cr}_{di}$. Eq. 16 thus takes the form,

$Bi^{cr}_{di}=\dfrac{We}{2+We{\overline{R}}_{a}}$ (17)

Note that $Bi^{cr}_{di}$ and $Bi^{cr}_{s}$ are related to each other as,
$Bi^{cr}_{s}=(k_{di}/k_{s})Bi^{cr}_{di}$. Equating, $k_{r}=k_{di}/k_{s}$ and
using Eq. 17 in Eq. 14 we obtain the dependence of $\delta$ on $We$ in the
time the drop is in contact with the surface as,

$\delta=\sqrt{\left(\dfrac{2D_{0}+WeR_{a}}{k_{r}We}\right)^{2}+(2\alpha_{s}Ste)\;t}-\left(\dfrac{2D_{0}+WeR_{a}}{k_{r}We}\right)$
(18)

For a roughness $R_{a}=10\;\mu m$ (see Fig. 2(ii)) contact time,
$t_{con}\approx 5D_{0}/V_{0}$ and $D_{0}$, $We$ and $Ste$ given in Section II
we plot the variation of $\delta$ with $We$ obtained in expression 18 as shown
in Fig. 6(i). Since, Eq. 18 is cumbersome to use we see it equivalence with
the commonly used form containing $\sqrt{t}$. Normally, it contains a
prefactor $\lambda$ which physically represents the role of solidification in
particular liquid (see Sec IV.1, Eq. 3) when contact resistance effects are
not included. For finite $Bi^{cr}_{s}$ we consider a form analogous to the
traditional form but includes a multiplicative factor $\lambda_{c}$ to account
for the difference caused due to a non-negligible contact resistance,

$\delta=2\lambda_{c}\lambda\sqrt{\alpha_{s}t}\textrm{ with
}\lambda=\sqrt{Ste/2}\textrm{ and }\lambda_{c}=f(Bi^{cr}_{s})\textrm{ or
}f(We)$ (19)

Our results for $\delta$ using Eq. 18 are now plotted as function of $We$ in
Fig. 6(i). The curve so obtained can now be compared with Eq. 19 to determine
$\lambda_{c}$, our correction to the standard expression which does not
include effects of contact resistance. We notice from Fig. 6(i) that $\delta$
decreases by a factor 5 at very low We or $Bi^{cr}_{s}$ such that
$\lambda_{c}$ is $\mathcal{O}(10^{-1})$ at lower $We$ with an average value of
0.5 at these lower values and close to 1 at higher $We$ or $Bi^{cr}_{s}$. We
specifically plot $\lambda_{c}$ in Fig. 6(ii) and see that it yields a
dependence of the form, $\lambda_{c}\sim We^{2/5}$ or $V_{0}^{4/5}$. As
$Bi^{cr}_{s}\to\infty$, (representing no contact resistance) we recover the
standard result, $\delta=\sqrt{2Ste\alpha_{s}\;t}$ which is shown by dotted
lines in Fig. 6(i) and implies $\lambda_{c}\to 1$. We observe that water has
lower values for $\delta$ compared to the alkanes. This is expected since it
has the lowest $Ste$ which implies slow solidification and thereby its
effective thermal diffusivity ($\alpha_{\textrm{eff}}=\alpha_{s}Ste^{2}$) [1,
34] is the least, differing by almost factor of 4 from water to the alkanes
which is seen as a difference in solidification thickness by a factor of 2 at
high $Bi^{cr}_{s}$ or $We$.

##  V Regime boundaries

###  V.1 Transition boundary between fragmentation and no-bounce

At large We, a (flattened) drop close to its maximum spread ($D_{max}$ at
time, $t=t_{max}$) displays finger-like formation as a consequence of
Rayleigh-Taylor instability arising at its rim due to the inertia overcoming
the stabilizing force of surface tension [4, 35, 36] ultimately leading to its
fragmentation as smaller droplets. On frigid surfaces as the drop equilibrates
to the contact temperature the solidified layer additionally contracts which
in the absence of pinning leads to bending [2] as opposed to fracture when
there is adhesion [17]. Therefore, inertia of the spreading drop encounters
additional resistance due to bending of the solidified layer, besides surface
tension and fragmentation is delayed.

For modeling such a scenario, we begin by approximating the volume of the
spreading drop as a cylindrical pancake with diameter $D$ and height $h$
where, $D_{max}/2$, $D_{max}/3$ and $2V_{0}$ are chosen as the approximate
average length scales (for $D$ and $h$) and horizontal spreading velocity
scale ($V$) from the moment inertia begins to compete with solidification and
surface tension just upon impact (at $t=0$) until fingers begin to form (at
$t=t_{max}$). To obtain these scales, we consider time averaged values from
$t=0$ to $t_{max}$. Thus, $D=D_{max}/2$ (average of $D=0$ at $t=0$ and
$D=D_{max}$ at $t=t_{max}$) and $V=4V_{0}/2$ (average of $V=4V_{0}$ at $t=0$
and $V<<V_{0}$ at $t=t_{max}$) where, the value $4V_{0}$ is motivated by the
fact that the drop upon impact experiences an initial acceleration
horizontally such that $V$ is related to the vertical impact velocity as,
$|V_{0}\textrm{cot}\;(\theta_{a})|$ [37] with the apparent contact angle
$\theta_{a}$ being approximately $166^{\textrm{o}}$ for our (and
superhydrophobic) surfaces.

Our experiments indicate that near fragmentation-bounce transition at
$t=t_{max}$, $h\approx D_{0}/3$, and related to $D_{max}$ as
$D_{max}/D_{0}\approx 2.5$ (more precisely, for water and decane this ratio is
2.1 and 2.8 respectively), from where we get the time-averaged pancake
thickness as $h\approx(D_{0}+D_{0}/3)/2=D_{max}/3$ which is the more
appropriate form compared to the one ($h_{pan}$) used later using mass
conservation as the drop flattens more and develops corrugation on the rim.
This implies that the kinetic energy is given by,
$(1/2)(\rho_{l})(\frac{\pi}{4}\frac{D_{max}^{2}}{4}\frac{D_{max}}{3})(4V_{0}^{2})$
which, for fragmentation to occur needs to exceed the sum of, (i) elastic
bending energy stored in the solidified layer of average thickness $\delta$
(in m), $Y\delta^{3}/12\left(1-\gamma^{2}\right)$ where, $Y$ is the Young’s
modulus (in N$\cdot$m-2) and $\gamma$ is Poisson’s ratio as well as (ii) the
surface energy of the cylinder,
$\sigma_{l}(\frac{\pi}{4})(\frac{D_{max}^{2}}{4})$ finally resulting in the
following criterion for fragmentation (scaled by $D_{0}^{3}$),

$\dfrac{D_{max}^{3}}{D_{0}^{3}}>\dfrac{2Y\delta^{3}}{\pi\rho_{l}V_{0}^{2}D_{0}^{3}\left(1-\gamma^{2}\right)}+\dfrac{3\sigma_{l}D_{max}^{2}}{2\rho_{l}V_{0}^{2}D_{0}^{3}}$
(20)

Note that the term (i), elastic bending energy signifies the energy associated
with the unbending of the solidified layer of the lower portion of area $A$ of
the curved drop (dark blue shaded region in schematic Fig. 8(ii)) and is also
multiplied by $D_{max}^{2}\kappa^{2}\approx 1$ as, $\kappa\sim D_{max}^{-1}$
and area, $A\sim D_{max}^{2}$ shown in the schematic Fig. 8(iii).

Figure 7:  Variation of dimensionless spreading diameter, $D/D_{0}$ with
dimensionless time, $tV_{0}/D_{0}$ for drop bouncing, (i) at constant
$We\approx 40$, different liquids and, (ii) for hexadecane and different $We$,
the dotted vertical line at $t=t_{max}$ denotes the time at maximum spread,
$D=D_{max}$ (iii) ratio of the experimental contact time, $t_{con,exp}$ with
the inertio-capillary time, $t_{in}$ showing its scaling with $\sqrt{We}$.

Solving the above equation requires determining the temporal evolution of
solidification, $\delta\left(t\right)$ as the drop spreads from $0\rightarrow
D_{max}\rightarrow D_{0}$ and evaluating it at the time, $t_{max}$
corresponding to $D\approx D_{max}$ which is when fingers begin to appear (see
Fig. 3 (iii)).

To estimate $t_{max}$, we analyzed the temporal evolution of the spreading
droplet diameter ($D$) variation with time ($t$) for all the liquids. Examples
of this analysis in the form of a non dimensional drop diameter ($D/D_{0}$) as
a function of non-dimensional time ($tV_{0}/D_{0}$) for all liquids at the
same $We$, and for the same liquid (hexadecane) at different $We$ are shown in
Figs. 7(i) and (ii) respectively. For all the liquids, we find that the total
contact time of drops, $t_{con}=5D_{0}/V_{0}$ and $t_{max}=5D_{0}/4V_{0}$
representing $1/4^{th}$ of its oscillation [38]. Further analysis of the ratio
of the experimental contact time, $t_{\textrm{{con,exp}}}$ with the inertio-
capillary time scale, $t_{in}(=\sqrt{\rho_{l}D_{0}^{3}/\sigma_{l}})$ as a
function of $\sqrt{We}$ showed that $t_{\textrm{{con,exp}}}$ scales as
$We^{-1/2}$ (see Fig. 7 (iii)). Using simple algebra this scaling simplifies
to $t_{con,exp}\sim D_{0}/V_{0}$ (replaced by $t_{con}$ hereon), supporting
the choice of this time scale in subsequent analysis.

We emphasize that the choice of $t_{con}$ scaling as $D_{0}/V_{0}$ is neither
counter-intuitive nor does it contradict existing works. Impact of dry ice
which have been likened to impacts on superhydrophobic surfaces have expressed
the rebound (or contact) time as $\sqrt{\rho_{l}D_{0}^{3}/\sigma_{l}}$ [31]
which seems to be odds with $D_{0}/V_{0}$. On closer inspection, we see that
both are in essence the same with $D_{0}/V_{0}$ being the more generalized
form. To understand this in detail we note that the inertio-capillary velocity
scale, $V_{0}$ is given by $\sqrt{\sigma_{l}/\rho_{l}D_{0}}$ for a length
scale given by the drop diameter, $D_{0}$ which means that $D_{0}/V_{0}$
produces $\sqrt{\rho_{l}D_{0}^{3}/\sigma_{l}}$ as the time inertio-capillary
time scale, $t_{in}$. A natural query that arises from this explanation is:
why we do not use $t_{in}$ for our analysis, consistent with literature. The
reason for it is that solidification makes rebound calculations more
complicated. Surface energy required to achieve rebound needs to not only
equal the diminished inertia due to solidification but also overcome bending
stresses (see Section VI for details). In earlier works[22], effects of
solidification in rebound from dry ice have been largely unexplored as the
choice of liquids (for instance, water) used for experimentation experience
minimal solidification upon impact and therefore obviating the need to account
for dissipation arising due to it. That being said, research on rebound of
viscous drops does exist [39], and a contact time,
$t_{con}=\left(1+\tfrac{1}{8}Oh^{2}\right)t_{in}$ which corrects $t_{in}$ by
the prefactor, $\left(1+\tfrac{1}{8}Oh^{2}\right)$ for small $Oh$ to
accommodate effects of viscous dissipation. In our work, similar correction
could be made to $t_{in}$ to account for dissipation due to solidification
however since the scale, $D_{0}/V_{0}$ absorbs all these effects we choose
that as the contact time. In a future study, an explicit relation between
$t_{con}$ and $t_{in}$ for non-isothermal drops impacts in general, could be
explored and derived.

Returning to deriving our criterion for fragmentation we invoke Stefan’s
condition [29] to evaluate $\delta$ at $t_{max}=5D_{0}/4V_{0}$, wherein the
latent heat of fusion equates to the conduction heat transfer across the
solidified layer with a temperature difference, $\Delta T=T_{di}-T_{m}$ where,
$T_{di}$ is the substrate temperature. From Eq. 19 this results in
$\delta=2\lambda\lambda_{c}\sqrt{\alpha_{s}t}$ where, $\lambda(=\sqrt{Ste/2})$
accounts for the contact resistance limited solidification,
$\lambda_{c}(=0.2We^{2/5})$ is a constant multiplier to the prefactor
$\lambda$. Using the defintion of Ste and Pe we obtain an expression for non-
dimensional solidified thickness,
$\overline{\delta}=\delta/D_{0}=\lambda_{c}\sqrt{5Ste/2Pe}$ where, $\lambda$
is replaced by $\sqrt{Ste/2}$. Physically, higher $Ste$ implies faster
solidification and higher $Pe$ indicates a smaller depth to which the effects
of the substrate temperature penetrate. Although low velocity (i.e. smaller
$Pe$) would mean a thicker thermal boundary layer, the contact resistance of
the drop/dry ice interface (accounted by $\lambda_{c}$) limits its growth. The
overall consequence of this competition is an increase in $\delta\sim
V_{0}^{4/5}$resulting from, $\overline{\delta}=0.32\;We^{2/5}\sqrt{Ste/Pe}$
which includes correction for contact resistance of the drop/dry ice interface
as shown in Section IV.

Experiments show that the transition from bouncing to fragmentation We for
water and decane are approximately 30 and 70 while the ratio of
$D_{max}/D_{0}$ are 2.1 and 2.8 respectively (see Section III, Fig. 3 (v) and
(vi)). This means the corresponding term, $(D_{max}/D_{0})^{3}$ is
approximately 10 and 22 for water and decane. At $t=t_{max}=t_{con}/4\approx
5D_{0}/4V_{0}$ (see Fig. 8(i) - (iii)), $\delta$ = 15 $\mu$m (water), 50
$\mu$m (decane) for a drop of $D_{0}$ = 2.3 mm impacting at $V_{0}$ = 1
m$\cdot$s-1. Using $Y$ = 9 (ice), 0.2 (paraffin wax for solid decane) $\times$
109 Pa and $\gamma$ = 0.5, the term
$2Y\delta^{3}/\pi\rho_{l}V_{0}^{2}D_{0}^{3}\left(1-\gamma^{2}\right)$
evaluates to 10 for water and 16 for decane. On the other hand, at the
transition We, we find that the term
$3\sigma_{l}D_{max}^{2}/2\rho_{l}V_{0}^{2}D_{0}^{3}$ amounts to 0.22 for water
and 0.16 for decane. From these we conclude that the second term,
$3\sigma_{l}D_{max}^{2}/2\rho_{l}V_{0}^{2}D_{0}^{3}$ of Eq. 20 can be dropped
for our impact conditions close to fragmentation. Consequently, inequality 1
transforms to,

$\dfrac{D_{max}}{D_{0}}>\chi_{1}\left(\lambda_{c}\sqrt{\dfrac{Ste}{Pe}}\right)$
(21)

where,
$\chi_{1}={\color[rgb]{0,0,0}\sqrt{5/2}}\left[2Y/\pi\left(1-\gamma^{2}\right)\rho_{l}V_{0}^{2}\right]^{1/3}$.
Inequality 21 suggests that as the thickness of the solidified layer
increases, fragmentation of the drop is delayed, in agreement with our
experimental observations for water and decane (see Fig. 8 (iv)).

Note that our criterion excludes splashing considerations based on lifting of
the lamella and a difference between the tip speed and the rate of increase of
the wetted area [14, 15]. Solidification of the liquid layer close to the
supercooled surface forestalls detachment of the lamella as its tip speed
reduces drastically [40, 41] thereby preventing its lifting. Details regarding
this are also described in the introduction, I. So Eq. 21 predicts a simpler
criterion for fragmentation. Further note that, other interfacial energies
like that of the solidified liquid with the dry ice substrate and solidified
liquid with the unsolidified liquid do not play a part in our considerations
for transition to fragmentation as they are balanced internally by the latent
heat which equals the free surface energy required to create these new
interfaces. Additionally, axial strain in the solidified splat is minimal and
only the energy required to bend it is significant which is provided by the
liquid surface tension ($\sigma_{l}$). Also, surface energy of the solid
($\sigma_{s}$) is insignificant due to smaller curved surface area. Therefore,
in the spreading process, only changes in surface energy of the liquid
manifests itself as a major factor along with elastic bending energy of the
solidified layer which is balanced with the kinetic energy.

Figure 8:  (i) Drop just before and after impact from height, $H_{0}$, with
initial diameter, $D_{0}$ and, impact velocity, $V_{0}$ (ii) Sketch of the
flattened drop of total thickness, $h_{pan}$ showing the surface tension force
acting on it (iii) The final shape of the solidified layer of thickness,
$\delta$ (with radius of curvature $\kappa^{-1}$) before rebound where
$\delta<<h_{pan}$. (iv) Regime map showing three distinct impact outcomes
affected by partial solidification - (a) Fragmentation (in light yellow) (b)
No-bounce (in light purple) (c) Bouncing with solidification (in light green)
. The dotted vertical line represents the region beyond which the solidified
layer thickness, $\delta$, is $\approx$ 14 $\mu m$ and bold lines correspond
to the regime boundaries for fragmentation and no-bounce given by the
inequalities 21 and 22.

###  V.2 Transition boundary between rebound and no-bounce

Delayed fragmentation implies bouncing is observed over a larger range of
impact velocities or $We$ when solidification is limited but significant. But,
an increase in We leads tetradecane, pentadecane and hexadecane drops to
transition from bounce to a no-bounce state without undergoing fragmentation
(see Fig. 2(ii), SM Movie 3 for hexadecane [1] where this is pronounced). We
suggest that as solidification increases, the energy required to deform, bend
and wrap the solidified layer around the drop also increases such that the
drop instead of bouncing (see Fig. 2(ii)) rests on the surface (see Fig.
2(iv)) without pinning.

To determine the conditions under which no-bounce would occur, we consider the
flattening out of the drop into a pancake (see Fig. 8(ii)) with diameter,
$D_{max}$ and thickness $h_{pan}\approx D_{0}^{3}/D_{max}^{2}$, obtained by
conserving mass before and after impact. The mass of the solidified layer
($\delta$) is a small fraction of the total mass and hence ignored. Bouncing
with drop solidification would require the available bending capillary torque
due to surface tension, ${\color[rgb]{0,0,0}\pi}\sigma_{l}D_{max}h_{pan}$ [42,
43] to overcome the flexural rigidity which is mathematically same as the
elastic bending energy, $Y\delta^{3}/12\left(1-\gamma^{2}\right)$. The
solidified layer thickness,
$\overline{\delta}{\color[rgb]{0,0,0}(=\lambda_{c}\sqrt{10Ste/Pe})}$ is
determined at the drop contact time, $t_{con}\approx 5D_{0}/V_{0}$
demonstrated in Fig. 8(i), (ii) and, (iii). and used in the remainder of this
paper where we are concerned with rebound. In non-dimensional terms, the above
arguments lead to,

$\dfrac{D_{max}}{D_{0}}<\chi_{2}\left(\lambda_{c}\sqrt{\dfrac{Ste}{Pe}}\right)^{-3}$
(22)

where, $\chi_{2}=\frac{3\pi\sqrt{10}}{25}\sigma_{l}(1-\gamma^{2})/YD_{0}$.
Plotting Eq. 22 in Fig. 8 (iv) we see that it predicts the transition from no-
bounce to bounce well. In development of the above expression contribution of
surface tension of the solidified liquid and interfacial tension of the
solidified liquid with the unsolidified liquid are contingent to their
dominant contributions in creating a torque to bend the solidified layer. Due
to the small thickness of the solidified layer, they act at shallow angles
with a small vertical component and therefore do not create a substantial
torque compared to the liquid surface tension which acts dominantly around the
periphery of the spreading drop. The lines of surface tension of the
solidified liquid drop and the dry ice substrate pass through the center of
the axisymmetric splat about which bending is considered. This creates no
torque as the moment arm is zero thereby allowing us to neglect it.

##  VI Drop spreading

Solidification affects both spreading and rebound of drops which can be
quantified using - maximum spread ($D_{max}$) of the drop on impact [31, 23]
and, its rebound height ($H_{1}$). To determine the arrest in the spread of
the diameter of the deforming drop upon impact, we consider the Padé
approximant that interpolates between limits where a drop bounces in absence
of solidification and the other when its completely solidified yet capable of
bouncing back.

Figure 9:  Scaling using (i) $Pe^{1/5}$ and $We^{1/2}$, (ii) $Re^{1/5}$ and
$We^{1/2}$ as the asymptotic limits for Padé approximant interpolation (dotted
lines).

While, $We^{1/2}$ is surely the limit for bouncing with no solidification the
other limit can have potentially other options. To demonstrate the success of
scaling choosing $We^{1/2}$ and $WePe^{1/2}Ste^{-1/2}$ as the asymptotic
limits, we consider other asymptotic limits and show their relatively poor
performance. The unsatisfactory scaling of data using
$Pe^{1/5}\;\textrm{or}\;Re^{1/5}$ as one of the asymptotic limits besides
$We^{1/2}$ is tested here. Before we proceed it is important to note that the
role of surface tension in our scaling model for drop spreading is significant
and the scaling relations, $D_{max}/D_{0}\sim Pe^{1/5}\;\textrm{or}\;Re^{1/5}$
only consider solidification and sticking [1, 30] however we also test them as
potential options to confirm the correct scaling as we show later in this
section.

(a) Interpolation between $We^{1/2}$ and $Pe^{1/5}$: The rationale behind the
scaling for $D_{max}/D_{0}\sim We^{1/2}$ [31] and $D_{max}/D_{0}\sim Pe^{1/5}$
[30] is documented in literature. Fig. 9(i) shows this scaling which has an
$R^{2}\approx 0.79$ and poor compared to the $R^{2}\left(\approx 0.97\right)$
obtained for the interpolation using $We^{1/2}$ and $WePe^{1/2}Ste^{-1/2}$ as
the asymptotic limits.

(b) Interpolation between $We^{1/2}$ and $Re^{1/5}$: Another candidate when
the drop is solidified and flattened is $Re^{1/5}$ [31]. Although it does not
explicitly include thermal effects it can still be viable candidate. Fig.
9(ii) shows this scaling which has an $R^{2}\approx 0.83$ and poor compared to
the $R^{2}\left(\approx 0.97\right)$ obtained for the interpolation using
$We^{1/2}$ and $WePe^{1/2}Ste^{-1/2}$ as the asymptotic limits.

Since we only consider conditions where complete drop rebound is observed, a
more restrictive limit which continues to account for surface tension is more
successful and is derived below. This is also confirmed by the fact that
$\alpha_{s}Pe/\alpha_{l}Ste>We^{5/2}$ for our test conditions, which follows
from the fact that the capillary time scale is indeed less than the time scale
for arrest of the spread of the drop [30].

Therefore, in scaling $D_{max}/D_{0}$ shown in Fig. 10 we choose $We^{1/2}$
and $WePe^{1/2}Ste^{-1/2}$ as the asymptotic limits for Padé’s approximant
which represent no solidification ($We^{1/2}$) and complete solidification
($WePe^{1/2}Ste^{-1/2}$). The first limit given by $D_{max}/D_{0}\sim
We^{1/2}$ is obtained from the complete conversion of kinetic energy into
surface energy and shown [44, 45] to be more applicable than
$D_{max}/D_{0}\sim We^{1/4}$ [31].

Figure 10:  Scaling for maximum non-dimensional spread, $D_{max}/D_{0}$ using
Padé approximant [45] and given by (dotted line), Eq. 23. The Padé approximant
closely follows the scaling relation, $D_{max}/D_{0}\sim We^{0.4}$ [22, 24]

For the second limit, we tested the possibility of $D_{max}/D_{0}\sim
Re^{1/5}$ and $D_{max}/D_{0}\sim Pe^{1/5}$ as possible options which have been
traditionally used for completely solidified splats [1, 30] however both yield
unsatisfactory scaling with $R^{2}\leq 0.85$ and do not depict a situation
where the drop rebounds. Consequently, we consider a hypothetical situation
where the drop solidifies almost instantly as it starts to spread on a
supercooled surface such that $D_{max}\approx D_{0}$ and
$h_{pan}\approx\delta$. In this limit, the energy balance between total energy
of the drop before and after contact reads as,
$\rho_{l}D_{0}^{3}V_{0}^{2}+\sigma_{l}D_{0}^{2}\sim\sigma_{s}D_{max}^{2}+\sigma_{s}h_{pan}D_{max}$,
where $\sigma_{s}$ (in N$\cdot$m-1) is the surface tension of the test liquids
in their solidified state. Ignoring the slight differences between
$\sigma_{s}$ and $\sigma_{l}$ (see [1] Section S1) it follows that
$\sigma_{l}D_{0}^{2}\approx\sigma_{s}D_{max}^{2}$ which results in the energy
balance, $\rho_{l}D_{0}^{3}V_{0}^{2}\sim\sigma_{s}\delta D_{max}$ that can be
recast as, $D_{max}/D_{0}\sim We({Pe/Ste})^{1/2}$.

The interpolating Padé approximant function corresponding to these limits is
given by,

$\dfrac{D_{max}}{D_{0}}We^{-1}({Pe/Ste})^{-1/2}=\dfrac{P^{1/2}}{0.0025+P^{1/2}}\textrm{
where, }P=We^{-1}Pe^{-1}Ste$ (23)

(see Fig. 10(i)) and corresponds to an $R^{2}\approx\textrm{0.98}$. The slight
preponderance of data to the left axis of the plot shows more cases with
partial solidification and bouncing. Note that, comparing our interpolation
with previous studies on drop impact and rebound on a gaseous cushion in non-
isothermal conditions [22, 24] we see that the scaling for the spread factor,
$D_{max}/D_{0}\sim We^{2/5}$ stated therein has a close overlap with our Padé
fit (see Fig. 10(ii)).

##  VII Drop Rebound

Figure 11:  Comparison of experimental data and theoretical prediction for
coefficient of restitution ($\epsilon$) for the first bounce of a partially
solidified drop.

In the last part of our discussion we investigate the role of drop
solidification (in terms of $Ste$) on the rebound height (in terms of $H_{1}$)
as shown in Fig. 4 (i) and (ii) (also see SM Movie 2 [1]). As discussed
before, the thickness of the solidified layer, $\delta$ is related to $Ste$
and the impact velocity through $Pe$ or $We$ which motivates these
investigations. To quantitatively understand this decrease in $H_{1}$ we
develop a theoretical expression for the coefficient of restitution, COR
($\epsilon$) which includes the effects of solidification. COR is represented
by the ratio of the rebound ($V_{1}$) and initial velocity ($V_{0}$) (see
inset of Fig. 11) and is indicative of the repellency of a surface such that
superhydrophobic surfaces with low contact angle hysteresis demonstrate the
highest $\epsilon\approx 1$ [38].

To evaluate $\epsilon$ we consider consider kinetic energy after
impact/rebound, $E_{r}(=mV_{1}^{2}/2)$ and the initial kinetic energy,
$E_{in}(=mV_{0}^{2}/2)$ and the energy loss during rebound, $\Delta E$, which
is the sum of contributions from loss of kinetic energy due to arrest of
movement of the solidified layer, $(1/2)(\rho_{s}\pi
D_{s}^{2}\delta/4)V_{0}^{2}$ (where, $D_{s}\approx D_{max}/2$ is the average
contact diameter as the drop spreads from 0 to $D_{max}$ and $\rho_{s}$ is the
density of the solidified layer, $\delta$) and energy expended in bending the
solidified layer, $Y\delta^{3}/12\left(1-\gamma^{2}\right)$.With these
considerations, $E_{r}$ can be simply written as $E_{in}-\Delta E$, which
leads to, $\epsilon^{2}={1-a_{1}\overline{\delta}-a_{2}\overline{\delta}^{3}}$
where,
$a_{1}=(3/8)\left(\rho_{s}/\rho_{l}\right)\left(D_{max}/D_{0}\right)^{2}$ and
$a_{2}=Y/\pi\rho_{l}V_{0}^{2}\left(1-\gamma^{2}\right)$. Denoting,
$\xi=a_{1}\overline{\delta}+a_{2}\overline{\delta}^{3}$ we may write
concisely, $\epsilon=\sqrt{1-\xi}$. Comparing our experimental results with
theoretical predictions in Fig. 11 evaluated at $t=t_{con}$ we find that $1:2$
scaling between $\epsilon$ and $1-\xi$ is recovered

##  VIII Summary and Conclusions

In summary, using a unique combination of ultra-low adhesive dry ice surface
and alkanes as liquids, we isolated the role of solidification during drop
impact. We further demonstrated that solidification within a drop, even though
partial, dissipates its initial kinetic energy to delay fragmentation, reduces
its spread, and decreases its coefficient of restitution – even suppressing
the rebound entirely, thus providing a strategy to control the drop deposition
by locally changing the substrate temperature in applications such as paint
spraying and additive manufacturing. The fragmentation/no-bounce criterion and
the maximum spreading diameter developed herein along with calculations on
contact resistance limited depth of solidification are expected to be
applicable to any liquid contact with supercooled substrates and especially
those with ultra-low adhesion. They also serve to provide new insights into
dissipation mechanisms in drop impact on supercooled, non-wetting surfaces.
The findings in this work expand our current understanding which has limited
itself to studies of drop rebound on dry ice to other previously unknown or
unexplained outcomes such as fragmentation and no-bounce. This makes dry ice a
versatile platform which can lead to a whole gamut of scenarios unlike
traditional supercooled surfaces which are restricted to pinning mediated
adhesion. The finding that despite partial solidification, drops can rebound
from a surface on ultra-low adhesive surfaces can have significant
implications for designing robust icephobic/anti-icing surfaces. Lastly, we
expect our results have a tremendous bearing on developing strategies to repel
or enhance adhesion of a wide range of liquids undergoing liquid-solid phase
change to the surfaces, including applications such as wax deposition, and
liquid transport in microfluidic channels besides scenarios where
fragmentation needs to be controlled.

## Acknowledgment

This work was supported by NSF (CAREER) award (no. 1847627) and Society in
Science Branco Weiss Fellowship. Suhas Tamvada, Nikhil Shirdade and Navid
Saneie contributed equally to this work. Additional data and material related
to this paper can be requested from Varun Kulkarni (varun14kul@gmail.com).

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