††thanks<EMAIL_ADDRESS>
# How does gravity influence freezing dynamics of drops on a solid surface

Hao Zeng  Sijia Lyu  Center for Combustion Energy, Key Laboratory for Thermal
Science and Power Engineering of Ministry of Education, Department of Energy
and Power Engineering, Tsinghua University, 100084 Beijing, China  Dominique
Legendre  Institut de Mécanique des Fluides de Toulouse (IMFT), Université de
Toulouse, CNRS -Toulouse, France  Chao Sun  Center for Combustion Energy, Key
Laboratory for Thermal Science and Power Engineering of Ministry of Education,
Department of Energy and Power Engineering, Tsinghua University, 100084
Beijing, China  Department of Engineering Mechanics, School of Aerospace
Engineering, Tsinghua University, Beijing 100084, China

###### Abstract

Water droplet freezing is a common phenomenon in our daily life. In both
natural scenarios and industrial production, different surface inclinations
bring distinctive deformation and freezing dynamics to frozen droplets. We
explore the freezing of pendent and sessile droplets at different Bond number
regimes. The effect of gravity on the droplet freezing process is analyzed by
considering droplet morphology, freezing front dynamics, and freezing time. It
is found that gravity can significantly influence droplet freezing processes
via shaping the initial droplet, resulting in the flattening or elongation of
pendent and sessile droplets, respectively. We show that the droplet initial
geometry is the most important parameter and it completely controls the
droplet freezing. Despite the significant difference in the initial droplet
shape several remarkable similarities have been found for pendent and sessile
droplets at small and large Bond numbers. The final height of a frozen droplet
is found to be linearly proportional to its initial height. The time evolution
of the ice-liquid-air contact line is found to reproduce the power-law
$t^{0.5}$, but noticeably faster than the Stefan 1-D icing front propagation.
As a consequence, the time to freeze a droplet is faster than predicted by the
Stefan model and it is found to be dependent on the initial droplet height and
base radius through a simple power-law.

††preprint: APS/123-QED

##  I Introduction

Water droplet freezing is an important and commonly seen phenomenon in our
daily life, especially in cold-weather regions. This process also
significantly affects many important industrial applications. In the power
delivery industry, the accumulated ice could cause damage to the electrical
wires and transformers[1]. In the aerospace industry, ice formation on the
aircraft wings is a central issue that may increase fuel consumption and
threaten aviation safety[2]. In the wind power plant, the ice accretion on the
wind turbine blade will drastically increase loads and reduce the system
performance[3]. On the other hand, the solidification of liquid droplets is
also dominant in many additive manufacturing processes, such as the metal
alloy droplet deposition[4], micro-droplet 3D printing[5], and laser
welding[6]. The solidification dynamics and morphology would lead to different
microstructures and divergence of material characteristics. To further
understand the mechanisms governing this very common phenomenon, prevent
undesirable ice formation and better utilizing droplet solidification in
productions, it is important to study the freezing features of water droplets.

The freezing processes are the focus of many current studies in a broad scope.
Since the first reported shapes of freezing droplets half a century ago[7],
the community has been trying to find the conditions that would affect the
droplet solidification process from many different aspects. The substrate
properties are proved to be one of the most important aspects that would
affect the freezing process via substrate temperature[8, 9] and
wettability[10, 11, 12, 13, 14]. The environment medium is also taken into
consideration of the main factors that influence the freezing process. The
still gaseous medium is always considered as an adiabatic condition for its
low thermal conductivity[15], while the interface evolution would be greatly
different when flow exists in the medium[16, 17, 18, 19]. The properties
related to the droplet itself are even much more complex. The initial
conditions of the droplet as well as the ambiant condition such as
humidity[20] largely determine the subsequent process of the freezing
kinetics. The supercooling [21] and impingment[22, 23, 9, 24] affect the
freezing process by changing the initial properties and morphology of the
droplet. Experiments[25, 26] indicate that the drop size is a critical factor
that influences the heat change efficiency in the freezing process. Different
initial configurations exhibit both universality and diversities in the
freezing process[27, 9]. The initial configuration of the droplet is
determined by the balance of surface tension, the intrinsic property of free
liquid surfaces, and gravitational effects on the Earth. The surface tension
force would change with the surface curvature and successively adapt the
droplet shape during the freezing process to minimize the surface energy[28].
The gravity puts many restrictions in the experiments on the Earth mainly due
to reduced droplet size and detachment from resolidified liquid-solid
systems[29, 30]. Several models are proposed to discuss the influence of
gravity on the solidification of droplets, but in such studies, the shape and
thermal problems are decoupled and density variation is not considered[31] for
simplicity. The effect of gravity in droplet freezing process has still not
been well studied.

In this study, we mainly investigate the influence of gravity in water drops
freezing on a copper plate cooled down below 0 ${}^{\circ}C$. After a short
description of the experimental setup and numerical model, we qualitatively
describe the freezing processes of pendent droplets and sessile droplets. The
gravitational effects on droplet morphology and freezing dynamics are
extensively explored with experimental and numerical results. Further more,
several remarkable similarities are found for pendent and sessile droplets,
and for both small and large Bond numbers. The results present intriguing laws
in the droplet freezing process with different gravity directions.

##  II  Method

###  II.1  Experiment setup

Figure 1:  Sketch of the experiment setup. The setup is flexible to turnover,
thus pendent droplet and sessile droplet can be studied with the same
apparatus

The gravitational effect on the freezing of singular water droplets can be
studied with the experimental setup shown in Fig. 1. In the experiment,
distilled water droplets are cooled on a copper substrate. A pipette is used
to create single droplet with a certain volume in the experiment. To better
control the droplet shape, concave outline is manufactured on the substrate to
fix the solid-liquid-vapor contact line at droplet base. Since the concave
depth is much less than the length scale of the droplet, the influence of the
structure on the freezing process can be neglected.

A semi-conductor cooler is embedded in the substrate to cool the substrate. A
heat exchanger connecting to a cold reservoir is attached to the opposite side
of the semiconductor to maintain energy balance. The surface temperature of
the substrate is measured with a thermocouple embedded 0.5 mm beneath the
upper surface. The thermocouple is connected with a PID controller to adjust
the power supply to the semiconductor cooler so that we can obtain a constant
surface temperature with a variation of less than 0.2 K. In the experiments,
the substrate is cooled down from room temperature to the desired temperature
after droplet deposition. The substrate cooling process takes about 3 minutes,
giving an average cooling rate of about 0.3 K per second. After the substrate
cooling down, there is sufficient time for the droplet to reach a thermal
equilibrium state before the freezing initiation. During droplet freezing, the
substrate surface temperature is controlled to $-10$ K below the freezing
temperature. The cooling area of the semi-conductor cooler fully covers the
contact region of the droplet with the substrate, thus the surface temperature
in contact with the droplet can be assumed uniform during the freezing
process.

The experiments are conducted in a highly transparent plexiglass chamber with
an inner size of 8 cm $\times$ 8 cm $\times$ 8 cm, and a wall thickness of 5
mm. To prevent frost formation on the freezing substrate, the chamber is
filled with pure nitrogen gas at a slight overpressure. At the beginning of
every experiment, the chamber is flushed with nitrogen gas, while the flow
rate is kept to the minimum during observations to reduce any possible
disturbance to the freezing process. The temperature outside of the chamber is
consistent with the temperature of the workspace at $298$ K.

The freezing process is recorded using a high-speed imaging system from the
side view with back-light illumination. The imaging system utilizing a high-
speed camera (Photron Nova S12) combined with a long-distance microscope (Zoom
6000 by Navitar) results in a maximum resolution of 5.7 $\mu$m/pixel. With
this imaging system, we can precisely acquire high-resolution images for
analyzing the droplet freezing features versus time.

###  II.2  Numerical model

Based on previous studies of the freezing process of droplets [26, 31, 32], a
simplified phenomenological model is developed in this study. The following
widely used assumptions are adopted in the model [21, 33]: Firstly, the gas-
liquid interface is assumed to be axi-symmetric, and the gravity direction
coincides with the axis of symmetric of the droplet. Secondly, the
solidification front remains flat in the freezing process, and heat transfer
in the ice layer is one-dimensional. Thirdly, the inner flow inside the
unfrozen liquid is negligible. Fourthly, the evaporation and heat transfer at
the gas-liquid interface is neglected. Finally, the temperature of ice-water
mixture remains constant in the freezing process. The temperature variation of
density and thermal diffusivity are not considered, and the change of
properties due to the phase change is assumed to happen suddenly at the
solidification front.

With these assumptions, the droplet freezing process can be resolved from mass
conservation of the liquid-solid system as shown in Fig. 2(2). The initial
profile as well as surface of the remaining liquid in the freezing process are
calculated by integrate the Young-Laplace equation along the liquid-gas
interface [34]. Dilatation is considered at the freezing front by the
conservation of total mass and receding contact angle [31]. With the presence
of a contact line slip at the late stage of freezing [35], dynamic ”growth
angle”, which is defined as the angle between the tangents to the solid-gas
and liquid-gas interface, is implied to the model to predict the movement
direction of tri-junction. The thermal problem can be characterized by the
energy conservation at the freezing front. Due to the second assumption we
take, the temperature is linear distributed between the substrate and the
front [33], and the front propagation can be described by the solution of one
phase Stefan problem at the beginning of the freezing process [36]. In the
later stage of freezing, considering that the freezing front takes spherical
shape [15], the conducted heat flux is calculated along a virtual spherical
cap (as shown in Fig. 2(a), the dashed line) confirmed by the freezing front
radius and liquid contact angle to the ice. The detailed derivation of the
model is provided in the appendix.

Figure 2:  (a) A schematic diagram showing the features of the simple model developed in this study to describe the solidification process of a droplet; (b) Experimental droplet profiles compared with the calculation results from the model. The open circle and dashed line represent the initial profile measured in the experiment and calculated from the model, respectively. The solid circle and solid line represent the final profile measured in the experiment and calculated from the model, respectively Table 1:  Specific values of physical properties used in the calculation. Parameters | Units | Water | Ice
---|---|---|---
Density | $\rm kg/m^{3}$ | 1000 | 917
Thermal diffusivity | $\rm mm^{2}/s$ | $0.132$ | $1.176$
Heat capacity | $\rm kJ/kg\cdot K$ | / | 2.028
Latent heat | $\rm kJ/kg$ | 333.4 | /
surface Tension | $\rm N/m$ | 0.0728 | /
Gravity | $\rm m/s^{2}$ | 9.81 | 9.81

With the physical properties given in Table 1 in the appendix, the calculation
results are compared to experiments to validate the reliability of this model.
Figure 2(b) shows typical initial and final profiles obtained experimentally.
These data correspond to a droplet with supercooling, which is the temperature
difference between the cold wall and the freezing point, $\Delta
T=T_{m}-T_{w}=10\ \rm K$ and volume of $V_{0}=7.8\ \rm{\mu L}$ in different
directions of gravity. Through experiments, we can get appropriate values of
$V_{0}$ to calculate the theoretical droplet profile with the model. Good
agreements are found between the theoretical and experimental results for both
the initial and final droplet profile.

##  III  Result and discussion

###  III.1  Qualitative description of the droplet solidification

Figure 3:  Experimental results of pendent and sessile water droplets freezing
on a cold substrate; (a-b): side-view snapshots for droplets at different
freezing stages; (c-d) initial and final droplet profiles extracted from (a-b)
respectively; (a) For $Bo<1$ case, the droplet base radius is $R_{0}=2$ mm and
its initial volume is $V_{0}=8.5\ \rm\mu L$ ; (b) For $Bo>1$ case, the droplet
base radius is $R_{0}=4$ mm and its initial volume is $V_{0}=78\ \rm\mu L$.

As shown in Fig. 3, both sessile and pendent droplets have similar freezing
processes, which can be divided into four stages : (1) supercooling, (2)
recalescence, (3) freezing, and (4) solid cooling[21]. During this process an
important shape change happens to the droplet, making a sphere-like water
droplet turns into an ice cone with a sharp tip. The freezing of water
droplets first starts with supercooling. In this process, the mean temperature
of the water droplet can decrease down to below the melt temperature. Then the
accumulated supercooling degree is released in the recalescence stage, leading
to rapid ice crystal formation and usually accomplished within several tens of
milliseconds [17]. After the recalescence stage, the droplet turns into a
mixed-phase state consisting of water-ice mixture with temperature equals to
the freezing temperature [20]. Droplet shape also changes a little bit due to
volume expansion. In the freezing stage, the ice layer starts to grow from the
wall to the droplet apex, forming a moving ice-water interface within the
droplet. During the advancement of this interface, which is also referred to
as the ”icing front”, the volume of the remaining liquid decreases and its
shape also changes due to surface tension effects. Finally, a sharp tip is
formed at the apex position of the droplet [15]. In the experiments, we varied
the direction of gravity subjected to the droplet by flipping the direction of
the cold substrate.

Figure 3 presents the side view snapshots acquired in the experiment of a
typical freezing process at different Bond numbers for the same volume
droplets at both sessile and pendent states. As shown in Fig.
3(a,b)(i.s-iv.s), when placed on the upper surface of the cold plate, the
droplet is usually referred to as a ”sessile droplet”. For a sessile droplet,
the direction of gravity is opposite to the direction of ice growth and tends
to flatten the droplets on the contact solid surface. When a droplet is
suspended on the lower surface of the cold plate, as shown in Fig.
3(a,b)(i.p-iv.p) the direction of gravity coincides with the direction of ice
growth and tends to repel the droplet away from the solid surface, known as
“pendent droplet”. The Bond number, which is defined as ${\Delta\rho
gD_{eq}^{2}}/{\sigma}$, is a dimensionless parameter used to characterize the
level of deformation of droplets in a gravity field by evaluating the
importance of gravitational forces compared to surface tension forces, in
which $\Delta\rho$ is the density difference between the liquid and gas
phases, $g$ is the gravity acceleration,
$D_{eq}=\left(6V_{0}/\pi\right)^{1/3}$ is the equivalent diameter defined with
the droplet initial volume $V_{0}$ and $\sigma$ is surface tension. When the
Bond number is greater than 1, gravitational forces dominate the shape of the
free surface, and the droplet is deformed due to gravity. When the Bond number
is smaller than 1, the shape of the droplet is controlled by surface tension
and its shape is a spherical cap [37].

The initial and final droplet profiles extracted from the experiment snapshots
are presented in Fig. 3(c,d). Figure 3(c) compares the results for $Bo<1$
cases, where the base radius of droplets is $R_{0}=2$ mm and droplet volume is
$V_{0}=8.5\ \rm\mu L$. The results show that the initial shape and the final
shape of the sessile droplet and pendent droplet with the same base radius and
volume are very similar in this regime. This comparison indicates that for
cases with $Bo<1$, the shape of the liquid-gas interface for both sessile and
pendent droplets remains controlled by surface tension during the icing and
the change of gravity direction does not induce a noticeable difference.
However, as shown in Fig. 3(d), for cases with $Bo>1$, both the initial and
final shape of droplets shows a noticeable difference. Here the base radius of
droplets is $R_{0}=4$ mm and droplet volume is $V_{0}=78\ \rm\mu L$. Before
freezing, the pendent droplet tends to have a larger initial height and a
smaller contact angle than the sessile droplet with the same volume. The
droplet thickness before freezing is 2.5 mm and 3.0 mm for the sessile case
and the pendent case, respectively, corresponding to a significant difference
of 20%. This significant difference is a result of different directions of
gravity on the free surface of the droplet. The gravity direction changes the
pressure distribution inside the droplet: according to Eq. (9), the pressure
outside the droplet being equal to the atmospheric pressure at the droplet
cusp, the pressure inside the pendent droplets is lower than in sessile
droplets at the droplet cusp due to the hydrostatic pressure distribution
inside the droplet. This pressure distribution results in smaller liquid
surface curvature at the same vertical distance to the apex. After freezing,
the difference in droplet profiles becomes more distinguishable between
sessile and pendent droplets. The droplet thickness after freezing is 3.0 mm
and 3.7 mm for the sessile case and the pendent case, respectively, indicating
a height difference of 0.7 mm comparing with 0.5 mm for droplet initial shape.
The difference in droplet height is further increased and the cone shape is
more pronounced for the frozen pendent droplet.

One further fact that emerged from these observations is that for all cases we
have investigated, a similar tip structure has been observed to appear on the
apex of the drop at the end of the freezing stage. The gravitational direction
obviously changes the initial geometry of large droplets ($Bo>1$) and results
in different frozen geometry as it will be discussed in the following.
However, the formation of the icing tip is not affected by gravity direction
because at the end of the icing, the liquid volume is becoming smaller and
smaller with a Bond number becoming rapidly much smaller than 1 so that the
final tip shape is controlled by surface tension and independent of any
gravity effect.

###  III.2  Droplet height

Figure 4:  Effects of gravity on droplet (a) initial and (b) final shapes for
various Bond numbers. The initial $H_{0}$ and final $H_{m}$ droplet heights
are normalized by their corresponding equivalent height $H_{eq,0}$ and
$H_{eq}$ (see the text), respectively. Vertical bars show $\pm 1$ s.d. The
horizontal black dashed line denotes the no-gravity case. The ”P” note and red
color on the legend mean pendent droplet, while the ”S” note and blue color
mean sessile droplet. Insert, the deformation strength versus the Bond number.

We now turn to the quantitative analysis of the effect of gravity direction on
the final droplet shape after freezing. As analyzed in the previous section,
the gravity direction has a great influence on the droplet shape. For Bond
numbers larger than 1, the height of pendent droplets measured in our
experiments [see Fig. 3(d)] are larger than that of sessile droplets for both
initial shape and final shape. In addition, the time it takes to freeze the
entire droplet is directly related to the height of the droplet, and the
freezing time is also expected to be impacted by the gravity direction. Figure
4 shows the normalized droplet heights $H_{0}/H_{eq,0}$ and $H_{m}/H_{eq}$ for
sessile and pendent droplets as a function of the Bond number. $H_{eq,0}$ is
the equivalent droplet height defined as the height of a spherical cap with
the same volume $V_{0}$ and based radius $R_{0}$, while $H_{eq}$ is the
droplet height obtained using the model when neglecting gravity. Both
$H_{0}/H_{eq,0}=1$ and $H_{m}/H_{eq}=1$ corresponds to the droplet height when
gravity effects are negligible. For droplets with Bond numbers smaller than 1
(droplet base radius $R_{0}=2$ mm), the droplet volume is varied from 4.2
$\rm\mu L$ to 12.2 $\rm\mu L$, and 6.2 $\rm\mu L$ to 18.2 $\rm\mu L$ for
sessile and pendent droplets, respectively. For Bond numbers larger than 1
(droplets with base radius fixed to $R_{0}=4$ mm), the droplet volume is
varied from 33 $\rm\mu L$ to 83 $\rm\mu L$ for both sessile and pendent
droplets. The experiment data points are averaged over a set of repeated
experiments, and the corresponding standard deviation is also presented using
error bars in the figure.

First, the comparison of the results between the experiments and the simple
model shows a satisfactory agreement for both the initial (Fig. 4(a)) and
final (Fig. 4(b)) shapes of the droplet. This indicates that the droplet shape
is always controlled by the balance of surface tension and gravity. Indeed,
the viscous-capillary time $t_{\sigma}=\mu D_{eq}/\sigma$ being several orders
of magnitude smaller than the freezing time, surface tension adjusts
instantaneously to gravity.

Gravity has an increasing effect on the initial droplet height when increasing
the Bond numbers as shown in Fig. 4. For Bond numbers smaller than 1, the
droplet initial shape and final shape satisfy $H_{0}\approx H_{eq,0}$ and
$H_{m}\approx H_{eq}$, respectively. For an initial Bond number larger than
one, the gravity direction induces a clear effect on the droplet shape and
final height. As long as the Bond number based on the remaining liquid volume
to ice is larger than one, a pendent liquid volume is elongated in the gravity
direction, while a sessile liquid volume is flattened.

To quantify the gravity impact on the freezing, we introduce the droplet
relative deformation as

$e_{p,s}=\frac{|H_{p,s}-H_{eq}|}{H_{eq}}$ (1)

where the $p$ and $s$ subscripts represent the pendent and sessile case,
respectively. $e_{p}$ and $e_{s}$ are reported in the insert of Fig. 4(a) and
Fig. 4(b). Concerning the effect of deformation on the initial shape we
observe that the impact of gravity is almost symmetric comparing the sessile
and the pendent droplets, and $e_{p}\approx e_{s}$. However, when considering
the final iced shape, there is a clear difference between the sessile and
pendent droplets, and $e_{p}\gg e_{s}$. For the sessile droplet icing dilation
and gravity deformation are acting in opposite directions and the gravity
effect is almost compensated for droplets considered here. For the pendent
droplet, the icing dilatation and the shape elongation in the gravity
direction are acting in the same way to increase the droplet final height, so
that a stronger deformation is observed, as clearly shown in Fig. 3.

We finally compare the final droplet height $H_{m}$ to the initial droplet
height $H_{0}$ in Fig. 5. The experiment results for all the cases: pendent
and sessile droplets as well as small and large Bond numbers, are reported. A
remarkable linear relation is observed between $H_{m}$ and $H_{0}$ as

$H_{m}\approx 1.27H_{0}$ (2)

which shows a homothetic translation of the drop height during the icing. As
shown in the figure, this relation is in good agreement with both the
experiment of Sebilleau et al. [20]for a water droplet of initial volume
$V_{0}=8.5\ \mu$L (Bond number $Bo=0.87$, $H_{0}=1.09$ mm, $H_{m}=1.26H_{0}$)
and the direct numerical simulation from Lyu et al. [28] of a spherical cap
water drop with an initial volume $V_{0}=2.09\ \rm\mu L$ (Bond number
$Bo=0.34$, $H_{0}=1$ mm, $H_{m}=1.23H_{0}$). This vertical translation of the
droplet height is dependent on the density ratio between the ice and the
liquid so that the pre-factor in Eq. (2) is varying with the density ratio
between the ice and the liquid.

Figure 5:  Maximum drop height $H_{m}$ as a function of $H_{0}$. The black
dashed line ’$--$’ shows $H_{m}=1.27H_{0}$ from relation (2).

###  III.3  Freezing front dynamics

Figure 6:  Temporal evolution of the freezing surface contact line $Z_{f}$ and
the freezing front radius $R_{s}$. (a) small droplets with $Bo=1.12$; (b)
Large droplets with $Bo=4.02$. The experimental data are presented with
symbols and colors for different gravity directions and front parameters. The
solution for the classic Stefan problem $z(t)=2\lambda\sqrt{\alpha_{i}t}$, in
black dashed line, is determined with the same freezing conditions.

To show how gravity influences the droplet freezing dynamics, Fig. 6 reports
the vertical position $Z_{f}$ of the ice-water-air contact line and the radius
of the freezing front $R_{s}$ (as defined in Fig. 2(a)) as a function of time
$t$ for both sessile and pendent droplets at different Bond numbers. For Bond
numbers smaller than 1, both the sessile droplet and the pendent droplet have
the same icing dynamics. As shown in Fig. 6(a) for $Bo=1.12$ the evolution of
$Z_{f}$ and $R_{s}$ are very close for the two cases. Due to a liquid
spherical cap shape controlled by surface tension, the front propagation and
shape evolution for pendent and sessile droplets are similar in the entire
freezing process, which further supports the idea of the limited effect of
gravity in the freezing of droplets at small Bond numbers.
However, for Bond numbers larger than 1, the dynamics of the icing are
different when comparing sessile and pendent droplets. This is clear in Fig.
6(b) when the case $Bo=4.02$ is reported. Pendent droplets have larger droplet
heights than sessile droplets due to gravity-induced deformation, and as a
consequence, the total freezing time for a pendent droplet is longer than that
of a sessile droplet. Surprisingly, despite a different freezing time, the
position $Z_{f}$ of the ice-water-air contact line follows the same evolution
for the two cases. The time evolution of the freezing front radius $R_{s}$
indicates that the radii of the freezing fronts also exhibit noticeable
difference between sessile and pendent droplets, a direct consequence of a
difference in droplet shape between the two cases. Fig. 6 also shows
comparison with the Stefan solution corresponding to a 1-D front propagation
as

$z(t)=2\lambda\left(\alpha_{i}t\right)^{0.5}$ (3)

where $\alpha_{i}$ is the thermal diffusivity of the solid phase and $\lambda$
is a parameter determined by the Stefan conditions [28]. The Stefan solution
can describe the experimental evolution at the beginning of the icing for both
small and large Bond numbers. But, a clear difference is then observed during
the icing. As observed, the freezing is faster than the Stefan 1-D solution
for both small and large $Bo$ cases. The departure happens at about 2 seconds
after the initiation and gradually deviates in the later stage of the freezing
process. As observed, the freezing is faster than the Stefan 1-D solution.

Figure 7:  Front height $Z_{f}$ as a function of freezing time $t$,
corresponding to the case previously presented in Fig. 6. The red and circular
marker represents the pendent droplets, while the blue and square marker
represents the sessile droplets. The dashed lines with slopes of 0.5 is to
guide the eye. Insert shows a zoom of the front propagation at the end of
freezing processes for droplets whose Bond number is 4.02.

For a detailed inspection of the difference with the Stefan evolution, the
evolution processes of $Z_{f}$ are reported using a log-log plot in Figure 7.
As shown, $Z_{f}$ follows the same power law evolution as given by the 1-D
Stefan solution (3) for both sessile droplets and pendent droplets:

$Z_{f}\propto t^{0.5}$ (4)

This result is consistent with most of the results from the literature.
However, we observe here that the pre-factor is larger than the one given by
the Stefan solution (3). This difference is clearly identified from the plot.
Very interestingly, all the cases considered, small and large Bond numbers as
well as sessile and pendent droplets, are following the same evolution.
Thus,is faster icing dynamics cannot be attributed to any gravity effect on
the droplet shape. Such difference was recently discussed in Refs. [38, 28,
20]. This fast evolution can be partially attributed to the constriction of
the heat flux in the droplet during the freezing process. The based radius in
contact with the cold surface remains constant while the icing front surface
is decreasing. In addition, the ambient humidity has been shown to
significantly impact the icing kinetics because of either condensation at the
droplet surface reducing the ice front propagation, or evaporation at the
droplet surface increasing the solidification process. Here the use of
nitrogen to prevent frost formation on the freezing substrate results in some
evaporation at the droplet surface. Thus the faster icing kinetics observed
here is attributed to a combination of droplet geometry and water evaporation
at the droplet surface.

###  III.4  Droplet freezing time

Figure 8:  Relation between the droplet freezing time $t_{frz}$ and droplet
shape. (a). The final droplet height $H_{m}$ is normalized using $R_{0}$ to
eliminate the influence of different base radii while $t_{frz}$ is normalized
using the Stefan and Fourier numbers according to Eq. 2. The dashed line with
slopes of 0.60 corresponds to Eq. 7 with $C=2.1$ and the dashed line with
slope 0.5 is to guide the eye. (b). Measured freezing time versus predicted
freezing time. The dash line with a slope of 0.43 represents the value
predicted with equation 8.

We finally consider the freezing time $t_{frz}$ i.e. the required time to
completely froze the droplet. From the 1-D Stefan solidification solution
given by Eq. (3), $t_{frz}$ can be related to the final drop height as

$H_{m}=2\lambda\left(\alpha_{i}t_{frz}\right)^{0.5}$ (5)

When the temperature of the liquid equals the freezing temperature as a result
of the recalescence and when the latent heat $L$ is much larger than the
sensible heat contribution, it yields $\lambda=\sqrt{{c_{i}\Delta T}/{2L}}$
where $c_{i}$ is the ice heat capacity. Introducing the Stefan number
$St={c_{i}\Delta T}/{L}$ and the Fourier number
$Fo={k_{i}t_{frz}}/{c_{i}\rho_{i}R_{0}^{2}}$, with $\rho_{i}$ the ice density,
to normalize the freezing time $t_{frz}$ in Eq. (5), the normalized final drop
height is expected to evolve with the normalized freezing time as

$H_{m}/R_{0}\propto St^{1/2}\ Fo^{1/2}$ (6)

Figure 8(a) reports the evolution of $H_{m}/R_{0}$ as a function of $St\cdot
Fo$ for all the experiments. The droplet freezing times and initial heights
are obtained in 62 independent freezing experiments with ten different Bond
numbers, for sessile and pendent droplet. The experimental points all collapse
on a single curve showing no significant effect of gravity. A clear power law
is also observed but with an exponent $\alpha=0.60$ larger than the one given
by the Stefan solution according to Eq. (6). From Fig. 8(a), a relation can be
proposed to relate the height of the iced drop $H_{m}$ and the freezing time
as

$H_{m}/R_{0}=C\ St^{0.6}\ Fo^{0.6}$ (7)

with the value $C\approx 2.1$ obtained by fitting the experimental points. The
difference with the 1-D Stefan problem is related to the geometrical
constriction of the liquid remaining to ice when pushed by the dilatation
induced by the water solidification, but also to the operating conditions
related to the recalescence stage and water evaporation at the droplet surface
in the ambient Nitrogen[20]. Combining relation (7) with relation (2) between
$H_{m}$ and $H_{0}$, the freezing time can be expressed as a function of the
initial drop height and base radius as

$t_{frz}\approx 0.43\ \frac{\rho_{i}L}{k_{i}\Delta T}\ H_{0}^{5/3}\
R_{0}^{1/3}$ (8)

As shown in Fig. 8(b), this simple relation allows determining the freezing
time as a function of the prior known parameters. In particular the freezing
time is shown to increase as $R_{0}^{1/3}\ H_{0}^{5/3}$ for both sessile and
pendent droplets, large and small Bond numbers. Thus, the longer time required
to freeze a pendent droplet compared to a sessile droplet of the same volume
is completely taken into account by considering their initial geometry.

##  IV Conclusions

The effect of gravity on water droplet freezing on a cold solid surface has
been investigated. The experiments with pendent and sessile droplets have been
conducted for Bond numbers smaller and larger than unity. For Bond numbers
smaller than unity, the freezing process of sessile and pendant droplets is
not affected by gravity because the shape of the liquid part of the droplet is
controlled by surface tension during the entire freezing process. For Bond
numbers larger than unity, gravity modifies the initial shape of the droplet:
sessile droplets are flattened while pendents droplet are elongated. It
results in a longer time to ice for a pendent droplet than for a sessile
droplet. Despite these differences in the initial droplet shape, several
remarkable similarities have been found for all the configurations, pendent
and sessile droplets, small and large Bond numbers. (i) The final height of a
frozen droplet is found to be linearly proportional to its initial height
$H_{m}=1.27H_{0}$. (ii) The time evolution of the ice-liquid-air contact line
is found to follow the classical power-law $t^{0.5}$, thus noticeably faster
than the Stefan 1-D icing front propagation because of the effect of the
initial recalescence and water evaporation at the droplet surface. (iii) As a
consequence, the time to freeze a droplet is faster than predicted by the
Stefan model and it has been found to evolve as $t_{frz}\propto H_{0}^{5/3}\
R_{0}^{1/3}$. Thus, the initial geometry of the droplet is completely
controlling the icing process.

###### Acknowledgements.

This work was financially supported by the Natural Science Foundation of China
under Grant Nos. 11988102 and 91852202, the National Key R&D Program of China
under Grant No. 2021YFA0716201 and Tencent Foundation through the XPLORER
PRIZE. We are grateful to Yuki Wakata for assistance with the code, and Mingbo
Li, Ziqi Wang, and Lei Yi for helpful conversations.

*

##  Appendix A Description on the simple model

Based on previous studies, we have proposed a simplified model here. The
geometry calculation in the model is first inspired by Ref. [31, 32], and the
heat transfer calculation is optimized by correcting the heat flux in the
later stage of freezing. The detailed descriptions on this model are given in
the following parts.

###  A.1  Liquid shape

For a droplet in the gravity field, the droplet shape is determined by the
balance of the gravity and surface tension. The gravity causes a downward
positive pressure gradient inside the droplet. However, with different
deposition directions, the gravity has different effects on the droplet shape.
For any point on a free surface of the water droplet, the curvature $\kappa$
can be described by the Young-Laplace equation as shown below:

$\kappa=\kappa_{0}\pm\frac{\rho_{f}g}{\sigma}z.$ (9)

Here, the choose of sign $\pm$ is related to the direction of gravity,
${}^{\prime}+^{\prime}$ sign for sessile droplets and ${}^{\prime}-^{\prime}$
sign for pendent droplet. $\rho_{f}$ is the liquid density, $\kappa_{0}$
refers to the curvature at the droplet apex, $z$ is the vertical distance to
the droplet apex and $\sigma$ is the surface tension for the water-air
interface. Normalized by the capillary length
$l=\sqrt{\frac{\sigma}{\rho_{f}g}}$, then Eq. (9) can be rewritten as:

$\kappa^{*}=\kappa_{0}^{*}\pm z^{*}.$ (10)

with $\kappa^{*}=\kappa l$, $z^{*}=z/l$. As shown in Fig. 2 (a), integrating
Eq. (10) with the arc length $s$ along with the liquid-gas interface from the
droplet apex to the droplet base, the equation can be solved by introducing
the tangential angle $\phi_{f}$ between arc length and horizontal line[34].

###  A.2  Freezing stage

According to the conservation of mass at the solidification front $Z=Z_{f}$.
The radius and remaining volume follows the relation as:

$\displaystyle\frac{dR_{s}}{dZ}=\textrm{tan}(\frac{\pi}{2}-\phi_{f}+\phi_{i}),$
(11) $\displaystyle\frac{dV}{dZ}=-\pi\rho R_{s}^{2},$ (12)

The sketch of the drop is shown in Fig. 2 (a), where $R_{s}$ is the radius of
the front, Z is the axial coordinate, V is the remaining liquid drop volume,
$\phi_{f}$ is the angle between the flat solidification front and the liquid
interface at the trijunction point, $\phi_{i}$ is the dynamic growth angle
referring to the angle between the tangential lines of gas-liquid interface
and gas-solid interface and $\rho=\rho_{s}/\rho_{l}$ is solid to liquid
density ratio. The initial conditions are:

$R_{s}(0)=R_{0},V(0)=V_{0},$ (13)

where $R_{0}$ is the radius of droplet base, and $V_{0}$ is the initial volume
of the droplet. When the above equation are normalized by
$r_{s}=R_{s}/R_{0},z=Z/R_{0}$ and $v=V/R_{0}^{3}$, we can obtain the
dimensionless differential equations at the icing front $z=z_{f}$ as:

$\displaystyle\frac{dr_{s}}{dz}=\textrm{tan}(\frac{\pi}{2}-\phi_{f}+\phi_{i}),$
(14) $\displaystyle\frac{dv}{dz}=-\pi\rho r_{s}^{2},$ (15)

with the appropriate initial conditions:

$r_{s}(0)=1,v(0)=v_{0}.$ (16)

The $\phi_{f}$ is determined by the shape of the remaining liquid droplet,
which can be derived from Eq.(9) with given base radius $r=r_{s}$ and
remaining volume $v$. As shown in previous studies, the contact angle between
frozen and liquid water can be influenced by the conditions at the contact
line[22]. With the flat ice front assumption, the ice profile growth not
always follows the tangential direction of the liquid-gas interface[32]. When
the liquid contact angle $\phi_{f}$ exceeds the critical receding angle, the
dynamic growth angle $\phi_{i}$ is nonzero, which is determined by the
conditions at the trijunction line. The contact line slips at a speed $U_{s}$,
which follows the geometrical relation as shown in the figure:

$U_{s}=U_{f}(\frac{1}{\textrm{tan}(\phi_{f}-\phi_{i})}-\frac{1}{\textrm{tan}(\phi_{f})}),$
(17)

where $U_{f}$ is the solidification front advancing speed. The slip speed also
associated with the contact angle, and the relation between them are given
by[33, 32]:

$U_{s}=\left\\{\begin{aligned}
&\frac{\eta(\phi_{r}-\phi_{f})}{\phi_{f}}&(\phi_{f}\geq\phi_{r}),\\\
&0&(\phi_{f}<\phi_{r}),\end{aligned}\right.$ (18)

where $\eta$ is the characteristic slip velocity related to the solidification
rate at the freezing front and $\phi_{r}$ is the critical receding angle. The
energy balance at the freezing front can be described as:

$\pi r_{s}^{2}\rho_{s}\frac{dz_{f}}{dt}=\int\frac{k_{i}\Delta T}{z}dA,$ (19)

Though the flat solidification front is observed in some experiments at the
beginning of the icing, the ice front is not flat near the interface and at
the end of the icing the front is more like a spherical cap[15]. Considering
the actual shape of the solidification front, in the later freezing process,
the actual area that brings heat away from the remaining liquid is increased
when compared with a flat interface. For simplicity, two stages approximation
is usually applied to models to simulate the unique change in front shape[39].
In this model, the heat transfer area is regarded as a spherical cap once the
ice layer thickness exceeds $h$, the height of the corresponding spherical
cap. The critical height is given by
$h={r_{s}}(1/{\rm{sin}\phi_{c}}-\rm{cos}\phi_{c})$, where $r_{s}$ is the ice
front radius and $\phi_{c}$ is the base angle of the spherical cap
$\phi_{c}=\pi/2-\phi_{f}$. Thus the effective heat transfer area can be
expressed as:

$A=\left\\{\begin{aligned} &\pi r_{s}^{2}&(z_{f}<h),\\\ &2\pi
r_{c}^{2}(1-\textrm{sin}(\phi_{f}))&(z_{f}\geq h),\end{aligned}\right.$ (20)

where $r_{c}=r_{s}/\text{cos}(\phi_{f})$ is the radius of curvature for the
spherical cap. Thus the relation between the solidification front height and
the freezing time can be given by:

$\frac{dz_{f}}{dt}=\left\\{\begin{aligned} &2\frac{k_{i}\Delta
T}{z_{f}\rho_{i}L_{m}}&(z_{f}<h),\\\ &\frac{2k_{i}\Delta
T}{r_{s}\textrm{cos}(\phi_{f})\rho_{i}L_{m}}\text{ln}\left(\frac{z_{f}}{z_{f}+r_{c}(\text{sin}(\phi_{f})-1)}\right)&(z_{f}\geq
h).\end{aligned}\right.$ (21)

Combining these equations, we can obtain a simple model that can be used to
describe the dynamic process of droplet freezing in different gravity
conditions. With a given set of conditions droplet volume, base radius, and
direction of gravity, the initial droplet shape can be calculated. The
equations give different solutions for the initial droplet profile with
different gravity directions. Then the initial droplet profile, as well as the
droplet volume and base radius, are used as input parameters of the
calculation for freezing. In the freezing stage, the droplet can be divided
into two parts: the ice part and the remaining liquid. Both of them are
coupled with the evolution of the liquid volume $v$. The shape of the ice part
is determined by the previously calculated evolution of the height of the
freezing front and the radial distance $r_{s}$ of the ice-water-air contact
line, while the shape of the remaining liquid is updated at each time step of
the calculation according to the remaining liquid volume $v$ and front radius
$r_{s}$. The prediction of the critical receding angle $\phi_{r}$ is based on
the inflection point in the droplet profile[32]. Thus we can regard $\phi_{r}$
as a variable related to the initial contact angle $\phi_{0}$ as
$\phi_{r}=\xi\phi_{0}$ for different droplets in the calculation. With the
conditions confirmed by the initial shape and prediction of the inflexion
point, it is find that $\rm\eta=1.1\rm mm/s$ is a reasonable value for all the
experiment cases, and $\xi=0.5$ agree well for most cases while $\xi=0.8$ for
the pendent droplet with a base radius $R_{0}=4\rm mm$ contact with the
substrate. The Eq. (14),(15),(21) are integrated using a fourth-order Runge-
Kutta scheme. For the stop criteria of the numerical resolution, the
integration procedure stops at $v=\epsilon$ ($\epsilon=1\times 10^{-6}\ll 1$
in this study).

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