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# Dual residence time for droplet to coalesce with liquid surface

Ting-Heng Hsieh1, Wei-Chi Li1,2, and Tzay-Ming Hong1<EMAIL_ADDRESS>1Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 30013,
Republic of China
2Department of Physics, Emory University, Atlanta, GA 30322

###### Abstract

When droplets approach a liquid surface, they have a tendency to merge in
order to minimize surface energy. However, under certain conditions, they can
exhibit a phenomenon called coalescence delay, where they remain separate for
tens of milliseconds. This duration is known as the residence time or the non-
coalescence time. Surprisingly, under identical parameters and initial
conditions, the residence time for water droplets is not a constant value but
exhibits dual peaks in its distribution. In this paper, we present the
observation of the dual residence times through rigorous statistical analysis
and investigate the quantitative variations in residence time by manipulating
parameters such as droplet height, radius, and viscosity. Theoretical models
and physical arguments are provided to explain their effects, particularly why
a large viscosity or/and a small radius is detrimental to the appearance of
the longer residence time peak.

##  I Introduction

Triggered by surface tension, examples of coalescence abound in our daily
life, e.g., when milk is poured into coffee or the raindrops fall on a puddle
[1, 2, 3, 4]. It is relatively less well-known that the liquid droplet in fact
does not necessarily merge immediately with the pool after touching down on
its surface. First reported by Lord Rayleigh almost one and a quarter
centuries ago [5], this delay in the coalescence is ascribed to the need for
air to flow out of the intermediate region. The ability to trap and create an
air film is made possible by the fact that the surfaces of both the droplet
and pool are deformable. Otherwise, a solid ball dropping on a pool or a
droplet plummeting onto a solid surface will only create a splash [6, 7, 8,
9].

As the air is drained out by the weight of the droplet during the non-
coalescence, the liquid molecules on both surfaces can ``feel'' each other as
soon as their distance becomes short enough to interact via the van der Waals
force. A recent simulation study [10] shows that the coalescence can actually
set in earlier if the thermal fluctuation of surfaces is taken into
consideration. Previous researchers have managed to prolong the residence time
by (1) renewing the film, e.g., via oscillating the pool surface vertically
[11, 12, 13] or replenishing the air via (2) creating a relative speed between
the droplet and the pool [14, 15, 16, 17] and (3) creating a temperature
difference between the droplet and pool [18]. The latter two methods are so
effective that they even work on a solid boundary, e.g., for dynamic
hydrophobicity [19, 20, 21] and the Leidenfrost effect [22, 23, 24].

If released from a medium height $H$, a non-coalescing droplet undergoes four
stages: free fall, oscillating vertically, floating, and coalescing, as shown
in Fig. 1(a). The oscillation time, defined as $\tau_{1}$ in Fig. 1(b), starts
from the emergence of ripple causing by the impact of droplet on the pool
surface and ends at the moment when it ceases to oscillate. The residence time
$\tau$, denoting the sum of $\tau_{1}$ and the remaining time until
coalescence $\tau_{2}$, is approximately 100 ms for the droplet radius
$R\approx$ 1.5 mm. It is worth mentioning that there exists a critical height
$H_{c}$ beyond which the droplet will simply smash onto the pool and render
$\tau=0$. Physically it can be estimated by requiring the kinetic energy $\rho
R^{3}gH_{c}$ to be comparable to the surface energy $\gamma R^{2}$. The ratio
of these energies is the Weber number.

Klyuzhin et al. [25] have reported that residual charges on the water droplets
can give rise to a second residence time. Encouraged by their nice work, we
set out to explore the effect of $H$ and $R$ on the dual residence time.
Rather than averaging the distribution of $\tau$ as in Ref.[25], we
distinguish different residence times and their corresponding probability by
the K-means clustering [26] which is an iterative algorithm capable of
objectively determining the most statistically meaningful number of peaks and
their corresponding weight from any distribution. As exemplified in Fig. 1(c),
it turns out to be the norm rather than the exception that there exist three
residence times, $\tau_{D}$ for death, and $\tau_{S}$ and $\tau_{L}$ for short
and long $\tau$. Droplets that coalesce within $\tau_{1}$ are designated in
the category of $\tau_{D}$. An interesting observation to bear in mind is
that, while $\tau_{L}\approx 2\tau_{S}$, their $\tau_{1}$ roughly equals.

[width=18cm]introduction.png

Figure 1: (a) Snap shots of a water droplet with radius $R$=1.79 mm and height
$H$=3 mm. Starting from above, three rows of possible outcomes are
distinguished by their length of residence time: coalescence, and short and
long residence. (b) For the latter two cases, the droplet survives the
oscillation stage and remains roughly still until suddenly collapsing. (c) A
typical histogram for the residence time.

##  II Experimental setup

[width=8.5cm]setup4.png

Figure 2:  (a) Schematic experimental setup for measuring the residence time
of water droplets falling on a water pool through the air, (b) Cutaway view of
droplet floating on the pool surface with the equilibrium depth $y_{0}$, the
radius of contact area $r$, and film thickness $h$.

Our goal is to understand when, why, and how the residence time for a droplet
to coalesce with the same fluid in the pool should adopt multiple values under
seemingly identical experimental conditions (e.g., temperature and humidity)
and parameters ($H$ and $R$). Schematic experimental setup can be found in
Fig. 2(a). The radius of the acrylic container is 15 cm, big enough to ensure
the reflected ripple that takes about 1 s to come back will not interfere with
the residence time $\sim$ 0.2 s. A syringe pump is used to make sure the
output speed of the droplet is negligible. The metallic needle of the syringe
is grounded and replaceable that comes in five sizes. The grounded conducting
table is to avoid an accumulation of static charge in the pool. Adjusted by a
translation stage with precision $\leq$ 0.01 mm, the height $H$ of the droplet
measures the displacement of its center of mass that ranges from 1 mm to 10
mm. The whole process of non-coalescence is captured by a high-speed camera
(Phantom Miro-ex4) with frame rate of 1000 fps. An acrylic cover is employed
throughout the experiment to prevent disturbance from the wind.

##  III Theory

From dimensional analysis, we can model the motion of droplet during
$\tau_{1}$ stage by an underdamped harmonic oscillator[27, 28, 29] where the
surface tension coefficient $\gamma$ plays the role of spring constant:

$\rho R^{3}\ddot{y}+\alpha\dot{y}+\gamma y=0.$ (1)

Setting $y=A\cdot\exp{(i\omega t)}$ gives

$\omega=\frac{i(\alpha/R)\pm\sqrt{\rho R\gamma-(\alpha/R)^{2}}}{2\rho R^{2}}.$
(2)

The term inside the square root represents the periodic oscillation and the
other term represents the decrease in amplitude. We assume that $\tau_{1}$
ends when the amplitude is smaller than some minimal discernible amplitude.
Incorporating the information that the initial amplitude equals $y_{1}-y_{0}$
where the maximum and equilibrium depths respectively obey $\rho
R^{3}gH\approx\gamma y_{1}^{2}$ and $\rho R^{3}g\approx\gamma y_{0}$ with
$g=9.8\ {\rm m/s^{2}}$ leads to

$\tau_{1}=\frac{\rho R^{3}}{\alpha}\ln\big{(}\sqrt{\frac{\rho
R^{3}gH}{\gamma}}-\frac{\rho R^{3}g}{\gamma}\big{)}.$ (3)

Note that in addition to the obvious source of dissipation from viscosity
$\mu$, the creation of ripples also takes away some mechanical energy from the
droplet. Therefore, the damping coefficient should include two terms as
$\alpha=\mu R+\sqrt{\gamma\rho R^{3}}$.

Now let's examine the residence time from the perspective of how long the film
can be sustained. By use of the Pythagorean theorem, the radius $r$ of contact
area in Fig. 2(b) can be derived by the knowledge of $y_{0}$ as

$r=\sqrt{\frac{\rho R^{4}g}{\gamma}-\frac{\rho^{2}R^{6}g^{2}}{\gamma^{2}}}.$
(4)

Note that the Bond number Bo=$\rho gR^{2}/\gamma\approx$ 0.15 for $R$=1.1 mm
which implies that the effect of gravity is small but not negligible compared
to the surface tension. Intuitively, we expect the gravity will squeeze the
droplet and lead to a roughly flat bottom surface, as depicted in Fig. 2(b).
The resulting increase in $r$ is equivalent to multiplying Eq. (4) by a
coefficient of (1+$\mathcal{O}$(Bo)). But since the original Eq. (4),
$r=r_{0}\sqrt{1-{\rm Bo}}$, can be treated as $r_{0}$ plus a leading term of
$\mathcal{O}$(Bo). This is of the same order of magnitude as the correction
from (1+$\mathcal{O}$(Bo)). Therefore, we expect the effect of gravity or the
deformation of the droplet not to alter the following theoretic predictions in
a qualitative way.

Drained by the difference $\Delta P\equiv P_{f}-P_{0}$ between the pressure
inside the air film $P_{f}$ and the ambient $P_{0}$, the air has to pass
through an opening whose area roughly equals $2\pi rh$. It will be estimated
in Sec. IV(E) that the Reynolds number roughly equals 0.5, i.e., the viscous
stress is not negligible compared with the inertia. For the sake of
simplicity, let's employ Bernoulli's principle for the time being to obtain an
analytic expression for the relation between the drift velocity $v$ and
$\Delta P$.

$\Delta P\approx\rho R^{3}g/r^{2}=\rho_{a}v^{2}$ (5)

where $\rho_{a}$ is the mass density of air. The continuity equation dictates
that

$\displaystyle-\frac{d(r^{2}h)}{dt}\sim rh\cdot v.$ (6)

where the film volume has been approximated by the product of its base area
$\pi r^{2}$ and thickness $h$. Since $h\ll y_{0}$, $r$ can be treated as a
constant when solving Eq. (6) to obtain

$\tau\approx\sqrt{\frac{\rho_{a}}{\rho g}}\big{(}\frac{\rho
R^{2.5}g}{\gamma}-\frac{\rho^{2}R^{4.5}g^{2}}{\gamma^{2}}\big{)}\ln(\frac{h_{i}}{h_{f}})$
(7)

where $h_{i}$ and $h_{f}$ are the initial and final thickness. It will be
shown in Appendix B to capture the same $\tau$ vs. $R$ behavior as when
viscosity is taken in account.

##  IV Experimental results

Presumably, the dual residence time has to derive from the dichotomy of some
experimental property, e.g., oblate vs. prolate shape for the droplet, zero
vs. nonzero triboelectric effect, inward vs. outward internal flow, and
symmetric vs. asymmetric geometry for the film. To sift through these
candidates, we shall systematically vary the height, shape, and radius of the
droplet, viscosity, and voltage between the syringe and pool to clarify their
roles.

###  IV.1 Flow field and particle image velocimetry

Previous research [16] has demonstrated that rotating droplets can exhibit a
significantly longer residence time, say, from 0.1 to 1 s for radius $R$= 1.5
mm and height $H$= 6 mm. Since the droplets are controlled to be stationary in
our experiment, the knowledge of Ref.[16] opens up a possible explanation for
the dual values of residence times, i.e., an inward vs. outward flow pattern.
To investigate this scenario, we employed Particle Image Velocimetry (PIV)
which is composed of a laser of 2.5 W and 532 nm and a convex lens passing
through the optical path, illuminating the titanium dioxide particles inside
the droplet. The reflected light signal is picked up by a high-speed camera
with 2000 fps. Afterward, data are converted to images via PIVlab of the
MATLAB toolbox for the analysis of flow direction. We measured the flow field
during the first ten millisecond of $\tau_{2}$ stage for both $\tau_{S}$ and
$\tau_{L}$. The reason why the $\tau_{1}$ stage was excluded was that
oscillations of the droplet made the measurement of PIV challenging. As shown
in Fig. 3 where the average velocity has been subtracted, there are no
internal circulation for $\tau_{S}$ and $\tau_{L}$. This rules out the pattern
of flow field as a source for the dual residence time.

[width=8.5cm]PIV.png

Figure 3: The averaged flow vectors for the first ten millisecond are
determined by PIV for (a) $\tau_{S}$ and (b) $\tau_{L}$. Both of $\tau_{S}$
and $\tau_{L}$ shows no specific direction of flow. The corresponding
vorticity in (c) and (d) rules out the possibility of inward vs. outward
internal flow.

###  IV.2 Coalescence point

Imagine throwing an upside-down bowl on water, it can sink either quickly with
a kerplunk sound or slowly with a series of gurgling blubs. The difference
lies in how level the bowl rim is with respect to the water surface. This led
us to an alternative theory for the dual residence time, i.e., the geometric
shape of the air film. We recorded the processes both prior to and during
coalescence by placing a high-speed camera under the pool. Within the scenario
of a sinking bowl, coalescence is expected to start nearer the middle of the
droplet for the symmetric or $\tau_{L}$ case. So, if we define $D$ as the
distance between the point of coalescence to the film center, it ought to be
negatively correlated to $\tau$. However, the data in Fig. 4 did not confirm
this picture, although they do support a negative rather than positive
correlation. This failure is partly ascribed to the fact that $D>r/2$ for both
$\tau_{S}$ and $\tau_{L}$. There is a simple explanation for this observation:
the air chamber is thicker in the middle than on the edge, according to
Ref.[30, 31].

[width=8cm]coalescencepoint.png

Figure 4: Distance of coalescence $D$, normalized by the film radius $r$, is
plotted against the residence time $\tau$ for $R$= 1.79 mm and $H$=4 mm. Note
that droplets always coalesce outside of $r/2$-range for both $\tau_{S}$ and
$\tau_{L}$.

###  IV.3 Net charge and external electric field

Klyuzhin et al. [25] have suggested that the electrical charge plays a
significant role in shortening the residence time. They observed that the
instances of shorter residence time could be erased by grounding the pool.
This inspired us to examine a third scenario for the dual residence time,
i.e., charged vs. neutral or with a different quantity of charges. The Kelvin
water dropper famously demonstrated the existence of and feasibility to
generate static charges by passing water through a metal ring on the hose. We
believe the detachment of droplets from the needle is no different.

To quantitatively measure possible charges accumulated on the droplets, we
employed two parallel vertical metal plates with a voltage difference of 3000
V. After being released from the needle, the droplets pass through the
junction of plates and get deflected horizontally with a displacement that
varies with the charges they carry. It turns out that all the droplets
appeared to exhibit a similar displacement in Fig. 5(a), namely, there are no
double peaks in their distribution. These droplets are released high above the
plates to avoid induction by the electric field. The price we paid was that
the droplets would experience the nonuniform field at the plate edges, which
complicated the determination of charges. However, since the data show that
all droplets share roughly the same displacement, we can at least be sure that
they carry similar charges. This excludes the static charge as a candidate to
explain the dual residence time.

It is worth mentioning that we also conducted experiments to apply an external
voltage $V$ up to 150 V to the needle by connecting it to a DC power supply,
while placing a metal plate under the water surface to ground the pool. As
depicted in Fig. 5(b), both $\tau_{L}$ and $\tau_{S}$ decrease with $V$. A
heuristic argument may be that the Coulomb attraction between the charged
droplet and the pool adds more pressure on the film and thus accelerates the
drainage of its air. It is interesting to note that both $\tau_{L}$ and
$\tau_{S}$ vanish simultaneously at 120 V when a Taylor cone [32, 33, 34]
emerges from the pool and ``penetrates'' the air film, as shown in Fig. 5(c,
d).

[width=8.5cm]V.png

Figure 5: (a) Horizontal displacement of droplets passing through two parallel
metal plates separated by 20 mm and with a voltage difference of 3000 V. The
$x$-parameter denotes the order of each trial. (b) Residence time $\tau$ vs.
voltage $V$ with $R=1.79$ mm and $H=3.00$ mm. Both $\tau_{S}$ and $\tau_{L}$
drop as $V$ increases. When $V\geq$ 120 V, all droplets immediately coalesce
with the pool due to the formation of the Taylor cone, as evidenced by the
snapshots in (c) and (d) taken before and after the development of the water
column. To enable better visualization of the Taylor cone, the photos were
taken with a much higher voltage at 1000 V.

###  IV.4 Height and shape of droplet

[ width=8.5cm]h-dep.png

Figure 6:  Residence time $\tau$, $\tau_{1}$, and $\tau_{2}$ are plotted
respectively in (a, b, c) as a function of height for water droplet of radius
$R=1.79$ mm in air. Snapshots of droplets upon touching the pool are inserted
in (a) to indicate how its shape alternates between oblate and prolate with
$H$. The straight line in the inset of (b) for $\exp{[\tau_{1}\alpha/(\rho
R^{3})]}$ vs. $\sqrt{H}$ is the fitting curve of Eq. (3) with an R-squared
value of 0.93.

If the droplet is released at $H=0$, we expect the coalescence to proceed
immediately[35] and $\tau=0$. In contrast, when $H$ is high enough to convert
a kinetic energy $mgH_{c}$ that is comparable to the surface energy
$4\pi\gamma R^{2}$, the droplet should either break up or ``penetrate'' the
water surface of the pool, i.e., $\tau$ is again 0. Therefore, a naive guess
is that $\tau$ ought to follow a convex function for $0\leq H\leq H_{c}$.
However, a different scenario was offered in Ref.[4, 36] that proposes $\tau$
should decrease monotonically with $H$. Their arguments are that, if $P_{f}$
increases with $H$, the drainage speed $v$ that results from $\Delta P$ should
go up and reduce $\tau$.

It turns out that the actual $\tau(H)$ deviates from both predictions, as
shown in Fig. 6(a). Three regions can be identified by their distinct
behavior. When $H$ is between $H_{d}$ and $H_{b}$, $\tau_{L}$ and $\tau_{S}$
are roughly independent of $H$. This is very different from the impact of
droplet on a flat solid surface where the film thickness was found to vary
sensitively with the impact velocity [9]. As $H$ decreases below $H_{d}$,
$\tau_{L}(H)$, still a plateau, adopts a larger value discontinuously, while
$\tau_{S}$ remains the same. For $H$ higher than $H_{b}$, both $\tau_{L}$ and
$\tau_{S}$ increase continuously by an increment of about 50 ms before $\tau$
becomes zero when $H\geq H_{c}$.

As soon as the pool surface shows signs of disruption, we start counting
$\tau$ for this sandwich structure of droplet, air film, and pool. When
detached from the syringe, we intuitively expect the droplet to be deformed
and switch from resembling rugby to a discus alternatively during free fall.
However, two observations rule out this dichotomy as the cause of dual
residence time. First, the shape of the droplet at $t$=0 is indistinguishable
for $\tau_{L}$, $\tau_{S}$, and even $\tau_{D}$ in Fig. 1(a). Second, the
middle two photos for $H_{d}<H<H_{b}$ in Fig. 6(a) do exhibit dissimilar
shapes, rugby vs. discus-like, but their $\tau_{L,S}$ are roughly the same.

Aided by the high-speed camera, we observe a ring of crest, created by the
recoil of the water filament after pinch-off, traveling down the droplet
surface for $H<H_{d}$. Video can be found in SM [37]. In our opinion, this
serves to shove and replenish the air into the film as the crest reaches the
bottom of the droplet, which results in a larger value for $\tau_{L}$. Since
$H_{d}$ corresponds to the free-fall distance within half a vibration period
in this scenario, it can be easily calculated as:

$H_{d}\approx(g/2)(\pi\sqrt{m/\gamma})^{2}\approx\rho R^{3}g/\gamma$ (8)

which roughly equals 2.55 mm. This is in agreement with the empirical value
estimated from the position of the orange dash line in Fig. 6(a). Note that
there is another prediction by this picture, i.e., $\tau$ ought to be zero at
$H=H_{d}$ because the crest will become an exact duplicate of the sharp end at
the top of the droplet, except it appears at the bottom and will ``perforate''
rather than help to retain the film. To search for the short interval around
$H_{d}$ where this physics should occur, we refined the increment of $H$ and
indeed found[37] a tiny window where the distribution of $\tau$ reduces to
only containing $\tau_{D}$. Since the amplitude of the surface wave is subject
to dissipation by the viscosity, its effect on $\tau$ is expected to diminish
as $H>H_{d}$.

If the syringe is raised even higher, Fig. 6(a) indicates that new physics
enters to lengthen $\tau$ beyond $H_{b}$. What happens is that the bottom of
the droplet is now capable of bouncing above the pool surface and allowing
ample fresh air to enter the film. Based on this picture, $H_{b}$ can be
estimated by setting $y_{1}-y_{0}=y_{0}$:

$H_{b}\approx 4\rho R^{3}g/\gamma$ (9)

which is about 6 mm for $R$=1.79 mm, consistent with Fig. 6(a).

The weak dependence on $H$ for $\tau_{1}$ in Fig. 6(b) is consistent with the
theoretical result in Eq. (3). The reason why the data points for $H>H_{b}$
seem to deviate from this prediction is that the maximum height after bouncing
is roughly the same for different $H$. For the sake of comparison, Fig. 6(c)
is obtained by subtracting Fig. 6(b) from (a).

###  IV.5 Drop radius

By adding minute amounts of surfactant, Amarouchene et al. [35] studied the
maximum height below which non-coalescence can occur. Perhaps due to the
presence of surfactant, it did not occur to them that the data might support
the dual residence time. After averaging, $\tau$ was claimed to vary linearly
with $R$ in accordance with the prediction of lubrication theory that is based
on the assumption that the viscous stress dominates the inertia. However, the
Reynolds number ${\rho_{a}vh}/{\mu_{a}}\approx 0.5$ for the drainage of air
film in our system does not meet such a criterion, where $\rho_{a}$ and
$\mu_{a}$ denote the mass density and viscosity of air, drift velocity
$v\approx$ 3 m/s, and film thickness $h\approx 10^{-6}$ m [35]. Aside from the
fact that we did not use any surfactant, our conclusion that there exist dual
residence times is also different from that of Ref.[35]. This justifies the
need for us to attempt alternative models in Sec. III which targets both
$\tau$ and $\tau_{1}$. Regarding $\tau$, the solid lines in Fig. 7(a) are the
prediction of Eq. (7), which enjoy excellent R-square values of 0.93 and 0.90
for $\tau_{L}$ and $\tau_{S}$.

[ width=8.5cm]R-dep.png

Figure 7:  (a) $\tau$ vs. $R$ for $H_{d}<H<H_{b}$. Solid lines are fitting
curves based on Eq. (7). Note that dual $\tau$ values only occur when $R\geq
R_{c}$, during which the probability weighting gradually shifts from
$\tau_{L}$ to $\tau_{S}$ as $R\rightarrow R_{c}$ in the inset. (b) $\tau_{1}$
varies as $R^{2}$ for $H_{d}<H<H_{b}$. (c) $H_{d}$ is found to be proportional
to $R^{2.8}$, consistent with Eq. (8). Solid lines in (b, c) are linear
fitting curves. The capillary length $L_{c}\equiv\sqrt{\gamma/\rho g}$ is used
to render the parameters $R$ and $H_{d}$ dimensionless.

Sharp-eyed readers may notice the scarcity of data for small $R$ in Fig. 7(a).
It is constrained by technical difficulty, i.e., to successfully pump fluid
out of a needle of radius $a$ involves overcoming the surface tension and thus
requires a pressure $\approx\gamma a/a^{2}$. The huge pressure to produce
really fine droplets thus may cause leaking from the joints. We came up with a
way to circumvent the problem, i.e., utilizing the phenomenon of coalescence
cascade or partial coalescence [38, 39, 40]. Each generation of daughter
droplets performs a few rebounds before partially coalescing, which is when we
measure its radius, height, and $\tau$, before the next generation repeats the
whole process. Although the trend of these data is equally consistent with our
prediction as those produced by the needle in Fig. 7(a), we were surprised to
find them all single-valued, i.e., small radius seems to disfavor the
appearance of the dual residence time. Via monitoring the change rate of
probability with $R$ in the inset of Fig. 7(a), we conclude that it was
$\tau_{L}$ that gets suppressed at small-$R$.

Now let's move on to the second prediction targeting $\tau_{1}$ in Sec. III,
i.e., Eq. (3). Its anticipation for $\tau_{1}$ to increase with $R$ is
intuitive because more kinetic energy released by a bigger droplet from the
same $H$ should take longer for the pool to dissipate. The fitting line in
Fig. 7(b) shows that $\tau_{1}$ is roughly proportional to $R^{1.5}$. We
checked that this is consistent with Eq. (3) due to two numerical
coincidences. First, the logarithmic term in Eq. (3) is insensitive to $R$
since our choice of $H_{d}<H<H_{b}$ is close to $H=H_{b}/2$ where its
derivative equals zero. Second, the inertia term is bigger than the viscous
term in $\alpha$.

Finally, we would like to examine the $R$-dependence of Eq. (8) for $H_{d}$.
It was not included in Sec. III because we did not foresee its existence until
we actually performed the experiments. As shown in Fig. 7(c), the prediction
of Eq. (8) based on simple physical arguments is a success with an R-square of
0.98.

###  IV.6 Viscosity

Intuitively, $\tau_{1}$ should be sensitive to the viscosity $\mu$ because the
droplet is oscillating on the pool surface. How about $\tau_{2}$? As explained
at the beginning of this section, one possible explanation for the dual
residence time is an inward/outward internal flow in the droplet that
replenishes/depletes the air in the film. If the flow indeed exists, we would
expect $\mu$ to also affect $\tau_{2}$. Unfortunately, this is ruled out by
our employment of Particle Image Velocimetry. How does $\mu$ help clarify the
plausibility of our final candidate, i.e., the uniformity or the lack of it
for the film thickness? A droplet with a large viscosity is slow to deform due
to its small velocity gradient. Therefore, it behaves like a lever with a
large bending modulus in the time span of $\tau_{2}$, about tens of
millisecond. In other words, once tilted, the droplet will remain lopsided and
create a large opening that allows the trapped air to rush out easily,
consistent with the disappearance of $\tau_{L}$ when $\mu\geq$ 10 mPa s in
Fig. 8(a).

[ width=8.5cm]vis-dep.png

Figure 8: (a) Residence time $\tau$, roughly independent of viscosity $\mu$
for $R$=1.79 mm and $H_{d}<H$=4 mm$<H_{b}$, returns to single value when $\mu$
$\geq$ 10 mPa s. The trend of probability shift in the inset indicates that it
is $\tau_{L}$ that gets suppressed at large $\mu$. (b) $\tau_{1}$ decreases as
$\mu$ increases, as predicted by Eq. (3). (c) For $H<H_{d}$, $\tau_{S}$
remains insensitive to $\mu$ as (a) for $R$=1.79 mm, while $\tau_{L}$
increases with $\mu$ before vanishing at $\mu$=400 mPa s, consistent with our
expectations in Sec. IV(F).

To investigate how viscosity affects $\tau$ and its distribution
experimentally, we modify the ratio of glycerol-water mixture. The viscosity
of glycerol is three orders larger than water, but its surface tension
coefficient and mass density are only 10$\%$ less and 25$\%$ higher,
respectively. Note that it is the mass $m$, not volume, of the droplet that is
determined by the syringe size due to the capillary force. So, when more
glycerol is added, the drop size $R$ shrinks, and the film pressure $mg/(\pi
r^{2})$ depleting the air goes up. Along this line of argument, $\tau_{2}$
should be shortened as $\mu$ increases. It is at odds with Fig. 8(a) which
shows that $\tau_{S}$ and $\tau_{L}$ are roughly independent of $\mu$, whereas
$\tau_{1}$ decreases in Fig. 8(b). This discrepancy can be explained by
considering the deformation of the droplet around the air outlet, similar to
the wind blowing open a window left ajar. As the air drains out from the film,
the outlet with a lower viscosity is more susceptible to the widening force
inflicted by the air current, which quickens the drainage process and results
in a lower $\tau_{2}$. How about $\tau_{1}$? According to Eq. (3), a higher
viscosity hastens the energy dissipation and results in a shorter $\tau_{1}$,
which is consistent with Fig. 8(b). Combined with the opposing trend of
$\tau_{2}$, Fig. 8(a) thus becomes featureless.

In Sec. IV(D), we ascribed the enhancement of $\tau_{L}$ for $H<H_{d}$ to the
replenishment of air in the film by surface waves triggered by the recoil of
liquid filament at pinch-off. Intuitively, we expect this effect to be
diminished as the filament becomes thinner and contains less volume if $\mu$
increases. The enhanced dissipation should also hinder the propagation of
surface waves. However, $\tau$ turns out to be lengthened in Fig. 8(c). A
back-of-the-envelope calculation gives the viscosity length
$\mu^{2}/\rho\gamma\leq$ 0.1 mm for $\mu$ as large as 100 mPa s. The fact that
this length is much shorter than the droplet radius $R$=1.79 mm means that the
surface waves have no trouble reaching the bottom of the droplet. What we
missed in our initial picture is the opposite effect after that, i.e., the
surface waves will push out some air when they leave the film. The lower crest
associated with a larger $\mu$ now becomes less harmful to the sustainment of
the film.

[width=8.6cm]phasediagram.png

Figure 9: Data of different liquid mixtures are compiled to demonstrate that
those exhibiting dual residence time and denoted by solid symbols, such as
water, glycerin, ethanol, and silicone oil, tend to fall below the blue dashed
line, separated from their single-valued counterparts in hollow symbols above.
The three small-$R$ points that defy such a rule are created via the
coalescence cascade, instead of a syringe like the rest of the samples.

Based on the findings in Figs. 7(a) and 8(a), it is tantalizing to assume that
there exists a dimensionless threshold that combines $R$ and $\mu$ and beyond
which the residence time becomes single-valued. The most obvious candidate is
the Laplace number, ${\rm La}=\rho\gamma R/\mu^{2}$, that characterizes the
free surface in fluid dynamics. By mixing DI water, glycerin, and ethanol, we
can prepare samples with different parameters to obtain Fig. 9 that pins the
threshold at ${\rm La}=1430$. The only data that are not consistent with this
theory are the three points with the smallest $R$ obtained by the coalescence
cascade described in Sec. IV(E). We suspect the discrepancy has to do with the
pool not completely calming down from the generation of daughter droplets when
the latter touchdown.

Noted that there is no solid triangles in Fig. 9. For those solid triangles,
the data are for liquid alkane, which is much more volatile than the silicone
oil denoted by hollow triangles. For the record, their vapor pressure at room
temperature is 57.9 and 0.6 KPa, respectively. What happened was that no
alkane droplet could survive the oscillating or $\tau_{1}$-stage. In other
words, no nonzero residence time, either single or dual, was observed. We
suspect the profuse alkane vapor above the pool contributes to and expedites
the formation of liquid bridge (between the droplet and pool surface), which
is known to be the precursor of coalescence[10].

###  IV.7 Symmetric vs asymmetric air film

In Sec. IV(F), we conclude that the short residence time is likely caused by
the tilt of the droplet. Three efforts were made to provide direct and
indirect evidence. Firstly, we tried to deduce the profile of film by
interference patterns. Due to the constant leakage of air, the profile is not
static so that a high-speed camera of 8000 fps equipped with a microscopic
lens must be called into action. Other than that, the employment of a beam
splitter while aiming the laser with wavelength 650nm from below the pool was
standard[30, 41, 42, 43]. Note that the pattern in Fig. 10(a) only covered
partially the base area near the drop center. By counting the number of bright
fringes, we know that the largest thickness occurs near the middle of the film
and is about 10 $\mu$m, while the opening in the perimeter is no more than 1
$\mu$m. This decrease in the film thickness as we move away from the center is
consistent with the observation in Sec. IV(b) that coalescence always happens
near the outskirt. We measured the distance $L$ between the centers of the
droplet and interference pattern and tried to see whether it could help us
distinguish the $\tau_{L}$ and $\tau_{S}$ cases. The negative correlation
between $L$ and $\tau$ revealed by Fig. 10(b) was consistent with our
expectation that an asymmetric film would more likely be associated with a
large $L$. The scatteredness of data is not surprising because the profile of
flim is subject to the influence of air flow which contains randomness in its
spatial distribution of speed.

So far, we were limited to pursuing consequences passively from the asymmetry
which occurred apparently by chance. Why don't we turn the tables and try to
create a tilt film proactively? After some thoughts, we came up with two
experiments by (1) making use of the tilt water surface near the wall of the
pool due to wetting and (2) tilting the needle that connects to the syringe.
The results of both attempts in Fig. 10(c, d) match our expectations and,
therefore, are in support of our theory. As the droplet moves closer to the
wall, i.e., when the distance $D_{w}$ decreases in Fig. 10(c), the degree of
tilt is intensified which suppresses the probability of $\tau_{L}$. In the
meanwhile, increasing the tilt angle $\theta$ of the needle in Fig. 10(d)
gives a similar effect.

[width=8.6cm]interference.png

Figure 10: (a) A snapshot of the interference pattern during the $\tau_{2}$
stage for $R$= 1.75 mm and $H$= 3.00 mm. The perimeter of the film is
highlighted by the red line whose center deviates from that of the
interference pattern by a distance $L$. (b) A negative correlation was
observed for $L$ vs $\tau$. (c) The probability of $\tau_{L}$ is found to
decrease as the droplet approaches the wall. (d) The probability of $\tau_{L}$
goes to zero when $\theta\geq$ 0.7.

##  V conclusion and discussions

In conclusion, we have investigated the distribution of non-coalescence time
for droplets falling on a pool of the same liquid. A strict statistical
algorithm helps us establish that there exist three distinct peaks for
droplets under identical initial conditions and controlled parameters. The
residence time can be divided into two parts: the oscillating time $\tau_{1}$
to dissipate off most of the kinetic energy and the remaining time $\tau_{2}$
when the droplet hovers stably over the pool surface. The drainage of air in
the film is believed to take place throughout $\tau$. We observed that the
height $H$ affects not only the impact velocity but also the shape of the
droplet upon contact with the pool. But, the existence and value of dual
residence time are not sensitive to both factors, except for $H\leq H_{d}$ and
$H\geq H_{b}$ when the surface wave created by pinch-off and rebound of
droplet respectively come into play to lengthen $\tau$. We managed to derive
analytic expressions for the important physical properties, such as
$\tau_{1}$, $\tau$, $H_{d}$, and $H_{b}$, which match well with the
experimental results. When theory failed, we did our best to provide heuristic
physical arguments to explain at least the qualitative behavior, e.g., why a
large $\mu$ or small $R$ suppresses the appearance of $\tau_{L}$.

After excluding possible origins, such as oblate vs. prolate shape for the
droplet, with vs. without the triboelectric effect, and inward vs. outward
internal flow, we attribute the dual residence time to how tilting the droplet
is. This is the only scenario that is consistent with all our experimental
findings. It is inspired by the ordinary experience of throwing a bowl upside-
down on water. We expect two outcomes: the bowl can sink either quickly with a
kerplunk sound or slowly with a series of gurgling blubs, depending on how
level the bowl rim is with respect to the water surface. Similar to the
experience from pinching the garden hose, it is thinkable that the side of the
film with a smaller opening will exert more force on the droplet to push it
back into a more balanced position. This feedback mechanism makes sure that
the symmetric or $\tau_{L}$ case is not a singular event, i.e., valid within a
range of small tilt angles.

To test how general our conclusions are and to gain more insights, it will be
interesting and recommendable to try different combinations of fluid and
medium, i.e., air bubble and DI water where the film composes of water rather
than air, and immiscible liquids such as oil and DI water. Potential
applications of our study include (1) the food industry, say, to create
distinctive flavor by mixing different oil, (2) the manufacturing, e.g., the
fusing of molten particles to form a continuous film, as in welding and
brazing, emulsion and coating, and (3) the material science of how microvoids
link together at the grain boundary that may lead to the fracture of metallic
alloys.

## Acknowledgement

We are grateful to Jow-Tsong Shy, Nan-Jung Hsu, and Chia-Wei Deng for
technical assistance in the early stage of this project, and to the Ministry
of Science and Technology in Taiwan for financial support under Grants No.
111-2112-M007-025 and 112-2112-M007-015.

##  Appendix A K-MEANS METHOD

To distinguish the number and position of peaks as well as their boundaries in
a histogram such as Fig. 1 in the main text, we introduced the K-means
clustering which is a method of vector quantization, originally from signal
processing. The algorithm involves three main steps: First, select the most
likely number of peaks in the distribution, say $K$, and then randomly place
$K$ points in the data and determine their Voronoi cells. Second, relocate the
$K$ points to the centroid of their residing clusters. Finally, repeat the
above processes until the algorithm converges. Presumably, a larger $K$ will
enable a smaller variance,

$\sum_{i=1}^{K}\sum_{x\in S_{i}}|x-S_{i}|^{2},$ (10)

which $S_{i}$ represents the centroid of each cluster and $x$ denotes the data
points. But we need a balance between the goodness of fit and the simplicity
of the model. The trade-off can be objectively determined by calculating the
silhouette coefficient which can help us select the best suitable $K$.

##  Appendix B EFFECT OF MEDIUM VISCOSITY FOR $\tau$

Since the Reynolds number is estimated to be 0.5 in Sec. IV(E), the viscous
pressure cannot be neglected. A quick way to modify Bernoulli's principle in
Eq. (5) is to deduct the heat from the gravitational energy before converting
it to the kinetic energy of the draining air:

$\rho R^{3}gh-\mu_{a}v^{2}h\tau\approx\rho_{a}r^{2}hv^{2}.$ (11)

where the form of $\mu v^{2}h$ can be argued from dimensional analysis.
Solving $v$ and plugging in the continuity equation of Eq. (6) give

$\tau\approx\bigg{(}\frac{R}{\gamma}-\frac{\rho
R^{3}g}{\gamma^{2}}\bigg{)}\bigg{(}\mu_{a}+\sqrt{\mu_{a}^{2}+\frac{\rho_{a}\rho
gR^{3}}{\big{(}\ln(\frac{h_{i}}{h_{f}})\big{)}^{2}}}\bigg{)}\big{(}\ln(\frac{h_{i}}{h_{f}})\big{)}^{2}$
(12)

which reproduces Eq. (7) if $\mu_{a}$ is set to be zero. Fitting Fig. 7(a) by
Eq. (12) gives R-Squared values of 0.91 and 095 for $\tau_{L}$ and $\tau_{S}$,
as opposed to 0.93 and 0.90 by Eq. (7).

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