When a droplet impacts a superhydrophobic surface, it undergoes
spreading and retraction, and eventually bounces off the
surface.1 The contact time, t c,
defined as a length of time interval for the droplet remaining in
contact with the solid surface, is a very important parameter to measure
the surface superhydrophobicity. A short contact time is in particular
favorable to serval applications, i.e.
self-cleaning,1,2 anti-corrosion,3anti-icing.4,5 On the contrary, a long contact time is
the most basic requirement for spray cooling.6
It has been widely recognized that, for low-viscosity fluids such as
water, the rebound dynamics on superhydrophobic surfaces is dominated by
inertial and capillary forces, so that the contact time depends only on
the Weber number, following a relationt c∼(D 0/V 0)f (We ),7,8hereWe =ρV 02D 0/σwith ρ , D 0, V 0, andσ being the liquid density, droplet diameter, impact velocity,
and liquid surface tension, respectively. In a low impact velocity
region with We <1, the impacting droplet can be treated
as an elastic ball because of its small deformation. Since the
deformation is described by a Hertz shock, the contact time scales ast c∼(D 0/V 0)We 2/5,
varying asV 0-1/5.8However, when We >1, despite the deformation
amplitude and details are found to depend strongly on the impact
velocity, the contact time is independent of the impact velocity in a
wide range of velocities. As a result, the contact time follows a scalet c∼(D 0/V 0)We1/2.8The prefactor is usually determined to be 2.6 for a water droplet
impacting superhydrophobic surfaces atWe >1,7,8 implying that there is a
limit of the contact time. Many efforts have been devoted to proposing
new methods to reduce the contact time, aiming at breaking the limit.
For example, using superhydrophobic surfaces with a macroscale ridge to
generate a non-axisymmetric retraction,7,9-11designing superhydrophobic surfaces with specific microstructures to
induce a pancake bouncing,12-15 and texturing
superhydrophobic surfaces with a point-like superhydrophobic
macrotexture16 or using convex superhydrophobic
surfaces17 to trigger a ring-bouncing.
It should be emphasized that the previous studies7-17focused only on a single droplet impact; however, multiple droplets
impact is more frequently encountered in practical applications. When
multiple droplets impact simultaneously a superhydrophobic surfaces,
they will coalesce with each other, forming a large droplet. Because of
complex interactions caused by coalescence, the rebuond dynamics may
become extremely abstruse. Moreover, coalescence also inevitably gives
rise to viscous dissipation even if on superhydrophobic
surfaces.18,19 As a result, the contact time should
depend on both the Weber and Reynolds numbers and thereby increasing as
compared to the single droplet impact. Unfortunately, how the
coalescence influnces the rebound dyanmics and resultant contact time of
the multiple droplets impact remians poorly understood. Recently,
several studies20,21 numerically studied impact
behaviors of multiple droplets on hydrophilic and hydrophobic surfaces.
These studies demonstrated that the droplet-to-droplet coalescence
dynamics significanlty alters spreading and retraction behaviors of the
droplets, leading to significantly different outcomes from the single
droplet impact.
In this letter, we ivestigate the reboud dynamics of two equally sized
droplets simultaneously impacting a superhydrophobic surface via LBM
simulations. We discover three rebound regimes depending on the distance
between the two droplets: a CCR regime, a PCR regime, a NCR regime. We
show that the liquid ridge or bridge formed between the two droplets
plays different role in coalescence dynamcis in the three regimes. We
also demonstrate that although partial coalescence takes places between
the two droplets, the contact time is still dramatically shortened in
the PCR regime, even smaller than that of the single droplet impact.
In this work, we employ a three-dimensional nineteen-velocity (D3Q19)
Shan-Chen model incorporated with the multi-relaxation-time (MRT)
collision operator, namely MRT pseudopotential lattice Boltzmann model
(MRT-LBM), due to the fact that the MRT model overcomes the numerical
instability and reduces spurious currents22,23 (see
the supplementary material S1 for Multi-relaxation-time pseudopotential
lattice Boltzmann model). The schematic of two equally sized droplets
simultaneously impacting a superhydrophobic surface is illustrated inFig. 1 . Dimensions of the computational box areNx ×Ny ×Nz =240×400×200
lattice units (lu). The box contains two kinds of immiscible fluids,
liquid droplets and surrounding gas. The physical properties of the
liquid and gas are listed in Table 1 . The two
side-by-side spherical droplets are placed just above the box bottom.
They have the same diameter of D 0=60 lu and
identical impact velocity of V 0 (lu
ts-1). The normalized distance between the two
droplets is L =l /D 0, where lis the center-to-center distance between the two droplets. The box
bottom is set as a flat superhydrophobic surface with a contact angle of0=157°. No slip boundary conditions are imposed on the
bottom and top of the box, whereas periodic boundary conditions are
specified to the box side walls. Because the droplet diameter is far
less than the capillary length,24 the influence of
gravity can be neglected. We validate the model by Laplace’s law of a
stationary droplet21 as well as the contact time of a
single droplet impacting a superhydrophobic
surface.10,14 (see the supplementary material S1 for
model validation).
We start our simulations with a high Weber number of 35. By varying the
distance L , we discover that after two droplets impact a flat
superhydrophobic surface, they always bounce off the surface and exhibit
three different rebound regimes depending on L : a CCR regime in a
small L range, a PCR regime in a moderate L range, and a
NCR regime in a large L range, as shown in Figs.2-4 .
For the CCR regime, the two droplets completely coalesce before bouncing
off the surface. Figure 2(a) illustrates the
morphology evolution of the two droplets with a small L =1.17.
After impacting the surface, the two droplets start to spread, forming
rims in their perimeters. The rims of the two droplets intensely collide
each other at the initial stage of spreading because of the smallL . Subsequently, the colliding rims develops into a connecting
ridge. The strong inertial force gives the rims large enough kinetic
energy, which facilitates the fluid flowing in the ridge and makes the
ridge move upwards, as shown at τ =0.48 in Fig.2(b) . Additionally, such an earlier collision also triggers the
fluid flowing toward to the two ends of the ridge, as shown atτ =0.48 in Fig. 2(c) , resulting in the
generation of two liquid lobes. Accompanied with continuous spreading,
the ridge and liquid lobes rapidly grow until the two droplets reach the
maximum spreading diameters. Meanwhile, two puddles and peripheral
bulges form on the two sides of the ridge. After that, the two droplets
start to retract due to the superhydrophobicity of the surface.
According to the Young-Laplace equation, the puddles have concave
interfaces and thereby have low pressures in the liquid phase. As a
result, the liquid transports from the peripheral bulges and the ridge
center to the puddles, as shown at τ =0.71 and 1.11 inFigs. 2(b) and 2(c) , which makes the ridge
height decrease, the two puddles shrink, and the two liquid lobes
expand, and finally the two droplets coalesce into a cross-like
morphology. Whereafter the cross-like droplet retracts mainly along two
intersecting directions of the cross, i.e., the x and ydirections, and the retraction in the y direction is faster than
that in the x direction, forming an upright hump at the center of
the cross. The hump height continuously increases with the retraction in
the x and y directions. Therefore, the droplet gradually
elongates in the vertical direction, i.e., the z direction, and
its footprint gradually decreases with the retraction. As a consequence,
the elongated droplet evolves into a parachute-shaped and finally
bounces off the surface.
For the PCR regime, the two droplets partially coalesce before they
bounce off the surface.Figure 3(a) shows
the morphology evolution of the two droplets with a moderateL =1.67. With such a value of L , the effect of inertial
force becomes weaker when the rims of the two droplets touch each other,
and hence, the touched rims possess less kinetic energy, which leads to
the formation of a shorter ridge. Compared with the CCR regime, the
length of the ridge is always less than the spreading diameter and no
liquid lobes are generated on the two ends of the ridge, which
significantly alter the coalescence dynamics. Two puddles and peripheral
bulges are still generated in the spreading stage, which are similar to
those in the CCR regime. Likewise, in the retraction stage, the liquid
flows from the peripheral bulges and the ridge center into the puddles,
as shown at τ =0.95 in Figs. 3(b) and3(c) . Accompanied with the shrinking of the puddles, the two
ends of the ridge become flat and the height of the ridge decreases, and
finally the ridge develops into a wider liquid bridge. After that, the
further retraction makes the fluid flow upward, forming two uprising
humps on the two sides of the bridge. In this period, because of the
presence of the two humps, the retraction in the x direction is
significantly stronger than that in the y direction, as shown atτ =1.43 in Figs. 3(b) and 3(c) .
Subsequently, the retraction in the y direction accelerates, so
that the two humps continue to rise and get close to each other. The
uprising humps pull the liquid bridge connecting them and makes the
bridge depart from the surface, forming a two contact zones between the
droplet and surface, as shown at τ =2.22 in Figs.3(b) and 3(c) . When the two contact zones shrink to
two points, the droplet evolves into a teeth-shaped and bounces off the
surface.
For the NCR regime, the two droplets fail to coalesce before bouncing
off the surface. Figure 4(a) presents the morphology
evolution of the two droplets with a large L =2.00. After the two
droplet impact the surface, they begin to spreads separately. With the
large L , the rims touch each other in the later stage of
spreading, and inertial force becomes therefore negligibly small.
Because the touched rims have extremely low kinetic energy, a very short
liquid bridge forms between the two droplets. Soon after the touch, the
two droplets start to retract, so that the liquid bridge almost dose not
grow up and no ridge is formed. Similar to the CCR and ICR regimes, two
puddles generate on the two sides of the liquid bridge. In the
retraction stage, the liquid flows from the peripheral bulges into the
puddles, leading to the shrinking of the puddles, as shown atτ =0.95 in Figs. 4(b) and 4(c) , and
eventually two uprising humps form on the two sides of the bridge, as
shown at τ =1.35 in Figs . 4(b) and4(c) . It is worth noting that the liquid bridge is stretched in
the y direction during the whole retraction. As the two humps
continuously rise, the coalescing droplet is elongated into two columns
and the liquid bridge gradually becomes thin, as shown at τ =2.22
shown in Fig. 4(b) and 4(c) . Owing to the
Rayleigh instability,25 the liquid bridge ruptures,
and hence, the coalescing droplet is split into the two elongated
droplets. Finally, the two elongated droplets separately bounce off the
surface.
Figure 5 shows the normalized contact time,τ c, as a function of L at We =35.
The dotted line in Fig. 5 represents the contact time
of a single droplet impacting a superhydrophobic surface, whose value is
found to beτ c=2.62 in our simulations. It can be
seen that τ c is closely dependent on L .
The yellow zone denotes the CCR regime with small L . In this
regime, the intense collision of rims generates a ridge with two growing
lobes, making the two droplets coalesce completely before bouncing off
the surface. The coalescence causes a large viscous dissipation and
delays the rebounding, and thereby increasing the contact time. This
effect becomes weaker at a larger L , so that the contact time
declines with L in the CCR regime. The green zone denotes the PCR
regime with moderate L . In this regime, because of the short
ridge and no lobes forming on the two ends of the ridge, two uprising
humps are generated in the retraction stage, which make the coalescing
droplet form two contact zones with the surface, so that only partial
coalescence takes places before bouncing off the surface. Intriguingly,
the contact time in this regime remains almost at a constant value ofτ c=2.46, lower than that of a single droplet
impact. The red zone denotes the NCR regime with large L . In this
regime, a liquid bridge forms between the two droplets, instead of the
formation of a ridge. The two droplet complete spreading and retraction
nearly separately. However, the elongated liquid bridge links the two
droplets and hinders their retraction, and hence, the contact time is
slightly larger than that of a single droplet impact.
We further study the rebound dynamics of two droplets impacting a
superhydrophobic surface at various Weber numbers. The rebound regimes
and contact time are illustrated in Fig. 6 . At two
high We =20 and 35, three rebound regimes all take place, with the
CCR regime in a small L range, the PCR regime in a moderateL range, and the NCR regime in a large L range. At a
moderate We =10, the CCR and PCR regimes take place, whereas the
NCR regime disappears. At two low We =1 and 5, only the CCR regime
takes place. In the PCR regime, the contact time is smaller than or
equal to that of a single droplet impact, regardless of the Weber
number. In the NCR and CCR regimes, the contact time is larger than that
of a single droplet impact. In particular, the contact time in the CCR
regime is significantly larger than that of a single droplet impact and
strongly depends on both We and L . As shown in the yellow
zone in Fig. 6 , at the moderate and high We =10,
20, and 35, the contact time declines with L , whereas an opposite
tendency occurs at the low We =1 and 5. This difference arises
from different coalescence and rebound dynamics. As shown inFig. 2 , at the moderate and high We , when two
droplets come into contact with each other, rims have been formed in
their periphery, so that intense collision of the rims dominates the
coalescence and rebound dynamics. However, at the low We , when
two droplets touch each other, they remain nearly spherical; therefore,
a liquid bridge produces between the two droplets, and the bridge growth
and impact dominates the coalescence and rebound dynamics, as shown inFig. S3 (see the supplementary material S2), which
resembles coalescence-induced self-jumping of droplets on
superhydrophobic surfaces.19,27 Thus, a small Lmakes the liquid bridge impact the surface earlier, which promotes the
droplet rebound and thereby reducing the contact time.
According to the above results, we speculate that the contact time in
the CCR regime for two droplets impact should be longer when the surface
wettability changes from superhydrophobic to hydrophobic. The rebound
may fail on hydrophobic surfaces with high Weber numbers and small
droplet distances. We add an extra simulation withθ 0=122°, We =35, and L =1. Because of
the intense collision of rims, the formed ridge is extremely strongly
elongated in the x direction, so that the coalescing droplet
undergoes several times oscillation and eventually adhere to the
surface. (see the supplementary material S3)
In conclusion, using MRT-LBM simulations, we investigate the rebound
dynamics of two equally-sized droplets simultaneously impacting a flat
superhydrophobic surface. Our emphasis is placed on revealing how the
Weber number and droplet distance influence the contact time. We
discover three rebound regimes at high Weber numbers depending on values
of L : a CCR regime in a small L range, a PCR regime in a
moderate L range, and a NCR regime in a large L range.
When Weber numbers fall into the moderate or low range, the NCR regime
or both the NCR and the PCR regime disappear. We demonstrate that the
contact time is longer than that of a single droplet impact in the CCR
and the NCR regime; however, intriguingly, it is less than that of a
single droplet impact in the PCR regime. We discuss the impact dynamics
in detail to understand the mechanism behind the three rebound regimes.
We believe that our simulations provide useful information for practical
applications such as self-cleaning, anti-corrosion, anti-icing, and so
forth.