Animation of Water Droplets on a Hydrophobic
The University of Tokyo
Tokyo University of Technology
The University of Tokyo
Animation of water drops on a windshield is used as a special effect in advanced driving games and simulators.
Existing water droplet animation methods trace the trajectories of the droplets on the glass taking into account
the hydrophilic or water-attracting nature of the glass material. Meanwhile, in the automobile industry, usage of
hydrophobic glass windshields has recently been a common solution for the drivers’ clear vision in addition to
cleaning the water with wipers. Water drops on a hydrophobic windshield behave differently from those on a
hydrophilic one. This paper proposes a real-time animation method for water droplets on a windshield taking
account of hydrophobicity. Our method assumes each relatively large droplet as a mass point and simulates its
movement using contact angle hysteresis accounting for dynamic hydrophobicity as well as other external forces
such as gravity and air resistance. All of a huge number of still, tiny droplets are treated together in a normal map
applied to the windshield. We also visualize the Lotus effect, a cleaning action by the moving droplets. Based on
the proposed simulation scheme, this paper demonstrates the motion of the virtual water droplets on the
windshield of a running vehicle model.
Water droplets, hydrophobicity, windshield, driving simulator, contact angle hysteresis
Water flow on the window or windshield surfaces
are commonly used as a rainy scene description in
film works and other types of motion pictures. More
recently, computer generated animations of water
flow on the windshields are realized for advanced
video games and driving simulators. Since the glass
material has hydrophilic or water-attracting nature,
water droplets move along irregular trajectories
seeking for water-attracting places of the surface, as
we often find on the windows in a rainy day. Most
of the existing water droplet animation methods
simulated these winding trajectories of the droplets.
In real driving situations, those water trajectories
or water-film on the windshields due to the
hydrophilicity seriously affect the visibility through
the glass. To clear the water, mechanical wipers
have been used since the beginning of the
automobile history. In addition, as auxiliary
measures, coating the windshield with water
repellent material became a solution a few decades
ago. In the year 2000, the first water-repellent
finished windshield became commercially available.
Nowadays such hydrophobic windshield products
are widely used in the automobile market.
A large amount of research literature on the
behaviour of water on hydrophobic surfaces is
published in chemical and mechanical engineering
fields. To the authors’ knowledge, however, little
work has been done on real-time simulation of
water droplets sliding across hydrophobic
windshields. In this paper, we address this problem
and propose a solution consisting of several
practical simulation models for use in games and
Water attracting or repelling feature of surface
material should be quantified differently in two
situations, static and dynamic. The static repellency
has been investigated for a long time and the
fundamentals have been established. For water
droplet animation, knowledge on the dynamic
repellency is more important, which is true in
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engineering analysis of water-shedding phenomena
on the windshield. While the dynamic water
repellency includes a number of unexplainable
phenomena, there are a couple of major factors and
indicators characterizing the dynamic repellency.
Those include contact angle hysteresis, falling angle,
falling velocity, and falling acceleration.
The relationship between the contact angle
hysteresis and the slope angle has long been
investigated. In case of an ideal water droplet shape,
the contact angle hysteresis is known to be in
proportion to the falling angle.
The falling velocity and acceleration vary by the
surface material even when the slope angle remains
constant. Although the standard methods for
evaluating and measuring the falling
velocity/acceleration were not established until
recently, it is known that the behaviour of a falling
water droplet on the hydrophobic surface is
explainable in terms of rolling and sliding.
In this paper, we take the knowledge on the
dynamic repellency into account and propose a real-
time animation method for water droplets on the
hydrophobic windshield. As the water-repellent
coated windshields become standard in the
automobile market, our contribution is to provide
video game and simulator developers with a means
of reproducing realistic and harmonious motions of
the water droplet cluster traveling across the
This paper is organized as follows. In the next
section we introduce related work on both
engineering analyses and animation techniques for
water droplets. Then our proposed method is
explained in a theoretical point of view in Section 3,
followed by more detailed descriptions on the
implementation and results in Section 4. Finally we
give conclusions and future work in Section 5.
2. RELATED WORK
In the computer graphics field, several methods
have been introduced for animating water droplets.
Kaneda et al. [Kan93a] [Kan96a] proposed methods
to describe the movement of the droplets by
defining each droplet as a particle and move it with
particle dynamics. Since the droplets travel seeking
for water-attracting places, their trajectories on the
glass surface form complex shapes. They also
simulated these motions by a random walk method
using random numbers [Kan99a]. Recently their
method was implemented as a real-time simulator
with a GPU computing technique [Tat06a].
Fournier et al. [Fou98a] depicted the trajectories of
droplets using the mass spring model. None of the
above methods took into account the
hydrophobicity of the inclined surface since they
assume hydrophilicity. Also, they do not
incorporate air resistance against the water drops or
rolling resistance of the drops.
Several researchers have developed fluid
dynamics based methods for the water droplet
simulation. Wang et al. [Wan05a] took into account
surface tension, contact angle, and contact angle
hysteresis. The surface tension is more dominant in
a water droplet than in regular large-scale fluid
forms. Thürey et al. [Thu10a] introduced the mean
curvature flow, which is known as a motion
equation for surface boundaries, and evaluated the
phenomena caused by the surface tension more
appropriately than Wang et al.
Zhang et al. [Zha11a] developed a faster
computation method for droplets using the mean
curvature flow without other fluid simulations.
They ignored the internal fluid flow of the droplets
but used the surface tension and other external
forces to give deformation, collision and division to
each droplet represented as a polygon mesh. They
achieved 10-50 fps in the experiment with 10K-50K
polygon mesh. However, due to the implicit method
for the mean curvature flow computation, the
stability of their solution depends highly on the
mesh quality and the time step, and the performance
optimization is limited.
In order to tackle the problem of the droplet
motion on the hydrophobic surfaces, we need to
understand dynamic repellency. The structure or the
behaviour of the surface molecules are considered
to be a source of the dynamic repellency. To figure
out the behaviour, Hirvi et al. [Hir08a] simulated a
droplet consisting of thousands of water molecules
using a molecular dynamics calculation technique.
Korlie [Kor93a] proposed a cluster model of quasi-
molecular particles on a horizontal plane and
introduced its dynamical equations which lead to
the value of the contact angle of the cluster.
Analyses of real water droplets have been done
by several research groups. For example, Sakai et al.
[Sak06a] measured the velocity and the acceleration
of a droplet sliding across water-repellent surfaces.
Droplets are known to run down either rolling or
slipping on the incline depending on the degree of
hydrophobicity [Ric99a] [Suz09a]. Hashimoto et al.
[Has08a] measured the relationship between the
volume and the velocity of a windswept droplet.
We address the problem of dynamic water-
repellency taking the contact angle hysteresis into
account. In addition, we use the knowledge of the
real water drop analyses to verify and compensate
our results. We avoided using the fluid dynamics
simulation, the mean curvature flow, or any type of
molecular forces since they are not suitable for real-
time visualization. Due to the computing load and
the time step limitations, those methods cannot
handle sufficient number of droplets on a car
In our method, each droplet is represented as a
mass point or a particle. Thus, we are able to
incorporate additional forces into the real-time
simulation loop; air resistance against the water
droplets and viscous dissipation which acts as a
rolling resistance of each drop. Although these
forces are crucial factors for the fast movement of
water drops, they have not been fulfilled in the
previous methods [Wan05a] [Thu10a] [Zha11a].
Particle dynamics are common in the real-time
simulation field. They are widely adopted in games
and interactive applications. Real-time physics
engines in the market are equipped with features of
particle dynamics and rigid body dynamics
including collision detections as fundamental
functions. We implemented our method on top of a
game engine ‘Unity’ and added unique behaviours
of water droplets running slowly or quickly, or
staying on the hydrophobic surfaces.
3. A PRACTICAL MODEL FOR
WATER DROPLETS ON
3.1 Water Droplet Geometry
When a droplet is on a solid surface, the contact
angle is defined as the angle between the solid
surface and the droplet surface. The contact angle is
determined by the Young equation, which describes
the balance of three surface tensions, as shown in
where, is the contact angle,
is the surface
tension of the water droplet,
is the surface
tension of the solid,
is the boundary tension
between the water and the solid (Figure 1).
Figure 1. Contact angle and tensions of a water
When the radius of the droplet on hydrophobic
surfaces is less than the radius of capillary (2.8mm),
the surface tensions are the dominant factors of the
water drop shape. Thus the droplet forms a near
spherical geometry. Meanwhile, the contact angle of
the glass becomes 90-100 when it is coated with
commercially available repellent material.
Based on the above two observations, we assume
that each rain droplet is rendered as a hemisphere.
In practice, the geometric shape is basically a disc-
like plane and the normal vectors for refraction are
controlled to make it look hemisphere. Details are
described in Section 4.3.
3.2 Contact Angle Hysteresis
When a thin pipe is inserted into water, the water
level in the pipe is raised by the capillary action.
This is caused by a force called the capillary force
which operates along the triple boundary line
among the water, the solid and the air. The capillary
force is determined by the Young-Laplace equation.
Figure 2. Advancing and receding contact angles of
a water droplet.
With regard to a droplet which lies on a solid
plane, the capillary forces along the circular triple
boundary cancel each other out if the contact angle
is constant along the circle. When some external
forces are put on the droplet and its shape is
deformed, the contact angles vary while the droplet
stands still until the contact angle variance reaches
at a certain value.
The contact angle hysteresis is defined as the
difference between the advancing and receding
contact angles (
, respectively). These two
angles are defined as the largest and the smallest
contact angles, respectively, at the moment that the
water droplet starts moving on the solid plane by
the sufficient external force. The slope angle at this
moment is called the falling angle. Figure 2
illustrates the advancing and receding contact
angles for an incline.
While the droplet is moving on the plane, a drag
operates on the droplet toward the reversed
direction against the proceeding direction. The
amount of drag is related to the contact angle
hysteresis. Assuming that the shape of the triple
Drag due to the
boundary is a circle, the drag
with the following equation [Car95a]
where, represents the radius of the water droplet.
are the receding and the advancing
contact angles, respectively.
3.3 Wind Drag
Automobile windshields meet with air resistance, or
wind drag, according to the velocity of the running
vehicle. The wind drag is defined as follows:
where, is the density of the air,
coefficient of resistance, is the projected size of
the droplet, and is the velocity relative to the air.
In Equation (3), the droplet is assumed to be
floating in the air. Since all droplets in our model
are placed on a solid windshield, the equation needs
to be modified. We assume that the wind is
weakened at places very close to the solid plane. It
is known that in such near-boundary layer, the wind
velocity changes in a complicated manner.
We employed a simplest compensation to
decrease the velocity in the near-boundary layer
using an exponential law as shown in the following
where, is the wind velocity out of the boundary
layer (relative to the solid plane), is the height of
the droplet, is a parameter representing the
thickness of the boundary layer, and
compensated wind velocity for the droplet.
3.4 Viscous Dissipation
When a droplet is moving or rolling, another drag is
caused by some in-bulk friction called viscous
dissipation [Bic05a]. The drag is in proportion to
the velocity of the droplet and represented as
where, is the degree of viscosity of the water, is
the radius of the droplet, is the velocity of the
is a factor dependent on the contact
3.5 Wind Speed and the Droplet
In the surface finishing engineering discipline,
Hashimoto et al. [Has08a] introduced an experiment
to measure the acceleration of various volumes of
water droplets placed on an angled hydrophobic
plane in a wind tunnel. Figure 3 quotes from the
literature and shows the result of the measured
descending or ascending acceleration of the droplets.
The contact angle, the slope angle, and the falling
angle are 105 , 35 and 10, respectively.
In the range where the wind velocity is relatively
low, moderate but more falling accelerations are
observed as the droplet size becomes greater. When
the wind velocity is raised beyond a certain value
(7m/s in Figure 3), the droplet stays still within
some range of wind velocities. When the velocity is
further raised beyond a higher value (11m/s), rapid
ascending accelerations are observed, which are
greater as the droplet becomes larger.
On the other hand, we simulated the sliding
accelerations of a droplet taking the following five
forces into account (Figure 4).
Wind drag (horizontal)
Perpendicular force (normal to windshield)
Figure 3. A measured relationship between the
wind velocity and the acceleration of droplets,
using a varying droplet size as a parameter
cerpt from [Has08a]).
Figure 4. External forces added to a droplet and
the resultant acceleration. In this example, the
gravity is more dominant than the wind drag and
thus the droplet slides down.
Viscous dissipation drag (tangential to
Contact angle hysteresis drag (tangential to
The wind drag
has been described in
Section 3.3. The contact angle hysteresis drag
behaves as a resistance force parallel to the
windshield, in the same way as the perpendicular
force normal to the windshield. The force
represented in Equation (2) defines the maximum
limit of the hysteresis drag.
In our implementation, the maximum limit is
specified by a dimensionless coefficient
Since the relationship between
the wind velocity and the contact angles is hard to
simulate, we approximate the value as a function
of the wind velocity . When the velocity is small,
we force the value to keep a minimum constant
which is typically 0.5.
where, is a constant parameter which controls the
saturation rate of
. When the wind is extremely
strong, the contact angles are assumed to be also as
, and thus
This is well accounted for by Equation (6).
Figure 5 shows a simulated result of the
accelerations for the varying droplet sizes. The
range of wind velocities in which the droplet stays
still is reproduced, and the range is very similar to
the measured result in Figure 3.
3.6 Collision between Droplets
The surface tension of the water droplet causes a
pressure difference in the droplet. This is known as
the Laplace pressure and is greater as the droplet
radius is smaller. Therefore, when two water
droplets of different sizes collide with each other,
the small droplet gets absorbed by the larger one.
We implemented this process and it is invoked on
droplet collision detection.
3.7 Distribution of Raindrop Radii and
the Lotus Effect
Lotus effect is a phenomenon which occurs when a
water droplet moves across a hydrophobic surface.
Lots of very small droplets and contamination
spread on the surface are removed by the moving
droplet along the trajectory. The same phenomenon
is observed on a windshield as demonstrated in the
snapshot of Figure 6.
Figure 7, an excerpt from [Fur02a], is a rain
droplet radius distribution under 1mm/h rainfall.
The graph is with the raindrop diameters as the
horizontal axis and the number of raindrops for each
diameter as the vertical logarithmic axis. The line
indicated as ‘MP’ is an exponential distribution
model called the Marshall-Palmer distribution
[Mar48]. Each graph legend is the place name of the
observing site. Some legends contain observing
periods in months.
7. Distribution of the number of raindrops for
each diameter (drop size
distribution). Each graph
legend indicates the name of the observ
6. Droplet trajectories caused by the Lotus
effect (image captured from a live
-action movie of a
Figure 5. Simulation results of the droplet
According to the model, the smaller the raindrop
diameter is, the greater the number of raindrops
becomes. Especially, tiny raindrops of below 1mm
are contained with an exponentially large numbers.
Therefore, it is impractical to simulate the motion of
every droplet. Fortunately, those tiny raindrops do
not move at all with our simulation model as shown
in Figure 5. Thus we apply a single large normal
map onto the windshield. The map contains the
normal vectors which represents all the small
droplets standing still on the windshield.
4. IMPLEMENTATION AND
This section describes implementation of our
method proposed in the previous section and
demonstrates some results.
4.1 Implementation Overview
We implemented the system on top of Unity, a
popular game engine. Although our method regards
each water droplet as a particle, we implemented
each droplet as a small rigid body which does not
rotate. Regarding the rigid body physics engine, we
used NVIDIA PHYSX embedded in the Unity
The flow of the whole process is outlined as
Updates of large droplet shapes
Update of windshield alpha map (Lotus
effect over small droplets)
4.2 Physics Simulation of Droplets
In each time step of the simulation, our system
calculates the external forces imposing on the water
droplets as illustrated in Figure 4.
Regarding the gravity, we added some random
noise to the force component parallel to the
windshield in order to realize natural motions of the
droplets caused by some assumed fluctuation of the
The implementation of viscous dissipation
(Section 3.4) is a heuristic matter since the factor
in Equation (5) is not determined. We used a
in the equation. The
important point is that the viscous dissipation drag
is in proportion to the droplet velocity. The
above constant value can be used to control the
maximum droplet speed.
While the droplets are moved by the external
forces, we obtain each collision point with its u-v
coordinates and the normal vectors of the colliders
from the collision detector of the physics engine.
For a droplet being regarded as to be on the
windshield, the windshield point corresponding to
the droplet is calculated and the refraction map
image for the Lotus effect is updated.
In case that a droplet collides with another
droplet, the Laplace pressure effect is applied. The
system compares the masses of the two droplets. If
the difference is greater than the pre-defined
threshold, these two will fuse together into one
4.3 Rendering Large, Movable
Each large water droplet (with over 1mm diameter)
is rendered as a disc-shape polygon mesh when it is
staying still on the windshield. The normal vectors
on the disc surface are controlled so that the
refracted environment appears to be mapped on a
While the droplet is moving across the
windshield, its shape is deformed to be longer along
the moving direction. The normal vectors are
controlled so that the lengthened transparent droplet
looks like a drug capsule sectioned by a screen-
parallel plane. The deformation is controlled so that
the assumed volume of the droplet is preserved.
Using its normal vectors, the pixel shader calculates
the refraction directions and maps the background
texture image as the environment. Figure 8 is a
close-up rendering image of a pseudo-hemisphere
water droplet and a deformed pseudo-hemisphere.
Those large droplets are generated with various
Figure 8. Droplets rendered as a pseudo-
hemisphere (left) and a deformed pseudo-
sizes according to the Marshall-Palmar distribution
shown in Figure 7. The number of large droplets
generated per frame is set to be five typically. They
are accumulated but eventually moved away out of
the windshield or collided and fused with others. As
a result, a couple of hundred to one thousand large
droplets reside in the steady-state situation.
4.4 Rendering Small and Still Droplets
Small droplets (with less than 1mm diameter) are
represented as perturbation in a normal map image
for the windshield, as described in Section 3.7. The
diameters of the generated small droplets vary also
according to the Marshall-Palmar distribution. The
number of small droplets in our implementation
amounts to approximately 800K
The outside scene image is refracted according to
the normal map. The trajectories of large droplets
(pseudo-hemispheres) are stored as an image
component which is used to suppress the normal
map. They are composed in the shader program and
the Lotus effect on the windshield surface is
rendered (Figure 9).
All results referred to in this section are captured
snapshots of real-time animations rendered from the
driver’s point of view toward the automobile
proceeding direction viewing the outside through
the windshield. The source of the outside image is a
motion picture shot with a video camera placed
between the two front seats of a running car when
no rain is falling. The pre-recorded image is
mapped as a video texture onto a billboard model
placed in front of the windshield model.
Figures 10 and 11 are the examples with a small
wind velocity. In Figure 10, a relatively large
contact angle hysteresis is specified and thus the
adherence is strong that the droplets do not move at
all. In Figure 11, the adherence is smaller and the
droplets move along the windshield curve.
Figure 12 is a result with stronger wind and the
large droplets climb straight up the windshield.
Since the adherence is strong and the boundary
layer is set to be thick, the small droplets are made
The frame rates for Figures 10, 11 and 12 are
134-153fps, 80-100fps, and 70-100fps, respectively.
The scene contains a windshield, large droplets and
the video texture billboard shapes, which total
approximately 17K vertices.
Figure 10. A result with low wind velocity
(11.3m/s) and a large contact angle hysteresis
Figure 9. The Lotus effect. Small and still
droplets are rendered as a normal map on the
windshield. Large and moving droplets are
rendered as pseudo-hemispheres.
Figure 11: A result with low wind velocity
(11.3m/s) and a small contact angle hysteresis
Figure 12. A result with high wind velocity
(15m/s) and a large contact angle hysteresis with
4.6 Rendering Conditions
For the rendering results, we used an Intel Core2
Extreme X9600 (3GHz), NVIDIA GeForce
GTX480 Graphics and 8GB main memory. The
horizontal field of view was 45 and the distance
between the viewpoint and the windshield was
approximately 0.5m. The horizontal curvature
radius of the windshield geometry was 5m constant
and the vertical curvature was 0 (flat). The slope of
the windshield was inclined at a 45 angle.
5. CONCLUSION AND FUTURE
We proposed a real-time animation method which
reproduces the behaviour of a group of water
droplets on a hydrophobic windshield. We modeled
each of large droplets as a mass point and took into
account dynamic hydrophobicity by employing the
contact angle hysteresis which causes appropriate
adherence for each droplet.
We also compared the accelerations of simulated
droplets with those of measured real water droplets
from literature of surface finishing engineering
analysis. By introducing a near boundary layer
where the wind is reasonably weakened, our result
matched the measured one and reproduced realistic
behaviours of the droplets.
For a huge number of tiny water droplets which
do not move in our model, we introduced a normal
map applied to the windshield. By using the image-
based droplets, the Lotus effect was effectively
For practical number of large droplets, our
method runs in real-time and can be easily adopted
as an effect for video games and vehicle simulators.
The performance is degraded when the large
droplets are not blown off and accumulated on the
windshield because the motion simulation is done
on a per large droplet basis.
Future work includes the performance
improvement for larger number of droplets, more
realistic deformation of the droplets, and handling
of uneven wind velocity distributions.
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