placement=top, position=8cm,2cm 1  This paper is under review for publication
at the IEEE International Conference on Robotics and Automation (ICRA),
London, 2023. ©IEEE gray 0 0.9

# WorldGen: A Large Scale Generative Simulator

Chahat Deep Singh1, Riya Kumari1, Cornelia Fermüller1, Nitin J. Sanket2,
Yiannis Aloimonos1 This work was supported in parts by Northrop Grumman
(N00014-17-1-2622, SMA 1540917), Army Research Laboratory under a cooperative
agreement with the University of Maryland (ArtIAMAS), Army Cooperative
Agreement (W911NF2120076) and NSF Grant (OISE 2020624). Chahat Deep Singh was
funded by Ann G. Wylie fellowship at the University of Maryland, College
Park.1Perception and Robotics Group, University of Maryland Institute for
Advanced Computer Studies, University of Maryland, College Park, MD 20742,
USA. Emails: {chahat, rkumari, fer<EMAIL_ADDRESS>Engineering,
Worcester Polytechnic Institute, MA 01609, USA. Email<EMAIL_ADDRESS>
###### Abstract

In the era of deep learning, data is the critical determining factor in the
performance of neural network models. Generating large datasets suffers from
various difficulties such as scalability, cost efficiency and photorealism. To
avoid expensive and strenuous dataset collection and annotations, researchers
have inclined towards computer-generated datasets. Although, a lack of
photorealism and a limited amount of computer-aided data, has bounded the
accuracy of network predictions.

To this end, we present WorldGen – an open source framework to autonomously
generate countless structured and unstructured 3D photorealistic scenes such
as city view, object collection, and object fragmentation along with its rich
ground truth annotation data. WorldGen being a generative model gives the user
full access and control to features such as texture, object structure, motion,
camera and lens properties for better generalizability by diminishing the data
bias in the network. We demonstrate the effectiveness of WorldGen by
presenting an evaluation on deep optical flow. We hope such a tool can open
doors for future research in a myriad of domains related to robotics and
computer vision by reducing manual labor and the cost of acquiring rich and
high-quality data. The project page is available at
http://prg.cs.umd.edu/WorldGen

##  I Introduction

High-quality image data is pivotal for accurate deep learning models. For
certain tasks, it is even more important than neural architectures and
hyperparameters. Data collection, cleaning, and annotation have become a
nightmare lately generally costing millions of dollars. This is majorly due to
challenges in data diversity, image quality, pixel-perfect manual annotations,
as well as licensing [1], and security concerns. This is exacerbated by the
fact that writing instructions for manual annotation are not a trivial task as
it is not always possible to explain concisely the requirements. To this end,
we propose a universal framework to generate countless 3D scenes in different
environments along with annotated images for various autonomous tasks for
cars, drones, and indoor robots. This would enable deep learning frameworks to
gather photorealistic data much more efficiently in terms of cost, speed,
labor and variety.

On the other side of the spectrum, synthetic data and simulators today are
commonly used but are very limited as they suffer from scalability and/or
photorealism. Researchers and practitioners have extensively used these data
as a benchmark to evaluate the depth, optical flow, etc. due to their perfect
ground truth annotation. Most of these simulators do not support a realistic
camera model or ray-tracing ability to render high-quality photorealistic
images. Although, few approaches have proven to narrow down the generalization
gap using synthetic data [2, 3] for geometric-based quantities such as depth
and optical flow as opposed to texture-centric approaches like semantic
segmentation.

Figure 1: Generative ability of WorldGen: (a) Comparison between Google Street
View (left) and the same street in WorldGen (right), (b) Comparison of Google
Maps satellite image vs. WorldGen top view, (c) Collection of 3D objects in
motion, (d) Object fragmentation,(e) Annotation from left to right: depth,
optical flow, surface normals, stereo anaglyph, image segmentation, event
frame. All the images in this paper are best viewed in color on a computer
screen at 200% zoom. TABLE I: Comparison of the different simulation
environments.

Name | Rendering | GI | Physics | Scaling
---|---|---|---|---
UnrealCV [4] | UE4 | $\times$ | UE4 | $\times$
iGibson [5] | PyRender | $\times$ | PyBullet | ✓
Omnidata [6] | Blender | ✓ | – | $\times$
Blenderproc [7] | Blender | ✓ | Bullet | $\times$
AirSim [8] | UE4 | $\times$ | AirSim | $\times$
CARLA [9] | UE4 | $\times$ | UE4 | $\times$
Kubric [10] | Blender | ✓ | PyBullet | ✓
WorldGen (Ours) | Blender | ✓ | Bullet | ✓

∗ Only Blender rendering engine supports ray-tracing with OptiX denoiser for
photorealistic images; others use rasterization. GI: Full Global Illumination
Support; Physics Engine Used; Scaling: Scalable to generate large datasets.

To reduce this sim-to-real gap even further, a large amount of synthetic scene
generation with wide variability is required. Creating a large number of
synthetic scenes suffers from either manually creating high-quality 3D assets
or bringing together these assets in a structured way to form a scene. Thus, a
scalable simulation environment becomes a necessity for neural networks to
improve the quality of the prediction models. [10] and [5] I demonstrated the
power of scalable data by providing an in-depth analysis of the prediction
using their data for training. In this work, we present a scalable framework
to automatically generate structured and unstructured environments for
perception and robotics applications without any need for manual effort. Fig.
1 demonstrates the ability of WorldGen to autonomously generate structured
cities, object datasets and object fragments without any requirement for
manual efforts.

###  I-A Related Work

In the last five years, there has been a boom in self-driving car simulators
for perception, planning and control [11, 8, 9]. One of the major drawbacks of
such simulators is the limitation in scalability for data generation. Such
simulators are shipped with a limited number of scenes (or towns). This is
because it requires intensive manual labor to generate these 3D environments
for better scalability and generalizability. Scalability in these simulators
is present for traffic simulations but not for perception tasks. For instance,
CARLA [9] library includes only 40 different building models. Our method
relies on 3D structures of buildings from satellite semantics and 3D maps to
build countless numbers of building models.

Learning geometric annotations such as an optical flow that relies
substantially on motion parameters (rather than textures) have been studied
extensively in the past decade. Unlike monocular depth estimation and instance
segmentation, the prediction of such geometric quantities through
convolutional neural networks generalizes well across different domains. For
such predictions, synthetic datasets have widely been used [3, 2, 12, 13] with
a limited number of objects and textures. Recent advancements in scalable
scene generation environments like Kubric [10] have shown improvements in
optical flow prediction due to more variability in data generation.

Another common problem in vision and robotics is segmenting and tracking
independently moving objects. It is a key process to understand the dynamics
of the scene structure for navigation tasks. Applications such as dodging any
malicious objects on drones [14, 15] and tracking objects using RGB and event
cameras in motion [16, 17] have been well examined. Also, from the virtue of
archaeology, it is an important task to reconstruct a broken pot from
unordered and mix fragment collection [18]. In this paper, we provide a
framework to generate countless 3D scenes with texture, structure and lighting
variability for the aforementioned applications. We summarize our key
contributions next.

###  I-B Key Contributions

  * •

We introduce WorldGen, a generative simulator for creating different
environments commonly encountered in real-world robotics and computer vision
tasks

  * •

WorldGen is scalable and can generate an infinite amount of photorealistic
data with variations to not only object placements but also object shape,
texture, lighting, camera and motion properties.

  * •

WorldGen supports realistic camera distortions such as barrel distortion,
chromatic aberration, and camera aperture among other computational
photographic properties.

  * •

We demonstrate WorldGen’s utility in learning more accurate optical flow due
to the virtue of better data generation ability.

The paper is organized as follows: Sec. II presents how our WorldGen
generative simulator is constructed, its design principles and details. Next,
Sec. III presents how WorldGen can be useful for a myriad of robotics and
computer vision applications. Finally, we conclude the paper in Sec. IV with
parting thoughts for future work.

Figure 2: An overview of WorldGen Framework: (a) Assets: Loads the assets such
as maps, objects, materials etc. into WorldGen environment, (b) Structural
Modification and Animation: Modifying the texture maps and applying physics
and motion models on different objects in the scene, (c) Rendering: Generates
rich ground truth data with the desired metadata (time, frame number, camera
intrinsic and extrinsic properties).

##  II WorldGen Generative Simulator

WorldGen is a high-level open-source python library to generate an unlimited
amount of synthetic data. It serves as an ease-of-use platform to generate
visual data for simulating self-driving cars, autonomous drones, object
segmentation, active vision, motion segmentation, tracking, computational
photography and more. Our key contribution is an API to build generative
environments and streamline the process of generating synthetic data by
lowering the usage difficulty barrier for researchers and practitioners alike.
WorldGen is built around Blender${}^{\text{TM}}$, a free and open-source 3D
creation suite, to generate synthetic data like city maps, a collection of
moving objects, and object fragmentation. Currently, our framework utilizes
Bullet physics engine mainly for object collisions, gravity, friction, and
other force fields. Fig. 2 shows an overview of WorldGen. We build WorldGen
around the central concept of scalability and speed. We discuss different
parts and a number of details that are used to build WorldGen next.

###  II-A Environment

Our framework is structured in three different stages: (a) Loader: loading
assets like objects, materials and textures, lights and/or map information to
generate a structured 3D environment, (b) Structural Modification and
Animation: this generates structurally different objects by varying texture
maps (explained in Sec. II-B), then applies physics and motion models to
objects, camera and lights and (c) Rendering: outputs rendered frames and its
annotation along with desired metadata (like time, frame number, camera
intrinsic, and extrinsic).

Rendering: Our framework utilizes Blender’s Cycles – a ray-trace based
production render engine. To avoid slow renders, the number of samples (or
path-traced for each pixel) is usually set to a lower value that produces
graininess in the output renders. We utilize NVIDIA OptiXTM [19], a recurrent
denoising autoencoder to reduce the graininess (noise) without a need for more
rendering iterations. Furthermore, our framework also utilizes both full
Global Illumination (GI) for photo-realistic renders and fast GI approximation
for faster renders.

###  II-B Texture Mapping

WorldGen utilizes UV mapping [20] which is a well-known process in computer
graphics that projects a 2D image onto a 3D model’s surface. Along with RGB
images, open-source textures are often composed of displacement and normal
maps in order to fake the lighting of bumps in the 3D model. These mapping
techniques are used to re-detail a simplified or low-poly mesh without
increasing the number of vertices, thus reducing the rendering time. By
varying the strength of the displacement and normal maps, a structurally
different variant of the same object can be rendered without a need to modify
the 3D object (See Fig. 3). Thus, we modify these maps with additive white
Gaussian noise before applying them to the 3D objects. Such variational
texture mapping can contribute to the generation of thousands of similar
object renders from a small set of object meshes. This can be used to generate
large amounts of datasets to predict geometric quantities (or annotations)
like optical flow for better generalizability across different domains. For
more details, read Sec. III-A.

Figure 3: Mapping textures to a round table. Top row: Rendered Output, Bottom
row: Sample textures projected on a sphere. (a) Barebone 3D model, (b)-(d)
Different Textures applied on (a). Note: Variational mapping models change the
structure of the 3D objects in different renders (notice the legs on the
chair). Here, the Gaussian noise in $(d)>(c)>(b)$.

###  II-C Generative Models

At the time of writing, WorldGen supports three different generative models:
(1) City Maps (2) Object Pile, and (3) Object Fragmentation.

####  II-C1 City Models

Figure 4: (a) OpenStreetView, (b) Depth Map, (c) 3D Model View Generated by
WorldGen and (d) Final Rendered View Figure 5: City environment in different
weather and time of the day: (a) Day, (b) Night with rain, (c) Dawn and (d)
Night without rain, (e) panoramic view of the city and (f) demonstrates the
generative ability of WorldGen by changing the textures of the entire scene
while keeping the same structure.

Our framework utilizes OpenStreetMaps (OSM), a crowd-sourced project that
contains semantic labels and 3D terrain maps from the satellite perspective.
Using latitude and longitude as the input, semantic and 3D terrain information
is imported into our framework using [21]. Using semantic information like
buildings, water bodies, forests, vegetation, roads, highways, pedestrian
pathways and railways, a set of relevant assets are imported from open source
libraries (see Sec. II-F) along with its appropriate material. Furthermore, we
can deploy different roof structures such as flat or gable on various
buildings. the semantic data from OSM also contains building roof information,
allowing our framework to deploy a realistic roof structures (flat or gable)
on the respective buildings. Assets like radio antennas, air-conditioning
vents, and chimneys are randomly distributed over the roof surface. For other
generative solid objects like roads, 3D meshes are created in the environment
using the vertices from the semantic data. Furthermore, lights are imported to
the scene with GI support for different types of weather and daytime scenarios
to get a photo-realistic visualization (Fig. 5). Using basic morphological and
vector operations, the road intersections are estimated to deploy traffic
lights and stop signs near these road crossings. Other commonly available
street objects like trees, benches, and street lights are structurally
distributed near the pedestrian pathways. Fig. 1 shows the final rendered
output of our city environment, complete with realistic distortions present in
camera images such as chromatic aberration.

####  II-C2 Object Pile

WorldGen’s ability to generate a collection of moving objects in space is
closest in scope to Kubric [10] and BlenderProc [22]. The key advantage of our
framework over Kubric is the flexibility to generate large-scale datasets due
to variational texture mapping that can generate different structural
variations of the same 3D models. Fig. 1b shows the output of an object pile
environment. WorldGen supports a variety of control in physics such as
collision shapes using convex hulls, 3D meshes, boxes, cylinders, etc. with
varying collision margins, friction, and coefficient of restitution for
allowing more control in the data generation process. Furthermore, apart from
the standard gravitational force, our framework deploys other force fields
such as wind and drag to influence the object’s motion which reflects reality
much more and would be useful in learning perception modules for realistic
scenarios.

####  II-C3 Object Fragmentation

Another generative model of our framework is to break or fragment any 3D mesh
into a user-defined number of Voronoi cell fractures. We utilize [23] to
generate the specified number of 3D meshes from the parent mesh. The object
fragments into child particles upon contact with another rigid body (active or
passive) or any other non-uniform force field. The rendered output is a video
sequence from the camera sensors with semantic information of each individual
fragment. Fig. 1c shows the simulation environment of the scene where a bullet
is colliding with a cup, thereby shattering it into a number of pieces. Such
an approach is potentially useful for assessing damage due to collisions.

###  II-D Sensors

WorldGen renders RGB images along with the annotation data such as depth map,
optical flow, stereo images, semantic map, and surface normal map (See Fig. 1)
which are rendered directly through the Blender Cycles engine. For event
camera renders, we utilize a simple event camera model [24, 25] to generate
events at a location $\mathbf{x}$ when

$\|\log\left(\mathcal{I}_{t}\left(\mathbf{x}\right)\right)-\log\left(\mathcal{I}_{t+1}\left(\mathbf{x}\right)\right)\|_{1}\geq\tau$
(1)

where $\tau$ is a user-defined threshold, $\mathbf{x}$ is the pixel location
and $\mathcal{I}_{t}$ represents a grayscale image captured at time $t$. An
event-frame is generated using the above model along with additive Gaussian
noise. Alternatively, a more realistic event model [26, 27] can be adapted to
our framework. For smaller integration time in generating event frames, we
remap the animation timeline to $\mathbb{N}$ times more frames to give a slow-
motion effect to simulate a visual high-speed camera from which events are
generated.

Multiple such sensors can be spawned in the environment to render outputs from
different camera pose at the same time. WorldGen takes 6 degree-of-freedom
camera extrinsic parameters to render outputs from different cameras. This
setup can be used to simulate a sensor suite like the ones typically found on
a self-driving car or an autonomous drone.

Camera Properties: WorldGen supports both dynamic camera trajectories as well
as variation in camera intrinsic properties – dynamic focal length, depth of
field, camera aperture radius, or variable camera baseline in the case of
stereo output to simulate and generate data for previous works like [28, 29].
These properties can be temporally modified by using simple linear
interpolation between defined key-frames or Bézier curves. WorldGen uses the
same method to dynamically move the solid objects in the scene. Also, due to
variability in camera focal length, our framework supports both fisheye and
equirectangular projection to render full 360-panorama images (Fig. 5e).
Furthermore, WorldGen also supports real-world camera distortions for a more
photorealistic render such as lens glare, chromatic aberrations, and barrel
distortion. This is particularly useful for handling corner case scenarios
which might aid in better sim-to-real generalization.

###  II-E Lighting and Climate Conditions

Currently, WorldGen supports three lighting conditions – midday, sunset and
night as well as 4 different weather conditions – rain, cloudy, clear, and fog
(Figs. 5a to 5d). Since WorldGen is Global Illumination (GI) enabled, the
sunlight diffusion and reflection are photorealistic. Along with GI, it
supports fast GI approximation for faster render time with a small loss in
photorealism. The diffused sky radiation; the angle, power, and color
temperature of the sun, and light sources such as street lights can be
temporally varied.

###  II-F Assets

The elegance of the WorldGen generative simulator is to use and modify
existing 3D meshes, and materials to create new 3D environments. For the
object pile and fragmentation scene generation, WorldGen imports objects, and
materials by reading a text file that contains a list of wavefront .obj file
path and material .mtl file path which makes it easier for the user to import
any 3D object mesh into the WorlGen environment. We utilize a myriad of 3D
object databases and image textures for incorporating a wide variety of
possibilities. Furthermore, users can also import any custom objects/textures
along with the ones described below:

ShapeNetCore.v2 [30]:  ShapeNetCore.v2 contains about 51,300 unique 3D models
across 55 object categories that are manually verified for classification and
alignment annotations. We utilize Kubric’s cleaned version of ShapeNet objects
that fixes issues regarding auto-smoothing and backface culling that are
present in the original ShapeNetCore objects.

Google Scanned Objects (GSO) [31]:  (CC-BY 4.0 License) GSO contains 1030
scanned objects and their associated metadata totaling ${\sim}13\text{GB}$.
Since these models are scanned and not hand crafted in a 3D modeling tool,
they realistically reflect real object properties.

MS-COCO [32]:  Microsoft’s COCO is a dataset that contains images of complex
everyday scenes containing common objects in their natural context containing
over 328k images across 91 object classes. We use these relatively lower-
resolution images to warp around objects to modify their natural appearance.

AmbientCG [33] is a large public domain resource with CC0 license for
Physically Based Rendering (PRB) with 3D objects, materials and High Dynamic
Range Images (HDRI) containing photo-scanned materials, displacement maps to
${\sim}1$ pixel accuracy with about 1760 assets with texture resolution up to
$8K$.

Polyhaven [34] is a public library with a CC0 license with $500+$ pre-
processed HDRI images which we resourced for backgrounds and lighting.

cgbookcase [35] is another CC0 license with $540+$ high-resolution PBR
textures for common city structures such as different walls, roads, buildings,
concrete, etc.

Apart from the aforementioned assets, we also utilized open-source libraries
and Blender add-ons for pre-processing and rendering operations: Blender-OSM,
MapBox and object_fracture_cell and bpycv.

###  II-G Design Principles

####  II-G1 Modularity

WorldGen is a highly modular framework as it is divided into three different
elements as discussed in Sec. II-A. Each of the modules can be treated
differently and can be replaced by a different third-party module. For
instance, we can extend this work to generate a variety of human motions on a
custom background using assets from 3D character rigging resources such as
Adobe’s Mixamo [36] to generate ground truth data annotations for human pose
estimation. We plan to add character-rigged motion scene generation and data
annotation with varying character motion to WorldGen.

####  II-G2 Ease of use

The design philosophy of WorldGen is to make it easier for researchers,
educators and practitioners to generate new scenes without needing any core
knowledge of computer graphics concepts, modeling, animating, and rendering
tools. WorldGen bridges this gap by presenting a simple high-level object-
oriented Python API with Blender (and Bullet) working in the background.

####  II-G3 Open Source

Scene and data generation codes will be released upon acceptance and will be
free to use in academia. All the third-party assets used in WorldGen with
Apache2.0, MIT and CC0 licenses enable researchers to generate different
environments, render annotations and share all the data with the community to
ensure reproducibility.

####  II-G4 Scalability

WorldGen brings generative methods to build new structured environments
without any need for manual labor. It can be used to generate environments
anywhere between a small town and a large city with a compromise on GPU and
access memory.

##  III Applications

WorldGen is designed to generate different 3D environments and annotated data
for robotics and computer vision applications. It is meant to be a one-stop
framework for zero-shot generalization due to its variability in shapes,
texture, and dynamic lighting; also minimizing the gap for sim-to-real
transfer. It would also serve as a simulator for developing and testing
planning and control algorithms due to its flexibility since it can use the
current state and annotation as input for the next time stamp. To demonstrate
the potential and flexibility of WorldGen, we discuss a set of common and
challenging problem statements in robotics and computer vision.

###  III-A Improvements in Optical Flow

Figure 6: High resolution views generated by WorldGen from views at different
altitudes with dynamic lighting, camera intrinsic, and extrinsic.

Optical flow is one of the most fundamental quantities in computer vision and
robotics which measures the 2D motion of each pixel between consecutive
frames. Contrary to high-level computer vision applications like instance
segmentation, ground truth cannot be reliably obtained from real-world data
with human annotation. Recent advances in optical flow methods – PWC-Net [37],
RAFT [38] and GMFlow [39] all rely on synthetic data like MPI Sintel [13],
FlyingChairs [2] and FlyingThings3D [3] for pre-training. Such datasets use
synthetic chairs and lack photorealism and realistic 3D motion. Recent
advances in dataset generation – AutoFlow [40] learns to render
hyperparameters for generating synthetic flow datasets reducing the End Point
Error (EPE). AutoFlow lacks 3D motion and photorealism as it utilizes a simple
2D layered model. Having photo-realistic data, Kubric [10] tends to outperform
AutoFlow, but only on Sintel Clean samples. This is the rendered output that
contains shading but no image degradation whereas the Final pass includes
motion blur, defocus blur and atmospheric effects. Since Kubric lacks both
volumetric features (like fog, mist, and rain) and advanced camera features
(such as depth of field and motion blur), AutoFlow performs better even
without photorealism as images are generated with motion blur and fog models
as well as data augmentation for visual effects.

We compare and analyze optical flow predictions using RAFT on Sintel Clean and
Final pass which are trained on different datasets. Table II shows the EPE in
optical flow. Note that WorldGen causes a significant increase in optical flow
accuracy as compared to FlyingChairs. Furthermore, WorldGen outperforms
AutoFlow due to the difference in photorealism. Note that WorldGen slightly
improves over Kubric. We speculate one of the reasons to be WorldGen rendering
multiple instances of the same object configuration by warping different
textures on the objects, ensuring the network avoids over-fitting to object
textures, and thereby learning the geometrical/motion properties of the scene.
Since WorldGen supports dynamic lighting, depth of field, and motion blur
modeling, we see a significant improvement over Kubric in the Final pass. Note
that AutoFlow performs better in Final pass (although only slightly) because
the hyperparameters of AutoFlow have been learned to optimize the performance
on the Sintel dataset which gives AutoFlow an unfair advantage.

TABLE II: Optical Flow EPE Comparison of Training RAFT [38] On Different Datasets. Lower Is better. Dataset | Dimensionality | Parameters | Sintel | Sintel
---|---|---|---|---
Clean | Final
FlyingChairs [2] | 2D | Manual | 2.27 | 3.76
Kubric [10] | 3D | Manual | 1.89 | 3.02
AutoFlow [40] | 2D | Learned | 2.08 | 2.75
WorldGen (Ours) | 3D | Manual | 1.86 | 2.87

###  III-B Computational Photography

Computational photography is one of the key areas that require a large
collection of image data for the prediction of depth maps, denoising image
data, etc. Most of the current photo-realistic synthetic data generators in
robotics and computer vision do not explore and support many features that are
offered by Blender, Unity, or Unreal Engines such as volumetric effects and
variability in sensor sizes, camera lens, and rolling shutter sensors. Since
WorldGen supports generating data for different light conditions, different
bokeh blur due to depth of field effect, motion blur, variable focal lens and
sensor size along with Albedo, Clean and Final renders, it can be used to
generate data for the prediction of various computational photography
applications such as HDR+ datasets [41], depth from defocus [42, 43],
recovering image from a rolling shutter blur [44, 45], extracting a video
sequence from a single blurred frame [46] and for learning synthesis motion
blur from a pair of images [47]. Furthermore, this work can be utilized to
generate datasets for learning depth estimators using a coded aperture [48,
49] camera. We intend to add a toolbox to generate coded blurred frames in the
WorldGen environment after release.

###  III-C View Synthesis using Neural Radiance Fields

Neural volume rendering has exploded in the last two years [50]. One of the
major challenges in [50] is to synthesize in the case of dynamic relighting.
These methods underperform with views ranging from satellite level capturing
the entire city to the ground level, majorly due to large camera displacement
and dynamic lighting. Recent advances in multi-scale scene rendering [51] and
dynamic irradiance view synthesis [52] have adapted to work in such adverse
conditions. In [52], the authors utilize multi-date images due to their
significant changes in appearance, mainly due to varying shadows and transient
objects like cars and vegetation using the WorldView-3 dataset. WorldGen can
generate thousands of 3D satellite views with dynamic lighting at a higher
resolution than the satellite images with known camera intrinsic and extrinsic
for more robust view synthesis in satellite images with larger displacement in
different lighting and weather conditions (See Fig. 6). Also, this can be
extended towards learning high-level planners for aerial cinematography [53].

###  III-D Active and Interactive Perception

Not only is WorldGen an automatic 3D scene and dataset generation tool but it
also serves as a framework for active and interactive perception applications.
Robots such as quadrotors are often required to use the current prediction and
annotations as input to the next state for its solution and testing.
Currently, there is no test bench to verify the robotics algorithms that rely
on active and interactive perception approaches [54, 55]. WorldGen will serve
as a benchmark pseudo-simulator to validate the performance of such algorithms
for better reproducibility. We hope WorldGen will become the OpenAI Gym [56]
for Active and Interactive perception.

###  III-E Generating Real World Traffic

The problem of generating realistic traffic scenes autonomously is a
challenging task. Existing methods typically are driven by planning and
reinforcement algorithms that require current and previous traffic states,
traffic signal conditions, and road/pathways vertices[57]. Since WorldGen has
the ability to export these parameters, neural autoregressive models such as
[58] can be deployed for realistic traffic distribution. The recent advances
in neural fields such as Panoptic Neural Fields [59] have pushed the boundary
to detect traffic in a 3D representation, allowing us to perform 3D scene
editing. Using such methods, WorldGen can also import real-life traffic into
its model for generating photorealistic real-world city traffic scenarios.

###  III-F Human Pose Estimation

In recent years, 3D human pose estimation has made great improvements which
have been driven by large-scale datasets. However, these datasets lack
accuracy in the pose as it mostly annotated by humans from open-world datasets
[60] that suffer from human labeling accuracy, mainly due to occlusions. On
the contrary, motion capture systems have helped to generate accurate human
pose data but they lack an open world environment due to motion capture space
restrictions [61]. Lastly, the simulation environment for dataset generation
lacks photorealism [62]. We plan to develop another generative environment
with WorldGen by utilizing Mixamo [36], an open platform with a variety of
character rig models and animations; and warping different 3D materials to
generate a large of photorealistic character animation even in the backgrounds
such as small towns and city. This will push the boundaries for monocular
human pose estimation and open new avenues for special sensors such as event
cameras.

##  IV Conclusion

We introduce WorldGen, a modular, generative, open-source Python API as a
large-scale generative simulator with a myriad of scenes. We can generate
accurate ground truth labels for optical flow, depth, surface normals, depth
cameras, semantic maps and event data for driving scenes, moving object piles
and object fragmentation. WorldGen also simulates camera properties, motion
properties for a photorealistic perception-centric simulation and data
generation. Not only can WorldGen change objects, lighting and scenes
dynamically, but it can also edit objects dynamically aiding in the generation
of data to better enable the generalization of neural networks such as optical
flow as shown in our experiments. Currently, sensors such as SONAR and LIDAR
are missing from WorldGen along with behavior modeling for self-driving cars,
rigging of human characters is not implemented and we see this as a potential
direction for future work.

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# Early Stages of Drop Coalescence

Antoine Deblais<EMAIL_ADDRESS>Van der Waals-Zeeman Institute, Institute of
Physics, University of Amsterdam, 1098XH Amsterdam, The Netherlands.  Kaili
Xie  Van der Waals-Zeeman Institute, Institute of Physics, University of
Amsterdam, 1098XH Amsterdam, The Netherlands.  Peter Lewin-Jones  Mathematics
Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.  Dirk
Aarts  Department of Chemistry, Physical and Theoretical Chemistry Laboratory,
University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom.
Miguel A. Herrada  Escuela Tecnica Superior de Ingeneria, Universidad de
Sevilla, Seville 41092, Spain  Jens Eggers  School of Mathematics, University
of Bristol, University Walk, Bristol BS8 1 TW, United Kingdom.  James E.
Sprittles  Mathematics Institute, University of Warwick, Coventry CV4 7AL,
United Kingdom.  Daniel Bonn  Van der Waals-Zeeman Institute, Institute of
Physics, University of Amsterdam, 1098XH Amsterdam, The Netherlands.

###### Abstract

Despite the large body of research on coalescence, firm agreement between
experiment, theory, and computation has not been established for the very
earliest times following the initial contact of two liquid volumes. Combining
a range of experimental and computational modelling approaches, we have been
able to elucidate the influence of the intervening gas, of the van der Waals
forces and of thermal fluctuations on coalescence. For simple liquids, the gas
influences both pre- and post-contact regimes, with a jump-to-contact the
primary mode of merging. Subsequently, wave-like air pockets are observed. For
a system with ultralow interfacial tension, that mimics nanodrop behaviour,
the very first moments are governed by thermal fluctuations at the interfaces,
with a nontrivial opening speed given by stochastic thermally driven motion.

Hydrodynamics, Coalescence

###### pacs:

Valid PACS appear here

The merging of liquid volumes is important for a plethora of fluid phenomena:
for example, drop-drop coalescence dynamics dictate the size of raindrops,
efficacy of virus transmission, accuracy of inkjet printing and efficiency of
spraying phenomena [1, 2, 3, 4, 5], whilst drop-surface interactions are key
for production of ocean mist, air-sea gas exchange and airborne salt particles
[6, 7].

When colliding drops are able to displace the surrounding gas to touch,
coalescence is initiated with the formation of a liquid bridge (or neck)
between the two drops, which subsequently grows, with surface tension driving
the expansion. Depending on the parameter regime, namely whether it is the
drop’s viscosity or inertia that resists capillary forces, different scaling
have been derived for this growth [8, 9].

In the conventional hydrodynamic description of coalescence, the growth is
initially inhibited by the viscosity $\eta$ of the liquid. The rate of growth
is then determined by a balance of capillary and viscous forces, and a simple
dimensional analysis then leads directly to a capillary velocity
$U_{cap}\approx\gamma/\eta$, with $\gamma$ the surface tension, which gives
the rate at which the fluid bridge increases. A more detailed analysis [8]
shows that there is a logarithmic correction to this, leading to a temporal
variation of the bridge radius $R\propto\tau\log\tau$, where $\tau$ = $t$ \-
$t_{0}$ is time from initial contact.

For the coalescence of water droplets ($\gamma\approx$ 70 mN/m and $\eta$ = 1
mPa s), $U_{cap}\approx$ 70 m/s. This means that very rapidly the fluid
inertia becomes important; the balance between inertial and viscous forces is
dictated by the local Reynolds number $Re(\tau)=\rho\gamma
R^{2}/R_{0}\eta^{2}$ with $\rho$ the density, $R(\tau)$ the neck/bridge radius
and $R_{0}$ the drop radius [10]. For mm-sized drops of water, $Re\approx 1$
when $R\approx 1\mu m$. Thus, for water the viscous regime is hard to observe,
and the coalescence is, from a practical point of view, mostly inertial. The
inertial dynamics is then given by $R(\tau)\approx A(\gamma
R_{0}/\rho)^{1/4}\tau^{1/2}$, where the prefactor $A$ derived from experiments
can take different values [1.1-1.25] [11, 9, 12].

For drop-drop coalescence, for sake of argument, these classical theories
assume that the initial state consists of two stationary undeformed spherical
drops joining at a point. But how do these drops actually displace the
intervening gas and come into contact?

Experimentally, this is a very difficult question as direct imaging is
evidently limited by both the optical resolution and the rate at which images
can be acquired [13, 9]. In addition, focusing on the plane of the coalescing
drops and ‘seeing into’ its cusp-like region may lead to spurious exponents
for the dynamics, as shown in [14]. To overcome these issues, Paulsen et al.
[10, 15] deployed an electrical method to infer the bridge radius in the early
stages and proposed a new initial regime in which inertia, capillarity and
viscosity are all important. However, simulations of the full Navier-Stokes
system [16, 17] show no evidence for this regime, so that links between
experiment and theory remain unclear.

In this Letter, we study the initial stages of drop coalescence at extremely
short ($\approx 0.1\mu$s) time and length ($\approx 2\mu$m) scales and in a
system with ultralow surface tension ($\approx$ 5 nN/m); the latter to
investigate the effect of thermal fluctuations. A new imaging arrangement
allows us to show that the rupture of an air film, the merging of a drop with
a bulk liquid happens with a jump into contact, and capillary waves are
generated at the liquid surface ahead of an entrapped air pocket. The ultralow
tension system shows two types of coalescence. For relatively high surface
tension, the coalescence is viscous; however for the lowest surface tension,
thermal-fluctuation-driven capillary waves are clearly visible, and these
govern the very early stages of coalescence, leading to a novel fluctuation
dominated coalescence regime with a nontrivial dependence of the neck radius
on time.

Drop-Drop. We use an electrical technique similar to those described in [18,
10, 19, 15] to measure the coalescence between two drops. In order to increase
the spatial and temporal resolution of this measurement, we use salt water
with a large conductivity, allowing us to reach submicron spatial and almost
nanosecond temporal resolution. In Fig. 1(a) we plot the measured conductivity
as a function of time. The shape of the liquid bridge is computed from a
simulation, allowing us to calculate the resistance $r$ of the neck in time.
We used a numerical scheme based on a volume-of-fluid (VOF) method provided by
the ‘Basilisk’ software [20] which uses QUADTREES [21] to allow efficient
adaptive grid refinement of the interface close to the coalescence region (see
more details in Supplemental Material, Sec. I and Sup. Fig. S1). The measured
resistance of the liquid bridge data is well described by the simulations at
early stage of the coalescence point.

Figure 1: Temporal coalescence dynamics of liquid droplets probed with an
electrical method. (a) Electrical resistance $r$ of a low-viscous saline
capillary bridge ($\eta$ = 1.8 mPa s) between two electrodes ($R_{0}$ = 1 mm)
as a function of time. The upper electrode moves at very low constant
downwards speed, initiating coalescence; the electrical resistance is obtained
from the complex impedance of the circuit, following the method detailed in
[15]. (b) From the electrical resistance shown in (a), the opening dynamics of
the neck can be obtained (Eq. 1 of Ref. [15]). Data from Ref. [10] are plotted
for a comparable low-viscous liquid ($\eta$=1.9 mPa s). Dots represent
averaged values, while the shaded area corresponds to standard deviations.
Figure 2: Temporal Coalescence Dynamics of Droplet-Bulk Probed Optically. (a)
Image sequences captured from the bottom view using the fast camera. The
photographs depict the very early stages of bath interface coalescence, where
the droplet (water, $R_{0}$ = 0.9 mm) contacts the lower bath. The artificial
green color is applied solely for improving image contrast. (b) Corresponding
spatio-temporal averaged intensity of (a). Black dot points indicate the
tracked minimum neck diameter $R$, and the dashed red line represents the
Basilisk simulation for the same droplet radius $R_{0}$ as in (a). As the
minimum neck diameter increases, capillary waves are excited ahead of the
front, as shown by the change in intensity. (c) Profiles of the interface
extracted from the simulation at $\tau$ = 64 $\mu$s after coalescence (left),
and its corresponding first derivative (right). We define the lengths
$\Lambda_{i}$ for which the local derivative is either positive or negative.
These lengths correspond to the change in intensity observed in (b) due to the
refraction of light at the bath interface. (d) Rescaled radius (with the
initial drop radius $R_{0}$ = [0.9;1.73] mm) versus rescaled time (with the
inertial time $\tau_{i}=\sqrt{\rho R_{0}^{3}/\gamma}$) for the different
liquids investigated and for three different initial drop sizes (for the fluid
with viscosity 5.7 mPa$\cdot$s).

Fig. 1(b) shows the minimum neck radius $R/R_{0}$ as a function of time after
coalescence as follows from the electrical measurements: the neck diameter is
obtained from the relation $R={2[(r-1/\pi R_{0}\sigma)3.62\sigma]}^{-1}$ (Eq.
1 of Ref. [10]), which we also confirmed from our simulation (see Sup. Fig. S4
in the Supplementary Material). The dynamics follow the form
$R(\tau)\propto\tau^{\alpha}$, and for which we recover $\alpha$ = 0.5 for the
inertial case. For the inertial regime, it was previously found that
$R(\tau)\approx 1.1(\gamma R_{0}/\rho)^{1/4}\tau^{1/2}$. We attribute the
different slope observed in Ref. [10] to be due to the initial deformation of
the interface due to the presence of charges (see Sup. Fig. S5). Our first
data point for water in this regime is $\approx$ 10 $\mu$m at $\approx$ 1 $\mu
s$, which is very close to what is expected from the equation. This is in
agreement with findings from [10]; however, in the absence of optical imaging
we have no information about how the initial contact occurs or the subsequent
shape of the interface.

To access this initial pre-contact approach stage, we deploy bespoke finite
element simulations which include the influence of the gas, with corrections
due to kinetic effects for microscale gas films, alongside van der Waals
effects, see [22] for details of the computational framework. In Sup. Fig. S2,
we see that for the slow approach speeds considered in our experiments, the
deformation of the drops caused by gas lubrication is relatively small so that
the interfaces meet at $r=0$, i.e. no bubble is entrapped. Notably, the van
der Waals force causes a ‘jump to contact’ when the separation of the drops is
$\approx 30$ nm over a timescale of $\sim\mu$s. This ‘jump into contact’ for
separations of $\sim$10’s of nm is consistent with the literature [23, 24, 25,
26] and was previously observed for AFM tips connecting to a fluid surface.

Drop-Bath. To access the initial stages of coalescence optically, we turn to
the drop-bath phenomenon and use a novel approach by viewing the event from
‘underneath’ (impossible for drop-drop) to overcome spurious effects due to
side imaging [27, 14], where it is not possible to measure small contact radii
(less than 20 $\mu$m for instance) due to the unfavorable spherical geometry
of the droplet and optical effects [27, 14, 26, 9]. The setup consists of a
drop on a needle to which liquid is supplied slowly by a syringe pump. This
drop is facing a bath of the same liquid (see Sup. Fig. S7). Our needle tip is
millimetric (diameter 1.37 mm) and a drop is slowly grown at its tip until it
coalesces with the bulk (growing velocity $\leq$ 20 $\mu$m/s, in an attempt to
minimize the effect of air drainage between the droplet and the bulk surface).
To change the properties of the fluids, different amounts of glycerol were
mixed with deionized water (Milli-Q). The viscosity, density and surface
tension are given in the Supplementary Table S1. All experiments were
performed at room temperature (22$\pm$1 ∘C) and atmospheric pressure.

The growth of the neck bridging the droplet with its bulk is recorded from
below through the bath using an inverted microscope (Zeiss Axiovert A1) at a
frame rate of up to 1,000,000 frames per second (fps) using a high-speed
camera (Phantom TMX 7510) allowing us to reach a spatial resolution of
typically $0.9\,\upmu$m/pix with a 20$\times$ microscope objective (Zeiss long
working distance WD = 20mm). The minimum neck radius $R$ is followed in time
with a simple homemade routine (Fig. 2).

Fig. 2(a) shows a sequence of the neck opening during coalescence for the
droplet-bulk geometry exhibiting two features. It is immediately clear that
(i) there is an abrupt change of intensity which corresponds to the
coalesced/bridge radius and (ii) there are also concentric rings ahead of the
bridge, see Fig. 2(b), which appear to be the signature of interfacial waves
that refract passing light. This is confirmed by examining the interface
profile obtained from the Basilisk simulations at a specific time ($\tau$ = 64
$\mu$s), as shown in Fig. 2(c), where waves are observed ahead of an air
pocket (see also Sup. Fig. S6). This combination of inertial coalescence
driving wave formation and gas lubrication preventing the pinch-off of
toroidal bubbles has been described previously [28, 16], and is also predicted
for viscous coalescence [8], but strong direct experimental evidence for it
has been lacking until now.

To further confirm this hypothesis, in Fig. 2(c) we plot the spatial
deformation ($dz/dr$) obtained from the simulations, in order to define
domains $\Lambda_{i}$ where the interface undergoes pronounced changes, and
show in Fig. 2(b) we show that this aligns with experimental changes in
intensity.

As shown in Fig. 2(d), different viscosities of fluids (range from 1
mPa$\cdot$s to 11.5 mPa$\cdot$s) were tested. In the late stage of
coalescence, the inertial-capillary regime is found, the same as we observed
in the electrical measurement $R(\tau)\sim\tau^{1/2}$. In the early stage, a
different regime is observed for all fluids, with a radius that is smaller
than would be expected from the inertial-capillary regime. For the earlier
regime, all the curves collapse when scaled with the inertial time: there is
no effect of viscosity, and the differences in [Fig. 2(d)] are due to
differences in density and surface tension (See Sup. Table 1).

Figure 3: Computed interface profiles of the final stages of a drop moving
towards a liquid bath at $5\times 10^{-6}$ m/s, before coalescence with first
contact occurring at $\tau=0$. It shows the deformation of the bath and the
‘jump to contact’. Model parameters are the same as those for the drop-drop
case in Sup. Fig. S2.

Basilisk simulations (red line) of the droplet-bulk coalescence for water,
agree quantitatively well with our experimental measurement overall the whole
range of times. However, to achieve this agreement, the time axis in the
simulation has been set to match the experiments [Fig. 2(b)] and we also
adjusted the minimum neck diameter $R$ by 50 $\mu$m to align the experiments
with the simulations. Adjusting the time is rather standard, and amounts to
fixing the coalescence time $t_{0}$, but the reasons for the spatial shift
remains unclear and motivated us to look into the pre-contact dynamics…

Figure 4: Temporal coalescence dynamics of colloidal liquid droplets. (a)
Sequence of pictures showing the dynamics of coalescence for $\gamma\approx$
160 nN/m (upper panel) and for $\gamma\approx$ 5 nN/m (bottom panel). Top row,
droplet of diameter 16.5 $\mu$m; bottom row, 21.8 $\mu$m. Data are extracted
from [29]. (b) Data scaled with the initial drop radius $R_{0}$ and viscous
time $t_{\nu}$. Black solid line shows the best fit, $R/R_{0}=(1.84\pm
0.13)\times{(t/t_{\nu})}^{0.55\pm 0.06}$. Blue dashed line is for $(0.57\pm
0.05)\times\gamma/\eta$. Black filled circles are the data of a silicone oil
drop.

In our experiments, we observe interference fringes between the two surfaces
in the last frames before coalescence (see Sup. Fig. S8), which suggest the
presence of a thin air film which acts as a lubricant between the two
approaching surfaces. This air film continually thins, and if it becomes thin
enough, the two surfaces jump into contact. This behaviour is also confirmed
by finite element simulations in the Fig. 3, showing the final moments of the
approach of a drop towards a liquid bath before coalescence. The gas is
modelled using a lubrication equation, and incorporates the van der Waals
driven disjoining pressure that causes the ‘jump to contact’. Fig. 3 shows
that the bath is pushed down vertically $\sim 100$ nm over the radial extend
of the drop $\sim$mm before the drop ‘jumps to contact’, over a radial extent
$\sim 3\ \mu$m in $\sim 0.1$ ms, similar to the mechanism seen in the drop-
drop case but with a vertical shift. Unfortunately, resolving the nanometric
scales associated with the jump-to-contact within Basilisk simulations is
currently unachievable, so we attempt to mimic the predicted deformation by
using the computed bath shape, vertically shift the sphere to account for the
‘jump’, and joining them with an artificial neck of 2.7 $\mu$m; further
assuming that all liquid is initially motionless. This adjustment improves
agreement between theory and experiment [Fig. 2, red dotted line], showing
clearly that pre-contact dynamics can affect coalescence, even in cases where
extreme care is taken to minimise the influence of the gas. However, a spatial
shift remains necessary and appears to be due to our inability to resolve the
very smallest scales, where there is a relaxation from pre- and post-contact
dynamics over some length scale – providing clear motivation for future
studies in this regime.

Coalescence of colloidal-polymer liquid droplets. Our system is a mixture of
colloids and polymers dispersed together in the same liquid. Entropy-driven
interactions create a phase separation between a colloid-rich and a colloid-
poor (polymer-rich) phase, akin to a liquid-gas phase separation in a
molecular system. For the latter, the order of magnitude of the surface
tension is $k_{B}T/d^{2}$, with $k_{B}$ the Boltzmann constant, $T$ the
temperature, and $d$ the molecular size. For water, taking $d$ = 2.5 $\AA$,
one finds a typical value for the surface tension of $\approx$ 60 mNm-1,
rather close to its actual value. For the colloidal system, the characteristic
size is given by the colloid size; for $d=1\mu m$, one finds $\gamma\approx$ 4
nNm-1, almost a million times smaller. According to the dimensional analysis
above, this slows down the coalescence dynamics by a factor of one million and
‘slow’ video microscopy is sufficient to track the coalescence. The latter is
done in a small flat capillary containing the colloidal liquid and gas phases,
which are studied by a confocal microscope tilted by 90 degrees, so that
gravity maintains the phase separation.

By shaking the vessel, we obtain colloidal liquid droplets that slowly fall
under the action of gravity allowing for coalescence with the bulk liquid
[Fig. 4(a)]. Because in this mixture the coalescence is much slower, inertial
effects only begin at a length scale of six meters. Hence, the coalescence is
entirely dominated by viscosity. In agreement with this, we observe that for
the first system ($\gamma\approx$ 160 nN/m), the liquid bridge grows linearly
in time. This observation is consistent with the coalescence of simple fluids
- namely, viscous silicone oil [Fig. 4(b)]. However, for the second system
which has an ultra-ultralow surface tension ($\gamma\approx$ 5 nN/m), one of
the most striking observations from the images is that in the formation of the
liquid bridge, the roughness of the interface between the two phases is
responsible for the initial merging of the drop with its bulk.

The coalescence is thus caused by the thermal fluctuations of the two
surfaces, with multiple points of contacts between the two surfaces [Fig.
4(a), bottom panel] and $R$ is measured from the total contact between the
drop and the bulk from the different connected regions. This leads to a new
fluctuation dominated regime in which the coalescence is much slower: the
liquid bridge radius varies as $R(t)\propto t^{\beta}$, with $\beta=0.55\pm
0.06$ from the best fit. This result is reminiscent of drop formation, the
inverse process of coalescence where a drop detaches from another one. Also
for such detachment, a fluctuation regime was uncovered, with an anomalous
exponent that was significantly smaller than that for the viscous breakup
regime [30].

Rather than such a linear time dependence, the fluctuation-driven breakup
shows an increasing rate of pinching, just like here, and satisfies the
predictions of a power law time dependence for fluctuation-driven pinch-off
with a power close to the theoretical prediction of 0.42 [31], again similar
to what is observed here. For the breakup the rationale for an exponent
$\beta$ smaller than unity with noise is when the bridge becomes smaller, the
motion speeds up. As a result, random fluctuations are biased toward smaller
bridge radii, consistent with a velocity $t^{\beta-1}$, which is singular.
Similarly, during coalescence random fluctuations might fill the gap more
quickly in the early stages, and then slow down disproportionally.

In summary, there are two ways in which the initial contact between two drops
can be established: interface fluctuations and jump-into-contact. The jump-
into-contact is undetectable for the drop-drop experiment, likely because the
air film can drain easily. However, it is very clearly present in the drop-on-
bath experiment, where it generates capillary waves that determine the initial
opening speed. Additionally, a combination of the two is also likely for
macroscopic drops: thermal fluctuations can initiate disturbance of the
interface, from which van der Waals forces can create a jump-into-contact. The
former is observed in a system with ultralow interfacial tension and extremely
small van der Waals forces, due to the vicinity of a critical point.
Interestingly, similar observations have been made for the coalescence of
water nanodroplets, where fluctuations become more comparable to the drop size
[32].

Acknowledgements. We are very grateful to the technology center of the
university of Amsterdam for technical assistance with the electrical setup. DB
thanks Jacco Snoeijer for very helpful discussions about the artifacts induced
by side imaging. JES acknowledges the support of the EPSRC under grants
EP/W031426/1, EP/S022848/1, EP/S029966/1, EP/P031684/1 and EP/V012002/1. Peter
Lewin-Jones is supported by a studentship within the UK EPSRC–supported Centre
for Doctoral Training in the Modeling of Heterogeneous Systems (HetSys),
EP/S022848/1. For the purpose of open access, the author has applied a CC BY
public copyright license to any Author Accepted Manuscript version arising
from this submission.

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## SUPPLEMENTARY MATERIALS

##  I Drop-drop

###  I.1 Basilisk Simulations.

Work has been carried out on dimensionless variables using the drop radius,
surface tension and density of the liquid as the basis for non-
dimensionalizing. In the numerical study, gravitational forces were neglected.
Axisymmetric simulations are performed in a cylindrical coordinate system
($\bar{r}$, $\bar{z}$) in a box of length $L$ where the axis of symmetry lies
along the z-axis (the left boundary of the box).

In the 2 drops case, the symmetry of the problem has been exploited to reduce
the computational cost. A numerical domain of length L=1.1 was chosen in which
one of the drops centered at the point $(0,-0.1)$ was placed [see figure 5
(a)]. A symmetry condition was imposed on the bottom boundary to model the
other drop. The two drops were brought into contact by an artificial neck with
a radius of $0.0028$. Initially, both fluids (liquid and gas) are at rest. The
cells in Basilisk are square, so an initial mesh with a maximum refinement of
18 was used in the contact zone to model the collapse of the droplets in the
early stages. As the meniscus formed and retracted, this maximum refinement
level was reduced.

Figure 5: Numerical domains. (a) Shows the numerical domain for the two drops
case while (b) shows the numerical domain used for the drop-bath case.

#### Numerical computation of the electrical resistance

The numerical resistance was computed using a combination of Basilisk
simulations and a finite element method. The methodology used is briefly
described below.

First, at the selected time at which the resistance is computed, the numerical
profile of the interface is extracted from the Basilisk simulation and saved
into a text file. This file is then imported to MATLAB to generate the closed
domain needed to solve the Laplace equation for electrical problem. The
geometry is normalized with $R_{min}$, and centered at the axial position
where the interface has its minimum radius. Neumann boundary conditions are
applied on the free surface and on the axis while Dirichlet boundary
conditions are applied on the solid supports. A normalized drop in the
electrical potential equal to one was considered. Then, the Laplace equation
is discretized and solved using a finite element method provided by MATLAB PDE
tool. Once we got the electrical potential, the constant and normalized
current flow (intensity) through the system is obtained by integrating the
axial derivative of electrical potential at any axial position, assuming an
electrical conductivity equal to one. Finally, the numerical electrical
resistance is simply computed by dividing the intensity by $R_{min}$. See [1]
for further details of the procedure.

###  I.2 Finite element simulations

Whilst Basilisk is able to accurately capture the post-merging coalescence
event, volume of fluid techniques are not well suited to resolving the pre-
contact ‘collision’ phase to determine the initial contact dynamics. Here, we
use the finite element approach described in detail in [22], focusing it on
low-speed collisions as opposed to its usual application to higher speed
impacts.

The computational model is based on solving the Navier-Stokes equations in the
bulk of the drop/bath coupled to a lubrication description of the gas phase.
Due to the small film heights encountered, gas kinetic effects are accounted
for, and are often crucial, alongside van der Waals forces between the two
distinct volumes (drop-drop or drop-bath) that will act to pull the interfaces
together. The governing equations are then solved using an open-source finite
element package oomph-lib [3] based on the arbitrary Lagrangian Eulerian
approach which, crucially, permits a high accuracy representation of the free
surface and is well suited to multiscale problems like this one. Complete
details are provided in [22].

Here, we take the experimental values, using material parameters for the
water-air system alongside a collision speed of $5\times 10^{-6}~{}$m/s and,
due to its negligible effect on the ‘jump to contact’, neglect gravity. In
Fig. 6 the profiles of the interface shape as the two drops approach one
another show that there is very little deformation due to the gas but that the
van der Waals forces between the two interfaces causes a ‘jump to contact’
when their separation is $\approx 30$nm which deformed the drop on a lateral
scale of $\sim\mu$m’s.

Figure 6: Computed interface profiles of a drop-drop case as the drops
approach and then make first contact at $\tau=0$.

###  I.3 Electrical setup

We measured the resistance of the opening of the neck neck with an electrical
circuit analogous to the one described in the references ([18, 10, 19]).

Figure 7: Electrical setup used for the measure of the electrical resistance
of the coalescence event. (a) The entire setup is placed in a chamber flushed
continuously with humid air to prevent evaporation. A close view of the two
electrodes receiving the two droplets. The upper electrode can be translated
downward thanks to a speed-controlled translation stage. The dynamics of
coalescence is followed, over a large time scale, by measuring the electrical
resistance (oscilloscope on the top) and by imaging (fast camera) the
capillary bridge. (b) Equivalent electrical circuit.

The voltages $V_{1}$ and $V_{2}$ were measured and sampled simultaneously at a
rate of 1 GHz. The frequency $f$ of the input sine wave from the function
generator is taken at a maximum value of $f$ = 10 MHz, allowing to reach sub-
microsecond time scales. Higher frequencies result is The voltages were read
in Matlab and analyzed with a homemade routine. The analysis averaged the
incoming signals over a single period to find the ratio of their amplitudes,
$V_{2}/V_{1}$, as a function of time $t$. In addition, the analysis compared
the input signals to a known sine wave and found the relative phase shift of
each signal versus time. We chose a maximum amplitude of voltage of 1 V to
prevent the deformation of the droplet’s interfaces when brought into contact
(see Sup. Fig. 9).

####  I.3.1 Relation between the electrical resistance and the neck diameter

The minimum neck diameter $R$ is obtained indirectly by measuring the
electrical resistance $r$ of the neck in time. To obtain the minimum neck
diameter, we use the equation 1 proposed by [10]:

$R=2{[(r-\frac{1}{\pi R_{0}\sigma})3.62\sigma]}^{-1}$ (1)

We confirmed the validity of this equation with our simulations by plotting
the electrical resistance obtained from our Basilisk simulation as a function
of the minimum neck diameter. The results presented in Figure 8 confirmed the
good agreement between the two. We thus use this equation to convert the
measured electrical resistance from our experiments to the physical minimum
neck diameter $R$.

Figure 8: Relation between electrical resistance and the minimum neck
diameter. The red solid line are results obtained from the Eq. 1 and the blue
solid line is obtained from our Basilisk simulations.

####  I.3.2 Effect of charges and high voltage on the drops’ interface and
coalescence dynamics.

Fig. 9 compares charged and non-charged water drops as they approach each
other. Initially, two water drops with similar sizes formed on the tips of two
vertical plastic nozzles (diameter 2.3 mm). Similar to the setup used in the
electric measurement, we fixed the bottom drop while moving the top drop
towards the bottom one. Fast imaging was applied from the side using an
optical zoom lens (20 $\times$ magnification). To charge the drops, we simply
used a charged plastic tube (through rubbing with dry hair) to get close to
the drops before coalescence occurred.

When the drops are charged, a sharp liquid neck appears in the early moments
of coalescence (Fig. 9b). This charge effect causes a deviation in dynamics
from the non-charged case, which follows the capillary-inertial scaling,
$R(t)\sim t^{1/2}$ (Fig. 9a). Instead, in the early stages, the liquid neck
linearly grows for the charged drops. This pointed shape could also occur in
drop-drop electric measurements where an external electric field is applied,
potentially explaining the observation of a linear regime in the work of
Paulsen et al. [10]. However, in our drop-drop electric measurement, before
triggering coalescence, we carefully used a zero-static gun to discharge the
drops, ruling out the pointed shapes.

Figure 9: Effect of surface charges on the dynamics of coalescence of water
droplets (imaging). (a) The presence of charges on water droplets (orange
symbols) is shown to influence the coalescence dynamics at early stages,
compared to the uncharged case (blue symbols). Note that, the initial drop
sizes in these two experiments are slightly different. (b) Snapshots of the
profiles at the first contact of coalescence, highlighting the difference
between the two cases.

##  II DROP-BULK

###  II.1 Basilisk Simulations

In this case, a computational domain of length L=1.5 has been chosen in which
the droplet is centred at point (0,0). A liquid bath with a free and flat
surface was added at a height $\hat{z}=z/R=-1$. The drop and the bath were
brought into contact by an artificial neck with a radius of $0.003$. An
initial mesh with a maximum refinement of 19 was used in the contact zone to
model the collapse of the droplet with the bath in the early stage. Again, as
the meniscus formed and retracted, this maximum refinement level was reduced.

Figure 10: Simulated (Basilisk) interface profiles for different times after
coalescence in the drop-bulk configuration. Note the formation and the growth
of the air pocket followed by waves.

An additional simulation (labeled ‘deformed’ in Fig. 2d) instead used the bath
profile computed at the moment of contact by the finite element simulation in
Fig. 3, where the bath has been forced down $\sim 100$ nm. The spherical drop
was also shifted vertically down so that at $r=0$ the vertical distance
between the drop and the bath is the same as that found by the finite element
simulation before the ‘jump to contact’ occurred ($t=-0.08$ ms in Fig. 2) An
artificial neck of radius 0.003 was then added (the same as the ‘undeformed’
case).

###  II.2 Finite element simulations

Fig. 2 for the drop-bath case in the main text shows that the drop exhibits a
rather similar shape to those recovered for the drop-drop case, with a notable
‘jump to contact’. However, the deformation of the bath is more prominent,
with a ‘dip’ of $\sim 100$nm below its initial height seen when contact
occurs. This is because Laplace pressure associated with the planar, or
slightly deformed, bath interface is significantly lower than the drop’s and
therefore the gas lubrication pressure has more of an influence on it – one
can see from the figure that this deformation has occurred well before the
final stages of contact and the inset shows that it extends radially to the
width of the drop,

###  II.3 Imaging setup

As mentioned in the main text, to access the initial stages of coalescence
optically, we deploy a novel approach by viewing the event from ‘underneath’
to overcome spurious effects due to the side imaging [27, 14]. The setup
consists of a drop on a needle to which liquid is supplied slowly by a syringe
pump (Harvard Apparatus). This drop is facing a bath of the same liquid (See
Fig. 11). Our needle tip is typically millimetric (diameter 1.37 mm) and a
drop is slowly grown at its tip until it coalesces with the bulk (growing
velocity $\leq$ 20 $\mu$m/s, in an attempt to minimize the effect of air
drainage between the droplet and the bulk surface). To change the properties
of the fluids, different amounts of glycerol were mixed with deionized water
(Milli-Q). The viscosity, density and surface tension are given in the
Supplementary Table S1. All experiments were performed at room temperature
(22$\pm$1 ∘C) and atmospheric pressure.

The growth of the neck bridging the droplet with its bulk is recorded from
below through the bath using an inverted microscope (Zeiss Axiovert A1) at a
frame rate of up to 1,000,000 frames per second (fps) using a high-speed
camera (Phantom TMX 7510) allowing us to reach a spatial resolution of
typically 0.9,$\mu$m/pix with a 20$\times$ microscope objective (Zeiss long
working distance WD = 20mm).

Figure 11: Experimental setup for drop-bulk coalescence. The setup combined a
microscope (Zeiss Observer) coupled to a fast camera with different
magnifications.

Before coalescence, we observe interference fringes between the two surfaces
in the last instances before coalescence (see Sup. Fig. 12), which suggest the
presence of a thin air film which acts as a lubricant between the two
approaching surfaces. This air film continually thins, and if it becomes thin
enough, the two surfaces jump into contact.

| Fluids
---
| Glycerol fraction
---
$\phi$ [wt%]
| Viscosity
---
$\eta$ [mPa$\cdot$s]
| Surf. Tension
---
$\gamma$ [mN/m]
| Density
---
$\rho$ [g/mL]
| Conductivity
---
$\sigma$ [S/m]
Water + NaCl | 0 | 1.78 | 82 | 1.12 | 30
Water | 0 | 1 | 72 | 1 | -
Water+Glycerol | 25 | 2.16 | 71 | 1.06 | -
Water+Glycerol | 50 | 5.71 | 69 | 1.13 | -
Water+Glycerol | 60 | 11.5 | 68 | 1.15 | -
Table 1: Liquid properties used in the electrical (drop-drop) and imaging
(drop-bulk) experiments. Figure 12: Fringes at the drop-bath interface before
the drop coalesces with the bulk.

Table 1 shows the properties of fluids used in this study. Different glycerol
fractions (by mass) were mixed with deionized water to tune the viscosity. The
density and surface tension of the mixture were estimated according to the
reference [2].

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