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DOE OSTI2022

Coalescence of GaP on V-Groove Si

In recent years, better understanding and control over the formation of crystalline defects during the direct epitaxy of III-V semiconductors on Si substrates via metal organic vapor phase epitaxy (MOVPE) has enabled large gains in III-V-on-Si solar cell efficiency, pointing to pathway to lower-cost, high-performance III-V solar cells. However, such results have only been achieved on costly chemo-mechanically-polished (CMP) Si wafers. The use of V-groove nanopatterned Si substrates has demonstrated similarly high-crystalline-quality III-V-on-Si epitaxy, but also can be combined with lower-cost polishing techniques. Although they offer a potential cost advantage, growth on V-groove substrates adds challenges not present for epitaxy on planar wafers (the III-V material must be coalesced into a thin film after an initial nucleation stage). MOVPE growth conditions that promote highly facet-selective lateral growth needed for coalescence are generally actively avoided for conventional thin fim growth, so growth conditions need to be re-developed for V-groove-based epitaxy. Additionally, coalescence add complexity to the dislocation dynamics related to lattice relaxation, so strategies used to keep threading dislocation density (TDD) low on planar substrates need to be re-tuned for V-grooves. We have studied the morphological evolution and dislocation dynamics of GaP grown on V-groove Si by MOVPE. Growth conditions of V/III=5,000 and Tg=800 degrees C were uniquely found to produce extremely smooth coalesced thin films, with an RMS roughness of 0.2 nm measured by AFM. Additionally, at this growth condition, we identified two regimes of growth determined by the width of the SiNx cap (a remnant of the nanopatterning process) at the top of the V-grooves. For narrow caps, the GaP coalesces into a thin film, and for wide caps, the GaP evolves into {1 1 1}-faceted diamonds that do not coalesce. We suggest the influence of Si from the sidewalls of the V-grooves on the surface reconstruction of the GaP as the mechanism for this effect, with reflection difference spectroscopy (RDS) and Si doping experiments supporting this theory. In addition to morphology, the dislocation dynamics of the system were studied with electron channeling contrast imaging (ECCI) and transmission electron microscopy (TEM). The TDD of the coalesced GaP films was found to be 5 x 10^7 cm^-2 after coalescence via ECCI, a level still too high for high-quality solar cells. However, misfit dislocations crossing multiple grooves greater than 20 micrometers long were observed in ECCI, suggesting that the V-grooves do not block dislocation glide. TEM prior to and after coalescence was used to distinguish between dislocation creation driven by growth conditions and coalescence. Finally, strategies to reduce the dislocation density to levels acceptable for solar cells will be discussed.

GaP↗

DOE OSTI2025

Atomic-Scale Dynamics of Five-Fold Twin Mediated Coalescence: Pathway-Dependent and Defect-Governed Nonclassical Growth Mechanisms

Defective crystals with distinct properties have been discovered in many systems. However, the growth mechanism of defective crystals is still poorly understood. Here, in this work, using a 5-fold twinned gold (Au) nanocrystal (NC) as a model system, three new coalescence pathways involving detwinning or twinning have been uncovered through atomic-scale dynamic observations in an aberration-corrected transmission electron microscope coupled with atomistic simulations. This demonstrates that beyond crystal size, coalescence growth dynamics involving 5-fold twins (5-FTs) are highly dependent on crystal defect density and the approach pathways of the crystals. When a 5-FT encounters a smaller 5-FT or a smaller NC in a face-to-face way, a new, larger 5-FT is produced at a relatively fast coalescence growth rate; while in a corner-to-corner way, the coalescence dynamics are more retarded and sluggish, which is conducive to the formation of complex multitwined structures rather than 5-FTs. This highlights that the planar defect density and crystal approach pathway influence the coalescence dynamics of crystals containing 5-FT. Moreover, a column-by-column grain boundary (GB) migration mechanism, which results in bent GBs, was also unveiled during the crystal coalescence process. These results enrich the general understanding of the crystal growth theory and provide new insights into the controllable fabrication of 5-FTs by crystal coalescence mechanisms.

crystallization↗

DOE OSTI2021

Identifying and tracking bubbles and drops in simulations: A toolbox for obtaining sizes, lineages, and breakup and coalescence statistics

Knowledge of bubble and drop size distributions in two-phase flows is important for characterizing a wide range of phenomena, including combustor ignition, sonar communication, and cloud formation. The physical mechanisms driving the background flow also drive the time evolution of these distributions. Accurate and robust identification and tracking algorithms for the dispersed phase are necessary to reliably measure this evolution and thereby quantify the underlying mechanisms in interface-resolving flow simulations. The identification of individual bubbles and drops traditionally relies on an algorithm used to identify connected regions. This traditional algorithm can be sensitive to the presence of spurious structures. A cost-effective refinement is proposed to maximize volume accuracy while minimizing the identification of spurious bubbles and drops. An accurate identification scheme is crucial for distinguishing bubble and drop pairs with large size ratios. The identified bubbles and drops need to be tracked in time to obtain breakup and coalescence statistics that characterize the evolution of the size distribution, including breakup and coalescence frequencies, and the probability distributions of parent and child bubble and drop sizes. An algorithm based on mass conservation is proposed to construct bubble and drop lineages using simulation snapshots that are not necessarily from consecutive time steps. These lineages are then used to detect breakup and coalescence events, and obtain the desired statistics. Accurate identification of large-size-ratio bubble and drop pairs enables accurate detection of breakup and coalescence events over a large size range. Accurate detection of successive breakup and coalescence events requires that the snapshot interval be an order of magnitude smaller than the characteristic breakup and coalescence times to capture these successive events while minimizing the identification of repeated confounding events. Together, these algorithms serve as a toolbox for detailed analysis of two-phase simulations, and enable insights into the mechanisms behind bubble and drop formation and evolution in flows of practical importance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Void coalescence and ductile failure in IN718 investigated via high-energy synchrotron X-ray tomography and diffraction

Ductile failure through the growth and coalescence of voids is of particular relevance for many engineering materials. Yet, the lack of experimental measurements of the mechanical state of the material at an appropriate length scale has limited further understanding of the regime of ductile failure. In this study, local grain-scale experimental measurements are used to determine their relevance for describing and predicting ductile failure. Additive manufactured materials, due to the presence of inherent porosity, as well as the ability to tailor internal porosity, provides a promising avenue to study ductile failure. Selective laser melting is used to manufacture a specially designed specimen with two large, internal voids, in addition to the natural porosity, which is characteristic to the production process. The initial porosity and its evolution upon tensile loading are characterized via micro tomography. Several locations of void coalescence are captured in the sample, revealing activity of multiple modes of failure. Finite element simulations, with a simplified J2 plasticity model, instantiated with the initial void structure, is deployed, in which geometric localizations of stress concentration corresponded with experimentally observed sites of coalescence but was inadequate in capturing the appropriate failure mechanism. Furthermore, from the experimental results of the far-field high-energy diffraction microscopy, the heterogeneous micromechanical state is identified and tracked around voids due to the local grain interactions. These experiments determine narrow bands of low stress triaxiality, at the onset of failure, which highlight the path of coalescence through intervoid shearing. Diffraction spot spreading analysis aided characterization of intragranular plasticity and strain heterogeneity, which can be coupled with high triaxiality to form the conditions inducive for coalescence through intervoid necking. In this study, the experimentally determined grain-scale description of the micromechanical state provides a physical basis that can accurately capture both the sites and the mechanism of void coalescence at the onset of ductile failure.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Designing a Convection‐Cloud Chamber for Collision‐Coalescence Using Large‐Eddy Simulation With Bin Microphysics

Abstract Collisional growth of cloud droplets is an essential yet uncertain process for drizzle and precipitation formation. To improve the quantitative understanding of this key component of cloud‐aerosol‐turbulence interactions, observational studies of collision‐coalescence in a controlled laboratory environment are needed. In an existing convection‐cloud chamber (the Pi Chamber), collisional growth is limited by low liquid water content and short droplet residence times. In this work, we use numerical simulations to explore various configurations of a convection‐cloud chamber that may intensify collision‐coalescence. We employ a large‐eddy simulation (LES) model with a size‐resolved (bin) cloud microphysics scheme to explore how cloud properties and the intensity of collision‐coalescence are affected by the chamber size and aspect ratio, surface roughness, side‐wall wetness, side‐wall temperature arrangement, and aerosol injection rate. Simulations without condensation and evaporation within the domain are first performed to explore the turbulence dynamics and wall fluxes. The LES wall fluxes are used to modify the Scalar Flux‐budget Model, which is then applied to demonstrate the need for non‐uniform side‐wall temperature (two side walls as warm as the bottom and the two others as cold as the top) to maintain high supersaturation in a tall chamber. The results of LES with full cloud microphysics reveal that collision‐coalescence is greatly enhanced by employing a taller chamber with saturated side walls, non‐uniform side‐wall temperature, and rough surfaces. For the conditions explored, although lowering the aerosol injection rate broadens the droplet size distribution, favoring collision‐coalescence, the reduced droplet number concentration decreases the frequency of collisions.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Impacts of Bulk Microphysics Scheme Structural Choices on Simulations of Rain Initiation Through Drop Coalescence

This study examines how different structural choices in bulk microphysics schemes impact the simulation of warm rain initiation. A single liquid category (SLC) approach prognosing up to four moments of a single drop size distribution (DSD) is compared to the traditional two-category, two-moment approach with separate DSDs for cloud and rain (four total prognostic variables). Different methods for calculating tendencies of the prognostic variables from drop collision-coalescence are also tested: a discretized numerical-integration approach, machine learning via neural networks, lookup tables, and traditional power law fits. Relative to simulations using a bin microphysics model, SLC gives smaller error overall than the two-category approach when numerical integration is used to calculate the collision-coalescence tendencies for both. Replacing the numerical integration with a pre-computed lookup table reduces computational cost with little loss of accuracy. However, using fitted power laws with SLC to represent the collision-coalescence tendencies substantially reduces accuracy and leads to an order of magnitude increase in error. It is also demonstrated that with SLC, reasonably accurate solutions are obtained using only three prognostic moments, while a two-moment SLC scheme leads to substantial error. Overall, both the choice of prognostic moments (e.g., SLC vs. two-category) and method to calculate the collision-coalescence tendencies are important to consider for minimizing errors in bulk schemes. SLC with a sufficiently detailed calculation of the collision-coalescence tendencies provides accurate solutions for a reasonable computational cost, providing a viable alternative to the traditional two-category, two-moment approach for bulk microphysics.

320 (cloud physics and chemistry)↗

DOE OSTI2024

Are turbulence effects on droplet collision–coalescence a key to understanding observed rain formation in clouds?

Rain formation is a critical factor governing the lifecycle and radiative forcing of clouds and therefore it is a key element of weather and climate. Cloud microphysics–turbulence interactions occur across a wide range of scales and are challenging to represent in atmospheric models with limited resolution. Based on past experiments and idealized numerical simulations, it has been postulated that cloud turbulence accelerates rain formation by enhancing drop collision–coalescence. We provide substantial evidence for significant impacts of turbulence on the evolution of cloud droplet size distributions and rain formation by comparing high-resolution observations of cumulus congestus clouds with state-of-the-art large-eddy simulations coupled with a Lagrangian particle-based microphysics scheme. Turbulent coalescence must be included in the model to accurately represent the observed drop size distributions, especially for drizzle drop sizes at lower heights in the cloud. Turbulence causes earlier rain formation and greater rain accumulation compared to simulations with gravitational coalescence only. The observed rain size distribution tail just above cloud base follows a power law scaling that deviates from theoretical scalings considering either a purely gravitation collision kernel or a turbulent kernel neglecting droplet inertial effects, providing additional evidence for turbulent coalescence in clouds. In contrast, large aerosols acting as cloud condensation nuclei (“giant CCN”) do not significantly impact rain formation owing to their long timescale to reach equilibrium wet size relative to the lifetime of rising cumulus thermals. Overall, turbulent drop coalescence exerts a dominant influence on rain initiation in warm cumulus clouds, with limited impacts of giant CCN.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Coalescence, the thermal model and multi-fragmentation: the energy and volume dependence of light nuclei production in heavy ion collisions

Here, we present results of a phase space coalescence approach within the UrQMD transport and -hybrid model for a very wide range of beam energies from SIS to LHC. The coalescence model is able to qualitatively describe the whole range of experimental data with a fixed set of parameters. Some systematic deviations are observed for very low beam energies where the role of feed down from heavier nuclei and multi-fragmentation becomes relevant. The coalescence results are mostly very close to the thermal model fits. However, both the coalescence approach as well as thermal fits are struggling to simultaneously describe the triton multiplicities measured with the STAR and ALICE experiment. The double ratio of tp/d 2 , in the coalescence approach, is found to be essentially energy and centrality independent for collisions of heavy nuclei at beam energies of E lab > 10A GeV. On the other hand the clear scaling of the d/p 2 and t/p 3 ratios with the systems volume is broken for peripheral collisions, where a canonical treatment and finite size effects become more important.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2023

Enhanced Deuteron Coalescence Probability in Jets

The transverse-momentum ($p_T$) spectra and coalescence parameters $B_2$ of (anti)deuterons are measured in p-p collisions at $\sqrt{s}=13$ $\text{TeV}$ for the first time in and out of jets. In this measurement, the direction of the leading particle with the highest $p_T$ in the event ($p^{lead}_T > 5 GeV/c$) is used as an approximation for the jet axis. The event is consequently divided into three azimuthal regions, and the jet signal is obtained as the difference between the toward region, that contains jet fragmentation products in addition to the underlying event (UE), and the transverse region, which is dominated by the UE. The coalescence parameter in the jet is found to be approximately a factor of 10 larger than that in the underlying event. This experimental observation is consistent with the coalescence picture and can be attributed to the smaller average phase-space distance between nucleons in the jet cone as compared with the underlying event. The results presented in this Letter are compared to predictions from a simple nucleon coalescence model, where the phase-space distributions of nucleons are generated using pythia8 with the Monash 2013 tuning, and to predictions from a deuteron production model based on ordinary nuclear reactions with parametrized energy-dependent cross sections tuned on data. The latter model is implemented in pythia8.3. Both models reproduce the observed large difference between in-jet and out-of-jet coalescence parameters, although the almost flat trend of the $B^{Jet}_2$ is not reproduced by the models, which instead give a decreasing trend.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Coalescence of GaP on V-Groove Si

With an increase of control over crystalline defects, metallorganic vapor phase epitaxy (MOVPE)-grown III-V-on-Si multijunction solar cells have seen rapid increases in efficiency in recent years, pointing to a promising path to lower cost III-V solar cells. However, the cost of chemo-mechanical polishing the Si wafers to prepare them for epitaxy is high. The use of V-groove nanopatterns enables similar defect reduction to that achieved on planar wafers, but the nanopatterns can be fabricated with a low-cost process. While V-grooves offer advantages over planar Si, they add complexity to the growth process. In particular, coalescence can cause the formation of threading dislocations, and the highly-directional growth conditions required for coalescence are unusual for MOVPE. We have studied the coalescence of GaP films nucleated directly on V-groove Si by MOVPE. We observed that for optimized growth conditions (V/III=5,000 and T=800 C) two growth modes were possible, and the resulting morphology depended on the exact geometry of the SiNx cap used to cover the (0 0 1)-oriented Si at the tops of the grooves. For caps with a width >100 nm, noncoalescing, nano ridge-like growth terminating in f1 1 1g facets was observed. For narrower caps, coalescence with an RMS roughness of 0.2 nm as measured by atomic force microscopy was observed. We will discuss mechanisms responsible for this phenomenon, including the role of Si from the substrate surface. The dislocation dynamics of this system were studied with electron channeling contrast imaging (ECCI) and transmission electron microscopy (TEM). We find that V-grooves do not block dislocation glide; ECCI measurements show misfit dislocations greater than 10 micrometers long observed to continue perpendicularly across neighboring V-grooves. In addition, all threading dislocations appear to lie on f1 1 1g planes, consistent with the GaP relaxing via glided-in glissile dislocations. The dislocation dynamics and morphological evolution of the coalescence of GaP on Si, possible mechanisms behind the observed phenomenons, and further dislocation mitigation strategies for these materials will be presented.

dislocation dynamics↗

DOE OSTI2025

Coalescence-Induced Spontaneous Shedding of Microdroplets on Superhydrophobic Surfaces Featuring Enclosed Micropillars with Hierarchical Roughness

This study investigated water vapor condensation on superhydrophobic surfaces (SHSs) featuring micropillars enclosed by wall lattices and having three-tier hierarchical roughness. A total of five samples were created with three (NW-J, W200-J and W400-J samples) having large micropillar depth (~6 μm) and two (NW-S and W200-S samples) having small micropillar depth (~1 μm). Two distinct condensate removal modes were observed during condensation: coalescence-induced jumping on samples with large micropillar depth and coalescence-induced shedding on samples with small micropillar depth. The results showed that the diameter of the shedding droplet on the W200-S sample having small micropillar depth could be as small as 107 μm, as compared to the theoretical critical diameter of 267 μm for gravitational shedding on the same sample. The enhanced functionality of the three-tier nanotextures on the W200-S sample could effectively suppress localized pinning of the three-phase contact line and Wenzel neck formation during the growth of condensate droplets. Consequently, during multidroplet coalescence, the released surface energy easily overcomes the solid–liquid adhesion, leading to spontaneous shedding of merged droplets. The inclusion of the wall lattice aids condensate growth by the droplet self-alignment along the walls and promoting coalescence. As a result, the W200-S sample exhibited the highest condensate collection as well. In conclusion, the proposed surface design has great potential for scaling up and implementation in heating, ventilation, and air-conditioning equipment due to the simplicity of the surface morphology and the facile spray-coating method used to achieve hierarchical roughness.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

The role of breakup and coalescence in fine-scale bubble-induced turbulence. I. Dynamics

We study the effect of bubble breakup and coalescence on fine-scale dynamics of bubbly turbulent flows using direct numerical simulations. We perform two different simulations of dilute bubbly flows of void fraction 0.5%: one with bubbles breaking up and coalescing and the other without these physical processes. The volume of the fluid method is used for simulating bubbles undergoing breakup and coalescence while the bubbles are treated as rigid spheres in the immersed boundary method simulation. The energy spectrum in both types of simulation, consistent with previous studies, exhibits a -3 slope. We follow a single infinitesimal fluid element as it evolves to understand velocity gradient dynamics using conditional mean trajectories. We note finite-time divergence when the fluid element evolves under the action of inertial and pressure dynamics. The inertial, pressure, and viscous velocity gradient dynamics, when considered individually, produce the same results for bubble-induced turbulence (BIT) as with the classical homogeneous isotropic turbulence (HIT). Yet when the overall velocity gradient dynamics is considered, BIT results in non-cyclic trajectories moving toward stable node and unstable saddle while classical HIT shows cyclic behavior in their trajectories that move toward the origin. Interestingly, both the volume of fluid and immersed boundary simulations produce similar results. Therefore, there are two main takeaways from this research. First, new velocity gradient models are needed for BIT as their velocity gradients behavior is entirely different from the HIT. Second, we can neglect the bubble topology, breakup, and coalescence while studying or modeling the fine-scale dynamics of BIT.

42 ENGINEERING↗

DOE OSTI2024

Numerical analysis of coalescence-induced bubble departure for enhanced boiling heat transfer

Boiling heat transfer plays a crucial role in a wide range of applications, such as power generation, refrigeration, electronics cooling, and pharmaceutics. Among the various factors that influence boiling heat transfer, the dynamics of vapor bubble nucleation, growth, and departure from the heated surface stand out as particularly important. An emerging phenomenon that can promote the departure of bubbles smaller than the Fritz diameter is coalescence-induced departure. If the dynamics of this process are fully understood, then surfaces can be engineered to promote faster bubble departure and substantially increase the performance of boiling heat transfer. Further, this work expands on published results by presenting a detailed numerical analysis of bubble coalescence and departure for a range of initial bubble diameters and size ratios between coalescing bubbles. Analysis of the results is focused on explaining how the release of surface energy and bubble surface dynamics lead to bubble departure, as well as fundamentally distinguishing capillary–inertial jumping and buoyant–inertial departure mechanisms across different bubble sizes and size ratios. The results show that both the initial sizes of the coalescing bubbles and the ratio between their sizes can determine whether the merged bubble will leave the surface through capillary–inertial jumping or buoyant departure. Below a certain bubble size, the release of surface energy by the merger is not sufficient to propel the merged bubble from the surface.

42 ENGINEERING↗

DOE OSTI2024

Rapidity dependence of antideuteron coalescence in pp collisions at $\sqrt{s}$ = 13 TeV with ALICE

The production yields of antideuterons and antiprotons are measured in pp collisions at a center-of-mass energy of $\sqrt{s}$ = 13 TeV, as a function of transverse momentum ( p T ) and rapidity ( y ), for the first time rapidity-differentially up to | y | = 0.7. The measured spectra are used to study the p T and rapidity dependence of the coalescence parameter B 2 , which quantifies the coalescence probability of antideuterons. The p T and rapidity dependence of the obtained B 2 is extrapolated for p T > 1.7 GeV/ c and | y | = 0.7 using the phenomenological antideuteron production model implemented in Pythia 8.3 as well as a baryon coalescence afterburner model based on EPOS 3. Such measurements are of interest to the astrophysics community, since they can be used for the calculation of the flux of antinuclei from cosmic rays, in combination with coalescence models.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Method and apparatus for front end gather/scatter memory coalescing

A system for processing gather and scatter instructions can implement a front-end subsystem, a back-end subsystem, or both. The front-end subsystem includes a prediction unit configured to determine a predicted quantity of coalesced memory access operations required by an instruction. A decode unit converts the instruction into a plurality of access operations based on the predicted quantity, and transmits the plurality of access operations and an indication of the predicted quantity to an issue queue. The back-end subsystem includes a load-store unit that receives a plurality of access operations corresponding to an instruction, determines a subset of the plurality of access operations that can be coalesced, and forms a coalesced memory access operation from the subset. A queue stores multiple memory addresses for a given load-store entry to provide for execution of coalesced memory accesses.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Method and apparatus for back end gather/scatter memory coalescing

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Molecular and structural basis of olfactory sensory neuron axon coalescence by Kirrel receptors

Projections from sensory neurons of olfactory systems coalesce into glomeruli in the brain. The Kirrel receptors are believed to homodimerize via their ectodomains and help separate sensory neuron axons into Kirrel2- or Kirrel3-expressing glomeruli. Here, we present the crystal structures of homodimeric Kirrel receptors and show that the closely related Kirrel2 and Kirrel3 have evolved specific sets of polar and hydrophobic interactions, respectively, disallowing heterodimerization while preserving homodimerization, likely resulting in proper segregation and coalescence of Kirrel-expressing axons into glomeruli. We show that the dimerization interface at the N-terminal immunoglobulin (IG) domains is necessary and sufficient to create homodimers and fail to find evidence for a secondary interaction site in Kirrel ectodomains. Furthermore, we show that abolishing dimerization of Kirrel3 in vivo leads to improper formation of glomeruli in the mouse accessory olfactory bulb as observed in Kirrel3 –/– animals. Our results provide evidence for Kirrel3 homodimerization controlling axonal coalescence.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Coalescence and splashing threshold for head-on collisions of liquid metal nanodroplets

Head-on collisions of liquid metal nanodroplets in a vacuum are investigated through molecular dynamics simulations in order to determine the transition threshold between the coalescing and splashing regimes for six different materials (aluminum, calcium, cerium, gold, platinum, and tin). Droplets of various sizes and initial speeds are simulated, and it is found that the Reynolds and Ohnesorge numbers are able to predict the transition between the coalescing and splashing regimes. An energy balance for coalescing droplets shows that the initial energy is mainly converted to thermal energy increasing the temperature of the combined droplets by several hundred to several thousand kelvin depending on the material, and this result is confirmed in the simulations. Furthermore, when splashing occurs, the number of smaller droplets formed and the spreading rate are found to be dependent on the initial size and initial speed of the original droplets.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗