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We present a theory on the coalescence of 2 spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid neck formed upon coalescence was formulated as an initial value problem and then solved to yield an exact solution without free parameters, with its 2 asymptotic approximations reproducing the well-known scaling relations in the inertially limited viscous and inertial regimes. The viscous-to-inertial crossover observed in previous research is also recovered by the theory, rendering the collapse of data of different viscosities onto a single curve.

We perform direct numerical simulations of surfactant-laden droplets in homogeneous isotropic turbulence with Taylor Reynolds number $Re_\\lambda \\approx 180$. The droplets are modelled using the volume-of-fluid method, and the soluble surfactant is transported using an advection–diffusion equation. Effects of surfactant on the droplet and local flow statistics are well approximated using a lower, averaged value of surface tension, thus allowing us to extend the framework developed by Hinze (AIChE J., vol. 1, no. 3, 1955, pp. 289–295) and Kolmogorov (Dokl. Akad. Navk. SSSR, vol. 66, 1949, pp. 825–828) for surfactant-free bubbles to surfactant-laden droplets. We find that surfactant-induced tangential stresses play a minor role in this set-up, thus allowing us to extend the Kolmogorov–Hinze framework to surfactant-laden droplets. The Kolmogorov–Hinze scale $d_H$ is indeed found to be a pivotal length scale in the droplets’ dynamics, separating the coalescence-dominated (droplets smaller than $d_H$) and the breakage-dominated (droplets larger than $d_H$) regimes in the droplet size distribution. We find that droplets smaller than $d_H$ have a rather compact, regular, spheroid-like shape, whereas droplets larger than $d_H$ have long, convoluted, filamentous shapes with a diameter equal to $d_H$. This results in very different scaling laws for the interfacial area of the droplet. The normalized area, $A/d_H^2$, of droplets smaller than $d_H$ is proportional to $(d/d_H)^2$, while the area of droplets larger than $d_H$ is proportional to $(d/d_H)^3$, where $d$ is the droplet characteristic size. We further characterize the large filamentous droplets by computing the number of handles (loops of the dispersed phase extending into the carrier phase) and voids (regions of the carrier fluid completely enclosed by the dispersed phase) for each droplet. The number of handles per unit length of filament scales inversely with surface tension. The number of voids is proportional to the droplet size and independent of surface tension. Handles are indeed an unstable feature of the interface and are destroyed by the restoring effect of surface tension, whereas voids can move freely in the interior of the droplets, unaffected by surface tension.

General particle dynamics (GPD), which is a novel meshless numerical method, is proposed to simulate the initiation, propagation and coalescence of 3D pre-existing penetrating and embedded flaws under biaxial compression. The failure process for rock-like materials subjected to biaxial compressive loads is investigated using the numerical code GPD3D. Moreover, internal crack evolution processes are successfully simulated using GPD3D. With increasing lateral stress, the secondary cracks keep growing in the samples, while the growth of the wing cracks is restrained. The samples are mainly split into fragments in a shear failure mode under biaxial compression, which is different from the splitting failure of the samples subjected to uniaxial compression. For specimens with macroscopic pre-existing flaws, the simulated types of cracks, the simulated coalescence types and the simulated failure modes are in good agreement with the experimental results.

Nuclear bodies are RNA and protein-rich, membraneless organelles that play important roles in gene regulation. The largest and most well-known nuclear body is the nucleolus, an organelle whose primary function in ribosome biogenesis makes it key for cell growth and size homeostasis. The nucleolus and other nuclear bodies behave like liquid-phase droplets and appear to condense from the nucleoplasm by concentration-dependent phase separation. However, nucleoli actively consume chemical energy, and it is unclear how such nonequilibrium activity might impact classical liquid–liquid phase separation. Here, we combine in vivo and in vitro experiments with theory and simulation to characterize the assembly and disassembly dynamics of nucleoli in earlyCaenorhabditis elegansembryos. In addition to classical nucleoli that assemble at the transcriptionally active nucleolar organizing regions, we observe dozens of “extranucleolar droplets” (ENDs) that condense in the nucleoplasm in a transcription-independent manner. We show that growth of nucleoli and ENDs is consistent with a first-order phase transition in which late-stage coarsening dynamics are mediated by Brownian coalescence and, to a lesser degree, Ostwald ripening. By manipulatingC. eleganscell size, we change nucleolar component concentration and confirm several key model predictions. Our results show that rRNA transcription and other nonequilibrium biological activity can modulate the effective thermodynamic parameters governing nucleolar and END assembly, but do not appear to fundamentally alter the passive phase separation mechanism.

Concentrated emulsions flowing through channels of varying widths are omnipresent in daily life, from dispensing mayonnaise in our kitchens to large-scale industrial processing of food, pharmaceuticals, etc. Local changes in channel geometry affect the stability of emulsions over length scales far beyond the droplet magnitude, for example through propagation of coalescence events called a coalescence avalanche. The underlying mechanisms are not well understood. In this work, we investigated the stability of concentrated emulsions flowing through microchannels featuring a constriction. We found that in this model geometry, the acceleration of the droplets induced near the entrance of the constriction triggers a coalescence event between the leading and the trailing droplet, but only above a critical droplet velocity. This separation-induced coalescence event, in turn, was found to trigger a coalescence avalanche in the upstream direction. Analysis of the flow behavior through particle image velocimetry and particle tracking velocimetry revealed that the propagation also follows a separation-induced coalescence mechanism, due to the retraction of the interface of the trailing droplet upon coalescence and the corresponding acceleration of the liquid inside the coalesced fluid thread. The constriction ratio was found to enhance the coalescence occurrence but did not affect the speed of coalescence propagation.

Carbon allotropes built from rings of two-coordinate atoms, known as cyclo[n]carbons, have fascinated chemists for many years, but until now they could not be isolated or structurally characterized because of their high reactivity. We generated cyclo[18]carbon (C18) using atom manipulation on bilayer NaCl on Cu(111) at 5 kelvin by eliminating carbon monoxide from a cyclocarbon oxide molecule, C24O₆. Characterization of cyclo[18]carbon by high-resolution atomic force microscopy revealed a polyynic structure with defined positions of alternating triple and single bonds. The high reactivity of cyclocarbon and cyclocarbon oxides allows covalent coupling between molecules to be induced by atom manipulation, opening an avenue for the synthesis of other carbon allotropes and carbon-rich materials from the coalescence of cyclocarbon molecules.

We study the effect of surfactants on the dynamics of a drop-interface coalescence using full three-dimensional direct numerical simulations. We employ a hybrid interface-tracking/level-set method, which takes into account Marangoni stresses that arise from surface-tension gradients, interfacial and bulk diffusion and sorption kinetic effects. We validate our predictions against the experimental data of Blanchette and Bigioni (Nat. Phys., vol. 2, issue 4, 2006, pp. 254–257) and perform a parametric study that demonstrates the delicate interplay between the flow fields and those associated with the surfactant bulk and interfacial concentrations. The results of this work unravel the crucial role of the Marangoni stresses in the flow physics of coalescence, with particular attention paid to their influence on the neck reopening dynamics in terms of stagnation-point inhibition, and near-neck vorticity generation. We demonstrate that surfactant-laden cases feature a rigidifying effect on the interface compared with the surfactant-free case, a mechanism that underpins the observed surfactant-induced phenomena.

Here we provide a theoretical framework describing the generation of the fast jet ejected vertically out of a liquid when a bubble, resting on a liquid–gas interface, bursts. The self-consistent physical mechanism presented here explains the emergence of the liquid jet as a consequence of the collapse of the gas cavity driven by the low capillary pressures that appear suddenly around its base when the cap, the thin film separating the bubble from the ambient gas, pinches. The resulting pressure gradient deforms the bubble which, at the moment of jet ejection, adopts the shape of a truncated cone. The dynamics near the lower base of the cone, and thus the jet ejection process, is determined by the wavelength $\\unicode[STIX]{x1D706}^{\\ast }$ of the smallest capillary wave created during the coalescence of the bubble with the atmosphere which is not attenuated by viscosity. The minimum radius at the lower base of the cone decreases, and hence the capillary suction and the associated radial velocities increase, with the wavelength $\\unicode[STIX]{x1D706}^{\\ast }$ . We show that $\\unicode[STIX]{x1D706}^{\\ast }$ increases with viscosity as $\\unicode[STIX]{x1D706}^{\\ast }\\propto Oh^{1/2}$ for $Oh\\lesssim O(0.01)$ , with $Oh=\\unicode[STIX]{x1D707}/\\sqrt{\\unicode[STIX]{x1D70C}R\\unicode[STIX]{x1D70E}}$ the Ohnesorge number, $R$ the bubble radius and $\\unicode[STIX]{x1D70C}$ , $\\unicode[STIX]{x1D707}$ and $\\unicode[STIX]{x1D70E}$ indicating respectively the liquid density, viscosity and interfacial tension coefficient. The velocity of the extremely fast and thin jet can be calculated as the flow generated by a continuous line of sinks extending along the axis of symmetry a distance proportional to $\\unicode[STIX]{x1D706}^{\\ast }$ . We find that the jet velocity increases with the Ohnesorge number and reaches a maximum for $Oh=Oh_{c}$ , the value for which the crest of the capillary wave reaches the vertex of the cone, and which depends on the Bond number $Bo=\\unicode[STIX]{x1D70C}gR^{2}/\\unicode[STIX]{x1D70E}$ . For $Oh>Oh_{c}$ , the jet is ejected after a bubble is pinched off; in this regime, viscosity delays the formation of the jet, which is thereafter emitted at a velocity which is inversely proportional to the liquid viscosity.

The partial coalescence dynamics of a compound drop in a liquid pool is numerically investigated. We study the effect of the ratio of the inner to outer radii $(R_{r})$ of the compound drop while maintaining a constant liquid volume in the outer shell of the compound droplet. It is observed that for small values of the radius ratio, the coalescence dynamics is similar to that of a ‘simple’ drop, but the partial coalescence is suppressed for large values of $R_{r}$ . Increasing the value of $R_{r}$ decreases the distance migrated by the inner bubble in the downward direction inside the pool. The location of the bubble after coalescence is found to play an important role in the pinch-off process of the satellite drop. The influence of the governing dimensionless parameters on the coalescence dynamics has also been investigated.

Coalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180°. The numerical simulations capture the spontaneous jumping process, which follows the capillary–inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary–inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary–inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.