Generation and stability of toroidal droplets in a viscous liquid

Ordered structures formed by nematic topological defects and their transformation with changing the Euler characteristics

Ordered chain structures from topological defects of opposite charges ("necklaces" of defects) were prepared and their dynamics and cooperative rearrangement were investigated. We studied topological defects in nematic films with change of the Euler characteristic induced by temperature. Topological defects emerged due to competing surface anchoring on the nematic-isotropic and nematic-solid interfaces. Transformation of the structure with a circular chain from topological defects to the structure with a single defect and then to a structure without defects takes place as the nematic geometry changes. The temporal evolution of the number of topological defects at their annihilation in the chains differs from coarsening in two-dimensional (2D) and 3D geometry.

Evolution dynamics of thin liquid structures investigated using a phase-field model

Liquid structures of thin-films and torus droplets are omnipresent in daily lives. The morphological evolution of liquid structures suspending in another immiscible fluid and sitting on a solid substrate is investigated by using three-dimensional (3D) phase-field (PF) simulations. Here, we address the evolution dynamics by scrutinizing the interplay of surface energy, kinetic energy, and viscous dissipation, which is characterized by Reynolds number Re and Weber number We. We observe special droplet breakup phenomena by varying Re and We. In addition, we gain the essential physical insights into controlling the droplet formation resulting from the morphological evolution of the liquid structures by characterizing the top and side profiles under different circumstances. We find that the shape evolution of the liquid structures is intimately related to the initial shape, Re, We as well as the intrinsic wettability of the substrate. Furthermore, it is revealed that the evolution dynamics are determined by the competition between the coalescence phenomenology and the hydrodynamic instability of the liquid structures. For the coalescence phenomenology, the liquid structure merges onto itself, while the hydrodynamic instability leads to the breakup of the liquid structure. Last but not least, we investigate the influence of wall relaxation on the breakup outcome of torus droplets on substrates with different contact angles. We shed light on how the key parameters including the initial shape, Re, We, wettability, and wall relaxation influence the droplet dynamics and droplet formation. These findings are anticipated to contribute insights into droplet-based systems, potentially impacting areas like ink-jet printing, drug delivery systems, and microfluidic devices, where the interplay of surface energy, kinetic energy, and viscous dissipation plays a crucial role.

When Bulk Matters: Disentanglement of the Role of Polyelectrolyte/Surfactant Complexes at Surfaces and in the Bulk of Foam Films

Foam films display exciting systems as on one hand they dictate the performance of macroscopic foams and on the other hand they allow studies of surface forces. With regard to surface forces, we attempt to disentangle the effect of the foam film surfaces and the foam film bulk. For that, we study the influence of salt (LiBr) on foam films formed by mixtures of oppositely charged polyelectrolyte and surfactant: anionic monosulfonated polyphenylene sulfone (sPSO₂-220) and cationic tetradecyltrimethylammonium bromide (C₁₄TAB). Adding a small amount of salt (≤10^-3 M) decreases the foam film stability due to a weakened electrostatic net repulsion. In contrast, a large amount of salt (10^-2 M) unexpectedly increases the foam film stability. Disjoining pressure isotherms reveal that the increased stability is due to an additional steric stabilization, which is attributed to sPSO₂-220/C₁₄TAB complexes in the film bulk. These bulk complexes also contribute to the measured apparent surface potential between the two air/water interfaces. We find, for the first time, the formation of Newton black films for mixtures of anionic polyelectrolytes and cationic surfactants.

Embedded Core-Shell 3D Printing (eCS3DP) with Low-Viscosity Polysiloxanes

Flexible core-shell 3D structures are essential for the development of soft sensors and actuators. Despite recent advancements in 3D printing, the fabrication of flexible 3D objects with internal architectures (such as channels and void spaces) remains challenging with liquid precursors due to the difficulty to maintain the printed structures. The difficulty of such fabrication is prominent especially when low-viscosity polysiloxane resins are used. This study presents a unique approach to applying direct ink writing (DIW) 3D printing in a three-phase system to overcome this limitation. We performed core-shell 3D printing using a low-viscosity commercial polysiloxane resin (Ecoflex 10) as shell inks combined with a coaxially extruded core liquid (Pluronic F127) in Bingham plastic microparticulate gels (ethanol gel). In the process termed embedded core-shell 3D printing (eCS3DP), we highlighted the dependence of the rheological characteristics of the three fluids on the stability of the printed core-shell filament. With the core liquid with a sufficiently high concentration of Pluronic F127 (30 w/w%; σ_y = 158.5 Pa), the interfacial instability between the shell liquid and core liquid was suppressed; the removal of the core liquid permitted the fabrication of perfusable channels. We identified the printing conditions to ensure lateral attachments of printed core-shell filaments. Interestingly, judicious selection of the rheological properties and flow rates of three phases allowed the formation of droplets consisting of core liquids distributed along the printed filaments. eCS3DP offers a simple route to fabricate 3D structures of a soft elastomeric matrix with embedded channels and should serve as a useful tool for DIW-based fabrication of flexible wearable devices and soft robotic components.

Experimental Dispersion Relation of Surface Waves along a Torus of Fluid

We report the observation of gravity-capillary waves on a torus of fluid. By means of an original technique, a stable torus is achieved by depositing water on a superhydrophobic groove with a shallow wedge-shaped channel running along its perimeter. Using a spatiotemporal optical measurement, we report the full dispersion relation of azimuthal waves propagating along the inner and outer torus borders, highlighting several branches modeled as varicose, sinuous, and sloshing modes. Standing azimuthal waves are also studied leading to polygonlike patterns arising on the two torus borders with a number of sides different when a tunable decoupling of the two interfaces occurs. The quantized nature of the dispersion relation is also evidenced.

Controlling the breakup of toroidal liquid films on solid surfaces

The breakup of a slender filament of liquid driven by surface tension is a classical fluid dynamics stability problem that is important in many situations where fine droplets are required. When the filament is resting on a flat solid surface which imposes wetting conditions the subtle interplay with the fluid dynamics makes the instability pathways and mode selection difficult to predict. Here, we show how controlling the static and dynamic wetting of a surface can lead to repeatable switching between a toroidal film of an electrically insulating liquid and patterns of droplets of well-defined dimensions confined to a ring geometry. Mode selection between instability pathways to these different final states is achieved by dielectrophoresis forces selectively polarising the dipoles at the solid-liquid interface and so changing both the mobility of the contact line and the partial wetting of the topologically distinct liquid domains. Our results provide insights into the wetting and stability of shaped liquid filaments in simple and complex geometries relevant to applications ranging from printing to digital microfluidic devices.

Frame-Guided Synthesis of Polymeric Colloidal Discs

Anisotropic colloidal particles are important building blocks for the studies of self-assembly, which are visualized models for basic research and can be used to construct structured materials. Discs are one of the most typical anisotropic colloids; however, the synthesis of monodisperse colloidal discs with well-defined shape remains a challenge. Here we report a novel strategy for synthesizing polymeric discs based on frame-guided droplet shrinkage. This was realized by creating frame/liquid core/shell rings and utilizing the shrinking instability of the liquid rings. The resulting disc's shape parameters are tunable. The method is general, is not limited to specific polymers, solvents, and frames, and therefore has the potential to afford a variety of polymer discs. We also demonstrate the possibility of tuning the surface chemistry of the discs through surface-initiated polymerization. The frame-guided droplet shrinkage method opens up a new way to design and fabricate colloidal particles.

Freeform 3D printing of soft matters: recent advances in technology for biomedical engineering

In the last decade, an emerging three-dimensional (3D) printing technique named freeform 3D printing has revolutionized the biomedical engineering field by allowing soft matters with or without cells to be printed and solidified with high precision regardless of their poor self-supportability. The key to this freeform 3D printing technology is the supporting matrices that hold the printed soft ink materials during omnidirectional writing and solidification. This approach not only overcomes structural design restrictions of conventional layer-by-layer printing but also helps to realize 3D printing of low-viscosity or slow-curing materials. This article focuses on the recent developments in freeform 3D printing of soft matters such as hydrogels, cells, and silicone elastomers, for biomedical engineering. Herein, we classify the reported freeform 3D printing systems into positive, negative, and functional based on the fabrication process, and discuss the rheological requirements of the supporting matrix in accordance with the rheological behavior of counterpart inks, aiming to guide development and evaluation of new freeform printing systems. We also provide a brief overview of various material systems used as supporting matrices for freeform 3D printing systems and explore the potential applications of freeform 3D printing systems in different areas of biomedical engineering.

Observation of the Resonance Frequencies of a Stable Torus of Fluid

We report the first quantitative measurements of the resonance frequencies of a torus of fluid confined in a horizontal Hele-Shaw cell. By using the unwetting property of a metal liquid, we are able to generate a stable torus of fluid with an arbitrary aspect ratio. When subjected to vibrations, the torus displays azimuthal patterns at its outer periphery. These lobes oscillate radially, and their number n depends on the forcing frequency. We report the instability "tongues" of the patterns up to n=25. These resonance frequencies are well explained by adapting to a fluid torus the usual drop model of Rayleigh. This approach could be applied to the modeling of large-scale structures arisen transiently in vortex rings in various domains.

Non-monotonic, lily-like twist distribution in toroidal nematics

Toroidal nematics are droplets of nematic liquid crystals in the form of a circular torus. When the nematic director is subject to planar degenerate boundary conditions, the bend-only director field with vector lines along the parallels of all nested torodial shells is an equilibrium solution for all values of the elastic constants. Local stability analyses have shown that an instability is expected to occur for sufficiently small values of the twist elastic constant. It is natural to wonder whether in this regime the global equilibrium would be characterized by a monotonic twist, or not. In the former case, the twist distribution over the torus' circular cross-section would be represented pictorially by a fennel-like surface emanating from the centre. We prove that instead the stable twist distribution is represented by a lily-like surface. Thus, generically the twist distribution is not monotonic and its maximum may fall well within the torus, far away from the boundary. To cope with the peculiar complexity of the elastic free-energy functional in the fully non-linear setting, we developed an ad hoc deep-learning optimization method, which here is also further validated and documented for it promises to be applicable to other similar problems, equally intractable analytically.

Thickness-Dependent Phase Transition Drives Nanoridge-to-Mesa Instability in Micellar Freestanding Films

Understanding fluxes and instabilities within freestanding ultrathin films is necessary for a better understanding of, and control over, the stability and lifetime of foams and emulsions. In micellar foam films, confinement-induced layering of micelles leads to stepwise thinning or stratification that occurs by the expansion of thinner, darker domains. Often, because of a nanoridge-to-mesa instability, one or more brighter white spots or "mesas" appear at the circular moving front between thinner domains and the thicker (less dark) surrounding film. Previous studies assume that the instability and the appearance of white spots are similar to the capillarity-driven Rayleigh instability that leads to the breakup of a coherent liquid jet. Using the IDIOM (interferometry digital imaging optical microscopy) protocols we recently developed, we characterize the nanoridge-to-mesa instability with exquisite spatiotemporal resolution (thickness <1 nm, time <1 ms). The instability could be classified as a Rayleigh instability if a similar sequence of thick and thin undulations is visualized around the expanding domains. However, quantitative analysis reveals that only mesas grow in size after the instability, whereas the rest of the nanoridge preserves its shape. By analogy to the phase separation into compositionally distinct regions, we show that the spontaneous nucleation of thicker mesas in stratifying films is a phase transition driven by the oscillatory nature of the free-energy functional.

Depletion zones and crystallography on pinched spheres

Understanding the interplay between ordered structures and substrate curvature is an interesting problem with versatile applications, including functionalization of charged supramolecular surfaces and modern microfluidic technologies. In this work, we investigate the two-dimensional packing structures of charged particles confined on a pinched sphere. By continuously pinching the sphere, we observe cleavage of elongated scars into pleats, proliferation of disclinations, and subsequently, emergence of a depletion zone at the negatively curved waist that is completely void of particles. We systematically study the geometrics and energetics of the depletion zone, and reveal its physical origin as a finite size effect, due to the interplay between Coulomb repulsion and concave geometry of the pinched sphere. These results further our understanding of crystallography on curved surfaces, and have implications in design and manipulation of charged, deformable interfaces in various applications.

One-dimensional reduction of viscous jets. II. Applications

In a companion paper [Phys. Rev. E 97, 043115 (2018)10.1103/PhysRevE.97.043115], a formalism allowing to describe viscous fibers as one-dimensional objects was developed. We apply it to the special case of a viscous fluid torus. This allows to highlight the differences with the basic viscous string model and with its viscous rod model extension. In particular, an elliptic deformation of the torus section appears because of surface tension effects, and this cannot be described by viscous string nor viscous rod models. Furthermore, we study the Rayleigh-Plateau instability for periodic deformations around the perfect torus, and we show that the instability is not sufficient to lead to the torus breakup in several droplets before it collapses to a single spherical drop. Conversely, a rotating torus is dynamically attracted toward a stationary solution, around which the instability can develop freely and split the torus in multiple droplets.

Breakup dynamics of toroidal droplets in shear-thinning fluids

We use toroidal droplets to study the breakup dynamics of a Newtonian liquid jet in the presence of rheologically nonlinear materials. We find that the droplets resist breakup for times that are longer than those in the presence of Newtonian liquids. More importantly, we show that our experiments can be explained by incorporating the nonlinearities into the linear treatment of the problem through the strain-rate-dependent viscosity. Finally, we show that the scaling factor required to relate the viscosity to the growth rate associated to unstable modes is given by the elastic modulus of the outside material, illustrating that both the viscoelastic and shear-thinning nature of the outside material play a crucial role in the dynamics of the problem.

Toroidal Droplets: Growth Rates, Dispersion Relations, and Behavior in the Thick-Torus Limit

Toroidal droplets in a viscous liquid are unstable and transform into single or multiple spherical droplets. For thin tori, this can happen via the Rayleigh-Plateau instability causing the breakup of cylindrical jets. In contrast, for thick tori, this can happpen via the shrinking of the "hole". In this work, we use the thin-torus limit to directly measure the growth rate associated with capillary disturbances. In the case of toroidal droplets inside a much more viscous liquid, we even obtain the full dispersion relation, which is in agreement with theoretical results for cylindrical jets. For thick tori, we employ particle image velocimetry to determine the flow field of a sinking toroidal drop inside another viscous liquid. We find that the presence of the "hole" greatly suppresses one of the circulation loops expected for sinking cylinders. Finally, using the flow field of a shrinking toroidal droplet and the time-reversal symmetry of the Stokes equations, we theoretically predict the expected shape deformation of an expanding torus and confirm the result experimentally using charged toroidal droplets.

Capillary flow and mechanical buckling in a growing annular bacterial colony

A growing bacterial colony is a dense suspension of an increasing number of cells capable of individual as well as collective motion. After inoculating Pseudomonas aeruginosa over an annular area on an agar plate, we observe the growth and spread of the bacterial population, and model the process by considering the physical effects that account for the features observed. Over a course of 10-12 hours, the majority of bacteria migrate to and accumulate at the edges. We model the capillary flow induced by imbalanced evaporation flux as the cause for the accumulation, much like the well-known coffee stain phenomenon. Simultaneously, periodic buckles or protrusions occur at the inner edge. These buckles indicate that the crowding bacteria produce a jam, transforming the densely packed population at the inner edge to a solid state. The continued bacterial growth produces buckles. Subsequently, a ring of packed bacteria behind the inner edge detach from it and break into pieces, forming bacterial droplets. These droplets slowly coalesce while they continually grow and collectively surf on the agar surface in the region where the colony had previously spread over. Our study shows a clear example of how fluid dynamics and elasto-mechanics together govern the bacterial colony pattern evolution.

Soft polyhedral particles based on cubic liquid crystalline emulsion droplets

Soft polyhedral particles based on variations of the cubic symmetry group are produced from a precursor emulsion by extracting solvent to grow facets on the droplets. The droplets transform into liquid crystals with solid-like rheology and controlled size and shape. Small-angle X-ray scattering confirms a bicontinuous cubic liquid crystalline phase forms from aqueous glycerol monoolein and is responsible for the particle faceting observed. Different polyhedra are produced by varying face growth rates through control of precursor droplet size, system temperature, and solubilization and adsorption of guest molecules. More exotic faceted shapes can be formed by the soft particles by applying asymmetric solvent removal gradients and by deforming the precursor droplets into, for example, ellipsoids before crystallization. The method is a powerful means to produce soft polyhedra, using continuous microfluidic or other approaches, or to act as templates for hard polyhedral particle synthesis.

Transition from turbulent to coherent flows in confined three-dimensional active fluids

Transport of fluid through a pipe is essential for the operation of macroscale machines and microfluidic devices. Conventional fluids only flow in response to external pressure. We demonstrate that an active isotropic fluid, composed of microtubules and molecular motors, autonomously flows through meter-long three-dimensional channels. We establish control over the magnitude, velocity profile, and direction of the self-organized flows and correlate these to the structure of the extensile microtubule bundles. The inherently three-dimensional transition from bulk-turbulent to confined-coherent flows occurs concomitantly with a transition in the bundle orientational order near the surface and is controlled by a scale-invariant criterion related to the channel profile. The nonequilibrium transition of confined isotropic active fluids can be used to engineer self-organized soft machines.

Toroidal drop under electric field: arbitrary drop-to-ambient fluid viscosity ratio

In the absence of external forces, a liquid toroidal drop freely suspended in another fluid shrinks towards its centre. It is shown that if the two phases are slightly conducting viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/ R , Q and λ , respectively, then the toroidal drop with volume 4 π /3 and having major radius ρ can become almost stationary when subjected to a uniform electric field aligned with the drop’s axis of symmetry. In this case, Q and electric capillary number Ca E that defines the ratio of electric stress to surface tension, are functions of R , ρ and λ and are found analytically. Those functions are relatively insensitive to λ , and for ρ ≥3, they admit simple approximations, which coincide with those obtained recently for λ =1. Streamlines inside and outside the toroidal drop for the same R and ρ but different λ are qualitatively similar.

Streaming from the Equator of a Drop in an External Electric Field

Tip streaming generates micron- and submicron-sized droplets when a thin thread pulled from the pointy end of a drop disintegrates. Here, we report streaming from the equator of a drop placed in a uniform electric field. The instability generates concentric fluid rings encircling the drop, which break up to form an array of microdroplets in the equatorial plane. We show that the streaming results from an interfacial instability at the stagnation line of the electrohydrodynamic flow, which creates a sharp edge. The flow draws from the equator a thin sheet which destabilizes and sheds fluid cylinders. This streaming phenomenon provides a new route for generating monodisperse microemulsions.

Charge-Induced Saffman-Taylor Instabilities in Toroidal Droplets

We show that charged toroidal droplets can develop fingerlike structures as they expand due to Saffman-Taylor instabilities. While these are commonly observed in quasi-two-dimensional geometries when a fluid displaces another fluid of higher viscosity, we show that the toroidal confinement breaks the symmetry of the problem, effectively making it quasi-two-dimensional and enabling the instability to develop in this three-dimensional situation. We control the expansion speed of the torus with the imposed electric stress and show that fingers are observed provided the characteristic time scale associated with this instability is smaller than the characteristic time scale associated with Rayleigh-Plateau break-up. We confirm our interpretation of the results by showing that the number of fingers is consistent with expectations from linear stability analysis in radial Hele-Shaw cells.

Liquid toroidal drop under uniform electric field

The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q =(2 R 2 +3 R +2)/(7 R 2 ), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca E , that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca E =0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca E >0. This work finds Q and Ca E such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius ρ and volume 4 π /3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca E depend on R and ρ , and for large ρ , e.g. ρ ≥3, they have simple approximations: Q ∼( R 2 + R +1)/(3 R 2 ) and Ca E ∼ 3 3 π ρ / 2   ( 6  ln  ⁡ ρ + 2  ln ⁡ [ 96 π ] − 9 ) / ( 12  ln  ⁡ ρ + 4  ln ⁡ [ 96 π ] − 17 )   ( R + 1 ) 2 / ( R − 1 ) 2 .

Toroidal-droplet instabilities in the presence of charge

Neutral toroidal droplets can break via the surface-tension-driven Rayleigh-Plateau instability. They can additionally exhibit a shrinking instability, which is also driven by surface tension, whereby the handle progressively disappears. We find that charging a toroidal droplet can qualitatively change the behavior and cause the droplet to expand. We successfully model the transition from shrinking to expanding, considering the pressure balance across the interface of the torus. However, despite the change in behavior, charged toroidal droplets end up breaking into spherical droplets. We quantify how the wavelength of the fastest unstable mode associated to this breakup depends on the applied voltage and compare the results with theoretical expectations for charged cylindrical jets.

Shrinking instability of toroidal droplets

Toroidal droplets are inherently unstable due to surface tension. They can break up, similar to cylindrical jets, but also exhibit a shrinking instability, which is inherent to the toroidal shape. We investigate the evolution of shrinking toroidal droplets using particle image velocimetry. We obtain the flow field inside the droplets and show that as the torus evolves, its cross-section significantly deviates from circular. We then use the experimentally obtained velocities at the torus interface to theoretically reconstruct the internal flow field. Our calculation correctly describes the experimental results and elucidates the role of those modes that, among the many possible ones, are required to capture all of the relevant experimental features.

Templated Solid-State Dewetting of Thin Silicon Films

Thin film dewetting can be efficiently exploited for the implementation of functionalized surfaces over very large scales. Although the formation of sub-micrometer sized crystals via solid-state dewetting represents a viable method for the fabrication of quantum dots and optical meta-surfaces, there are several limitations related to the intrinsic features of dewetting in a crystalline medium. Disordered spatial organization, size, and shape fluctuations are relevant issues not properly addressed so far. This study reports on the deterministic nucleation and precise positioning of Si- and SiGe-based nanocrystals by templated solid-state dewetting of thin silicon films. The dewetting dynamics is guided by pattern size and shape taking full control over number, size, shape, and relative position of the particles (islands dimensions and relative distances are in the hundreds nm range and fluctuate ≈11% for the volumes and ≈5% for the positioning).

Topological Symmetry Breaking in Viscous Coarsening

The crucial role of hydrodynamic pinch-off instabilities is evidenced in the coarsening stage of viscous liquids. The phase separation of a barium borosilicate glass melt is studied by in situ synchrotron tomography at high temperature. The high viscosity contrast between the less viscous phase and the more viscous phase induces a topological symmetry breaking: capillary breakups occur preferentially in the less viscous phase. As a result, contrasting morphologies are obtained in the two phases. This symmetry breaking is illustrated on three different glass compositions, corresponding to different volume fractions of the two phases. In particular, a fragmentation phenomenon, reminiscent of the end-pinching mechanism proposed by Stone and co-workers is evidenced in the less viscous phase.

Liquid toroidal drop in compressional flow with arbitrary drop-to-ambient fluid viscosity ratio

Existing experiments show that a sufficiently fat toroidal drop freely suspended in another liquid shrinks towards its centre to form a spherical drop. However, recent simulations reveal that if a liquid torus with circular cross section is embedded in a compressional same-viscosity flow that acts to expand the torus, then depending on the torus radius R and a capillary number Ca characterizing the balance between the viscous forces and the interfacial tension, the torus may either coalesce, expand indefinitely or attain a stationary shape. For each Ca less than 0.2, there is a single value of R , called the critical radius, for which the torus attains the stationary shape. Here, the drop-to-ambient fluid viscosity ratio, λ, is assumed to be arbitrary. The corresponding two-phase Stokes flow problem is solved for a liquid toroidal drop with circular cross section in terms of stream functions in the toroidal coordinates. When λ=1, the stream functions admit a closed-form integral representation for a drop of arbitrary axisymmetric shape. ‘Stationary’ circular tori minimize a certain measure of the normal velocity over the interface, and as in the case of λ=1, their radii are expected to predict the critical ones for arbitrary λ and Ca in a certain range (e.g. for Ca<0.2 when λ=1). Streamlines about ‘stationary’ circular tori are analysed for various Ca and λ.

Capillary rupture of suspended polymer concentric rings

We present the first experimental study on the simultaneous capillary instability amongst viscous concentric rings suspended atop an immiscible medium. The rings ruptured upon annealing, with three types of phase correlation between neighboring rings. In the case of weak substrate confinement, the rings ruptured independently when they were sparsely distanced, but via an out-of-phase mode when packed closer. If the substrate confinement was strong, the rings would rupture via an in-phase mode, resulting in radially aligned droplets. The concentric ring geometry caused a competition between the phase correlation of neighboring rings and the kinetically favorable wavelength, yielding an intriguing, recursive surface pattern. This frustrated pattern formation behavior was accounted for by a scaling analysis.

Doughnut-shaped soap bubbles

Soap bubbles are thin liquid films enclosing a fixed volume of air. Since the surface tension is typically assumed to be the only factor responsible for conforming the soap bubble shape, the realized bubble surfaces are always minimal area ones. Here, we consider the problem of finding the axisymmetric minimal area surface enclosing a fixed volume V and with a fixed equatorial perimeter L. It is well known that the sphere is the solution for V=L(3)/6π(2), and this is indeed the case of a free soap bubble, for instance. Surprisingly, we show that for V<αL(3)/6π(2), with α≈0.21, such a surface cannot be the usual lens-shaped surface formed by the juxtaposition of two spherical caps, but is rather a toroidal surface. Practically, a doughnut-shaped bubble is known to be ultimately unstable and, hence, it will eventually lose its axisymmetry by breaking apart in smaller bubbles. Indisputably, however, the topological transition from spherical to toroidal surfaces is mandatory here for obtaining the global solution for this axisymmetric isoperimetric problem. Our result suggests that deformed bubbles with V<αL(3)/6π(2) cannot be stable and should not exist in foams, for instance.

Particles with changeable topology in nematic colloids

We show that nematic colloids can serve as a highly variable and controllable platform for studying inclusions with changeable topology and their effects on the surrounding ordering fields. We explore morphing of toroidal and knotted colloidal particles into effective spheres, distinctively changing their Euler characteristic and affecting the surrounding nematic field, including topological defect structures. With toroidal particles, the inner nematic defect eventually transitions from a wide loop to a point defect (a small loop). Trefoil particles become linked with two knotted defect loops, mutually forming a three component link, that upon tightening transform into a two-component particle-defect loop link. For more detailed topological analysis, Pontryagin-Thom surfaces are calculated and visualised, indicating an interesting cascade of defect rewirings caused by the shape morphing of the knotted particles.

Domain expansion dynamics in stratifying foam films: experiments

The stability, rheology and applications of foams, emulsions and colloidal sols depend on the hydrodynamics and thermodynamics of thin liquid films that separate bubbles, drops and particles respectively. Thin liquid films containing micelles, colloidal particles, liquid crystals or polyelectrolyte-surfactant mixtures exhibit step-wise thinning or stratification, often attributed to the layer-by-layer removal of the aforementioned supramolecular structures. Stratification proceeds through emergence and growth of thinner circular domains within a thicker film, and the domain expansion dynamics are the focus of this study. Domain and associated thickness variation in foam films made from sodium dodecyl sulfate (SDS) micellar solutions are examined using a Scheludko-type cell with a novel technique we call Interferometry Digital Imaging Optical Microscopy (IDIOM). Below 100 nm, stratification and drainage cause a thickness-dependent variation in reflected light intensity, visualized as progressively darker shades of gray. We show that the domain expansion dynamics exhibit two distinct growth regimes with characteristic scaling laws. Initially, the radius of the isolated domains grows with square root time, and the expansion rate can be characterized by an apparent diffusion constant. In contrast, after a section of the expanding domain coalesces with the Plateau border, the contact line between domain and the surrounding thicker region moves a constant velocity. We show that a similar transition from a constant diffusivity to a constant velocity regime is also realized when a topological instability occurs at the contact line between the growing thinner isolated domain and the surrounding thicker film. Though several studies have focused on the expansion dynamics of isolated domains that exhibit a diffusion-like scaling, the change in expansion kinetics observed after domains contact with the Plateau border has not been reported and analyzed before.

Influence of slip on the Plateau-Rayleigh instability on a fibre

The Plateau-Rayleigh instability of a liquid column underlies a variety of fascinating phenomena that can be observed in everyday life. In contrast to the case of a free liquid cylinder, describing the evolution of a liquid layer on a solid fibre requires consideration of the solid-liquid interface. Here we revisit the Plateau-Rayleigh instability of a liquid coating a fibre by varying the hydrodynamic boundary condition at the fibre-liquid interface, from no slip to slip. Although the wavelength is not sensitive to the solid-liquid interface, we find that the growth rate of the undulations strongly depends on the hydrodynamic boundary condition. The experiments are in excellent agreement with a new thin-film theory incorporating slip, thus providing an original, quantitative and robust tool to measure slip lengths.

On the Stability of Rotating Drops

We consider the equilibrium and stability of rotating axisymmetric fluid drops by appealing to a variational principle that characterizes the equilibria as stationary states of a functional containing surface energy and rotational energy contributions, augmented by a volume constraint. The linear stability of a drop is determined by solving the eigenvalue problem associated with the second variation of the energy functional. We compute equilibria corresponding to both oblate and prolate shapes, as well as toroidal shapes, and track their evolution with rotation rate. The stability results are obtained for two cases: (i) a prescribed rotational rate of the system ("driven drops"), or (ii) a prescribed angular momentum ("isolated drops"). For families of axisymmetric drops instabilities may occur for either axisymmetric or non-axisymmetric perturbations; the latter correspond to bifurcation points where non-axisymmetric shapes are possible. We employ an angle-arc length formulation of the problem which allows the computation of equilibrium shapes that are not single-valued in spherical coordinates. We are able to illustrate the transition from spheroidal drops with a strong indentation on the rotation axis to toroidal drops that do not extend to the rotation axis. Toroidal drops with a large aspect ratio (major radius to minor radius) are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. Prolate spheroidal shapes occur if a drop of lower density rotates within a denser medium; these drops appear to be linearly stable. This work is motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013].

Equilibrium and stability of axisymmetric drops on a conical substrate under gravity

Motivated by recent investigations of toroidal tissue clusters that are observed to climb conical obstacles after self-assembly [Nurse et al., Journal of Applied Mechanics 79 (2012) 051013], we study a related problem of the determination of the equilibrium and stability of axisymmetric drops on a conical substrate in the presence of gravity. A variational principle is used to characterize equilibrium shapes that minimize surface energy and gravitational potential energy subject to a volume constraint, and the resulting Euler equation is solved numerically using an angle/arclength formulation. The resulting equilibria satisfy a Laplace-Young boundary condition that specifies the contact angle at the three-phase trijunction. The vertical position of the equilibrium drops on the cone is found to vary significantly with the dimensionless Bond number that represents the ratio of gravitational and capillary forces; a global force balance is used to examine the conditions that affect the drop positions. In particular, depending on the contact angle and the cone half-angle, we find that the vertical position of the drop can either increase ("the drop climbs the cone") or decrease due to a nominal increase in the gravitational force. Most of the equilibria correspond to upward-facing cones, and are analogous to sessile drops resting on a planar surface; however we also find equilibria that correspond to downward facing cones, that are instead analogous to pendant drops suspended vertically from a planar surface. The linear stability of the drops is determined by solving the eigenvalue problem associated with the second variation of the energy functional. The drops with positive Bond number are generally found to be unstable to non-axisymmetric perturbations that promote a tilting of the drop. Additional points of marginal stability are found that correspond to limit points of the axisymmetric base state. Drops that are far from the tip are subject to azimuthal instabilities with higher mode numbers that are analogous to the Rayleigh instability of a cylindrical interface. We have also found a range of completely stable solutions that correspond to small contact angles and cone half-angles.

Stability of toroidal droplets inside yield stress materials

We study the stability of toroidal droplets inside a yield stress material. Similar to toroidal droplets in a viscous liquid, the slenderness of the torus controls whether it breaks into spherical droplets or grows thicker towards its center to coalesce onto itself and form a single spherical droplet. However, unlike tori generated in a viscous liquid, the elasticity of the outer medium can prevent either or both of these instabilities; this depends on the slenderness of the torus. Interestingly, we find that the value of the tube radius needed to prevent breakup is always larger than the value of the radius of the handle to prevent growth. This reflects the different deformations experienced by the yield stress material in either process. A simple model balancing the surface tension stress, which drives the evolution of the torus, and the yield stress, which favors its stability, accounts for all of our observations.

Topological defects in flat geometry: the role of density inhomogeneity

Topological defects are found in particles confined to planar disks interacting via the 1/r Coulomb potential. The total interior topological charge is found to monotonically converge to a negative value as the energy decreases during the relaxation process regardless of initial configurations; it is more negative in a larger cluster. The comparison with a uniform hyperbolic tessellation reveals an underlying hyperbolic structure in a low-energy configuration where the particle density increases from the center of the disk to its boundary. An elliptic structure is identified in an opposite particle distribution where the particle density decreases from the center to the edge of the disk. The novel mechanism of density inhomogeneity driven topological defects as well as the underlying geometric structure may shed light in understanding a wide variety of relevant systems.

Instability of a rotating liquid ring

It is shown numerically that a rotating inviscid liquid ring has a temporally oscillating state, where the radius of the ring varies periodically because of the competition between the centrifugal force and the centripetal force caused by the surface tension. Stability analysis reveals that an enlarging or shrinking ring is unstable to a varicose-type mode, which is affected by both the radial velocity and the radius ratio between the cross section and the ring. Furthermore, uniform rotation of a ring leads to a traveling unstable mode, whose frequency is determined by a simple sinuous mode, while the surface shape is modulated by the varicose mode and twisted by the rotation-induced Coriolis force.

Crystalline particle packings on constant mean curvature (Delaunay) surfaces

We investigate the structure of crystalline particle arrays on constant mean curvature (CMC) surfaces of revolution. Such curved crystals have been realized physically by creating charge-stabilized colloidal arrays on liquid capillary bridges. CMC surfaces of revolution, classified by Delaunay in 1841, include the 2-sphere, the cylinder, the vanishing mean curvature catenoid (a minimal surface), and the richer and less investigated unduloid and nodoid. We determine numerically candidate ground-state configurations for 1000 pointlike particles interacting with a pairwise-repulsive 1/r(3) potential, with distance r measured in three-dimensional Euclidean space R(3). We mimic stretching of capillary bridges by determining the equilibrium configurations of particles arrayed on a sequence of Delaunay surfaces obtained by increasing or decreasing the height at constant volume starting from a given initial surface, either a fat cylinder or a square cylinder. In this case, the stretching process takes one through a complicated sequence of Delaunay surfaces, each with different geometrical parameters, including the aspect ratio, mean curvature, and maximal Gaussian curvature. Unduloids, catenoids, and nodoids all appear in this process. Defect motifs in the ground state evolve from dislocations at the boundary to dislocations in the interior to pleats and scars in the interior and then isolated sevenfold disclinations in the interior as the capillary bridge narrows at the waist (equator) and the maximal (negative) Gaussian curvature grows. We also check theoretical predictions that the isolated disclinations are present in the ground state when the surface contains a geodesic disk with integrated Gaussian curvature exceeding -π/3. Finally, we explore minimal energy configurations on sets of slices of a given Delaunay surface, and we obtain configurations and defect motifs consistent with those seen in stretching.

Stable nematic droplets with handles

We stabilize nematic droplets with handles against surface tension-driven instabilities, using a yield-stress material as outer fluid, and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy, resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ∼20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional -1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy.

Splash control of drop impacts with geometric targets

Drop impacts on solid and liquid surfaces exhibit complex dynamics due to the competition of inertial, viscous, and capillary forces. After impact, a liquid lamella develops and expands radially, and under certain conditions, the outer rim breaks up into an irregular arrangement of filaments and secondary droplets. We show experimentally that the lamella expansion and subsequent breakup of the outer rim can be controlled by length scales that are of comparable dimension to the impacting drop diameter. Under identical impact parameters (i.e., fluid properties and impact velocity) we observe unique splashing dynamics by varying the target cross-sectional geometry. These behaviors include (i) geometrically shaped lamellae and (ii) a transition in splashing stability, from regular to irregular splashing. We propose that regular splashes are controlled by the azimuthal perturbations imposed by the target cross-sectional geometry and that irregular splashes are governed by the fastest-growing unstable Plateau-Rayleigh mode.

The shrinking instability of toroidal liquid droplets in the Stokes flow regime

We analyze the stability and dynamics of toroidal liquid droplets. In addition to the Rayleigh instabilities akin to those of a cylindrical droplet there is a shrinking instability that is unique to the topology of the torus and dominates in the limit that the aspect ratio is near one (fat tori). We first find an analytic expression for the pressure distribution inside the droplet. We then determine the velocity field in the bulk fluid, in the Stokes flow regime, by solving the biharmonic equation for the stream function. The flow pattern in the external fluid is analyzed qualitatively by exploiting symmetries. This elucidates the detailed nature of the shrinking mode and the swelling of the cross-section following from incompressibility. Finally the shrinking rate of fat toroidal droplets is derived by energy conservation.