The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations

Bézier Curve Method to Compute Various Meniscus Shapes

This paper is an extension of our earlier paper in which it was shown that the meniscus shape in a cylindrical capillary could be computed by solving the Young-Laplace equation via optimization of a Bézier curve. This work extends the previous work by demonstrating that the method is applicable to predict the meniscus shape not only in a cylindrical capillary but also in other cases, such as at a tilted plate, between two plates, and for a sessile drop. Numerous works have attempted previously to solve the Young-Laplace equation, and their results all agree with this paper's validating its method. All the preceding approaches, however, used special techniques to solve the differential equation, while the Bézier curve method proposed in this work is more simple, which allows it to maintain greater computational simplicity. Moreover, the Bézier curve method can be applied to solve many other different differential equations in the same way as shown in this work. The effect of the Bézier curve degree on the precision of prediction was also thoroughly investigated. It was found that the 4th degree Bézier curve was required to predict the meniscus shape precisely in a cylindrical capillary, against a tilted plate, and between two plates, while the 5th degree was required for the shape of the sessile drop.

Orbiting Self-Organization of Filament-Tethered Surface-Active Droplets

Dissipative chemical systems hold the potential to enable life-like behavior in synthetic matter, such as self-organization, motility, and dynamic switching between different states. Here, out-of-equilibrium self-organization is demonstrated by interconnected source and drain droplets at an air-water interface, which display dynamic behavior due to a hydrolysis reaction that generates a concentration gradient around the drain droplets. This concentration gradient interferes with the adhesion of self-assembled amphiphile filaments that grow from a source droplet. The chemical gradient sustains a unique orbiting of the drain droplet, which is proposed to be driven by the selective adhesion of the filaments to the front of the moving droplet, while filaments approaching from behind are destabilized upon contact with the hydrolysis product in the trail of the droplet. Potential applications are foreseen in the transfer of chemical signals amongst communicating droplets in rearranging networks, and the implementation of chemical reactions to drive complex positioning routines in life-like systems.

Biophysical determinants of cancer organotropism

Metastasis remains the leading cause of cancer lethality. The 'seed/soil' hypothesis provides the framework to explain this cancer phenomenon where the concept of organotropism has been in part mechanistically explained by the properties of the tumor cells and their compatibility with the stromal environment of the distal site. The 'mechanical' hypothesis counters that non-random seeding is driven solely by the circulation patterns and vascular networks of organ systems. We incorporate concepts of mechanobiology and revisit the two hypotheses to provide additional insights into the mechanisms that regulate organ selection during metastatic outgrowth. We focus on the latter stages of the metastatic cascade and examine the role of the endothelium in regulating organ selectivity.

Hybrid Nanoparticles at Fluid-Fluid Interfaces: Insight from Theory and Simulation

Hybrid nanoparticles that combine special properties of their different parts have numerous applications in electronics, optics, catalysis, medicine, and many others. Of the currently produced particles, Janus particles and ligand-tethered (hairy) particles are of particular interest both from a practical and purely cognitive point of view. Understanding their behavior at fluid interfaces is important to many fields because particle-laden interfaces are ubiquitous in nature and industry. We provide a review of the literature, focusing on theoretical studies of hybrid particles at fluid-fluid interfaces. Our goal is to give a link between simple phenomenological models and advanced molecular simulations. We analyze the adsorption of individual Janus particles and hairy particles at the interfaces. Then, their interfacial assembly is also discussed. The simple equations for the attachment energy of various Janus particles are presented. We discuss how such parameters as the particle size, the particle shape, the relative sizes of different patches, and the amphiphilicity affect particle adsorption. This is essential for taking advantage of the particle capacity to stabilize interfaces. Representative examples of molecular simulations were presented. We show that the simple models surprisingly well reproduce experimental and simulation data. In the case of hairy particles, we concentrate on the effects of reconfiguration of the polymer brushes at the interface. This review is expected to provide a general perspective on the subject and may be helpful to many researchers and technologists working with particle-laden layers.

Inkjet-Assisted Electroformation of Magnetically Guidable Water Striders for Interfacial Microfluidic Manipulation

Gerridae, colloquially called water striders, are a peculiar class of insects characterized by the extraordinary ability to walk on the surface of water bodies. Owing to this capacity, they constitute an ideal source of inspiration for designing untethered microdevices capable of navigating the interface between two fluids. Such steerable micrometric objects can be of great interest for various applications, ranging from the handling of floating objects to the remote control of microreactions and the manipulation of self-assembled monolayers. This paper describes the realization of artificial water striders via an inkjet-assisted electroforming approach. Inkjet deposition patterns the negative mask, which is subsequently filled with different layers of metals through electroforming. One of such layers is the magnetic alloy NiFe, which allows wireless propulsion of the striders by means of externally applied magnetic fields. The magnetic actuation tests prove good maneuverability at the water-air and silicone oil-air interfaces, with superior control over the speed and position of the devices. The surface of the devices is modified to tune its superficial energy in order to maximize buoyancy on these different combinations of fluids. A magnetic field-controlled strider manipulates a droplet and demonstrates collecting oil microdroplets and synthesizing platinum nanoparticles by chemical microreactions. Finally, the remotely operated microrobot could be employed in laboratories as a real avatar of chemists.

Energies Associated with a Meniscus along a Flat Vertical Wall

In this theoretical work, existing equations for the height and shape of a liquid meniscus along a vertical flat wall are used to estimate the volume and energies associated with its formation. A mechanism and an associated energy balance are proposed. Equations for the work of wetting, surface energy, gravitational energy, and dissipation are derived. This analysis shows that the energy spontaneously released during wetting is either stored in the air-liquid interface of the newly formed meniscus, stored in its bulk liquid as gravitational energy, or dissipated as heat. The absolute magnitude of these energies depends on the surface tension and density of the liquid as well as the wettability of the wall.

Calculation of the Meniscus Shape Formed under Gravitational Force by Solving the Young-Laplace Differential Equation Using the Bézier Curve Method

This work presents a method to calculate the meniscus shape by solving the differential equation based on the Young-Laplace equation. More specifically, the differential equation is solved by applying the cubic Bézier curve. A complicated nonlinear differential equation is solved using the Bézier control points and the least-squares method while maintaining computational simplicity. The results show all of the expected features of the meniscus under the gravitational force. A brief discussion is also made on the effect of the errors on the results. The method is further validated by its agreement with the numerical solutions reported in the existing literature.

Self-Organization Emerging from Marangoni and Elastocapillary Effects Directed by Amphiphile Filament Connections

Self-organization of meso- and macroscale structures is a highly active research field that exploits a wide variety of physicochemical phenomena, including surface tension, Marangoni flow, and (elasto)capillary effects. The release of surface-active compounds generates Marangoni flows that cause repulsion, whereas capillary forces attract floating particles via the Cheerios effect. Typically, the interactions resulting from these effects are nonselective because the gradients involved are uniform. In this work, we unravel the mechanisms involved in the self-organization of amphiphile filaments that connect and attract droplets floating at the air-water interface, and we demonstrate their potential for directional gradient formation and thereby selective interaction. We simulate Marangoni flow patterns resulting from the release and depletion of amphiphile molecules by source and drain droplets, respectively, and we predict that these flow patterns direct the growth of filaments from the source droplets toward specific drain droplets, based on their amphiphile depletion rate. The interaction between such droplets is then investigated experimentally by charting the flow patterns in their surroundings, while the role of filaments in source-drain attraction is studied using microscopy. Based on these observations, we attribute attraction of drain droplets and even solid objects toward the source to elastocapillary effects. Finally, the insights from our simulations and experiments are combined to construct a droplet-based system in which the composition of drain droplets regulates their ability to attract filaments and as a consequence be attracted toward the source. Thereby, we provide a novel method through which directional attraction can be established in synthetic self-organizing systems and advance our understanding of how complexity arises from simple building blocks.

Insights into the wetting phenomenon induced by scaling of calcium sulfate in membrane distillation

Development of water/wastewater treatment based on membrane distillation (MD) suffers from the drawback that the hydrophobic membrane could be wetted for various reasons. Despite significant efforts, there is uncertainty in addressing the wetting induced by scaling of calcium sulfate, which is ubiquitous and recalcitrant in MD processes. This study made the first attempt to analyze the interplay between the growing crystals and the porous structures in the framework of Stoney's equation. Optical coherence tomography (OCT) was exploited to measure the membrane shift, whereby the scaling-induced deformation was correlated with the variation in stress created in the crystal-containing layer. Along with the stress analysis, the OCT-based characterization was combined with conventional approaches to ascertain the dependence of the scaling-induced wetting on the rate of concentrating the crystallizing species when arriving at a high degree of supersaturation in the feed. This study would refine the physical picture for better understanding crystal-membrane interactions that result in not only the wetting phenomenon but also the irreversible damage of membrane structures, thereby lending itself to the development of strategies for MD-based applications with improved efficiency.

Chains of cubic colloids at fluid-fluid interfaces

Inspired by recent experimental observations of spontaneous chain formation of cubic particles adsorbed at a fluid-fluid interface, we theoretically investigate whether capillary interactions can be responsible for this self-assembly process. We calculate adsorption energies, equilibrium particle orientations, and interfacial deformations, not only for a variety of contact angles but also for single cubes as well as an infinite 2D lattice of cubes at the interface. This allows us to construct a ground-state phase diagram as a function of areal density for several contact angles, and upon combining the capillary energy of a 2D lattice with a simple expression for the entropy of a 2D fluid we also construct temperature-density or size-density phase diagrams that exhibit large two-phase regions and triple points. We identify several regimes with stable chainlike structures, in line with the experimental observations.

Unravelling three-dimensional adsorption geometries of PbSe nanocrystal monolayers at a liquid-air interface

The adsorption, self-organization and oriented attachment of PbSe nanocrystals (NCs) at liquid-air interfaces has led to remarkable nanocrystal superlattices with atomic order and a superimposed nanoscale geometry. Earlier studies examined the NC self-organization at the suspension/air interface with time-resolved in-situ X-ray scattering. Upon continuous evaporation of the solvent, the NC interfacial layer will finally contact the (ethylene glycol) liquid substrate on which the suspension was casted. In order to obtain structural information on the NC organization at this stage of the process, we examined the ethylene glycol/NC interface in detail for PbSe NCs of different sizes, combining in-situ grazing-incidence small-and-wide-angle X-ray scattering (GISAXS/GIWAXS), X-ray reflectivity (XRR) and analytical calculations of the adsorption geometry of these NCs. Here, we observe in-situ three characteristic adsorption geometries varying with the NC size. Based on the experimental evidence and simulations, we reveal fully three-dimensional arrangements of PbSe nanocrystals at the ethylene glycol-air interface with and without the presence of rest amounts of toluene.

Mechanical phase inversion of Pickering emulsions via metastable wetting of rough colloids

The possibility to invert emulsions from oil-in-water to water-in-oil (or vice versa) in a closed system, i.e. without any formulation change, remains an open fundamental challenge with many opportunities for industrial applications. Here, we propose a mechanism that exploits particle surface roughness to induce metastable wetting and obtain mechanically-responsive Pickering emulsions. We postulate that the phase inversion is driven by an in situ switch of the particle wettability from metastable positions at the interface following the input of controlled mechanical energy. Oil-in-water emulsions can be prepared at low energy using mildly hydrophobic rough colloids, which are dispersed in water and weakly pinned at the interface, and switched to water-in-oil emulsions by a second emulsification at higher energy, which triggers the relaxation of the particle contact angle. The same principle is demonstrated for the complementary emulsions using mildly hydrophilic colloids initially dispersed in oil. Our experiments and simulations support that the delicate interplay between particle surface design during synthesis and the energy of the emulsification process can encode a kinetic pathway for the phase inversion. Both organic and inorganic nanoparticles can be used, allowing for the future implementation of our strategy in a broad range of smart industrial formulations.

Equilibrium configurations and capillary interactions of Janus dumbbells and spherocylinders at fluid-fluid interfaces

We numerically investigate the adsorption of a variety of Janus particles (dumbbells, elongated dumbbells and spherocylinders) at a fluid-fluid interface by using a numerical method that takes into account the interfacial deformations. We first determine the equilibrium configuration of a single adsorbed particle, and we find that the overall shape of the induced deformation field has a strong hexapolar mode while non-Janus particles of the same shape do not induce any interfacial deformation. We then calculate the capillary interactions between two Janus spherocylinders adsorbed at an interface. The hexapolar deformation field induces capillary attractions for laterally aligned Janus spherocylinders and repulsions for laterally anti-aligned ones. We also experimentally synthesize micrometer-sized charged Janus dumbbells and let them adsorb at a water-decane interface. After several hours we observe the formation of aggregates of dumbbells predominantly induced by interactions that appear to be capillary in nature. Our Janus dumbbells attach laterally and are all aligned, as predicted by our numerical calculations.

Equilibrium Drop Shapes on a Tilted Substrate with a Chemical Step

We calculate the equilibrium shape of a droplet sitting on a tilted substrate with a "chemical step", that is, different lypophilicity at the two sides of the step. This problem can be generalized to that of a droplet experiencing a body force, pushing it from the lyophilic part to the lyophobic part of the substrate. We present phase diagrams, in which we show for which droplet sizes there are dynamically inaccessible equilibrium shapes. We also identify what determines the threshold volume. While this given system was studied previously in the literature using contact angle hysteresis laws, we present the full static thermodynamical solution of the interfacial energy including the contact energy, while omitting the hysteresis effects from the contact line.

Colloidal particles at fluid interfaces: behaviour of isolated particles

The adsorption of colloidal particles to fluid interfaces is a phenomenon that is of interest to multiple disciplines across the physical and biological sciences. In this review we provide an entry level discussion of our current understanding on the physical principles involved and experimental observations of the adsorption of a single isolated particle to a liquid-liquid interface. We explore the effects that a variation of the morphology and surface chemistry of a particle can have on its ability to adhere to a liquid interface, from a thermodynamic as well as a kinetic perspective, and the impact of adsorption behaviour on potential applications. Finally, we discuss recent developments in the measurement of the interfacial behaviour of nanoparticles and highlight open questions for future research.

Modeling the surface of fast-cured polymer droplet lenses for precision fabrication

Optical lenses with diameter in the millimeter range have found important commercial use in smartphone cameras. Although these lenses are typically made by molding, recent demonstration of fast-cured polymer droplets by inkjet printing has gained interest for cost-effective smartphone microscopy. In this technique, the surface of a fast-cured polydimethylsiloxane droplet obtains dynamic equilibrium via the interplay of surface tension, gravity, thermalization, and a steep viscosity hike. The nature of surface formation involves multiple physical and chemical domains, which represent significant challenges in modeling with the Young-Laplace theory, assuming constant surface tension and viscosity. To overcome these challenges, we introduce the concept of effective surface tension, which allows fast-cured polymer droplets to be modeled as normal liquid droplets with constant viscosity.

Emanating Jets As Shaped by Surface Tension Forces

We show that emanating jets can be regarded as growing liquid towers, which are shaped by the twofold action of surface tension: first the emanated fluid is being accelerated back by surface tension force, herewith creating the boundary conditions to solve the shape of the liquid tower as a solution of an equation mathematically related to the hydrostatic Young-Laplace equation, known to give solutions for the shape of pending and sessile droplets, and wherein the only relevant forces are gravity g and surface tension γ. We explain that for an emanating jet under specific constraints all mass parts with density ρ will experience a uniform time dependent acceleration a( t). An asymptotic solution is subsequently numerically derived by making the corresponding Young-Laplace type equation dimensionless and by dividing all lengths by a generalized time dependent capillary length λ_c( t) = [Formula: see text]. The time dependent surface tension γ( t) can be derived by measuring both time dependent acceleration a( t) and time dependent capillary length λ_c( t). Jetting experiments with water and coffee show that the dynamic surface tension behavior according to the emanating jet method and with the well-known maximum bubble pressure method are the same, herewith verifying the proposed model.

Self-assembly of cubic colloidal particles at fluid-fluid interfaces by hexapolar capillary interactions

Colloidal particles adsorbed at fluid-fluid interfaces can self-assemble, thanks to capillary interactions, into 2D ordered structures. Recently, it has been predicted by theoretical and numerical calculations [G. Soligno et al., Phys. Rev. Lett., 2016, 116, 258001] that cubes with smooth edges adsorbed at a flat fluid-fluid interface generate hexapolar capillary deformations that cause the particles to self-assemble into honeycomb and hexagonal lattices, at equilibrium and for Young's contact angle π/2. Here we extend these results. Firstly, we show that capillary interactions induced by hexapolar deformations can drive the particles at the interface to form also thermodynamically-stable square lattices, in addition to honeycomb and hexagonal lattices. Then, we study the effects of tuning the particle shape on the particle self-assembly at the interface, considering, respectively, smooth-edge cubes, sharp-edge cubes, slightly truncated-edge cubes, and highly truncated-edge cubes. In our calculations, both capillary and hard-particle interactions are taken into account. We show that such variations in the particle shape significantly affect both qualitatively and quantitatively the self-assembly of the particles at the interface, and we sum up our results in the form of temperature-density phase diagrams. For example, using typical experimental parameters, our results show that only 4-to-5 nm sized sharp-edge and smooth-edge cubes can self-assemble into a honeycomb lattice, while slightly and highly truncated-edge cubes can form a honeycomb lattice only if they have a 8-to-12 and 10-to-16 nm size, respectively, for the same experimental parameters. Also, our results show that the capillarity-induced square lattice phase is stable only for the smooth-edge and truncated-edge cubes, but not for the sharp-edge cubes.

Regiospecific Nucleation and Growth of Silane Coupling Agent Droplets onto Colloidal Particles

Nucleation-and-growth processes are used extensively in the synthesis of spherical colloids, and more recently regiospecific nucleation-and-growth processes have been exploited to prepare more complex colloids such as patchy particles. We demonstrate that surface geometry alone can be made to play the dominant role in determining the final particle geometry in such syntheses, meaning that intricate chemical surface patternings are not required. We present a synthesis method for "lollipop"-shaped colloidal heterodimers (patchy particles), combining a recently published nucleation-and-growth technique with our recent findings that particle geometry influences the locus of droplet adsorption onto anisotropic template particles. Specifically, 3-methacryloxypropyl trimethoxysilane (MPTMS) is nucleated and grown onto bullet-shaped and nail-shaped colloids. The shape of the template particle can be chosen such that the MPTMS adsorbs regiospecifically onto the flat ends. In particular, we find that particles with a wider base increase the range of droplet volumes for which the minimum in the free energy of adsorption is located at the flat end of the particle compared with bullet-shaped particles of the same aspect ratio. We put forward an extensive analysis of the synthesis mechanism and experimentally determine the physical properties of the heterodimers, supported by theoretical simulations. Here we numerically optimize, for the first time, the shape of finite-sized droplets as a function of their position on the rod-like silica particle surface. We expect that our findings will give an impulse to complex particle creation by regiospecific nucleation and growth.

Sessile Nanodroplets on Elliptical Patches of Enhanced Lyophilicity

We theoretically investigate the shape of a nanodroplet on a lyophilic elliptical patch in lyophobic surroundings on a flat substrate. To compute the droplet equilibrium shape, we minimize its interfacial free energy using both Surface Evolver and Monte Carlo calculations, finding good agreement between the two methods. We observe different droplet shapes, which are controlled by the droplet volume and the aspect ratio of the ellipse. In particular, we study the behavior of the nanodroplet contact angle along the three-phase contact line, explaining the different droplet shapes. Although the nanodroplet contact angle is constant and fixed by Young's law inside and outside the elliptical patch, its value varies along the rim of the elliptical patch. We find that because of the pinning of the nanodroplet contact line at the rim of the elliptical patch, which has a nonconstant curvature, there is a regime of aspect ratios of the elliptical patch in which the nanodroplet starts expanding to the lyophobic part of the substrate, although there is still a finite area of the lyophilic patch free to be wetted.

Self-Assembly of Cubes into 2D Hexagonal and Honeycomb Lattices by Hexapolar Capillary Interactions

Particles adsorbed at a fluid-fluid interface induce capillary deformations that determine their orientations and generate mutual capillary interactions which drive them to assemble into 2D ordered structures. We numerically calculate, by energy minimization, the capillary deformations induced by adsorbed cubes for various Young's contact angles. First, we show that capillarity is crucial not only for quantitative, but also for qualitative predictions of equilibrium configurations of a single cube. For a Young's contact angle close to 90°, we show that a single-adsorbed cube generates a hexapolar interface deformation with three rises and three depressions. Thanks to the threefold symmetry of this hexapole, strongly directional capillary interactions drive the cubes to self-assemble into hexagonal or graphenelike honeycomb lattices. By a simple free-energy model, we predict a density-temperature phase diagram in which both the honeycomb and hexagonal lattice phases are present as stable states.

Planar equilibrium shapes of a liquid drop on a membrane

The equilibrium shape of a small liquid drop on a smooth rigid surface is governed by the minimization of energy with respect to the change in configuration, represented by the well-known Young's equation. In contrast, the equilibrium shape near the line separating three immiscible fluid phases is determined by force balance, represented by Neumann's Triangle. These two are limiting cases of the more general situation of a drop on a deformable, elastic substrate. Specifically, we have analyzed planar equilibrium shapes of a liquid drop on a deformable membrane. We show that to determine its equilibrium shape one must simultaneously satisfy configurational energy and mechanical force balance along with a constraint on the liquid volume. The first condition generalizes Young's equation to include changes in stored elastic energy upon changing the configuration. The second condition generalizes the force balance conditions by relating tensions to membrane stretches via their constitutive elastic behavior. The transition from Young's equation to Neumann's triangle is governed by the value of the elasto-capillary number, β = TRo/μh, where TRo is twice the surface tension of the solid-vapor interface, μ is the shear modulus of the membrane, and h is its thickness.

Water-in-Water Emulsions Stabilized by Nanoplates

Ultrathin plate-like colloidal particles are effective candidates for Pickering stabilization of water-in-water emulsions, a stabilization that is complicated by the thickness and ultralow tension of the water-water interface. Plate-like particles have the advantage of blocking much of the interface while simultaneously having a low mass. Additionally, the amount of blocked interface is practically independent of the equilibrium contact angle θ at which the water-water interface contacts the nanoplates. As a result, the adsorption of nanoplates is stronger than for spheres with the same maximal cross section, except if θ = 90°.