Erratum: Pairing-induced motion of source and inert particles driven by surface tension [Phys. Rev. E 106, 024604 (2022)].

This corrects the article DOI: 10.1103/PhysRevE.106.024604.

Droplet duos on water display pairing, autonomous motion, and periodic eruption.

Under non-equilibrium conditions, liquid droplets dynamically couple with their milieu through the continuous flux of matter and energy, forming active systems capable of self-organizing functions reminiscent of those of living organisms. Among the various dynamic behaviors demonstrated by cells, the pairing of heterogeneous cell units is necessary to enable collective activity and cell fusion (to reprogram somatic cells). Furthermore, the cyclic occurrence of eruptive events such as necroptosis or explosive cell lysis is necessary to maintain cell functions. However, unlike the self-propulsion behavior of cells, cyclic cellular behavior involving pairing and eruption has not been successfully modeled using artificial systems. Here, we show that a simple droplet system based on quasi-immiscible hydrophobic oils (perfluorodecalin and decane) deposited on water, mimics such complex cellular dynamics. Perfluorodecalin and decane droplet duos form autonomously moving Janus or coaxial structures, depending on their volumes. Notably, the system with a coaxial structure demonstrates cyclic behavior, alternating between autonomous motion and eruption. Despite their complexity, the dynamic behaviors of the system are consistently explained in terms of the spreading properties of perfluorodecalin/decane duplex interfacial films.

Self-emergent vortex flow of microtubule and kinesin in cell-sized droplets under water/water phase separation.

By facilitating a water/water phase separation (w/wPS), crowded biopolymers in cells form droplets that contribute to the spatial localization of biological components and their biochemical reactions. However, their influence on mechanical processes driven by protein motors has not been well studied. Here, we show that the w/wPS droplet spontaneously entraps kinesins as well as microtubules (MTs) and generates a micrometre-scale vortex flow inside the droplet. Active droplets with a size of 10-100 µm are generated through w/wPS of dextran and polyethylene glycol mixed with MTs, molecular-engineered chimeric four-headed kinesins and ATP after mechanical mixing. MTs and kinesin rapidly created contractile network accumulated at the interface of the droplet and gradually generated vortical flow, which can drive translational motion of a droplet. Our work reveals that the interface of w/wPS contributes not only to chemical processes but also produces mechanical motion by assembling species of protein motors in a functioning manner.

Pairing-induced motion of source and inert particles driven by surface tension.

We experimentally and theoretically investigate systems with a pair of source and inert particles that interact through a concentration field. The experimental system comprises a camphor disk as the source particle and a metal washer as the inert particle. Both are floated on an aqueous solution of glycerol at various concentrations, where the glycerol modifies the viscosity of the aqueous phase. The particles form a pair owing to the attractive lateral capillary force. As the camphor disk spreads surface-active molecules at the aqueous surface, the camphor disk and metal washer move together, driven by the surface tension gradient. The washer is situated in the front of the camphor disk, keeping the distance constant during their motion, which we call a pairing-induced motion. The pairing-induced motion exhibited a transition between circular and straight motions as the glycerol concentration in the aqueous phase changed. Numerical calculations using a model that considers forces caused by the surface tension gradient and lateral capillary interaction reproduced the observed transition in the pairing-induced motion. Moreover, this transition agrees with the result of the linear stability analysis on the reduced dynamical system obtained by the expansion with respect to the particle velocity. Our results reveal that the effect of the particle velocity cannot be overlooked to describe the interaction through the concentration field.

Edge current and pairing order transition in chiral bacterial vortices.

Bacterial suspensions show turbulence-like spatiotemporal dynamics and vortices moving irregularly inside the suspensions. Understanding these ordered vortices is an ongoing challenge in active matter physics, and their application to the control of autonomous material transport will provide significant development in microfluidics. Despite the extensive studies, one of the key aspects of bacterial propulsion has remained elusive: The motion of bacteria is chiral, i.e., it breaks mirror symmetry. Therefore, the mechanism of control of macroscopic active turbulence by microscopic chirality is still poorly understood. Here, we report the selective stabilization of chiral rotational direction of bacterial vortices in achiral circular microwells sealed by an oil/water interface. The intrinsic chirality of bacterial swimming near the top and bottom interfaces generates chiral collective motions of bacteria at the lateral boundary of the microwell that are opposite in directions. These edge currents grow stronger as bacterial density increases, and, within different top and bottom interfaces, their competition leads to a global rotation of the bacterial suspension in a favored direction, breaking the mirror symmetry of the system. We further demonstrate that chiral edge current favors corotational configurations of interacting vortices, enhancing their ordering. The intrinsic chirality of bacteria is a key feature of the pairing order transition from active turbulence, and the geometric rule of pairing order transition may shed light on the strategy for designing chiral active matter.

Spontaneous deformation and fission of oil droplets on an aqueous surfactant solution.

We investigated the spontaneous deformation and fission of a tetradecane droplet containing palmitic acid (PA) on a stearyltrimethylammonium chloride (STAC) aqueous solution. In this system, the generation and rupture of the gel layer composed of PA and STAC induce the droplet deformation and fission. To investigate the characteristics of the droplet-fission dynamics, we obtained the time series of the number of the droplets produced by fission and confirmed that the number has a peak at a certain STAC concentration. Since the fission of the droplet should be led by the deformation, we analyzed four parameters which may relate to the fission dynamics from the spatiotemporal correlation of the droplet-boundary velocity. We found that the parameter which corresponds to the expansion speed had the strongest positive correlation among them, and thus we concluded that the faster deformation would be the key factor for the fission dynamics.

Pattern of a confined chemical garden controlled by injection speed.

Pattern of confined chemical garden was controlled by the speed of injected fluid, and their mechanism is discussed. A confined chemical garden system was constructed where an aqueous solution of cobalt chloride was injected into a cell filled with sodium silicate solution. The reaction of these two solutions resulted in the formation of precipitation. The viscosities of the prepared aqueous solutions were set to be similar in order to rule out the possibility of Saffman-Taylor instability. The injection front showed three distinctive patterns: algaes, shells, and filaments, which were dependent on injection speed. The injection pressure and the spatio-temporal pattern of the injected fluid were measured, and a significant increase in the injection pressure was observed when the filament pattern appeared, which indicated the existence of thin lubrication layer between the precipitation and the substrate. The filament pattern was further analyzed quantitatively, and the number of active filaments was determined to be proportional to the injection speed. A mathematical model was constructed that considered both the viscous effect from the thin lubrication layer and the Laplace pressure. This model successfully reproduced the characteristic filament dynamics.

Relationship between the size of a camphor-driven rotor and its angular velocity.

We consider a rotor made of two camphor disks glued below the ends of a plastic stripe. The disks are floating on a water surface and the plastic stripe does not touch the surface. The system can rotate around a vertical axis located at the center of the stripe. The disks dissipate camphor molecules. The driving momentum comes from the nonuniformity of surface tension resulting from inhomogeneous surface concentration of camphor molecules around the disks. We investigate the stationary angular velocity as a function of rotor radius ℓ. For large ℓ the angular velocity decreases for increasing ℓ. At a specific value of ℓ the angular velocity reaches its maximum and, for short ℓ it rapidly decreases. Such behavior is confirmed by a simple numerical model. The model also predicts that there is a critical rotor size below which it does not rotate. Within the introduced model we analyze the type of this bifurcation.

Selection of the Rotation Direction for a Camphor Disk Resulting from Chiral Asymmetry of a Water Chamber.

Self-motion of a camphor disk rotating inside a water chamber composed of two half-disks was investigated. The half-disks were joined along their diameter segments, and the distance between their midpoints (ds) was considered as the control parameter. Various types of camphor disk motions were observed depending on ds. When ds = 0, the chamber had a circular shape, so it was symmetric. A camphor disk showed either a clockwise (CW) or counterclockwise (CCW) rotation with the direction determined by its initial state. The symmetry of the chamber was broken for ds > 0. For moderate distances between the midpoints, a unidirectional orbital motion of the disk was observed. The preferred rotation direction was determined by the shape of the chamber, and it did not depend on the initial rotation direction. For yet larger ds, the unidirectional circular motion was no longer observed and the trajectory became irregular. A mathematical model coupling the camphor disk motion with the dynamics of the developed camphor molecular layer on water was constructed, and the numerical results were compared with the experimental results. The selection of motion type can be explained by considering the influence of camphor concentration on the disk trajectory through the surface tension gradient.

Self-oscillating Gel Accelerated while Sensing the Shape of an Aqueous Surface.

The reciprocating motion of a self-oscillating square gel induced by the Belousov-Zhabotinsky (BZ) reaction was investigated on an aqueous surface. The chemical wave propagated from the side at which the oxidation of the Ru catalyst in the gel started. As the chemical wave propagated, the gel moved in either the opposite (mode I) or the same (mode II) direction as the chemical wave propagation. The gel then went back as the Ru catalyst in the gel was slowly reduced. We examined the relationship between the modes of motion (mode I or II) and the shape of the aqueous BZ solution surface. The mode selection was discussed in relation to the contact angle around the gel which was changed by the BZ reaction, i.e., the lateral imbalance of surface tension and the capillary interaction.

Mechanism of Spontaneous Blebbing Motion of an Oil-Water Interface: Elastic Stress Generated by a Lamellar-Lamellar Transition.

A quaternary system composed of surfactant, cosurfactant, oil, and water showing spontaneous motion of the oil-water interface under far-from-equilibrium condition is studied in order to understand nanometer-scale structures and their roles in spontaneous motion. The interfacial motion is characterized by the repetitive extension and retraction of spherical protrusions at the interface, i.e, blebbing motion. During the blebbing motion, elastic aggregates are accumulated, which were characterized as surfactant lamellar structures with mean repeat distances d of 25 to 40 nm. Still unclear is the relationship between the structure formation and the dynamics of the interfacial motion. In the present study, we find that a new lamellar structure with d larger than 80 nm is formed at the blebbing oil-water interface, while the resultant elastic aggregates, which are the one reported before, have a lamellar structure with smaller d (25 to 40 nm). Such transition of lamellar structures from the larger d to smaller d is induced by a penetration of surfactants from an aqueous phase into the aggregates. We propose a model in which elastic stress generated by the transition drives the blebbing motion at the interface. The present results explain the link between nanometer-scale transition of lamellar structure and millimeter-scale dynamics at an oil-water interface.

Collective motion of self-propelled particles with memory.

We show that memory, in the form of underdamped angular dynamics, is a crucial ingredient for the collective properties of self-propelled particles. Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular velocity, we uncover a rich variety of collective phases not observed in usual overdamped systems, including vortex lattices and active foams. In a model with strictly nematic interactions the smectic arrangement of Vicsek waves giving rise to global polar order is observed. We also provide a calculation of the effective interaction between vortices in the case where a telegraphic noise process is at play, explaining thus the emergence and structure of the vortex lattices observed here and in motility assay experiments.

Rotational motion of a droplet induced by interfacial tension.

Spontaneous rotation of a droplet induced by the Marangoni flow is analyzed in a two-dimensional system. The droplet with the small particle which supplies a surfactant at the interface is considered. We calculated flow field around the droplet using the Stokes equation and found that advective nonlinearity breaks symmetry for rotation. Theoretical calculation indicates that the droplet spontaneously rotates when the radius of the droplet is an appropriate size. The theoretical results were validated through comparison with the experiments.

Drift instability in the motion of a fluid droplet with a chemically reactive surface driven by Marangoni flow.

We theoretically derive the amplitude equations for a self-propelled droplet driven by Marangoni flow. As advective flow driven by surface tension gradient is enhanced, the stationary state becomes unstable and the droplet starts to move. The velocity of the droplet is determined from a cubic nonlinear term in the amplitude equations. The obtained critical point and the characteristic velocity are well supported by numerical simulations.

Asymmetry-symmetry transition of double-sided adhesive tapes.

We report on the debonding process of a double-sided adhesive tape sandwiched between two glass plates. When the glass plates are separated from each other at a constant rate, a highly asymmetric extension of top and bottom adhesive layers and bending of the inner film are observed first. As the separation proceeds, the elongation of both layers becomes symmetric, and the inner film becomes flat again. When this happens, there appears a local maximum in the force-displacement curve. We explain this asymmetry-symmetry transition and discuss the role of the bimodal force-displacement relation of each adhesive layer. We also discuss the effect of the inner film thickness and the separation rate on the debonding behavior, which causes undesirable early detachment of the double-sided adhesive tape in a certain condition.

Failure of film formation of viscoelastic fluid: dynamics of viscoelastic fluid in a partially filled horizontally rotating cylinder.

The dynamics of a viscoelastic Maxwell fluid is studied in a partially filled cylinder rotating around a horizontal axis. At low rotational velocity, the fluid behaves in the same manner as a viscous fluid. A thin fluid film is pulled up from the edge of a fluid bump at the bottom of the cylinder, and it covers the inner wall of the cylinder completely. As a result, a steady state is the coexistence of the film and the bump of the fluid. When the rotational velocity of the cylinder is increased, the film formation fails and the bump of fluid rolls steadily at the bottom of the cylinder. This failure of film formation has never been observed in the case of a viscous fluid. At higher rotational velocity, the bump of the fluid starts to oscillate at the bottom of the cylinder. Then, the fluid bump again rolls steadily with a further increase in the rotational velocity. The failure of film formation is explained in terms of the elastic behavior of the viscoelastic fluid near the boundary between the film and the bump regions. The theoretical prediction shows good agreement with the experimental results. We further estimate the condition for which a viscoelastic fluid displays dynamically nonwetting behavior; i.e., the absence of fluid film at any value of rotational velocity.

Large-scale vortex lattice emerging from collectively moving microtubules.

Spontaneous collective motion, as in some flocks of bird and schools of fish, is an example of an emergent phenomenon. Such phenomena are at present of great interest and physicists have put forward a number of theoretical results that so far lack experimental verification. In animal behaviour studies, large-scale data collection is now technologically possible, but data are still scarce and arise from observations rather than controlled experiments. Multicellular biological systems, such as bacterial colonies or tissues, allow more control, but may have many hidden variables and interactions, hindering proper tests of theoretical ideas. However, in systems on the subcellular scale such tests may be possible, particularly in in vitro experiments with only few purified components. Motility assays, in which protein filaments are driven by molecular motors grafted to a substrate in the presence of ATP, can show collective motion for high densities of motors and attached filaments. This was demonstrated recently for the actomyosin system, but a complete understanding of the mechanisms at work is still lacking. Here we report experiments in which microtubules are propelled by surface-bound dyneins. In this system it is possible to study the local interaction: we find that colliding microtubules align with each other with high probability. At high densities, this alignment results in self-organization of the microtubules, which are on average 15 µm long, into vortices with diameters of around 400 µm. Inside the vortices, the microtubules circulate both clockwise and anticlockwise. On longer timescales, the vortices form a lattice structure. The emergence of these structures, as verified by a mathematical model, is the result of the smooth, reptation-like motion of single microtubules in combination with local interactions (the nematic alignment due to collisions)--there is no need for long-range interactions. Apart from its potential relevance to cortical arrays in plant cells and other biological situations, our study provides evidence for the existence of previously unsuspected universality classes of collective motion phenomena.

Formation of a multiscale aggregate structure through spontaneous blebbing of an interface.

The motion of an oil-water interface that mimics biological motility was investigated in a Hele-Shaw-like cell where elastic surfactant aggregates were formed at the oil-water interface. With the interfacial motion, millimeter-scale pillar structures composed of the aggregates were formed. The pillars grew downward in the aqueous phase, and the separations between pillars were roughly equal. Small-angle X-ray scattering using a microbeam X-ray revealed that these aggregates had nanometer-scale lamellar structures whose orientation correlated well with their location in the pillar structure. It is suggested that these hierarchical spatial structures are tailored by the spontaneous interfacial motion.

Spontaneous motion of a droplet coupled with a chemical wave.

We propose a framework for the spontaneous motion of a droplet coupled with internal dynamic patterns generated in a reaction-diffusion system. The spatiotemporal order of the chemical reaction gives rise to inhomogeneous surface tension and results in self-propulsion driven by the surrounding flow due to the Marangoni effect. Numerical calculations of internal patterns together with theoretical results of the flow fields at low Reynolds number reproduce well the experimental results obtained using a droplet of the Belousov-Zhabotinsky reaction medium.