Nonlinear dynamics of a chemically-active drop: From steady to chaotic self-propulsion

Interplay of chemotactic force, Péclet number, and dimensionality dictates the dynamics of auto-chemotactic chiral active droplets

In living and synthetic active matter systems, the constituents can self-propel and interact with each other and with the environment through various physicochemical mechanisms. Among these mechanisms, chemotactic and auto-chemotactic effects are widely observed. The impact of (auto-)chemotactic effects on achiral active matter has been a recent research focus. However, the influence of these effects on chiral active matter remains elusive. Here, we develop a Brownian dynamics model coupled with a diffusion equation to examine the dynamics of auto-chemotactic chiral active droplets in both quasi-two-dimensional (2D) and three-dimensional (3D) systems. By quantifying the droplet trajectory as a function of the dimensionless Péclet number and chemotactic strength, our simulations well reproduce the curling and helical trajectories of nematic droplets in a surfactant-rich solution reported by Krüger et al. [Phys. Rev. Lett. 117, 048003 (2016)]. The modeled curling trajectory in 2D exhibits an emergent chirality, also consistent with the experiment. We further show that the geometry of the chiral droplet trajectories, characterized by the pitch and diameter, can be used to infer the velocities of the droplet. Interestingly, we find that, unlike the achiral case, the velocities of chiral active droplets show dimensionality dependence: its mean instantaneous velocity is higher in 3D than in 2D, whereas its mean migration velocity is lower in 3D than in 2D. Taken together, our particle-based simulations provide new insights into the dynamics of auto-chemotactic chiral active droplets, reveal the effects of dimensionality, and pave the way toward their applications, such as drug delivery, sensors, and micro-reactors.

Singular bifurcations and regularization theory

Nonlinear sciences are present today in almost all disciplines, ranging from physics to social sciences. A major task in nonlinear science is the classification of different types of bifurcations (e.g., pitchfork and saddle-node) from a given state to another. Bifurcation analysis is traditionally based on the assumption of a regular perturbative expansion, close to the bifurcation point, in terms of a variable describing the passage of a system from one state to another. However, it is shown that a regular expansion is not the rule due to the existence of hidden singularities in many models, paving the way to a new paradigm in nonlinear science, that of singular bifurcations. The theory is first illustrated on an example borrowed from the field of active matter (phoretic microswimers), showing a singular bifurcation. We then present a universal theory on how to handle and regularize these bifurcations, bringing to light a novel facet of nonlinear sciences that has long been overlooked.

Shaping active matter from crystalline solids to active turbulence

Active matter drives its constituent agents to move autonomously by harnessing free energy, leading to diverse emergent states with relevance to both biological processes and inanimate functionalities. Achieving maximum reconfigurability of active materials with minimal control remains a desirable yet challenging goal. Here, we employ large-scale, agent-resolved simulations to demonstrate that modulating the activity of a wet phoretic medium alone can govern its solid-liquid-gas phase transitions and, subsequently, laminar-turbulent transitions in fluid phases, thereby shaping its emergent pattern. These two progressively emerging transitions, hitherto unreported, bring us closer to perceiving the parallels between active matter and traditional matter. Our work reproduces and reconciles seemingly conflicting experimental observations on chemically active systems, presenting a unified landscape of phoretic collective dynamics. These findings enhance the understanding of long-range, many-body interactions among phoretic agents, offer new insights into their non-equilibrium collective behaviors, and provide potential guidelines for designing reconfigurable materials.

Buoyancy-driven attraction of active droplets

For dissolving active oil droplets in an ambient liquid, it is generally assumed that the Marangoni effect results in repulsive interactions, while the buoyancy effects caused by the density difference between the droplets, diffusing product and the ambient fluid are usually neglected. However, it has been observed in recent experiments that active droplets can form clusters due to buoyancy-driven convection (Krüger et al. Eur. Phys. J. E, vol. 39, 2016, pp. 1-9). In this study, we numerically analyze the buoyancy effect, in addition to the propulsion caused by Marangoni flow (with its strength characterized by Péclet number Pe). The buoyancy effects have their origin in (i) the density difference between the droplet and the ambient liquid, which is characterized by Galileo number Ga, and (ii) the density difference between the diffusing product (i.e. filled micelles) and the ambient liquid, which can be quantified by a solutal Rayleigh number Ra. We analyze how the attracting and repulsing behaviour of neighbouring droplets depends on the control parameters Pe, Ga, and Ra. We find that while the Marangoni effect leads to the well-known repulsion between the interacting droplets, the buoyancy effect of the reaction product leads to buoyancy-driven attraction. At sufficiently large Ra, even collisions between the droplets can take place. Our study on the effect of Ga further shows that with increasing Ga, the collision becomes delayed. Moreover, we derive that the attracting velocity of the droplets, which is characterized by a Reynolds number Red, is proportional to Ra1/4/(ℓ/R), where ℓ/R is the distance between the neighbouring droplets normalized by the droplet radius. Finally, we numerically obtain the repulsive velocity of the droplets, characterized by a Reynolds number Rerep, which is proportional to PeRa-0.38. The balance of attractive and repulsive effect leads to Pe ~ Ra0.63, which agrees well with the transition curve between the regimes with and without collision.

Motility and swimming: universal description and generic trajectories

Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that for a particle driven by a concentration field of an active species, straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the particle or that of suspending medium. Our proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations, leading to spontaneous symmetry breaking in the concentration field driving the particle motility. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium, or influence of bounding walls are not necessary.

Self-Solidifying Active Droplets Showing Memory-Induced Chirality

Most synthetic microswimmers do not reach the autonomy of their biological counterparts in terms of energy supply and diversity of motions. Here, this work reports the first all-aqueous droplet swimmer powered by self-generated polyelectrolyte gradients, which shows memory-induced chirality while self-solidifying. An aqueous solution of surface tension-lowering polyelectrolytes self-solidifies on the surface of acidic water, during which polyelectrolytes are gradually emitted into the surrounding water and induce linear self-propulsion via spontaneous symmetry breaking. The low diffusion coefficient of the polyelectrolytes leads to long-lived chemical trails which cause memory effects that drive a transition from linear to chiral motion without requiring any imposed symmetry breaking. The droplet swimmer is capable of highly efficient removal (up to 85%) of uranium from aqueous solutions within 90 min, benefiting from self-propulsion and flow-induced mixing. These results provide a route to fueling self-propelled agents which can autonomously perform chiral motion and collect toxins.

Simple model for self-propulsion of microdroplets in surfactant solution

We propose a simple active hydrodynamic model for the self-propulsion of a liquid droplet suspended in micellar solutions. The self-propulsion of the droplet occurs by spontaneous breaking of isotropic symmetry and is studied using both analytical and numerical methods. The emergence of self-propulsion arises from the slow dissolution of the inner fluid into the outer micellar solution as filled micelles. We propose that the surface generation of filled micelles is the dominant reason for the self-propulsion of the droplet. The flow instability is due to the Marangoni stress generated by the nonuniform distribution of the surfactant molecules on the droplet interface. In our model, the driving parameter of the instability is the excess surfactant concentration above the critical micellar concentration, which directly correlates with the experimental observations. We consider various low-order modes of flow instability and show that the first mode becomes unstable through a supercritical bifurcation and is the only mode contributing to the swimming of the droplet. The flow fields around the droplet for these modes and their combined effects are also discussed.

Non-Brownian diffusion and chaotic rheology of autophoretic disks

The dynamics of a two-dimensional autophoretic disk is quantified as a minimal model for the chaotic trajectories undertaken by active droplets. Via direct numerical simulations, we show that the mean-square displacement of the disk in a quiescent fluid is linear at long times. Surprisingly, however, this apparently diffusive behavior is non-Brownian, owing to strong cross correlations in the displacement tensor. The effect of a shear flow field on the chaotic motion of an autophoretic disk is examined. Here, the stresslet on the disk is chaotic for weak shear flows; a dilute suspension of such disks would exhibit a chaotic shear rheology. This chaotic rheology is quenched first into a periodic state and ultimately a steady state as the flow strength is increased.

Motion of a swimming droplet under external perturbations: A model-based approach

Microdroplets driven by the Marangoni effect are known to continue to swim for hours despite their simple composition. This swimming microdroplet changes its motion from straight to curvilinear and further to chaotic as the Péclet number increases. In this study, we investigate the effect of external perturbations on the three-dimensional axis-asymmetric model of a droplet driven by the Marangoni effect. The aim here is to elucidate the contribution of external perturbation to the complex motion of the droplet and the change in its effect according to the droplet size. In this paper, first we provide a detailed explanation on the derivation of the model introduced in our previous work, which is next used to describe the motion of the droplet in the numerical study of the angular response to random perturbations. The numerical simulation of droplet motion with different types of noise indicates that the model does not converge them into a certain type of motion but rather helps to reflect the external perturbations. The obtained results suggest that the types and properties of external perturbation have a considerable effect on the droplet motion.

Chemotactic self-caging in active emulsions

The out-of-equilibrium dynamics of chemotactic active matter—be it animate or inanimate—is closely coupled to the environment, a chemical landscape shaped by secretions from the motile agents, fuel uptake, or autochemotactic signaling. This gives rise to complex collective effects, which can be exploited by the agents for colony migration strategies or pattern formation. We study such effects using an idealized experimental system: self-propelled microdroplets that communicate via chemorepulsive trails. We present a comprehensive experimental analysis that involves direct probing of the diffusing chemical trails and the trail–droplet interactions and use it to construct a generic theoretical model. We connect these repulsive autochemotactic interactions to the collective dynamics in emulsions, demonstrating a state of dynamical arrest: chemotactic self-caging.

A common feature of biological self-organization is how active agents communicate with each other or their environment via chemical signaling. Such communications, mediated by self-generated chemical gradients, have consequences for both individual motility strategies and collective migration patterns. Here, in a purely physicochemical system, we use self-propelling droplets as a model for chemically active particles that modify their environment by leaving chemical footprints, which act as chemorepulsive signals to other droplets. We analyze this communication mechanism quantitatively both on the scale of individual agent–trail collisions as well as on the collective scale where droplets actively remodel their environment while adapting their dynamics to that evolving chemical landscape. We show in experiment and simulation how these interactions cause a transient dynamical arrest in active emulsions where swimmers are caged between each other’s trails of secreted chemicals. Our findings provide insight into the collective dynamics of chemically active particles and yield principles for predicting how negative autochemotaxis shapes their navigation strategy.

Solute effects on the dynamics and deformation of emulsion droplets during freezing

Soft or rigid particles, suspended in a liquid melt, interact with an advancing solidification front in various industrial and natural processes, such as fabrication of particle-reinforced-composites, growth of crystals, cryopreservation, frost heave, and growth of sea ice. The particle dynamics relative to the front determine the microstructure as well as the functional properties of the solidified material. Previous studies have extensively investigated the interaction of foreign objects with a moving solid-liquid interface in pure melts while in most real-life systems, solutes or surface active impurities are almost always present. Here we study experimentally the interaction of spherical oil droplets with a moving planar ice-water interface, while systematically increasing the surfactant concentration in the bulk liquid, using in situ cryo-confocal microscopy. We demonstrate that a small amount of surfactant in the bulk liquid can instigate long-range droplet repulsion, extending over a length scale of 40 to 100 μm, in contrast to the short-range predicted previously (<1 μm). We report on the droplet deformation, while they are in contact with the ice-water interface, as a function of the bulk surfactant concentration, the droplet size, and the crystal growth rate. We also depict the dynamic evolution of solute-enriched premelted films (≈5 μm). Our results demonstrate how an increasing concentration of surfactant in the bulk and its subsequent segregation during solidification can dramatically alter the solidification microstructures. We anticipate that our experimental study can aid in the development of theoretical models incorporating solute effects.

Oscillatory rheotaxis of artificial swimmers in microchannels

Biological microswimmers navigate upstream of an external flow with trajectories ranging from linear to spiralling and oscillatory. Such a rheotactic response primarily stems from the hydrodynamic interactions triggered by the complex shapes of the microswimmers, such as flagellar chirality. We show here that a self-propelling droplet exhibits oscillatory rheotaxis in a microchannel, despite its simple spherical geometry. Such behaviour has been previously unobserved in artificial swimmers. Comparing our experiments to a purely hydrodynamic theory model, we demonstrate that the oscillatory rheotaxis of the droplet is primarily governed by both the shear flow characteristics and the interaction of the finite-sized microswimmer with all four microchannel walls. The dynamics can be controlled by varying the external flow strength, even leading to the rheotactic trapping of the oscillating droplet. Our results provide a realistic understanding of the behaviour of active particles navigating in confined microflows relevant in many biotechnology applications.

Spontaneously rotating clusters of active droplets

We report on the emergence of spontaneously rotating clusters in active emulsions. Ensembles of self-propelling droplets sediment and then self-organise into planar, hexagonally ordered clusters which hover over the container bottom while spinning around the plane normal. This effect exists for symmetric and asymmetric arrangements of isotropic droplets and is therefore not caused by torques due to geometric asymmetries. We found, however, that individual droplets exhibit a helical swimming mode in a small window of intermediate activity in a force-free bulk medium. We show that by forming an ordered cluster, the droplets cooperatively suppress their chaotic dynamics and turn the transient instability into a steady rotational state. We analyse the collective rotational dynamics as a function of droplet activity and cluster size and further propose that the stable collective rotation in the cluster is caused by a cooperative coupling between the rotational modes of individual droplets in the cluster.

Straight-to-Curvilinear Motion Transition of a Swimming Droplet Caused by the Susceptibility to Fluctuations

In this Letter, a water-in-oil swimming droplet's transition from straight to curvilinear motion is investigated experimentally and theoretically. An analysis of the experimental results and the model reveal that the motion transition depends on the susceptibility of the droplet's direction of movement to external stimuli as a function of environmental parameters such as droplet size. The simplicity of the present experimental system and the model suggests implications for a general class of transitions in self-propelled swimmers.

Spontaneous Mode Switching of Self-Propelled Droplet Motion Induced by a Clock Reaction in the Belousov-Zhabotinsky Medium

Interfacial chemical dynamics on a droplet generate various self-propelled motions. For example, ballistic and random motions arise depending on the physicochemical conditions inside the droplet and its environment. In this study, we focus on the relationship between oxidant concentrations in an aqueous droplet and its mode of self-propelled motion in an oil phase including surfactant. We demonstrated that the chemical conditions inside self-propelled aqueous droplets were changed systematically, indicating that random motion appeared at higher concentrations of oxidants, which were H₂SO₄ and BrO₃^-, and ballistic motion at lower concentrations. In addition, spontaneous mode switching from ballistic to random motion was successfully demonstrated by adding malonic acid, wherein the initially observed reduced state of the aqueous solution suddenly changed to the oxidized state. Although we only observed one-time transition and have not yet succeeded to realize alternation between ballistic (reduced state) and random motion (oxidized state), such spontaneous transitions are fundamental steps in realizing artificial cells and understanding the fundamental mechanisms of life-like behavior, such as bacterial chemotaxis originating from periodical run-and-tumble motion.

Alignment and scattering of colliding active droplets

Active droplets emit a chemical solute at their surface that modifies their local interfacial tension. They exploit the nonlinear coupling of the convective transport of solute to the resulting Marangoni flows in order to self-propel. Such swimming droplets are by nature anti-chemotactic and are repelled by their own chemical wake or their neighbours. The rebound dynamics resulting from pairwise droplet interactions was recently analysed in detail for purely head-on collisions using a specific bispherical approach. Here, we extend this analysis and propose a reduced model of a generic collision to characterise the alignment and scattering properties of oblique droplet collisions and their potential impact on collective droplet dynamics. A systematic alignment of the droplets' trajectories is observed for symmetric collisions, when the droplets interact directly, and arises from the finite-time rearrangement of the droplets' chemical wake during the collision. For more generic collisions, complex and diverse dynamical regimes are observed, whether the droplets interact directly or through their chemical wake, resulting in a significant scattering.

Adsorption inhibition by swollen micelles may cause multistability in active droplets

Experiments indicate that microdroplets undergoing micellar solubilization in the bulk of surfactant solution may excite Marangoni flows and self-propel spontaneously. Surprisingly, self-propulsion emerges even when the critical micelle concentration is exceeded and the Marangoni effect should be saturated. To explain this, we propose a novel model of a dissolving active droplet that is based on two fundamental assumptions: (a) products of the solubilization may inhibit surfactant adsorption; (b) solubilization prevents the formation of a monolayer of surfactant molecules at the droplet interface. We use numerical simulations and asymptotic methods to demonstrate that our model indeed features spontaneous droplet self-propulsion. Our key finding is that in the case of axisymmetric flow and concentration fields, two qualitatively different types of droplet behavior may be stable for the same values of the physical parameters: steady self-propulsion and steady symmetric pumping. Although stability of these steady regimes is not guaranteed in the absence of axial symmetry, we argue that they will retain their respective stable manifolds in the phase space of a fully 3D problem.

Chaotic Swimming of Phoretic Particles

The swimming of a rigid phoretic particle in an isotropic fluid is studied numerically as a function of the dimensionless solute emission rate (or Péclet number Pe). The particle sets into motion at a critical Pe. Whereas the particle trajectory is straight at a small enough Pe, it is found that it loses its stability at a critical Pe in favor of a meandering motion. When Pe is increased further, the particle meanders at a short scale but its trajectory wraps into a circle at a larger scale. Increasing even further, Pe causes the swimmer to escape momentarily the circular trajectory in favor of chaotic motion, which lasts for a certain time, before regaining a circular trajectory, and so on. The chaotic bursts become more and more frequent as Pe increases, until the trajectory becomes fully chaotic, via the intermittency scenario. The statistics of the trajectory is found to be of the run-and-tumble-like nature at a short enough time and of diffusive nature at a long time without any source of noise.

Orientational instability and spontaneous rotation of active nematic droplets

In experiments, an individual chemically-active liquid crystal (LC) droplet submerged in the bulk of a surfactant solution may self-propel along a straight, helical, or random trajectory. In this paper, we develop a minimal model capturing all three types of self-propulsion trajectories of a drop in the case of a nematic LC with homeotropic anchoring at the LC-fluid interface. We emulate the director field within the drop by a single preferred polarization vector that is subject of two reorientation mechanisms, namely, the internal flow-induced displacement of the hedgehog defect and the droplet's rotation. Within this reduced-order model, the coupling between the nematic ordering of the drop and the surfactant transport is represented by variations of the droplet's interfacial properties with nematic polarization. Our analysis reveals that a novel mode of orientational instability emerges from the competition of the two reorientation mechanisms and is characterized by a spontaneous rotation of the self-propelling drop responsible for helical self-propulsion trajectories. In turn, we also show that random trajectories in isotropic and nematic drops alike stem from the advection-driven transition to chaos. The succession of the different propulsion modes is consistent with experimentally-reported transitions in the shape of droplet trajectories as the drop size is varied.