Self-propulsion of pure water droplets by spontaneous Marangoni-stress-driven motion

Recirculatory Solvotaxis of a Nematic Droplet on Water Surface Enabling Miniaturization

Self-organized contact line instabilities (CLI) of a macroscopic liquid crystal (LC) droplet can be an ingenious pathway to generate a large collection of miniaturized LC drops. For example, when a larger drop of volatile solvent (e.g., hexane) is dispensed near a smaller LC drop resting on a soft and slippery surface of a nonsolvent (e.g., water), unique self-organized locomotion in the form of a twin vortex has been observed within the droplets. This phenomenon is driven by the rapid counter diffusion of hexane and LC between the two droplets, resulting in the formation of a pair of vortices within the droplets before instigating a CLI at the three-phase contact line (TPCL) of the LC droplet. Initially, the higher Laplace pressure inside the LC droplet (PL,5CB) due to a net pressure gradient, PL,5CB > PL,Hex, drives the LC toward hexane. However, as the volatile solvent droplet shrinks due to rapid evaporation, a flow reversal happens owing to PL,5CB < PL,Hex. Subsequently, the diffusion of hexane into the LC droplet and its subsequent evaporation manifest a periodic oscillatory CLI expansion and retraction at the TPCL, which in turn form periodic finger-like structures. Following this, the fingers with a higher aspect ratio break into an array of miniaturized satellite LC droplets undergoing Rayleigh-Plateau instability (RPI). The observed deviation in the normalized satellite droplet spacingλ / R 5 CB ∼ 3.15 2 π compared to theoretical value ∼ 2 2 π affirm the stabilizing influence of LC elasticity in such fingers, where λ and R5CB are experimentally calculated droplet spacing and 5CB droplet radius. Control experiments elucidate the specific contributions of capillary, drag, solutal Marangoni, and osmotic forces to the 5CB droplet locomotion phenomena. The experimentally and analytically consistent demonstration also supports and predicts pressure drop-induced droplet velocities as v ∼ t1.16.

Active droplet driven by collective chemotaxis

Surfactant-laden fluid interfaces of soft colloids, such as bubbles and droplets, are ubiquitously seen in various natural phenomena and industrial settings. In canonical systems where microparticles are driven in hydrodynamic flows, convection of the surfactant changes local surface tension. Subsequently, the interplay of Marangoni and hydrodynamic stresses leads to rich interfacial dynamics that directly impact the particle motions. Here we introduce a new mechanism for self-propelled droplets, driven by a thin layer of chemically active microparticles situated at the interface of a suspended droplet, which is a direct extension of the planar collective surfing model by Masoud and Shelley (H. Masoud and M. J. Shelley, Phys. Rev. Lett., 2014, 112, 128304). These particles can generate chemicals locally, leading to spontaneous Marangoni flows that drive the self-aggregation of microparticles. This process, in turn, creates a polarized surfactant distribution, which induces collective chemotaxis and dipolar bulk flows, ultimately breaking the symmetry. By assuming the local surfactant production to be either proportional to particle density or saturated at a high particle density, we observe that the system can be chemotactically diverging or approach a steady state with constant migration velocity. The system is studied analytically in the linear region for the initial transient dynamics, yielding critical numbers and familiar patterns, as well as numerically for larger amplitudes and over a long time using spectral methods.

Spontaneous emergence of motion of an isotropic active particle in a Carreau fluid

Active particles are self-propelling in nature due to the generation of a fore-aft asymmetry in the concentration of solutes around their surface. Both the surface activity and mobility play an important role in the particle dynamics. The solutes are the products of the chemical reaction between the active particle surface and suspending medium. Unlike Janus particles, isotropic active particles have been shown to undergo spontaneous self-propulsion beyond a critical particle size (or the Péclet number). Compared to Janus active particles, there is a third ingredient, namely, advection-induced instability that dictates the dynamics of such particles. The present study numerically investigates the role played by shear rate-dependent viscosity of a suspending medium in the self-phoretic dynamics of such isotropic active particles. Towards this, a non-Newtonian Carreau fluid is taken as the suspending medium. One of the important findings of this study is the presence of a second critical Péclet number beyond which the spontaneous motion of the particle ceases to exist. Even though this critical Péclet number had been previously investigated for Newtonian fluids, strong dependence of the former on the rheology of the suspending medium was not explored. The analysis also shows that a shear thinning fluid significantly reduces the maximum velocity of the particle. In addition, confinement is found to have a significant effect on the axial propulsive velocity of the particle suspended in a Carreau fluid.

Self-propulsion of a droplet induced by combined diffusiophoresis and Marangoni effects

Chemically active droplets display complex self-propulsion behavior in homogeneous surfactant solutions, often influenced by the interplay between diffusiophoresis and Marangoni effects. Previous studies have primarily considered these effects separately or assumed axisymmetric motion. To understand the full hydrodynamics, we investigate the motion of a two-dimensional active droplet under their combined influences using weakly nonlinear analysis and numerical simulations. The impact of two key factors, the Péclet number (P e $Pe$ ) and the mobility ratio between diffusiophoretic and Marangoni effects ( m $m$ ), on droplet motion is explored. We establish a phase diagram in theP e - m $Pe-m$ space, categorizing the boundaries between four types of droplet states: stationary, steady motion, periodic/quasi-periodic motion, and chaotic motion. We find that the mobility ratio does not affect the criticalP e $Pe$ for the onset of self-propulsion, but it significantly influences the stability of high-wavenumber modes as well as the droplet's velocity and trajectory. Scaling analysis reveals that in the highP e $Pe$ regime, the Marangoni and diffusiophoresis effects lead to distinct velocity scaling laws:U ∼ P e - 1 / 2 $U\sim Pe^{-1/2}$ andU ∼ P e - 1 / 3 $U\sim Pe^{-1/3}$ , respectively. When these effects are combined, the velocity scaling depends on the sign of the mobility ratio. In cases with a positive mobility ratio, the Marangoni effect dominates the scaling, whereas the negative diffusiophoretic effect leads to an increased thickness of the concentration boundary layer and a flattened scaling of the droplet velocity.

Dynamics of phase-separated microdroplets near the contact line of evaporating all-aqueous drops

Evaporation of multicomponent drops can induce liquid-liquid phase separation and spatial reconfiguration of phases. Here, we unveil several novel dynamics near the contact line of evaporating multicomponent drops containing polyethylene glycol and dextran. The interplay between background Marangoni flow and self-migration of nucleated microdroplets creates both unstable and stable equilibrium points. This leads to either continuous migration or stepwise advancement of microdroplets, influenced by random coalescence events. Tiny dextran microdroplets nucleating at the contact line can migrate toward the bulk only by growing in size with coalescence events. Our findings offer new insights into the fundamental understanding of evaporating multicomponent drops and factors influencing the spatial segregation of phases in evaporative liquid-liquid phase separation with implications in prebiotic biomolecular reactions to industrial applications.

Streaming electric field, electroviscous effect, and electrokinetic liquid flows in the induced pressure-driven transport of active liquids in narrow capillaries

In this paper, we develop a theory for studying the electrokinetic effects in a charged nanocapillary filled with active liquid. The active particles present within the active liquid are self-driven, demonstrate vortex defects, and enforce a circumferentially arranged polarization field. Under such circumstances, there is the development of an induced pressure-gradient-driven transport dictated (similar to diffusioosmotic transport) by the presence of an axial gradient in the activity (or the concentration of the active particles). This pressure-driven transport has a profile different from the standard Hagen-Poiseuille flow in a nanocapillary. Also, this induced pressure-driven flow drives electrokinetic effects, which are characterized by the generation of a streaming electric field, associated electroosmotic (EOS) transport opposing pressure-driven flow, and electroviscous effect. We quantify these effects as functions of dimensionless parameters that vary inversely as the strength of the activity-induced pressure-driven flow and salt concentrations. Overall, we anticipate that this paper will draw immense attention toward a new type of activity-induced pressure-driven flow and associated electrokinetic phenomena in charged nanoconfinements.

Quorum Sensing in Emulsion Droplet Swarms Driven by a Surfactant Competition System

Quorum sensing enables unicellular organisms to probe their population density and perform behavior that exclusively occurs above a critical density. Quorum sensing is established in emulsion droplet swarms that float at a water surface and cluster above a critical density. The design involves competition between 1) a surface tension gradient that is generated upon release of a surfactant from the oil droplets, and thereby drives their mutual repulsion, and 2) the release of a surfactant precursor from the droplets, that forms a strong imine surfactant which suppresses the surface tension gradient and thereby causes droplet clustering upon capillary (Cheerios) attraction. The production of the imine-surfactant depends on the population density of the droplets releasing the precursor so that the clustering only occurs above a critical population density. The pH-dependence of the imine-surfactant formation is exploited to trigger quorum sensing upon a base stimulus: dynamic droplet swarms are generated that cluster and spread upon spatiotemporally varying acid and base conditions. Next, the clustering of two droplet subpopulations is coupled to a chemical reaction that generates a fluorescent signal. It is foreseen that quorum sensing enables control mechanisms in droplet-based systems that display collective responses in contexts of, e.g., sensing, optics, or dynamically controlled droplet-reactors.

Motility and pairwise interactions of chemically active droplets in one-dimensional confinement

Self-propelled droplets serve as ideal model systems to delve deeper into understanding the motion of biological microswimmers by simulating their motility. Biological microorganisms are renowned for showcasing a diverse array of dynamic swimming behaviors when confronted with physical constraints. This study aims to elucidate the impact of physical constraints on swimming characteristics of biological microorganisms. To achieve this, we present observations on the individual and pairwise behavior of micellar solubilized self-propelled 4-cyano-4'-pentyl-biphenyl (5CB) oil droplets in a square capillary channel filled with a surfactant trimethyl ammonium bromide aqueous solution. To explore the effect of the underlying Péclet number of the swimming droplets, the study is also performed in the presence of additives such as high molecular weight polymer polyethylene oxide and molecular solute glycerol. The capillary confinement restricts droplet to predominantly one-dimensional motion, albeit with noticeable differences in their motion across the three scenarios. Through a characterization of the chemical and hydrodynamic flow fields surrounding the droplets, we illustrate that the modification of the droplets' chemical field due to confinement varies significantly based on the underlying differences in the Péclet number in these cases. This alteration in the chemical field distribution notably affects the individual droplets' motion. Moreover, these distinct chemical field interactions between the droplets also lead to variations in their pairwise motion, ranging from behaviors like chasing to scattering.

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.

Self-Timed and Spatially Targeted Delivery of Chemical Cargo by Motile Liquid Crystal

A key challenge underlying the design of miniature machines is encoding materials with time- and space-specific functional behaviors that require little human intervention. Dissipative processes that drive materials beyond equilibrium and evolve continuously with time and location represent one promising strategy to achieve such complex functions. This work reports how internal nonequilibrium states of liquid crystal (LC) emulsion droplets undergoing chemotaxis can be used to time the delivery of a chemical agent to a targeted location. During ballistic motion, hydrodynamic shear forces dominate LC elastic interactions, dispersing microdroplet inclusions (microcargo) within double emulsion droplets. Scale-dependent colloidal forces then hinder the escape of dispersed microcargo from the propelling droplet. Upon arrival at the targeted location, a circulatory flow of diminished strength allows the microcargo to cluster within the LC elastic environment such that hydrodynamic forces grow to exceed colloidal forces and thus trigger the escape of the microcargo. This work illustrates the utility of the approach by using microcargo that initiate polymerization upon release through the outer interface of the carrier droplet. These findings provide a platform that utilizes nonequilibrium strategies to design autonomous spatial and temporal functions into active materials.

Chemotactic Interactions Drive Migration of Membraneless Active Droplets

In nature, chemotactic interactions are ubiquitous and play a critical role in driving the collective behavior of living organisms. Reproducing these interactions in vitro is still a paramount challenge due to the complexity of mimicking and controlling cellular features, such as tangled metabolic networks, cytosolic macromolecular crowding, and cellular migration, on a microorganism size scale. Here, we generate enzymatically active cell-sized droplets able to move freely, and by following a chemical gradient, able to interact with the surrounding droplets in a collective manner. The enzyme within the droplets generates a pH gradient that extends outside the edge of the droplets. We discovered that the external pH gradient triggers droplet migration and controls its directionality, which is selectively toward the neighboring droplets. Hence, by changing the enzyme activity inside the droplet, we tuned the droplet migration speed. Furthermore, we showed that these cellular-like features can facilitate the reconstitution of a simple and linear protometabolic pathway and increase the final reaction product generation. Our work suggests that simple and stable membraneless droplets can reproduce complex biological phenomena, opening new perspectives as bioinspired materials and synthetic biology tools.

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.

Phase-separated droplets swim to their dissolution

Biological macromolecules can condense into liquid domains. In cells, these condensates form membraneless organelles that can organize chemical reactions. However, little is known about the physical consequences of chemical activity in and around condensates. Working with model bovine serum albumin (BSA) condensates, we show that droplets swim along chemical gradients. Active BSA droplets loaded with urease swim toward each other. Passive BSA droplets show diverse responses to externally applied gradients of the enzyme's substrate and products. In all these cases, droplets swim toward solvent conditions that favor their dissolution. We call this behavior "dialytaxis", and expect it to be generic, as conditions which favor dissolution typically reduce interfacial tension, whose gradients are well-known to drive droplet motion through the Marangoni effect. These results could potentially suggest alternative physical mechanisms for active transport in living cells, and may enable the design of fluid micro-robots.

Emergence of inertia in the low-Reynolds regime of self-diffusiophoretic motion

For isotropic swimming particles driven by self-diffusiophoresis at zero Reynolds number (where particle velocity responds instantaneously to applied force), the diffusive timescale of emitted solute can produce an emergent quasi-inertial behavior. These particles can orbit in a central potential and reorient under second-order dynamics, not the first-order dynamics of classical zero-Reynolds motion. They are described by a simple effective model that embeds their history-dependent behavior as an effective inertia, this being the most primitive expression of memory. The system can be parameterized with dynamic quantities such as particle size and swimming speed, without detailed knowledge of the diffusiophoretic mechanism.

Pair Interactions of Self-Propelled SiO2-Pt Janus Colloids: Chemically Mediated Encounters

Driven by the necessity to achieve a thorough comprehension of the bottom-up fabrication process of functional materials, this experimental study investigates the pairwise interactions or collisions between chemically active SiO2-Pt Janus colloids. These collisions are categorized based on the Janus colloids' orientations before and after they make physical contact. In addition to the hydrodynamic interactions, the Janus colloids are also known to affect each other's chemical field, resulting in chemophoretic interactions, which depend on the degree of surface anisotropy in reactivity of Janus colloid and the solute-surface interaction at play. Our study reveals that these interactions lead to a noticeable decrease in particle speed and changes in orientation that correlate with the contact duration and yield different collision types. Distinct configurations of contact during collisions were found, whose mechanisms and likelihood are found to be dependent primarily on the chemical interactions. Such estimates of collision and their characterization in dilute suspensions shall have a key impact in determining the arrangement and time scales of dynamical structures and assemblies of denser suspensions and potentially the functional materials of the future.

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.

Spontaneous Multi-scale Supramolecular Assembly Driven by Noncovalent Interactions Coupled with the Continuous Marangoni Effect

Reported herein is the multi-scale supramolecular assembly (MSSA) process along with redox reactions driven by supramolecular interactions coupled with the spontaneous Marangoni effect in ionic liquid (IL)-based extraction systems. The black powder, the single sphere with a black exterior, and the single colorless sphere were formed step by step at the interface when an aqueous solution of KMnO4 was mixed with the IL phase 1-(2-hydroxyethyl)-3-methylimidazolium bis(trifluoromethylsulfonyl) imide (C2OHmimNTf2) bearing octyl(phenyl)-N,N-diisobutylcarbamoylmethylphosphine oxide (CMPO). The mechanism of the whole process was studied systematically. The phenomena were related closely to the change in the valence state of Mn. The MnO4- ion could be reduced quickly to δ-MnO2 and further to Mn2+ slowly by the hydroxyl-functionalized IL C2OHmimNTf2. Based on Mn2+, Mn(CMPO)32+, elementary building blocks (EBBs), and [EBB]n clusters were generated step by step. The [EBB]n clusters with the large enough size that were transferred to the interface, together with the remaining δ-MnO2, assembled into the single sphere with a black exterior, driven by supramolecular interactions coupled with the spontaneous Marangoni effect. When the remaining δ-MnO2 was used up, the mixed single sphere turned completely colorless. It was found that the reaction site of C2OHmim+ with Mn(VII) and Mn(IV) was distributed mainly at the side chain with a hydroxyl group. The MSSA process presents unique spontaneous phase changes. This work paves the way for the practical application of the MSSA-based separation method developed recently. The process also provides a convenient way to observe in situ and characterize directly the continuous Marangoni effect.

Reconfigurable Droplet-Droplet Communication Mediated by Photochemical Marangoni Flows

Droplets are attractive building blocks for dynamic matter that organizes into adaptive structures. Communication among collectively operating droplets opens untapped potential in settings that vary from sensing, optics, protocells, computing, or adaptive matter. Inspired by the transmission of signals among decentralized units in slime mold Physarum polycephalum, we introduce a combination of surfactants, self-assembly, and photochemistry to establish chemical signal transfer among droplets. To connect droplets that float at an air-water interface, surfactant triethylene glycol monododecylether (C12E3) is used for its ability to self-assemble into wires called myelins. We show how the trajectory of these myelins can be directed toward selected photoactive droplets upon UV exposure. To this end, we developed a strategy for photocontrolled Marangoni flow, which comprises (1) the liquid crystalline coating formed at the surface of an oleic acid/sodium oleate (OA/NaO) droplet when in contact with water, (2) a photoacid generator that protonates sodium oleate upon UV exposure and therefore disintegrates the coating, and (3) the surface tension gradient that is generated upon depletion of the surfactant from the air-water interface by the uncoated droplet. Therefore, localized UV exposure of selected OA/NaO droplets results in attraction of the myelins such that they establish reconfigurable connections that self-organize among the C12E3 and OA/NaO droplets. As an example of communication, we demonstrate how the myelins transfer fluorescent dyes, which are selectively delivered in the droplet interior upon photochemical regulation of the liquid crystalline coating.

Emergent Collective Motion of Self-Propelled Condensate Droplets

Recently, there is much interest in droplet condensation on soft or liquid or liquidlike substrates. Droplets can deform soft and liquid interfaces resulting in a wealth of phenomena not observed on hard, solid surfaces (e.g., increased nucleation, interdroplet attraction). Here, we describe a unique collective motion of condensate water droplets that emerges spontaneously when a solid substrate is covered with a thin oil film. Droplets move first in a serpentine, self-avoiding fashion before transitioning to circular motions. We show that this self-propulsion (with speeds in the 0.1-1  mm s^{-1} range) is fueled by the interfacial energy release upon merging with newly condensed but much smaller droplets. The resultant collective motion spans multiple length scales from submillimeter to several centimeters, with potentially important heat-transfer and water-harvesting applications.

Launching a Drop via Interplay of Buoyancy and Stick-Jump Dissolution

According to Archimedes' principle, a submerged object with a density lower than that of aqueous acid solution is more buoyant than a smaller one. In this work, a remarkable phenomenon is reported wherein a dissolving drop on a substrate rises in the water only after it has diminished to a much smaller size, though the buoyancy is smaller. The drop consisting of a polymer solution reacts with the acid in the surrounding, yielding a water-soluble product. During drop dissolution, water-rich microdroplets form within the drop, merging with the external aqueous phase along the drop-substrate boundary. Two key elements determine the drop rise dynamics. The first is the stick-jump behavior during drop dissolution. The second is that buoyancy exerts a strong enough force on the drop at an Archimedean number greater than 1, while the stick-jump behavior is ongoing. The time of the drop rise is controlled by the initial size and the reaction rate of the drop. This novel mechanism for programmable drop rise may be beneficial for many future applications, such as microfluidics, microrobotics, and device engineering where the spontaneous drop detachment may be utilized to trigger a cascade of events in a dense medium.

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.

Isotropic active colloids: explicit vs. implicit descriptions of propulsion mechanisms

Modeling the couplings between active particles often neglects the possible many-body effects that control the propulsion mechanism. Accounting for such effects requires the explicit modeling of the molecular details at the origin of activity. Here, we take advantage of a recent two-dimensional model of isotropic active particles whose propulsion originates from the interactions between solute particles in the bath. The colloid catalyzes a chemical reaction in its vicinity, which results in a local phase separation of solute particles, and the density fluctuations of solute particles cause the enhanced diffusion of the colloid. In this paper, we investigate an assembly of such active particles, using (i) an explicit model, where the microscopic dynamics of the solute particles is accounted for; and (ii) an implicit model, whose parameters are inferred from the explicit model at infinite dilution. In the explicit solute model, the long-time diffusion coefficient of the active colloids strongly decreases with density, an effect which is not captured by the derived implicit model. This suggests that classical models, which usually decouple pair interactions from activity, fail to describe collective dynamics in active colloidal systems driven by solute-solute interactions.

Effect of speed fluctuations on the collective dynamics of active disks

Numerical simulations are performed on the collective dynamics of active disks, whose self-propulsion speed (U) varies in time, and whose orientation evolves according to rotational Brownian motion. Two protocols for the evolution of speed are considered: (i) a deterministic one involving a periodic change in U at a frequency ω; and (ii) a stochastic one in which the speeds are drawn from a power-law distribution at time-intervals governed by a Poissonian process of rate β. In the first case, an increase in ω causes the disks to go from a clustered state to a homogeneous one through an apparent phase-transition, provided that the direction of self-propulsion is allowed to reverse. Similarly, in the second case, for a fixed value of β, the extent of cluster-breakup is larger when reversals in the self-propulsion direction are permitted. Motility-induced phase separation of the disks may therefore be avoided in active matter suspensions in which the constituents are allowed to reverse their self-propulsion direction, immaterial of the precise temporal nature of the reversal (deterministic or stochastic). Equally, our results demonstrate that phase separation could occur even in the absence of a time-averaged motility of an individual active agent, provided that the rate of direction reversals is smaller than the orientational diffusion rate.

From autocatalysis to survival of the fittest in self-reproducing lipid systems

Studying autocatalysis - in which molecules catalyse their own formation - might help to explain the emergence of chemical systems that exhibit traits normally associated with biology. When coupled to other processes, autocatalysis can lead to complex systems-level behaviour in apparently simple mixtures. Lipids are an important class of chemicals that appear simple in isolation, but collectively show complex supramolecular and mesoscale dynamics. Here we discuss autocatalytic lipids as a source of extraordinary behaviour such as primitive chemical evolution, chemotaxis, temporally controllable materials and even as supramolecular catalysts for continuous synthesis. We survey the literature since the first examples of lipid autocatalysis and highlight state-of-the-art synthetic systems that emulate life, displaying behaviour such as metabolism and homeostasis, with special consideration for generating structural complexity and out-of-equilibrium models of life. Autocatalytic lipid systems have enormous potential for building complexity from simple components, and connections between physical effects and molecular reactivity are only just beginning to be discovered.

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.

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.

Inverted Sedimentation of Active Particles in Unbiased ac Fields

Gaining control over the motion of active particles is crucial for applications ranging from targeted cargo delivery to nanomedicine. While much progress has been made recently to control active motion based on external forces, flows, or gradients in concentration or light intensity, which all have a well-defined direction or bias, little is known about how to steer active particles in situations where no permanent bias can be realized. Here, we show that ac fields with a vanishing time average provide an alternative route to steering active particles. We exemplify this route for inertial active particles in a gravitational field, observing that a substantial fraction of them persistently travels in the upward direction upon switching on the ac field, resulting in an inverted sedimentation profile at the top wall of a confining container. Our results offer a generic control principle that could be used in the future to steer active motion, direct collective behaviors, and purify mixtures.

Multiple iodide autocatalysis paths of chemo-hydrodynamical patterns in the Briggs-Rauscher reaction

Autocatalytic feedback is often regarded as the core step for the chemo-hydrodynamical patterns in the nonlinear reaction system. The Briggs-Rauscher (BR) reaction shows sequential chemo-hydrodynamical patterns with three states, i.e. labyrinth, high iodine state, and rotating dendritic patterns. The short-lived labyrinth patterns, depending on [Mn²⁺]₀, the ratio of [CH₂(COOH)₂]₀ and [KIO₃]₀ and light intensities, result from iodide autocatalytic loop, which has three paths (involving Mn²⁺-induced radical reactions, the oxidation of iodomalonic compounds, and light-induced radical reactions, respectively). The high iodine state appears in a high ratio of [CH₂(COOH)₂]₀ and [KIO₃]₀, relating to the autocatalytic path involving the oxidation of iodomalonic compounds. The light-induced radical autocatalytic path can act as a convenient control parameter to modulate the patterns in the first stage by increasing the iodine radicals. The dendritic patterns in the third stage result from the Marangoni effect caused by the evaporation of the solutions and reactions between H₂O₂ and iodine-containing species, which is independent of [CH₂(COOH)₂]₀ and [Mn²⁺]₀. This work contributes to a better understanding of the complex spatiotemporal patterns in the chemo-hydrodynamical system.

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.

Most probable paths for active Ornstein-Uhlenbeck particles

Fluctuations play an important role in the dynamics of stochastic systems. In particular, for small systems, the most probable thermodynamic quantities differ from their averages because of the fluctuations. Using the Onsager Machlup variational formalism we analyze the most probable paths for nonequilibrium systems, in particular, active Ornstein-Uhlenbeck particles, and investigate how the entropy production along these paths differs from the average entropy production. We investigate how much information about their nonequilibrium nature can be obtained from their extremum paths and how these paths depend on the persistence time and their swim velocities. We also look at how the entropy production along the most probable paths varies with the active noise and how it differs from the average entropy production. This study would be useful to design artificial active systems with certain target trajectories.

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.

New types of complex motion of a simple camphor boat

We discuss the motion of a rectangular camphor boat, considering the position of a camphor pill in relation to the boat's stern as the control parameter. The boat moves because the pill releases surface active molecules that decrease the surface tension and support the motion. We introduce a new experimental system in which the boat rotates on a long arm around the axis located at the centre of a Petri dish; thus, the motion is restricted to a circle and can be studied under stationary conditions for a long time. The experiments confirmed two previously reported modes of motion: continuous motion when the pill was located at the boat edge and pulsating (intermittent) motion if it was close to the boat centre (Suematsu et al., J. Phys. Chem. C, 2010, 114(21), 9876-9882). For intermediate pill locations, we observed a new, unreported type of motion characterised by oscillating speed (i.e. oscillating motion). Different modes of motion can be observed for the same pill location. The experimental results are qualitatively confirmed using a simple reaction-diffusion model of the boat evolution used in the above-mentioned paper.

Gravitational settling of active droplets

The gravitational settling of oil droplets solubilizing in an aqueous micellar solution contained in a capillary channel is investigated. The motion of these active droplets reflects a competition between gravitational and Marangoni forces, the latter due to interfacial tension gradients generated by differences in filled-micelle concentrations along the oil-water interface. This competition is studied by varying the surfactant concentration, the density difference between the droplet and the continuous phase, and the viscosity of the continuous phase. The Marangoni force enhances the settling speed of an active droplet when compared to the Hadamard-Rybczynski prediction for a (surfactant free) droplet settling in Stokes flow. The Marangoni force can also induce lateral droplet motion, suggesting that the Marangoni and gravitational forces are not always aligned. The decorrelation rate (α) of the droplet motion, measured as the initial slope of the velocity autocorrelation and indicative of the extent to which the Marangoni and gravitational forces are aligned during settling, is examined as a function of the droplet size: correlated motion (small values of α) is observed at both small and large droplet radii, whereas significant decorrelation can occur between these limits. This behavior of active droplets settling in a capillary channel is in marked contrast to that observed in a dish, where the decorrelation rate increases with the droplet radius before saturating at large values of droplet radius. A simple relation for the crossover radius at which the maximal value of α occurs for an active settling droplet is proposed.

Two-dimensional diffusiophoretic colloidal banding: optimizing the spatial and temporal design of solute sinks and sources

Diffusiophoresis refers to the phenomenon where colloidal particles move in response to solute concentration gradients. Existing studies on diffusiophoresis, both experimental and theoretical, primarily focus on the movement of colloidal particles in response to one-dimensional solute gradients. In this work, we numerically investigate the impact of two-dimensional solute gradients on the distribution of colloidal particles, i.e., colloidal banding, induced via diffusiophoresis. The solute gradients are generated by spatially arranged sources and sinks that emit/absorb a time-dependent solute molar rate. First we study a dipole system, i.e., one source and one sink, and discover that interdipole diffusion and molar rate decay timescales dictate colloidal banding. At timescales shorter than the interdipole diffusion timescale, we observe a rapid enhancement in particle enrichment around the source due to repulsion from the sink. However, at timescales longer than the interdipole diffusion timescale, the source and sink screen each other, leading to a slower enhancement. If the solute molar rate decays at the timescale of interdipole diffusion, an optimal separation distance is obtained such that particle enrichment is maximized. We find that the partition coefficient of solute at the interface between the source and bulk strongly impacts the optimal separation distance. Surprisingly, the diffusivity ratio of solute in the source and bulk has a much weaker impact on the optimal dipole separation distance. We also examine an octupole configuration, i.e., four sinks and four sources, arranged in a circle, and demonstrate that the geometric arrangement that maximizes enrichment depends on the radius of the circle. If the radius of the circle is small, it is preferred to have sources and sinks arranged in an alternating fashion. However, if the radius of the circle is large, a consecutive arrangement of sources and sinks is optimal. Our numerical framework introduces a novel method for spatially and temporally designing the banded structure of colloidal particles in two dimensions using diffusiophoresis and opens up new avenues in a field that has primarily focused on one-dimensional solute gradients.

Achieving Control in Micro-/Nanomotor Mobility

Unprecedented opportunities exist for the generation of advanced nanotechnologies based on synthetic micro/nanomotors (MNMs), such as active transport of medical agents or the removal of pollutants. In this regard, great efforts have been dedicated toward controlling MNM motion (e.g., speed, directionality). This was generally performed by precise engineering and optimizing of the motors' chassis, engine, powering mode (i.e., chemical or physical), and mechanism of motion. Recently, new insights have emerged to control motors mobility, mainly by the inclusion of different modes that drive propulsion. With high degree of synchronization, these modes work providing the required level of control. In this Minireview, we discuss the diverse factors that impact motion; these include MNM morphology, modes of mobility, and how control over motion was achieved. Moreover, we highlight the main limitations that need to be overcome so that such motion control can be translated into real applications.

Marangoni-induced reversal of meniscus-climbing microdroplets

Small water droplets or particles located at an oil meniscus typically climb the meniscus due to unbalanced capillary forces. Here, we introduce a size-dependent reversal of this meniscus-climbing behavior, where upon cooling of the underlying substrate, droplets of different sizes concurrently ascend and descend the meniscus. We show that microscopic Marangoni convection cells within the oil meniscus are responsible for this phenomenon. While dynamics of relatively larger water microdroplets are still dominated by unbalanced capillary forces and hence ascend the meniscus, smaller droplets are carried by the surface flow and consequently descend the meniscus. We further demonstrate that the magnitude and direction of the convection cells depend on the meniscus geometry and the substrate temperature and introduce a modified Marangoni number that well predicts their strength. Our findings provide a new approach to manipulating droplets on a liquid meniscus that could have applications in material self-assembly, biological sensing and testing, or phase change heat transfer.

Conditions for the propulsion of a colloid surrounded by a mesoscale phase separation

We study a two-dimensional model of an active isotropic colloid whose propulsion is linked to the interactions between solute particles of the bath. The colloid catalyzes a chemical reaction in its vicinity, that yields a local phase separation of solute particles. The density fluctuations of solute particles result in the enhanced diffusion of the colloid. Using numerical simulations, we thoroughly investigate the conditions under which activity occurs, and we establish a state diagram for the activity of the colloid as a function of the parameters of the model. We use the generated data to unravel a key observable that controls the existence and the intensity of activity: The filling fraction of the reaction area. Remarkably, we finally show that propulsion also occurs in three-dimensional geometries, which confirms the interest of this mechanism for experimental applications.

Spatial programming of self-organizing chemical systems using sustained physicochemical gradients from reaction, diffusion and hydrodynamics

Living organisms employ chemical self-organization to build structures, and inspire new strategies to design synthetic systems that spontaneously take a particular form, via a combination of integrated chemical reactions, assembly pathways and physicochemical processes. However, spatial programmability that is required to direct such self-organization is a challenge to control. Thermodynamic equilibrium typically brings about a homogeneous solution, or equilibrium structures such as supramolecular complexes and crystals. This perspective addresses out-of-equilibrium gradients that can be driven by coupling chemical reaction, diffusion and hydrodynamics, and provide spatial differentiation in the self-organization of molecular, ionic or colloidal building blocks in solution. These physicochemical gradients are required to (1) direct the organization from the starting conditions (e.g. a homogeneous solution), and (2) sustain the organization, to prevent it from decaying towards thermodynamic equilibrium. We highlight four different concepts that can be used as a design principle to establish such self-organization, using chemical reactions as a driving force to sustain the gradient and, ultimately, program the characteristics of the gradient: (1) reaction-diffusion coupling; (2) reaction-convection; (3) the Marangoni effect and (4) diffusiophoresis. Furthermore, we outline their potential as attractive pathways to translate chemical reactions and molecular/colloidal assembly into organization of patterns in solution, (dynamic) self-assembled architectures and collectively moving swarms at the micro-, meso- and macroscale, exemplified by recent demonstrations in the literature.

A review on control of droplet motion based on wettability modulation: principles, design strategies, recent progress, and applications

The transport of liquid droplets plays an essential role in various applications. Modulating the wettability of the material surface is crucial in transporting droplets without external energy, adhesion loss, or intense controllability requirements. Although several studies have investigated droplet manipulation, its design principles have not been categorized considering the mechanical perspective. This review categorizes liquid droplet transport strategies based on wettability modulation into those involving (i) application of driving force to a droplet on non-sticking surfaces, (ii) formation of gradient surface chemistry/structure, and (iii) formation of anisotropic surface chemistry/structure. Accordingly, reported biological and artificial examples, cutting-edge applications, and future perspectives are summarized.

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.

Onsager reciprocal relations and chemo-mechanical coupling for chemically active colloids

Similar to cells, bacteria, and other micro-organisms, synthetic chemically active colloids can harness the energy from their environment through a surface chemical reaction and use the energy to self-propel in fluidic environments. In this paper, we study the chemo-mechanical coupling that leads to the self-propulsion of chemically active colloids. The coupling between chemical reactions and momentum transport is a consequence of Onsager reciprocal relations. They state that the velocity and the surface reaction rate are related to mechanical and chemical affinities through a symmetric matrix. A consequence of Onsager reciprocal relations is that if a chemical reaction drives the motion of the colloid, then an external force generates a reaction rate. Here, we investigate Onsager reciprocal relations for a spherical active colloid that catalyzes a reversible surface chemical reaction between two species. We solve the relevant transport equations using a perturbation expansion and numerical simulations to demonstrate the validity of reciprocal relations around the equilibrium. Our results are consistent with previous studies and highlight the key role of solute advection in preserving the symmetry of the Onsager matrix. Finally, we show that Onsager reciprocal relations break down around a nonequilibrium steady state, which has implications for the thermal fluctuations of the active colloids used in experiments.

Chemically Tuning Attractive and Repulsive Interactions between Solubilizing Oil Droplets

Micellar solubilization is a transport process occurring in surfactant-stabilized emulsions that can lead to Marangoni flow and droplet motility. Active droplets exhibit self-propulsion and pairwise repulsion due to solubilization processes and/or solubilization products raising the droplet's interfacial tension. Here, we report emulsions with the opposite behavior, wherein solubilization decreases the interfacial tension and causes droplets to attract. We characterize the influence of oil chemical structure, nonionic surfactant structure, and surfactant concentration on the interfacial tensions and Marangoni flows of solubilizing oil-in-water drops. Three regimes corresponding to droplet "attraction", "repulsion" or "inactivity" are identified. We believe these studies contribute to a fundamental understanding of solubilization processes in emulsions and provide guidance as to how chemical parameters can influence the dynamics and chemotactic interactions between active droplets.

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.

Collective motion of epithelial cells along a wrinkled 3D-buckled hydrogel

Epithelial cells migrate autonomously by aligning and inducing a collective motion. Controlling the collective motion of epithelial cells in geometrically confined environments is important for understanding physiological processes such as wound healing and self-organized morphogenesis. However, collective migration under a three-dimensional (3D) curved surface resembling living epithelial tissue has not yet been explored. In this study, we investigated the collective motion of a 3D-buckled polyacrylamide (PAAm) gel that mimics the shape of folds and wrinkles of epithelial tissue to understand the geometric effects of collective motion. We found that the velocity correlation in the space near the hydrogel boundary showed a periodic change that correlated with the wrinkled folding of the hydrogel pattern. Furthermore, the characteristic length of the velocity correlation increased proportionally with the wavelength of wrinkled folding. These observations indicated that the hydrogel pattern could steer the collective motion of epithelial cells over long distances. Our study also suggests that the wrinkled design of the hydrogel is a versatile platform for studying the geometric effect of a curved surface on complex epithelial cell dynamics.

Motion of a self-propelled particle with rotational inertia

Overdamped active Brownian motion of self-propelled particles in a liquid has been fairly well studied. However, there are a variety of situations in which the overdamped approximation is not justified, for instance, when self-propelled particles move in a low-viscosity medium or when their rotational diffusivity is enhanced by internal active processes or external control. Examples of various origins include biofilaments driven by molecular motors, living and artificial microflyers and interfacial surfers, field-controlled and superfluid microswimmers, vibration-driven granular particles and autonomous mini-robots with sensorial delays, etc. All of them extend active Brownian motion to the underdamped case, i.e., to active Langevin motion, which takes into account inertia. Despite a rich experimental background, there is a gap in the theory in the field where rotational inertia significantly affects the random walk of active particles on all time scales. In particular, although the well-known models of active Brownian and Ornstein-Uhlenbeck particles include a memory effect of the direction of motion, they are not applicable in the underdamped case, because the rotational inertia, which they do not account for, can partially prevent "memory loss" with increasing rotational diffusion. We describe the two-dimensional motion of a self-propelled particle with both translational and rotational inertia and velocity fluctuations. The proposed generalized analytical equations for the mean kinetic energy, mean-square displacement and noise-averaged trajectory of the self-propelled particle are confirmed by numerical simulations in a wide range of self-propulsion velocities, moments of inertia, rotational diffusivities, medium viscosities and observation times.