Abstract
Condensation of objects into stable clusters occurs naturally in equilibrium^{1} and driven systems^{2,3,4,5}. It is commonly held that potential interactions^{6}, depletion forces^{7}, or sensing^{8} are the only mechanisms which can create longlived compact structures. Here we show that persistent motile structures can form spontaneously from hydrodynamic interactions alone, with no sensing or potential interactions. We study this structure formation in a system of colloidal rollers suspended and translating above a floor, using both experiments and largescale threedimensional simulations. In this system, clusters originate from a previously unreported fingering instability, where fingers pinch off from an unstable front to form autonomous ‘critters’, whose size is selected by the height of the particles above the floor. These critters are a stable state of the system, move much faster than individual particles, and quickly respond to a changing drive. With speed and direction set by a rotating magnetic field, these active structures offer interesting possibilities for guided transport, flow generation, and mixing at the microscale.
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Main
We have identified a new instability in one of the most basic systems of lowReynoldsnumber (steady Stokes or overdamped) flow, a collection of spheres rotating near a wall. This system has been well studied analytically and numerically^{9,10}, since it is considered a base model for understanding many microbial and colloidal flows. The instability visually resembles wet paint dripping down a wall or individual droplets sliding down a windshield^{11}—examples of Rayleigh–Taylor instabilities^{12}. However, in those and other clustering phenomena, what holds things together is surface tension or other forces deriving from an interaction potential. Here we use a model system to explore whether hydrodynamic interactions alone, without particle collisions, attractions or sense/response redirection, can lead to stable finite clusters.
The experimental system consists of polymer colloids with radius a = 0.66 μm which have a small permanent magnetic moment (m ∼ 5 × 10^{−16} A m^{−2}) from an embedded haematite cube^{13} (see schematic in Fig. 1a). Interparticle magnetic interactions are small compared to thermal energy (<0.1k_{B}T). A rotating magnetic field () with magnitude B_{0} and frequency f = ω/2π is applied, causing all the particles to rotate about the yaxis at the same rate ω. The particles rotate synchronously with the field for ω < ω_{c}, where ω_{c} is the critical frequency above which the applied magnetic torque is not enough to balance the viscous torque on the particle (see Supplementary Section I for details of the rotation mechanism). In all of our experiments, ω < ω_{c}. In contrast with recent experiments on Quincke rollers^{14}, the rotation direction is prescribed and does not arise from the system dynamics. Hydrodynamics is the dominant interparticle interaction in this system, which is distinctly different from many other systems of rotating magnetic particles, where dynamics is found to be a strong function of interparticle magnetic interactions^{15,16,17,18}. Many ferromagnetic particles with a small remnant moment could produce the same behaviour. Gravity plays an unusual role in our instability. Rather than a driving force partially compensated by surface tension or viscosity, as in Rayleigh–Taylor instabilities, here gravity is completely compensated by thermal motion and sets the average particle height, h = a + k_{B}T/mg, where mg is the particle buoyant weight. The particles are contained in glass capillary chambers, with a depth ≥100 μm. The particles have a density of 2,000 kg m^{−3}, and are thus suspended at h ≍ 1.0 μm above the capillary floor, interacting essentially with only one wall.
We model the system as particles near an infinite wall driven at a constant rotational frequency. The manybody simulations are done using an accelerated stressletfree variant of Stokesian dynamics^{19} that represents each particle as a single regularized stokeslet and rotlet singularity. The hydrodynamic interactions between the particles, and the particles and the wall, are explicitly modelled (see Methods). This is a minimally resolved, and thus lowaccuracy method. The accuracy of the numerical calculations can be improved by using a ‘multiblob’ approach where particles such as our rotating spheres are represented as a rigid cluster of regularized stokeslet singularities or ‘blobs’^{20}. The accuracy and resolution are set by the number of blobs per particle. Improved resolution comes at the computational expense of a reduction in the number of particles and the time span that can be simulated. In Supplementary Fig. 1b we compare a wellresolved sphere model containing 2,562 blobs, which is used in Fig. 1c, d, with the minimally resolved model used elsewhere, which produces qualitatively similar results. The simulations include neither the effects of thermal motion nor magnetic forces as they are small compared to viscous forces (see Supplementary Section II). The particle–wall separation is set by creating a repulsive wall potential. In addition to the hydrodynamic interactions between particles, steric repulsion between particles is used for some simulations (for example, to model the experiment) and dropped for others (for example, to test the role of pure hydrodynamics).
Hydrodynamic coupling plays a crucial role in the dynamics of this system. A sphere rotating near a wall about the yaxis will move in the xdirection (see Fig. 1c). This motion is a result of the unequal drag force on the top and the bottom of the particle. Its translation speed v_{0} is set by the scaled distance to the wall and the rotation rate^{10}: v_{0} = ωaf(h/a), and vanishes as the height increases—that is, f(h/a) → 0 as h → ∞. However, as shown in Fig. 1c, v_{0} is orders of magnitude slower than the fluid velocity at the particle’s surface (ωa). The velocity field around a particle decays slowly (as 1/r^{2}) in the xyplane, where r is the distance to the particle centre. Thus, even at moderate area fractions, φ, particle motion is mainly a result of being advected in the flow of neighbouring particles. This collective effect can be described by a densitydependent mean velocity, , where α is found to be 50 ± 2 in our experimental system (see Fig. 1b). Putting v ≍ αφv_{0} into the continuity equation,
results in the inviscid Burger’s equation, which is well known to lead to the development of a shock front^{21}. Although the development of a shock is a simple consequence of having a densitydependent velocity, to our knowledge, this feature has been observed only in lowdensity flowing emulsion systems^{22,23}, and not other colloidal roller systems^{14}. As in Burger’s solution, we need a leading edge density gradient to observe the shock.
The propagating shock front quickly becomes unstable in the direction transverse to propagation, leading to the appearance of density fluctuations which continue to grow as fingers (see Fig. 2a and Supplementary Movies 1 and 2). This fingering instability does not occur in a planar Burger’s shock^{24}. Both the experiments and simulations show a qualitatively similar evolution of the shock, the shock instability, and the fingering, at the same relative times. Despite its deceptively similar appearance, this fingering instability is distinct from other previously reported viscous^{11}, granular^{2,25} and colloidal instabilities^{26,27,28}. In a Rayleigh–Taylorlike instability, the fingering dynamics is controlled by a balance of viscous damping and a bodyforce driving term (like gravity). In contrast, the instability wavelength in this system is independent of both viscosity and driving torque (that is, rotation rate); see Fig. 2b, c. We define the instability wavelength, λ_{max}, as the wavelength associated with the fastest growing normal mode, as is typical for a linear instability (see Supplementary Section III for details). The control parameter for this instability, in both the experiments and the simulations, is h, as illustrated in Fig. 2d, e. As shown in that figure, when the particle–wall distance increases, the instability wavelength increases linearly. The numerical results are obtained by confining the particles to a plane parallel to the wall at a given height h (see Supplementary Movie 3). As discussed later in the text, the same dynamics are observed in this configuration as in the fully threedimensional (3D) simulations. In the experiments, the height is adjusted by changing the solvent density, and hence the colloids’ buoyant weight mg. Although we change the particle–wall separation in quite different ways in the experiments and simulations, in both cases h is the key control parameter for λ_{max}.
Due to the increased density in the shock region, the fingertips are much denser and, due to collective effects, that is, , move much faster than the rest of the suspension. In the simulations, if the particles are maintained high enough away from the wall, the fingertips break off to form selfsustained, compact clusters made of hundreds of particles, which we term ‘critters’ (see Fig. 3a and Supplementary Movie 4). These critters rotate around their centre of mass and translate with a speed 15,000 times faster than a single roller would at the same centre of mass height (Fig. 3b inset). Critters form a natural stable state of the system: they move at a constant speed, do not lose particles, and are not observed to dissolve (see Fig. 3b and Supplementary Movie 4). We further explore their stability by changing the direction (but not the magnitude) of ω periodically in time. As shown in Fig. 3c and Supplementary Movie 5, the critters follow the prescribed circular trajectory. Somewhat similar structures to the critters were obtained experimentally when h was increased (Fig. 2e and Supplementary Movie 6), although in the experiment critters continually lost some particles as they moved. In the simulations, the compact critters are extremely robust, suggesting they may be an attractor in the system dynamics—similar critters appear regardless of the initial conditions (see Supplementary Movie 7).
The velocity field in Fig. 1d suggests that the transverse instability of the shock originates from the lateral hydrodynamic attraction and repulsion in the xyplane; this lateral flow shows the same qualitative features for a rigid sphere and for a point rotlet above a noslip boundary^{9}. To test the assumption that this is a planar instability, we simplify the system in our simulations by restricting the rollers to a fixed plane above the wall. This simplified system reproduces the instability: the shock forms, the transverse instability develops, and autonomous clusters with selected size detach and translate much faster than individual particles, as shown in Fig. 4a. We further remove all nonhydrodynamic effects and simplify the system to its bare minimum by simulating instead a collection of point rotlet singularities without any steric repulsion. Figure 4c and Supplementary Movie 8 show that both the fingering instability and clustering are reproduced with only this one ingredient: hydrodynamic interactions in the vicinity of a noslip boundary.
A closer look at the flow field around a cluster in the frame moving with its centre of mass shows a welldefined recirculation zone whose size matches the cluster size, as shown in Fig. 4b. The closed streamlines in this flow field are responsible for the selfsustained and compact clusters. As seen in studies of sedimenting particle clouds^{29}, the chaotic nature of the flow inside a cluster can lead to the loss of particles. However, in our 3D simulations, additional circulation in the xzplane prevents particle loss; the critters are stabilized by the closed streamlines. Critters smaller than the size of the recirculation zone, which is proportional to the height above the wall, can form and remain stable, while larger ones break up by shedding excess particles.
Clustering is seen in many other lowReynoldsnumber systems, from sedimentation to active colloidal particles. What is notable about the critters that emerge from this instability is that they are a persistent state which can be produced from hydrodynamic interactions. Other kinds of hydrodynamic clusters, such as those seen in sedimentation^{30,31}, are always transient and not longlived structures. Almost all active matter systems exhibit some kind of clustering instability^{3,4}, but it is usually a consequence of particle–particle interactions, either directly through an attractive potential, sensing, or via selftrapping, which is a consequence of a repulsive particle potential. Here we have demonstrated that the same instability observed in the experiments is preserved in the simulations, even when all interactions except hydrodynamics are removed.
In this study, we isolated the role of hydrodynamics. We note that this instability is generic and should be found in any system of particles rotating parallel to a floor, provided that hydrodynamics is the dominant particle–particle interaction. The addition of particle–particle potentials can strongly modify the instability structure and dynamics. For example, varying the Mason number (relative strength of hydrodynamic and magnetic interactions) completely changes the dynamics of the system (see Supplementary Section II and Supplementary Movie 9). The structures and flow patterns formed in our model system suggest a number of possible applications. Collections of rollers create strong advective flows, and their motion and direction can be externally powered and controlled. As shown in Supplementary Section IV, we have found, both experimentally and in simulations, a number of promising ways to transport passive particles by microrollers in either homogeneous suspensions, fingers, or critters.
Methods
Experiments.
The colloidal particles are TPM (3methacryloxypropyl trimethoxysilane) spheres (a = 0.66 μm) with haematite cubes embedded into them (see Sacanna et al. ^{13} for details of the synthesis). Haematite is a canted antiferromagnet, thus the particles possess a small permanent moment, m ∼ 5 × 10^{−16} A m^{−2}, which can be oriented in an applied magnetic field. The particles are dispersed in either water or aqueous glycerol solutions (dynamic viscosity η = 1 mPa s, η = 4 mPa s, or η = 10 mPa s). To increase buoyancy, additional samples are created with particles dispersed in a 410 mM sodium polytungstate solution, with a small amount of tetramethylammonium hydroxide (TMAH) added as an additional stabilizer (1.2% v/v).
In all cases, the particles are contained in glass capillary tubes of depth 100 μm or greater (VitroCom VitroTubes), which are sealed with ultraviolet epoxy (Norland NOA63). To create the initial density gradient in particle concentration, the chambers are tilted so that particles gather to one side, then are laid flat to ensure a monolayer is formed as the initial condition. Distance of this initial gradient to the vertical chamber wall does not affect the instability wavelength.
The rotating magnetic field is created using custom triaxial coils. A bipolar current supply (KEPCO BOP 502M) is used to apply the current to the coils and create a rotating magnetic field. The waveforms for the rotating field are generated using a DAQ (MCC USB3101FS) controlled via Matlab. The field generated by the coils is measured with a Hall sensor (Ametes MFS3A). For all experiments described in this work, the magnitude of the field is 2.94 mT, and the frequency is varied from 0.2–25 Hz.
All observations are made using a Nikon TiU inverted microscope. Roller velocity is calculated in two ways. At low area fractions (φ < 0.1), the velocity is computed using the instantaneous velocity calculated from particle tracking^{32}. The velocity was computed for small segments of the individual particle tracks, and the results were then binned to calculate the mean roller velocity. At high area fractions, individual particle velocities cannot be measured, and a custom Python code was used to process the images and perform particle image velocimetry (PIV) analysis. The roller velocity was then taken to be the mean suspension velocity computed from the PIV analysis. Using a range of area fractions where both particle tracking and PIV could be used, φ = 0.10 − 0.20, we validated that the mean suspension velocity matched the individual particle velocity—that is, when hydrodynamic collective effects are predominant, the mean suspension velocity is equivalent to the individual particle velocity.
Simulations.
The flow fields around one roller shown in Fig. 1 are obtained by using the rigid multiblob method developed by Usabiaga and colleagues^{20}. The surface of the roller is discretized with 2,562 blobs which are rigidly connected with constraint forces. Pairwise hydrodynamic interactions between blobs are modelled with the Rotne–Prager–Blake tensor with wall corrections^{19}.
The multiparticle simulations are performed using the Stokesian Dynamics method developed by Swan and Brady^{19}, omitting stresslets and thermal fluctuations (Brownian motion). In brief, the hydrodynamic response is computed by replacing each sphere with a regularized singularity (stokeslet and rotlet) and accounting for the hydrodynamic interaction with the wall in an approximate but selfconsistent way by applying Rotne–Prager corrections to the Blake image construction^{9}. This modelling includes only leadingorder corrections for the finite size of the particles to limit the computational cost required to simulate large numbers (O(10^{4})) of particles. Even though this lowresolution model overestimates the particle mobility, it remains physically consistent, and its accuracy can be controlled and quantified. A betterresolved multiblob model^{20} would increase the hydrodynamic accuracy but incur a higher computational cost. We compare both the minimally resolved model^{19} and the wellresolved multiblob model^{20} with experiments in Supplementary Section I.
As in the experiments, the rotation rate of the particles is prescribed. This is ensured by applying the appropriate torquesT obtained by solving the following resistance problem^{19}
where M^{rr} is the mobility matrix coupling the prescribed particle rotations Ω to the unknown particle torques T. M^{rt} is the mobility matrix coupling Ω to the known external forces F acting on the particles, which are a combination of particle–particle and particle–wall repulsive forces and gravity. Once the torques are obtained, the translational velocities V are found with the mobility relation
where M^{tr} = (M^{rt})^{T} couples V to T and M^{tt} couples V to F. When the particles are restricted to a plane at fixed height z = h, forces and motion in the zdirection are discarded. Particle trajectories are integrated with the twostep Adams–Bashforth–Moulton predictor–corrector scheme. The time step Δt is chosen so that a particle does not travel more than 5% of its size per time step: vΔt < 0.05a. Typically, Δt = 0.005 s in most of the simulations. Mobilityvector products and steric interactions are computed with PyCUDA on an Nvidia K40 GPU. The typical simulation time is 7 h for 20,000 time iterations with O(10^{4}) particles.
When included, steric interactions between the particles are modelled with a pairwise softcore repulsive potential U_{part} of Yukawa type,
where r is the centretocentre distance between particles, S_{p} is the strength of the potential (S_{p} = 2.43 × 10^{8} mga) and the interaction range is D_{p} = 0.1a. Since the simulations do not include Brownian motion, to balance gravity forces and set the equilibrium height of the particles we use a repulsive potential from the wall
where z is height of the particle centre. The strength S_{w} and the range D_{w} are changed between the simulations to investigate the effect of the particle height on the instability: S_{w} = 0.05–25.1 mga and D_{w} = 0.1a–7a. The total force on the particles F is given by the gradient of the combination of the repulsive potentials, U_{part} and U_{wall}, and the gravitational potential mgz, where m = 1.27 × 10^{−15} kg is the excess mass of a roller.
Data availability.
The data that supports the plots within this paper and other findings of this study are available from the corresponding author on request.
Change history
10 March 2017
In the version of this Letter orignally published, the Acknowledgements was missing the following sentence: 'We gratefully acknowledge the support of the NVIDIA Corporation with the donation of GPU hardware for performing some of the simulations reported here'. This has been corrected in all versions of this Letter.
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Acknowledgements
This work was supported primarily by the Gordon and Betty Moore Foundation through Grant GBMF3849 and the Materials Research Science and Engineering Center (MRSEC) program of the National Science Foundation under Award Number DMR1420073. A. Donev and B. Delmotte were supported in part by the National Science Foundation under award DMS1418706. P. Chaikin was partially supported by NASA under Grant Number NNX13AR67G. We gratefully acknowledge the support of the NVIDIA Corporation with the donation of GPU hardware for performing some of the simulations reported here.
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M.D. performed the experiments. B.D. performed the simulations. M.Y. and S.S. synthesized the colloidal particles. M.D., B.D., A.D. and P.C. conceived the project, analysed the results and wrote the paper.
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Driscoll, M., Delmotte, B., Youssef, M. et al. Unstable fronts and motile structures formed by microrollers. Nature Phys 13, 375–379 (2017). https://doi.org/10.1038/nphys3970
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