Letter pubs.acs.org/Langmuir
Self-Organized Cooperative Swimming at Low Reynolds Numbers Alexander Reinmüller, Hans Joachim Schöpe, and Thomas Palberg* Johannes Gutenberg-Universität Mainz, Institut für Physik, Staudingerweg 7, 55128 Mainz, Germany S Supporting Information *
ABSTRACT: Investigations of swimming at low Reynolds numbers (Re < 10−4) so far have focused on individual or collectively moving autonomous microswimmers consisting of a single active building unit. Here we show that linear propulsion can also be reproducibly generated in a self-assembled dynamic complex formed from a granular, HCl-releasing particle settled on a charged quartz wall and a swarm of micrometer-sized negatively charged colloids. In isolation, none of the constituents shows motion beyond diffusion. When brought together, they self-assemble into a complex capable of directed swimming. It is stabilized by toroidal solvent flow centered about the granular particle. Propulsion is then launched by an asymmetric distribution of the colloids. Motion is selfstabilizing and continues for up to 25 min with velocities of 1−3 μm/s. Although the details of the mechanisms involved pose a formidable experimental and theoretical challenge, our observations offer a conceptually new, well-reproduced, versatile approach to swimming and transport at low Reynolds numbers.
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INTRODUCTION Swimming on your own at low Reynolds numbers is a difficult task. Purcell’s fundamental scallop theorem states that reciprocal configuration changes of a microswimmer (i.e., reversible changes following the same configuration space path back and forth rather than following a closed loop) do not lead to locomotion.1 This affects bacteria and other microorganisms.2−4 It also poses a challenge to microfluidics and medical engineering, whether it be to transport cargo or to generate solvent flow.5−7 Purcell’s theorem is circumvented by employing flexible filaments for propulsion allowing different phases for different subunits.8,9 Furthermore, the theorem affects neither the osmophoretic nor diffusiophoretic motion10,11 of shape-retaining objects generated by the difference in chemical potential along the microswimmer contour. The latter can be self-generated using a local concentration gradient of a charge neutral or ionic, adsorbing solute generated by chemical reactions of some fuel substance.12−17 Likewise, nonadsorbing reaction products can induce purely osmotic propulsion.18,19 In principle, the presence of any long-ranged gradient in solute concentration may facilitate interactions between a swimmer and its surroundings, including a container wall, other inactive (nonswimming) particles, and/or other swimmers. Typically, this leads to pattern formation or collective behavior.20,21 Ibele et al.22 used AgCl particles releasing HCl under UV illumination. Small particles showed phoretic motion, while sedimenting agglomerates induced diffusioosmotic solvent flow along the negatively charged substrate converging at each agglomerate and creating small toroidal convection cells.23 These in turn could be used to reversibly attract both equally sized, inactive particles and additional active swimmers (“schooling”). In this Letter, a long-ranged electrolyte concentration gradient allows the self-organization of similar dynamic © 2013 American Chemical Society
complexes. Under suitably altered boundary conditions, however, we can for the first time demonstrate the emergence of directed propulsion for individual complexes. In our study, we combine individual large HCl-releasing but phoretically inactive cation-exchange resin particles24 (CIEX − CatIon EXchange resin) sedimented loosely on a negatively charged substrate (glass or quartz) with a small but finite concentration of inactive colloidal spheres (anionic polystyrene spheres of 5.2 μm diameter at initial volume fractions φ < 0.01%) in pure water. Our observations show that the former induce diffusioosmotic solvent flow (cf. ref 23) as a result of different diffusivities of the H+ and Cl− ions involved. The flow is strong enough to collect and partially confine the latter within the accompanying convection cells. The resulting dynamic complex circumvents Purcell’s theorem.25 In fact, upon asymmetric loading with colloids we regularly observed the propulsion of the whole complex along the substrate (cf. movies S1−S4). Although the details of the actual driving forces of this cooperative motion are not yet fully clarified, we find that the propulsion as such is very robust and occurs with varying effectiveness over a large range of experimental parameters. We anticipate that our observations open an important alternative approach to transporting cargo or sorting colloids at low Reynolds numbers.
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EXPERIMENTAL SECTION
In a series of preliminary experiments involving several combinations of experimental boundary conditions, we could establish the following minimum protocol for the preparation of self-propulsive complexes. Untreated cation exchange resin (CIEX) spheres (Amberlite K306, Received: November 21, 2012 Revised: January 23, 2013 Published: January 23, 2013 1738
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Roth GmbH, Germany) were crushed, and tiny fragments (lateral extension 15 μm corresponding to a background area fraction of less than 0.005/μm2. Prior to measurement, the stock suspension was in contact with mixed-bed ion exchangers for several weeks. During the experiments, samples were in contact with ambient air, providing a residual electrolyte concentration of c ≈ 5 μmol/L resulting from contamination with carbon dioxide. In some cases, a cover glass was used; the fluid height in each case amounted to several hundred micrometers. When we observed these conditions, self-propulsion was reproducibly induced, but the CIEX fragment size and shape, colloid size, and concentration were all found to alter the efficiency of propulsion (cf. Supplementary Note B and Figures S1−S2). Smaller particles not leading to propulsion were anioniccarboxylate-modified polystyrene spheres with 2a = 1.1 μm, batch no. 2-1200 2149 by Interfacial Dynamics Corporation, USA. Furthermore, cationic melamin resin particles (2a = 5.2 μm, batch no. MF-F-S2616 by Microparticles GmbH Berlin, Germany) were used on cationic PDADMAC (poly(diallyldimethylammonium chloride) in H2O; Mw = 100 000−200 000 by Sigma-Aldrich) dipcoated quartz substrates. Samples were observed with a standard scientific microscope (DMIRBE by Leica, Germany) at low magnification using a standard CMOS camera.
solvent convection (cf. Figure 2). At the same time, the whole complex moves linearly in the direction of the arrow with a
Figure 2. Schematic drawing (side view) of the convective particle circulation: particles (gray) close to the CIEX fragment (brown) are lifted up vertically by diffusio-osmotically driven solvent flow (blue arrows). Subsequently, sedimentation (red arrow) back to the substrate surface occurs, and colloids are either recirculated or left behind. Although for stationary CIEX fragments the convection cell is radially symmetrically populated, it is strongly asymmetrically loaded in the case of propulsion (open arrow). Smaller colloidal particles are only weakly subjected to gravity and thus follow the solvent flow to much higher altitudes z so that particles cannot accumulate.
velocity of v = 1.3 ± 0.1 μm/s. For comparison, the freediffusion coefficient of a single colloidal particle was calculated to be 0.09 μm2/s. A dense trail of settled particles is deposited behind the CIEX fragment outside the immediate convection cell where it slowly and diffusely dissolves. As shown in Figure 3, this allows for tracing the motion of the complexes. Linear, slightly curved, and even circular trajectories were observed.
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RESULTS AND DISCUSSION Figure 1 shows a typical moving complex. An elongated, irregularly shaped CIEX fragment is loosely but asymmetrically
Figure 3. Optical micrographs of complexes with differently curved trajectories. (A, B) The same swimmer at different times. (C) The left complex has launched from the meanwhile stationary assembly on the right. Its thickening trace indicates acceleration; the thinning trace of the right assembly indicates previous deceleration. (Scale = 1150 × 860 μm2.)
Figure 1. Optical micrograph (top view) of a linearly self-propelling complex of colloidal particles accumulated at a CIEX fragment (bright elongated object in the center of the image), with the direction of motion indicated by the arrow. Sedimented colloidal particles (sharp circular objects) laterally converge in the substrate plane. Close to the CIEX fragment, these are vertically lifted by the convective currents (blurred circular objects), settle again, and form a prominent trail behind the CIEX. (scale = 270 × 200 μm2; snapshot taken from movie S1).
The motion of the complexes shows a characteristic temporal evolution on longer time scales (movies S2−S4). After propulsive motion is launched, an acceleration period follows that features a continuous transition from a particle number increase within the back convective circulation together with a negligible loss of particles lagging behind to a stage of maximum speed, during which an increasingly pronounced trail of colloidal particles, similar to a comet’s tail, is left behind (cf. the left particle in Figure 3C). Up to moderate colloid concentrations (area densities 10−4 mol/L. This convection is attributed to electrolyte-gradient-induced diffusio-osmotic solvent flow along the anionically charged substrate surface, which induces transport and the local accumulation of colloidal particles at stationary CIEX.22−24 Alternative diffusio-phoretic particle transport is ruled out as a major contribution because it would repel anionic colloid particles for the assumed electrolyte distribution23 and because it would not account for circular particle motion. We also tried NaOH-releasing anion exchanger resin fragments. On a negative substrate, diverging convection creates a void around the resin fragments. On a positively charged, polyelectrolyte dip-coated substrate in combination with positively charged colloids (melamin resin spheres of 5.2
Figure 5. Optical micrographs of typical colloid assemblies around stationary CIEX fragments on a glass substrate. The fragment size and convection strength increase from A to D. (A) Loosely crowded monolayer, (B) crystalline double layer, (C, D) marked convective circulation. (Scale = 265 × 265 μm2.)
increasing fragment size an increasing number of colloids are assembled. Figure 5A exhibits fluidlike ordering, and Figure 5B exhibits crystalline ordered sheets of sedimented colloids. Crystallization is due to the repulsive interaction between the size-monodisperse colloid particles at elevated concentrations.24 In panels C and D, a significant fraction of colloids appear blurred, indicating that the convection is strong enough to lift them above the focal plane. With increasing fragment size, the range over which colloids were assembled also increased. Although convection cells were regularly formed, the propulsion of isolated complexes set in only when dense and markedly asymmetric colloid distributions formed. In fact, the assemblies shown in Figure 5 are more or less symmetrical and are stationary. By contrast, moving CIEX fragments always had an asymmetrically loaded convection cell. This could be correlated to shape anisotropy and/or other irregularities of the fragments. Typically, the strongest convection (and assembly) was observed at pronounced grooves and/or thorns in the fragment contour (cf. movie I in ref 24), and it was less pronounced at straight or only slightly curved surfaces. Hence, 1740
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nonequilibrium conditions were externally induced by ac magnetic fields. Our self-assembled complexes show a remarkable degree of task sharing. CIEX acts as fuel reservoir, and the colloidal assembly is realized via the convection cell set in motion by the combination of HCl fuel with a negatively charged substrate. At present, our complexes transport and redistribute colloids. In principle, this task sharing may be further extended: an on/off switch may be realized by employing photosensitive HCl release (e.g., with suitably shaped and sized AgCl particles), and additional cargo may be adhered to the CIEX fragments by phoretic means.23 Also, the adherence of small magnetic beads to the latter may be of interest in testing steering possibilities. At present, steering could be realized only in a very crude fashion (i.e., scattering off stationary assemblies).
all swimmers had irregular shapes and moved with their longest side facing the direction of motion. Most of the asymmetric colloid distributions formed during the initial stage through fluctuations in the density of arriving colloids (movies S2, S3). In a few cases, however, we also observed that a CIEX fragment with an asymmetric colloid distribution launched from a larger stationary agglomerate of several complexes (e.g., the left complex in Figure 3C). Finally, the choice of colloid species influences the efficiency of loading the convection cell and in turn the propulsion efficiency. In general, large colloids or colloids with a large mass density populated only a single sedimented layer next to the substrate, and small colloids were lifted to very large heights and settled over extended times at large distances from the CIEX fragments. Polystyrene beads of mass density ρ = 1.05 kg/m3 and diameters of about 5 μm were observed to be lifted not much higher than the vertical extension of the fragments (some 30 μm) and to settle again in the immediate vicinity. This situation was found to create the largest 3D colloid concentration in the immediate surroundings of the fragments and turned out to be the most efficient for propulsion. A background concentration of colloids of
