Living Crystals of Light-Activated Colloidal Surfers

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Science  22 Feb 2013:
Vol. 339, Issue 6122, pp. 936-940
DOI: 10.1126/science.1230020


Spontaneous formation of colonies of bacteria or flocks of birds are examples of self-organization in active living matter. Here, we demonstrate a form of self-organization from nonequilibrium driving forces in a suspension of synthetic photoactivated colloidal particles. They lead to two-dimensional "living crystals," which form, break, explode, and re-form elsewhere. The dynamic assembly results from a competition between self-propulsion of particles and an attractive interaction induced respectively by osmotic and phoretic effects and activated by light. We measured a transition from normal to giant-number fluctuations. Our experiments are quantitatively described by simple numerical simulations. We show that the existence of the living crystals is intrinsically related to the out-of-equilibrium collisions of the self-propelled particles.

Self-organization often develops in thermal equilibrium as a consequence of entropy and potential interactions. However, there are a growing number of phenomena where order arises in driven, dissipative systems, far from equilibrium. Examples include "random organization" of sheared colloidal suspensions (1) and rods (2), nematic order from giant-number fluctuations in vibrated rods (3), and phase separation from self-induced diffusion gradients (4). Biological (57) and artificial active particles (811) also exhibit swarm patterns that result from their interactions (1215).

In order to study active, driven, collective phenomena, we created a system of self-propelled particles where the propulsion can be turned on and off with a blue light. This switch provides rapid control of particle propulsion and a convenient means to distinguish nonequilibrium activity from thermal Brownian motion. Further, the particles are slightly magnetic and can be stabilized and steered by application of a modest magnetic field. Our system consists of an active bimaterial colloid. A polymer sphere, 3-methacryloxypropyl trimethoxysilane (TPM), encapsulates most of a canted antiferromagnetic hematite cube (16), but with part exposed to the solvent (Fig. 1A). The particles are immersed in a basic solution (pH ∼ 8.5) containing hydrogen peroxide [0.1 to 3% weight/weight (w/w)], 5 mM tetramethylammonium hydroxide, and 3.4 mM sodium dodecyl sulfate (SDS) (16). Under normal bright-field illumination, the colloids are at equilibrium with the solvent and thus sediment toward the bottom glass surface of the observation cell. When illuminated through the microscope objective with blue-violet light (λ ∼ 430 to 490 nm), the particles exhibit self-propulsion (movie S1). The motion, with a velocity up to 15 μm/s, only takes place at the cell surface, whether it is the bottom, the vertical side walls, or the cell top. Individual particles undergo a random walk with a persistence length determined by the reorientation time τr = 8.0 ± 1.5 s, consistent with Stokes-Einstein rotational diffusion.

Fig. 1

(A) Scanning electron microscopy (SEM) of the bimaterial colloid: a TPM polymer colloidal sphere with protruding hematite cube (dark). (B) Living crystals assembled from a homogeneous distribution (inset) under illumination by blue light. (C) Living crystals melt by thermal diffusion when light is extinguished: Image shows system 10 s after blue light is turned off (inset, after 100 s). (D to G) The false colors show the time evolution of particles belonging to different clusters. The clusters are not static but rearrange, exchange particles, merge (D→F), break apart (E→F), or become unstable and explode (blue cluster, F→G). For (B) to (G), the scale bars indicate 10 μm. The solid area fraction is Φs ≈ 0.14.

In equilibrium, with no blue light, the particles diffused and were disordered (Fig. 1B, inset). At surface area fractions Φs ≳ 7%, cooperative behavior of the light-activated colloids began to emerge. Crystallites started to form in the sample 25 s after the light was turned on. An image of the crystals after 350 s is shown in the main part of Fig. 1B. Immediately after the light was extinguished, the crystallites began "dissolving" because of thermal diffusion (Fig. 1C); after 100 s, there was no trace of the crystals (Fig. 1C, inset). Although the particles formed organized crystallite structures under illumination, those structures were not static (movie S2). The crystallites actively translated and rotated, collided, joined, and split (Fig. 1, D to G). These "living crystals" reached a dynamic steady state of creation and self-destruction. They did not grow to incorporate all available particles, as would be the case for an equilibrium system with attractive interactions. We measured an average cluster size of ∼35 particles, which did not seem to depend on Φs > 10%. The lifetime of the crystals was finite and broadly distributed. The typical time for a cluster to change its size by 50% is 100 ± 75 s. Fluctuations in the local number N of particles follows a power law ∆NNα. There is a transition at ΦS ∼ 7% from normal α = 1/2 to giant fluctuations α ≅ 0.9, in line with recent predictions for disordered clusters in a system of polar isotropic active particles (17) and observed in a granular system (18).

To understand the mechanisms involved in the self-propulsion and crystallization, we performed a series of experiments on the separate components of our composite colloid, the hematite cube, and the polymer sphere. First, we attached a hematite cube to a glass substrate and immersed it in our solution of surfactants, buffer, and H2O2. Then, 1.5-μm-diameter colloidal tracer particles of polystyrene, silica, or TPM were added. The tracers were observed to diffuse randomly under normal bright-field illumination. When illuminated with blue light, however, the tracers all moved toward the immobilized hematite cube (movie S3), converging on it from all directions, as indicated by the cartoon in Fig. 2A. This observation rules out advection, because advective fluid flow must have zero divergence. Therefore, the motion of the colloids toward the hematite particle must be caused by a phoretic motion (19) induced by some gradient generated by the cube. Under blue-light illumination, hematite catalyzes the exothermic chemical decomposition of H2O2, creating thermal and chemical (H2O2 and O2) gradients. Heating studies of the system suggest that diffusiophoresis is more important than thermophoresis in our system. The motion of the tracers toward the cube can be quantified by monitoring their position versus time and calculating their velocity as a function of distance from the particle (Fig. 2B). The dashed line through the data is a fit to A/r2 (where A is a fitting parameter and r the center-to-center distance between the tracer and the hematite particle), consistent with a diffusive concentration profile CC(1 − B/r).

Fig. 2

Out-of-equilibrium driving forces. (A) A hematite cube, indicated by an arrow, is immobilized on a surface and immersed in a solution of colloidal tracers. At t = 0 s, the blue light is switched on, triggering the decomposition of hydrogen peroxide on the hematite surface. The tracers are attracted to the hematite until they contact the cube. The attraction is isotropic with particles coming from all directions, thus discounting advective flow that must exhibit zero divergence. When the light is turned off, the attraction ceases and the tracers diffuse away. (B) The attraction is quantified by the radial velocity Vp extracted from the ensemble average of the tracer drift (inset, black symbols) and is consistent with the r−2 behavior (red dashed line) expected for phoretic attraction to a reaction source. Error bars represent the uncertainty in the determination of the velocity from the position measurement. (C) A hematite cube protruding from a TPM polymer sphere moves on fixed glass substrate when exposed to blue light (red part of trace) and diffuses when the light is off (black part of trace). Initially, with no light, the hematite cube is oriented randomly (image, right) but rotates and faces downward toward the glass substrate when the light is turned on (image, left). The particle then surfs on the osmotic flow it induces between the substrate and itself. (Inset) A superposition of the trajectories of many particles with their origins aligned. (D) The particle velocity V increases with light intensity P and follows Michaelis-Menten law (red dashed line). The black arrow indicates the point of zero velocity for P = 0. (E) The particle velocity is also strongly dependent on the Debye length λD of the system and asymptotically follows the V ∝ λD2 scaling expected for osmotic mechanisms (red dashed line). The Debye length is varied by adding NaCl to the buffer solution except for the blue symbol, for which the SDS surfactant is suppressed to reach higher λD. The error bars in (D) and (E) are given by the standard deviation of the velocity measured for 10 to 20 different particles.

Phoresis and osmosis are complementary interfacial phenomena: In a gradient, a free colloid will exhibit a phoretic migration, whereas a fixed surface of the same material will exhibit an osmotic flow at its surface in the opposite direction (19). Therefore, a particle phoresing to the right has an osmotic flow at its surface to the left. Just as a silica colloid is attracted to a hematite cube, a free hematite particle is attracted to a stationary silica surface. Therefore, when we add free cubes to a sample cell, the silica surface of the cover slip attracts the cubes. Indeed, we observed that hematite cubes were quickly drawn to the glass substrate as soon as blue light was turned on.

Unexpectedly, once on the glass substrate, the hematite cubes continue to move on the glass surface when illuminated with blue light. Naïvely, one might expect the cubes to remain stationary, because the osmotic flow on the cover slip surface is away from the cube and ideally should be symmetric. However, the symmetry is broken either by imperfections on the cube or spontaneously by an instability where the motion of the cube induces different gradients fore and aft. Thus, in a solution of free hematite cubes we see attraction of the cubes to the surface followed by self-propulsion of the cubes surfing on the substrate when the light is turned on.

When using a suspension of our composite particles, a hematite cube in a TPM sphere, we observed a similar scenario. When illuminated with blue light, the composite particle reoriented so that the exposed hematite sat on the glass substrate (Fig. 2C, insets) and then began to move at speeds comparable to that of the hematite alone. Figure 2C shows the trajectory of a single composite particle, with the light turned on then off, whereas the inset shows a superposition of many trajectories with their origins aligned. The self-propelled motions are isotropic and diffusive with a persistence length (15 to 100 μm) determined by the rotational diffusion time and the velocity of the particle. The velocity of the particles depends weakly on the H2O2 concentration but strongly on the light intensity and the Debye screening length. The velocity versus light intensity, P, follows Michaelis-Menten law (20) behavior characteristic of a catalytic reaction (Fig. 2D). Figure 2E suggests that the composite particle velocity asymptotes to a quadratic behavior with Debye length λD (21), a behavior expected from osmotic effects within a Debye length of a surface where the driving force ∝ λD and the drag force is ∝ velocity/λD (19, 22).

If we now consider a solution of composite particles activated by light, two effects have to be taken into account: (i) the collisions between our self-propelled particles surfing on the osmotic flow they set up and (ii) the phoretic attraction between the particles. In order to see whether these effects explain the formation of our living crystals, we performed simulations guided by our experimentally determined parameters.

We considered a minimal numerical model (16) in which the self-propelled colloids are represented by self-propelled hard disks that move with a constant velocity in a direction that changes randomly on a time scale τr determined by rotational diffusion. We modeled the phoretic attraction between particles as a pairwise attraction between nearby particles consistent with the phoretic velocity (red line in Fig. 2B). If a displacement makes two disks overlap, the particles are separated by moving each one half the overlap distance along their center-to-center axis. We tried various approximations to account for the effect of (hydrodynamic) lubrication forces in the crystals, for example, increase of the apparent viscosity, and found little qualitative difference.

In Fig. 3 and movie S4, we present the results of simulations in which the attractive phoretic effects are taken into account. For our experimental conditions Φs ∼ 3 to 20% and ∼ 300 to 1500, the simulations reproduce the crystallite formation as well as the size and lifetime of the crystallites quite well. The simulations (Fig. 4, A and B) also capture the transition observed experimentally from normal to giant fluctuations of number, above a critical density ΦSC 7% ± 1%. If we turned off the pairwise phoretic attraction, we observed large spatiotemporal fluctuations of density with normal number fluctuations, α ∼ 0.5 (Fig. 4B). The formation of clusters that grow and decay is recovered at much higher particle concentration, Φs > ~35 to 45% (fig. S2).

Fig. 3

Numerical simulations of self-propelled disks coupled by a phoretic attraction. Simulation parameters were defined to mimic the experimental conditions. (A) Starting from a homogeneous distribution (B) (~t = 8~τr), the disks assemble in mobile crystalline clusters with faceted edges. (C to E) The false colors show the time evolution of particles belonging to different clusters. The crystals are mobile [(C) to (E)], can merge [(C) and (D)], and can break apart or dissolve [(D) and (E)]. This minimal model reproduces the experimental dynamics of the living crystals and demonstrates the relevance of the proposed mechanism. We used parameters consistent with the experimental conditions of Fig. 1: Φs = 0.14, Embedded Imager = 16 and Embedded Image = 0.87 [see (16) for a definition of the reduced parameters, indicated by tildes].

Fig. 4

(A) Number fluctuations measured in the simulations for varying Φs values, in the range 1 to 15% for N = 600 particles [Embedded Imager = 16 and Embedded Image = 0.87 from the experiment, see (16)]. The system exhibits a transition from normal to giant fluctuations for ΦsC ∼ 7%. (B) Scaling α of the number fluctuations, ∆NNα, for various Φs values measured in the experiment at equilibrium (black symbols), under activation by the light (blue symbols), and in the simulations with (Embedded Image = 0.87, red open symbols) and without (Embedded Image = 0, magenta open symbols) attraction. We observed normal fluctuations, α = 1/2, at equilibrium. The driven system exhibited a transition from normal fluctuations, α = 1/2, to giant-number fluctuations, α ∼ 0.9, at Φs ∼ 7% in both the experiment and the simulations. The slight decay of the exponent observed in experiments and simulations is a finite size effect. Error bars indicate the standard deviation in the measurement of the exponent. (C) Scaling α of the number fluctuations for N = 1000 (blue symbols), N = 600 (red), and N = 400 (magenta) particles in the simulations. For N = 400, the curve after 50τp (square) collapses with 35τp (circles), showing that the scaling is steady. (D to I) Investigating the crystal mechanism. We used an external magnetic field B0 ∼ 1 mT to orient the particles and direct their motion. The red arrow is the orientation of B0. (D and E) The magnetic field is turned on, and the light is on; the crystal is self-propelled in the direction of the magnetic field, and crystal breakup is suppressed. (F) The light is turned off, and the magnetic field B0 is left; the crystal dissolves. (G) The magnetic field is turned off, and the light is turned on; particles collide, and the crystal re-forms. (H) The light is turned off, and the magnetic field remains off; the crystal dissolves. (I) The magnetic field is turned on first, and then the light is turned on. The particles all move in the the field direction; they do not collide and do not crystallize.

Our understanding of the living crystals comes from the idea that active particles undergo diffusive motion with a large persistence length when not in "contact" but slow down, translate, and diffuse more slowly when they are in contact. The slowing down results from the inability of particles to penetrate their neighbors when encountered.

Several recent papers have pointed out that such density-dependent diffusion can lead to giant fluctuations, clustering, and phase separation in nonequilibrium systems (17, 2326), however, at much larger surface density. The formation of noncrystalline clusters of active particles has been reported with bacteria coupled through short-range depletion interaction (27) and Janus particles with chemical attraction (28).

We always observed the intermittent formation and breakage of the large crystals. This differs from the equilibrium nucleation and growth of a crystal of attractive colloids or the asymptotic formation of a single cluster from an assembly of self-propelled disks reported by Fily and Marchetti (17) (at higher number and density of disks than in our experiment).

In order to investigate the underlying mechanisms, we took advantage of the magnetic properties of the embedded hematite cube. Under an external magnetic field (B ∼ 1 mT), the alignment of the hematite slighty tilts the orientation of the particle, and the self-propulsion proceeds in a direction parallel to the field, suppressing the rotational diffusion.

We test two crucial aspects of our scenario in Fig. 4, D to I, and movie S5. We suggest that the crystal breakup depends on the velocity redirection by rotational diffusion of the particles in the crystal. If the direction of all the particles are aligned by using an external magnetic field, then the crystal will not break up. This is shown in Fig. 4, D and E, and movie S5. Also note the suppression of the breakup in the magnetically steered crystal in movie S6. We also claim that collisions are required for crystal formation. In Fig. 4F we turned off the light and the crystal dissolved. With the magnetic field off, we turned on the light, the particles collided, and the crystal re-formed (Fig. 4G). We turned off the light and redissolved the crystal in Fig. 4H. We then first turned on the magnetic field and then the light (Fig. 4I). The particles all moved in the same direction and did not collide, and there was no crystallization.

We have demonstrated a form of self-assembly from nonequilibrium driving forces leading to living crystals with complex dynamics. The osmotically driven motion and steric hindrances lead to the formation of dynamic aggregates. The introduction of a small attractive interaction, in this case arising from phoresis, orders the aggregates into periodic crystals at low surface fraction. Phoretic and osmotic effects in our system can conveniently be switched on and off by light. Rotational diffusion of the particles reorients their motions, leading to a finite persistence length, crystal evaporation and breakup, and finite size and lifetime of the crystals. The use of active particles and nonequilibrium forces for directed self-assembly opens a new area for design and production of novel and moving structures. The fact that they can be turned on and off with visible light adds control to this system, as does the ability to use external magnetic steering.

Supplementary Materials

Materials and Methods

Figs. S1 and S2

References (29, 30)

Movies S1 to S6

References and Notes

  1. Materials and methods are available as supplementary materials on Science Online.
  2. Acknowledgments: We thank J. Layné, K. Hanson, A. Grosberg, R. Dreyfus, E. Lerner, A. Baskaran, and L. Bocquet for fruitful discussions. This work was supported by the Materials Research Science and Engineering Centers program of the NSF under award number DMR-0820341 and by the U.S. Army Research Office under grant award no. W911NF-10-1-0518. We acknowledge partial support from the NASA under grant award NNX08AK04G.
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