Convective Self-Sustained Motion in Mixtures of Chemically Active

Jul 25, 2017 - Diffusiophoretically induced interactions between chemically active and inert particles. Shang Yik Reigh , Prabha Chuphal , Snigdha Tha...
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Convective Self-Sustained Motion in Mixtures of Chemically Active and Passive Particles Oleg E. Shklyaev, Henry Shum,† Victor V. Yashin, and Anna C. Balazs* Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States S Supporting Information *

ABSTRACT: We develop a model to describe the behavior of a system of active and passive particles in solution that can undergo spontaneous self-organization and self-sustained motion. The active particles are uniformly coated with a catalyst that decomposes the reagent in the surrounding fluid. The resulting variations in the fluid density give rise to a convective flow around the active particles. The generated fluid flow, in turn, drives the self-organization of both the active and passive particles into clusters that undergo selfsustained propulsion along the bottom wall of a microchamber. This propulsion continues until the reagents in the solution are consumed. Depending on the number of active and passive particles and the structure of the self-organized cluster, these assemblies can translate, spin, or remain stationary. We also illustrate a scenario in which the geometry of the container is harnessed to direct the motion of a self-organized, self-propelled cluster. The findings provide guidelines for creating autonomously moving active particles, or chemical “motors” that can transport passive cargo in microfluidic devices.

1. INTRODUCTION Gradients in external fields can be harnessed to propel micrometer-sized particles in solution; this motion is commonly termed electro-, diffusio-, osmo-, or thermophoresis, depending on the specific nature of the applied field.1 When the particle itself generates the gradient, it can undergo autonomous self-propulsion.2−8 This self-phoresis gives rise to molecular-scale interactions at the particle−solution interface that drive the movement of the surrounding fluid relative to the particle surface. The interfacial slip velocity1,3 causes the motion of the particle, which also experiences friction from the surrounding fluid. As the particle size increases, the surface area-to-volume ratio decreases, and phoretic mechanisms become less effective as modes of propulsion.9 To overcome this limitation, new strategies for transporting microscale particles are needed. Here, we use theory and simulation to show how chemical reactions on the surface of “active” particles can induce buoyancy-driven convective flows. Distinct from the passive particles, the surfaces of the active species are coated with a catalyst. As described below, the flow generated at the surface of the active particles leads to the self-organization and self-propulsion of both active and passive particles in the solution. “Chemical pumps”10−13 provide a novel approach for generating fluid flows that can transport particles (or submersed cargo) in solutions.14,15 These pumps consist of catalytic patches on the wall of a microchamber; chemical reagents in the solution react with the anchored catalysts, changing the chemical composition of the solution. Gradients in chemical concentrations give rise to variations in fluid © XXXX American Chemical Society

density, which can generate significant buoyancy-driven convection.16,17 Because buoyancy forces act on the bulk of the fluid, the characteristic flow velocities can increase with the system size. In shallow chambers, for example, a simple scaling argument suggests that the flow velocities are proportional to the cube of the chamber height.12 To date, the buoyancy effect has been analyzed only in the context of immobile chemical pumps; this effect, however, could be utilized to drive the selfpropulsion of catalytically active microparticles. That is, anchoring enzymes to the surface of a particle can create a mobile version of a catalytic pump, namely, a buoyancy-driven “chemical motor.” Through computational modeling, we investigate the behavior of systems that encompass both active and passive particles immersed in a reagent-containing solution. The surface of the active particles is coated with catalysts that chemically decompose the dissolved reagent, whereas the surface of passive particles is not coated and, thus, does not produce chemical reactions. For clarity, we focus on a specific catalytic reaction: the decomposition of hydrogen peroxide into water and oxygen catalyzed by catalase. (Other chemical reactions causing changes in the fluid density or temperature of the solution could also be used to drive fluid motion.) This decomposition reaction generates variations in the fluid density.12,17 These density variations, in turn, generate a flow that promotes the organization of the active and passive particles into mobile clusters, such as dimers, trimers, and larger Received: June 1, 2017 Published: July 25, 2017 A

DOI: 10.1021/acs.langmuir.7b01840 Langmuir XXXX, XXX, XXX−XXX

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centers. We define this line as the ξ axis and denote the positions of the passive and active particles by ξp and ξa, respectively. Neglecting the hydrodynamic influence of the passive particle and assuming that the fluid flow is sufficiently slow that it can be regarded as quasisteady, the velocity u of the flow generated by the active particle along the ξ axis can be expressed in the form u(ξ) = u(ξ − ξa). The passive particle is dragged with a velocity u(ξp − ξa) along the ξ axis toward the active particle until the two particles form a dimer (Figure 1). A repulsive force, FP−P(ξp − ξa), acts between the passive and active particles when they come sufficiently close to each other. Consequently, the dynamics of the two-particle system in the overdamped limit is described as

aggregates. The specific dynamic behavior of the clusters depends on the number, ratio, and relative positions of active and passive particles. Similarly to the propulsion of a camphor boat18 where local changes in surface tension create fluid motion, in the proposed mechanism, the variations of the fluid density localized around the active particle drive flows that impose a net fluid drag on clusters of particles, enabling them to translate. In the next section, we describe our modeling approach, which allows for the determination of the fundamental physical principles governing the motion of the chemically active particles and the self-organization and propulsion of the clusters.

2. METHODOLOGY 2.1. Phenomena Controlling the Dynamic Behavior of the System. The mechanism behind convective propulsion in this system can be explained by examining the behavior of a pair of particles, one active and one passive, resting on the bottom wall of an infinitely large, fluid-containing chamber (Figure 1). Assume that a certain physical mechanism generates

dξp

= u(ξp − ξa) − μF P−P(ξp − ξa) dt dξa = μF P−P(ξp − ξa) dt

(1)

where μ is the particle mobility. The system can reach a dynamic equilibrium if μ and FP−P exhibit the appropriate magnitudes and functional forms in the vicinity of ξa. In particular, at ξp < ξa, u is positive and decreasing and FP−P is positive and increasing as ξp − ξa → −0. The steady separation, d(ξp − ξa)/dt = 0, is reached when u(ξp − ξa) = 2μFP−P(ξp − ξa), and then the two particles move at the same velocity dξ1 dξ = dt2 ≡ v = μF P−P(ξp − ξa). At the steady state, the force dt from the fluid drag is balanced by the repulsive force between the particles. Consider now the leading-order term in a Taylor expansion for a flow that draws fluid radially inward toward ξa along the wall, u(ξp − ξa) ≈ − α(ξp − ξa). As we use the quadratic form of the interparticle repulsion potential (see SI for details), the repulsive force between the particles is FP−P(ξp − ξa) = −(ξp − ξa)−3. Then, the steady motion of the dimer is described as ξa(t) = (2 μ/α)−3/4μt and ξp(t) = ξa(t) − (2 μ/ α)−1/4, and the two particles are separated by the distance (2μ/ α)−1/4. Here, α and μ are the constants introduced above. It can also be shown that this steady state is stable with respect to small perturbations. Hence, through eq 1, we demonstrate the ability of a dimer to translate with a finite constant velocity as a result of the coupling between the particle positions and the convective flow. Note that the described mechanism of steady dimer motion is quite generic, as it is insensitive to the details of the model. The appropriate convective flows, which initially rise above the particle and return to the wall, can be generated by a density variation in the fluid around particles that are thermally or chemically active. In particular, thermally driven buoyancy flow can be produced around a particle that is warmer than the surrounding fluid. Chemically driven flow can be created by catalyst-coated particles that chemically decompose the reagents in the solution into less dense products; the resulting density variations cause the fluid to rise upward around the particle. In either case, the generated fluid flow is qualitatively similar to the flow uS of a vertically oriented Stokeslet located above an infinite horizontal solid plate19 that drives the fluid upward; the numerical solution of eq 1 with the flow u = uS associated with the position of an active particle to produce constant-speed dimer propulsion is presented in the SI. The same mechanism can drive multiple, randomly placed active and inert particles to self-organize into larger clusters. The active particles again generate the flow, and the passive

Figure 1. Fluid flow generated by a chemically active spherical particle (red) imposes fluid drag on the passive particles (blue). The drag force (black arrow) is transmitted to the active particle by steric repulsion, moving the dimer.

a fluid flow that rises above the active particle and, because of the continuity of the flow, returns to the particle along the bottom wall (as depicted by a green loop in Figure 1). An isolated active particle will remain motionless because the generated flow is azimuthally symmetric about the vertical η axis passing through the center of the active particle and imposes equal drag on the particle from all lateral directions. The particle will only move if the symmetry of the flow field is broken; in the present case, this symmetry breaking is accomplished by introducing a passive particle that does not actively generate fluid flow. The flow generated by the active particle drags this passive sphere toward it until the two particles form a dimer, as shown in Figure 1. Because the flow is generated solely by the active particle (red in Figure 1), the flow field is not symmetric with respect to the dimer structure and imposes a net drag force that translates the entire dimer. The dynamic behavior of the system observed in the simulations described further below can be described through the following heuristic considerations [see the Supporting Information (SI) for more details]. Because of the symmetry of the dimer and the assumption that gravity localizes the particles on the bottom wall, any motion of the particles should be confined to the one-dimensional line that passes through their B

DOI: 10.1021/acs.langmuir.7b01840 Langmuir XXXX, XXX, XXX−XXX

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Langmuir particles move toward the active spheres. Upon reaching the active spheres, the passive particles transmit the fluid drag to the assembled cluster so that the entire group can translate along the surface. The dynamic behavior of the cluster is affected by both self-generated and external stimuli. That is, the structure of the convective flow defines the dynamics of the cluster and is, in turn, determined by the structure of the cluster; this constitutes the self-generated influence. The convective flow (and particle dynamics) is also affected by the geometry of the container, which constitutes an external influence. To separate the impacts of these different factors and analyze the physics underlying the propulsion, we initially consider a simplified situation where the side walls are sufficiently far apart (modeled through periodic boundary conditions) that their influence on the dynamics of the cluster can be neglected. We then determine how to exploit the presence of these walls to control the motion of the particles. In the next section, we describe the theoretical and computational models that we developed to capture the behavior of this multicomponent, dynamic system. Using this model, we then probe the rich collective behavior that can emerge in this system. 2.2. Modeling of the Dynamic Behavior of the System. To fully characterize this buoyancy-driven propulsion, we numerically model the dynamics of spherical particles of density ρπ and radius R placed in a finite-sized chamber filled with a solution of chemical reagents characterized by a density ρ and a kinematic viscosity ν. Some spheres are assumed to be chemically active, with the whole surface S of the particle being coated with a catalyst; others are passive because they are not coated with the catalyst. Decomposition of the reagent that occurs on the surface of an active particle is assumed to follow the Michaelis−Menten kinetics. Hence, the rate of decomposition per active particle, Kd, depends on the concentration of reagent, C, and is calculated as Kd(C) =

k maxSC , KM + C

It has been shown that the density change as a result of heat generated by the decomposition reaction is negligible.17 Noting that the expansion coefficient of hydrogen peroxide βHC 2O2 is substantially larger15 than that of oxygen βOC 2, we simplify the expression for the density of the fluid to ρ = ρ0(1 + βHC 2O2CH2O2) = ρ0(1 + βCC). The motion of fluid in a rectangular simulation domain Ω = {(x, y, z): 0 ≤ x, z ≤ L, 0 ≤ y ≤ H} is described by the velocity u = (ux, uy, uz) and pressure p. The reagent concentration field is given as C. The positions of Na active and Np passive particles (Na + Np = N) are specified by the vectors ri(t) = (xi(t), yi(t), zi(t)). The equations governing the behavior of the system are the continuity, Navier−Stokes (in the Boussinesq approximation20), and reagent reaction and diffusion equations. These equations are given by (2)

∂u 1 + (u ·∇)u = − ∇p + ν∇2 u − egβCC ∂t ρ0

(3)

N

a ∂C + (u ·∇)C = D∇2 C − Kd(C) ∑ δ(r − rj) ∂t j=1

(4)

respectively, where ∇ is the spatial gradient operator, ν is the kinematic viscosity of water, D is the diffusivity of the reagent, and e = (0, 1, 0) specifies the direction of the gravitational force. For solid walls that bound the domain, we require zero velocity and zero flux of the reagent concentration normal to the walls. For periodic boundary conditions in the horizontal directions, we set the following boundary conditions y = 0, H:

u = 0,

∂C =0 ∂y

(5)

where kmax is the maximal

reaction rate per unit particle surface area and KM is the Michaelis constant. In turn, the maximal areal reaction rate kmax = kcat[E] (mol s−1 m−2) depends on the rate of degradation of a reagent molecule, kcat (s−1), and the surface concentration of the catalyst, [E] (mol m−2). In the absence of flow, particles denser than the solution sediment along the direction of gravity g to an impenetrable bottom wall with velocity V =

∇·u = 0

2 (ρπ − ρ) 2 R g. 9 ρν

x , z:

u(0) = u(L),

C(0) = C(L)

(6)

We solve this system of equations in a computational domain that is 80Δx × 20Δx × 80Δx units in size, where Δx corresponds to 50 μm. The lattice Boltzmann algorithm21,22 is applied to simulate the continuity equation, eq 2, and the Navier−Stokes equation, eq 3. To reproduce the kinematic viscosity ν of water, the time step Δt = (1/6)Δx2/ν was used. All times quoted below are obtained by multiplying Δt by the number of numerical iterations. A second-order finite-difference scheme is used to solve the diffusion equation, eq 4. Each particle is described by a Lagrangian mesh of nodes distributed over a spherical shell and connected by elastic bonds. The evolution of this mesh is coupled to the fluid flow u by the immersed boundary method (IBM).23,24 Immersed boundary nodes have finite hydrodynamic size,25 giving the simulated particles an effective diameter of 2R = nΔx, where n = 6 provides R = 150 μm. Additionally, the particles experience repulsive interactions from each other (denoted P−P) and from the solid walls (denoted P−W) that are described by the following Morse potentials

Particles that are

less dense than the solution localize at the top wall. The aqueous solution consists of water of density ρ0 and NC distinct species of solute, characterized by concentrations Cj and diffusivities Dj, j = 1, 2, ..., NC. The fluid flow is generated by a catalytic reaction, which changes the chemical content of the solution and the fluid density.12 In response to variations in the chemical concentration and solution temperature, the fluid density can be approximated as ρ = ρ0 [1 + ∑βjCCj + βT(T − T0)]. The expansion coefficients βjC and βT characterize the magnitude of the density variations in response to the respective changes in solute concentrations Cj and temperature (T − T0). As noted above, we focus on one particular reaction: the decomposition of hydrogen peroxide by catalase into oxygen and water

⎧ ε P−P{1 − exp[−ω P−P(r − 2R )]}2 , r ≤ 2R U P−P(r ) = ⎨ ⎪ r > 2R ⎩ 0, ⎪

catalase

2H 2O2 ⎯⎯⎯⎯⎯⎯→ 2H 2O + O2

(7) C

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Langmuir ⎧ ε P−W (1 − exp[ −ω P−W (r − R )])2 , r ≤ R U P−W(r ) = ⎨ r>R ⎩ 0,

sphere (top panel in Figure 2) until the particles form a dimer. The drag force directed from the passive to the active particle enables translation of the dimer along the bottom wall (middle and bottom panels in Figure 2). The depletion of the reagent in the domain (as indicated by the progressively lighter colors in the sequential panels in Figure 2) fuels the propulsion of these particles. The panels also show that the upward flow generated by the active particle translates with the moving particle, revealing a coupling between the particle positions, ri, and the flow, u(ri). In contrast to phoretic modes of self-propulsion, the flow field u is global (although it decays with the distance from the source) and involves fluid motion throughout the entire domain. Multiple particles interacting through the fluid flow can assemble into a cluster defined by the particle coordinates and coupled to the self-consistent global flow, which is generated by the cluster and defines its structure. The dynamics of such a self-propelled cluster depends on the geometry of the channel, the properties of the particles, the configuration of the clusters of particles, and the chemical reaction used to generate the fluid motion. In the case of the dimer, the chemical reaction rates and the particle size are among the most important properties affecting the dynamics of the system. Figure 3a shows the dimer velocity for various maximum reaction rates per unit surface area of the particle, kmax. For a fixed catalyst (characterized by a specific value of kcat), the value of kmax = kcat[E] can be altered by changing the areal concentration [E] on the surface of the particle. Alternatively, at a fixed value of [E], kmax can be altered by changing the catalyst (and, hence, kcat). As expected, larger values of kmax provide higher propulsion velocities, as well as faster consumption of the reagent. (The initial high-velocity transitional regime describes the dynamics of the separate particles until they come into contact and form a dimer.) In wide channels of thickness H, the dimensionless Rayleigh number, which scales as H3, controls the velocity of the buoyancy-driven flow; some portion of this velocity can be transmitted as the fluid drag that acts on the particles. Small particles (with relative diameters of 2R ≪ H) that are localized near the walls experience a small drag due to the no-slip (zerovelocity) boundary conditions. Particles with diameters larger than H/2 are slowed down by the return flow, which results from the conservation of flux and is directed opposite to the dimer propulsion. In addition, larger particles that carry larger amounts of catalyst can provide higher reaction rates Kd (mol s−1), thus increasing the fluid velocity. The competition among these factors suggests that an optimal particle diameter of ∼H/ 2 should provide the fastest dimer propulsion. This trend is observed in Figure 3b, where the dimer velocities for particles of various radii R are shown for fixed kmax. The larger particles drive faster flows and experience larger fluid drag, enabling faster dimer propulsion and faster consumption of the reagent; the latter factor slows down the dimer. The highest dimer velocity was indeed achieved for particles with diameters close to H/2. The fluctuations of the dimer velocity observed in Figure 3 arise as a result of deformations of the particle elastic shell coupled to the fluid flow and can be changed (but not eliminated) by adjusting the flow parameters or the stiffness of the shells.26,27 By comparing the maximal horizontal fluid velocities umax xz = max{(ux2 + uz2)1/2} with the speed of dimer propulsion, we conclude that only a fraction of the fluid motion is converted into particle translation. For the cases presented in Figure 3a

⎪

⎪

(8)

Here, the parameters ε and ω that characterize the respective potential strength and width were set to εP−P = εP−W = 1.2 × 10−14J and ωP−P = ωP−W = 2 × 104 m−1. The friction between the particles and the walls is ignored. To obtain a micrometerper-second propulsion velocity, the size of the active particles should be on the order of hundreds of micrometers; for particles of this size, we ignore Brownian motion.

3. RESULTS AND DISCUSSION 3.1. Motion of Two Particles in the Absence of Side Walls. We first discuss the self-propulsion of an active particle and a passive particle when the motion is not affected by the presence of side walls, as simulated by using the periodic boundary conditions (eq 6). The graphical output from the simulations (Figure 2) shows the behavior of the active (red)

Figure 2. Fluid flow (arrows), generated by the chemical decomposition of a reagent around a mobile active particle (red), drags a passive particle (blue), which transmits momentum to the active particle and moves the dimer to the right. Depletion of the reagent concentration fueling the reaction around the active particle is shown with a color bar (M). A movie demonstrating the translation of the dimer is provided in the SI.

and passive (blue) particles, which are both denser than the solution and, thus, sediment to the bottom of the domain. (We set the particle density to ρπ = 1.1ρ0, where ρ0 is the density of water; the initial hydrogen peroxide concentration is set to be 1 M.) The arrows in these figures specify the direction of the fluid flow; the size of the arrows indicates the relative magnitude of the fluid velocity. The reagent in solution is decomposed into the less dense products by the catalysts on the surface of the active particle. (The color bar in the figure indicates the local concentration of reagent.) The resulting density variation in the solution drives a net fluid flow. This flow rises above the active particle and forms vortexes that drag the passive sphere toward the active D

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Figure 3. (a) Simulated velocity of a dimer consisting of active and passive particles (R = 150 μm) for different reaction rates kmax. (b) Dimer velocities for particle sizes of R = 125, 150, 200, and 250 μm and fixed kmax = 0.002 mol s−1 m−2. The channel height is H = 1000 μm.

Figure 4. Averaged (over all particles) speed |∑Ni ui|N of clusters formed by Na active (red) and Np passive (blue) particles. The radius of all particles is R = 150 μm, and the reaction rate is kmax = 0.003 mol s−1 m−2. Arrows next to insets show the directions of the propulsion relative to the cluster structures. Arrow colors match the corresponding evolution curves. Clusters without arrows are stationary. The structures and dynamics of the clusters are shown in snapshots and videos provided in the SI.

with reaction rates of kmax = 0.001, 0.002, and 0.02 mol s−1 m−2, the values of umax xz observed at t = 45 min were 19, 36, and 45 μm s−1, respectively. For increasing particle radii R = 125, 150, 200, and 250 μm (at kmax = 0.002 mol s−1 m−2) (Figure 3b), the −1 values of umax xz were 28, 36, 40, and 41 μm s , respectively. As can be seen from Figure 3, in all presented cases, the values of umax xz are greater than the speed of the dimers. Notably, the fluid velocities umax xz generated by motors, which can be viewed as mobile catalytic pumps, are comparable to the horizontal fluid velocities produced by the stationary pumps.15 The highest values of the velocities of dimer propulsion, ∼3 μm s−1 (green lines in Figure 3), were achieved for the following two cases: dimers with a smaller radius of R = 150 μm and a larger value of kmax = 0.02 mol s−1 m−2 and larger particles with a radius of R = 250 μm and a smaller value of kmax = 0.002 mol s−1 m−2. The latter case, however, more efficiently converts the fluid drag because the values of umax xz are smaller than in the case where R = 150 μm and kmax = 0.02 mol s−1 m−2. Given the following characteristic scales of velocity U ≈ 102 μm s−1 and length W ≈ 103 μm, the typical Reynolds number becomes UW/ν ≈ 0.1. It is worth noting, however, that the mechanism

of convective propulsion is not limited to such small Reynolds numbers. 3.2. Dynamics of Clusters Containing Multiple Particles. Having isolated the basic mechanism that propels the dimers, we now examine the behavior of multiple active and passive particles. As in the above scenario, the structure of the self-organized mobile cluster affects the fluid flow, and this flow affects the drag that propels the particles. In the case of multiple particles, this self-consistent behavior can lead to distinctive forms of convective self-organization. To investigate the rich dynamics that can emerge in these systems, we consider clusters denoted as (Na, Np) that include Na active (red) and Np passive (blue) particles. The particle arrangement {ri, i = 1, 2, ..., N} gives rise to the generated fluid flow u({ri}). (Recall that N = Na + Np.) The cases described below exemplify how the relative number and arrangement of these spheres play crucial roles in the evolution of the system. That is, some initial configurations remain stable and translate as a cluster along the surface. Other configurations reorganize into stable structures that can then translate or even rotate. Notably, some configurations remain motionless. E

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Langmuir In the ensuing simulations, particles with ρπ > ρ are placed on the bottom plane. If an initially prescribed configuration of the cluster remains unchanged during the entire simulation, it is considered to be stable; if the configuration spontaneously reorganizes into a different shape, it is taken to be metastable. Representative examples of the cluster dynamics are presented in Figure 4. As a natural progression, we first consider the behavior of representative trimers. Figure 4a reveals how an initially metastable collinear trimer (1a, 2p) transforms into a stable triangular configuration (brown line). The green line shows the transition of an initially metastable stationary (2a, 1p) trimer into a traveling triangular structure. The blue curve serves as a reference, indicating the motion of the dimer (1a, 1p) discussed above. The dynamical behavior of the system becomes more complex when the total of particles within the simulation domain is greater than three. The brown line in Figure 4b indicates the behavior when two stable traveling (1a, 1p) dimers collide and recombine into a stationary (2a, 2p) tetramer. The blue and green lines show propulsion of tetramer (1a, 3p) and pentamer (2a, 3p), respectively. We observed that all collinear clusters with a long tail (Np > 1) can experience larger fluid drag and be propelled faster, but because these clusters are metastable, they reorganize into more compact, slower-moving configurations, such as the one associated with the (1a, 3p) tetramer indicated by the blue curve. Asymmetry in the cluster configuration can result in a circular motion, as in the case of the spinning (2a, 3p) cluster shown with the black line in Figure 4b. The average speed is not zero because the center of the circle does not coincide with the center of mass of the cluster. Movies demonstrating the details of these evolutions are provided in the SI. The fact that the velocities of a traveling dimer (1a, 1p), trimer (1a, 2p), and pentamer (2a, 3p) differ by less than 3% (at a time of 35 min) indicates that the speed of propulsion does not scale with the number of active particles or constituting dimers (each including one active particle and one passive particle). This also implies that particles can have different roles within the cluster: Some are essential for the propulsion of the cluster, whereas others are the effective cargo, entrapped by the generated convective vortex associated with the cluster. The simulated cluster configurations28 and the corresponding types of dynamics are summarized in Figure 5. The arrangements of active and passive particles that have mirror

symmetry enable the clusters to translate or remain stationary. Transformations of metastable to stable clusters that are observed in the simulations are indicated by green arrows. The absence of mirror symmetry can result in a rotation, as shown in Figure 5 for (2a, 3p) and (2a, 4p) clusters that contain odd (N = 5) and even (N = 6) numbers of particles, respectively. The direction of the rotation is indicated by red arrows. The above findings show that active particles can be harnessed to autonomously transport a cargo of passive spheres in the microchamber. The ability to perform this operation does depend on the initial arrangement of the active “engine” and the load. Nonetheless, for a stable traveling system, the assembly will continue to move until all of the reagent in the solution is depleted. 3.3. Influence of the Side Walls. The presence of solid side walls in a finite-sized domain is another factor that can break the symmetry of the fluid flow generated around an active particle. Because of the walls, the unbalanced fluid drag imposed on the particle from different lateral directions changes the particle dynamics so that even a single active particle can move within the domain. If an active particle is placed precisely in the center on the bottom (ρπ > ρ) of a square container, equidistant from all solid vertical walls, then the generated symmetric flow imposes equal drag at each side and the particle does not move. Nevertheless, the particle’s position is unstable as any small displacement gives rise to an asymmetry in the flow field that drives the particle toward the closest wall, as shown in Figure 6a. Upon reaching the wall, the unbalanced forces of the fluid drag continue to drive the particle toward the closest corner, where it remains until the reagent is depleted. We isolate one situation where the presence of the walls can be exploited to regulate the continuous motion of particles. Speciifically, an active particle cannot be arrested at a corner if these architectural features are removed from the chamber, and instead, the particle is introduced into a circular cage, as in Figure 6b. To create an unbalanced fluid drag that would move the active particle, we again introduce a passive sphere, which breaks the flow symmetry along the circumference of the cage. Figure 6b demonstrates the dynamics of a dimer within a circular, solid cage that is permeable to the reagent; no-slip boundary conditions are imposed on the velocity at this circular wall. In this case, the dimer reaches the wall and then translates along the circumference in a clockwise manner. This motion will continue until the reagent is depleted from the solution.

4. CONCLUSIONS In summary, we have developed a model to describe the behavior of active particles that act as chemical motors, which generate convective fluid flows. The flows, in turn, can organize particles into clusters and can drive them along the bottom plane of microchannels. To perform as a chemical motor, the particle was coated with a catalyst that decomposed reagent dissolved in the solution. The variation in fluid density generated by the changing fluid composition gave rise to bulk buoyancy forces. As a representative example of a chemical reaction driving fluid motion, we considered the decomposition of hydrogen peroxide (of concentration C) into less dense products that modify the density of the solution ρ ≈ (1 + βCC). This choice dictates that particles denser than the solution (ρπ > ρ) aggregate at the bottom plane of a microchannel where the flow is directed toward the active particle and then moves upward

Figure 5. Schematics of (Na, Np) clusters containing Na active (red) and Np passive (blue) particles, which, depending on the composition and particle arrangement, show different behaviors. Transitions from metastable to stable isomers are shown with green arrows. Propulsion directions are shown with red arrows. Symmetric clusters propel or remain stationary, whereas asymmetric clusters rotate. F

DOI: 10.1021/acs.langmuir.7b01840 Langmuir XXXX, XXX, XXX−XXX

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Figure 6. (a) Motion of a single active particle into a corner of a domain with solid side walls. (b) Motion of a dimer consisting of active and passive particles along the circumference of a solid circular cage shown with a green dashed line. Transitions between positions are shown with black arrows. The reagent concentration is shown with color for the last simulation moment. The particle velocities are ∼1 μm s−1. Movies demonstrating the details of these evolutions are provided in the SI.

above the particle. Other chemical reactions12 can be used as well. It is worth mentioning that particles less dense than the solution (ρπ > ρ) localize at the top plane of the container where the flow is directed away from the active particles and causes particle separation. Chemical reactions involving reagent transformations to denser products, ρ ≈ (1 − βCC), reverse the particle behavior because the flow generated around the active particles is always directed downward. Therefore, when the products are denser than the reagents, particles that are denser than the solution spread away from the active particles at the bottom plane, whereas less dense particles aggregate into clusters at the top of the container. Movies demonstrating this behavior are provided in the SI. We emphasize that, unlike the self-diffusiophoretic motion of Janus spheres, where the velocity field is generated locally at the particle surface, the proposed chemically activated, buoyancydriven mechanism of particle self-propulsion generates a global convective fluid flow throughout the chamber that is harnessed to transport particles (cargo). The speed of the particle propulsion is limited by the hydrodynamic properties of the system (defined by the geometry of the chamber, the particles, and the fluid viscosity) and the efficiency of chemical exchange between the particle surface and the solution that controls the buoyancy force through variations in fluid density. In the propulsion of clusters, particles perform different functions: Active particles generate the flow, whereas passive particles are dragged by the flow and transmit the momentum to the cluster. As a result of the collective interactions, some initially metastable cluster configurations that have a relatively high velocity of propulsion transform into clusters with slower propulsion but a stable structure. Notably, these findings can be used to design clusters with a fixed, immutable geometry so that they exhibit slow, fast, stationary, or rotating dynamics. In simulations of clusters of particles with diameters on the order of hundreds of micrometers moving in a millimeterheight chamber, we obtained micrometer-per-second propulsion velocities. The maximal velocity was achieved for particles with a diameter that was approximately one-half of the channel height. The proposed mechanism is not restricted to the low-

Reynolds-number regime. Moreover, in contrast to the diffusiophoresis of Janus spheres, where the velocity of a particle of radius R decreases as 1/R,9 the velocities of the convectively propelled spheres increase with the radius (for 2R < H/2). The absence of limitations on the upper size of the self-propelled particles allows the system to be scaled up and, thus, achieve higher velocities. For larger systems, however, we do expect a greater influence of the interparticle friction, which was ignored in the present model. Finally, we note that thermally active particles, which are heated to temperatures higher than the surrounding fluid, can also generate buoyancy-driven or thermoosmotic flows that can be harnessed to organize and manipulate particles in microchannels. For example, suitably coated particles can catalyze exothermic chemical reactions and generate heat production.12 Alternatively, particles coated with gold and heated by electromagnetic radiation can generate thermoplasmonic convection.29,30 The treatment of thermally driven flows that transport particles can be realized within the framework developed here by replacing the equation of reagent diffusion with the heat conduction equation and adjusting the corresponding boundary conditions.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b01840. Details of dynamics of active and passive particles in the overdamped limit and examples of simulated types of particle dynamics (PDF) Convective propulsion of a (1a, 1p) dimer, consisting of an active (coated with a catalyst) particle and a passive (no catalyst) particle (AVI) Transition of a (1a, 2p) metastable collinear trimer to a stable triangular configuration (AVI) Propulsion of a (1a, 3p) tetramer (AVI) Collision of two travelling (1a , 1p) dimers with subsequent transformation into a stationary (2a, 2p) tetramer (AVI) G

DOI: 10.1021/acs.langmuir.7b01840 Langmuir XXXX, XXX, XXX−XXX

Article

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(15) Das, S.; Shklyaev, O. E.; Altemose, A.; Shum, H.; Ortiz-Rivera, I.; Valdez, L.; Mallouk, T. E.; Balazs, A. C.; Sen, A. Harnessing Catalytic Pumps for Directional Delivery of Microparticles in Microchambers. Nat. Commun. 2017, 8, 14384. (16) Ortiz-Rivera, I.; Shum, H.; Agrawal, A.; Sen, A.; Balazs, A. C. Convective Flow Reversal in Self-Powered Enzyme Micropumps. Proc. Natl. Acad. Sci. U. S. A. 2016, 113 (10), 2585−2590. (17) Valdez, L.; Shum, H.; Ortiz-Rivera, I.; Balazs, A. C.; Sen, A. Solutal and Thermal Buoyancy Effects in Self-Powered Phosphatase Micropumps. Soft Matter 2017, 13, 2800−2807. (18) Rayleigh, L. Measurements of the Amount of Oil Necessary in Order to Check the Motions of Camphor upon Water. Proc. R. Soc. London 1889, 47, 364−367. (19) Blake, J. R. A Note on the Image System for a Stokeslet in a Noslip Boundary. Math. Proc. Cambridge Philos. Soc. 1971, 70, 303−310. (20) Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Clarendon Press: Oxford, U.K., 1961). (21) Qian, Y. H.; D’Humieres, D.; Lallemand, P. Lattice BGK Models for Navier-Stokes Equation. Europhys. Lett. 1992, 17, 479−484. (22) Guo, Z.; Zheng, C.; Shi, B. Discrete Lattice Effects on the Forcing Term in the Lattice Bolzmann Method. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 65, 046308. (23) Peskin, C. S. The Immersed Boundary Method. Acta Numer. 2002, 11, 479−517. (24) Shum, H.; Yashin, V. V.; Balazs, A. C. Self-assembly of Microcapsules Regulated Via the Repressilator Signaling Network. Soft Matter 2015, 11, 3542. (25) Bringley, T. T.; Peskin, C. S. Validation of a Simple Method for Representing Spheres and Slender Bodies in an Immersed Boundary Method for Stokes Flow on an unbounded Domain. J. Comput. Phys. 2008, 227, 5397−5425. (26) Simulations of cluster propulsions where particles were modeled as material points advected with the fluid flow u revealed no fluctuations of the particle velocities but showed much faster cluster propulsion velocities because of an incorrect description of the particle-flow hydrodynamic interactions. (27) The stiffness of the elastic shells was chosen to preserve the integrity of the particles but allow for their small deformations. Elastic shells that are too stiff require small time steps and can potentially lead to numerical instabilities. Simulations with different stiffnesses looked qualitatively similar. (28) Although the mechanism of self-organization is different, some structures of small clusters are similar to those reported elsewhere; see: Soto, R.; Golestanian, R. Self-Assembly of Catalytically Active Colloidal Molecules: Tailoring Activity Through Surface Chemistry. Phys. Rev. Lett. 2014, 112, 068301. (29) Roxworthy, B. J.; Bhuiya, A. M.; Vanka, S. P.; Toussaint, K. C. Understanding and Controlling Plasmon-Induced Convection. Nat. Commun. 2014, 5, 3173. (30) Ndukaife, J. C.; Kildishev, A. V.; Nnanna, A. G. A.; Shalaev, V. M.; Wereley, S. T.; Boltasseva, A. Long-Range and Rapid Transport of Individual Nano-Objects by a Hybrid Electrothermoplasmonic Nanotweezer. Nat. Nanotechnol. 2016, 11, 53−59.

Rotation of a (2a, 3p) pentamer (AVI) Translation along the top wall of a (2a, 3p) pentamer where the particles are less dense than the solution and the reaction product is heavier than the reagent (AVI) Attraction of an active (1a, 0p) monomer (that is denser than the solution) from the middle of a rectangular domain with solid walls to the solid side wall and then to the corner of the domain (AVI) Translation of a (1a, 1p) dimer consisting of active and passive particles moving along the circumference of a solid circular cage (AVI)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Anna C. Balazs: 0000-0002-5555-2692 Present Address †

Henry Shum, Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the Department of Energy through Grant DE-FG02 90ER45438. REFERENCES

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DOI: 10.1021/acs.langmuir.7b01840 Langmuir XXXX, XXX, XXX−XXX