Artificial Chemotaxis of Self-Phoretic Active Colloids: Collective

Oct 16, 2018 - A particular appealing example is dynamic clustering in dilute suspensions first reported by a group from Lyon. A subtle balance of att...
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Artificial Chemotaxis of Self-Phoretic Active Colloids: Collective Behavior Published as part of the Accounts of Chemical Research special issue “Fundamental Aspects of Self-Powered Nano- and Micromotors”. Holger Stark*

Acc. Chem. Res. Downloaded from pubs.acs.org by UNIV OF SUNDERLAND on 10/17/18. For personal use only.

Technische Universität Berlin, Institute of Theoretical Physics, Hardenbergstrasse 36, D-10623 Berlin, Germany CONSPECTUS: Microorganisms use chemotaxis, regulated by internal complex chemical pathways, to swim along chemical gradients to find better living conditions. Artificial microswimmers can mimic such a strategy by a pure physical process called diffusiophoresis, where they drift and orient along the gradient in a chemical density field. Similarly, for other forms of taxis in nature such as photo- or thermotaxis the phoretic counterpart exists. In this Account, we concentrate on the chemotaxis of self-phoretic active colloids. They are driven by self-electro- and diffusiophoresis at the particle surface and thereby acquire a swimming speed. During this process, they also produce nonuniform chemical fields in their surroundings through which they interact with other colloids by translational and rotational diffusiophoresis. In combination with active motion, this gives rise to effective phoretic attraction and repulsion and thereby to diverse emergent collective behavior. A particular appealing example is dynamic clustering in dilute suspensions first reported by a group from Lyon. A subtle balance of attraction and repulsion causes very dynamic clusters, which form and resolve again. This is in stark contrast to the relatively static clusters of motility-induced phase separation at larger densities. To treat chemotaxis in active colloids confined to a plane, we formulate two Langevin equations for position and orientation, which include translational and rotational diffusiophoretic drift velocities. The colloids are chemical sinks and develop their long-range chemical profiles instantaneously. For dense packings, we include screening of the chemical fields. We present a state diagram in the two diffusiophoretic parameters governing translational, as well as rotational, drift and, thereby, explore the full range of phoretic attraction and repulsion. The identified states range from a gaslike phase over dynamic clustering states 1 and 2, which we distinguish through their cluster size distributions, to different types of collapsed states. The latter include a full chemotactic collapse for translational phoretic attraction. Turning it into an effective repulsion, with increasing strength first the collapsed cluster starts to fluctuate at the rim, then oscillates, and ultimately becomes a static collapsed cloud. We also present a state diagram without screening. Finally, we summarize how the famous Keller−Segel model derives from our Langevin equations through a multipole expansion of the full one-particle distribution function in position and orientation. The Keller−Segel model gives a continuum equation for treating chemotaxis of microorganisms on the level of their spatial density. Our theory is extensible to mixtures of active and passive particles and allows to include a dipolar correction to the chemical field resulting from the dipolar symmetry of Janus colloids.

1. INTRODUCTION

using chemical sensors and an internal biochemical signaling pathway12−15 or light sensing algae alter the breast stroke of their flagella, which is triggered by light receptors.10 The rapidly evolving field of artificially designed microswimmers16,17 is also driven by the idea of mimicking different taxis strategies, now based on pure physical principles, to control their swimming paths and explore a wide field of intriguing applications. Indeed, in colloidal systems phoretic motion induced by field gradients (phoresis) is well established.18 Combined with activity, colloids exhibit novel

In nature microorganisms have developed the ability to sense their environment to direct their motion. An important behavioral response is called taxis. Microorganisms can sense and move along the gradient of an external stimulus or field and thereby are able to find better living conditions.1 The most prominent example is chemotaxis, where the gradient is formed by the density of a chemical species,2 and many microorganisms employ this strategy.2−6 Other forms of taxis found for living organisms are gravitaxis,7 rheotaxis,8 magnetotaxis9 phototaxis,10 or thermotaxis.11 To implement a taxis strategy, microorganisms change their swimming mode in response to field gradients. For example, bacteria modify their tumble rate © XXXX American Chemical Society

Received: June 3, 2018

A

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research complex behavior while they sense temperature gradients,19−22 gravitational fields,23 perform phototaxis,24 or show diffusiophoresis,25−29 where they drift and orient along chemical gradients. In this Account, we concentrate on how chemical fields influence the collective motion of self-phoretic active colloids,28,30−33 which produce nonuniform chemical fields by consuming chemical reactants and generating products. Prominently, they show dynamic clustering25−27,29,34 or stimulate the assembly of passive colloids.35 In suspensions of active AgCl particles a colloidal collapse and cluster oscillations were observed.36,37 Continuum theories reveal a wealth of pattern formation.28,38,39 A detailed diffusiophoretic and hydrodynamic modeling reproduces the motion of selfphoretic colloids near boundaries with step-like topography.40 On the molecular level enzymes are established as nanomotors, which undergo chemotaxis.41−44 Hydrodynamic modeling of chemical nanomotors either of Janus type or sphere dimers show different kinds of clustering45−47 or orientational order when pinned to a substrate.48 Finally, also emulsion droplets exhibit chemotaxis.49 This Account summarizes our work on the chemotaxis of self-phoretic active colloids.27,29 It was very much inspired by experiments of ref 25. Gold particles half covered with platinum, which catalyzes the reaction of H2O2 into water and oxygen, self-propel through a combination of self-diffusioand electrophoresis. They create nonuniform chemical fields in their surroundings with chemical gradients, along which neighboring particles drift (translational diffusiophoresis) or reorient (rotational diffusiophoresis). In combination with selfpropulsion this gives rise to an effective phoretic attraction and repulsion. A subtle balance of these effective interactions then causes dynamic clustering in dilute suspensions, where clusters form and resolve again.25,26 This is in stark contrast to the relatively static clusters of motility-induced phase separation at larger densities.50,51 In the following we summarize our particle-based Langevin model for addressing collective motion of self-phoretic active colloids. In formulating the model, we wanted to concentrate on the essential features. Exploring the full range of phoretic attraction and repulsion, we illustrate the collective dynamics of self-phoretic colloids ranging from a gas-like state, over dynamic clustering 1 and 2, to different types of collapsed states including a full chemotactic collapse. The latter can be rationalized by the Keller−Segel equation,52,53 which derives from our Langevin model.

concentration c, in which nearby colloids perform translational and rotational diffusiophoretic motion. Hydrodynamic interactions, which have a reduced range close to a no-slip surface, are neglected against these effective interactions. The influence of hydrodynamic flow has been studied in different systems and is reviewed in ref 17. 2.1. Translational and Rotational Diffusiophoresis

The molecules or solutes of the chemical field interact with the surface of a colloid. This results in a body force on the fluid, which influences fluid pressure. Ultimately a gradient in the concentration c along the particle surface causes a pressure gradient, which then drives a slip velocity vs = ζ∇c along the colloidal surface, where the slip velocity coefficient ζ depends on the surface interaction potential between the chemical and the colloidal surface. Averaging vs over the particle surface gives the translational diffusiophoretic velocity18 vD = [⟨ζ ⟩1 − ⟨ζ(3n ⊗ n − 1)/2⟩]∇c

(1)

where ∇c is evaluated at the particle center, n is the local surface normal, and ⟨...⟩ means average over the particle surface. For particles with uniform surface properties but also for half-coated Janus colloids the quadrupolar term in eq 1 vanishes and we obtain the diffusiophoretic drift velocity, which we use in the following vD = −ζtr ∇c

(2)

Here, ζtr ≔ −⟨ζ⟩ is the translational diffusiophoretic parameter. The slip velocity field also causes a diffusiophoretic rotational velocity18 ωD =

9 ⟨ζ n⟩ × ∇c = −ζrot ei × ∇c 4a

(3)

where the rotational diffusiophoretic parameter ζrot is defined by (9/4a)⟨ζn⟩ = −ζrote. Spherical particles with a uniform surface (constant ζ) do not rotate. However, for half-coated Janus particles ωD is nonzero, as long as the solutes interact differently with the two sides, and symmetry dictates that swimming direction e is parallel to ⟨ζn⟩. Since the slip velocity coefficient ζ can be controlled by choosing appropriate materials for the Janus colloids and their caps, but also by the geometry and number of the caps covering the colloidal surface,28 we expect the phoretic parameters ζrot and ζtr to be tunable to either positive or negative values. This opens the possibility to induce and explore a variety of collective colloidal dynamics. For example, since each active particle acts as a chemical sink in our setting, neighboring particles will drift toward the sink for positive translational diffusiophoretic parameter ζtr in eq 2, which corresponds to an effective attraction, while ζtr < 0 gives rise to an effective repulsion. Similarly, a positive rotational parameter ζrot in eq 3 rotates the swimming direction of an active colloid toward a neighboring chemical sink and the colloid moves toward the sink. Hence, rotational phoresis also acts like an attractive colloidal interaction while it becomes repulsive for ζrot < 0.

2. MODEL To explore the collective dynamics of self-phoretic colloids induced by diffusiophoretic or chemotactic interactions, we set up a simplified model system. We consider a colloidal monolayer close to a bounding plate with fluid in the infinite half-space above it. The active colloids are Janus particles partially covered by a catalyst, which catalyzes a chemical reaction in the surrounding fluid. In general, through a combined process of self-electro- and diffusiophoresis, the colloids move with a velocity v0 along the unit vector e, which gives a direction fixed in the particle. In the following, we assume that v0 and e are not changed by the presence of other nearby particles. Instead of taking into account all reactants and products of the chemical reaction, we simply consider the self-phoretic colloid as a chemical sink. It consumes a chemical and thereby produces a nonuniform chemical field with

2.2. Langevin Dynamics and Chemical Field

At the micron scale inertia is negligible and position ri and orientation ei of the ith Janus colloid obey overdamped Langevin equations. Adding up the deterministic translational velocities from self-propulsion and diffusiophoretic drift (v0ei + B

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research 2.3. Essential Parameters

vD) and using the kinematic relation for the orientation vector, ėi = ωD × ei, we obtain ri̇ = v0 ei − ζtr ∇c(ri) + ξi

(4)

ei̇ = −ζrot(1 − ei ⊗ ei)∇c(ri) + μi × ei

(5)

Rescaling the dimensions of all quantities in the equations of motion can be used to reduce the number of system parameters and thereby reveals the essential parameters. One possibility is to set the rescaled diffusion constants, which determine the noise strengths in the Langevin equations 4 and 5, to one. This is achieved when rescaling time by tr = 1/ (2Drot) and length by lr = Dtr /Drot = 2.33a. The thermal diffusion coefficients in a bulk fluid would give Dtr /Drot = 1.15a, while the experiments of ref 25 measure lr = 1.79a for colloids moving close to a bottom wall. The higher value used for all reported results in this Account does not change the qualitative behavior of our system. Ultimately, one obtains four essential system parameters: the Peclet number Pe = v0/(2 Dtr Drot ), the rescaled translational

The additional vectors represent translational (ξi) and rotational (μi) white noise of thermal origin with zero mean and respective time correlation functions ⟨ξi(t) ⊗ ξi(t′)⟩ = 2Dtr1δ(t − t′) and ⟨μi(t)⊗μi(t′)⟩ = 2Drot1δ(t − t′), where Dtr and Drot are the translational and rotational diffusion coefficients, respectively. All the results presented in this review are generated by a two-dimensional version of eqs 4 and 5, since the Janus colloids move in a monolayer. An effective hard-core repulsion between the colloids is implemented. Whenever they overlap during the simulations, we separate them along the line connecting their centers to the point of contact. Finally, hydrodynamic flow fields are not included here. Close to a bounding plane they are less long-ranged and we also assume that chemical interactions dominate. According to our model assumption the colloids consume a chemical with rate k, which typically diffuses much faster on the micron scale than the Janus colloids swim. Therefore, when the colloids move, they carry around with them a static concentration field c, which obeys the Poisson equation [for a discussion of this approximation and its impact, see refs 38 and 39]

diffusiophoretic parameter ζtrkh/(8πDc) Drot /Dtr3 → ζtr , the rescaled rotational diffusiophoretic parameter ζrotkh/(8πDcDtr) → ζrot, and the area fraction σ defined as the ratio of projected area of all Janus colloids to the area of the simulation box. Note that the factor kh/(4πDc) from eq 7 is already subsumed into the rescaled diffusiophoretic parameters, when using ∇c2D(ri) in the Langevin eqs 4 and 5. 2.4. Numerical Implementation

The Langevin eqs 4 and 5 are solved by a typical Euler scheme in a two-dimensional square simulation box. Always 800 particles are used and different area fractions σ are realized by adjusting the size of the simulation box. To implement the hard-core interactions mentioned earlier, a sufficiently small time step has to be chosen, which makes them the numerically most expensive part of the simulations. Therefore, a neighbor list is implemented such that the search for overlapping particles could be restricted to eight immediate neighbors, at most. Furthermore, due to the small translational steps of the colloids it is sufficient to update the chemical concentration field every 50th time step, which further saves computational time. Since only bulk properties are of interest here, boundary conditions are implemented, which keep the colloids away from the boundary of the simulation box. Thus, whenever hitting the boundary, the particles are reflected into a randomly chosen direction.

N

0 = Dc ∇2c − k ∑ δ(r − ri) i=1

(6)

where we approximate the Janus colloids as point-like chemical sinks. We neglect here higher-order contributions that would break the radial symmetry of the concentration field because of the dipolar character of the Janus colloid. The solution is given by c 2D(r) = hc0 −

kh 4πDc

N

∑ i=1

1 |r − ri|

(7)

Here, c0 is the concentration field far away from the colloids, and we have multiplied with the thickness of the colloidal monolayer h = 2a to obtain a two-dimensional density in the plane of the monolayer. Note that according to our model the chemical field still diffuses in an infinite three-dimensional halfspace. A no-flux boundary condition at the bounding surface does not change the principal 1/r dependence in eq 7. If the chemotactic attraction between the active colloids is sufficiently strong, they form compact clusters, where the chemical substance cannot diffuse freely between the colloids. We roughly take this into account by implementing a screened chemical field, whenever a colloid is surrounded by six closely packed neighbors with distances below rs = 2a(1 + ϵ). Then, we replace the algebraic decay 1/r = 1/|r − ri| in eq 7 by exp[−(r − ξ)/ξ]/r, where we introduce the screening length ξ ≔ rs. We set ϵ = 0.3 but have checked that varying ϵ by 50% does not change the results. [In the Supporting Information of ref 26, the authors also mention a cutoff of the particle attraction at distances larger than three particle diameters. So, in particular, dynamic clustering should also be visible for larger screening lengths.]

3. RESULTS AND DISCUSSION 3.1. Overview: State Diagram

Figure1 shows a typical state diagram at low area fractions of the colloids for the two phoretic parameters ζtr and ζrot. It is roughly divided by a diagonal line, which separates collapsed states, where all colloids form one cluster with different properties, from states with varying cluster size. In particular, for ζtr and ζrot both positive meaning that translational and rotational phoretic motion act like an effective attraction between the colloids, a sharp transition from a gaslike to a collapsed state occurs, where all particles are packed into one single static cluster (see snapshot in Figure 2, bottom right). This behavior is reminiscent of the chemotactic collapse in bacterial systems.52,53 Making both phoretic interactions repulsive (ζtr < 0 and ζrot < 0), the gaslike state is realized, where small transient clusters are possible. From here two directions are possible. On the one hand, inreasing ζtr to sufficiently large ζtr > 0, so that translational C

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research

rim due to particles leaving and rejoining it. Then the cluster develops nearly regular oscillations and, finally, for strongly negative ζtr a static collapsed cloud appears, where colloids do not touch and their mutual distance increases toward the rim of the cloud. It is possible to derive the famous Keller−Segel equation from the Langevin eqs 4 and 5 (see section 3.5 and ref 27). As in bacterial systems it here predicts a chemotactic collapse of the active colloids at 8πσ(ζtr + ζrotPe)/(1 + 2Pe2) = b, where b is a positive constant. In the ζrot−ζtr state diagram, the linear separation line between the gaslike and collapsed state has a negative slope and it is shifted along the vertical to positive values. Thus, it gives the right trend for the state diagram of Figure 1. The Keller−Segel equation is a mean-field equation, it only contains the density of the colloids. Therefore, it cannot describe subtle states such as dynamic clustering 1 and 2. For example, if ζtr is sufficiently large so that the phoretic drift velocity vD exceeds the swim speed v0, the active colloids will always stay attached to a cluster, regardless how negative ζrot becomes. Hence, the separation line in Figure 1 deviates from the simple straight line. Finally, the state diagram of Figure 1 does not change qualitatively with Pe and σ, as long as Pe is well above one and σ sufficiently low so that phase separation does not occur. In the following two sections we characterize the dynamic clustering states and collapsed states in detail, report on the state diagram, when screening of the chemical field in dense colloid clusters is switched off, and comment on how to derive the Keller−Segel equation from our Langevin model.

Figure 1. Full state diagram ζtr versus ζrot at Pe = 19 and surface fraction σ = 0.05. The mean cluster size Nc for the gaslike and dynamic-clustering state 1 are color-coded. A full discussion is provided in the main text. Reprinted with permission from ref 29. Copyright 2015 European Physical Journal.

3.2. Gas Phase and Dynamic Clustering

3.2.1. Signature of Dynamic Clustering. Dynamic clustering states are distinguished from the gaslike state by the occurrence of larger cluster and a pronounced increase in cluster size between states 1 and 2. This is already pictured in Figure 1, where the dynamic clustering state 1 shows an increase of the mean cluster size Nc in a small region. To better characterize these states and justify, why there are two clustering states, Figure 3 shows the cluster size distribution for increasing ζtr for fixed rotational phoretic parameter ζrot = −0.38. The curves until ζtr = 15.4 are well fitted by P(n) = c0n−β exp( −n/n0)

(8)

where the cutoff size n0 is a measure how large the clusters can become. While for pure steric interaction (ζtr = 0, blue curve)

Figure 2. Snapshots of colloid configurations for increasing ζtrans at ζrot = −0.38 and Pe = 19. Top left: Gas-like state. Top right: Dynamic clustering 1. Bottom left: Dynamic clustering 2. Bottom right: Collapsed state. Nc is the mean cluster size. Reprinted with permission from ref 27. Copyright 2014 American Physical Society.

diffusiophoresis acts attractive, dynamic clustering occurs in two different states 1 and 2. Motile clusters form that strongly fluctuate in shape and size and may ultimately dissolve again (see snapshots in Figure 2, top right and bottom left). The strongly dynamic clusters form due to a delicate balance of the attractive translational phoresis (holding the clusters together) and the fact that active colloids turn and swim away from the cluster (effective repulsion). This behavior is decisively different from the motility-induced phase separation of hardcore active particles.50,51 On the other hand, keeping ζtr < 0 (effective repulsion) but making ζrot sufficiently large so that active colloids orient and therefore swim toward each other (effective attraction), different types of collapsed states occur. With decreasing ζtr first the collapsed cluster fluctuates at the

Figure 3. Cluster size distributions P(n) for increasing translational phoretic parameter ζtr at Pe = 19 and ζrot = −0.38. The transition between dynamic clustering states 1 and 2 occurs between the red and green curves. Inset: mean cluster size Nc versus ζtr. The transition is indicated. Reprinted with permission from ref 27. Copyright 2014 American Physical Society. D

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research and small ζtr (gray curve) an exponential decay is predominant, the distributions for larger ζtr (orange and red curves) follow first the power law before they are cut off by the exponential. The exponent β = 2.1 ± 0.1 gradually decreases for more negative ζrot and the fit is robust against increasing particle number.27 So there is a smooth transition from the gaslike to the dynamic clustering state 1. Further increasing ζtr, the distribution P(n) develops an inflection point (green curves), which marks the onset of the dynamic clustering state 2. At the transition the mean cluster size strongly increases as the inset indicates. Now, the sum of two power-law-exponential curves have to be used for fitting the distributions, P(n) = c1n

−β1

exp( −n/n1) + c 2n

−β2

exp(−n/n2)

Figure 5. State diagram ζtr versus Pe at ζrot = −0.38 and surface fraction σ = 0.05. The mean cluster size Nc for the gaslike and dynamic-clustering state 1 are color-coded. Reprinted with permission from ref 27. Copyright 2014 American Physical Society.

(9)

with β1 = 2.1 ± 0.2 and β2 ≈ 1.5. Both dynamic clustering states are observed for all negative ζrot. However, turning off screening of the chemical field within clusters, dynamic clustering is less pronounced and the clustering state 2 does not occur at all.27 Finally, all the exponents βi decrease for larger area fraction σ and the sharp increase of Nc at the transition from state 1 to 2 vanishes (see Figure 4). In

phoretic parameters ζtr, ζrot ∝ k introduced in section 2.3. Assuming Michaelis−Menten kinetics for the reaction rate in the linear regime well before saturation, k ∼ c0, we not only find Pe ∝ c055 but also ζtr ∝ ζrot ∝ c0. Therefore, varying c0 defines a straight line in the ζtr−ζrot−Pe parameter space. In this three-dimensional space the dynamic clustering states 1 and 2 are separated by a plane. We choose different lines, which always hit the transition plane and plot in Figure 6 the

Figure 4. Mean cluster size Nc plotted against ζtr for different areal fractions σ. The dashed line indicates the transition between dynamic clustering states 1 and 2. The rotational diffusiophoretic parameter is ζrot = −0.38. Reprinted with permission from ref 29. Copyright 2015 European Physical Journal.

Figure 6. Mean cluster size Nc versus Pe for different lines in the ζtr−ζrot−Pe parameter space. The lines are defined via a parametrization with x ∈ [0, 1], where Pe varies as in the experiments of,25 Pe = 9.5 + 11.5x, ζtr = 4.8 + 16.6x, and ζrot = −0.16 − ζ0x. The parameter ζ0 defines the different graphs. The transition between clustering states 1 and 2 roughly occurs at the intersection of the fitted two straight lines. Reprinted with permission from ref 27. Copyright 2014 American Physical Society.

literature similar cluster-size distributions including the transition indicated by the occurrence of an inflection point were observed in experiments on gliding bacteria, when the bacterial density was varied.54 While this system shows pure hard-core interactions causing nematic alignment, we instead vary the strength of the diffusiophoretic coupling. 3.2.2. Dynamic Clustering for Varying Péclet Number. In Figure 5, we plot the state diagram ζtr versus Péclet number Pe. Dynamic clustering becomes more pronounced for larger ζtr and then also occurs at larger Pe, which is necessary to establish the delicate balance between translational phoretic attraction (ζtr > 0) and effective repulsion (ζrot < 0), where the active colloids swim away from the clusters. However, for constant ζtr large clusters disappear with increasing Pe. This is in contrast to the experiments with diffusiophoretic coupling, which showed a linear scaling of the mean cluster size with Pe: Nc ∼ Pe, when increasing fuel concentration.25 To rationalize the experimental scaling prediction, we note that increasing the fuel concentration c0 means higher reaction rate k on the colloidal surface and therefore more self-activity but also larger phoretic forces, which is encoded in the reduced

mean cluster size Nc versus Pe along the lines. The blue and purple curves show the strong increase of Nc when the clustering state 2 is entered, since the respective lines are nearly normal to the transition plane. Tilting the lines more, the increase of Nc becomes smoother. In particular, the green graph shows an almost linear increase of Nc in the range Pe = 10−20. Thus, Figure 6 demonstrates that the relation of cluster size and swimming speed might take different forms depending on the relation between fuel concentration c0, activity Pe, and phoretic strengths ζtr and ζrot. 3.3. Clustering States

To classify the different collapsed states in Figure 1 and to describe the hexagonal order in the N-particle cluster, we introduce the global 6-fold bond orientational parameter E

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

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Accounts of Chemical Research q6 ≔ with q6(k) ≔

1 N 1 6

3.4. State Diagram without Screening

N

∑ q6(k)

∈ [0, 1]

Some of the states in the full state diagram of Figure 1 discussed so far depend on the screening of the chemical field, which is implemented when the colloids are densely packed. To demonstrate its influence, we show in Figure 8 the state

k=1

∑

ei6αkj

j ∈ 5 (6k)

(10)

5 (6k)

The second sum goes over the set of six nearest neighbors of particle k and αkj is the angle between the line connecting particle k to j and some prescribed axis.17 The local bond parameter q(k) 6 becomes one if all six nearest neighbors form a regular hexagon around the central colloid and the global order parameter is one in a hexagonal lattice. The temporal evolution of q6 is plotted in Figure 7 for several ζtr at constant ζrot = 4.5. At positive ζtr the order

Figure 8. Full state diagram ζtr versus ζrot at Pe = 19 and σ = 0.05 in the absence of screening. Reprinted with permission from ref 29. Copyright 2015 European Physical Journal.

diagram in the absence of any screening for the same parameters. A comparison reveals that without screening dynamic clustering 1 is less pronounced producing on average smaller clusters and the dynamic clustering state 2 disappears completely. The rich phenomenology of the collapsed states at negative ζtr is replaced by a single core−corona state, where a densely packed core is surrounded by a cloud of colloids (see Figure 9). The extension of the core decreases, when ζtr becomes more negative.

Figure 7. Time evolution of the bond orientational parameter q6 for different ζtr. Further parameters are Pe = 19, ζrot = 4.5, and σ = 0.05. Reprinted with permission from ref 29. Copyright 2015 European Physical Journal.

parameter is nearly constant in time indicating a static crystalline cluster, while q6 < 1 results from the colloids at the rim of the cluster, which do not have six neighbors on a hexagon. At ζtr = 0 and especially for negative ζtr, where particles effectively repel each other due to translational phoretic motion, the order parameter shows increasingly strong fluctuations. The cluster fluctuates close to the rim, where particles leave and rejoin it frequently (ζtr = −6.4 in Figure 7). A video of the fluctuating collapsed state is attached to ref 29. Further decreasing ζtr the order parameter develops nearly regular oscillations, where the cluster oscillates between densely packed and a cloud of confined colloids (ζtr = −12.8 in Figure 7). When the dense packing develops, the diffusiophoretic interaction becomes strongly screened and the particles lose their orientations toward the cluster center. Repulsion due to translational phoresis takes over and the cluster expands until the particles are oriented back to the center. This initiates the collapse of the cloud and the cycle starts again. The pulsating cluster is visualized in a video attached to ref 29. The power spectrum of q6 shows a broad peak at a nonzero frequency and confirms the regular oscillations.29 Finally, further decreasing ζtr, the oscillations abruptly stop and a static collapsed cloud forms, where the strong effective repulsion prevents direct contact between the particles (ζtr = −16.0 in Figure 7). In this state the hexagonal bond order is small since q6 ≈ 0.35 is close to the value q6 = 1/3 of uniformly distributed particles.

Figure 9. Snapshots of the colloidal configuration in the core−corona state: (a) ζtr = −6.4 and (b) ζtr = −16. Other parameters are Pe = 19 and ζrot = 4.1. Reprinted with permission from ref 29. Copyright 2015 European Physical Journal.

3.5. Relation to Keller−Segel Model

In the end, we illustrate here how to derive the Keller-Segel equation starting from our model for the chemically interacting active particles. More details are presented in ref 27. The Langevin eqs 4 and 5 are equivalent to the Smoluchowski equation for the full one-particle distribution function P(e, r, t), where we neglect direct interactions between the colloids: ∂tP(e, r, t ) = − v0 ∇·(Pe) + ζtr ∇·(P ∇c) + Dtr ∇2P + ζrot ∂φ[(∂φe) ·(∇c)P ] + Drot ∂φ 2P

(11)

To formulate the Smoluchowski equation, we used e = (sin φ, cos φ) and rewrote eq 5 to ∂tφi = ζrot∂φ ei·∇c + μi. One then derives dynamic equations for the colloidal density P0(r,t) = ∫ P(e, r, t)dφ, as well as the polar [P1(r,t) = ∫ e P dφ ] and F

DOI: 10.1021/acs.accounts.8b00259 Acc. Chem. Res. XXXX, XXX, XXX−XXX

Accounts of Chemical Research ÄÅ ÉÑ 1 the nematic ÅÅÅÅP2(r, t ) = ∫ e ⊗ e − 2 P dφÑÑÑÑ order parameÅÇ ÑÖ ter. The resulting hierarchy of coupled equations is closed by setting P3 to zero. Neglecting time derivatives of P1 and P2 on time scales much larger than the rotational diffusion time 1/ Drot and also higher-order spatial derivatives, one arrives ultimately at the Keller−Segel equation for the colloidal density

(

Article

their swimming paths and explore a wide field of intriguing applications.

)

2

∂tP0 = ζeff ∇(P0 ∇c) + Deff ∇ P0

■

Corresponding Author

*E-mail: [email protected]. ORCID

Holger Stark: 0000-0002-6388-5390

(12)

Notes

with renormalized phoretic coefficient and effective translational diffusion constant: ζeff = ζtr +

The author declares no competing financial interest.

v02

ζrotv0 and Deff = Dtr + 2Drot 2Drot

AUTHOR INFORMATION

Biography (13)

Holger Stark received his Ph.D. from University of Stuttgart and pursued postdoctoral studies at the University of Pennsylvania in Philadelphia. After staying as a Heisenberg fellow at the University of Konstanz and a group leader at the Max-Planck-Institute for Dynamics and Self-Organization in Göttingen, he became a Professor of Theoretical Physics at the Technical University Berlin. His research interests lie in the areas of nonequilibrium statistical physics of soft matter and biological systems.

From the Keller−Segel equation the condition for the chemotactic collapse mentioned in section 3.1 can be derived.

4. CONCLUSIONS This Account reviews our work on self-phoretic active colloids interacting by self-generated chemical gradients and thereby mimicking chemotaxis well-known from the biological world. The combination of translational and rotational diffusiophoretic drift velocities with self-propulsion gives rise to effective phoretic attraction and repulsion. Exploration of the full range of the drift velocities reveals that a variety of dynamic states occur that range from a gas-like state, over dynamic clustering 1 and 2, to different types of collapsed states including a full chemotactic collapse. The latter can be rationalized by the Keller−Segel equation, which derives from our Langevin model. We shortly discuss one point. In our state diagram in Figure 1, the dynamic clustering states only appear in a narrow region, while in the experiments dynamic clustering seems to be a generic feature.25,26 One reason could be that further attractive/repulsive forces between the active colloids are present in the experiments, which are not included in our model. Indeed, in simulations with a Lennard-Jones potential dynamic clustering with cluster size distributions similar to the ones discussed in this article are reported.56,57 Possible extensions of the Langevin model presented in this article are mixtures of active and passive particles. An experimental realization was published recently in ref 35. Furthermore, Janus particles do not just produce a monopole disturbance in the chemical environment but also a dipolar contribution due to their polar character. We are currently exploring the consequences of such a contribution. Ultimately, a full solution of the chemical field with appropriate boundary conditions at the colloid surface is needed for being able to treat particles, which are really close to each other. However, this will also directly influence the swimming speed of the colloids. In addition, a full hydrodynamic treatment needs to be included, which has been addressed in recent publications.45−47 Finally, the slip velocity coefficient ζ in eq 1 determines the two diffusiophoretic parameters ζrot and ζtr. It strongly depends on material properties. So, as a challenge to the fabrication of self-phoretic colloids, one can ask if it is possible to purposely tune the phoretic parameters by choosing appropriate materials, catalysts, cap sizes, and physical mechanisms for the phoretic process? All this will help to further develop the idea of how artificial microswimmers can mimic biological taxis strategies to control

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ACKNOWLEDGMENTS

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REFERENCES

The author thanks Oliver Pohl and Julian Stürmer for collaborating on the topic and acknowledges funding from the DFG within the research training group GRK 1558 and the priority program SPP 1726, project number STA 352/11. The author also thanks Julian Stürmer for preparing the picture for the Conspectus.

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