Effective Interactions between Chemically Active Colloids and Interfaces

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Cite This: Acc. Chem. Res. 2018, 51, 2991−2997

Effective Interactions between Chemically Active Colloids and Interfaces Published as part of the Accounts of Chemical Research special issue “Fundamental Aspects of Self-Powered Nano- and Micromotors”. Mihail N. Popescu,*,†,‡ William E. Uspal,†,‡ Alvaro Domínguez,§ and Siegfried Dietrich†,‡ †

Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, D-70569 Stuttgart, Germany IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany § Física Teórica, Universidad de Sevilla, Apdo. 1065, 41080 Sevilla, Spain

Acc. Chem. Res. 2018.51:2991-2997. Downloaded from pubs.acs.org by UNIV OF RHODE ISLAND on 12/19/18. For personal use only.

‡

CONSPECTUS: Chemically active colloids can achieve force- and torque-free motility (“self-propulsion”) via the promotion, on their surface, of catalytic chemical reactions involving the surrounding solution. Such systems are valuable both from a theoretical perspective, serving as paradigms for nonequilibrium processes, as well as from an application viewpoint, according to which active colloids are envisioned to play the role of carriers (“engines”) in novel lab-on-a-chip devices. The motion of such colloids is intrinsically connected with a “chemical field”, i.e., the distribution near the colloid of the number densities of the various chemical species present in the solution, and with the hydrodynamic flow of the solution around the particle. In most of the envisioned applications, and in virtually all reported experimental studies, the active colloids operate under spatial confinement (e.g., within a microfluidic channel, a drop, a free-standing liquid film, etc.). In such cases, the chemical field and the hydrodynamic flow associated with an active colloid are influenced by any nearby confining surfaces, and these disturbances couple back to the particle. Thus, an effective interaction with the spatial confinement arises. Consequently, the particle is endowed with means to perceive and to respond to its environment. Understanding these effective interactions, finding the key parameters which control them, and designing particles with desired, preconfigured responses to given environments, require interdisciplinary approaches which synergistically integrate methods and knowledge from physics, chemistry, engineering, and materials science. Here we review how, via simple models of chemical activity and self-phoretic motion, progress has recently been made in understanding the basic physical principles behind the complex behaviors exhibited by active particles near interfaces. First, we consider the occurrence of “interface-bounded” steady states of chemically active colloids near simple, nonresponsive interfaces. Examples include particles “sliding” along, or “hovering” above, a hard planar wall while inducing hydrodynamic flow of the solution. These states lay the foundations for concepts like the guidance of particles by the topography of the wall. We continue to discuss responsive interfaces: a suitable chemical patterning of a planar wall allows one to bring the particles into states of motion which are spatially localized (e.g., within chemical stripes or along chemical steps). These occur due to the wall responding to the activity-induced chemical gradients by generating osmotic flows, which encode the surface-chemistry of the wall. Finally, we discuss how, via activity-induced Marangoni stresses, long-ranged effective interactions emerge from the strong hydrodynamic response of fluid interfaces. These examples highlight how in this context a desired behavior can be potentially selected by tuning suitable parameters (e.g., the phoretic mobility of the particle, or the strength of the Marangoni stress at an interface). This can be accomplished via a judicious design of the surface chemistry of the particle and of the boundary, or by the choice of the chemical reaction in solution.

1. INTRODUCTION Recently, there has been increasing interest in the development of chemically active colloids which are capable of moving autonomously within a fluid environment by promoting chemical reactions involving their surrounding solution. Such particles exhibit motility in the absence of external forces or torques acting on them or on the fluid. A wide variety of such particles has been proposed and studied experimentally.1−12 The mechanisms of motility have been the topic of numerous theoretical studies;13−26 thorough and insightful reviews of the © 2018 American Chemical Society

developments in the area of man-made motile colloids are provided by refs 27−30. Similarly to the case of classical phoresis,31 the motion of chemically active particles and the corresponding hydrodynamic flow of the solution result from the distinct interactions between the particle and the various molecular species, i.e., solvent, reactants, and reaction products.31,32 However, distinct from classical phoresis, the Received: May 29, 2018 Published: November 7, 2018 2991

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Accounts of Chemical Research spatial inhomogeneities in the distribution of species in solution (which we shall denote as “chemical field”) are selfgenerated (rather than being externally imposed), e.g., by the chemical reactions being promoted only on parts of the surface of the particle. Recent studies have shown that when active particles move near walls, fluid interfaces, or in the vicinity of other (active or inert) particles (i.e., situations which typically do occur in experiments1,7,9,14,33,34), they may exhibit complex behaviors, such as surface-bound steady states,35−38 long-ranged effective interactions,39,40 “guidance” by topographical or chemical features,41−45 enhanced velocity under geometrical confinement,46,47 Marangoni flow driven motion,39,40,48 the formation of “living” clusters,7,49,50 and collective effects.38,40,51−54 If in addition they are exposed to external flows or force fields, a very rich and interesting phenomenology appears, including, e.g., rheotaxis,55,56 cross-stream rheotaxis,57 gravitaxis,9,58,59 or photogravitaxis.60 The complex phenomenology discussed above can be rationalized in terms of effective interactions of the active particle with the spatial confinement. These arise because the chemical field and the hydrodynamic flow of an active colloid are modified by nearby confining surfaces. Since these disturbances reflect back to the particle, it becomes endowed with means to perceive and to respond to its environment. However, the exact nature of these effective interactions is not transparent, due to the intricate coupling between the chemical and hydrodynamic fields induced by the particle. An in-depth understanding of these effective interactions would allow one to find the key parameters which control them and eventually design particles with desired, preconfigured responses to given environments. Here we review the recent progress made in understanding, via considering simple models of chemical activity and selfphoretic motion, the basic physical principles which determine the complex behavior exhibited by an active particle near interfaces; the latter can be inert (i.e., only confining the solution) or responsive to the activity of the particle.

Figure 1. Schematic illustration of a chemically active, spherical particle of radius R and orientation d. At the catalyst side (dark gray area), solute molecules (black circles) are released into the surrounding solution, while the remaining surface (light gray) is chemically inactive. The vector rP denotes a point on the surface. Adapted with permission from ref 62. Copyright 2016 The European Physical Journal (EPJ).

orientation of the particle via a unit vector d pointing along the axis of symmetry toward the noncatalytic pole (see Figure 1). A simple model for chemical activity of the particle13 is the release (with a constant rate 2 per area) into the solution of a molecular solute at the catalytic part of the surface of the colloid. The solute diffuses freely with diffusion constant D. catalyst

(This can apply, e.g., for a chemical reaction fuel ⎯⎯⎯⎯⎯⎯→ solute in the case of reaction-limited kinetics and a well-stirred solution with abundant fuel.24) At steady state, the number density c(r) of solute (i.e., the chemical field) obeys the Laplace equation, while the hydrodynamic flow u(r) is the solution of the incompressible Stokes equations.

2. MODEL CHEMICALLY ACTIVE PARTICLE Typical experimental studies of chemically active colloids involve aqueous solutions, molecular-sized, fast diffusing reactants and products, and micrometer-sized particles moving at speeds of the order of a few particle diameters per second. Therefore, an adequate theoretical framework is that of a Newtonian fluid with a very small Reynolds number of the hydrodynamic flow, and with a very small Péclet number of the reactant and product species.14,29,30,61 Accordingly, viscous friction dominates over inertial effects, as far as hydrodynamics is concerned, and the diffusion of reactant and product species dominates the transport by advection. Here, we focus on steady state motion in the absence of external forces or torques. (This means, inter alia, that effects due to, e.g., buoyancy or bottom-heaviness, are disregarded. These are not intrinsically connected to the chemical activity, and thus can be addressed separately and included by superposition.35) We consider basic geometries relevant for experiments, e.g., spherical colloids which are partially decorated by a catalyst over a spherical cap region of opening angle θ0 (see Figure 1), with a focus on the cases θ0 = 90° (Janus particle) and θ0 = 180° (uniformly active sphere). For θ0 ≠ 0°, 180°, the distinct material identity of the two parts of the surface defines an

3. SELF-PHORESIS NEAR A NONRESPONSIVE PLANAR WALL One mechanism of motion for such active particles is that of self-phoresis. The variations in the solute number density c(rP) along the surface of the particle, occurring due to the fact that the release of solute occurs only on a part of the surface, give rise to a “phoretic slip”14,31,32 us(rP) ≔ −b(rP)∇ c(rP)

(1)

This serves as a boundary condition for the incompressible hydrodynamic Stokes flow u(r) of the solution. The prefactor b(rP) is the so-called “phoretic mobility”, while ∇∥ is the projection of the gradient onto the tangent plane at rP. The mobility b(rP) originates from the excess interaction of the solute molecules (over that of the solvent molecules) with the surface of the particle near rP. It has the property that b(rP) < 0 (b(rP) > 0) if that interaction is repulsive (attractive). This implies that, e.g., for repulsive interactions, the phoretic slip points toward the regions of increased density of solute, i.e., there is solvent flow tending to “dilute” that region of the solution; vice versa, for attractive interactions the solvent is “squeezed” out. 2992

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Accounts of Chemical Research Experimentally, for particles assumed to be moving via a selfphoretic mechanism, it has been often observed that during their motion near walls they exhibit a stable preference for orienting their director d basically parallel to the wall.41,42,60 For example, in ref 41, this has been reported for Janus particles made out of silica spheres and with half of their surfaces covered by a Pt film, upon immersion in an aqueous hydrogen peroxide solution (Figure 2).

Figure 2. Pt/silica Janus particles sedimented near a wall (view from above): (a) suspended in water (inactive particles), Pt caps facing downward (due to the density mismatch with the silica core); (b) upon addition of H2O2 (leading to active particles), the symmetry axis is quasi-parallel to the wall (dark half-disk corresponds to Pt). Adapted with permission from ref 41. Copyright 2016 Springer Nature.

Figure 3. Chemically active Janus sphere in sliding (a) and hovering (b) states above a planar wall located at z = 0. The active part is indicated in black. The phoretic mobility b(rP) takes different, but constant, values bc < 0 and bi for the active and inactive regions, respectively, with bi/bc = 0.7 (a) and −0.7 (b), respectively. In (a) the magenta arrow shows the translational velocity of the particle. The white streamlines correspond to the flow field in the laboratory frame. The chemical field c(r) (given in units of the characteristic density C0 = 2R /D ) is color coded. Adapted with permission from ref 62. Copyright 2016 The European Physical Journal (EPJ).

This behavior can be captured and rationalized by the model of a self-phoretic, chemically active particle as described above via following the collision between an incoming particle and the wall,35,41,60 for various choices of the phoretic mobilities of the two halves of the particle. The wall is confining the solution and is nonresponsive with respect to the distribution of the solute. These two properties are imposed via suitable boundary conditions for the chemical and hydrodynamic fields: vanishing normal current of the diffusing solute and vanishing hydrodynamic flow, respectively. The observed direction of motion being parallel to d suggests assigning a negative phoretic mobility bc < 0 to the chemically active part. By varying the constant value of the phoretic mobility bi at the chemically inert part, it has been shown35,41 that there is a broad range of values bi with |bi/bc| < 1 for which indeed a steady state of motion parallel to the wall emerges. It exhibits a fixed height of the particle above the wall and the orientation of d being slightly tilted toward the wall. An example of such a “sliding” state is shown in Figure 3a. This state is an attractor for most of the configurations in which the particle, starting far away from the wall, is on a collision course with the wall. The emergence of such a steady state is a complex phenomenon because it requires that the phoretic slip at the particle varies around its surface in such a way that it simultaneously cancels the vertical translation and the hydrodynamic rotation (spinning), which usually accompanies a translation of the particle parallel to the wall. First, the change in the solute distribution, the wall-induced distortion of which is evident from Figure 3a, leads to changes in the phoretic slip at the particle surface; these translate into changes in the hydrodynamic flow induced by the particle. Second, in terms of hydrodynamics, this flow is reflected by the wall due to the no-slip boundary condition and couples back to the particle. In particular, in the absence of the changes in the phoretic slip, the hydrodynamics would tend to rotate the catalytic cap toward the wall.35 It is the modulation of the

phoretic slip, due to the increased solute density in between the wall and the particle, which gives rise to a rotation of the catalytic cap away from the wall. Thus, the steady state emerges once these opposite effects balance.35,41 Upon turning the phoretic mobility on the inactive part into positive values, the direction of the phoretic slip along the inert part reverses and points away from the equator, which is the region with increased solute density. Due to the modulation of the phoretic slip by the distortion of the chemical field, the phoretic rotation becomes so large that it always dominates the hydrodynamic rotation. Therefore, the sliding state is no longer possible. But a different steady state emerges, which is also an attractor (in the same sense as above) for the dynamics of the active colloid: the particle is oriented with the cap up and hovers at a certain height above the wall, while it pumps the solution.35,41 An example of the hovering state is shown in Figure 3b. While this theoretically predicted state has proved to be difficult to evidence experimentally, recently it has been inferred as possibly playing a role in the behavior exhibited by photochemically active titania/Ti/silica colloids near walls.60

4. SELF-PHORESIS NEAR A CHEMIOSMOTIC RESPONSIVE WALL In the preceding section, the wall was considered to play the strictly passive role of confining the solution. However, the same physical mechanisms which drive “phoretic” flow in a boundary layer surrounding the particle surface are expected to also drive “chemiosmotic” flow in a layer at the wall. If present, chemiosmotic flow at the wall drives flow in the bulk solution, coupling back to the particle and contributing to its motion. Indeed, for chemically active particles which are sufficiently 2993

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Figure 4. (a) Schematic illustration of a catalytic Janus particle near a “chemical step” (top view). The wall beneath the particle is composed of two different materials. The diffusing solute molecules (green spheres) have repulsive interactions with both materials, but the interaction with the right (gray) material is stronger than the interaction with the left (orange) material. The chemiosmotic flow induced at the wall causes the particle to rotate (magenta arrow) and to be dragged toward the chemical step (blue arrow). In addition, the particle self-propels (green arrow). (b) Trajectory of a catalytic Janus particle which spontaneously “docks” at such a chemical step. In the docking state, the particle is motionless and its director d is perpendicular to the step. (c) Side view of a catalytic Janus particle, self-propelling toward its catalytic cap near a chemical stripe. The particle spontaneously focuses at the center of the stripe and aligns with the axial direction of the stripe. Panels (a) and (b) Reproduced with permission from ref 43. Copyright 2016 APS.

propulsion and chemiosmosis to the translation and the rotation of the particle. The interplay between these contributions can lead to various outcomes. For example, in the case shown in Figure 4b, the particle spontaneously “docks” at the step, aligning its director d with the step normal, and comes to a halt near the step. In this steady state, self-propulsion away from the catalytic cap (i.e., to the right) is balanced by the chemiosmotic flow directed from the location of the step to the left. This docking state is an attractor for the dynamics of the particle. A more complex situation is illustrated in Figure 4c for a particle near a “chemical stripe”. The distinction between the gray and orange materials is the same as for the chemical step, but here the self-propulsion of the particle is directed toward its catalytic cap. For a broad range of phoretic mobilities on the stripe and on the wall, one finds that the particle is attracted to the center of the stripe and aligns its director d parallel to the stripe edges.43,45 In general, for spherical particles, the docking state is stable only for “inertforward” propulsion, and the stripe-following state is stable only for “catalyst-forward” propulsion.43,45

close to a solid wall, chemiosmotic (and electro-osmotic) surface flows can be significant, as can be inferred from experimental observations49,63,64 and theoretical calculations.43,63,65 Within the continuum model discussed in section 3, one can include chemiosmotic flow by taking the boundary condition expressed in eq 1 to apply not only on the particle surface, but also on the wall. If the wall is chemically uniform, chemiosmotic flow does not qualitatively change the general scenarios of motion  reflecting, sliding, and hovering  but contributes quantitatively to the configuration of such states.41,60 New scenarios of motion arise, however, if the chemical composition of the wall varies spatially. For example, the wall can consist of distinct regions or patches (Figure 4a), with the surface chemistry varying from patch to patch,43,45 or it could feature a continuous gradient in the chemical composition.44 The active particle can “sense” and respond to such variations of the surface chemistry because the interaction between the solute molecules and the wall, which depends on the local chemical composition of the wall, is encoded in the surface mobility parameter b(rs) (where rs denotes a point on the wall). In turn, b(rs) determines the sign and the magnitude of the chemiosmotic flow at rs (eq 1). This spatial modulation of the chemiosmotic flow significantly and qualitatively affects the flow in the bulk liquid. For instance, it can induce, at the location of the particle, a nonzero component of the flow vorticity ∇ × u along the wall-normal, i.e., the vertical direction. This causes the particle to rotate inplane and induces changes of the angle φ, as schematically depicted by the magenta arrow in Figure 4a. We provide some concrete illustrations of this rotation and of its effect on particle trajectories.43,45 In Figure 4a, we show a top-down view of a Janus sphere near a “chemical step”. Both sides of the step have a repulsive interaction with the solute, but the interaction is stronger on the right-hand side (gray material) than on the left one (orange material). The intrinsic self-phoretic motion of the particle is pointing away from its catalytic cap. For simplicity, we assume that the particle orientation d is “locked” into the plane of the wall (i.e., dz = 0) and that the particle has a constant height h above the wall. The arrows illustrate the leading order contributions from self-

5. DYNAMICS DRIVEN BY MARANGONI FLOWS Another important case of effective interaction between active particles and a responsive boundary concerns the interface between two fluid phases, e.g., water and air or water and an oil. This fluid interface is responsive because its physical properties, most notably its surface tension γ, depend on the chemical environment. The excess interaction of the solute molecules produces a spatially varying surface tension, which in turn induces the analogue of a “phoretic mobility” (see eq 1) along the interface. More precisely, at each point at the interface a jump arises in the normal-tangent fluid stress, which is given by discontinuity of η∇⊥ u = −∇ γ = b0∇ c

(2)

Here, η is the viscosity of the corresponding fluid, while b0 = −dγ/dc quantifies the susceptibility of the surface tension to changes in the local chemical environment. These so-called Marangoni stresses are responsible for inducing a Marangoni f low in the two adjacent fluid phases. This flow acts back on 2994

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that of a 2D self-gravitating fluid66 or of 2D chemotactic aggregation.54 If any other interparticle forces are negligible, one predicts “surfing” of the particles along the fluid interface due to the drag by the superposition of the Marangoni flows.54 Although the ambient collective Marangoni flow is incompressible in 3D, its projection onto the plane of the interface gives a 2D compressible flow.67 As a consequence, an “attractive” collective Marangoni flow can destabilize a homogeneous particle distribution and lead to particle clustering within the monolayer. An even richer phenomenology can be expected if the Marangoni flow competes with interparticle forces. If the particles are sufficiently heavy and trapped within the interface, they can deform the latter and experience a mutual attraction due to capillary forces.66,68 This particular form of interaction is interesting because both the capillary forces and the “effective Marangoni interaction” are given in terms of the interfacial tension and decay in the same manner with separation. A homogeneous distribution is unstable either because of capillary collapse69 or because of the Marangoni flow. The competition between both effects has been studied in ref 52. This analysis shows that the monolayer can reach an inhomogeneous steady state, in which the direct interparticle forces and the drag by the collective Marangoni flow balance. Given the analogy with a 2D self-gravitating fluid, one can envisage an “onionlike” pattern at the interface, much like in astrophysical objects: distinct thermodynamic phases (related to the direct interparticle forces) coexist arranged in concentric layers, while the whole structure is globally confined by the mutual “attractive” Marangoni flow. This scenario has been explored theoretically in ref 53, where an interparticle repulsive force decaying as d−4 is considered: this is a realistic model (at least for chemically inert particles70) for describing ionizable particles at a fluid interface, paramagnetic particles in an external, tunable magnetic field, and polarizable particles in an external, tunable electric field. Upon increasing density and in 2D, this soft repulsion renders a liquid, a hexatic, and a crystal phase, which have indeed been observed in experiments with inert colloids.71,72 Theory predicts53 that Marangoni flows induced by chemical activity can promote the coexistence of these phases in a spatial pattern as described above.

the particle responsible for the chemical gradients, thus providing another example of activity-induced particle dynamics. (In general, there is also a self−phoretic contribution due to chemical gradients induced by the presence of the interface. As discussed in section 3, this can be dealt with separately.47) In order to isolate the effects due to the response of the fluid interface, one considers the configuration of a single, uniformly active spherical colloid (Figure 1 with θ0 = 180°) near a macroscopically extended, flat fluid interface. In the absence of the interface, the particle is motionless. In the presence of the interface, the particle will drift toward (V < 0) or away (V < 0) from it with a velocity (within the point particle approximation) V=−

4πR2 2b0 64πη+D+L

(3)

where L is the distance of the particle center from the interface, η+ quantifies the average viscosity of the fluids, and D+ quantifies the average diffusion coefficient of the solute in the fluids.39,40 Thus, if the particle produces a solute (2 > 0) which reduces the surface tension (b0 > 0), or if it consumes a solute (2 < 0) which tends to increase it (b0 < 0), the particle is attracted toward the interface; otherwise it is repelled. The Marangoni flow induced by a particle with the former properties is shown in Figure 5.

Figure 5. Streamlines of an “attractive” Marangoni flow u(r) induced by the activity of a point particle located at the position of the gray dot (streamlines are reversed for a “repulsive” flow). The fluid interface coincides with the plane z = 0. The color code for |u(r)| is given in units of 4 |V| (eq 3). Adapted with permission from ref 40. Copyright 2016 APS.

6. CONCLUSIONS AND OUTLOOK The paradigmatic examples discussed above highlight the role played by parameters such as the phoretic mobility of an active particle, or the strength of the Marangoni stress at an interface. They can be tuned via a judicious choice of the surface chemistry of the particle or of the boundary, as well as via the choice of chemical reaction, reactants, and products in solution. Such simple models capture a number of the features observed experimentally, such as sliding states near walls41,60 (section 3). Significantly, they also lead to a number of predictions, as discussed in sections 4 and 5, which await experimental tests. For example, as discussed in ref 53, the effects of the Marangoni flow may be undetectable at the single particle level, yet they will be observable at the collective level, e.g., encompassing the phenomenology of “surfing”54 or the one of layered phase coexistence.53 Such cross-checks between theory and experiments will promote the development of more detailed descriptions (see, e.g., ref 25) and pave the way toward the aim of the rational design of active particles,

It is clear that such flows will also lead to an effective interaction between neighboring particles, as each one will be dragged by the Marangoni flows induced by the others, thus giving rise to collective dynamics. This effect will be particularly important when the particles form a monolayer at or near the fluid interface. Since symmetry prevents net lateral motion of an isolated particle, the only possible dynamics is of collective nature, driven by a long−ranged, “effective Marangoni pair interaction” which decays laterally as d−1, where d is the distance between the centers of the two particles in the pair.40,54 This is formally identical with a two-dimensional (2D) Newtonian potential, so that the activity-induced collective dynamics is expected to bear resemblance with 2995

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endowed with preprogrammed response capabilities. This will require integrated approaches by the physical sciences.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 711 689 1904. Fax: +49 711 689 1922. ORCID

Mihail N. Popescu: 0000-0002-1102-7538 Notes

The authors declare no competing financial interest. Biographies Mihail N. Popescu received his Ph.D. from Emory University and is currently a scientist at the Max-Planck-Institut für Intelligente Systeme (MPI-IS) in Stuttgart. His current research interests are in the area of active matter. William E. Uspal received his Ph.D. from MIT. After postdoctoral research at the MPI-IS in Stuttgart, he is currently starting an Assistant Professor position at the University of Hawai’i, Department of Mechanical Engineering. His research interests are in the areas of fluid dynamics, active colloids, and swimmers. Alvaro Domıń guez received his Ph.D. from the Universidad Autónoma de Madrid. Since 2009 he is Professor of Physics at the Universidad de Sevilla. His research interests lie in the areas of Statistical Physics and Cosmology. Siegfried Dietrich received his Ph.D. from the Ludwig-MaximiliansUniversität in Munich. Since 2000 he is Director at the MPI-IS Stuttgart, heading the department Theory of Inhomogeneous Condensed Matter, and holds a Chair in Theoretical Physics at the Universität Stuttgart. His research interests are in Statistical Physics, with a focus on condensed matter systems which are spatially inhomogeneous on mesoscopic length scales.

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ACKNOWLEDGMENTS A.D. acknowledges support by the Ministerio de Economı ́a y Competitividad del Gobierno de España through Grant FIS2017-87117-P (partially financed by the European Regional Development Fund). The authors thank Dr. Z. Eskandari for the help in producing the TOC graphic.

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