Active Janus colloids at chemically structured surfaces

J. Chem. Phys. 150, 204904 (2019); https://doi.org/10.1063/1.5091760 150, 204904 © 2019 Author(s). Active Janus colloids at chemically structured surfaces Cite as: J. Chem. Phys. 150, 204904 (2019); https://doi.org/10.1063/1.5091760 Submitted: 05 February 2019 • Accepted: 02 May 2019 • Published Online: 30 May 2019 W. E. Uspal, M. N. Popescu, S. Dietrich, et al. COLLECTIONS Paper published as part of the special topic on Chemical Physics of Active Matter ARTICLES YOU MAY BE INTERESTED IN Theory of light-activated catalytic Janus particles The Journal of Chemical Physics 150, 114903 (2019); https://doi.org/10.1063/1.5080967 Which interactions dominate in active colloids? The Journal of Chemical Physics 150, 061102 (2019); https://doi.org/10.1063/1.5082284 Light-controlled assembly of active colloidal molecules The Journal of Chemical Physics 150, 094905 (2019); https://doi.org/10.1063/1.5079861 The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp Active Janus colloids at chemically structured surfaces Cite as: J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 Submitted: 5 February 2019 •Accepted: 2 May 2019 • Published Online: 30 May 2019 Corrected: 2 August 2019 W. E. Uspal,1M. N. Popescu,2S. Dietrich,2and M. Tasinkevych3,a) AFFILIATIONS 1Department of Mechanical Engineering, University of Hawai’i at M¯ anoa, 2540 Dole Street, Holmes Hall 302, Honolulu, Hawaii 96822, USA 2Max-Planck-Institut für Intelligente Systeme, Heisenbergstr. 3, D-70569 Stuttgart, Germany and IV. Institut für Theoretische Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany 3Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande P-1749-016 Lisboa, Portugal and Centro de Física Teórica e Computacional, Universidade de Lisboa, Campo Grande P-1749-016 Lisboa, Portugal Note: This article is part of the Special Topic “Chemical Physics of Active Matter” in J. Chem. Phys. a)Author to whom correspondence should be addressed: [email protected] ABSTRACT If catalytically active Janus particles are dispersed in certain liquid solutions, they can create a gradient in the chemical composition of this solution along their surfaces, as well as along any nearby confining surfaces. This gradient drives self-propulsion via a self-phoretic mechanism, while the compositional gradient along a wall gives rise to chemiosmosis, which additionally contributes to self-motility. In this study, we analyze theoretically the dynamics of an active colloid near chemically patterned walls. We use a point-particle approximation combined with a multipole expansion in order to discuss the effects of pattern geometry and chemical contrast on the particle trajectories. In particular, we consider planar walls patterned with chemical steps and stripes. We investigate in detail the changes in the topology of the corresponding phase portraits upon varying the chemical contrast and the stripe width. Published under license by AIP Publishing. https://doi.org/10.1063/1.5091760 ., s I. INTRODUCTION Active colloidal particles can perform autonomous directional motion within a liquid medium.1–7 This offers the possibility to use such particles as autonomous micromotors, which perform useful tasks such as delivering drugs, sensing a specific substance, or assembling structures via the autonomous local deposition of materials.8–11 Catalytic colloids form a large class of active particles which employ self-phoretic effects6,7,12–16 for their motion. For example, a colloidal particle partially covered by a catalyst will activate a chemical reaction in the surrounding solution.1,2,13,17 This, in turn, will generate gradients of solute concentration or of an electric potential, leading to a directed motion of the particle by self-diffusiophoretic13,14,18 or self-electrophoretic7,16,19–21 mechanisms, respectively. Another distinct class of active particles is formed by gold-capped Janus colloidal spheres which are suspended in near-critical fluids such as water-lutidine mixtures.22 These particles can perform active Brownian motion when illuminated by light. The particle motion has been attributed to selfthermophoresis,15 self-diffusiophoresis,23 or self-phoresis due to a local solvent demixing, and associated self-generated chemical potential gradients.24 Most of the experiments carried out so far have used Pt catalyzed2,3,13,17 decomposition of hydrogen peroxide into water and oxygen. Self-propelled particles frequently employed in experiments are, e.g., gold-platinum nanorods1,3 or silica particles partially coated by a catalyst.8,13,25,26 Theoretically, aside from using molecular dynamics (MD) simulations,27–31 mesoscopic continuum approaches have been employed,12,14,18 based on the classical theory of diffusiophoresis.32,33 Using this method, the dependence of the self-propulsion velocity on the shape and surface chemistry of microswimmers,18,34–36 their chemotactic behavior,37,38 and how active colloids can be used as cargo carriers26,39 have been studied. J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-1 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp A few recent studies have begun to address self-phoresis in bounded systems. In Ref. 29, MD simulations have been used to study self-diffusiophoresis within a square channel. For a highly idealized confining geometry, an increase in the self-diffusiophoretic velocity was predicted theoretically in Ref. 40. The role of the boundaries was also highlighted in Ref. 26, where the motion of carriercargo composites has been studied near a single wall. Translocation of a Janus active particle through cylindrical pores has been addressed in Ref. 41, which reported a decrease in the particle velocity induced by the confinement and emphasized the importance of the “chemical effects” in addition to the “standard” hydrodynamic interaction. Experimentally, wall-bounded motion of active Janus particles was observed in Refs. 20 and 42. Theoretical models based on a far-field multipole description of the microswimmers43 or models, in which the surface slip is specified a priori,44 showed that the hydrodynamic interaction may generate particlewall attraction and sliding steady states along the wall.44 However, as shown recently,45–47 a more realistic description, explicitly taking into account the propulsive mechanism, is required to fully understand the particle dynamics near boundaries. Indeed, novel sliding and hovering steady states were predicted in Refs. 45 and 48–50 for a Janus microswimmer near a wall by numerically solving the coupled hydrodynamic transport equations for chemical fluxes and momentum. The sliding states may be exploited further in order to achieve desired particle behaviors in two-dimensional (2D) space by, e.g., endowing the confining surface either with topographical or chemical contrasts. For example, the occurrence of wall-bounded sliding states explains experimentally reported directional guidance of chemically active microswimmers by shallow steplike topographical features.46 In the case of chemically patterned walls, it was shown recently51,52 that under certain conditions a stable motion along a one-dimensional (1D), macroscopically long chemical stripe may be obtained. More specifically, Ref. 51 demonstrates that (i) a catalystforward spherical particle can robustly follow certain chemical stripes; (ii) an inert-forward spherical particle can achieve a motionless docking steady state at a chemical step with sufficiently strong chemical contrast. This proof of concept study employed, for illustration only, a single realization of the chemical step (i.e., only one value of the chemical contrast between the right and the left side of the surface), a single realization of the chemical stripe (i.e., one value of the chemical contrast between the two chemical entities of the surface), and a single value of the distance between the particle and the surface. Additionally, since Ref. 51 is focused on the bare possibility of guided motion, it does not systematically explore the very rich landscape of other dynamical behaviors, e.g., reflection from a chemical step, or focusing off the particle orientation as a particle crosses a step. Reference 52 discusses in detail the effect of the particle shape on the dynamics of active, half-covered spherocylinders at chemical steps and stripes. The guiding of catalytic sphere-dimer nanoparticles by highly idealized chemical stripes has been reported in Ref. 53 based on using the hybrid molecular dynamics multiparticle collision dynamics method. In this study, we further explore this idea of controlling the motion of Janus microswimmers via chemically structured, planar surfaces. By using the analytical method developed in Ref. 51, here we address the following questions: (i) How does the dynamics of an active Janus colloid depend on the chemical contrast between the right and the left side of the chemical step and on the particle-surface separation? In particular, we want to understand how the topology of the phase portrait [in the (xp,ϕ) plane, see Fig. 1(b)] depends on the aforementioned parameters. (ii) Concerning a self-propelled particle at a chemical stripe, how does this phase portrait depend on the chemical contrast of the stripe and on the particle-surface separation? (iii) In general, what dynamical behaviors can be found beyond stripe-guided or step-guided motion? FIG. 1. (a) A catalytic Janus sphere above a planar wall at height h. The black region on the particle indicates the catalytic cap which emits solute molecules (green discs, O2) at a constant rate per area κ. The solute molecules interact with the colloidal particle, and due to the inhomogeneous solute distribution, a gradient of osmotic pressure emerges in the direction parallel to the surface (for the orientation ˆ dshown) which drives a surface flow as indicated by two blue arrows. The direction of the surface flow corresponds to the repulsive nature of the solute-particle interaction, resulting in an inert-forward self-diffusiophoretic motion in direction ˆ d. If the solute-particle interaction is attractive, the flow is away from the cap, leading to catalyst-forward motion in direction −ˆ d. The solute gradient also drives chemiosmotic flow at the wall, schematically indicated by purple lines, assuming a repulsive nature of the solute-wall interaction. (b) Schematic top-down view of a Janus particle above a chemically patterned substrate. The gray region, which is characterized by the surface phoretic mobility br w, is more repulsive for the solute than the orange one with the surface phoretic mobility bl w. The orientation vector ˆ dand the x-axis form the angle ϕ;xpis the distance between the chemical step and the particle center. (c) A top-down view of the wall patterned with a rectangular chemical stripe (orange) of width W. The stripe extends in the y-direction and carries the surface phoretic mobility bc w. The rest of the wall (gray) exhibits the surface phoretic mobility bw. J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-2 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp The paper is organized as follows: In Sec. II, we present our theoretical approach which is based on the so-called point-particle approximation and a multipole expansion for the concentration of the reaction product in the solution. Concerning some of the material of this section, we also refer to the supplementary material of Ref. 51, parts of which are also presented in Appendixes A–C. In Secs. III–V, we describe and discuss our results. For the case of a chemical step, we show how changing the chemical contrast of the substrate affects the topology of the corresponding phase portraits for both inert-forward and catalyst-forward Janus colloids. It is demonstrated that inert-forward particles can dock at the boundary between the two regions of the substrate forming a chemical step and the conditions for docking to occur are identified. In the case of a chemical stripe, only catalyst-forward microswimmers are considered because only such particles can stably follow a chemical stripe (see Ref. 51). Finally, in Sec. VI, we summarize our findings. II. MODEL AND THEORETICAL APPROACH Assuming a stationary reference frame in which the instantaneous position of the Janus particle is x0= (xp,yp,h), we consider an impermeable spherical colloid of radius Rwith a caplike region covered by a catalyst, which promotes the chemical reaction in the surrounding solution, as schematically depicted in Fig. 1. The orientation ˆ dof the particle is directed along the axis of symmetry from the catalytic region to the particle center (Fig. 1). In this study, we focus on the particular case in which the net result of the chemical reaction can be approximated as the generation of a solute, i.e., the catalytic cap acts effectively as an emitter of the solute, and in which the reaction does not lead to solvent depletion near the catalytic cap. This scenario corresponds approximately to Pt-catalyzed decomposition of hydrogen peroxide (in aqueous solution) into water and oxygen,13,54 where oxygen plays the role of the product (solute) and the solvent is a binary liquid mixture of H2O and H2O2, altogether forming a solution. Nonsymmetric coverage of the particle by the catalyst, acting as a source of solute, gives rise to a nonuniform and time dependent composition of the solution, which is, in general, characterized by the solute (O2) number density c(x,t). We assume that the Péclet number Pe ≡U0R/D, where U0is a characteristic velocity of the particle with radius Rand Dis the diffusion coefficient of the solute molecule (O2), is small such that the convection of the solute molecules can be disregarded compared to its diffusive transport. Consequently, at each instantaneous position of the particle, the solute number density can attain its steady-state, which is governed by the Laplace equation ∇2c= 0. We assume that c(|x|→∞) = 0 and employ the following boundary conditions at the particle surface with outward normal ˆ n: −Dˆ n⋅∇c={κ,xs∈catalytic cap, 0, xs∈inert face, where κis a constant rate per area of the emission of the solute and xsis the location on the surface. At the planar wall placed at z= 0, we impose the no-flux boundary condition ˆ z⋅∇c=0, where ˆ zis the surface normal pointing into the solution. According to the classical theory of neutral diffusiophoresis,33 the nonuniform solute number density c(x) gives rise to lateral gradients of the osmotic pressure along the surface of the particle as well as along the surface of the confining wall. The lateral gradient, which is concentrated only within a thin interfacial layer of molecular thickness, drives interfacial flow around the particles and along the surface of the wall, which are accounted for as an effective slip boundary condition vs(xs) = −b(xs)∇||cfor the outer hydrodynamic flow (i.e., outside of the diffuse interfacial region). ∇|| denotes the projection of the gradient operator onto the corresponding local tangential plane of the surface of the particle or the wall, and b(xs) is a material dependent parameter which encapsulates the details of the effective solute-surface interactions. Furthermore, we assume that for typical velocities U0in selfphoresis the Reynolds number obeys Re ≡ρU0R/η≪1, where ρand ηare the mass density and the viscosity, respectively, of the solution. With this the flow field u(x) and the pressure field P(x) in the outer region satisfy the incompressible, force free Stokes equations −∇P+η∇2u=0 (1) and ∇⋅u=0, (2) subject to the boundary conditions u(xs)=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ U+Ω×(xs−x0)+vs(xs),xs∈on the particle, vs(xs),xs∈on the wall, 0,∣xs∣→∞. (3) In the above equations, Uand Ωare unknown translational and angular velocities of the particle, respectively, which are determined by imposing that the particle is force- and torque-free. Exploiting the linearity of Eqs. (1)–(3), we make the ansatz U=Uws+Usd and Ω =Ωws+Ωsd, where the superscripts indicate the respective velocity contributions stemming from wall slip and self-diffusiophoresis (particle slip). Physically, Uwsand Ωwsoriginate in the chemiosmotic flows induced at the surface of the confining wall by the gradient of the solute concentration. These interfacial flows drive flow in the bulk solution, coupling back to the particle and affecting its motion. The quantities Usd and Ωsd in the presence of an inert uniform wall have been studied in detail in Ref. 45, where it was shown that Usd depends only on hand ˆ d⋅ˆ z. Therefore, in order to simplify the analysis, we restrict ˆ dto the x–yplane (ˆ d⋅ˆ z=0) and take hto be constant. In principle, such wall-bounded 2D motion may also be achieved by using magnetic fields and particles containing a magnetic core.8In the following, we set Usd =Usd ˆ d, with Usd treated as an input parameter. We recall that for Usd >0 (Usd <0), the particle moves away from (toward) its cap when it is in the bulk fluid due to the repulsive (attractive) interaction between the particle and the solute.33 Additionally, by symmetry, one has Ωsd z=0. The trajectory of the particle is obtained by numerical integration of the coupled equations ˙ xp=Usd cos(ϕ)+Uws x(xp,ϕ), ˙ yp=Usd sin(ϕ)+Uws y(xp,ϕ),(4) J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-3 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp and ˙ ϕ=Ωws z(xp,ϕ). (5) We calculate Uwsand Ωwsby employing the Lorentz reciprocal theorem, expressed in Eq. (A1).55 The details of these calculations are provided in Appendix A. As the “unprimed” problem, we chose the one for the six unknowns V≡(Uws,Ωws), requiring six “primed” subproblems. For the primed subproblems (u′(j),σ′(j)), indexed by j, we consider a passive colloid subject to an external normalized force in the ˆ x,ˆ y, or ˆ zdirection for j= 1, 2, 3, or an external normalized torque in the ˆ x,ˆ y, or ˆ zdirection for j= 4, 5, 6. For each of the cases j= 1, ..., 6, we impose no-slip boundary conditions at the planar wall and u′(j)= 0 at infinity. At the particle surface, there is a no-slip condition u′(j)=U′(j)+Ω′(j)×(x−x0), where U′(j)and Ω′(j)are the translational and angular velocities, respectively, of the particle driven by the external force or the external torque in subproblem j. The expressions for the fluid stresses σ′(j) and for the velocity field u′(j)(j= 1, ..., 6) of the auxiliary solution of the Stokes equation (required by the Lorentz theorem) are provided in Appendix B. Finally, the approximation used for determining c(xs), which, in turn, is needed to calculate the phoretic slip boundary conditions vs(xs) = −bw(xs)∇||c(xs), is presented in Appendix C. Taking into account only the contributions of the monopole (mp) and dipole (dp) terms of the number density [see Eqs. (C3) and (C5)] to the particle velocity, we write Uws≈Ump +Udp and Ωws≈Ωmp +Ωdp. Because of the assumption that the self-propulsion proceeds in the x–yplane, only the components Ux,Uy, and Ωzare of interest here, corresponding to the components V1,V2, and V6of the generalized velocity V. Below, the monopole and dipole contributions to these velocity components are calculated for a homogeneous substrate, a chemical step, and a chemical stripe. In each case, the components of the stress tensor σ′(j)|z=0 required for the calculation of Ux(j= 1), Uy(j= 2), and Ωz(j= 6) are provided by Eqs. (B5), (B6), and (B12), respectively. III. SURFING ABOVE A UNIFORM SUBSTRATE For a uniform substrate with surface mobility bw, Eq. (A4) yields Vj=bw∫−∞ −∞ dx ∫∞ −∞ dy (∇∣∣c⋅σ′(j)⋅ˆ z)∣z=0. (6) The monopole contributions are obtained by replacing ∇||cin Eq. (6) with the corresponding expression in Eq. (C3). The result is Ump x=Ump y=0, Ωmp z=0. (7) This result is expected because in the plane of the wall the number density distribution due to a monopole above the wall is radially symmetric around (xp,yp). Therefore, the flow it induces cannot drive translations parallel to the plane or in-plane rotations of ˆ d. Similarly, the dipole contributions are obtained by replacing ∇||cin Eq. (6) by the corresponding expression in Eq. (C5). After carrying out the ensuing integrals, we obtain Udp x=−bwR3∣α1∣ 16Dh3cos(ϕ), (8) Udp y=−bwR3∣α1∣ 16Dh3sin(ϕ), (9) and Ωdp z=0, (10) with α1=−3κ/4 [see Eq. (C1)], where ϕis the angle between ˆ dand ˆ x. Accordingly, these results show that, above a uniform substrate, the dipole contribution drives chemiosmotic “surfing,” i.e., translation in the −sgn(bw)ˆ ddirection: Udp =−bwR3∣α1∣ 16Dh3ˆ d. We note that “surfing” can change the inert-forward or catalystforward character of the motion in direction ˆ dnear a surface as compared with that observed in the bulk; here, the bulk motion is represented by the parameter Usd. For such a change to occur, one must have |bw|≫|bp|, where b(xs) = bpat the particle surface; we recall that Usd ∝bp. The occurrence of this special case requires that the particle and substrate materials have very distinct strengths of interaction with the solute. IV. CHEMICAL STEPS We first consider a wall with a chemical step between two materials such that b(xs)=bl wfor x<0 and b(xs)=br wfor x>0. We introduce ζ≡br w/bl win order to quantify the strength of the chemical contrast between two regions of the wall. Accordingly, Eq. (A4) is evaluated piecewise Vj=bl w∫0 −∞ dx ∫∞ −∞ dy (∇∣∣c⋅σ′(j)⋅ˆ z) +br w∫∞ 0dx ∫∞ −∞ dy (∇∣∣c⋅σ′(j)⋅ˆ z). (11) Following the line of the derivations in Subsection III, after straightforward but cumbersome algebra, we obtain the contributions from the monopole term, Ump x=3hR2α0 16D(br w−bl w)(h2+ 2x2 p) (h2+x2 p)5/2(12) {with α0=κ/2 [see Eq. (C1)]} and Ump y=0, Ωmp z=0, (13) and from the dipole term, Udp x=∣α1∣R3 256Dh3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣−8(br w+bl w)+(bl w−br w) ×(−25 h4+ 28h2x2 p+ 8x4 p)x3 p (h2+x2 p)7/2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦cos(ϕ), (14) Udp y=∣α1∣R3 256Dh3 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣−8(br w+bl w)+(bl w−br w) ×(8x4 p+ 20h2x2 p+ 3h4)xp (h2+x2 p)5/2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦sin(ϕ), (15) J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-4 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp and Ωdp z=−3hR3∣α1∣ 64D(bl w−br w) (h2+x2 p)5/2sin(ϕ). (16) As expected, the results do not depend on ypbecause the system exhibits translational symmetry along the y-direction. We emphasize that Ωdp zis mirror symmetric with respect to xp= 0, which is a general property of Ωws z, valid for any contribution from higher multipole moments.51 It turns out that precisely the same arguments can be applied to Ump xin order to infer that it is an even function of xp. Further details are provided in Appendix D. The x-component of the monopole term Ump xvanishes for |xp| →∞, while limxp→+∞Udp x=−∣α1∣R3br wcos(ϕ)/(16Dh3)≠0 and limxp→−∞ Udp x=−∣α1∣R3bl wcos(ϕ)/(16Dh3)≠0, i.e., the particle “senses” an approximately uniform surface beneath itself. This asymptotic expression coincides with the one in Eq. (8) giving the dipole contribution at a uniform surface. Figure 2 reports on Uws x≃Ump x+Udp xat fixed orientations ϕ= 0○[Fig. 2(a)] and ϕ= 180○[Fig. 2(b)], as well as on Uws y ≃Udp y[Fig. 2(c)] and on Ωdp z[Fig. 2(d)], both at fixed orientation ϕ = 90○, as functions of xp. The velocities are presented in units of U0≡2∣bl w∣α0/D. The leading order monopole contribution always drives translation of the particle toward that side of the wall which exhibits the larger value of bw, which in the present case is the left (less repulsive) side. The motion away from the step {i.e., Uws x>0 as observed for xp>0 and at ϕ= 0○[Fig. 2(a)]} is due to the dipole term in Eq. (14), which is independent of xp. Furthermore, the dipole contribution to the angular velocity in Eq. (16) drives rotation of the catalytic cap of the particle toward the side of the wall with the larger value of bw(i.e., the left, less repulsive side). A. Phase portraits in the (ϕ,xp) plane for inert-forward colloids Figure 3 reports the effects of the chemical contrast ζat fixed Usd =(Usd ⋅ˆ d)>0 (inert-forward) on the topology of the phase portraits in the (ϕ,xp) plane. We assume that br w<bl w<0. For strong enough chemical contrast, we find that inert-forward particles can dock at the chemical step. The corresponding stable fixed points are depicted in Figs. 3(a),3(b), and 3(d) as white triangles. The docking occurs at xp=xd p<0 and ϕ= 0○, i.e., the particle is stationary above the left (i.e., less repulsive) side of the surface, with ˆ dperpendicular to the chemical step. The dependence of xd pon ζ>0 is shown in Fig. 4 by red lines representing various values of the separation hbetween the colloid center and the wall. The black curves in Fig. 4 show the dependence on ζof the location of the saddle fixed points, indicated as white circles in Figs. 3(a),3(b), and 3(d). At a certain value of ζ>0, the two fixed points collide and “annihilate” each other [see Fig. 3(c)]. The location of this bifurcation point, as a function of ζ, may be tuned by varying h. The docking state results from the interplay between the dipolar rotation term Ωdp z[Eq. (16)] and the sum of the monopole term Ump x [Eq. (12)] and the self-diffusiophoresis contribution Usd. The rotation induced by the chemical step orients the particle at ϕ= 0○such that the black cap completely faces the region of weaker repulsion (orange region in Fig. 1). With this orientation, Ump xdrives the particle away from the step into the orange region, while Usd is directed toward the step. At a certain condition, the three contributions balance each other, leading to the motionless docking state. As a condition for docking, we identify the relation Utot x≡Usd x+Ump x0+Udp x0≲0, which implies that the particle cannot cross the step; Ump x0and Udp x0 are the monopole [Eq. (12)] and dipole [Eq. (14)] contributions, respectively, to Uws xat xp= 0, and Usd x=Usd cos(ϕ), with Usd FIG. 2. [(a) and (b)] Chemiosmotic contribution Uws x≃Ump x+Udp x[see Eqs. (12) and (14)] to the particle translational velocity in the x-direction as a function of xpfor particle orientation ϕ= 0○and ϕ = 180○, respectively. (c) Chemiosmotic contribution Uws y≃Udp y[see Eq. (15)] to the particle translational velocity in the y-direction as a function of xpfor a particle oriented parallel (ϕ= 90○) to the chemical step. (d) Angular velocity, calculated from Eq. (16) of a particle oriented parallel (ϕ= 90○) to the chemical step as a function of xp. The results are obtained for br w/bl w=4, bl w<0, and are presented in units of U0≡2∣bl w∣α0/D. The two regions of the chemical step are color-coded. J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-5 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. 3. Inert-forward Janus colloid with Usd ⋅ˆ d=0.1 ×U0at a chemical step, where Usd is the self-diffusiophoretic component of the particle velocity and ˆ dis the direction of the self-diffusiophoretic motion. Effects of the wall chemical contrast on the topology of the phase portrait in the (ϕ,xp) plane. The wall contrast is varied by varying ζ=br w/bl w>0at constant negative bl w. Here, br wand bl ware the surface mobility coefficients for the right and the left part of the wall, respectively. The top row corresponds to h= 1.1 ×Rwith (a) ζ= 4, (b) ζ= 3.5, and (c) ζ= 3. The bottom row has h= 1.25 ×Rwith (d) ζ= 4, (e) ζ= 3.5, and (f ) ζ= 3. The white triangles in (a), (b), and (d) at ϕ= 0○and for xpslightly below zero depict the stable fixed points which we call “docking” states. The white circles in (a), (b), and (d) at ϕ= 0○and for xpslightly above zero depict the saddle fixed points. The five diamond symbols in (c) give the initial conditions corresponding to the trajectories shown in Fig. 6. The background color encodes Ux/U0, where U0≡2∣bl w∣α0/D. being an input parameter. The docking condition Utot x=0 yields the following estimate for the threshold value Usd c: −Usd ccos(ϕ)=3α0R2(br w−bl w) 16Dh2−∣α1∣R3(br w+bl w) 32Dh3cos(ϕ). (17) If bl w>br w, one has ϕ= 0○for the docking configuration (see Fig. 3), whereas for bl w<br w, one has ϕ= 180○at the docking configuration. Accordingly, we obtain Usd c=3α0R2∣bl w−br w∣ 16Dh2+∣α1∣R3(br w+bl w) 32Dh3, (18) which renders a line in the (Usd,h) plane, separating docking and crossing particle states (see Fig. 5). It is instructive to take a closer look at the structure of the phase portraits and at the shape of some representative particle trajectories in the (ϕ,xp) plane. For sufficiently strong chemical contrast ζ, as in Figs. 3(a),3(b), and 3(d), all trajectories can be classified into three groups. The first group contains the trajectories leading to the docking fixed point, marked by white triangles in Figs. 3(a),3(b), and 3(d). All particles approaching the step from the left and with orientation ϕ<π/2 will dock. The first group also contains a bundle of trajectories with π/2 <ϕ≲2 and which approach the step from the right. In the second group, which includes all trajectories with ϕ≳2, trajectories cross over the step from the right. Finally, the third group consists of trajectories according to which the particle is initially located on the right side of the step and is oriented away from the step, as in the upper left corner of Figs. 3(a),3(b), and 3(d). In this case, the particle simply moves directly away from the step and escapes. Decreasing the chemical contrast ζ, the docking state and the saddle point approach each other in the (xp,ϕ) plane, and eventually collide and annihilate. For ζbelow the annihilation value, as in J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-6 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. 4. The dependence of the x-location of fixed points on the chemical contrast ζ≡br w/bl wfor an inert-forward Janus colloid with Usd =Usd ⋅ˆ d=0.1×U0and for several values of the distance hfrom the wall. Here, Usd is the self-diffusiophoretic component of the particle velocity and ˆ dis the direction of the self-diffusiophoretic motion. The surface mobility coefficient of the left side of the substrate bl wis kept constant and negative: U0≡2∣bl w∣α0/D. The open triangles and open circles on the curve h= 1.1 ×Rcorrespond to the systems shown in Figs. 3(a) and 3(b), while the black solid circle corresponds to the system in Fig. 3(c). Figs. 3(c),3(e), and 3(f ), the phase portraits reveal four groups of trajectories: reflected trajectories, trajectories which cross from the right, direct escape trajectories, and trajectories which cross from the left. Reflection occurs for particles approaching the step from the right and with orientations in a narrow band π/2 ≤ϕ≲2. One FIG. 5. Upper critical self-diffusiophoretic velocity Usd c, for which a particle can dock at the step, as a function of h/R. The curves are obtained from Eq. (18): U0≡2∣bl w∣α0/Dand ζ=br w/bl w. example of a reflecting trajectory is shown in Fig. 6(a); the corresponding initial position and orientation are marked by the white diamond in Fig. 3(c). For this particular initial condition, the particle penetrates slightly into the orange (less repulsive) region of the substrate before turning around and moving back into the gray region. By increasing the initial orientation ϕas compared to that of the white diamond, we may increase the extent of the penetration into the orange side, up to a maximum of two particle diameters. With a FIG. 6. Representative trajectories of an inert-forward (Usd ⋅ˆ d=0.1 ×U0,U0≡2∣bl w∣α0/D) particle near a chemical step with contrast br w/bl w=3,bl w<0. The diamond symbols match with the initial position xp=x0 pand orientation ϕ=ϕ0of the trajectories in Fig. 3(c): (a) x0 p=10 ×R,ϕ0≃1.7; (b) x0 p=10 ×R,ϕ0≃2.2; (c) x0 p=10 ×R, ϕ0≃3.1; (d) x0 p=−10 ×R,ϕ0≃1.1; and (e) x0 p=−10 ×R,ϕ0≃0.5. The trajectories are shown to scale with the colloid size, i.e., the penetration depth in (a) is indeed given in units of the particle radius. J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-7 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp further increase in the orientation, we find the group of trajectories which cross from the right to the left. This group includes the initial conditions indicated by the brown and yellow diamonds in Fig. 3(c), which correspond to the trajectories shown in Figs. 6(b) and 6(c), respectively. The exit orientations ϕof these trajectories are slightly smaller than the initial orientations. If, starting from the white diamond, we decrease the initial orientation, we encounter the group of directly escaping trajectories [upper left corner of Figs. 3(c),3(e), and 3(f )]. Finally, for trajectories which start in the lower left corner of Figs. 3(c),3(e), and 3(f ), the particle crosses over the step with exit orientation ϕ≪1. In other words, the step operates as a trajectory focusing device for particles which cross from the left. This group includes the initial conditions indicated by the blue and red diamonds in Fig. 3(c), corresponding to the trajectories shown in Figs. 6(d) and 6(e). B. Phase portraits in the (ϕ,xp) plane for catalyst-forward colloids Figure 7 shows how the topology of the phase portraits for catalyst-forward (Usd =Usd ⋅ˆ d<0) particles varies with the chemical contrast ζ=br w/bl w. This system has no stable fixed point. The physical reason for this is that now both the monopole and the self-diffusiophoretic components drive the particle away from the step into the orange, less repulsive region of the surface. The dipole contribution to the angular velocity still rotates the particle into the configuration ϕ= 0. We find only saddle points, marked by white circles in Figs. 7(a),7(b), and 7(d), and unstable fixed points, marked by black circles in Figs. 7(c) and 7(d). The dependence of the spatial locations of the saddle points x○ pand of the unstable fixed points x● pon the chemical contrast ζis presented in Fig. 8. The location of the saddle point within the orange area (x○ p<0) with ϕ=π(the catalytic cap facing the gray, more repulsive side of the substrate) depends only weakly on ζwithin the studied range (see the blue curve in Fig. 8). The location x○ pis determined by the balance between the chemiosmotic contribution, which drives the particle away from the step, and the self-diffusiophoretic velocity, which is directed toward the step. The orientation ϕ=πis not stable with respect to Ωdp zwhich favors ϕ= 0. At small enough ζ=ζbif (from bif urcation), this point collides with the upper (x● p >0) unstable fixed point, and both disappear upon further decrease in ζ. FIG. 7. Catalyst-forward Janus colloid with Usd ⋅ˆ d=−0.1 ×U0at a chemical step, where Usd is the self-diffusiophoretic component of the particle velocity and ˆ dis the direction of the self-diffusiophoretic motion. Influence of the wall chemical contrast ζ=br w/bl w>0, with bl w<0, on the topology of the phase portrait in the (ϕ,xp) plane at h= 1.1 ×Rand at (a) ζ= 3, (b) ζ= 2.86, (c) ζ= 2.8, (d) ζ= 2, and (e) ζ= 1.5. Here, br wand bl ware the surface mobility coefficients for the right and the left part of the wall, respectively, and his the particle-wall distance. The white circles in (a)–(d) depict the saddle fixed points; the black circles in (c) and (d) denote the unstable fixed points. The brown and blue diamonds in (a) and the pink diamond in (e) indicate the initial conditions corresponding to the trajectories in Fig. 9. The background color encodes Ux/U0 with U0≡2∣bl w∣α0/D. J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-8 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp each subproblem j, the component F′(j) j, such as F′(1) xor τ′(4) x, must exactly cancel the imposed force or torque because the motion of the particle is overdamped (Re ≪1). It is therefore known a priori that, e.g., F′(1) x=−F′ext and τ′(4) x=−τ′ext. The other, off-diagonal components, such as τ′(1) y, are unknown prior to finding the solution of subproblem j. However, the off-diagonal terms are significant only if the particle is very close to the wall, i.e., if h/R≈1.58 We therefore neglect the off-diagonal terms and obtain F′(j) jVj≈−∫vs⋅σ′(j)⋅ˆ zdS,j=1, ..., 6, (A4) where Vjare the components of the generalized velocity V. [We note that the left-hand side of Eq. (A4) is not a sum over j.] Due to the linearity of the Stokes equation, σ′(j)contains as a prefactor either F′ext or τ′ext, i.e., −F′(j) jTherefore, the arbitrary amplitudes F′ext or τ′ext drop out of the problem. In order to avoid a clumsy notation, in the following, we shall denote σ′(j)/F′(j) jas σ′(j). In order to perform the integral in Eq. (A4) analytically, two expressions are needed: one for vsand the other for σ′(j). For both quantities, we use the “point-particle” approximation in order to obtain analytically tractable expressions. In Appendices B and C, we derive approximate expressions for the surface concentration gradient c(xs) and for the primed fluid stresses σ′(j). The knowledge of c(xs) permits the determination of the phoretic slip boundary conditions vs(xs)=−bw(xs)∇||c(xs). APPENDIX B: AN APPROXIMATION FOR σ′(j)AT THE WALL We obtain an approximate analytical expression for the shear stress σ′(j)by using image solutions for a point force or point torque above a planar wall.59 This approximation neglects the finite size of the particle, i.e., the no slip condition at the particle surface. For the “primed” problems j= 1, 2, 3, we replace the particle by a point force (Stokeslet) pointing into the directions ˆ x,ˆ y, and ˆ z, respectively, located at x0= (xp,yp,h) (i.e., the center of the particle) above a planar wall located at z= 0. The fluid satisfies incompressibility and the Stokes equations. As shown by Blake,59 the governing equations and the no-slip condition at the wall can be satisfied by locating a system of images, which consists of a Stokeslet, a force-dipole, and a source-doublet (see Ref. 59), at the point xI= (xp,yp,−h). For the sake of clarity, we introduce the mapping (1, 2, 3, 4, 5, 6)s →(x,y,z,x,y,z)connecting the indices of the “primed” problem and the corresponding directions of the unit force or torque (see also below), as well as the shorthand notation sj∶=s(j) [i.e., s1∶=s(1) = x,s5∶=s(5) = y, etc.]. The corresponding fluid velocity at an observation point x= (x,y,z) is given by 8πη u′(j) i=(1 r−1 X)δisj+rirsj r3−XiXsj X3+ 2h(δsjαδαl−δsjzδzl) ×∂ ∂Xl[hXi X3−(δiz X+XiXz X3)], (B1) where i∈{x,y,z}, α∈{x,y}, r≡x−x0,X≡x−xI,r≡|r|, X≡|X|, and the Einstein convention of summation over repeated indices is used (here and in the following). We note that the first product of Kronecker delta symbols vanishes if either sjor ltakes the value z, while the second such product contributes only if both sjand ltake the value z. Therefore, the index lin the partial derivative with respect to Xlis taken to be x,y, or z. The pressure is given by 4πP′(j)=rsj r3−Xsj X3−2h(δsjαδαl−δsjzδzl)∂ ∂Xl(Xz X3). (B2) (Although pressure is a scalar quantity, sjappears in the expression for the pressure because its functional form depends on the direction of the point force.) From the velocity and pressure, it follows that the stress tensor σ′=−P′𝟙+η(∇u′+∇u′T)(B3) in the fluid is given by the following expression: 4πη 3σ′(j) ik =rirsjrk r5−XiXsjXk X5−2h(δsjαδαl−δsjzδzl) ×(−h X5δikXl+z X5(Xiδlk +Xkδil) +XiXk X5δzl −5zXiXlXk X7). (B4) This expression approximately recovers the stress from a sphere dragged by an external point force in the presence of a wall at z = 0, although it neglects the finite size of the sphere (represented, concerning hydrodynamics, by a no-slip condition at the surface of the sphere.) Concerning the substitution into the reciprocal theorem, we are interested in the quantities σ′(j) iz ∣z=0, where i∈{x,y}. These quantities are components of the shear stress evaluated at the wall. We obtain σ′(1) iz ∣z=0=−3h 2π rirx r5, (B5) σ′(2) iz ∣z=0=−3h 2π riry r5, (B6) and σ′(3) iz ∣z=0=3h2 2π ri r5. (B7) Now we turn to the “primed problems” j= 4, 5, 6. For these three problems, we consider a point torque oriented into the directions ˆ x,ˆ y, and ˆ z, respectively, located at x0. For a point torque above a planar wall at z= 0, Blake found,59 via the method of images, that the velocity field is 8πη u′(j) i=𝜖isjk(rk r3−Xk X3)+ 2h𝜖ksjz(δik X3−3XiXk X5)+ 6𝜖ksjz XiXkXz X5, (B8) where 𝜖αβγ denotes the Levi-Civita symbol (with the convention that its indices are interpreted as x→1, y→2, and z→3), and as mentioned above, summation over repeated indices is employed here and the following. The pressure is59 J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-15 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp P′(j)=−4η∂ ∂Xk(𝜖ksjzXz X3). (B9) By using the definition [Eq. (B3)] for the stress tensor σ′(j), we obtain the corresponding matrix elements iz (i∈{x,y}) evaluated at the wall, which are needed for substitution into the reciprocal theorem σ′(4) iz ∣z=0=−3 4π δiyh2−riry r5, (B10) σ′(5) iz ∣z=0=−3 4π δixh2−rirx r5, (B11) and σ′(6) iz ∣z=0=3h 4π 𝜖izkrk r5. (B12) APPENDIX C: MULTIPOLE EXPANSION FOR c(xs) For a spherical particle with axisymmetric catalyst coverage in free space (i.e., far from bounding surfaces), the solute number density can be expanded in terms of Legendre polynomials14 cfs(r,θ′)=c∞+R D ∞ ∑ l=0 αl l+ 1(R r)l+1 Pl(cos(θ′)), (C1) where αlare the multipole coefficients of the surface activity (i.e., areal density per time): α(θ′)=∑∞ l=0αlPl(cos(θ′)),ris the vector from the center of the particle to an observation point, r= |r|, and θ′is the angle between rand the vector ˆ doriented along the axis of symmetry of the particle. As defined in the main text, ˆ dpoints from the catalytic cap of the particle to the inert region. For any specification α(θ′) of the particle activity, the coefficients αlare easily calculated. For instance, for the constant-flux model of the activity presented in the main text, one has the monopole coefficient α0 =κ/2 and the dipole coefficient α1=−3κ/4. (The sign of α1is negative due to our choice of the direction of ˆ d.) Now we consider an active particle near a planar wall in a configuration in which ˆ dis parallel to the wall (which occupies the xy plane); thus, one has ˆ d=(cos(ϕ), sin(ϕ), 0). (The conditions, under which such configurations can be realized, are discussed in Refs. 51 and 52.) In order to obtain approximate analytical expressions for the surface gradient ∇||c(xs), we make two approximations: (i) We truncate the multipole expansion for the activity of the particle, and we consider only the monopole and dipole terms. (ii) In order to model the effect of confinement of the solute field by the wall, we place the mirror images of the monopole and the dipole below the wall. We neglect additional reflections of these two images across the particle surface. In Refs. 51 and 52, these assumptions have been validated via comparisons of the theoretically predicted dynamical behavior of the particle with that obtained from numerical solutions of the full problem. We therefore write c(xs)≈cmp(xs) + cdp(xs). Here, c(xs) is the sum of two terms: the field due to a point source (monopole) of the number density located at the center x0= (xp,yp,h) of the particle, plus the field due to an image point source of the number density located at xI= (xp,yp,−h). The second term, cdp(xs), is a contribution from a dipole and its image. The real and image dipoles are likewise located at x0and at the image point xI, respectively, and both have the strength p=−∣α1∣ˆ d. For the monopole term, we obtain cmp(xs)=2α0R2 Dr (C2) and ∇∣∣cmp(xs)=−2α0R2rs Dr3. (C3) Here, xs= (x,y, 0) denotes a point at the wall, rs= (x−xp,y−yp, 0) and r=√(x−xp)2+(y−yp)2+h2. For the dipole term, we obtain cdp(xs)=−∣α1∣R3 D ˆ d⋅rs r3(C4) and ∇∣∣cdp(xs)=−∣α1∣R3 Dr3(𝟙−3rs⊗rs r2)⋅ˆ d. (C5) More details concerning the calculation of the chemiosmotic contribution to the linear and angular velocities of the particle, as well as a discussion of the effect of the nonvanishing size of the particle, can be found in the supplementary material of Ref. 51. APPENDIX D: EVENNESS OF Ump x(xp) FOR CHEMICAL STEPS We provide here a general symmetry argument according to which Ump x(xp)for a chemical step is mirror symmetric with respect to xp= 0 [see Eq. (12)]. By linearity, we can express the monopolar contribution to the velocity as Ump x(xp)=br wgr(xp)+bl wgl(xp)+C. (D1) Without osmotic response (br w=0 and bl w=0), one has Ump x=0, and therefore, C= 0. Furthermore, we know that Ump x(xp)=0 for a uniform wall, i.e., if br w=bl w, gl(xp)=−gr(xp)(D2) and Ump x(xp)=(br w−bl w)gr(xp). (D3) Additionally, the mirror symmetry about x= 0 requires that gl(xp) =−gr(−xp) (see Fig. 14). When this result is combined with Eq. (D2), it yields that gr(xp) is even gr(xp)=gr(−xp). (D4) Applying the mirror symmetry (i.e., the parity) operator with respect to the y-axis one obtains bl w→br w,br w→bl w,xp→−xp, and Ump x(xp)→−Ump x(−xp)so that −Ump x(−xp)=−(br w−bl w)gr(−xp). (D5) Consequently, J. Chem. Phys. 150, 204904 (2019); doi: 10.1063/1.5091760 150, 204904-16 Published under license by AIP Publishing The Journal of Chemical Physics ARTICLE scitation.org/journal/jcp FIG. 14. (a) Schematic illustration of the contribution Ul mp =bwgl(xp)of the left-hand side of the substrate surface z= 0 (shaded area) to the translational velocity of a particle at x=xpfor bl w≡bw. This contribution can be calculated by taking br w=0on the right-hand side of the substrate surface (white area). Ul mp can be either positive or negative; here, Ul mp >0is chosen for illustration. (b) Upon applying a mirror transformation about x= 0 to (a), we find that the resulting contribution has changed sign. 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