Toward Understanding of Self-Electrophoretic Propulsion under

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Cite This: Acc. Chem. Res. 2018, 51, 2998−3005

Toward Understanding of Self-Electrophoretic Propulsion under Realistic Conditions: From Bulk Reactions to Confinement Effects Published as part of the Accounts of Chemical Research special issue “Fundamental Aspects of Self-Powered Nano- and Micromotors”. Michael Kuron,*,‡ Patrick Kreissl,*,‡ and Christian Holm*

Acc. Chem. Res. 2018.51:2998-3005. Downloaded from pubs.acs.org by UNIV OF RHODE ISLAND on 12/19/18. For personal use only.

Institute for Computational Physics (ICP), University of Stuttgart, Allmandring 3, 70569 Stuttgart, Germany CONSPECTUS: Active matter concerns itself with the study of particles that convert energy into work, typically motion of the particle itself. This field saw a surge of interest over the past decade, after the first micrometer-sized, man-made chemical motors were created. These particles served as a simple model system for studying in a well-controlled manner complex motion and cooperative behavior as known from biology. In addition, they have stimulated new efforts in understanding out-of-equilibrium statistical physics and started a revolution in microtechnology and robotics. Concentrated effort has gone into realizing these ambitions, and yet much remains unknown about the chemical motors themselves. The original designs for self-propelled particles relied on the conversion of the chemical energy of hydrogen peroxide into motion via catalytic decomposition taking place heterogeneously over the surface of the motor. This sets up gradients of chemical fields around the particle, which allow it to autophorese. That is, the interaction between the motor and the heterogeneously distributed solute species can drive fluid flow and the motor itself. There are two basic designs: the first relies on redox reactions taking place between the two sides of a bimetal, for example, a gold−platinum Janus sphere or nanorod. The second uses a catalytic layer of platinum inhomogeneously vapor-deposited onto a nonreactive particle. For convenience’s sake, these can be referred to as redox motors and monometallic half-coated motors, respectively. To date, most researchers continue to rely on variations of these simple, yet elegant designs for their experiments. However, there is ongoing debate on the exact way chemical energy is transduced into motion in these motors. Many of the experimental observations on redox motors were successfully modeled via self-electrophoresis, while for half-coated motors there has been a strong focus on self-diffusiophoresis. Currently, there is mounting evidence that self-electrophoresis provides the dominant contribution to the observed speeds of half-coated motors, even if the vast majority of the reaction products are electroneutral. In this Account, we will summarize the most common electrophoretic propulsion model and discuss its strengths and weaknesses in relation to recent experiments. We will comment on the possible need to go beyond surface reactions and consider the entire medium as an “active fluid” that can create and annihilate charged species. This, together with confinement and collective effects, makes it difficult to gain a detailed understanding of these swimmers. The potentially dominant effect of confinement is highlighted on the basis of a recent study of an electro-osmotic pump that drives fluid along a substrate. Detailed analysis of this system allows for identification of the electro-osmotic driving mechanism, which is powered by micromolar salt concentrations. We will discuss how our latest numerical solver developments, based on the lattice Boltzmann method, should enable us to study collective behavior in systems comprised of these and other electrochemical motors in realistic environments. We conclude with an outlook on the future of modeling chemical motors that may facilitate the community’s microtechnological ambitions.

1. INTRODUCTION

simple artificial model systems. In this Account, we will focus on two such models. Paxton et al.3 introduced a redox motor that was capable of autonomous motion, the desired biological aspect, by converting a simple chemical fuel, hydrogen peroxide, on a gold−platinum bimetallic particle as shown in Figure 1. This motion was quickly ascribed to self-electrophoresis,4 a process

Nano- and micromachines promise a bright future for technical and medical applications, but hold many technical challenges for today’s scientists. Inspired by the many examples found in nature, an interdisciplinary community of biologists, chemists, and physicists strives to understand, recreate, and control the complexity and cooperative behavior exhibited by spermatozoa, bacteria, and flocks of birds or schools of fish.1,2 To understand the underlying nonequilibrium physics, we need © 2018 American Chemical Society

Received: June 15, 2018 Published: November 12, 2018 2998

DOI: 10.1021/acs.accounts.8b00285 Acc. Chem. Res. 2018, 51, 2998−3005

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Accounts of Chemical Research

may become instrumental in determining the origin of emergent collective behavior in self-electrophoretic systems, and how one can extend the theory to be applicable for nanosized systems where fluctuations might become relevant.

2. MODELING ELECTROPHORESIS In this section, we introduce the electrokinetic equations for bulk electrolytes, add a model for chemical self-propulsion, and discuss their application to describe a simple redox motor. For each dissolved species k, the flux is given by

Figure 1. Schematic of a catalytic bimetallic nanorod indicating electrochemical reactions. Pt acts as the anode and Au as cathode, with the resulting positive and negative charge density indicated by color. The electric field follows the approximated field lines. Reproduced with permission from ref 10. Copyright 2010 American Physical Society.

D jk (r, t ) = −Dk ∇ρk (r , t ) − k zkeρk (r, t )∇Φ(r, t ) kBT≠ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÆ ´ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖ diff jk

+ ρk (r, t )u(r, t ) ´ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖadv ≠ÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÖÆ

wherein the surface reactions create a heterogeneous distribution of ions in the solution, leading to an electric field which propels the charged swimmer.5 Soon afterward, Howse et al.6 succeeded in creating an alternative, also powered by hydrogen peroxide, where the decomposition took place on a platinum-coated hemisphere of an otherwise chemically inert particle. The theoretical work by Golestanian et al.7,8 on particles driven by gradients of neutral solutes, so-called self-diffusiophoresis, provided the background for this experiment. There are other motors and mechanisms that we will not discuss further.1,2,9 There is certainly a need to fully understand the way these two chemical motors achieve their motion. Many advances were made in this regard, with the theory for the propulsion of redox motors having the most predictive power early on.4,10 For half-coated motors, the state is not as clear, especially since new experimental evidence pointed in the direction of selfelectrophoresis; their speed proved to depend strongly on salt concentration. Currently, it is widely believed that the selfpropulsion of both redox and monometallic half-coated motors involves ionic species.11−13 Self-diffusiophoresis, where the heterogeneous solute distribution created by surface reactions leads to propulsion due to molecular interactions between solute and surface, generally does not suffice to explain the propulsion observed in experiment.11 A contribution by selfelectrophoresis, on the other hand, can never be ruled out as water always contains ionic impurities14 and most surfaces in aqueous solutions acquire charges. However, the specifics of how self-electrophoresis comes about in half-coated swimmers continues to challenge experiment and theory.15 When modeling even smaller swimmers, for example, to explain the enhanced diffusion of nanomotors16 and enzymes17 when brought into contact with their chemical substrate, a self-electrophoretic description fails to capture experimental observations, possibly due to the influence of thermal effects18 or increased catalytic activity.13 Finally, many of the experimental observations can be captured qualitatively or even semiquantitatively by models that only account for self-diffusiophoresis.6,19,20 In what follows, we give an overview of the theory of selfelectrophoresis, show working examples, and demonstrate how the inclusion of the bulk fluid as an active system can produce self-electrophoretic propulsion even for half-coated motors. We elucidate how the presence of surfaces can induce subtle effects on large length scales, even in the presence of only micromolar salt concentrations, for the case of an electroosmotic pump. We also point out state-of-the-art methods by which the theoretical models can be solved numerically, which

(1)

jk

which corresponds to a change in density via the continuity equation ∂ρk ∂t

(r, t ) = −∇·jk (r, t )

(2)

The total flux consists of a diffusive and an advective (jkadv) contribution, where ρk(r, t) is the kth species’ concentration, Dk is the diffusion coefficient, and zk is the valency; Φ(r, t) is the electrostatic potential, u(r, t) is the fluid flow velocity, kBT is thermal energy, and e is the unit charge. The electrostatic potential is determined by Poisson’s equation (jdiff k )

∇2 Φ(r, t ) = −

1 ε

∑ zkeρk k

(3)

where ε = ε0εr, with relative dielectric permittivity εr and vacuum permittivity ε0, is assumed to be spatially homogeneous. The Debye length describes the characteristic length of electrostatic screening: κ −1 =

εkBT ∑k zk 2e 2ρk

(4)

Synthetic chemical motors typically fall into the nano- to micrometer size range; therefore, the fluid dynamics are welldescribed by the incompressible Stokes equations: η∇2 u(r, t ) = ∇p(r, t ) − fext(r, t )

(5)

∇·u(r, t ) = 0

(6)

with η the dynamic viscosity, p(r, t) the pressure, and fext(r, t) the force density applied to the fluid by the solutes:21 fext(r, t ) = kBT ∑ k

jdiff (r , t ) k Dk

(7)

which accounts for the electrostatic force acting on the fluid through the presence of charge excess. The above equations, together with appropriate boundary conditions, can be used to model the behavior of the simple redox motor4,10 shown in Figure 1. On the Pt side, H2O2 is decomposed, producing both electrons and protons: H 2O2 → O2 + 2H+ + 2e− 2999

(8a) DOI: 10.1021/acs.accounts.8b00285 Acc. Chem. Res. 2018, 51, 2998−3005

Article


theory in the next section by taking the activity of the bulk into account.

Electrons are conducted through the rod to the gold cap, while protons move through the fluid. On the gold cap, both participate in a different H2O2-decomposing reaction: H 2O2 + O2 + 6H+ + 6e− → 4H 2O

3. THE IMPORTANCE OF BULK REACTIONS Self-electrophoresis has been shown to be the appropriate propulsion mechanism for the bimetallic rod of section 2. However, recent results on spherical particles half-coated with Pt suggest that ionic currents are part of their driving mechanism too.11,12 In a recent paper,13 some of us argued that in fact all H2O2-powered swimmers in aqueous solutions are at least partially self-electrophoretically driven due to the presence of inevitable ionic gradients arising from the selfdissociation of the hydrogen peroxide solution, which we refer to as bulk reactions. These couple to surface reactions and can drastically modify the self-electrophoretic propulsion mechanism. Here we discuss only the main results and refer the interested reader to the original publication, ref 13. Hydrogen peroxide autodissociates more strongly than H2O according to

(8b)

Moran et al.10 provided a theoretical description of the system by incorporating the catalytic reaction as a boundary condition to eq 2. They prescribe H+ fluxes perpendicular to the surface, out of the Pt cap and into the Au cap, while the electron flux inside the rod is not explicitly considered: n̂ ·jH+ Pt ‐ surface = +jH+

(9a)

n̂ ·jH+ Au ‐ surface = −jH+

(9b)

n̂ ·jk surface = 0

k ≠ H+

(9c)

Furthermore, the rod’s surface is a no-slip velocity boundary, u surface = 0, and has a constant zeta potential, Φ surface = ζ. This implies that one works in the comoving frame of the swimmer and that the propulsion speed is obtained at infinity, u∞ = −Uswimmer. There, the ion concentrations assume their bulk ∞ values, ρk = ρ∞ k , and p = p , the atmospheric pressure, with the electrostatic potential Φ decaying to zero. While an analytical treatment of the electrokinetic equations is possible for spherical bimetallic swimmers via linearization,8,12,22 for the geometry of a Au−Pt nanorod, this has to be done numerically. Figure 2 shows experimental values and

H 2O2 F H+ + HO2−

(10) −1

The Debye screening length of a typical 3 mol L H2O2 solution is κ−1 ≈ 100 nm and violates the thin-Debye layer assumption common in analytical calculations. Since surface reactions on the Pt side create gradients in hydrogen peroxide, these will inevitably also produce gradients in the ionic dissociation products. Surface and bulk reactions together produce a new screening length, dubbed reactive screening.23 As an illustrative example, a simple 1D diffusion model with a source term at the surface and a sink term in the bulk, together with its exponentially decaying solution, is shown in Figure 3.

Figure 3. Schematic of the 1D molecular screening model introduced in the text. The wall acts as source for molecules (red triangles), which diffuse with D (black arrow). In the bulk, molecules are consumed with a rate γ (blue symbol), which leads to an exponential decay of the concentration. Reproduced with permission from ref 13. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

Figure 2. Velocity of the nanorod as a function of H 2 O 2 concentration (or equivalently, the ion flux in units of Nernst’s diffusion-limited current) for different surface potentials ζ. Open symbols show simulation results. Reproduced with permission from ref 10. Copyright 2010 American Physical Society.

The swimmer radius a and the reactive screening length q−1 give a new control parameter qa for reactive screening. Typical experimental values yield q−1 ≈ 74 nm for 3 mol L−1 H2O2, demonstrating that experimental systems are between the reactionless and the fully reactive limits. The surface reactions drive the bulk reaction out of equilibrium, and the system responds by reducing the effect of the perturbation according to Le Chatelier’s principle. Incorporating this effect not only modifies the self-electrophoretic propulsion of the bimetallic motor, but even adds a self-electrophoretic contribution to the half-coated motor. The corresponding effects for three types of model swimmers are illustrated in Figure 4, where So describes

simulation results for the swimming speeds at different fuel concentrations and surface potentials ζ, where the latter were obtained using the finite element method (FEM). Experiments and simulations show good agreement for ζ = −25 mV, which is a reasonable value for Au/Pt particles in aqueous solutions.10 The swimming behavior of this catalytic bimetallic swimmer seems to be quantitatively explained by a simple selfelectrophoretic mechanism. We will provide a more refined 3000


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Figure 5. (a) Boundary conditions for external electrophoresis (upper) and self-electrophoresis (lower) of positively charged particles. The direction of propulsion is indicated by thick arrows. (b) Henry’s function f(κa) determines the particle mobility in an external electric field (black solid line). The equivalent function for self-electrophoresis F(κa) is drawn as red dashed line. Reproduced with permission from ref 13. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

Figure 4. Upper half-pictures (a, b, c) show the system without and the lower ones (d, e, f) with bulk reactions. Colored arrows represent fluxes of chemical species H2O2 (purple), H+ (red), and HO2− (blue). The fluxes’ intensities are roughly indicated by the arrows’ thickness. Speed and direction of particle propulsion are indicated by white arrows (length corresponds to relative speed; x = no propulsion). The approximate extent of the reactive screening length, q−1, is indicated by a dashed line. Reproduced with permission from ref 13. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

a nominally neutral swimmer like a half-coated sphere, which would not self-electrophorese without bulk reactions, S+ models a bimetallic sphere, and S= is a swimmer powered by ionic diffusiophoresis, that is, it is itself composed of the fuel and the electric field arises due to the different diffusivities of the ions into which it dissociates; for details refer to the figure caption. The theory predicts an altered self-electrophoretic propulsion speed that can conveniently be cast in the form of U = U SM(j s , c salt , σ , ...)F(κa)B(qa , ...)

Figure 6. (a) Propulsion speed of a swimmer in the presence of chemical reactions. The swimmer is powered by a proton current. Parameters were chosen to match typical experiments on microparticles (red solid, theory; blue circle, experiment25) and on a nanoswimmer16 (red dashed, theory; black triangle, experiment). Surface parameters were kept the same. (b) Speed of a S= swimmer plotted against experimental data (green squares).19 Reproduced with permission from ref 13. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

(11)

where U SM is the “standard self-electrophoretic model” propulsion speed, assuming the thin-Debye layer limit without bulk reactions.8 USM depends on, among other parameters, the surface reaction rates js, the salt concentration csalt, and the surface charge density σ. The factors F and B account for realistic electrostatic screening (κa) and bulk reactions (qa), respectively. An interesting detail is the generalization of the Henry function,24 F, from external electrophoresis to selfelectrophoresis, see Figure 5. One result of this self-electrophoresis description is the swimming speed as shown as a function of the radius a in Figure 6, where panel a displays the speed for a bimetallic spherical swimmer and panel b the speed of Pt-half-coated microswimmers. It is noteworthy that a scaling of a3 is predicted for sufficiently small particles; however, this is a purely electrostatic effect and independent of the bulk reactions. Unfortunately experimental data for smaller swimmers in bulk that could test this prediction is scarce. Reference 16 observes that nanoswimmers move much faster than equivalent microswimmers, contrary to the theoretical prediction, suggesting that either the surface reaction rates are larger on nanoswimmers or that additional propulsive effects are present.13 This analysis demonstrates that not only might self-propulsion be a simple surface-driven phenomenon but bulk reactions may significantly modify the swimming mechanisms, even to the point where neutral swimmers can generate ionic gradients and thereby electric fields (model So).

Nevertheless we should keep in mind that the theory describes bulk swimmers, while all experiments are performed in quasi2D confinement, introducing additional chemical, electrical, and hydrodynamical surface effects. We will see in the next section how the presence of surfaces can noticeably change the swimmers’ motion.

4. INFLUENCE OF SURFACES AND CONFINEMENT Most theoretical descriptions of phoretic swimmers make the assumption of a bulk environment. However, this idealized situation is rarely reproduced experimentally, where swimmers usually reside near (charged) surfaces. These can strongly impact the flow fields and electric potentials, as has previously been investigated for self-diffusiophoresis26 and self-electrophoresis.27 Such investigations are typically restricted to the case of a single wall, while we will show a microfluidic pump confined between two walls. This additional confinement has a significant impact on the flow. The considered microfluidic pump is made from an ion-exchange resin glued to the glass 3001


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Accounts of Chemical Research surface of a cylindrical sample cell and operates in close-to fully deionized water,14 see Figure 7a.

Figure 8. Experimentally measured values for the tracer speed, UPS, as a function of the distance r from the ion-exchange resin’s center. Changing the sample cell height H modifies the exponent of the observed power-law decay. Reproduced with permission from ref 14. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

using FEM simulations, exploiting axisymmetry, on a quasitwo-dimensional domain. Full details of the model are given in ref 14. It includes three types of ions: protons (H+), potassium (K+), and chloride (Cl−) ions, with bulk concentrations ρ∞ K+ = 1 μ mol L−1 (in line with estimates from the experiment), ρ∞ H+ = ∞ ∞ 0.1 μ mol L−1 (pH = 7), and ρ∞ − = ρ + + ρ + (charge neutrality Cl H K in bulk). Ion exchange at the pump’s surface is modeled via flux boundary conditions. The rate constant for the K+−H+ exchange was determined by fitting to experimental data. Nopenetration conditions are applied at the bottom and top walls. For the Poisson eq 3, a constant surface charge density condition is used at all surfaces, choosing the charge density such that the surface potential matches the experimentally measured zeta potential ζ ≈ −0.1 V. For Stokes eq 5, no-slip is prescribed on all solid surfaces, whereas at the outer edge of the domain, a no-normal-stress boundary condition is used. Our results confirm that the exchange mechanism combined with the micromolar concentration of ionic impurities leads to fluid flow of the experimentally observed magnitude of micrometers per second, while reproducing the experimentally observed change in the power-law due to confinement. Furthermore, the induced fluid flow is long-ranged, see Figure 9 for a visualization.

Figure 7. (a) Top and side view sketching the ion-exchange resin pump in the cylindrical sample cell of radius R and height H. A zoomin on the resin of radius rR (red) also indicates a polystyrene tracer (green) following fluid flow along a (blue) flow line. Radial distance r is measured from the pump’s center. (b) Top-view microscopy image of tracers flowing toward an ion-exchange resin (center of image). Experimentally measured tracer trajectories are indicated by blue arrows. Reproduced with permission from ref 14. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0.

While there are no decomposition-type surface reactions present like in section 3, the exchange mechanism has a comparable effect: fuel (cations) is consumed, while producing (releasing) an equally large amount of product (protons). Consequently, the effects observed in the pump system are likely also applicable to catalytic swimmers in close proximity to a wall or in confinement. In close-to deionized water, fluid flow directed toward the pump evolves close to the lower wall, as can be seen by the trajectories of polystyrene (PS) tracer particles, compare Figure 7b. The induced fluid flow on the order of micrometers per second is acting for periods exceeding 24 h and over hundreds of micrometers. Trace amounts of cations present in the fluid are exchanged for protons by the resin bead, creating an electro-osmotically driven pumping mechanism. The difference in mobility between the contaminant ions and the protons induces a diffusion potential that counteracts the resulting charge separation, inducing an electro-osmotic flow. The tracer velocities, UPS, in Figure 8 display a far-field power-law decay, which is 3D for large sample cell heights, H (UPS ∝ r−2), and approaches 2D scaling for smaller H that are still ≫rR (UPS ∝ r−1). To verify that electo-osmosis can indeed explain the experimentally observed fluid flow on the order of micrometer per second, the electrokinetic equations of section 2 are solved

Figure 9. FEM simulation result for the fluid flow in a H = 1 mm sample cell. The resin is located in the bottom-left corner. Color indicates the magnitude of the local fluid velocity, while arrows show the direction of flow. Adapted with permission from ref 14. Published by Royal Society of Chemistry 2017. Licensed under CC BY 3.0. 3002


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Figure 10. LB simulation result for the fluid flow in a H = 1 cm sample cell. The resin is located in the bottom center. Color indicates the magnitude of the local fluid velocity, while white stream lines show the direction of flow. The black grid lines indicate blocks of the same number of lattice cells, corresponding to a higher resolution where flow gradients are stronger.

5. MODELING APPROACH TO STUDY COLLECTIVE EFFECTS For three-dimensional geometries, time-dependent problems, and swimmers moving with respect to the simulation grid, FEM is too expensive in terms of computational effort. This makes it generally unsuited to study problems beyond the interaction of two swimmers, like the bimetallic, self-electrophoretically propelled nanorods of ref 28 that were observed to form pairs from two swimmers moving in the same direction. Extension from pairwise interactions to entire swarms of catalytic swimmers that show phenomena like clustering29 requires more versatile numerical methods. One alternative method is based on the lattice Boltzmann (LB) algorithm, see ref 30 for a comprehensive overview. Other simulation techniques with explicit solvent and implicit ions include dissipative particle dynamics and multi-particle collision dynamics.31 LB has multiple different ways of coupling particles to the fluid,32−34 and simulations with thousands of individually resolved swimmers have already been performed.35 LB solves for the fluid flow velocity u, given an applied force density f and boundary conditions. These quantities exist on a uniform grid, and each cell is connected to a certain number of neighbors by lattice vectors ci. Here we will only describe the extension to charged species, an approach we call lattice electrokinetics (EK).21,36 The ion dynamics per eqs 1−3 is calculated via a finite difference scheme. For example, the diffusive term of the flux eq 1 becomes21,36

This is already sufficient to model the system of section 3, a single self-electrophoretic colloid. Chemical concentration fields agree with FEM results to within several percent. To achieve the same precision in the flow field and swimming speed, one needs to take into account that LB typically uses periodic boundary conditions, which introduces finite size effects. To use varying grid resolutions like in FEM, LB grid refinement is needed as suggested at the end of this section. To study the interaction of multiple colloids or between a colloid and a wall, the colloid needs to be able to freely move with respect to the grid. For plain LB, these moving boundaries are well established.32 For EK, we have developed a scheme37 with the required exact solute charge conservation and use eq 12 with all solute densities ρk conveniently rescaled. Simulations of a sphere undergoing electrophoresis in an external field37 demonstrated that a sphere radius of 4 lattice cells is sufficient to get mobilities correct to 2%, with little dependence on parameters like the sphere’s charge or the salt concentration. Overall, LB combined with EK and reactions works well for the study of a single self-electrophoretic microswimmer. However, all relevant length scales need to be resolved, both in LB/EK and in the FEM model: these can range from nanometers for the electric double layer to millimeters for the sample cell size. As computational effort scales with the number of grid cells, systems with low salt concentrations or small swimmers are difficult to handle. The obvious solution, increasing the grid resolution only where there are strong gradients in flow, potential, or concentration, comes naturally to FEM. For LB, grid refinement is also possible:38 Figure 10 shows an LB-equivalent to Figure 9; an effective surface slip velocity14 is employed on the bottom plane to be able to neglect the charge and reaction details. The difference between Figures 9 and 10 comes from the fact that an effective slip on the top plane was not considered in LB. Compared to a uniform grid, this specific simulation uses 99% less CPU time and 95% less memory, though limitations on the speed-up are imposed by the data structures employed. For EK, refinement is the subject of ongoing research but will have similar speedups and limitations. These methods are available or currently being integrated into the LB software packages waLBerla39 and ESPResSo.40

jkidiff (r → r + τ ci , t ) = Dk

ρk (r, t ) − ρk (r + τ ci , t ) |τ c i |

−

Dk zke ρk (r, t ) + ρk (r + τ ci , t ) kBT 2

×

Φ(r, t ) − Φ(r + τ ci , t ) |τ c i |

(12)

where the index i runs over the same 18 neighbor cells used by the LB. In a similar fashion, expressions for the continuity eq 2 and the force acting on the fluid via eq 7 are obtained. To calculate the electrostatic potential of eq 3, the linear system of equations consisting of the discrete Laplace operator and the discrete charge distribution can be solved by any available method. To incorporate chemical reactions into the model, eq 2 needs to be modified with a source term. For bulk reactions, this extra term converts solute densities of one or more species into other species. For surface reactions and constant-flux surface conditions, the source term replaces the fluxes perpendicular to the surface.

6. OUTLOOK AND PERSPECTIVE We have summarized our progress in understanding selfelectrophoresis as a mechanism for self-propulsion of artificial swimmers. Theory has taught us that dissociation reactions in the bulk together with surface reactions, even neutral ones, can lead to self-electrophoretic propulsion, with a rich phenomenology but an analytically tractable interplay. The development of advanced numerical algorithms has enabled us not only to investigate electrophoretic individual swimmers efficiently at high resolutions but also to tackle 3003



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irregularly shaped swimmers41 and swimmers moving in complicated confined geometries.42 We will also be able to go to the bulk limit (simulations without boundary effects, matching theory), and to study collective effects of hundreds of particles with fully resolved hydrodynamic, electrostatic, and phoretic interactions. However, our mesoscopic approach relies on the selection of appropriate input parameters that capture the relevant details of the chemistry involved in the catalytic motors. Here, further careful experimentation and modeling is needed. Scaling down the results obtained for microswimmers to nanomotors or even enzymes18,43 will prove challenging since thermal fluctuations might affect the propulsion mechanisms and would need to be incorporated into the model. Combining these efforts will enable a better understanding of selfpropelled particles, both natural and man-made ones, and their pairwise, collective, and environmental interactions.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Christian Holm: 0000-0003-2739-310X Author Contributions ‡

M.K. and P.K. contributed equally to this work.

Notes

The authors declare no competing financial interest. Biographies Michael Kuron is a Ph.D. candidate in physics at the University of Stuttgart. His research focuses on new simulation methods for and the study of self-propulsion of man-made swimmers and the influence of reaction anisotropies thereon. Patrick Kreissl is a Ph.D. candidate in physics at the University of Stuttgart. His research topics are magnetic soft matter and the selfpropulsion mechanisms of active swimmers based on shape and reaction anisotropies. Christian Holm earned his Ph.D. in physics from the Georgia Institute of Technology in Atlanta (USA) in 1987. After a postdoctoral stay at the TU Clausthal, he obtained his habilitation in physics at the Free University of Berlin in 1996. Subsequently, he worked as a group leader at the Max Planck Institute for Polymer Research in Mainz and as a fellow at the Frankfurt Institute for Advanced Studies. In 2009 he was appointed to the physics faculty of the University of Stuttgart as the director of the ICP. His research focuses on the development of numerical simulation algorithms to study active, charged, and magnetic soft matter systems.

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ACKNOWLEDGMENTS

We thank the Deutsche Forschungsgemeinschaft (DFG) for funding through the SPP 1726 “Microswimmers: from single particle motion to collective behavior” (HO1108/24-1 and HO1108/24-2). We are very grateful to J. de Graaf, A. Brown, and D. Sean for discussions and suggestions on the manuscript. 3004


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