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# Shape-directed dynamics of active colloids powered by induced-charge electrophoresis

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 18, 2017 (received for review June 28, 2017)

## Significance

Despite recent advances in the ability to “program” the self-assembly of colloidal components, the resulting structures are often static and therefore incapable of performing dynamic functions such as the ability to actuate, heal, replicate, and compute. The realization of colloidal machines that organize in space and time to perform such functions requires new strategies for encoding the dynamic behaviors of colloidal components. Focusing on active colloids powered by induced-charge electrophoresis, we use theory and simulation to show how the shape of a colloidal particle can be rationally tailored to specify complex motions powered by simple energy inputs.

## Abstract

The symmetry and shape of colloidal particles can direct complex particle motions through fluid environments powered by simple energy inputs. The ability to rationally design or “program” the dynamics of such active colloids is an important step toward the realization of colloidal machines, in which components assemble spontaneously in space and time to perform dynamic (dissipative) functions such as actuation and transport. Here, we systematically investigate the dynamics of polarizable particles of different shapes moving in an oscillating electric field via induced-charge electrophoresis (ICEP). We consider particles from each point group in three dimensions (3D) and identify the different rotational and translational motions allowed by symmetry. We describe how the 3D shape of rigid particles can be tailored to achieve desired dynamics including oscillatory motions, helical trajectories, and complex periodic orbits. The methodology we develop is generally applicable to the design of shape-directed particle motions powered by other energy inputs.

The creation of colloidal machines (1)—that is, dynamic assemblies of colloidal components that perform useful functions—requires advances in our ability to rationally engineer the dynamics of active colloids (2, 3) operating outside of thermodynamic equilibrium. Owing to their small size (nanometers to micrometers), such machines must assemble spontaneously and operate autonomously in response to simple energy inputs due to chemical fuels or external fields. Achieving nontrivial dynamical behaviors and ultimately function demands the use of complex components, into which the desired behaviors can be effectively encoded. The challenge is conceptually similar to that of programmable self-assembly (4), whereby assembly information encoded in the building blocks directs their organization into a specific structure. For equilibrium assemblies, this information takes the form of colloidal interactions, which can be designed by controlling particle shape (5, 6) and surface chemistry (7⇓⇓⇓–11). Extending this approach to design colloidal machines will require control over particle organization in time as well as space—that is, over dynamics as well as structure. In this context, it is instructive to consider first the dynamics of a single particle and how it might be programed to perform increasingly complex tasks [e.g., the weaving of microscopic braids (12)]. Understanding the complex motions of individual units is a critical prerequisite to the design of dynamic assemblies of active particles.

The motion of colloidal particles relative to their fluid surroundings can be driven by a variety of different physicochemical mechanisms. Self-phoretic particles (13) induce local gradients (e.g., in the electric potential) that propel particle motions through interfacial “phoretic” effects (e.g., electrophoresis) (14). By engineering the shape and symmetry of such particles, different dynamical behaviors have been realized, including linear (15⇓–17), rotational (18, 19), and circular (20) motions. The variety of possible dynamics for self-phoretic particles in isotropic media is significantly limited by the translational and rotational invariance of particle motions (only helical trajectories have yet to be reported). By contrast, rigid particles within uniform shear flows move relative to the fluid at velocities that depend on their orientation, which can lead to complex—even chaotic—rotational and translational motions (21, 22). Similarly, asymmetric polarizable particles within uniform electric fields are known to swim through conductive fluids by means of induced-charge electrophoresis (ICEP) (23). Such motions are well understood (24) and depend on the symmetry of the particle and its orientation in the applied field (25, 26). Notably, metallodielectric Janus particles (

Here, we systematically investigate the ICEP dynamics of particles of different symmetries and discuss how particle shape can be used to encode a variety of complex dynamical behaviors. We consider particles from each point group in 3D (31) and identify the rotational and translational motions allowed by symmetry. For each qualitatively distinct motion, we create specific realizations of the dynamics, using rigid polarizable particles of a particular shape. The dynamics of such particles are computed numerically using a boundary integral formulation of the electrostatic and hydrodynamic problems governing ICEP motion. In addition to linear, rotational, and circular motions reported previously, our analysis suggests that particles of appropriate shapes are capable of oscillatory motions, helical trajectories, and complex periodic orbits. We show how the complexity of the dynamics grows as the symmetry of the particle is reduced. We discuss how complex particle trajectories can be rationally engineered into asymmetric particles through a careful combination of simpler shapes and their accompanying motions. In particular, we demonstrate how ICEP particles can be designed to achieve effective “foraging” motions within liquid environments for applications in chemical sensing or remediation (32, 33). While the present focus is ICEP, our approach is readily extended to other shape-directed particle motions powered by self-phoresis, hydrodynamic shear, and ultrasonic actuation (34⇓–36). Overall, this work represents one step toward a broader goal of programing the organization of active particles in space and time to create colloidal machines with bio-inspired function.

## Results and Discussion

### ICEP Dynamics.

We consider the motion of an ideally polarizable particle immersed in an unbounded electrolyte and subject to an oscillating electric field,

The frequency of the ac field is assumed to be slower than the rate of ion accumulation at the particle–electrolyte interface—that is,

To describe the rigid-body motion of the particle, we introduce two coordinate systems: a stationary system and a moving system, which is fixed to the particle and participates in its motion (Fig. 1). A vector **v** in the stationary system as ** R** is an orthogonal rotation matrix that depends on the orientation of the particle (e.g., on the Euler angles ϕ, θ, ψ; Fig. 1

*A*). Similarly, the components of the shape tensors, 𝕮 and 𝕯, in the stationary system are related to those in the moving system as

**1**and

**2**and the kinematics of rigid-body motion (

*Materials and Methods*). In general, there are 18 quantities associated with each shape tensor (27 components less 9 relations of the form

### Particle Symmetry and Shape.

The symmetry of a particle is characterized by the set of operations such as reflections, rotations, and inversions that leave the particle unchanged; each such operation can be specified by an orthogonal matrix ** Q**. Invariance of the particle shape with respect to the operation

**implies the following relationships among the components of the shape tensors:**

*Q***5**implies that

**6**implies

We consider particles from each of the possible point groups in three dimensions, visualized in Fig. 1*B* and represented by their Schoenflies notation (31) in Table 1 (*SI Appendix*, Fig. S1). For each point group, we define a representation of the associated shape tensors

While the symmetry of a particle can significantly constrain its dynamics, the detailed particle shape is needed to uniquely specify the shape tensors *Materials and Methods*). Finally, we compute the particle dynamics by integrating Eqs. **1** and **2** starting from a specified orientation. We neglect effects due to Brownian motion, such that particle dynamics are fully deterministic. Physically, this approximation is appropriate for sufficiently large particles and fields such that *A* shows the computed trajectory for a particle with

### Possible Particle Dynamics.

We now survey the variety of possible ICEP motions for particles of different symmetries and shapes. Owing to the translational invariance of the particle dynamics, it is possible to fully describe the rotational motion of a given particle independent of any translational motions. It is therefore convenient to organize particles of different point groups into classes that share common rotational dynamics. Within each rotation class, the point groups can be further divided based on the different possible translational motions. This organizational scheme is illustrated in Table 1 and serves to guide our exploration through the space of different particle shapes. Each rotation class is denoted by a prototypical member that possesses inversion symmetry and therefore exhibits no translational motion.

### Sphere Rotation Class.

The electrokinetic flows induced around a spherical particle are highly symmetric and result in no translation or rotational motion of the particle. Similarly, particles with octahedral (

𝑫 ∞ 𝒉 Rotation Class.

Particles in the *A* and *B*). More generally, we find that a particle’s aspect ratio is key to specifying its preferred orientation in the field.

Members of this rotation class that exhibit no translational motion include the point groups *C* and *D*). Such steady translational motion has been explored in experimental studies on Janus particles, which belong to the

While the members of the *E*). We refer to such particles as shuttles, which are capable of bidirectional motion. When aligned perpendicular to the field, *F*). We refer to such particles as gliders.

To summarize, particles in this rotation class are all capable of aligning parallel or perpendicular to the applied field. We further distinguish three particle types capable of steady translation: Rockets move parallel to their primary axis at a constant velocity; shuttles move parallel to their axis with a velocity that depends on the orientation ϕ; and gliders move perpendicular to their axis in a ϕ-dependent direction.

D 2 h Rotation Class.

Particles in the

D 3 d Rotation Class.

Particles in the *A*). Particles in the

When *B*). The net motion is reminiscent of a rotating corkscrew that swims through viscous surroundings. This example shows how similar motions are achieved using very different particles—e.g., *C* and 3*B*). In another example, *C*). The size of the circular orbit is determined by the relative rates of rotation and translation, which can be tuned by varying the particle shape.

C 4 h Rotation Class.

Particles in the *A* and *B*). In contrast to flipping, spinning refers to rotational motion about the primary axis of the particle, which can be oriented either parallel or perpendicular to the field. Particles of

𝑺 𝟔 Rotation Class.

Particles in the *A*). This motion—termed precessing—is similar to flipping but with the particle axis tilted with respect to the field. Finally, some *B*). The specific conditions for which each motion occurs are detailed in *SI Appendix*.

Breaking the fore–aft symmetry of the *C*). For *D* shows one example where a

C 2 h Rotation Class.

Particles in the *SI Appendix*.

Within this rotation class, particles of *A*). Particles with one plane of mirror symmetry (*B*). Tall particles orient parallel to the field and trace circular orbits in the plane perpendicular to the field (Fig. 6*C*). Both

### Programing Particle Motion.

So far, we have shown how the symmetry of a particle can constrain its dynamics to permit certain translational and rotational motions; the detailed shape of the particle further specifies its unique trajectory. We now consider the inverse problem: Given a desired dynamical behavior, we seek to determine the particle shape that “encodes” those dynamics. We limit our discussion to those motions described in the previous section such that the necessary particle symmetry is implied by the desired particle motion. For example, to achieve helical motions along the field axis, one would select particles of *C*).

The desired motion is characterized by a set of features ** F** such as the radius, pitch, and speed of the helical trajectory. More generally, particle trajectories can be well approximated by truncated Fourier series. The particle shape is specified by some weighted combination of basis functions, which are chosen to preserve the desired symmetry of the particle. Here, we use linear combinations of spherical harmonics; however, other choices are possible. In the forward problem, the basis function weights

**are used to specify the particle shape, compute the shape tensors, integrate the particle motion, and determine the features of the particle trajectory. The features are therefore a function of the weights,**

*B*In designing a

In the inverse problem, we are given the desired features *C* shows the helical trajectory of a

It is important to note that there are particle motions which cannot be accessed by ICEP. Owing to the invariance of the dynamics with respect to rotation about the axis of the field, the angular velocity of the particle depends on only two variables (e.g., Euler angles ϕ and θ). Chaotic motions are therefore prohibited by the Poincare–Bendixson theorem (40); the particle orientation evolves in time to a constant value (a fixed point) or to a periodic function of time (a limit cycle).

In experimental practice, the “encoding” of colloidal motions into particle shapes remains challenging. Recent advances in colloidal synthesis offer routes to low-symmetry particles such as colloidal doublets, trimers, and tetramers (5, 6, 41); however, most of the particles described here are currently inaccessible to bottom–up synthetic approaches. By contrast, top–down fabrication techniques such as two-photon lithography (42, 43) now enable one to “print” micrometer-scale particles of arbitrary, 3D shapes with features on the scale of tens of nanometers. As a demonstration, we printed a *SI Appendix*, Fig. S20). Such polymeric particles could be coated with a metal layer by electroless deposition to create the kinds of polarizable particles studied here (44, 45).

### Sensitivity on Particle Shape.

The ability to precisely prescribe a particle’s motion demands similar precision over the particle’s shape, which may be difficult to achieve in experiment. It is therefore important to consider the sensitivity of such motions with respect to perturbations in the particle shape. Specifically, we consider how the shape tensors, ** B** (here, the coefficients

For a spherical particle (*SI Appendix*, Fig. S16).

A particle’s sensitivity to small perturbations depends on its shape (i.e., on

In the linear regime, the effects of individual perturbations can be added together to describe the particle’s response. Higher-order mixing of two or more perturbations is prohibited. For example, there exists no small defect that will cause a

For low-symmetry shapes, just about any perturbation will alter the shape tensors at first order but by different amounts. The addition of spherical harmonics of low order ℓ has a larger impact on particle motions than the addition of those of higher order. This observation suggests that large-scale defects accompanying particle fabrication are more likely to disrupt the desired particle motions than small-scale defects due to surface roughness. The sensitivity of particle motions to defects could likely be mitigated by altering the objective function

### Brownian Motion.

The effects of Brownian motion may be significant for smaller particles or weaker fields. The relative importance of electrokinetic vs. diffusive particle motions is quantified by dimensionless Péclet numbers for translation and rotation:

We therefore simulated the Brownian dynamics of particles subject to applied fields at various dimensionless temperatures *Materials and Methods*). At long times, particles exhibit anisotropic enhancements in their translational diffusion due to the applied field. Such motions are characterized by diffusion coefficients *C*). In the absence of Brownian motion, such particles align and translate parallel to the field. At finite temperatures *SI Appendix*, Fig. S18). Motions perpendicular to the field can be similarly enhanced (*SI Appendix*, Fig. S19). The specific values of

### Boundaries and Interactions.

Realizing the 3D motions described here would likely require density-matched materials to avoid sedimentation of the microparticles onto system boundaries. The presence of such boundaries—neglected in the present analysis—can alter the dynamics of active colloids by modifying the electric field and the accompanying fluid flows (46). For a single planar wall oriented normal to the applied field (z direction), the particle velocity is still described by Eqs. **1** and **2** such that

Such motions are constrained by the symmetry of the particle–wall system, not just the particle itself. For example, a sphere near a wall has

The symmetry of the particle–wall system can change, depending on the particle orientation. A *B*). Interestingly, the same particle may even switch its preferred orientation as a function the surface separation, thereby changing the symmetry of the system in time.

Within bulk dispersions, long-ranged particle interactions can also influence ICEP motions. In the far field, each particle creates an electrostatic disturbance like that of a charge dipole as well as a hydrodynamic disturbance like that of a force dipole [a so-called stresslet of the “puller” variety (49)]. These disturbances cause particles to attract one another along the direction of the field and then repel along a perpendicular direction (50, 51). Combined with linear self-propulsion, such hydrodynamic interactions can lead to stable “flocks” of particles moving in a common direction (52). These and other collective motions should depend on the individual particle trajectories and their near-field interactions, both of which are controlled by particle shape. It may therefore be possible to extend the present concept of shape-based programing to direct colloidal dynamics within ensembles of active particles. Recent experimental results highlight opportunities for creating complex dynamic assemblies using shape-directed ICEP motions (29, 30, 53, 54).

## Conclusions

Low-symmetry particles can exhibit complex dynamics powered by induced-charge electrophoresis in three dimensions. In contrast to motions of self-phoretic particles, the ICEP velocity depends on the particle orientation relative to the applied field. The field can therefore serve to guide particle motions along intricate cycles of rotation and translation. These motions are largely dictated by particle symmetry and can be uniquely prescribed by engineering particle shape. The diversity of particle trajectories described here may offer useful functions as colloidal clocks (

## Materials and Methods

The formulation of the ICEP problem below follows closely that of Squires and Bazant (26). Additionally, we introduce a boundary integral formulation of the problem and describe its numerical solution.

### Electrostatics.

The electric potential ** x** is chosen as the center of the particle. This expression assumes that the electric double-layer thickness is much smaller than the size of the particle, such that there is no free charge in solution. At the particle surface

**is the unit normal vector directed out from the surface. Far from the particle, the potential approaches the externally applied potential**

*n***10**–

**12**imply that the electric potential on the surface of the particle is governed by the integral equation

*SI Appendix*for details) (55). As detailed below, the integral Eq.

**13**is solved numerically to determine the potential and the potential gradient on the particle surface.

### Hydrodynamics**.**

The fluid flows around the particle are described by the Stokes equations for creeping flow,** U** and angular velocity 𝛀. Here, we adopt a moving frame of reference centered on the particle. In this frame, there is no flow normal to the particle surface

**is a unit vector tangent to the surface, and**

*t**SI Appendix*for details). The velocity far from the particle is that due to pure translation and/or rotation

**20**numerically gives the stress

**and 𝛀 are computed using Eq.**

*U***19**.

### Numerical Solution.

The integral equations above are solved numerically using Lebedev quadrature (57) over surfaces parameterized by the spherical angles θ and ϕ. In this approach, integrals are approximated as**13**, we introduce an alternative Green’s-like function ** y** due to a Gaussian charge distribution of width

**. This function approaches the standard Green’s function for a point charge in an unbounded medium in the limit as**

*x***13**for the potential is then divided into two components: (

*i*) a far-field contribution using the Gaussian-modulated Green’s function and (

*ii*) a near-field correction that contains the singularity. The first component is nonsingular and can be computed numerically using Lebedev quadrature and a linear solver [MATLAB’s gmres() function]; the second one is nonzero only in the vicinity of the singularity and can be approximated analytically. We use an analogous approach for computing the hydrodynamic integral Eq.

**20**with a similarly filtered Green’s function (58) (see

*SI Appendix*for details).

For a given orientation of the applied field, the above approach was used to compute the translational and rotational velocity of the particle. This process was repeated for multiple orientations—typically, the 38 points of the ninth-order Lebedev grid. The nonzero tensor coefficients were then estimated by linear regression of Eqs. **1** and **2**. Given the shape tensors, the equations of motion were integrated numerically using MATLAB’s ode113() Adams–Bashforth–Moulton solver (59); particle orientation was represented and integrated using unit quaternions (60).

### Effects of Brownian Motion.

To describe the effects of Brownian motion on the shape-directed dynamics of particles moving by ICEP, we start from the Langevin equation for translational and rotational motion,** m** is a generalized mass/moment-of-inertia tensor,

**1**and

**2**for the ICEP velocity, the electric force is equal and opposite to the hydrodynamic force,

Like the shape tensors **5** and **6**. With these constraints, the resistance tensors were computed numerically, using the boundary integral formulation detailed in the previous section. The Langevin equation Eq. **23**) was integrated numerically in the overdamped regime, using Fixman’s midpoint scheme with a constant time step of

## Acknowledgments

This work was supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0000989. A.M.B. was supported in part by the National Science Foundation Graduate Research Fellowship Program under Grant DGE1255832.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: kyle.bishop{at}columbia.edu.

Author contributions: A.M.B. and K.J.M.B. designed research; A.M.B. and K.J.M.B. performed research; A.M.B., S.S., and K.J.M.B. analyzed data; and A.M.B., S.S., and K.J.M.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1711610115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- Spellings M, et al.

- ↵
- Ebbens SJ

- ↵
- Dey KK,
- Wong F,
- Altemose A,
- Sen A

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Kim Y,
- Macfarlane RJ,
- Jones MR,
- Mirkin CA

- ↵
- Tian Y, et al.

- ↵
- ↵
- Goodrich CP,
- Brenner MP

- ↵
- ↵
- ↵
- ↵
- ↵
- Michelin S,
- Lauga E

- ↵
- ↵
- ↵
- ↵
- Makino M,
- Doi M

- ↵
- Hermans TM,
- Bishop KJM,
- Stewart PS,
- Davis SH,
- Grzybowski BA

- ↵
- ↵
- Bazant MZ,
- Squires TM

- ↵
- Yariv E

- ↵
- ↵
- ↵
- Boymelgreen A,
- Yossifon G,
- Park S,
- Miloh T

- ↵
- Ma F,
- Wang S,
- Wu DT,
- Wu N

- ↵
- Zhang J,
- Yan J,
- Granick S

- ↵
- Cotton FA

- ↵
- ↵
- ↵
- ↵
- Nadal F,
- Lauga E

- ↵
- Ahmed S,
- Wang W,
- Gentekos DT,
- Hoyos M,
- Mallouk TE

- ↵
- Davidson SM,
- Andersen MB,
- Mani A

- ↵
- ↵
- Nocedal J,
- Wright SJ

- ↵
- Strogatz SH

- ↵
- ↵
- Maruo S,
- Fourkas JT

- ↵
- Hashemi SM, et al.

- ↵
- ↵
- ↵
- Kilic MS,
- Bazant MZ

- ↵
- ↵
- Mozaffari A,
- Sharifi-Mood N,
- Koplik J,
- Maldarelli C

- ↵
- ↵
- Saintillan D,
- Darve E,
- Shaqfeh ESG

- ↵
- Saintillan D

- ↵
- ↵
- Ma F,
- Yang X,
- Zhao H,
- Wu N

- ↵
- Yan J, et al.

- ↵
- Pozrikidis C

- ↵
- ↵
- Lebedev VI

- ↵
- ↵
- ↵
- Diebel J

- ↵
- ↵
- Brenner H

- ↵
- Delong S,
- Balboa Usabiaga F,
- Donev A

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