# Chiral motion in driven, shaped colloids

## Our Papers

A review of shaped colloidal particles in fluids: Anisotropy and chirality
Authors: Thomas A. Witten and Haim Diamant
This review treats asymmetric colloidal particles moving through their host fluid under the action of some form of propulsion. The propulsion can come from an external body force or from external shear flow. It may also come from externally-induced stresses at the surface, arising from imposed chemical, thermal or electrical gradients. The resulting motion arises jointly from the driven particle and the displaced fluid. If the objects are asymmetric, every aspect of their motion and interaction depends on the orientation of the objects. This orientation in turn changes in response to the driving. The objects' shape can thus lead to a range of emergent anisotropic and chiral motion not possible with isotropic spherical particles. We first consider what aspects of a body's asymmetry can affect its drift through a fluid, especially chiral motion. We next discuss driving by injecting external force or torque into the particles. Then we consider driving without injecting force or torque. This includes driving by shear flow and driving by surface stresses, such as electrophoresis. We consider how time-dependent driving can induce collective orientational order and coherent motion. We show how a given particle shape can be represented using an assembly of point forces called a Stokeslet object. We next consider the interactions between anisotropic propelled particles, the symmetries governing the interactions, and the possibility of bound pairs of particles. Finally we show how the collective hydrodynamics of a suspension can be qualitatively altered by the particles' shapes. The asymmetric responses discussed here are broadly relevant also for swimming propulsion of active micron-scale objects such as microorganisms.

Chiral motion in colloidal electrophoresis
Authors: Lara Braverman, Aaron Mowitz, Thomas A. Witten
Asymmetrically charged, nonspherical colloidal particles in general perform complex rotations and oblique motions under an electric field. The interplay of electrostatic and hydrodynamic forces complicate the prediction of these motions. We demonstrate a method of calculating the body tensors that dictate translational and rotational velocity vectors arising from an external electric field. We treat insulating, rigid bodies in the linear-response regime, with indefinitely small electrostatic screening length. The method represents the body as an assembly of point sources of both hydrodynamic drag and surface electric field. We demonstrate agreement with predicted electrophoretic mobility to within a few percent for several shapes with uniform and nonuniform charge. We demonstrate strong chiral twisting motions for colloidal bodies of symmetrical realistic shapes. The method applies more generally to active colloidal swimmers.

Adapting the Teubner reciprocal relations for stokeslet objects
Authors: Thomas A. Witten and Aaron Mowitz
Self-propelled colloidal swimmers move by pushing the adjacent fluid backwards. The resulting motion of an asymmetric body depends on the profile of pushing velocity over its surface. We describe a method of predicting the motion arising from arbitrary velocity profiles over a given body shape, using a discrete-source "stokeslet" representation. The net velocity and angular velocity is a sum of contributions from each point on the surface. The contributions from a given point depend only on the shape. We give a numerical method to find these contributions in terms of the stokeslet positions defining the shape. Each contribution is determined by linear operations on the Oseen interaction matrix between pairs of stokeslets. We first adapt the Lorentz Reciprocal Theorem to discrete sources. We then use the theorem to implement the method of Teubner[1] to determine electrophoretic mobilities of nonuniformly charged bodies.

Predicting tensorial electrophoretic effects in asymmetric colloids
Authors: Aaron J. Mowitz, Thomas A. Witten
We formulate a numerical method for predicting the tensorial linear response of a rigid, asymmetrically charged body to an applied electric field. This prediction requires calculating the response of the fluid to the Stokes drag forces on the moving body and on the countercharges near its surface. To determine the fluid's motion, we represent both the body and the countercharges using many point sources of drag known as stokeslets. Finding the correct flow field amounts to finding the set of drag forces on the stokeslets that is consistent with the relative velocities experienced by each stokeslet. The method rigorously satisfies the condition that the object moves with no transfer of momentum to the fluid. We demonstrate that a sphere represented by 1999 well-separated stokeslets on its surface produces flow and drag force like a solid sphere to one-percent accuracy. We show that a uniformly-charged sphere with 3998 body and countercharge stokeslets obeys the Smoluchowski prediction \cite{Morrison} for electrophoretic mobility when the countercharges lie close to the sphere. Spheres with dipolar and quadrupolar charge distributions rotate and translate as predicted analytically to four percent accuracy or better. We describe how the method can treat general asymmetric shapes and charge distributions. This method offers promise as a way to characterize and manipulate asymmetrically charged colloid-scale objects from biology (e.g. viruses) and technology (e.g. self-assembled clusters).

Screening, hyperuniformity, and instability in the sedimentation of irregular objects
Authors: Tomer Goldfriend, Haim Diamant, Thomas A. Witten
Abstract:We study the overdamped sedimentation of non-Brownian objects of irregular shape using fluctuating hydrodynamics. The anisotropic response of the objects to flow, caused by their tendency to align with gravity, directly suppresses concentration and velocity fluctuations. This allows the suspension to avoid the anomalous fluctuations predicted for suspensions of symmetric spheroids. The suppression of concentration fluctuations leads to a correlated, hyperuniform structure. For certain object shapes, the anisotropic response may act in the opposite direction, destabilizing uniform sedimentation.

#### Swirling a liquid from within

Pouring a powder into a glass of water sets off a swirling motion as the powder grains settle under gravity. The heavy grains entrain water as they drift down, dragging one another into chaotic motion. Even a uniformly stirred, resting liquid starts to swirl as gravity begins to act. Now scientists in Israel and the US have identified a way to control this swirling by adding shaped particles to the powder. They found shapes that reduce the swirling and shapes that enhance it. The shaped particles work by gliding sideways as they fall, thus reducing or increasing the slight variations of weight that cause the swirling. Industrial reactors use suspended powders to clean up water and promote chemical reactions. Shaped powders give engineers a new handle to improve the performance of these reactors.

Hydrodynamic Interactions between Two Forced Objects of Arbitrary Shape: II Relative Translation
Authors: Tomer Goldfriend, Haim Diamant, Thomas A. Witten
Abstract: We study the relative translation of two arbitrarily shaped objects, caused by their hydrodynamic interaction as they are forced through a viscous fluid in the limit of zero Reynolds number. It is well known that in the case of two rigid spheres in an unbounded fluid, the hydrodynamic interaction does not produce relative translation. More generally such an effective pair-interaction vanishes in configurations with spatial inversion symmetry, for example, an enantiomeric pair of arbitrary shape does not develop relative translation instantaneously. We show that the breaking of inversion symmetry by boundaries of the system accounts for the interactions between two spheres in confined geometries, as observed in experiments. The same general principle also provides new predictions for interactions in other object configurations near obstacles. We examine the time-dependent relative translation of two self-aligning objects, extending the numerical analysis of our preceding publication [Goldfriend, Diamant and Witten, arXiv:1502.00221 (2015)]. The interplay between the orientational interaction and the translational one, in most cases, leads over time to repulsion between the two objects. The repulsion is qualitatively different for self-aligning objects compared to the more symmetric case of uniform prolate spheroids. The separation between the two objects increases with time t as t1/3 in the former case, and more strongly, as t, in the latter.

Hydrodynamic Interactions between Two Forced Objects of Arbitrary Shape: I Effect on Alignment
Authors: Tomer Goldfriend, Haim Diamant, Thomas A. Witten
Abstract: We study the properties and symmetries governing the hydrodynamic interaction between two identical, arbitrarily shaped objects, driven through a viscous fluid. We treat analytically the leading (dipolar) terms of the pair-mobility matrix, affecting the instantaneous relative linear and angular velocities of the two objects at large separation. We find that the ability to align asymmetric objects by an external time-dependent drive [Moths and Witten, Phys. Rev. Lett. 110, 028301 (2013)] is degraded by the hydrodynamic interaction. The effects of hydrodynamic interactions are explicitly demonstrated through numerically calculated time-dependent trajectories of model alignable objects composed of four stokeslets. In addition to the orientational effect, we find that the two objects generally repel each other, thus restoring full alignment at long times.
21 pages, 7 figures

Brian Moths and T. A. Witten.   This paper extends and amplifies the results of the PRL below.  It proves conditions for alignment in greater detail and shows that the ability to align extends much beyond the range of these proofs.  Physical Review E 88, 022307 (2013) [15 pages] .

, Brian Moths, T. A. Witten.  In this paper, we examine the rotation exhibited by an asymmetric colloidal object as it falls through a viscous medium. Given other physical systems where the orientational degrees of freedom of an ensemble can be intimately controlled by a uniform external forcing (one example being NMR), we ask if the same is possible for an ensemble of colloids. The first task one would attempt might be to align the members of an ensemble. Previous work has shown that orientational alignment of an ensemble up to rotations about an axis is possible using constant forcing. In this work, we describe two methods of aligning by a time-dependent but spatially uniform forcing.  We go on to briefly discuss the possible realizations and limitations of these methods.Physical Review Letters, vol. 110, Issue 2, id. 028301 (2012)  DOI: 10.1103/PhysRevLett.110.028301

Chiral sedimentation of extended objects in viscous media We study theoretically the chirality of a generic rigid object's sedimentation in a fluid under gravity in the low Reynolds number regime. We represent the object as a collection of small Stokes spheres or stokeslets, and the gravitational force as a constant point force applied at an arbitrary point of the object. For a generic configuration of stokeslets and forcing point, the motion takes a simple form in the nearly free draining limit where the stokeslet radius is arbitrarily small. In this case, the internal hydrodynamic interactions between stokeslets are weak, and the object follows a helical path while rotating at a constant angular velocity $\omega$ about a fixed axis. This $\omega$ is independent of initial orientation, and thus constitutes a chiral response for the object. Even though there can be no such chiral response in the absence of hydrodynamic interactions between the stokeslets, the angular velocity obtains a fixed, nonzero limit as the stokeslet radius approaches zero. We characterize empirically how $\omega$ depends on the placement of the stokeslets, concentrating on three-stokeslet objects with the external force applied far from the stokeslets. Objects with the largest $\omega$ are aligned along the forcing direction. In this case, the limiting $\omega$ varies as the inverse square of the minimum distance between stokeslets. We illustrate the prevalence of this robust chiral motion with experiments on small macroscopic objects of arbitrary shape.

## Illustration

Trajectories of an irregular object sinking in a viscous fluid, showing chiral sedimentation [Nathan Krapf, unpublished]. The object, based on the molecular structure of the protein CD2AP from the NCBI Entrez Gene data base, is pictured in the inset. The cyan trajectory at left shows the trajectory expected for the actual object, pulled at its center of mass. Successive trajectories show the effect of displacing the center of mass from its actual location along the rainbow line in the inset, keeping a fixed initial orientation . Typically the object quickly reaches a state of periodic twisting motion. In narrow regions this regular twisting gives way to irregular tumbling. This chiral sedimentation response is a potential tool for characterizing chiral shapes of colloidal objects. Nathan Krapf, working with Prof. Tom Witten, explored the connection between shape and chiral sedimentation in N. Krapf and T. A. Witten, Phys. Rev. E 79, 056307 (2009)

Click on the picture to see higher resolution (4.8 meg)