Going in circles Slender body analysis of a self-propelling bent rod Arkava Ganguly1and Ankur Gupta1 1Department of Chemical and Biological Engineering University of Colorado Boulder

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Going in circles: Slender body analysis of a self-propelling bent rod

Arkava Ganguly1and Ankur Gupta1, ∗

1Department of Chemical and Biological Engineering, University of Colorado, Boulder

(Dated: October 21, 2022)

We study the two-dimensional motion of a self-propelling asymmetric bent rod. By employing

slender body theory and the Lorentz reciprocal theorem, we determine particle trajectories for diﬀer-

ent geometric conﬁgurations and arbitrary surface activities. Our analysis reveals that all particle

trajectories can be mathematically expressed through the equation for a circle. The rotational

speed of the particle dictates the frequency of the circular motion and the ratio of translational and

rotational speeds describes the radius of the circular trajectory. We ﬁnd that even for uniform sur-

face activity, geometric asymmetry is suﬃcient to induce a self-propelling motion. Speciﬁcally, for

uniform surface activity, we observe (i) when bent rod arm lengths are equal, the particle only trans-

lates, (ii) when the length of one arm is approximately four times the length of the other arm and

the angle between the arms is approximately π

2, the rotational and translational speeds are at their

maximum. We explain these trends by comparing the impact of geometry on the hydrodynamic

resistance tensor and the active driving force. Overall, the results presented here quantify self-

propulsion in composite-slender bodies and motivate future research into self-propulsion of highly

asymmetric particles.

Keywords: slender body theory, diﬀusiophoresis, self-propulsion, active matter

I. INTRODUCTION

Biological entities typically propel by the beating of cellular appendages like ﬂagella or cilia in asymmetric, wave-

like patterns [1–6]. To mimic biological motion, synthetic propellers have garnered attention due to their promising

applications in medicine [7–10], microﬂuidic devices [11, 12], environmental remediation [13, 14], and the fabrication

of self-repairing surfaces [15, 16]. Broadly speaking, there are two categories of propulsion mechanisms in synthetic

particles. The ﬁrst category of motion is externally actuated, where the propulsion is driven through an external

ﬁeld. For instance, magnetophoresis due to a magnetic ﬁeld [17–20], acoustic propulsion through ultrasound [21–29],

electrophoresis driven by constant electric ﬁelds [30–35], induced-charged electrophoresis due to AC electric ﬁelds

[36–43], diﬀusiophoresis due to concentration gradients of solute(s) [44–49], and thermophoresis [50–52] because of

temperature gradients are all examples of externally driven phoretic motion. The second category of propulsion in

synthetic particles is self-actuated, where the ﬁelds are generated by the particles themselves. Typical examples include

self-diﬀusiophoresis and self-thermophoresis, among others [53–63]. The focus of this work is self-diﬀusiophoresis,

though the results outlined here are readily extended to self-thermophoresis as well.

The most common example of self-diﬀusiophoresis reported in literature consists of a Janus sphere, where the

motion is induced through an asymmetric reaction [60, 64–68]. However, several studies have argued that asymmetry

in reaction is not a necessary requirement for self-diﬀusiophoresis. Instead, geometric asymmetries also induce a

self-diﬀusiophoretic motion, even for a uniform surface activity. Existing theoretical analyses have largely focused on

speciﬁc particle geometries such as spheroidal [69–73] and cylindrical [74, 75]. However, the work by Shklyaev et al.

[76] and Daddi-Moussa-Ider et al. [77] demonstrates that a perturbation to these shapes can modify the direction

and speed of the propulsion. Clearly, geometry plays a key role in self-diﬀusiophoretic propulsion.

To go beyond these typical shapes, recent literature utilized slender body theory (SBT) [78–80] to predict the

motion of self-diﬀusiophoretic particles. Schnitzer and Yariv [75], and Yariv [81] studied the motion of a straight

slender rod with an arbitrary cross-section, arbitrary surface activity, and ﬁrst-order reaction kinetics. Poehnl and

Uspal [82] investigated catalytic helical particles to obtain a good agreement between their SBT prediction and

boundary element calculations. Katsamba et al. [83, 84] outlined a comprehensive SBT framework that can predict

the motion for arbitrary surface activity and an arbitrary three-dimensional axisymmetric geometry.

While the studies described above advance our understanding of self-propulsion in slender bodies, they focus on a

slender body with a single axis. In this work, we analyze the self-diﬀusiophoretic motion of a composite slender body,

i.e., a bent-rod geometry. Our motivation to study a bent-rod is twofold. First, such an asymmetric geometry has

been experimentally studied by K¨ummel et al. [85], who reported a circular motion in L-shaped particles, which was

∗Corresponding author: ankur.gupta@colorado.edu

arXiv:2210.10894v1 [physics.flu-dyn] 19 Oct 2022

FIG. 1. Non-dimensional schematic of the problem setup. We consider a rigid bent rod composed of two cylindrical

arms of equal radius a, aligned at an angle θ. The length of the two arms are 1

2+q`and 1

2−q`. Therefore, the total

length of the bent rod is `. We focus on the slender limit, i.e., =a

`1. The rod self-propels due to solute ﬂux on the rod in

the e1-e2plane and can rotate about the e1×e2plane. er-etrepresent the directions normal and tangential to the bent rod.

To non-dimensionalize our problem setup, we scale all the lengths by `.srepresents the dimensionless coordinate along the

rod. s= 0 is the hinge and s=1

2±qare the ends of two arms. The dimensionless solute ﬂux is represented by j(s). Both e1-e2

and er-etare deﬁne in reference frame of particle. e1is deﬁned such that is aligned with the positive arm, i.e., 0 ≤s≤1

2+q.

e2is perpendicular to e1.erand etare expressed as a function of e1and e2; see Eq. (1a). The lab reference frame is given

by ex-ey; see section II D.

later extended by Rao et al. [86] who studied slender rods bent at diﬀerent angles. Here, we describe the motion of

similar geometries through SBT and do not invoke an external force and torque [87]. Second, the hydrodynamics of

a passive bent-rod have been studied in detail by Roggeveen and Stone [88]. The authors calculate hydrodynamic

mobility for such a geometry, which we utilize to predict the motion of a self-propelling bent rod. In section II, we

calculate the excess solute concentration and obtain the slip velocity at the particle surface. Next, we evaluate the

particle motion by using the Lorentz reciprocal theorem [82, 89, 90]. Subsequently, we ﬁnd that the particle trajectory

is always circular. In section III, we validate our predictions with the experimental results of K¨ummel et al. [85]

and obtain good quantitative agreement without any ﬁtting parameters. Next, we investigate the scenario of uniform

surface ﬂux. Our model reveals the impact of geometry on the circular motion of particles. For speciﬁc geometric

parameters, the translation-rotation coupling is counteracted by the rotation arising from surface activity, causing

the particles to move in a straight line. We show that the translation and rotation speeds are maximum when one

arm is approximately 4 times longer than the other and the arms are at right angles to each other. In section IV, we

summarize our results, discuss the implications of our ﬁndings, and outline future directions.

II. THEORETICAL FRAMEWORK

A. Particle Geometry

We follow the geometric description of a bent-rod outlined in Roggeveen and Stone [88]. The bent-rod is composed

of two cylindrical arms aligned at an angle θ; see Fig. 1. The lengths of the two arms are assumed to be 1

2+q`

and 1

2−q`, where `is the total length and qis the length asymmetry parameter. We note that q∈−1

2,1

2. Both

the arms are assumed to be of the same radius asuch that a

`=1. The rod self-propels due to diﬀusiophoresis,

induced by a surface reaction. We note that though the analysis presented here focuses on a diﬀusiophoretic process

[75, 81, 91], the results are also readily extendable to thermophoretic propulsion [52, 85].

We non-dimensionalize the coordinate system by `. We introduce the arc-length parameter sto describe the position

along the centerline of the rod such that −1

2+q≤s≤1

2+q.s= 0 represents the hinge, whereas s=q±1

2denote

the end of the two arms. For consistency, we refer to the arm where 0 ≤s≤q+1

2as the positive arm and the arm

where −1

2+q≤s < 0 as the negative arm. The shape of the bent rod is thus dictated by qand θ.

We assume that the rod only propels in the e1-e2plane and can rotate about the e3=e1×e2axis. The direction

e1is always assumed to be aligned with the positive arm. For convenience, we also deﬁne etand eras the tangential

and normal directions to the rod, respectively, such that

et=(−cos θe1−sin θe2s < 0

e1s≥0,(1a)

er=(−sin θe1+ cos θe2s < 0

e2s≥0.(1b)

Note that both e1-e2and et-erare in the particle frame of reference and moves with the particle. In section II D, we

deﬁne ex-eyas our universal frame of reference to obtain equations for the particle trajectories; see Eq. (16). The

position of a point on the particle centerline is xh(s) = set. The center of mass of the bent-rod is denoted as xcom. To

model self-propulsion through catalytic activity, we follow the common practice in literature [58, 75, 81, 82, 92–94],

and assume a solute ﬂux j(s) on the particle surface (the mathematical deﬁnition of j(s) is provided later). The

induced translation and rotation velocities of the particle are denoted by Uand Ω, respectively. The objective of this

paper is to determine the particle trajectory in the limit 1 for a given q,θ, and j(s). The limit 1 enables

us to invoke ﬁrst-order slender body theory to evaluate Uand Ω. We follow the approach outlined in Schnitzer

and Yariv [75] and Poehnl and Uspal [82] to obtain the excess solute concentration proﬁle and eﬀective slip velocity.

Next, we employ the geometric resistance coeﬃcients obtained from Roggeven and Stone [88] and use the Lorentz

reciprocal theorem to obtain the particle trajectory for a self-propelling bent rod. Since we utilize ﬁrst-order slender

body theory, we superpose the concentration and hydrodynamic eﬀects of the two arms and neglect the higher-order

interactions between them. Therefore, our analysis becomes less applicable for cases where the interaction between

the arms become important. We also acknowledge that our analysis ignores the circumferential variations in the solute

ﬂux, discussed in-depth by Kastamba et al. [83, 84].

B. Concentration Proﬁle

We seek to evaluate the concentration of the solute at the particle surface for a given geometry and surface ﬂux.

To do so, we deﬁne dimensionless surface ﬂux j(s) scaled by reference ﬂux Jref. We deﬁne c(s, r) to be dimensionless

solute concentration, scaled by aJref

D, where Dis the solute diﬀusivity. Mathematically, our objective is to evaluate

concentration at the slip plane cs(s) for a given q,θand j(s). We note that cs(s) is equivalent to the surface

concentration from the outer solution cout(s, ) [75, 82, 95] (also see Appendix A). We deﬁne the P´eclet number of

the rod as Pe= Uref `

D, where Uref is a typical velocity scale. For representative values, we focus on ref. [96]. Here,

catalytic spheres with 2µm diameters were driven in H2O2solutions. D=O(10−8) m2/s, `ref =O(10−6) m, and

Uref =O(10−5) m/s. Therefore, Pe = O(10−3). This helps us justify neglecting convection and unsteady terms in

Eq. (2) (additional justiﬁcation is provided below Eq. (17)). Therefore, we write

∇2c= 0, r ≥. (2)

The diﬀusiophoretic activity is represented with a surface ﬂux boundary condition,

−ˆ

n· ∇c=j(s), r =, (3a)

where ˆ

nis the surface normal vector. The far ﬁeld boundary condition for solute concentration reads

c= 0, r → ∞.(3b)

Since 1, we use boundary-layer theory [95] to evaluate cs(s). As outlined in Appendix A, we divide the ﬂuid

volume into an inner and an outer region in the radial direction er. In the inner region, we stretch the coordinates

such that ρ=r−1and evaluate cin(s, ρ) with the boundary condition in Eq. (3a). Subsequently, in the outer region,

we evaluate cout(s, r) as a line integral of diﬀusive sources of strength α(s). We determine α(s) via an asymptotic

matching cin(s, ∞) = cout(s, ). From the leading order behavior, we obtain α(s) = j(s)

2, which gives

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Goingincircles:Slenderbodyanalysisofaself-propellingbentrodArkavaGanguly1andAnkurGupta1,1DepartmentofChemicalandBiologicalEngineering,UniversityofColorado,Boulder(Dated:October21,2022)Westudythetwo-dimensionalmotionofaself-propellingasymmetricbentrod.ByemployingslenderbodytheoryandtheLorentzrecipro...

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Going in circles Slender body analysis of a self-propelling bent rod Arkava Ganguly1and Ankur Gupta1 1Department of Chemical and Biological Engineering University of Colorado Boulder

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