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

Arkava Ganguly  Ankur Gupta  Corresponding author<EMAIL_ADDRESS>Department of Chemical and Biological Engineering, University of Colorado,
Boulder

###### Abstract

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 different 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 surface
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 translates, (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 $\frac{\pi}{2}$, 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.

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

††preprint: APS/123-QED

##  I Introduction

Biological entities typically propel by the beating of cellular appendages
like flagella or cilia in asymmetric, wave-like patterns [1, 2, 3, 4, 5, 6].
To mimic biological motion, synthetic propellers have garnered attention due
to their promising applications in medicine [7, 8, 9, 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, 18, 19, 19,
20], acoustic propulsion through ultrasound [21, 22, 23, 24, 25, 26, 27, 28,
29], electrophoresis driven by constant electric fields [30, 31, 32, 33, 34,
35], induced-charged electrophoresis due to AC electric fields [36, 37, 38,
39, 40, 41, 42, 43], diffusiophoresis due to concentration gradients of
solute(s) [44, 45, 46, 47, 48, 49], and thermophoresis [50, 51, 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, 54,
55, 56, 57, 58, 59, 60, 61, 62, 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, 65, 66, 67, 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, 70, 71, 72, 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, 79, 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 Kümmel
et al. [85], who reported a circular motion in L-shaped particles, which was
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 [89, 90, 82]. Subsequently, we find that the particle
trajectory is always circular. In section III, we validate our predictions
with the experimental results of Kümmel 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.

Figure 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 $\theta$. The length of the two arms are
$\left(\frac{1}{2}+q\right)\ell$ and $\left(\frac{1}{2}-q\right)\ell$.
Therefore, the total length of the bent rod is $\ell$. We focus on the slender
limit, i.e., $\epsilon=\frac{a}{\ell}\ll 1$. The rod self-propels due to
solute flux on the rod in the $\mathbf{e}_{1}$-$\mathbf{e}_{2}$ plane and can
rotate about the $\mathbf{e}_{1}\times\mathbf{e}_{2}$ plane.
$\mathbf{e}_{r}$-$\mathbf{e}_{t}$ represent the directions normal and
tangential to the bent rod. To non-dimensionalize our problem setup, we scale
all the lengths by $\ell$. $s$ represents the dimensionless coordinate along
the rod. $s=0$ is the hinge and $s=\frac{1}{2}\pm q$ are the ends of two arms.
The dimensionless solute flux is represented by $j(s)$. Both
$\mathbf{e}_{1}$-$\mathbf{e}_{2}$ and $\mathbf{e}_{r}$-$\mathbf{e}_{t}$ are
define in reference frame of particle. $\mathbf{e}_{1}$ is defined such that
is aligned with the positive arm, i.e., $0\leq s\leq\frac{1}{2}+q$.
$\mathbf{e}_{2}$ is perpendicular to $\mathbf{e}_{1}$. $\mathbf{e}_{r}$ and
$\mathbf{e}_{t}$ are expressed as a function of $\mathbf{e}_{1}$ and
$\mathbf{e}_{2}$; see Eq. (1a). The lab reference frame is given by
$\mathbf{e}_{x}$-$\mathbf{e}_{y}$; see section II.4.

##  II Theoretical framework

###  II.1 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 $\theta$; see Fig. 1. The lengths of the two arms are assumed to be
$\left(\frac{1}{2}+q\right)\ell$ and $\left(\frac{1}{2}-q\right)\ell$, where
$\ell$ is the total length and $q$ is the length asymmetry parameter. We note
that $q\in\left[-\frac{1}{2},\frac{1}{2}\right]$. Both the arms are assumed to
be of the same radius $a$ such that $\frac{a}{\ell}=\epsilon\ll 1$. 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 [85, 52].

We non-dimensionalize the coordinate system by $\ell$. We introduce the arc-
length parameter $s$ to describe the position along the centerline of the rod
such that $-\frac{1}{2}+q\leq s\leq\frac{1}{2}+q$. $s=0$ represents the hinge,
whereas $s=q\pm\frac{1}{2}$ denote the end of the two arms. For consistency,
we refer to the arm where $0\leq s\leq q+\frac{1}{2}$ as the positive arm and
the arm where $-\frac{1}{2}+q\leq s<0$ as the negative arm. The shape of the
bent rod is thus dictated by $q$ and $\theta$.

We assume that the rod only propels in the $\mathbf{e}_{1}$-$\mathbf{e}_{2}$
plane and can rotate about the
$\mathbf{e}_{3}=\mathbf{e}_{1}\times\mathbf{e}_{2}$ axis. The direction
$\mathbf{e}_{1}$ is always assumed to be aligned with the positive arm. For
convenience, we also define $\mathbf{e}_{t}$ and $\mathbf{e}_{r}$ as the
tangential and normal directions to the rod, respectively, such that

$\mathbf{e}_{t}=\begin{cases}-\cos\theta\ \mathbf{e}_{1}-\sin\theta\
\mathbf{e}_{2}&s<0\\\ \mathbf{e}_{1}&s\geq 0\end{cases},$ (1a)

$\mathbf{e}_{r}=\begin{cases}-\sin\theta\ \mathbf{e}_{1}+\cos\theta\
\mathbf{e}_{2}&s<0\\\ \mathbf{e}_{2}&s\geq 0\end{cases}.$ (1b)

Note that both $\mathbf{e}_{1}$-$\mathbf{e}_{2}$ and
$\mathbf{e}_{t}$-$\mathbf{e_{r}}$ are in the particle frame of reference and
moves with the particle. In section II.4, we define
$\mathbf{e}_{x}$-$\mathbf{e}_{y}$ as 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 $\mathbf{x}_{h}(s)=s\mathbf{e}_{t}$. The
center of mass of the bent-rod is denoted as $\mathbf{x}_{\rm com}$. To model
self-propulsion through catalytic activity, we follow the common practice in
literature [75, 81, 82, 58, 92, 93, 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 $\mathbf{U}$ and $\mathbf{\Omega}$, respectively. The objective of
this paper is to determine the particle trajectory in the limit $\epsilon\ll
1$ for a given $q$, $\theta$, and $j(s)$. The limit $\epsilon\ll 1$ enables us
to invoke first-order slender body theory to evaluate $\mathbf{U}$ and
$\mathbf{\Omega}$. 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].

###  II.2 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 $J_{\textrm{ref}}$. We define
$c(s,r)$ to be dimensionless solute concentration, scaled by
$\frac{aJ_{\textrm{ref}}}{D}$, where $D$ is the solute diffusivity.
Mathematically, our objective is to evaluate concentration at the slip plane
$c_{s}(s)$ for a given $q$, $\theta$ and $j(s)$. We note that $c_{s}(s)$ is
equivalent to the surface concentration from the outer solution
$c^{\textrm{out}}(s,\epsilon)$ [82, 75, 95] (also see Appendix A). We define
the Péclet number of the rod as Pe$=\frac{U_{\textrm{ref}}\ell}{D}$, where
$U_{\textrm{ref}}$ is a typical velocity scale. For representative values, we
focus on ref. [96]. Here, catalytic spheres with $2\mu$m diameters were driven
in H2O2 solutions. $D=O(10^{-8})$ m2/s, $\ell_{\textrm{ref}}=O(10^{-6})$ m,
and $U_{\textrm{ref}}=O(10^{-5})$ 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

$\nabla^{2}c=0,r\geq\epsilon.$ (2)

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

$-\epsilon\hat{\mathbf{n}}\cdot\nabla c=j(s),\ r=\epsilon,$ (3a) where
$\hat{\mathbf{n}}$ is the surface normal vector. The far field boundary
condition for solute concentration reads $c=0,r\rightarrow\infty.$ (3b)

Since $\epsilon\ll 1$, we use boundary-layer theory [95] to evaluate
$c_{s}(s)$. As outlined in Appendix A, we divide the fluid volume into an
inner and an outer region in the radial direction $\mathbf{e}_{r}$. In the
inner region, we stretch the coordinates such that $\rho=r\epsilon^{-1}$ and
evaluate $c^{\textrm{in}}(s,\rho)$ with the boundary condition in Eq. (3a).
Subsequently, in the outer region, we evaluate $c^{\textrm{out}}(s,r)$ as a
line integral of diffusive sources of strength $\alpha(s)$. We determine
$\alpha(s)$ via an asymptotic matching
$c_{\textrm{in}}(s,\infty)=c_{\textrm{out}}(s,\epsilon)$. From the leading
order behavior, we obtain $\alpha(s)=\frac{j(s)}{2}$, which gives

$c_{s}(s)=\frac{1}{2}\displaystyle\int_{q-\frac{1}{2}}^{q+\frac{1}{2}}\frac{j(s^{\prime})}{\left|\mathbf{x}_{h}(s)+\epsilon\mathbf{e}_{r}(s)-\mathbf{x}_{h}(s^{\prime})\right|}\rm
ds^{\prime}.$ (4)

The expression for $c_{s}(s)$ in Eq. (4) is consistent with the results of
Schnitzer and Yariv [75] and Poehnl and Uspal [82]. Our analysis deviates from
previous studies since we account for a composite slender body where
$\mathbf{e}_{t}$ and $\mathbf{e}_{r}$ are different for the two arms. We
define the concentration differences between the junction and the respective
end-points as $\Delta
c^{q+\frac{1}{2}}_{0}=c_{s}\left(q+\frac{1}{2}\right)-c_{s}(0)$, and $\Delta
c_{q-\frac{1}{2}}^{0}=c_{s}(0)-c_{s}\left(q-\frac{1}{2}\right)$. As we show
later, $\Delta c^{q+\frac{1}{2}}_{0}$ and $\Delta c_{q-\frac{1}{2}}^{0}$ drive
particle motion.

###  II.3 Particle Velocity

We define a dimensionless fluid velocity $\mathbf{u}$ around the particle,
scaled by a reference velocity $U_{\rm ref}$, where
$U_{\textrm{ref}}=\frac{k_{B}TaJ_{\textrm{ref}}\lambda^{2}}{\mu D\ell}$, where
$k_{B}$ is the Boltzmann constant, $T$ is the absolute temperature, and
$\lambda$ is the interaction length scale. Typically,
$\lambda_{\textrm{ref}}=O(1)-O(10)$ nm [92]. Following the analyses in
literature [75, 93, 83], we represent the interaction between the solute and
the particle through a diffusiophoretic slip velocity [97, 92], such that

$\mathbf{u}_{\textrm{slip}}=M(s)\left(\mathbf{I}-\mathbf{e}_{r}\mathbf{e}_{r}\right)\cdot\nabla
c_{s}=M(s)\frac{dc_{s}}{ds}\mathbf{e}_{t},$ (5)

where $M(s)$ is a non-dimensional lumped mobility parameter scaled by
$\frac{k_{B}T\lambda^{2}}{\mu}$. For simplicity, we consider $M(s)=1$. To
justify $M(s)=1$, we note
$M=O\left(\frac{U_{\textrm{ref}}}{k_{B}Tc_{\textrm{ref}}\lambda^{2}/(\mu\ell_{\textrm{ref}})}\right)$.
For $k_{B}=O(10^{-23})\ $J/K, $T=300$ K, $c_{\textrm{ref}}=O(10^{24})$ m-3,
$\lambda=O(10^{-9})$ m [92], $\mu=O(10^{-3})$ Pa$\cdot$s,
$\ell_{\textrm{ref}}=O(10^{-6})$ m, we get $M=O(1)$. Before proceeding with
hydrodynamic calculations, we highlight that while Eq. (4) is derived for a
diffusiophoretic system, the results can also be extended to a thermophoretic
system. Specifically, the surface concentration $c_{s}$ can be replaced by the
dimensionless surface temperature (appropriately scaled by subtracting far-
field temperature), and the point sources of solute flux can be replaced by
point sources of heat flux.

The fluid velocity at the particle surface for a phoretic particle with
translational velocity $\mathbf{U}$ (scaled by $U_{\rm ref}$) and rotational
velocity $\mathbf{\Omega}$ (scaled by $\frac{U_{\rm ref}}{\ell}$) is described
as

$\mathbf{u}(s,\epsilon)=\mathbf{U}+\mathbf{\Omega}\times\left(\mathbf{x}_{h}-\mathbf{x}_{\textrm{com}}\right)+\mathbf{u}_{\textrm{slip}}.$
(6)

In addition, the fluid velocity vanishes in the far-field,
$\mathbf{u}\left(s,r\to\infty\right)\to 0$. Instead of solving for the
velocity field, we employ the Lorentz reciprocal theorem to estimate
$\mathbf{U}$ and $\mathbf{\Omega}$ [89, 58]. To this end, we relate the fluid
velocity $\mathbf{u}$ to an auxiliary Stokes flow around the same particle
geometry, with velocity $\mathbf{u}^{j}$ as follows [82],

$\displaystyle\int_{S}\mathbf{u}\cdot\mathbf{\sigma}^{j}\cdot\hat{\mathbf{n}}\
\rm
dS=\displaystyle\int_{S}\mathbf{u}^{j}\cdot\mathbf{\sigma}\cdot\hat{\mathbf{n}}\
\rm dS,$ (7)

where the surface stresses $\mathbf{\sigma}$ and $\mathbf{\sigma}^{j}$ are
scaled by $\frac{4\pi\mu U_{\rm ref}}{\ell}$. The auxiliary problem has a no-
slip condition at the rigid particle boundary, or

$\mathbf{u}^{j}(s,\epsilon)=\mathbf{U}^{j}+\mathbf{\Omega}^{j}\times\left(\mathbf{x}_{h}-\mathbf{x}_{\textrm{com}}\right).$
(8)

The fluid velocity vanishes in the far-field,
$\mathbf{u}^{j}(s,r\rightarrow\infty)=\mathbf{u}^{j,\infty}\to\mathbf{0}$.
Since the particle is constrained to move in the
$\mathbf{e}_{1}$-$\mathbf{e}_{2}$ plane,
$\mathbf{U}=U_{1}\mathbf{e}_{1}+U_{2}\mathbf{e}_{2}$ and
$\mathbf{\Omega}=\Omega_{3}\mathbf{e}_{3}$, implying we need three different
auxiliary problems to obtain $\mathbf{U}$ and $\mathbf{\Omega}$. We consider
the auxiliary problems to be classified as $j\in\\{1,2,3\\}$ such that

$\displaystyle\mathbf{U}^{1}=V_{0}\mathbf{e}_{1},\
\mathbf{\Omega}^{1}=\mathbf{0},$ (9a)
$\displaystyle\mathbf{U}^{2}=V_{0}\mathbf{e}_{2},\
\mathbf{\Omega^{2}}=\mathbf{0},$ (9b)
$\displaystyle\mathbf{U}^{3}=\mathbf{0},\
\mathbf{\Omega^{3}}=\Omega_{0}\mathbf{e}_{3}.$ (9c)

We substitute Eqs. (6) and (8) into Eq. (7) to obtain

$\displaystyle\mathbf{U}\cdot$
$\displaystyle\int_{S}\mathbf{\sigma^{j}}\cdot\hat{\mathbf{n}}\ \rm
dS\it+\mathbf{\Omega}\cdot\int_{S}\left(\mathbf{x}_{h}(s)-\mathbf{x}_{\rm
com}\right)\times\mathbf{\sigma}^{j}\cdot\hat{\mathbf{n}}\ \rm
dS+\it\int_{S}\mathbf{u}_{\rm slip}\it\cdot\sigma^{j}\cdot\hat{\mathbf{n}}\
\rm dS$ (10)
$\displaystyle\it=\mathbf{U}^{j}\cdot\int_{S}\mathbf{\sigma}\cdot\hat{\mathbf{n}}\
\rm dS+\it\Omega^{j}\cdot\int_{S}\left(\mathbf{x}_{h}(s)-\mathbf{x}_{\rm
com}\right)\it\times\mathbf{\sigma}\cdot\hat{\mathbf{n}}\ \rm dS.$

Since a self-propelling particle is force and torque free [58, 89, 82], the
right hand side of Eq. (10) vanishes. Further, we can represent the surface
integral of the hydrodynamic stress $\mathbf{\sigma}^{j}$ in terms of the line
integral of the hydrodynamic force density $\mathbf{f}^{j}$,
$\mathbf{\sigma}^{j}\cdot\hat{\mathbf{n}}\ \rm{dS}=\it\mathbf{f}^{j}\ \rm ds$.
In slender body theory [90, 80], the force density $\mathbf{f}^{j}$ (scaled by
$4\pi\mu U_{\rm ref}$) is defined as,

$\mathbf{f}^{j}(s)=-\mathscr{A}\left(\mathbf{I}-\frac{1}{2}\mathbf{e}_{t}\mathbf{e}_{t}\right)\cdot\left(\mathbf{u}^{j}-\mathbf{u}^{j,\infty}\right),$
(11)

where $\mathscr{A}=\frac{1}{\ln\left(1/\epsilon\right)}$. The first two
integrals in Eq. (10) are the non-dimensional hydrodynamic force
$\mathbf{F}^{j}$ and torque $\mathbf{T}^{j}$ on the particle in the auxiliary
flow problem. We thus obtain

$\mathbf{U}\cdot\mathbf{F}^{j}+\mathbf{\Omega}\cdot\mathbf{T}^{j}+\displaystyle\int_{q-\frac{1}{2}}^{q+\frac{1}{2}}\mathbf{u}_{\textrm{slip}}\cdot\mathbf{f}^{j}(s)\
\rm ds=0.$ (12)

To evaluate $\mathbf{F}^{j}$ and $\mathbf{T}^{j}$ for given $\mathbf{U}^{j}$
and $\mathbf{\Omega}^{j}$, we invoke the results of Roggeveen and Stone [88].
The resistance coefficients, as provided in Roggeveen and Stone [88],
$\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ (see Appendix B), allow us to
write

$\begin{bmatrix}F^{j}_{1}\\\ F^{j}_{2}\\\
T^{j}_{3}\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}&\tilde{B}_{13}\\\
A_{12}&A_{22}&\tilde{B}_{23}\\\
\tilde{B}_{13}&\tilde{B}_{23}&C_{33}\end{bmatrix}\begin{bmatrix}U^{j}_{1}\\\
U^{j}_{2}\\\ \Omega^{j}_{3}\end{bmatrix}.$ (13)

Substituting Eqs. (9), (11), and (13) in Eq. (12), we obtain a system of
linear equations to calculate phoretic motion of the particle

$\displaystyle\begin{bmatrix}U_{1}\\\ U_{2}\\\ \Omega_{3}\end{bmatrix}=$
$\displaystyle\mathscr{A}\mathbf{R}^{-1}\begin{bmatrix}\displaystyle\int_{\left(q-\frac{1}{2}\right)}^{\left(q+\frac{1}{2}\right)}\mathbf{u}_{\rm
slip}\cdot(\mathbf{I}-\frac{1}{2}\mathbf{e}_{t}\mathbf{e}_{t})\cdot\mathbf{e}_{1}\
\rm ds\\\
\displaystyle\int_{\left(q-\frac{1}{2}\right)}^{\left(q+\frac{1}{2}\right)}\mathbf{u}_{\rm
slip}\cdot(\mathbf{I}-\frac{1}{2}\mathbf{e}_{t}\mathbf{e}_{t})\cdot\mathbf{e}_{2}\
\rm ds\\\
\displaystyle\int_{\left(q-\frac{1}{2}\right)}^{\left(q+\frac{1}{2}\right)}\mathbf{u}_{\rm
slip}\cdot(\mathbf{I}-\frac{1}{2}\mathbf{e}_{t}\mathbf{e}_{t})\cdot\mathbf{e}_{3}\times\left(\mathbf{x}_{h}\left(s\right)-\mathbf{x}_{\rm
com}\right)\ \rm ds\end{bmatrix},$ (14)

where $\mathbf{R}$ is the scaled resistance matrix. Note that the velocities
are zero if $\mathbf{u}_{\rm slip}=0$. Substituting the expression of
$\mathbf{u}_{\rm slip}$ from Eq. (5) in the above equation and integrating, we
obtain

$\begin{bmatrix}U_{1}\\\ U_{2}\\\
\Omega_{3}\end{bmatrix}=\mathscr{A}\begin{bmatrix}a_{11}&a_{12}&\tilde{b}_{13}\\\
a_{12}&a_{22}&\tilde{b}_{23}\\\
\tilde{b}_{13}&\tilde{b}_{23}&c_{33}\end{bmatrix}\begin{bmatrix}\frac{1}{2}\Delta
c_{0}^{q+\frac{1}{2}}-\frac{\cos\theta}{2}\Delta c_{q-\frac{1}{2}}^{0}\\\
-\frac{\sin\theta}{2}\Delta c_{q-\frac{1}{2}}^{0}\\\
\frac{1}{4}\left(q-\frac{1}{2}\right)^{2}\sin\theta\Delta
c_{0}^{q+\frac{1}{2}}+\frac{1}{4}\left(q+\frac{1}{2}\right)^{2}\sin\theta\Delta
c_{q-\frac{1}{2}}^{0}\end{bmatrix}.$ (15)

The coefficients of the mobility, i.e., $\mathbf{a}$, $\mathbf{b}$, and
$\mathbf{c}$, can be obtained by inverting the resistance matrix in Eq. (13)
(see Appendix B). We underscore that Eq. (15) is valid for an arbitrary
$j(s)$, which modifies the values of $\Delta c_{0}^{q+\frac{1}{2}}$ and
$\Delta c_{q-\frac{1}{2}}^{0}$. Additionally, Eq. (15) demonstrates that
$\mathbf{U}$ and $\mathbf{\Omega}$ are dictated by concentration differences
between the end-points and the hinge, i.e., $\Delta c_{0}^{q+\frac{1}{2}}$ and
$\Delta c_{q-\frac{1}{2}}^{0}$, and the spatial variation of the concentration
profile does not influence the particle motion.

###  II.4 Particle Trajectories

$\mathbf{U}$ and $\mathbf{\Omega}$ given by Eq. (15) are in the reference
frame of the particle, i.e., $\mathbf{e}_{1}$-$\mathbf{e}_{2}$. To obtain an
equation of motion for the center of mass of the particle, i.e.,
$\left[x(t),y(t)\right]$, in a universal frame of reference
$\mathbf{e}_{x}$-$\mathbf{e}_{y}$, we employ the rotation matrix as

$\frac{d}{dt}\begin{bmatrix}\vspace{0.08in}x(t)\\\ \vspace{0.08in}y(t)\\\
\phi(t)\end{bmatrix}=\begin{bmatrix}\vspace{0.08in}\cos\phi(t)&-\sin\phi(t)&0\\\
\vspace{0.08in}\sin\phi(t)&\cos\phi(t)&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}\vspace{0.08in}U_{1}\\\
\vspace{0.08in}U_{2}\\\ \Omega_{3}\end{bmatrix},$ (16)

where $\phi$ is the angle between $\mathbf{e}_{1}$ and $\mathbf{e}_{x}$, and
$t$ is dimensionless time scaled by $\ell/U_{\textrm{ref}}$. Without any loss
of generality, we define $x(0)=y(0)=\phi(0)=0$. By integrating (16) in time,
we get

$\begin{aligned}
\left(x(t)-\mathscr{X}\right)^{2}+\left(y(t)-\mathscr{Y}\right)^{2}=\mathscr{R}^{2}\end{aligned},$
(17)

where $\mathscr{X}=-\frac{U_{2}}{\Omega_{3}}$ and
$\mathscr{Y}=\frac{U_{1}}{\Omega_{3}}$. Eq. (17) demonstrates that the
particle moves in a circular trajectory where the center of the trajectory
depends on the particle velocities. The radius of curvature of the trajectory
is given by $\mathscr{R}=\sqrt{\frac{U_{1}^{2}+U_{2}^{2}}{\Omega_{3}^{2}}}$.
The turn frequency of the particle is simply $f=\frac{\Omega_{3}}{2\pi}$. We
note that Eq. (17) implicitly assumes that diffusion is significantly faster
than the particle motion for Pe = $O(10^{-3})$, as previously discussed in
section II.2. At this point, we will like to further justify the pseudosteady
state assumption. Since our predictions suggest a circular trajectory, one
might argue that the continuous release of solute could leave a trace behind
that can build up over time. To quantify this effect, we briefly restore
dimensions and perform an order of magnitude analysis. The total amount of
solute released by the particle for time $\tau$ is given by $m\sim J_{0}(2\pi
a\ell)\tau$, where $J_{0}$ is the dimensional surface flux. The trace
concentration $C_{\textrm{trace}}$ can be estimated as $C_{\textrm{trace}}\sim
m/\left(\frac{\pi}{6}\delta^{3}\right)$, where $\delta\sim\sqrt{4D\tau}$ is
the diffusive boundary layer thickness. Upon rearrangement, it is easy to
obtain that
$\frac{C_{\textrm{trace}}}{C_{\textrm{ref}}}\sim\frac{3}{2}j_{0}\sqrt{\frac{\ell^{2}}{D\tau}}$.
Since Eq. (17) is calculated for $\tau\sim\ell/U_{\textrm{ref}}$,
$\frac{C_{\textrm{trace}}}{C_{\textrm{ref}}}\sim\frac{3}{2}j_{0}\textrm{Pe}^{1/2}\ll
c_{s}\sim O\left(j_{0}\log\left(\frac{1}{\epsilon}\right)\right)$. Therefore,
solute concentration generated due to surface activity is significantly higher
than the trace solute concentration, which enables us to employ the quasi-
steady state assumption.

##  III Results and Discussion

Figure 2:  Surface concentration $c_{s}(s)$ for different combinations of $q$,
$\theta$ and $j(s)$. (a, b) $j(s)=H(s)$ and (c,d) $j(s)=1$. (a, c)
$\theta=\frac{\pi}{2}$ and $q=-0.25$ (pink), $q=0$ (blue), and $q=0.25$
(yellow). (b, d) $q=0$ and $\theta=\frac{\pi}{18}$ (pink),
$\theta=\frac{\pi}{4}$ (blue), and $\theta=\frac{3\pi}{4}$ (yellow).
$\epsilon=10^{-3}$ for all cases.

###  III.1 Validation

For a given $j(s)$, $q$, and $\theta$, Eqs. (4), (15), and (17) can be
employed to calculate the particle trajectory. To test the validity of our
approach, we compare our predictions with the experimental trajectories of a
L-shaped thermophoretic particle, as described in Kümmel et al. [85]. We note
that the L-shaped particle utilized by the authors does not consist of
cylindrical arms. However, at the slender limit, the cross-sectional geometry
will have a minor impact on the calculations. Therefore, we are able to use
the methodology described above.

The geometry of L-shaped particle in Kümmel et al. [85] yields $\ell=12$
$\mu$m, $\epsilon=0.125$, $q=-\frac{1}{8}$, and $\theta=\frac{\pi}{2}$. Since
only the positive arm of the L-shaped geometry in Kümmel et al. [85] is
thermophoretically active, $j(s)=H(s)$, where $H$ is the Heaviside function.
Substituting geometric values and $j(s)=H(s)$ in Eq. (4), we evaluate
$\mathbf{U}$ and $\mathbf{\Omega}$ by using Eq. (15), and obtain the
trajectory through Eq. (17). First, we are able to recover the circular
trajectory observed experimentally by Kümmel et al. [85]. Further, we obtain
$\ell\mathscr{R}=8.1\ \mu$m, which is in surprisingly good quantitative
agreement with the experimentally reported radius of $7.5\ \mu$m. We emphasize
that no fitting parameter was employed for this comparison. Finally, while the
values of $\mathbf{U}$ and $\mathbf{\Omega}$ are dependent on the magnitude of
$j$ and $\epsilon$, the value of $\mathscr{R}$ is independent of the magnitude
$j$ and $\epsilon$, and is only a function of $q$ and $\theta$. Physically,
this result implies that the radius of the circular trajectory of the particle
is less sensitive to the magnitude of the reactive flux and the slenderness of
the geometry. However, both the speed of the particle and the frequency of
rotation increase with an increase in slenderness, i.e., lower $\epsilon$, and
an increase in reactive flux.

Interestingly, for $j(s)=H(s)$, $\Delta c_{0}^{q+\frac{1}{2}}=0$ and $\Delta
c_{q-\frac{1}{2}}^{0}\neq 0$, see Fig. 2(a,b). This finding implies that for
$j(s)=H(s)$, the concentration about the active arm is symmetric and the
concentration difference across it is negligible. The $3\times 1$ matrix in
the RHS of Eq. (15) is the effective force and torque generated by the active
propulsion mechanism. Since $\Delta c_{0}^{q+\frac{1}{2}}=0$ for $j(s)=H(s)$,
the effective force and torque are driven solely by the diffusive tail on the
negative arm, i.e., the passive arm. Furthermore, our analysis reveals that
the effective force is parallel to the passive arm. This result is in contrast
to the findings of Kümmel et al. [85], who assumed that a force is applied
perpendicular to the active arm. In fact, our analysis enables us to move away
from the artificial force and torque approach [87], as we are able to evaluate
the effective force and torque by utilizing the Lorentz reciprocal theorem.
Our analysis suggests that, for multiple arms, the effective force
$\mathbf{F}_{\textrm{eff}}\propto\sum_{i}\Delta c_{0}^{i}\mathbf{e}_{ti}$,
where $\Delta c_{0}^{i}$ is the concentration difference along the
$i^{\textrm{th}}$ arm and $\mathbf{e}_{ti}$ is the parallel vector to the arm.

Figure 3: Particle trajectories for $j(s)=1$ and for different $q$ and
$\theta$. All the trajectories are calculated up to the time period of
rotation for the particle with $q=0.35$ and $\theta=\frac{\pi}{2}$. The
trajectories of individual particles are not drawn to a fixed scale; only the
total length of each particle is $1$. $\epsilon=10^{-3}$ for all trajectories.

###  III.2 Uniform Surface Flux

To gain deeper insight into the impact of $q$ and $\theta$ on the motion of
particles, we consider the scenario of $j(s)=1$. Under this assumption, the
particle motion is solely driven by geometrical asymmetry since the entire
particle is uniformly active. In Fig. 2(c), when the angle between the arms is
fixed to $\theta=\frac{\pi}{2}$, $c_{s}\left(s+\frac{1}{2}-q\right)$ is
relatively insensitive to $q$. This is because the concentration at a
particular location is dictated by the contributions of the point sources
($\alpha(s)$ defined in Appendix A) spread along the particle. For a fixed
angle, the position of the hinge does not have a significant impact on these
contributions and thus the concentration is relatively insensitive to $q$. In
contrast, for equal arm lengths ($q=0$), a change in $\theta$ has significant
impact on $c_{s}(s)$, see Fig. 2(d). This occurs because the relative distance
between any point on the particle surface with respect to the other arm
changes significantly with $\theta$. For smaller angles, the relative distance
between the arms decreases and thus the surface concentration is higher.

By employing Eqs. (4), (15), and (17), we obtain particle trajectories for a
combination of different $q$ and $\theta$ values; see Fig. 3 (also see
Supplementary Video 1). We find that when the two arms are of equal lengths,
i.e., $q=0$, the particle moves in a straight line irrespective of $\theta$.
For $q=0$, due to symmetry, $\Delta c_{0}^{q+\frac{1}{2}}+\Delta
c_{0}^{q-\frac{1}{2}}=0$; see Fig. 2(d). Therefore, the effective torque on
the particle, as mathematically described in Eq. (15), is zero. In addition,
the rotational-translational coupling terms also cancel out such that the net
turning induced by geometric asymmetry is zero and $\Omega_{3}=0$. Therefore,
the particles only translate. For $q\neq 0$, we observe that the particle
turns; see Fig. 3. As is evident from the figure, the degree and speed of
turning depend on $q$ and $\theta$.

Figure 4: Phase plots for $j(s)=1$ and $\epsilon=10^{-3}$. (a) The speed of
the particle $\sqrt{U_{1}^{2}+U_{2}^{2}}$. (b) The angular velocity of the
particle $\Omega_{3}$. A maximum in $U_{\textrm{tot}}$ and $\Omega_{3}$ is
observed for $q\approx 0.3$ and $\theta\approx\frac{\pi}{2}$.
$\epsilon=10^{-3}$ for both plots.

To understand the quantitative dependence of particle trajectory on
geometrical parameters, we construct phase plots for
$U_{\textrm{tot}}=\sqrt{U_{1}^{2}+U_{2}^{2}}$ and $\Omega_{3}$; see Fig. 4.
First, we observe that $U_{\textrm{tot}}$ is symmetric about $q=0$, i.e.,
$U_{\textrm{tot}}(q,\theta)=U_{\textrm{tot}}(-q,\theta)$. This result is
consistent with the expectation, i.e., for $j(s)=1$, switching the positive
and negative arms should not influence the speed. In contrast,
$\Omega_{3}(q,\theta)=-\Omega_{3}(-q,\theta)$, which indicates that the
direction of turning switches, but the magnitude remains the same. This
dependence agrees with expectation, since changing the sign of $q$ changes the
initial orientation, which causes the particle to turn in the opposite
direction.

One of the key features that emerges from Fig. 4 is the presence of a maximum
in $U_{\textrm{tot}}$ and $\Omega_{3}$ around $\theta\approx\frac{\pi}{2}$ and
$q\approx 0.3$. Therefore, the maximum $U_{\textrm{tot}}$ and $\Omega_{3}$ is
observed for an L-shaped particle with one arm 4 times as long as the other
one. We write Eq. (15) as follows

$\begin{bmatrix}\mathbf{U}\\\
\mathbf{\Omega}\end{bmatrix}=\mathscr{A}\mathbf{M}(q,\theta)\cdot\begin{bmatrix}\mathbf{F}_{\textrm{eff}}(q,\theta)\\\
\mathbf{T}_{\textrm{eff}}(q,\theta)\end{bmatrix},$ (18)

where $\mathbf{M}$ is the hydrodynamic mobility and
$\mathbf{F}_{\textrm{eff}}$ and $\mathbf{T}_{\textrm{eff}}$ are the effective
driving propulsion force and torque. The geometric parameters $q$ and $\theta$
influence $\mathbf{M}$, $\mathbf{F}_{\textrm{eff}}$, and
$\mathbf{T}_{\textrm{eff}}$, but the impact of $q$ and $\theta$ is different
on $\mathbf{F}_{\textrm{eff}}$ and $\mathbf{T}_{\textrm{eff}}$. For instance,
$\mathbf{F}_{\textrm{eff}}$ is dependent on $\Delta c_{0}^{q+\frac{1}{2}}$ and
$\Delta c_{q-\frac{1}{2}}^{0}$, which are less sensitive to $q$ and more
sensitive to $\theta$; see Fig 2(c,d). $\mathbf{T}_{\textrm{eff}}$ has a
similar dependence on $\theta$ but also varies with $q$. In contrast, the
mobility parameter $\mathbf{M}$ is more sensitive to $q$ than $\theta$; see
Appendix B. The interplay of mobility and driving force yields the optimum
values of $q$ and $\theta$. We note that these optimal values will change if a
different $j(s)$ is used, since that will directly impact the values of
$\mathbf{F}_{\textrm{eff}}$, and $\mathbf{T}_{\textrm{eff}}$.

We note that the dependence of $\epsilon$ in Eq. (18) only comes from
$\mathscr{A}$. This occurs because we ignore the higher order corrections in
$\mathbf{M}$, $\mathbf{F}_{\textrm{eff}}$ and $\mathbf{T}_{\textrm{eff}}$. To
this end, we acknowledge that our analysis will be less applicable when the
interactions between the two rods are significant. Additionally, our analysis
does not focus on circumferential variation of chemical activity, which could
be important in certain scenarios, as discussed in refs. [83, 84, 86]. Even
with these limitations, our analysis provides a convenient starting point to
predict trajectories and design self-propelling composite slender bodies.

##  IV Concluding Remarks

In summary, this article presents a theoretical analysis to predict the two-
dimensional motion of self-propelling bent rods. By employing slender body
theory and the Lorentz reciprocal theorem, we derive Eqs. (4), (15), and (17)
to predict the particle trajectories for given $q$, $\theta$, and $j(s)$. Our
analysis reveals that the trajectory of the particle is circular, such that
the radius is dependent on $q$ and $\theta$, and is insensitive to the
slenderness parameter $\epsilon$. The speed and the frequency of the particle
are dependent on the values of $q$, $\theta$, and $\epsilon$, and display an
optimum with the phase plot of $q$ and $\theta$. The results outlined here
provide a convenient method to design self-propellers for micro-bot
applications in diagnostics and drug delivery [7, 8]. For instance, if the
propeller is required to travel in a straight line, the propeller should have
symmetric arms. If the propeller is required to move and turn faster, optimum
values of $q$ and $\theta$ and a smaller $\epsilon$ should be utilized.
Physically, our analysis also helps to clarify the role of the particle
geometry in hydrodynamic mobility as well as the self-propelling driving
force. As such, we can move away from the external force and torque approach,
and show that the self-propulsion is driven by the concentration differences
across each arm. In the future, our work can be extended to more complex
geometries, such as slender bodies with multiple arms and activity profiles
with circumferential variation.

## Acknowledgements

The authors acknowledge Robert Davis, Bhargav Rallabandi, James Roggeveen, and
Howard Stone for their helpful input in the preparation of this manuscript. We
would also like to thank Filipe Henrique, Nathan Jarvey, Ritu Raj, and Gesse
Roure for their helpful discussions and feedback leading to the culmination of
this work. A. Ganguly and A. Gupta also acknowledge the donors of the American
Chemical Society Petroleum Research Fund for partial support of this research.

## Appendix A: Derivation of surface concentration

Ignoring any circumferential dependence, we rewrite Eqs. (2) and (3a) in
cylindrical coordinates as

$\displaystyle\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial
c}{\partial r}\right)+\frac{\partial^{2}c}{\partial s^{2}}=0,\ r>\epsilon,$
(A1a)

$\epsilon\frac{\partial c}{\partial r}=-j(s),\ r=\epsilon,$ (A1b)

and $c\left(r\to\infty\right)\to 0$. To solve our system of equations and
obtain the surface concentration $c_{s}\left(s\right)$, we divide the fluid
volume into an inner region and an outer region in the radial direction
$\mathbf{e}_{r}$. In the inner region, we introduce a stretched radial
coordinate $\rho=r/\epsilon$. The concentration in the inner region is denoted
by $c^{i}$. Eqs. (A1a) and (A1b) in the inner region can be written as,

$\displaystyle\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial
c^{i}}{\partial\rho}\right)+\epsilon^{2}\frac{\partial^{2}c^{i}}{\partial
s^{2}}=0,\ \rho>1,$ (A2a)

$\displaystyle\frac{\partial c^{i}}{\partial\rho}=-j(s),\ \rho=1.$ (A2b)

We expand the solute concentration in the inner region in orders of
$\epsilon$;
$c^{i}=\log\left(\frac{1}{\epsilon}\right)c^{i}_{0}+c^{i}_{1}+O(\epsilon)$ and
solve for the leading order and first order solutions. The resultant
concentration profile in the inner region is found to be

$c^{i}=-j(s)\log\rho+K(z;\epsilon)+O(\epsilon^{2}),$ (A3)

where $K$ in Eq. (A3) is obtained by matching with the outer solution.

In the outer region, we represent the concentration $c^{o}$ as being due to a
distribution of sources $\alpha(s)$ along an infinitesimally thin space curve
along the particle centerline. The position of any point in the fluid volume
$\mathbf{x}(r,s)=s\mathbf{e}_{t}(s)+r\mathbf{e}_{r}(s)$. The outer region
concentration is represented as

$c^{o}=\displaystyle\int_{q-\frac{1}{2}}^{q+\frac{1}{2}}\frac{\alpha(s^{\prime})}{\left|\mathbf{x}(r,s)-s^{\prime}\mathbf{e}_{t}(s^{\prime})\right|}\
\textrm{d}s^{\prime}.$ (A4)

Note that the concentration field in the outer region, proposed in equation
(A4), decays to zero in the far-field. By using Eq. (1), we evaluate the
concentration profile to be

$\displaystyle
c^{o}=\begin{cases}\displaystyle\int_{q-\frac{1}{2}}^{0}\frac{\alpha(s^{\prime})}{\left[\left(s-s^{\prime}\right)^{2}+r^{2}\right]^{1/2}}\
\
\textrm{d}s^{\prime}+\int_{0}^{q+\frac{1}{2}}\frac{\alpha(s^{\prime})}{\left[\left(s+s^{\prime}\cos\theta\right)^{2}+\left(r+s^{\prime}\sin\theta\right)^{2}\right]^{1/2}}\
\ \textrm{d}s^{\prime}&s<0,\\\
\displaystyle\int^{0}_{q-\frac{1}{2}}\frac{\alpha(s^{\prime})}{\left[\left(s+s^{\prime}\cos\theta\right)^{2}+\left(r+s^{\prime}\sin\theta\right)^{2}\right]^{1/2}}\
\textrm{d}s^{\prime}+\displaystyle\int_{0}^{q+\frac{1}{2}}\frac{\alpha(s^{\prime})}{\left[\left(s-s^{\prime}\right)^{2}+r^{2}\right]^{1/2}}\
\textrm{d}s^{\prime}&s\geq 0.\end{cases}$ (A5)

Due to the singularity in Eq. (A5), we propose a substitution of variables
$\zeta=s^{\prime}-s$ and integrate. For $s\geq 0$

$\displaystyle c^{o}(r,s\geq 0)$
$\displaystyle=\alpha(s)\int_{-s}^{q+\frac{1}{2}-s}\frac{\textrm{d}\zeta}{\left[\zeta^{2}+r^{2}\right]^{1/2}}+\int_{-s}^{q+\frac{1}{2}-s}\frac{\alpha(s+\zeta)-\alpha(s)}{\left[\zeta^{2}+r^{2}\right]^{1/2}}\
\textrm{d}\zeta+$ (A6)
$\displaystyle+\int_{q-\frac{1}{2}-s}^{-s}\frac{\alpha(s+\zeta)}{\left[\left(s+\zeta\right)^{2}+2\left(s\cos\theta+r\sin\theta\right)\left(s+\zeta\right)+s^{2}+r^{2}\right]^{1/2}}\
\textrm{d}\zeta.$

Similarly, we can separate out the singularity for $s<0$. The integrals
demonstrate that $c^{o}(r,s)=2\alpha(s)\log r+c^{o}_{1}+O(\epsilon)$. To match
the outer solution to the inner solution from Eq. (A3), we write
$\lim_{\rho\to\infty}c^{i}=\lim_{r\to 0}c^{o}$, which shows
$\alpha(s)=\frac{j(s)}{2}$. The surface concentration thus at the slip plane
is written as Eq. (4). To evaluate the integrals in Eq. (4), one needs to
remove the singularity, as shown in Eq. (A6).

We note that the concentration profiles provided in Fig. 2 display a
discontinuity in gradients at the hinge. This is expected since the first-
order slender body theory ignores the interaction between the two arms and
superposes the corresponding contributions to $c_{s}$. The interactions are
more pronounced when $\theta\rightarrow 0$, and thus our analysis would need
to be appropriately corrected.

## Appendix B: Determining the mobility coefficients for a slender bent rod

The resistance coefficients described Eq. (13) are given as (see Roggeveen and
Stone [88])

$\begin{bmatrix}A_{11}\\\ A_{12}\\\ A_{22}\\\ \tilde{B}_{13}\\\
\tilde{B}_{23}\\\
C_{33}\end{bmatrix}=\begin{bmatrix}\frac{1}{8}\left(5-2q+(2q-1)\cos
2\theta\right)\\\ \frac{1}{8}\left(2q-1\right)\sin 2\theta\\\
1+\frac{1}{4}\left(2q-1\right)\sin^{2}\theta\\\
-\frac{1}{8}\left(1-2q\right)^{2}\sin\theta\\\
\frac{1}{8}\left(\left(2q+1\right)^{2}+\left(1-2q\right)^{2}\cos\theta\right)\\\
\frac{1}{12}+q^{2}\end{bmatrix}.$ (A7)

We invert $\mathbf{R}$ and obtain the following mobility coefficients

$\begin{bmatrix}a_{11}\\\ a_{12}\\\ a_{22}\\\ \tilde{b}_{13}\\\
\tilde{b}_{23}\\\
c_{33}\end{bmatrix}=\begin{bmatrix}\dfrac{-\frac{1}{64}\left(\left(1+2q\right)^{2}+\left(1-2q\right)^{2}\cos\theta\right)^{2}+\left(\frac{1}{12}+q^{2}\right)\left(1+\frac{1}{4}\left(-1+2q\right)\sin^{2}\theta\right)}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\\\\[25.0pt]
\dfrac{-\frac{1}{64}\left(1-2q\right)^{2}\left(\left(1+2q\right)^{2}+\left(1-2q\right)^{2}\cos\theta\right)\sin\theta-\frac{1}{8}\left(-1+2q\right)\left(\frac{1}{12}+q^{2}\right)\sin
2\theta}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\\\\[25.0pt]
\dfrac{\frac{1}{8}\left(\frac{1}{12}+q^{2}\right)\left(5-2q+\left(-1+2q\right)\cos
2\theta\right)-\frac{1}{64}\left(1-2q\right)^{4}\sin^{2}\theta}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\\\\[25.0pt]
\dfrac{\frac{1}{32}\left(-1+2q\right)\left(-3+4q+4q^{2}+\left(1+2q\right)^{2}\cos\theta\right)\sin\theta}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\\\\[25.0pt]
\dfrac{\frac{1}{64}\left(-\left(\left(\left(1+2q\right)^{2}+\left(1-2q\right)^{2}\cos\theta\right)\left(5-2q+\left(-1+2q\right)\cos
2\theta\right)\right)-2\left(-1+2q\right)^{3}\cos\theta\sin^{2}\theta\right)}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\\\\[25.0pt]
\dfrac{\frac{1}{32}\left(17-4q^{2}+\left(-1+4q^{2}\right)\cos
2\theta\right)}{\frac{1}{768}\left(19+88q^{2}-144q^{4}-12\left(1-4q^{2}\right)^{2}\cos\theta+\left(1+8q^{2}-48q^{4}\right)\cos
2\theta\right)}\end{bmatrix}$ (A8)

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††thanks: These authors contributed equally.††thanks: These authors
contributed equally.

# Epitaxial Growth and Domain Structure Imaging of
Kagome Magnet Fe3Sn2

Shuyu Cheng  Department of Physics, The Ohio State University, Columbus, Ohio
43210, USA  Igor Lyalin  Department of Physics, The Ohio State University,
Columbus, Ohio 43210, USA  Alexander J. Bishop  Department of Physics, The
Ohio State University, Columbus, Ohio 43210, USA  Roland K. Kawakami
<EMAIL_ADDRESS>Department of Physics, The Ohio State University,
Columbus, Ohio 43210, USA

###### Abstract

Magnetic materials with kagome crystal structure exhibit rich physics such as
frustrated magnetism, skyrmion formation, topological flat bands, and
Dirac/Weyl points. Until recently, most studies on kagome magnets have been
performed on bulk crystals or polycrystalline films. Here we report the
synthesis of high-quality epitaxial films of topological kagome magnet Fe3Sn2
by atomic layer molecular beam epitaxy. Structural and magnetic
characterization of Fe3Sn2 on epitaxial Pt(111) identifies highly ordered
films with c-plane orientation and an in-plane magnetic easy axis. Studies of
the local magnetic structure by anomalous Nernst effect imaging reveals in-
plane oriented micrometer size domains. The realization of high-quality films
by atomic layer molecular beam epitaxy opens the door to explore the rich
physics of this system and investigate novel spintronic phenomena by
interfacing Fe3Sn2 with other materials.

††preprint: APS/123-QED Figure 1:  Left: Top view of an individual Fe3Sn layer
with kagome structure (top) and Sn2 layer with honeycomb structure (bottom),
respectively. Right: Side view of the crystal structure of Fe3Sn2 consisting
of alternating stacking of two Fe3Sn kagome layers and one Sn2 layer.

In recent years, studies on magnetic topological materials with kagome
lattices have become one of the hottest frontiers of condensed matter
research, due to their exotic physical properties in both real space and
momentum space [1, 2]. In momentum space, angle-resolved photoemission
spectroscopy (ARPES) experiments on Mn3Sn, Fe3Sn2, FeSn, and CoSn [3, 4, 5, 6]
show that kagome lattices give rise to Dirac cones and flat bands which are
topologically protected and are of particular interest. In addition, scanning
tunneling spectroscopy finds evidence for topological flat bands as a sharp
peak in the local density of states [7]. These topologically nontrivial
features result in signatures of anomalous transport (e.g. chiral anomaly) in
magnetotransport experiments [3, 8]. Furthermore, it is theoretically
predicted that the band structures of the kagome topological magnets can be
controlled by tuning of their magnetic structures [3, 1]. In real space, the
kagome topological magnets have layered structures with spins occupying
corner-sharing triangular lattices, which leads to geometrical spin
frustration [9, 10]. A surprisingly large anomalous Hall effect (AHE) and
magneto-optic Kerr effect (MOKE) have been reported in noncollinear
antiferromagnet Mn3Sn, even with vanishingly small net magnetization [10, 11].
Skymrmion spin textures have been observed in ferromagnetic Fe3Sn2 resulting
from the competition of exchange, dipolar, and Zeeman energies [12, 13].
However, most of the studies on the kagome magnets have been done on bulk
materials [14, 15, 10, 16, 12, 4, 11, 13, 5] with a few papers reporting the
growth and characterization of epitaxial films [17, 18, 19, 20, 21]. Looking
forward, epitaxial films will provide new opportunities for exploring
fundamental physics and potential applications by tuning the dimensionality in
thin films and interfacing Fe3Sn2 with different materials. In addition,
preparing and visualizing a well-defined domain structure is crucial for
studying the spatially resolved physics of kagome magnets, such as spin
transfer torque and spin-orbit torque induced domain wall and skyrmion motion
[22, 23]. While the local magnetic structure of bulk Fe3Sn2 has been recently
studied and room temperature magnetic skyrmions were revealed [12, 13], there
have been no studies probing the domain structure of Fe3Sn2 in thin films.

In this paper, we report the epitaxial growth and magnetic domain imaging of
high-quality Fe3Sn2 thin films on Pt(111)/Al2O3(0001) substrates. We utilize
atomic layer molecular beam epitaxy (MBE) to synthesize Fe3Sn2 films by
sequentially depositing Fe3Sn kagome layers and Sn2 layers (see Figure 1).
Structural characterization by in situ reflection high energy electron
diffraction (RHEED) and X-ray diffraction (XRD) confirm the crystalline
structure of Fe3Sn2. The magnetic properties of Fe3Sn2 are investigated using
MOKE and the anomalous Nernst effect (ANE) and consistently observe easy-plane
magnetic anisotropy with square hysteresis loops. Using a microscopy technique
based on ANE, we successfully image the in-plane oriented domain structure of
the epitaxial Fe3Sn2 films and investigate the magnetization reversal as a
function of applied field.

(a)  (b)

Figure 2: (a) RHEED patterns for the Al2O3(0001) substrate, 5 nm Pt film, and
20 nm Fe3Sn2 film measured with the beam along $[1\bar{1}00]$ (left column)
and $[11\bar{2}0]$ (right column) directions of the substrate. (b)
Oscillations in the normalized RHEED intensity as a function of time. The
RHEED intensity is measured within the red box and normalized by the
background in the blue box.

Fe3Sn2 is a ferromagnet with a high Curie temperature, $T_{C}$ = 670 K [24],
and saturation magnetization of 1.9 $\mu_{B}$ per Fe at low temperature [4].
Fig. 1 shows the crystal structure of Fe3Sn2 (space group R$\bar{3}$m, with
lattice constants a = 5.338 Å and c = 19.789 Å [24]) which consists of Fe3Sn
kagome layers and Sn spacer layers. In each Fe3Sn monolayer, the Fe atoms form
corner-sharing equilateral triangles surrounding hexagons, with Sn atoms
sitting in the center of the hexagons. The alternating sequence of one Sn2
monolayer with honeycomb lattice and two Fe3Sn kagome layers produces the
layered crystal structure of Fe3Sn2.

Based on this layered structure, we synthesized Fe3Sn2 thin films on top of
epitaxial Pt(111) buffer layers on Al2O3(0001) substates by atomic layer MBE.
The epitaxial growth was performed in an MBE chamber with a base pressure of
$4\times 10^{-10}$ Torr. Films were deposited on Al2O3(0001) substrates (MTI
Corporation) prepared by annealing in air at 1000 ∘C for 3 hours followed by
annealing in ultrahigh vacuum (UHV) at 500 ∘C for 30 minutes. A 5 nm Pt(111)
buffer layer was deposited from an e-beam evaporator (Pt: 99.99%, Kurt J.
Lesker) onto the Al2O3 (0001) substrate by growing the first 0.6 nm at 440 ∘C
and the rest 4.4 nm while cooling down from 140 ∘C to 80 ∘C. The Pt buffer
layer was post-annealed at 300 ∘C to improve the crystallinity and surface
roughness. The Fe3Sn2 layer was grown on Pt(111) at room temperature using the
following atomic layer MBE sequence: deposit two atomic layers of Fe3Sn with a
Fe:Sn flux ratio of 3:1, deposit one atomic layer of Sn2 with the growth time
same as two Fe3Sn layers, then repeat. The Fe and Sn fluxes were generated
from Knudsen cells (Fe: 99.99%, Alfa Aesar; Sn: 99.998%, Alfa Aesar) and the
growth rates were determined using a quartz deposition monitor that was
calibrated by x-ray reflectometry. Typical growth rates were $\sim$ 0.85
Å/min, $\sim$ 0.67 Å/min, and $\sim$ 0.45 Å/min for Fe, Sn, and Pt,
respectively. To protect the sample from oxidation, a 3 nm Pt or 5 nm CaF2
capping layer was deposited on top of the Fe3Sn2.

RHEED patterns were measured during growth to characterize the epitaxial
growth and determine the in-plane lattice constants. Figure 2a shows the RHEED
patterns for the Al2O3(0001) substrate (top row), 5 nm Pt buffer layer (middle
row), and the Fe3Sn2 layer after 20 nm of growth (bottom row). The left and
right columns show patterns taken for the beam along the $[1\bar{1}00]$ and
$[11\bar{2}0]$ directions of the substrate, respectively. With in-plane
rotation of the sample, RHEED exhibits the same pattern every 60∘ (i.e. six-
fold rotation symmetry) with the patterns alternating between
$[1\bar{1}00]$-type and $[11\bar{2}0]$-type every 30∘. For the in-plane
epitaxial alignment, the Pt(111) and Fe3Sn2(0001) surface unit cells are
aligned with each other and rotated 30∘ with respect to the the Al2O3(0001)
substrate, as
$a_{Al_{2}O_{3}}\approx(\sqrt{3}/2)2a_{Pt}\approx(\sqrt{3}/2)a_{Fe_{3}Sn_{2}}$
(bulk values for in-plane lattice constant are $a_{Al_{2}O_{3}}=4.759$ Å,
$2a_{Pt}=5.549$ Å, $a_{Fe_{3}Sn_{2}}=5.338$ Å).

The streaky patterns observed during Fe3Sn2 growth signify diffraction from a
two-dimensional surface. In addition, we observe oscillations (Fig. 2b) in the
normalized RHEED intensity where the maxima occurs for the Fe3Sn termination
and the minima occurs for the Sn2 termination. The normalization is performed
by dividing the intensity of the background and is helpful for canceling
variations in the incident beam intensity and background lighting. Except for
the change in RHEED intensity, we did not observe any other significant
differences in the RHEED pattern between Sn2 and Fe3Sn terminations.
Nevertheless, the presence of RHEED oscillations in atomic layer MBE confirms
the modulation of the surface termination during growth.

(a)  (b)

Figure 3: (a) 2$\theta$-$\omega$ scan of a 20 nm Fe3Sn2 film grown on
c-sapphire. (b) A triple axis $\omega$-relative scan (rocking curve) of the
Fe3Sn2 (009) peak (red) with a Gaussian fitting (blue).

Films grown by this method were studied with XRD to analyze their crystal
structure. A representative $\omega$-$2\theta$ scan of a 20 nm film is shown
in Figure 3a and includes the Fe3Sn2 (0009) peak with several Laue-
oscillations, indicating a high degree of film smoothness. The out-of-plane
lattice parameter extracted from analysis of this scan is 19.84 Å which agrees
well with previous reports of 19.789 Å [24]. A peak from the 5 nm Pt(111)
buffer layer produces a shoulder on the Fe3Sn2 peak. Larger range scans do not
show additional peaks from the Fe3Sn2 or any other materials. To further
analyze the crystallinity of our films, rocking curve scans about the Fe3Sn2
(0009) peak were taken and analyzed. By scanning the sample angle $\omega$
while keeping the detector fixed, this characterizes the angular distribution
of the (0001) orientation relative to the film normal within the x-ray spot.
Figure 3b contains a rocking curve or $\omega$-relative scan of the Fe3Sn2
using a triple-axis analyzer to achieve high resolution of the desired peak.
Fitting the peak with a standard Gaussian yields a full width half maximum
(FWHM) of 0.0027∘. Such a sharp peak indicates that films grown by atomic
layer MBE have excellent mosaicity.

(a)  (b)

Figure 4: (a). Longitudinal (red) and polar (blue) MOKE hysteresis loops of a
20 nm Fe3Sn2 film (Sample I). (b). Zoomed-in scans for five different samples
with thicknesses ranging from 5 nm to 20 nm, showing similar magnetic
properties. The data are normalized to the saturation values and offset for
clarity.

(a)  (b)  (c)  (d)

Figure 5:  (a) Schematics of thermal gradient microscopy. The laser beam is
scanned over the sample surface, the induced local ANE voltage reflects the
local magnetic properties. (b) Microscope image of a typical device, (the
dashed rectangle corresponds to the area imaged in (d)) (c) ANE hysteresis
loop of a 20 nm Fe3Sn2 film. (d) Magnetization reversal of a 20 nm film
through multidomain state imaged by ANE at a series of magnetic fields
$\mu_{0}$Hx.

To investigate the in-plane and out-of-plane magnetic properties of the Fe3Sn2
films, we measured longitudinal and polar MOKE hysteresis loops. The samples
were probed using a linearly-polarized He-Ne laser (633 nm wavelength, $\sim
100~{}\mu$W power, $\sim 100~{}\mu$m spot size) and a polarizing beamsplitter,
photodiode bridge, and lock-in amplifier (463 Hz intensity modulation) to
detect the Kerr rotation. The laser beam had a $\sim$45∘ angle of incidence
for longitudinal MOKE and normal incidence for polar MOKE. Figure 4a shows a
representative longitudinal hysteresis loop (red curve) measured on a 20 nm
thick Fe3Sn2 sample (Sample I). The square hysteresis loop with a coercivity
of 2.4 mT indicates ferromagnetic order with in-plane magnetization. In
contrast, the polar hysteresis loop (blue curve) shows a small Kerr rotation
with slight variation with out-of-plane magnetic field. Together, the
longitudinal and polar MOKE loops show that the Fe3Sn2 samples have an easy-
plane magnetic anisotropy. This agrees with a previous study of Fe3Sn2 films
grown by sputter deposition [20]. To check the consistency of the synthesis
and magnetic properties, we investigated additional samples grown by the same
method with thickness varying from 5 to 20 nm. The results are shown in Fig.
4b. All the samples show very similar magnetic properties, with square in-
plane hysteresis loops and similar coercive fields.

The magnetic domain structure of Fe3Sn2 films are of interest due to the
observation of skyrmions in bulk Fe3Sn2, but has not yet been studied in thin
films. Longitudinal MOKE microscopy with oblique angle incidence can detect
the in-plane magnetization and therefore determine in-plane domain structure
of our Fe3Sn2 films. However, in this manuscript, we choose to use thermal
gradient microscopy (TGM) [25, 26, 27] over longitudinal MOKE to image domain
structure because we found that it has a better signal-to-noise ratio in our
experimental setup.

TGM is based on moving a laser spot over the sample surface, and recording a
voltage induced by the local laser heating. The thermal gradient generated in
the out-of-plane direction $Z$ and a component of magnetization in the $X$
direction give rise to the anomalous Nernst effect, which is detected as a
voltage along the $Y$ direction, $V_{ANE}\sim\left[\nabla
T\times\mathbf{M}\right]$ (see Fig. 5a).

For the ANE imaging, we fabricated 10 $\mu$m wide Hall bar devices by a
combination of photolithography and argon ion milling (Fig. 5b). The laser
excitation for the thermal gradient was produced by a frequency-doubled
(BaB2O4 crystal) mode-locked Ti:Sapphire laser for a wavelength of 400 nm. The
laser beam with 0.7 mW power was focused by a 50$\times$ objective lens (NA of
0.6) to a spot size of 0.9 $\mu$m, and a fast steering mirror in the 4f
alignment scheme was used for scanning the laser spot over the sample surface.
The intensity of the beam was modulated at a frequency of 120 kHz and the
generated ANE voltage was detected using a lock-in amplifier.

We first utilized the ANE microscope to measure a detailed hysteresis loop at
a fixed position. As shown in Fig. 5c for magnetic field along the $X$
direction, the hysteresis loop shows a gradual reversal followed by a sharp
switching behavior with coercivity of 1.9 mT. This has a similar coercivity
but more gradual initial reversal than the in-plane hysteresis loops obtained
by MOKE (Fig. 4b).

The origin of the different hysteresis properties is revealed by imaging the
magnetic domain structure of Fe3Sn2 films at a series of magnetic fields. A
representative sequence during the magnetization reversal is shown in Fig. 5d.
Starting at -5.0 mT, the magnetization is in a saturated state along $-X$
(blue). The reversal initiates with the nucleation of white regions with
$M_{x}\approx 0$, mainly at the edges of the sample. This can be explained by
the minimization of domain wall energy as the edge boundary does not
contribute a domain wall energy cost. The nucleation at the edges initiates
magnetization reversal which results in a more rounded hysteresis loop
compared to the uniform films. With increasing magnetic field, domains of
opposite polarity grow inward and coalesce across the channel. At about +1.0
mT, the magnetic structure is in a multidomain state with characteristic
features (e.g. blue and red regions) ranging from 1 to 10 microns in size. By
+2.0 mT, most of the magnetic moments have switched to $+X$ direction, with
only a few regions remaining along $-X$. Finally, at +5.0 mT the magnetization
reversal is complete and the films is fully saturated along $+X$.

In conclusion, we report the growth and characterization of kagome ferromagnet
Fe3Sn2 thin films on Pt(111)/Al2O3(0001) by atomic layer molecular beam
epitaxy. Structural characterization by in situ RHEED and XRD confirm the high
quality of the epitaxial Fe3Sn2 films. The magnetic properties were
investigated by magneto-optical Kerr effect and anomalous Nernst effect,
confirming the easy-plane magnetic anisotropy of the thin films. Finally, the
local magnetic structure was probed by ANE microscopy revealing the presence
of in-plane oriented micrometer size domains during magnetization reversal.
These results highlight the potential for epitaxial growth to enable new
scientific research in kagome magnets at the intersection of topology and
magnetism.

## Acknowledgements

S.C., I.L., and R.K.K. acknowledge support from DARPA Grant No. D18AP00008.
A.J.B. and R.K.K. acknowledge support from DOE Grant No. DE-SC0016379. This
research was partially supported by the Center for Emergent Materials, an NSF
MRSEC, under award number DMR-2011876.

## Author contributions

S.C., I.L., and R.K.K. conceived the experiments. S.C. conducted the MBE
growth and MOKE measurements. I.L. conducted the ANE measurements. A.J.B.
conducted the XRD measurements. All authors participated in data analysis and
preparation of the manuscript.

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