# Orbits, spirals, and trapped states:
Dynamics of a phoretic Janus particle in a radial concentration gradient

Parvin Bayati  Department of Chemistry, The Pennsylvania State University,
University Park, Pennsylvania, 16802, USA  Stewart A. Mallory<EMAIL_ADDRESS>Department of Chemistry, The Pennsylvania State University, University Park,
Pennsylvania, 16802, USA

###### Abstract

A longstanding goal in colloidal active matter is to understand how gradients
in fuel concentration influence the motion of phoretic Janus particles. Here,
we present a theoretical description of the motion of a spherical phoretic
Janus particle in the presence of a radial gradient of the chemical solute
driving self-propulsion. Radial gradients are a geometry relevant to many
scenarios in active matter systems and naturally arise due to the presence of
a point source or sink of fuel. We derive an analytical solution for the Janus
particle’s velocity and quantify the influence of the radial concentration
gradient on the particle’s trajectory. Compared to a phoretic Janus particle
in a linear gradient in fuel concentration, we uncover a much richer set of
dynamical behaviors, including circular orbits and trapped stationary states.
We identify the ratio of the phoretic mobilities between the two domains of
the Janus particle as a central quantity in tuning their dynamics. Our results
provide a path for developing novel protocols for tuning the dynamics of
phoretic Janus particles and mixing fluid at the microscale. In addition, this
work suggests a method for quantifying the surface properties of phoretic
Janus particles, which have proven challenging to probe experimentally.

###### keywords:

colloidal active matter, phoretic Janus particles, phoretic motion,
micro/nanomotors, active particles, self-propulsion, external gradients,
chemotaxis, diffusiophoresis

Department of Chemical Engineering, The Pennsylvania State University,
University Park, Pennsylvania, 16802, USA

##  1 Introduction

Phoretic Janus particles are a novel class of colloids that show promise for
applications related to mixing, sorting, and chemical delivery 1, 2, 3, 4.
These typically micron-sized particles self-propel due to their unique ability
to generate and sustain chemical gradients across their surface 5, 6. The
particles’ Janus nature, where their surface is composed of two chemically
distinct regions, is a common design feature for introducing self-propulsion.
Experimentally, it is possible to tune both the size and composition of these
regions, and there now exists a sizeable catalog of phoretic Janus particles,
including bimetallic and platinum-coated colloids, biodegradable Janus
micromotors, and colloidal particles coated in two different enzymes 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17. An appropriate fuel source is a second design
element nearly universal to these systems. Hydrogen peroxide is a popular
choice as one of the particle regions is usually metallic and catalytic.
However, the fuel choice is flexible depending on the particle’s composition.
For example, enzyme-coated particles use the corresponding substrate as the
fuel source 18. With their diverse compositions and fuel sources, phoretic
Janus particles offer a robust design platform for engineering behavior at the
microscale.

Notably, the ability of these particles to autonomously navigate complex
microfluidic environments makes them an ideal candidate for a range of
chemical delivery and sensing applications, including targeted drug delivery
and environmental remediation 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.
For example, recent in situ work shows CaCO3 Janus particles exhibit a pH
sensitivity that can induce a chemotactic response toward HeLa cancer cells
31. Regarding environmental remediation, Wang et al. 28 recently proposed a
strategy for removing microplastics using photocatalytic TiO2 Janus particles.
Additionally, phoretic Janus can serve as a tool for fluid mixing and
directing self-assembly at the microscale 32, 33. Examples include
restructuring colloidal gels by incorporating a small fraction of phoretic
Janus particles into the gel network 34, 35, 36, 37, powering primitive
micromachines 38, 39, 40, and generating bulk fluid flow by fabricating self-
pumping walls patterned with Janus micropillar arrays 41 or by trapping
phoretic Janus particles near boundaries or surfaces 42, 43.

Any application utilizing phoretic Janus particles requires a deep
understanding of how they explore and respond to their environment. A
ubiquitous feature of these systems is the role of gradients in chemical fuel
concentration, which can dramatically affect their single particle and
collective behavior 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60. Several studies have focused on the behavior of a single phoretic
Janus particle in a linear fuel concentration gradient and shown phoretic
Janus particles will undergo a chemotactic response where the particle will
reorient to move parallel or anti-parallel to the gradient 61, 62, 63, 64.
However, the characterization of the motion of phoretic Janus particles in
chemical gradients of other geometries is limited. A critical case that has
received lesser attention and the focus of this study is radially symmetric
gradients generated from the presence of a point sink or source of the
chemical fuel self-propelling the particle.

Using a combination of analytical theory and numerical simulation, we quantify
the influence of the radial concentration field on the Janus particle’s
trajectory. The particle can exhibit a rich array of behaviors that strongly
depend on its surface properties, initial configuration, and the strength of
the sink or source. The particle can migrate toward or away from the sink or
source, similar to a passive particle in an external chemical gradient. In
addition, we identify conditions that trap the particle in a stationary state
at a fixed distance from the sink or source. Furthermore, the particle’s
motion is no longer rectilinear as in the case of a particle in a linear fuel
concentration gradient but can undergo a spiraling motion. We identify
conditions that stabilize the spiraling trajectories leading to a circular
orbit about the sink or source. As stationary and orbiting states offer an
innovative way to blend and pump fluid continually, our findings suggest
potential applications in fluid mixing and microscale transport. In addition,
the characteristic dynamics of a phoretic Janus particle in the presence of a
point sink or source can serve as a diagnostic tool to identify its surface
properties, which has been an experimental challenge.

Figure 1: Schematic of a spherical Janus particle with Janus balance
$\chi=-\cos\Phi$ immersed in solution near a source or sink of the same
chemical solute driving self-propulsion. The distance of the particle from the
source $R$ and its orientation
$\Theta=\cos^{-1}(\bm{\hat{R}}\cdot\bm{\hat{d}})$ fully specify the particle
configuration.

Model System – We consider a phoretic Janus particle of diameter $a$ with
bilateral symmetry along a predefined orientation unit vector $\bm{\hat{d}}$
as shown in Fig. 1. We implement a standard generic model for the phoretic
self-propulsion mechanism where one region of the particle (gray) emits a
particular chemical solute and the other region absorbs the solute (blue) 65,
66. The chemical solute represents the fuel driving self-propulsion. The
relative ratio of the absorbing to the emitting region is defined via the
Janus balance $\chi$ such that a half-covered particle has $\chi=0$, a
particle emitting solute over its entire surface has $\chi=1$, and for a fully
absorbing particle $\chi=-1$. To preserve mass balance, the rate of emission
$Q_{e}$ and absorption $Q_{a}$ of the solute are constant, and there is no net
change of solute such that $S_{e}Q_{e}-S_{a}Q_{a}=0$, where $S_{e}$ and
$S_{a}$ are the surface areas of the emitting and absorbing regions of the
particle, respectively. Under these steady-state conditions, a simple
relationship exists between the emission and absorption rates of the two
regions and the Janus balance $\chi$ given by $Q_{a}/Q_{e}=(1+\chi)/(1-\chi)$.

Here, we focus on the athermal low Reynolds number limit, where Brownian
motion is negligible, and particle motion is dominated solely by the phoretic
self-propulsion mechanism. In this regime, the particle’s motion is
deterministic and, as we demonstrate, confined to a two-dimensional plane. The
position of the particle relative to the source is given by a vector
$\bm{R}=R\cos\varphi\,\bm{\hat{x}}+R\sin\varphi\,\bm{\hat{y}}$ where $R$ is
the radial distance from the source and $\varphi$ is the angle between
$\bm{R}$ and the positive $x$-axis as shown in Fig. 1. We obtain the
trajectory of the particle by integrating the equations of motion:

$\frac{dR}{dt}=U_{R}$ (1a) $\frac{d\varphi}{dt}=\frac{U_{\varphi}}{R}$ (1b)
$\frac{d\gamma}{dt}=\Omega_{z}$ (1c)

where $U_{R}=\bm{U}\cdot\bm{\hat{R}}$ and
$U_{\varphi}=\bm{U}\cdot\bm{\hat{\varphi}}$ are the radial and tangential
components of the translation velocity $\bm{U}$, respectively. The
corresponding radial and tangential unit vectors are given by
$\bm{\hat{R}}=\cos\varphi\,\bm{\hat{x}}+\sin\varphi\,\bm{\hat{y}}$ and
$\bm{\hat{\varphi}}=\bm{\hat{z}}\times\bm{\hat{R}}=-\sin\varphi\,\bm{\hat{x}}+\cos\varphi\,\bm{\hat{y}}$,
respectively. The orientational dynamics of the particle are given by Eq. (1c)
where $\gamma$ is the angle between the orientation vector $\bm{\hat{d}}$ and
the positive $x$-axis and $\Omega_{z}$ is the only nonzero component of the
angular velocity of the particle. It is useful to define an auxiliary angle
$\Theta=\gamma-\varphi$ which is the angle between $\bm{\hat{R}}$ and the
orientation vector $\bm{\hat{d}}$, and from Eq. (1b,c) it follows that
$d\Theta/dt=d\gamma/dt-d\varphi/dt=\Omega_{z}-U_{\varphi}/R$. A particle’s
configuration relative to the singularity is fully specified by $\Theta$ and
$\bm{R}$.

The Stokes equations prescribe the dynamics of the fluid and are given by

$\eta\nabla^{2}\bm{u}-\nabla p=0$ (2a) $\nabla\cdot\bm{u}=0$ (2b)

where $\eta$, $\bm{u}$ and $p$ are the dynamic viscosity, fluid velocity, and
pressure, respectively. In the laboratory frame, the boundary conditions are
$\bm{u}=0$ at infinity and on the surface of the particle
$\bm{u}=\bm{U}+\bm{\Omega}\times(\bm{r}-\bm{r}_{0})+\bm{u}_{s}$ where
$\bm{u}_{s}$ is the slip velocity of the fluid at the particle’s surface.
Using the force and torque-free condition on the particle and the Lorentz
reciprocal theorem 67, the translation and angular velocity of a particle in
an unbounded domain are 68

$\bm{U}=-\frac{1}{4\pi a^{2}}\int_{s}\bm{u}_{s}~{}dS$ (3a)
$\bm{\Omega}=-\frac{3}{8\pi a^{3}}\int_{s}\bm{\hat{n}}\times\bm{u}_{s}~{}dS$
(3b)

where $dS$ is a differential element of the surface, and the domain of
integration is over the entire surface of the particle $S$.

Equation (3) relates the slip velocity on the particle surface to the net
motion of the particle. The slip velocity arises from the interaction between
the solute molecule and the particle’s surface. It is well-known that a solute
gradient along the particle’s surface induces an osmotic pressure gradient,
generating a fluid flow within the Debye layer of the particle’s surface. In
the thin Debye layer limit 5, this slip velocity is assumed to be located on
the Janus particle surface and given by

$\bm{u}_{s}=-b(\bm{\hat{n}})\nabla_{s}C\big{|}_{s}$ (4)

where $C$ is the fuel concentration field,
$\nabla_{s}=(\mathbb{I}-\bm{\hat{n}}\bm{\hat{n}})\cdot\nabla$ the tangential
projection of the surface gradient operator, and $\bm{\hat{n}}$ the normal
unit vector on the Janus particle surface directed into the bulk solution. The
phoretic mobility $b(\bm{\hat{n}})$ can be either positive or negative and is
determined by the details of the molecular interaction between the Janus
particle and the solute particles 5, 69. We assume the particle’s phoretic
mobility is constant in a given surface region and denote the particle’s
absorbing and emitting sides as $b_{a}$ and $b_{e}$, respectively. A central
quantity in this study is the ratio of the phoretic mobilities of the two
regions, which we call the phoretic mobility ratio and denote by
$\beta=b_{a}/b_{e}$.

We treat the solute flux as being purely diffusive such that the fuel
concentration field evolves according to the continuity equation
$\partial_{t}C=D\nabla^{2}C+\alpha~{}\delta(\bm{r})$ where $D$ is the
diffusivity of the solute and $\alpha$ is the strength of the singularity,
which can be positive or negative. Furthermore, it is reasonable to assume the
fuel concentration field relaxes rapidly with respect to the particle’s motion
such that we can neglect its time dependence and assume $\partial C/\partial
t=0$. Under these conditions, the fuel concentration $C$ is given by Laplace’s
equation with a point singularity at the origin

$D\nabla^{2}(C-C_{\infty})+\alpha~{}\delta(\bm{r})=0\,.$ (5)

The boundary condition on the surface of the Janus particle is given by

$-D\bm{\hat{n}}\cdot\nabla
C(\bm{r})=Q_{e}H(\bm{\hat{n}}\cdot\bm{\hat{d}}+\chi)-Q_{a}\left[1-H(\bm{\hat{n}}\cdot\bm{\hat{d}}+\chi)\right]$
(6)

where $H(x)$ is the Heaviside function. The boundary condition for the fuel
concentration is assumed to be constant at infinity and denoted by
$C_{\infty}$.

To summarize our workflow for obtaining the particle’s trajectory, we first
solve Eq. (5) with the appropriate boundary conditions to determine the
concentration gradient along the surface of the Janus particle. Once the
gradient of the concentration field is known, we can compute the slip velocity
along the surface of the particle via Eq. (4), which in turn furnishes the
translation and angular velocities via Eq. (3). The final step is to obtain
the particle’s trajectory by integrating Eq. (1) with the known translational
and angular velocities.

Figure 2: Classification of the different types of fixed points that can arise
as a function of the ratio of the phoretic mobilities $\beta$ and singularity
strength $\tilde{\alpha}$ for a phoretic Janus particle with $\chi=0$ and
$U_{0}=1$. For clarity, the fixed point state diagrams are organized based on
the particle’s orientation: (a) $\Theta=0$, (b) $\Theta=\pi$, and (c)
$0<\Theta<\pi$. The purple points correspond to representative values in the
$\tilde{\alpha}\beta$-phase space where the $R\Theta$-phase portrait and real-
space trajectories are given in Fig. 3 and Fig. 4. The dashed line at
$\beta=-1$ corresponds to where the velocity vanishes for an isolated Janus
particle, demarcating whether a particle will propel along or opposite its
orientation vector.

##  2 Results

A central outcome of this work is an analytical solution for the translational
and rotational velocity of the Janus particle as a function of the ratio of
the phoretic mobilities $\beta=b_{a}/b_{e}$, the intrinsic velocity of the
particle $U_{0}=Q_{e}b_{e}/(2D)$, and the effective strength of the
singularity $\tilde{\alpha}=\alpha/(4\pi a^{2}Q_{e})$. We present the final
result here for brevity, but a detailed derivation is available in the
Supporting Information 70. The radial, tangential, and angular velocities of
the particle are given by

$\displaystyle\frac{U_{R}}{U_{0}}$
$\displaystyle=\Gamma[\chi,\beta]\cos\Theta-\tilde{\alpha}\left(2\frac{a^{2}}{R^{2}}-(1-\beta)M[\chi,R,\Theta]\right)$
(7a) $\displaystyle\frac{U_{\varphi}}{U_{0}}$
$\displaystyle=\sin\Theta\bigg{(}\Gamma[\chi,\beta]-\tilde{\alpha}(1-\beta)N[\chi,R,\Theta]\bigg{)}$
(7b)
$\frac{\Omega_{z}}{U_{0}/a}=-\frac{3}{2}\tilde{\alpha}(1-\beta)\sin\Theta~{}\omega[\chi,R,\Theta]$
(7c)

where $\Gamma[\chi,\beta]=1+\chi-(1-\beta)B[\chi]$ is a positive dimensionless
parameter that modulates the particle’s intrinsic velocity. The functional
form of $B[\chi]$ is given in the Supporting Information 70 along with a plot
illustrating its dependence on the Janus balance $\chi$, which shows
$B[\chi]=0$ for $\chi=-1$ and $\chi=1$, and obtains its maximum value of
$B[\chi]=0.5$ for $\chi=0$. The dimensionless parameters
$M[\chi,R,\Theta],N[\chi,R,\Theta]$, and $\omega[\chi,R,\Theta]$ are given as
infinite series whose explicit form are in the Supporting Information 70. Each
dimensionless parameter is a positive monotonically decreasing function of $R$
where the leading order term decays as $1/R^{2}$. For the equal Janus balance
case $\chi=0$, we show the radial dependence of these dimensionless parameters
for different values of the orientation $\Theta$ in the Supporting Information
70. The analytical solution for the velocities and the resulting trajectories
are in excellent agreement with Boundary Element Method simulations (see
Supporting Information 70).

The particle’s motion is confined to the plane containing its orientation
vector $\bm{\hat{d}}$ and radial vector $\bm{\hat{R}}$ and the only nonzero
component of the angular velocity is normal to this plane. The first terms of
Eqs. (7a) and (7b) correspond to the translational velocity of an isolated
phoretic Janus particle. The second terms in each expression correspond to the
velocity induced by the radial gradient generated by the singularity. Far from
the singularity (i.e., $R\rightarrow\infty$), these terms vanish, and Eq. (7)
reduces to that of a phoretic Janus particle in free space. The angular
velocity of the particle $\Omega_{z}$ [Eq. (7c)] is due exclusively to the
presence of the singularity and is strictly zero when $\beta=1$. For other
values of $\beta$, the presence of the singularity can induce particle
rotation.

Figure 3: Dynamics near a chemical source: Phase portrait (top) and real-space
trajectories (bottom) for phoretic Janus particles with different ratios of
the phoretic mobilities $\beta$ in the presence of a point source
$\tilde{\alpha}=10$. For each plot, particles have an equal Janus balance
$\chi=0$, and the intrinsic velocity of the particle is $U_{0}=1$. The fixed
points are shown as purple circles, and the $R\Theta$-phase space trajectories
and real space trajectories are color-coded accordingly. Videos of real-space
trajectories are provided in the Supporting Information 70.

As the motion of the Janus particle is deterministic and completely specified
by its initial position and orientation, we find that unless there exists an
orbiting trajectory or a stationary point where the velocities vanish, the
particle will either eventually collide with the singularity or move off to
infinity. Thus, a natural scheme for classifying the dynamical behavior is
identifying when fixed points occur as a function of the singularity strength
$\tilde{\alpha}$ and the particle’s surface properties.

The analytical solution for the particle’s velocity facilitates determining
the location of fixed points and the particle’s trajectory for any
$\tilde{\alpha}$, $\beta$, $\chi$, or $U_{0}$. However, for simplicity, we
restrict our discussion to a particle with equal Janus balance $\chi=0$ and
$U_{0}=1$. The equal Janus balance case is representative of many experimental
systems and qualitatively illustrates the main features of the dynamics,
including the emergence of fixed points. The most critical parameters
impacting a particle’s dynamics are the phoretic mobility ratio $\beta$ and
the singularity strength $\tilde{\alpha}$. The singularity strength
$\tilde{\alpha}$ is the only direct and easily controllable parameter of the
problem, whereas the material properties of the Janus particle will determine
the phoretic mobility ratio $\beta$. Hence, we investigate the occurrence of
fixed points as a function of $\tilde{\alpha}$ and $\beta$, which we refer to
as the $\tilde{\alpha}\beta$-phase space.

For each point in the $\tilde{\alpha}\beta$-phase space, we conducted an
exhaustive search as a function of $R$ and $\Theta$ to identify where the
different components of the velocities vanish. In our fixed point
classification scheme, we recognize a trapped or stationary state when
$U_{R}$, $U_{\varphi}$ and $d\Theta/dt$ vanish and an orbiting state when only
$U_{R}$ and $d\Theta/dt$ vanish with $U_{\varphi}$ remaining finite. As shown
in Fig. 2, by analyzing the eigenvalues of the Jacobian about a fixed point
71, we identify three types of stationary states: stable, unstable, and
saddle, as well as an orbiting state. Stable fixed points are stationary
states where any small perturbation in $R$ or $\Theta$ will result in the
particle returning to the initial fixed point. Unstable fixed points exhibit
opposite behavior where any small perturbation in $R$ or $\Theta$ will lead to
the particle moving away from the fixed point. Saddle points are of a mixed
character where it is stable for small perturbations in $R$ and unstable for
perturbations in $\Theta$, or vice versa. The defining feature of an orbiting
state is that the particle’s trajectory executes a closed circular path about
the singularity. We use a similar classification for the stability of orbiting
states.

For clarity, the fixed point state diagram is divided based on the particle’s
orientation where Fig. 2(a,b,c) corresponds to $\Theta=0$, $\Theta=\pi$ and
$0<\Theta<\pi$, respectively. For the values of $\tilde{\alpha}$ and $\beta$
located in the white regions of Fig. 2, no fixed point exists for any position
and orientation. An important landmark in the $\tilde{\alpha}\beta$-phase
space is the value of $\beta$, where the particle’s intrinsic velocity
vanishes. From Eq. (7), the intrinsic velocity vanishes for values of $\beta$
that satisfy $1+\chi-(1-\beta)B[\chi]=0$. In the equal Janus balance case,
$B[\chi=0]=\,0.5$, and thus the intrinsic velocity is zero for $\beta=\,-1$
(dashed line in Fig. 2), For this value of $\beta$, any particle motion is due
solely to the presence of the radial gradient. For values of $\beta<-1$, an
isolated particle with $U_{0}>0$ moves with the absorbing side of the particle
in front (i.e., along the $-\bm{\hat{d}}$ direction in Fig. 1), and for
$\beta>-1$, the particle moves with the emitting side of the particle in
front.

Figure 4: Dynamics near a chemical sink: Phase portrait (top) and real-space
trajectories (bottom) for phoretic Janus particles with different ratios of
the phoretic mobilities $\beta$ in the presence of a point sink $\alpha=-10$.
The other parameters are the same as Fig. 3, i.e., $\chi=0$ and $U_{0}=1$.
Videos of real-space trajectories are provided in the Supporting Information
70.

Figure 2(a) highlights the region in the $\tilde{\alpha}\beta$-phase space
where fixed points exist for $\Theta=0$. In this case, the orientation vector
$\bm{\hat{d}}$ is parallel to the radial vector $\bm{R}$ (i.e., the emitting
side of the particle is furthest from the source) and from Eq. (7),
$U_{\varphi}=\Omega_{z}=0$. Most fixed points are saddle points and occur for
a point source $\tilde{\alpha}>0$. However, there is a narrow region between
$-1<\beta<0$ where we find saddle points for a point sink. Additionally, there
is a narrow region $0\leq\beta\leq 1$ where it is possible to have an unstable
fixed point. A wider variety of fixed points occur for $\Theta=\pi$, where the
emitting side of the particle is closest to the source [see Fig. 2(b)]. As the
phoretic mobility ratio increases, there are regions in the
$\tilde{\alpha}\beta$-phase space with an unstable stationary point, both an
unstable point and saddle point, a saddle point, and a stable point. When a
particle is in the presence of a point source, only saddle points are observed
for $\Theta=\pi$, and the unstable and stable fixed points arise in the
presence of a point sink. For intermediate orientations $0<\Theta<\pi$, we
find all fixed points are orbiting states and are more prevalent when the
particle is in the presence of a point source [see Fig. 2(c)].

As a function of the source strength $\tilde{\alpha}$, the occurrence and
disappearance of a particular combination of fixed points can be used to
deduce the approximate range of a particle’s phoretic mobility ratio $\beta$.
The physical explanation for the emergence of a fixed point is a particle
adopts an orientation and distance from the source such that the relevant
components of its intrinsic velocity are equal and opposite to the velocities
induced by the presence of the singularity, resulting in those components of
the velocity vanishing. In general, fixed points are a robust feature of a
phoretic Janus particle in a radially symmetric gradient. Similar behavior was
observed for particles with different Janus balances $\chi$.

We now survey in more detail the different dynamical behaviors that arise in
the $\tilde{\alpha}\beta$-phase space by examining the $R\Theta$-phase
portrait and the corresponding real space trajectories for selected values of
$\tilde{\alpha}$ and $\beta$. For a known source strength $\tilde{\alpha}$,
the location and type of the fixed point and, more generally, the
$R\Theta$-phase portrait serve as a fingerprint for the particle’s surface
properties. In Fig. 3, we consider the case of a point source and choose a
representative source strength of $\tilde{\alpha}=10$ and explore various
values of $\beta$, which span the different regions outlined in Fig. 2. In the
Supporting Information 70, we include figures that illustrate how the location
of the fixed points and orbits change as a function of the strength of the
singularity $\tilde{\alpha}$. The general trend is that the location of the
fixed point moves further away from the singularity as its strength increases.
Similarly, the orbit radius increases for orbiting states as the singularity
strength increases, and the particle’s orientation asymptotically approaches
$\Theta\rightarrow\pi/2$. These trends agree with our previously mentioned
physical interpretation, whereas the strength of the singularity increases, so
does the contribution to the particles’ velocity from the presence of the
singularity. Thus, for a given intrinsic velocity of the particle $U_{0}$, a
fixed point or orbiting state will occur at a further distance from the
singularity or its orientation closer to $\pi/2$ to cancel the increased
contribution to the velocity from increasing the strength of the singularity.

In Fig. 3(a), we start with the most negative value of the phoretic mobility
ratio, $\beta=-10$, where there is a single saddle point at $\Theta/\pi=0$ and
$R/a\approx 5.1$. The saddle point is stable in $R$, as shown by the red and
blue trajectories, where any displacement will lead to the particle moving
back toward the fixed point and unstable for $\Theta$, where any slight
variation from $\Theta/\pi=0$ will lead the particle to move toward infinity
in a clockwise spiraling trajectory (green trajectory). As the phoretic
mobility increases, we enter a region where two saddle points emerge in the
$R\Theta$-phase portrait. For example, for $\beta=-2$, there are saddle points
for $\Theta/\pi=0$ and $\Theta/\pi=\pi$ at a distance of $R/a\approx 5.7$ and
$R/a\approx 2.5$, respectively [see Fig. 3(b)]. The further saddle point is
similar to that identified for $\beta=-10$ and is stable in $R$. However, the
saddle point that emerges closer to the source exhibits the opposite behavior,
where it is stable for changes in $\Theta$.

As $\beta$ transitions from negative to positive values, in addition to the
saddle point for $\Theta=0$, a saddle orbiting state emerges at
$\Theta/\pi\approx 0.5$ and $R/a\approx 22$ [see Fig 3(c) for the
$R\Theta$-phase portrait and real space trajectories for $\beta=0$]. Particles
with this initial orientation and position will execute a clockwise orbit
about the source (green trajectory). For $\beta\gtrsim 1$, we enter a region
with only a single saddle point. The phase portrait resembles the case for
large negative values of the ratio of the phoretic mobilities (i.e.,
$\beta<-10$). However, the direction of the trajectories is reversed in the
$R\Theta$-phase portrait as shown in Fig. 3(d) for $\beta=5$. The saddle point
occurs for $\Theta/\pi=0$ and $R/a\approx 4.8$ and is unstable in $R$ and
stable for $\Theta$. The green trajectory in Fig 3(d) is associated with the
saddle point, and the particle undergoes a counterclockwise spiraling motion
until it reaches the saddle point.

Figure 4 illustrates the different dynamical behaviors that can arise for a
point sink. We select a representative sink strength of $\tilde{\alpha}=-10$
to demonstrate the variation in the $R\Theta$-phase portrait as a function of
the phoretic mobility ratio $\beta$. In Fig. 4(a), we begin with the most
negative value of the phoretic mobility ratio $\beta=-15$ where there are two
fixed points: an unstable point for $\Theta/\pi=1.0$ at $R/a\approx 3.5$, and
an orbiting state for $\Theta/\pi\approx 0.51$ and $R/a\approx 25.6$. This
point is within one of the few regions in the $\tilde{\alpha}\beta$-phase
space where we observe an unstable fixed point. The other is a narrow region
for $\beta\approx 0.2$ and $\tilde{\alpha}>0$. Interestingly, unstable fixed
points will become stable under the reversal of the intrinsic velocity
$U_{0}$. It is important to note the sign of $U_{0}$ does not alter the
location of the fixed points in the $\tilde{\alpha}\beta$-phase space. From
Eq. 7, reversing the sign of $U_{0}$ will only lead to a reversal of the
velocities. Thus, the $R\Theta$-phase portraits given in Fig. 3 and Fig. 4 for
$U_{0}=1$ will have the same topology as $U_{0}=-1$. However, the arrows
indicating the direction of motion are reversed. Thus, a stable point for
$U_{0}>0$ will become an unstable point for $U_{0}<0$. Similarly, a saddle
point stable in $R$ for $U_{0}>0$ will be unstable in $R$ for $U_{0}<0$.

As the phoretic mobility ratio increases, the next region has three fixed
points: an unstable stationary state, a saddle point, and an orbiting state. A
representative example is shown in Fig. 4(b) for $\beta=-8$. The emergent
saddle point occurs for $\Theta/\pi=1.0$, is close to the singularity at
$R/a\approx 1.4$, and is stable for variations in $R$ but not $\Theta$. The
unstable point occurs at $R/a\approx 3.8$ also for $\Theta/\pi=1.0$. The next
region represented by $\beta=-2$ [Fig. 4(c)] contains no fixed points, and the
particle eventually collides with the singularity unless the initial
orientation is $\Theta/\pi=1$ where it moves off to infinity. Interestingly,
particles starting with their orientation closer to $\Theta/\pi=1$ can undergo
a trajectory that will wind several times about the singularity before being
drawn toward the center (blue trajectory). In Fig. 4(d), we show the case of
$\beta=-0.5$, where there exist two saddle points one located at
$\Theta/\pi=0$ and $\Theta/\pi=1.0$ The phase portrait is similar to
$\beta=-2$ and $\tilde{\alpha}=10$ [see Fig. 3(b)], except for it being
reflected about the line $\Theta/\pi=0.5$.

Figure 5: Infinite number of orbiting states: (a) $R\Theta$-phase portrait and
(b) real-space trajectory in the region of the $\tilde{\alpha}\beta$-phase
space where a continuous and infinite number of orbiting fixed points exists
about the source. These plots correspond to the parameter values
$\tilde{\alpha}=10$, $\beta=0.644$, $\chi=0$ and $U_{0}=-1$. Videos of the
real-space trajectories of these orbiting states are provided in the
Supporting Information 70.

In Fig. 4(e), we show the case of $\beta=10$, which is representative of the
only region of the $\tilde{\alpha}\beta$-phase space with a stable fixed
point, which occurs for $\Theta/\pi=1.0$ and has a distance of $R/a\approx
3.9$ from the singularity. A unique feature of this region of the
$\tilde{\alpha}\beta$-phase space is the $R\Theta$-phase portrait shows an
exclusion region between the location of the singularity and the fixed point.
Any particle initially in the region will either migrate toward the fixed
point or off to infinity. This behavior mirrors the movement pattern observed
in a recent study of P. aeruginosa bacteria in response to a CO2 point source,
where the cells form an accumulation front at a distance from the source 72.
Stable fixed points of this character have potential applications in surface
cleaning from bio-contaminants or preventing the accumulation of active
particles near surfaces.

We conclude by highlighting a peculiar region of the
$\tilde{\alpha}\beta$-phase space not categorized in Fig. 2, where we observed
a continuum of orbiting states. A representative example of this behavior
occurs for $U_{0}=-1$, $\tilde{\alpha}=10$ and $\beta\approx 0.64$, and the
corresponding phase portrait and real space trajectories are given in Fig. 5.
The magenta line in the $R\Theta$-phase portrait represents an infinite number
of orbiting states that are stable with variations in $R$ and neutrally stable
for $\Theta$. Thus, any small changes in $\Theta$ or $R$ will lead to the
particle finding a new orbit with a different radius. This behavior, where
there was observed to be a continuum of orbiting states, only occurs in a very
small region of the $\tilde{\alpha}\beta$-phase space and will be further
characterized in future work.

##  3 Conclusion

In this study, we quantified the dynamics of a phoretic Janus particle in a
radially symmetric gradient generated by a point source or sink of the fuel
driving self-propulsion. We derived an analytical expression for the phoretic
Janus particle’s velocity and found that its motion is highly sensitive to its
surface properties and can exhibit various dynamical behaviors. In addition to
positive and negative chemotaxis, we identify system parameters that give rise
to circular orbits and trapped stationary states. We show that both types of
fixed points are a robust feature of the $\tilde{\alpha}\beta$-phase space.
The sensitivity of the location of the fixed points and, more generally, the
topology of the $R\Theta$-phase portrait that characterizes their trajectories
suggests a method for quantifying the surface properties of phoretic Janus
particles.

In addition, circular orbits and trapped stationary states offer a mechanism
for pumping fluid and mixing at the microscale, particularly the stable
stationary states, which are resistant to small fluctuations usually present
in an experimental setting. Even when a particle is trapped in a stationary or
orbiting state, its surface is still chemically active and will pump fluid
across its surface. Our results demonstrate how to tune the location of a
trapped or orbiting state via the strength of the source and the particle’s
phoretic mobility ratio. This ability to localize particles at a particular
distance from the singularity suggests the possibility of achieving controlled
fluid mixing at a desired rate, which is challenging at the microscale.

Future research related to this work includes investigating the role of
chemical solute convection and the effect of Brownian motion on the dynamics
in a radial chemical gradient. In addition, to better align the model to many
experimental systems, we are currently investigating the role of a confining
boundary, as many phoretic Janus particles are confined to move at a two-
dimensional surface.

See Supporting Information at [URL] for complete derivation of the velocity of
phoretic Janus particle, plot illustrating the radial dependence of
dimensionless parameters in solution of Janus particles velocity, plot
characterizing the location of fixed points as a function of the strength of
the singularity, and movies illustrating the real space trajectories for
different system parameters.

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