text:
\section{Introduction}

Over the past two decades, significant effort has been invested in the development of
nano- and micro-particles that can propel themselves through an aqueous
environment \MP{\cite{ebbens10,gompper15,bechinger16}}. These synthetic swimmers have myriad potential
applications in, e.g.,
cell sorting and manipulation \cite{baraban12b,solovev12}, micromanufacturing \cite{goodrich17},
and the assembly of dynamic and programmable materials \cite{stenhammar16}. For these and other
applications, a longstanding challenge is to endow the synthetic swimmers with a semblance of
``intelligence'', i.e., the ability to autonomously sense the local environment (e.g., ambient flow
fields \cite{palacci15,uspal15b}, or their position with respect to confining
surfaces \cite{das15,simmchen16}) and to respond according to their design, \MP{i.e., to exhibit some
form of ``taxis.'' For example, recent studies have reported, experimentally and theoretically, on
the gravitaxis of asymmetrically shaped colloids \cite{hagen14}, as well as on phototaxis of light-activated
colloids in a spatially modulated light intensity landscape \cite{lozano16,geiseler17}.}

Among artificial swimmers, catalytically active particles constitute a large and important
class \cite{paxton06,moran17,aubret17}. These particles typically move by
self-phoresis, i.e.,  by consuming molecular ``fuel'' -- available in the surrounding
solution -- they generate gradients of one or more thermodynamic variables (e.g., chemical
composition or electrical potential) in the solution. These gradients, in conjunction with the
interaction of the particle surface with the molecules of the solution, drive flow of the
solution past the particle, powering directed motion \cite{golestanian05,golestanian07}.  For
example, bimetallic Au/Pt rods consume hydrogen peroxide and release charged molecules
into the solution, and hence move by self-electrophoresis \cite{paxton04}. Janus spheres with an
inert core and a platinum cap, when exposed to hydrogen peroxide, move by self-diffusiophoresis
\cite{howse07} or, as proposed recently, by a more complex mechanism involving self-
electrophoretic contributions \cite{brown14,ebbens14,brown17}. In addition to spheres and rods,
catalytically active particles with more complex shapes have been fabricated, e.g., dimers
of two connected particles \cite{valadares10,palacci15}, ``matchsticks,'' \cite{morgan14} and
microstars \cite{simmchen17}.

Self-phoretic particles exhibit complex dynamics when moving in the vicinity of a confining boundary.
As with other swimmers, e.g., those moving due to mechanical surface deformations, self-phoretic
particles create flow fields that extend into the surrounding solution, are reflected from nearby
boundaries, and couple back to the particles \cite{spagnolie12}. These hydrodynamic interactions
can lead to bound states of motion in which the swimming particle always remains in the vicinity
of the boundary \cite{ishimoto13}. Notably, an additional level of complexity arises for self-phoretic
particles, because the thermodynamic gradients that drive particle motion will  also be modified
by confining boundaries. For instance, if the particle creates gradients of chemical composition,
a solid boundary will present a no-flux boundary condition for the number density fields of the
various molecular species diffusing in the solution. In a previous study, we have shown that a
no-flux boundary condition, in combination with hydrodynamic interactions, can lead to novel
bound states, such as motionless ``hovering'' of a catalytic Janus sphere above a planar solid
boundary (or  ``wall'') \cite{uspal15}.

\WEU{All of these phenomena are enriched even further if the boundary itself is ``responsive,'' i.e.,
if thermodynamic gradients along the boundary translate into tangential surfaces stresses driving
flow of the solution.} Indeed, in analogy with phoresis, ``chemi-osmotic'' surface flows can occur if
gradients in chemical composition occur along the surface of a solid boundary
\cite{anderson89,derjaguin:47}. For a catalytic particle, chemi-osmotic surface flows, \textit{locally}
induced by the particle,  will drive flow in the bulk solution, coupling back to the particle. Recently,
the propulsion of catalytic ``surfers'' near a solid boundary was attributed to such an effect
\cite{palacci13}. Significantly, chemi-osmotic flows depend on the material identity of the solid
boundary. This naturally leads to the question of whether, by patterning the chemical composition of
the responsive wall, one can guide the motion of a catalytically active particle. Recently,
we investigated the interplay of chemi-osmosis and self-diffusiophoresis for a catalytically active
Janus sphere above a patterned wall \cite{uspal16,popescu17}. By analytical and numerical calculations
we demonstrated that a sphere that swims \textit{away} from its catalytic cap can ``dock'' at a
chemical step between two materials. In contrast, a sphere that swims \textit{towards} its cap can
align with and follow a chemical stripe. We identified the physical mechanisms driving these
behaviors. One key finding is that a chemically patterned wall can create fluid \textit{vorticity} in
the bulk solution, and that this wall-induced vorticity completely accounts for rotation of the
spherical particle.

\begin{figure}[t]
\includegraphics[scale=0.8]{./Fig1.pdf}
\caption{\label{fig:schematic} (a, b) Schematic views of a Janus sphere and a Janus spherocylinder, respectively, at a distance $h$ from a solid planar wall. Each particle produces solute molecules (green spheres) over the area of its catalytic cap (black). The effective interaction between the solute and the particle surface drives a self-diffusiophoretic flow (purple arrows) at the surface of the particle. Similarly, the solute drives chemi-osmotic
surface flow at the wall (blue arrows). The orientation vector $\mathbf{\hat{d}}$ is defined to point from the catalytic pole of the particle to the inert pole. For repulsive
interactions between the solute molecules and the wall (which corresponds to the direction of the flow shown here), the surface flow converges towards the region of maximum density of solute. This surface flow entrains flow in the bulk solution (not shown). Near the point of convergence, due to fluid incompressibility the bulk flow must lift off the surface into the volume of the solution, coupling back to the particle. (c) Top-down schematic view of a spherocylinder near a chemically patterned surface. In this case, the pattern is a chemical step, as defined by Eq. \ref{eq:step_surface_mob}. The orientation vector $\mathbf{\hat{d}}$ and the $\hat{x}$ axis form the angle $\phi$.}
\end{figure}

The orientational dynamics of a solid particle in a spatially varying ambient flow field is a classical problem in microhydrodynamics \cite{kim1991}. A natural starting point is to perform a Taylor expansion of the flow velocity with respect to the particle centroid (i.e., its geometrical center). At first order in such an expansion, the vorticity can be mathematically identified with the antisymmetric component of the first derivative of the flow velocity. The symmetric component of this tensor, the so-called \textit{rate of strain} tensor, does not contribute to the rotation of a sphere.
However, it does contribute to rotation of a non-spherical or rod-like particle, such as a spheroid or spherocylinder. Consequently, the orientational dynamics of a rod-like particle in a spatially varying flow field is typically much more complex than for a sphere. For instance, a sphere in a linear shear flow will simply rotate around the axis of vorticity at a rate proportional to the vorticity, while any other solid body of revolution will execute a complex three-dimensional periodic motion known as a Jeffery orbit \cite{jeffery22,bretherton62}.

These previous findings naturally raise the question of how a rod-like catalytically active particle will behave near a chemically patterned wall. The present study addresses
this question. Via detailed numerical calculations, we show that a rod-like particle, as compared with a sphere, exhibits significantly richer dynamics near a patterned wall. We find a new ``edge-following'' steady state, in which a rod-like particle moves along the edge of a chemical step at a steady distance from the step and with a steady orientation. Moreover, within a certain range of system parameters, the edge-following state co-exists with a docking state, i.e., the dynamics is bistable. We develop approximate analytical expressions that isolate the contributions of fluid vorticity and the rate of strain to particle rotation. We show that the rich phenomenology of rod-like particles is indeed a
consequence of the competition between fluid vorticity and rate of strain.

\section{Formulation of the model}

We consider a catalytic Janus particle in the vicinity of a hard planar wall. The particle
is a solid body of revolution with axially symmetric surface chemistry. Two
examples of particles of this kind -- a sphere and a spherocylinder -- are shown
schematically in Fig. \ref{fig:schematic}.  We assume a stationary reference frame in
which the wall is located
at $z = 0$ and the instantaneous position of the particle centroid is
$\mathbf{x}_{p}(t) = (x_p(t), y_p(t), z_p(t))$; for brevity of notations, in the following we shall
indicate the dependence on time only if it is explicitly required. We denote by $\mathbf{x}$ a point
in the liquid surrounding the particle. Over half of its surface (i.e., the catalytic ``cap''), the particle
consumes ``fuel'' molecules, taken to be present in abundance in the solution, and
emits product molecules (``solute''). We additionally assume that the reaction accounting
for the conversion of the fuel molecules into solute molecules is such that the system
operates within the so-called regime of reaction limited kinetics (see, e.g., Ref. \cite{oshanin17} and references therein). Under these
assumptions, the chemical activity of the particle can be modeled simply as a release of
solute molecules at a constant rate $\kappa$ (areal number density per time) over half
of its surface (the catalytic ``cap''), which corresponds to a so-called
``constant-flux'' model \cite{golestanian05}.
The orientation of the particle is defined by a unit vector $\mathbf{\hat{d}}$, which
is aligned with the axis of symmetry of the particle, and points from the catalytic cap to the inert ``face'' of the particle. The solute diffuses in the surrounding liquid with a diffusion constant $D$. We define the P\'{e}clet number $Pe \equiv U_{0} R_{1}/D$, where $U_0$ and $R_1$ are the characteristic velocity and the characteristic length scale of the particle, respectively. If the P\'{e}clet number is small, the solute number density field
$c(\mathbf{x})$ is quasi-static, and can be taken to obey the Laplace equation $\nabla^{2} c = 0$. The wall and the inert face are taken to be impenetrable to solute, so that the number density field is subject to the boundary conditions  $- D [\mathbf{\hat{n}} \cdot \nabla c] = \kappa$ on the catalytic cap, and $- D [\mathbf{\hat{n}} \cdot \nabla c] = 0$ on the inert face and on the wall, where the normal $\mathbf{\hat{n}}$ is defined to point into the liquid. Additionally, the number density field is required to attain a fixed value $c^{\infty}$
far away from the particle, i.e., $c(|\mathbf{x} - \mathbf{x}_{p}| \rightarrow \infty) = c^{\infty}$.

The suspending solution, consisting of solvent, fuel, and solute molecules, is
an incompressible Newtonian fluid with dynamic viscosity $\eta$ and mass density
$\rho$.  We use the classical theory of charge neutral diffusiophoresis in order to model
how concentration gradients drive particle motion through the liquid \cite{anderson89}.
Within this framework, the interaction between the solute molecules and a confining
surface (i.e., the wall or one of the particle faces) drives a surface flow which is
described by an effective slip velocity $\mathbf{v}_{s}(\mathbf{x}_{s}) = -b(\mathbf{x}_{s})
\nabla_{||} c(\mathbf{x})|_{\mathbf{x}=\mathbf{x}_{s}}$, where $\nabla_{||} \equiv
(\mathcal{I} - \mathbf{\hat{n}} \mathbf{\hat{n}}) \cdot \nabla$, $\mathcal{I}$ is the
identity tensor, and $\mathbf{x}_{s}$ denotes a position on that surface. The so-called
surface mobility $b\mathbf(\mathbf{x}_{s})$ is a material-dependent parameter which
encodes the molecular interaction between the solute and that surface. We consider
small Reynolds numbers $Re \equiv \rho U_0 R_{1} / \eta$, so that the hydrodynamic
flow $\mathbf{u}(\mathbf{x})$ and the pressure $P(\mathbf{x})$ of the solution obey
the Stokes equation $-\nabla P + \eta \nabla^{2} \mathbf{u} = 0$ and the incompressibility
condition $\nabla \cdot \mathbf{u} = 0$.  The fluid velocity obeys the boundary conditions
$\mathbf{u}\lvert_{wall}=\mathbf{v}_{s}(\mathbf{x}_{s})$ on the wall, and
$\mathbf{u}\lvert_{part}=\mathbf{U}+\bm{\Omega}\times(\mathbf{x}_{s}-\mathbf{x}_{p})+
\mathbf{v}_{s}(\mathbf{x}_{s})$ on the particle surface; note that the dependence on
$\mathbf{x}_{s}$ accounts also for the, in general, distinct surface mobilities.
Additionally, the fluid is quiescent far away from the particle, so that $\mathbf{u}(|
\mathbf{x} - \mathbf{x}_{p}| \rightarrow \infty) = 0$.  Here, $\mathbf{U}$ and $\bm{\Omega}$
are unknown translational and angular velocities, respectively, of the particle.  In order
to close the system of equations for $\mathbf{U}$ and $\bm{\Omega}$, we invoke the force balance, $\int \, \bm{\sigma} \cdot \mathbf{\hat{n}} \, dS = -\mathbf{F}^{ext}$, and the torque balance, $\int \, (\mathbf{x} - \mathbf{x}_{p}) \times \bm{\sigma} \cdot \mathbf{\hat{n}} \, dS = -\bm{\tau}^{ext}$, for the particle.
The fluid stress tensor is $\bm{\sigma} = -P \mathcal{I} + \eta \left[ \nabla \mathbf{u} + \nabla^{T} \mathbf{u} \right]$. Here, $\mathbf{F}^{ext}$ and $\bm{\tau}^{ext}$ are the net external force and net external torque, respectively; these can represent (for instance) particle buoyancy, bottom-heaviness, and any external forces and torques of constraint. The above two integrals are taken over the surface of the particle.

The above considerations completely specify how to determine the instantaneous
particle velocities $\mathbf{U}$ and $\bm{\Omega}$ as functions of the instantaneous
position $\mathbf{x}_{p}$ and the orientation $\mathbf{\hat{d}}$ of the particle. This
scheme includes various contributions to the particle velocities, such as from slip on
the surface of the particle (self-diffusiophoresis); slip on the surface of the wall
(chemi-osmosis), which drives hydrodynamic flow in the volume of the solution; and
external forces and torques. At this point, we make some conceptual distinctions which
will prove to be helpful later. We note that the Stokes equations are linear. Therefore, the
full problem can be split into three subproblems which isolate the \MP{individual contributions
of wall slip (\textbf{ws}), self-diffusiophoresis (\textbf{sd}), and external forces and
torques (\textbf{ext}).}  In subproblem
\textbf{(sd)}, we take the surface velocity on the wall to be zero, i.e., $\mathbf{v}_{s}
(\mathbf{x}_{s})\lvert_{wall} \,= 0$, as well as $\mathbf{F}^{ext} = 0$ and $\bm{\tau}^{ext} =
0$, but otherwise the subproblem is posed as given above.  This subproblem can be solved for
$\mathbf{U}^{sd}$ and $\bm{\Omega}^{sd}$. Similarly, in subproblem \textbf{(ws)}, we take
$\mathbf{v}_{s}(\mathbf{x}_{s})\lvert_{part} \,= 0$, $\mathbf{F}^{ext} = 0$ and $\bm{\tau}^{ext} = 0$.
We obtain $\mathbf{U}^{ws}$ and $\bm{\Omega}^{ws}$ by solving subproblem \textbf{(ws)}. Finally,
if there are external forces and torques, we consider a subproblem \textbf{(ext)}, in which
$\mathbf{v}_{s}(\mathbf{x}_{s})\lvert_{wall} \,= 0$ and
$\mathbf{v}_{s}(\mathbf{x}_{s})\lvert_{part} \,= 0$, and which can be solved for $\mathbf{U}^{ext}$
and $\bm{\Omega}^{ext}$.  We superpose the individual solutions to  obtain the total
particle velocities $\mathbf{U} = \mathbf{U}^{sd} + \mathbf{U}^{ws} + \mathbf{U}^{ext}$ and
$\bm{\Omega} = \bm{\Omega}^{sd} +  \bm{\Omega}^{ws} + \bm{\Omega}^{ext}$.

For a sphere, the self-diffusiophoretic contributions $\mathbf{U}^{sd}$ and $\bm{\Omega}^{sd}$ had
been considered in Ref. \cite{uspal15}. The very same framework developed in that study can be
applied to spherocylinders. Additionally, the contributions $\mathbf{U}^{ext}$ and $\bm{\Omega}^{ext}$
of the external forces and torques can be solved with standard methods (e.g., see
Ref. \cite{simmchen16}.) In the remainder of the
present study, we focus on obtaining the contributions from chemi-osmotic wall slip.

At this stage we make some simplifying assumptions. We take the particle height $z_p(t)$ to
be a fixed quantity $z_p(t) = h$, and assume that $\mathbf{\hat{d}}$ is always oriented within
the plane of the wall, i.e., $\mathbf{\hat{d}} \cdot \hat{z} = 0$, where $\hat{z}$ is
the unit vector in $z$-direction. In this case the orientation is completely
described by the angle $\phi$, as shown in Fig. \ref{fig:schematic}(c). For a
spherocylindrical particle, these assumptions can be viewed as realistic. Catalytic
particles are typically heavy, and they will sediment to the bottom surface of a
bounding container. Concerning the restriction on $\mathbf{\hat{d}}$, if $h$ is small
(compared to the pole-to-pole length of the particle), ``rocking'' of the particle orientation
$\mathbf{\hat{d}}$ out of the $xy$ plane is restricted to small angles by the presence of the
hard wall; this is apparent from inspecting Fig. \ref{fig:schematic}(b). For spheres, the restriction
on $\mathbf{\hat{d}}$ could be imposed by an external magnetic torque. \MP{Furthermore, we note that
for heavy colloids the surface-to-surface distance is expected to be small (when compared to
the radius of the particle): it is set mainly by the balance between the weight of the particle and
the DLVO repulsion. If the vertical component of the osmotic flow is not very large, which is
expected to be the case for our system, it is a reasonable assumption that the change in height due
to the particle being dragged by the vertical component of the osmotic flow at the location of the
particle is small compared to $h$, i.e., $h/R$ remains practically unchanged.} (For
a further discussion of this assumption in relation to spheres, the reader is invited to peruse
Ref. \cite{uspal16}.)  We note that based on these assumptions in Ref.
\cite{uspal15} it was shown via symmetry considerations that $\mathbf{U}^{sd}$ and $\bm{\Omega}^{sd}$
are functions of $h$ and $\mathbf{\hat{d}} \cdot \hat{z}$ only, and that ${\Omega}^{sd}_{z} = 0$.
Since we take $h$ to be constant and
$\mathbf{\hat{d}} \cdot \hat{z} = 0$, one finds
$\mathbf{U}^{sd} = U^{sd} \, \mathbf{\hat{d}}$ and ${\Omega}^{sd}_{z} = 0$. In the present
study, we treat $U^{sd}$ as an input parameter; however, for a given particle geometry and
surface chemistry, it can be calculated via the methods outlined in Ref. \cite{uspal15}.

The position and orientation of the particle evolve according to the equations
\begin{equation}
\label{eq:particle_dynamics}
\begin{split}
\dot{x}_{p} &= U^{sd} \cos(\phi) + U_{x}^{ws}(x_p, y_p, \phi), \\
\dot{y}_{p} &= U^{sd} \sin(\phi) + U_{y}^{ws}(x_p, y_p, \phi),~\mathrm{and}\\
\dot{\phi} &= \Omega^{ws}_{z}(x_p, y_p, \phi),
\end{split}
\end{equation}
where the over-dot denotes the time derivative of the corresponding observable. These equations can be integrated to obtain complete particle trajectories \footnote{Note that
we have assumed that external forces and torques do not contribute to the in-plane motion of the particle. In some cases, this assumption represents an additional approximation. For instance, for a sphere subject to a magnetic torque of constraint, there is a contribution to the in-plane velocity of the particle which is due to wall-induced translational-rotational coupling. As discussed at length in Ref. \cite{popescu17}, this contribution is negligible for the particle-wall separations considered here.}.

We consider two types of surface patterns: a ``chemical step'' and a ``chemical stripe.'' For the chemical step, the surface mobility $b(\mathbf{x}_{s})$ is
\begin{equation}
\label{eq:step_surface_mob}
b(\mathbf{x}_{s}) =
\begin{cases}
b_{w}^{l},~ x_s < 0,\\
b_{w}^{r},~ x_s \geq 0,\hspace*{1cm}
\end{cases}
\end{equation}
where $b_{w}^{l}$ and $b_{w}^{r}$ hold on the \textit{l}eft and \textit{r}ight hand sides
of the \textit{w}all, respectively. The chemical stripe is defined by
\begin{equation}
\label{eq:stripe_surface_mob}
b(\mathbf{x}_{s}) =
\begin{cases}
b_{w},~ x_s < -W,\\
b_{w}^{c},~ -W \leq x_s \leq W, \\
b_{w},~ x_s > W, \hspace*{1cm}
\end{cases}
\end{equation}
i.e., the \textit{c}entral stripe has $b=b_{w}^{c}$, while $b=b_{w}$ on the rest of the wall. Both the chemical step and the chemical stripe have translational symmetry in the $\hat{y}$-direction. Therefore, the dependence of $U_{x}^{ws}$, $U_{y}^{ws}$, and $\Omega_{z}^{ws}$ on ${y}_p$ drops out of Eq. (\ref{eq:particle_dynamics}), and the dynamics is completely described by two degrees of freedom, ${x}_{p}$ and $\phi$.

We express lengths in units of $R_{1}$, where $2 R_{1}$ is the largest dimension of the particle. The solute density field has a characteristic number density $\kappa R_{1}/D$. Combining this with the surface mobility gives a characteristic velocity $U_0 = |b_0 \kappa/D|$, where $b_0$ is a characteristic value of the surface mobility. For the chemical step, we choose $b_0 = b_w^{l}$, and for the stripe, we choose $b_0 = b_w^{c}$. \WEU{We express $U^{sd}$ in units of $U_0$ so that, for a given particle geometry and height, the dimensionless parameter $U^{sd}/U_{0}$ characterizes the relative strengths of the self-diffusiophoretic and the chemi-osmotic contributions to particle motility.} For reasons of space, in this study we restrict our considerations to
$U^{sd} > 0$, i.e., to a particle which tends to move away from its catalytic cap via self-diffusiophoresis. This ``inert-forward'' direction of motion is more typically seen in experiments than the ``catalyst-forward'' direction of motion. (Nevertheless, a catalyst-forward particle can also exhibit an interesting behavior near a chemically patterned surface, as we demonstrated for spheres in Ref. \cite{uspal16}.) An additional dimensionless parameter characterizes the surface: $b_{w}^{r}/b_{w}^{l}$ for the chemical step and $b_{w}/b_{w}^{c}$
for the stripe, respectively.\newline

\section{Numerical results and discussion}

In order to numerically solve the problem posed in the previous section, we resort to the
so-called Boundary Element Method (BEM) \cite{uspal15}. The effect of a non-spherical shape
is revealed by considering two particle geometries: a Janus sphere
and a spherocylinder, each of them half covered by catalyst. These geometries are shown
schematically in Fig. \ref{fig:schematic}(a) and (b). The sphere has a radius $R$, and
the height $h$ of the sphere is fixed as $h/R = 1.1$. The major axis of the spherocylinder
has a tip-to-tip length $2c$, and the radius of the cylindrical part of the particle is
chosen to be  $a = 0.4 c$.  The height of the spherocylinder is fixed as
$h = 0.6 c$.

\begin{figure}[!htb]
\includegraphics[scale=0.45]{./Fig2.pdf}
\caption{\label{fig:sphere_traj_portraits} (a-b) Trajectories of a
self-diffusiophoretic sphere, half-covered by catalyst, at height $h/R = 1.1$.
For (a) and (b), the self-propulsion velocity is $U^{sd}/U_{0} = 0.1$ and
$U^{sd}/U_{0} = 0.4$, respectively. Star symbols match the trajectories to the
initial conditions shown in the phase portraits in (c) and (d), respectively.
(c-d) Phase portraits for a half-covered sphere at a height $h/R = 1.1$ for two
values of the self-diffusiophoretic velocity $U^{sd}$. The white circle indicates
an attractor, whereas the white triangle indicates a saddle point. The color
field encodes the component $U_{x}$ of the particle velocity in $\hat{x}$ direction.}
\end{figure}
The motion of a sphere near a chemical step had previously been studied in
Ref. \cite{uspal16}. We briefly revisit those findings here, choosing $b_{w}^{r}/b_{w}^{l} = 4$
and $b_{w}^{l} < 0$. Therefore, both regions of the substrate have a repulsive
effective interaction with the solute molecules, but the region to the right of the step ($x_p > 0$) has a stronger repulsive interaction. At low self-diffusiophoretic velocity $U^{sd}$, and depending on the initial orientation $\phi$ and distance $x_{p}$ from
the step, the sphere can spontaneously ``dock'' at the step. In the docking state, the
particle is trapped and motionless near the step (i.e., $|x_p/R| \ll 1$), with the
orientation vector $\mathbf{\hat{d}}$ pointing towards the region where the substrate
is more repulsive to the solute. A particle trajectory with $U^{sd}/U_0 = 0.1$, which
is attracted to the docking state, is illustrated in Fig. \ref{fig:sphere_traj_portraits}(a). The initial condition of this trajectory is matched by the magenta star symbol to Fig. \ref{fig:sphere_traj_portraits}(c), which is a  phase portrait showing the time evolution of any initial state $(x_{p}, \phi)$. From the phase portrait, it is clear that if the particle approaches the edge from the left ($x_p < 0$), the particle will dock for nearly any angle of incidence $0 \leq \phi < \pi/2$.
If the particle approaches from the right ($x_p > 0$),
the angle of incidence must be oblique ($\phi \gtrsim \pi/2$) for the particle to
dock; otherwise, the particle will pass over the step. At higher speeds $U^{sd}$,
the particle always passes over the step. Figure \ref{fig:sphere_traj_portraits}(b)
shows the trajectory of a particle with $U^{sd}/U_0 = 0.4$; the corresponding phase
portrait is shown in Fig. \ref{fig:sphere_traj_portraits}(d).  Interestingly, if the
particle passes from the left $(x_p < 0$, $\phi < \pi/2$), there is a narrow range of
exit angles $\phi \ll 1$ for a broad range of angles of approach $0 \leq \phi < \pi/2$.
In other words, in this case the chemical step operates
as an ``angular focusing'' device. If the particle passes from the right ($x_p > 0$,
$\phi > \pi/2$), the deviation of the exit angle from the initial angle is modest.

\begin{figure}[!htb]
\includegraphics[scale=0.5]{./Fig3.pdf}
\caption{\label{fig:traj_spherocyl} Trajectories of a half-covered spherocylinder with height
$h/c = 0.6$ near a chemical step. For (a), the self-propulsion velocity is
$U^{sd}/U_{0} = 0.12$. For (b)-(c), the speed is $U^{sd}/U_{0} = 0.15$, and for
(d) it is $U^{sd}/U_{0} = 0.35$. The star symbols match the trajectories to the
initial conditions in Fig. \ref{fig:phase_portrait_spherocyl}.}
\end{figure}
Now we consider the spherocylindrical particle near the same chemical step
(Fig. \ref{fig:traj_spherocyl}). We find that the phenomenology is considerably enriched by
the elongated shape. At low speed $U^{sd}$, there is a stable ``edge-following'' state in
which the particle remains at a fixed distance $|x_p/c| \ll 1$ and angle $\phi \lesssim
\pi/2$. Accordingly, the particle orientation $\mathbf{\hat{d}}$ aligns almost parallel
to the step edge, but tilted slightly towards the more repulsive region of the substrate,
and the particle moves steadily along the edge. A phase portrait for $U^{sd}/U_0 = 0.12$
is shown in Fig. \ref{fig:phase_portrait_spherocyl}(a); the white circle represents the
edge-following attractor. The cyan star in the phase portrait gives the initial condition
for the trajectory in Fig. \ref{fig:traj_spherocyl}(a). At a slightly higher speed
$U^{sd}/U_0 = 0.15$, the edge-following state (Fig. \ref{fig:traj_spherocyl}(b)) co-exists
with a stable docking state (Fig. \ref{fig:traj_spherocyl}(c)), i.e., the dynamics is
bistable. For the speed $U^{sd}/U_0 = 0.15$ edge-following and docking trajectories
are illustrated in Figs. \ref{fig:traj_spherocyl}(b) and (c); the complete phase
portrait corresponding to $U^{sd}/U_0 = 0.15$ is shown in Fig.
\ref{fig:phase_portrait_spherocyl}(b). The bistability occurs within a finite range of speed
values $U^{sd}$. As $U^{sd}$ is increased within this range, the attractors and the saddle
points of the phase diagram move in the phase space $(\phi, x_{p}/c)$. Within this
process, at the upper threshold of this range, the saddle point and the edge-following state
collide and annihilate, leaving only the docking state. The docking state persists for a
certain range of high speeds $U^{sd}$. In Fig. \ref{fig:phase_portrait_spherocyl}(c), we show
the phase portrait for $U^{sd}/U_0 = 0.25$. As $U^{sd}$ is increased even further, the docking
state attractor also collides with a saddle point and annihilates. Therefore, for very high
speed $U^{sd}$, the spherocylinder passes over the step for any inital angle with
$\mathbf{\hat{d}}$ pointing towards the step. A corresponding phase portrait for
$U^{sd}/U_{0} = 0.35$ is shown in Fig. \ref{fig:phase_portrait_spherocyl}(d). As in the
case of the sphere, a spherocylinder passing from the left has a narrow band of exit angles,
while a spherocylinder passing from the right shows only modest deviations of the exit angles
$\phi$ from the incoming ones. Such a trajectory, which passes from the left, is shown in
Fig. \ref{fig:traj_spherocyl}(d).
\begin{figure}[!htb]
\includegraphics[scale=0.5]{./Fig4.pdf}
\caption{\label{fig:phase_portrait_spherocyl}Phase portraits of a half-covered spherocylinder
near a chemical step for various self-diffusiophoretic velocities $U^{sd}$. The particle is at
a height $h/c = 0.6$. The star symbols indicate the initial conditions corresponding to the
trajectories shown in Fig. \ref{fig:traj_spherocyl}. White circles
indicate attractors, whereas white triangles indicate saddle points. The color field
encodes the component $U_{x}$ of the particle velocity in $\hat{x}$ direction.}
\end{figure}

We note that we have obtained a very similar phenomenology for a prolate spheroid, i.e.,
edge-following at low speed $U^{sd}$, bistable edge-following and docking at
intermediate speeds $U^{sd}$, etc. This agreement indicates that the enrichment of the
phenomenology as compared to that for a sphere is a generic consequence of non-spherical
shapes, rather than being specific for spherocylinders. We omit
these calculations in order to avoid an overly repetitive presentation. Additionally, we
note that for both spheres and spherocylinders the phenomenology we have discussed above
is observed for a range of other values of the surface mobility contrast
$(b_{w}^{r}/b_{w}^{l} - 1)$ which are of $\mathcal{O}(1)$.

\begin{figure}
\includegraphics[scale=0.9]{./Fig5.pdf}
\caption{\label{fig:stripe_figure} (a) Phase portrait for a half-covered spherocylinder
near a chemical stripe. The particle is at height $h/c = 0.6$ and has a
self-diffusiophoretic velocity $U^{sd}/U_0 = 0.1$. Star symbols indicate initial
conditions corresponding to the trajectories shown in panels (b) and (c). White circles
indicate attractors, whereas white triangles indicate saddle points. The color field
in (a) encodes the component $U_x$ of the particle velocity in $\hat{x}$ direction.
Panels (b) and (c) show a docking and a stripe-following trajectory, respectively, of
a spherocylinder near a chemical stripe. }
\end{figure}
We briefly discuss the behavior of a rod-like particle near a chemical stripe. As one
might expect, if the stripe width $2W$ is large, the dynamics near the stripe resembles
the one near two isolated chemical steps. However, as the width is decreased and
becomes comparable to the particle size, the effects of the two edges start to interfere with
each other. We consider a stripe with $2W = 3 c$, $b_{w}/b_{w}^{c} = 3$, and $b_{w}^{c} < 0$.
Both the sides and the center of the stripe have a repulsive effective interaction with
the solute, but this interaction is stronger for the sides than for the center.  In
Fig. \ref{fig:stripe_figure}(a), we show the phase portrait for a spherocylinder near
this stripe with $U^{sd}/U_{0} = 0.1$. The dynamics is multistable. There are two docking
states, associated with the two edges. But instead of two edge-following states, we obtain
a ``stripe-following'' attractor in which the particle is positioned at the center of the
stripe ($x_p = 0$) and is perfectly aligned with it ($\phi = \pi/2$). Docking and stripe-
following trajectories are shown in Fig. \ref{fig:stripe_figure}(b) and (c).  For
comparison, in previous studies we had obtained a stripe-following state for Janus spheres,
but only for $U^{sd} < 0$, i.e., for a particle which tends to move towards its
catalytic face.

\WEU{We also briefly consider the effect of changing the particle aspect ratio  $e = a/c$. In order to address this issue, we consider spherocylinders with various aspect ratios located near the same chemical step as in Figs. \ref{fig:sphere_traj_portraits}-\ref{fig:phase_portrait_spherocyl}. We fix the ratio $h/a$, i.e., the ratio of the height of the particle center to the radius of the spherical end-caps of the particle, to $h/a = 1.5$. By decreasing $e = a/c$ from $e = 1$, the shape of the particle is smoothly deformed from spherical ($e = 1$) to quasi-spherical ($e \lesssim 1$) to rod-like ($e < 1$), while maintaining a constant thickness of the gap between the particle and the wall (as measured by the smallest dimension of the particle.) For each aspect ratio, we study particle behavior, in terms of the presence or absence of steady states, as we increase the self-propulsion velocity $U^{sd}$ from $U^{sd} = 0$. Our results are shown in Fig. \ref{fig:phase_map}. We find that even quasi-spherical particles ($e = 0.95$) have edge-following steady states, but these are restricted to a very small band near $U^{sd} = 0$. Decreasing the aspect ratio significantly increases the range of $U^{sd}$ over which edge-following states can be obtained. We also observe a region of bistability for $e \leq 0.9$, further extending the range of propulsion speeds for edge-following. For each aspect ratio considered, there are no steady states localized near the step for very large $U^{sd}$. However, decreasing the aspect ratio increases the threshold value of $U^{sd}$ for the disappearance of steady states.}

\begin{figure}
\includegraphics[scale=0.45]{./Fig6.pdf}
\caption{\label{fig:phase_map} \WEU{Phase map showing the presence or absence of steady states for spherocylinders of various aspect ratios $e = a/c$ and self-propulsion speeds $U^{sd}/U_{0}$. The ratio of the particle height to the radius of the spherical end-caps is fixed to $h/a = 1.5$. Blue squares indicate that there is a monostable edge-following steady state at the corresponding combination of parameters $e$ and $U^{sd}$. Green circles indicate that there is a monostable docking state, and purple diamonds indicate that there are bistable edge-following and docking states.  Red crosses indicate that there are no steady states localized near the step.}}
\end{figure}

\MP{The results in Figs. \ref{fig:sphere_traj_portraits} - \ref{fig:phase_map} summarize the
various phenomenology emerging from self-diffusiophoresis near chemically patterned walls.
They clearly show that the shape of the particle is a relevant parameter for the dynamics.} Based on these observations, we seek to understand which new physical aspects are
introduced by the non-spherical shape, as well as the mechanism by which they
enrich the particle dynamics. We particularly focus on the rotation of the particle angle
$\phi$. For a sphere, we note that a chemical step always drives rotation of the catalytic
face towards the region of the substrate which is less repulsive to the solute. This fact
can be clearly seen in the phase portraits of Fig. \ref{fig:sphere_traj_portraits},
because all trajectories in phase space always exhibit $\dot{\phi} \leq 0$, i.e.,
they always move from right to left in the diagrams. This observation suggests that
there is a single dominant physical effect which drives the rotation of a
sphere. For the spherocylinders, instead, the existence of edge-following attractors with
$\phi$ between $\phi = 0$ and $\phi = \pi$ suggests that there might be multiple competing
contributions to the angular velocity of the particle, and that these contributions balance
for a particle in the edge-following state.

\WEU{In order to understand} the interplay of the physical effects
introduced by a non-spherical shape, in the following section we take an approximate analytical
approach. We seek to identify and isolate the various competing contributions to translational
and rotational velocities of a non-spherical particle.

\section{Analytical results and discussion}

\subsection{Point-particle approach for the fluid velocity}

In order to obtain analytical expressions for $\mathbf{U}^{ws}$ and $\bm{\Omega}^{ws}$,
we neglect the finite size of the particle under consideration, and treat it as a point-like object located at $\mathbf{x}_{p}$. Within this approximation, the fluid domain
can be regarded as an infinite half-space
$\lbrace \mathbf{x}_{f} \in \mathbb{R}^3 | z_f > 0 \rbrace$ bounded by a planar wall;
the hydrodynamic boundary condition on the surface of the particle is neglected. There is a
slip  velocity $\mathbf{v}_{s}(\mathbf{x}_{s})$ on the wall, where $\mathbf{x}_{s}$
denotes a position on the wall. The wall slip drives flow in the volume of the
solution, and there are no external forces or torques applied to the system. We seek an
expression for the fluid velocity $\mathbf{u}(\mathbf{x}_{f})$ at an \textit{arbitrary} point
$\mathbf{x}_{f}$ in the fluid, i.e., $\mathbf{x}_{f}$  is not necessarily identical to
$\mathbf{x}_{p}$.

We use the Lorentz reciprocal theorem, which relates the fluid stresses $(\bm{\sigma}, \bm{\sigma}')$ and velocity fields $(\mathbf{u}, \mathbf{u}')$ of two solutions of the
Stokes equations which share the same geometry \cite{lorentz}:
\begin{equation}
\label{eq:reciprocal_thm}
\int \mathbf{u} \cdot \bm{\sigma}' \cdot \mathbf{\hat{n}} \, dS = \int \mathbf{u}' \cdot \bm{\sigma} \cdot \mathbf{\hat{n}} \, dS \,,
\end{equation}
where the integrals are performed over the wall with the surface normal
$\mathbf{n} = \hat{\mathbf{z}}$ pointing into the fluid. We take the ``unprimed'' problem
to be the one under consideration: the velocity on the wall is $\mathbf{u}(\mathbf{x}_{s}) = \mathbf{v}_{s}(\mathbf{x}_{s})$, and there are no external forces or torques. We consider three ``primed'' or auxiliary subproblems. For each of them, we take the wall to be a no-slip surface (i.e., $\mathbf{u}'(\mathbf{x}_{s}) = 0$). Additionally, there is an external force exerted on the fluid at point $\mathbf{x}_{f}$. The force has a magnitude $F_{0}$ and is oriented in $j$ direction, where $j \in \{x, y, z\}$ designates the auxiliary subproblem. The force creates a stress $F_0 \, \bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \, \mathbf{x}_{f})$
in the fluid. Reformulating Eq. (\ref{eq:reciprocal_thm}) leads to \cite{uspal16}
\begin{equation}
\label{eq:velocity_in_fluid}
{u}_{j}(\mathbf{x}_{f}) = - \int \mathbf{v}_{s} (\mathbf{x}_{s}) \cdot \bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \, \mathbf{x}_{f}) \cdot \mathbf{\hat{n}} \, dS \,.
\end{equation}
For each $j$ the integration kernel $\bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \, \mathbf{x}_{f}) \cdot \mathbf{\hat{n}}$ is \cite{uspal16}:
\begin{equation}
\begin{split}
\label{eq:sigma_all_modes}
{\sigma}'^{(x)}_{iz}(\mathbf{x}_{s}; \, \mathbf{x}_{f}) &= - \frac{3 z_{f}}{2 \pi} \frac{r_i r_x}{r^{5}} \,,\\
{\sigma}'^{(y)}_{iz}(\mathbf{x}_{s}; \, \mathbf{x}_{f}) &= - \frac{3 z_{f}}{2 \pi} \frac{r_i r_y}{r^{5}} \,,~\mathrm{and}\\
{\sigma}'^{(z)}_{iz}(\mathbf{x}_{s}; \, \mathbf{x}_{f}) &=  \frac{3 z_{f}^{2}}{2 \pi} \frac{r_i}{r^{5}} \,,
\end{split}
\end{equation}
where $\mathbf{r} \equiv \mathbf{x}_{s} - \mathbf{x}_{f}$ and $r \equiv |\mathbf{r}|$. We
note that although textbook statements of the reciprocal theorem are often restricted to
Eq. (\ref{eq:reciprocal_thm}), an expression equivalent to Eq. (\ref{eq:velocity_in_fluid})
was already given in the original paper by Lorentz \cite{lorentz}.

Now we consider how the fluid velocity in Eq. (\ref{eq:velocity_in_fluid}) will affect the
motion of the particle.  We apply Fax\'{e}n's Laws for the velocity of a point-like tracer
particle in an ambient flow field \cite{spagnolie12}:
\begin{equation}
\label{eq:U_ws}
\mathbf{U}^{ws} = \mathbf{u}(\mathbf{x}_{p})
\end{equation}
and
\begin{equation}
\label{eq:Omega_ws}
\bm{\Omega}^{ws} = \frac{1}{2} \bm{\omega}(\mathbf{x}_{p}) +
\Gamma \, \mathbf{\hat{d}} \times (\mathbf{E}(\mathbf{x}_{p}) \cdot \mathbf{\hat{d}}).
\end{equation}
The {vorticity} vector evaluated at the particle position is
\begin{equation}
\bm{\omega}(\mathbf{x}_{p}) = \nabla \times \mathbf{u}(\mathbf{x}_{f}) \big|_{\mathbf{x}_{f} = \mathbf{x}_{p}},
\end{equation}
where the derivative is taken with respect to $\mathbf{x}_{f}$. The {strain rate tensor}
evaluated at the particle position is
\begin{equation}
\mathbf{E}(\mathbf{x}_{p}) = \frac{1}{2} \left( \nabla \mathbf{u}(\mathbf{x}_{f}) + \nabla^{T} \mathbf{u}(\mathbf{x}_{f}) \right) \big|_{\mathbf{x}_{f} = \mathbf{x}_{p}}.
\end{equation}
In Eq. (\ref{eq:Omega_ws}), the parameter $\Gamma$, with $0 \leq \Gamma \leq 1$,
characterizes the elongation of the particle. It is zero for a sphere and close to unity
for a long, slender object. For a prolate spheroid, there is an exact expression for
$\Gamma$:
\begin{equation}
\label{eq:gamma_prolate}
\Gamma_{prolate} = \frac{1 - e^2}{1 + e^2}\,,
\end{equation}
where the spheroid aspect ratio $e = a/c$ is the ratio of the semi-minor axis length $a$
and the semi-major axis length $c$. For spherocylinders, there is no exact expression for
$\Gamma$, but
theoretical \cite{bretherton62} and experimental \cite{trevelyan51,snook14} studies have
established that Eq. (\ref{eq:gamma_prolate}) holds as an excellent approximation if $e$
is replaced by an empirically determined effective aspect ratio $e_{eff} \approx 1.25 e$:
\begin{equation}
\label{eq:gamma_spherocyl}
\Gamma_{spherocyl} \approx \frac{1 - e_{eff}^2}{1 + e_{eff}^2}.
\end{equation}

In order to facilitate the application of Fax\'{e}n's Laws, by using Eq. (\ref{eq:velocity_in_fluid})
we have obtained
\begin{equation}
\label{eq:velocity_at_xp}
u_j(\mathbf{x}_{p}) =  - \int \mathbf{v}_{s} (\mathbf{x}_{s}) \cdot \bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \, \mathbf{x}_{s}) \cdot \mathbf{\hat{n}} \, dS \,,
\end{equation}
\begin{equation}
\label{eq:vorticity_at_xp}
\omega_i(\mathbf{x}_{p}) =  - \int \left[ \epsilon_{ijk} \frac{\partial}{\partial x_{f,j}} (\mathbf{v}_{s}(\mathbf{x}_{s})  \cdot \bm{\sigma}'^{(k)}(\mathbf{x}_{s}; \mathbf{x}_{f}) \cdot \mathbf{\hat{n}} ) \right]_{\mathbf{x}_{f} = \mathbf{x}_{p}} \, dS \,,
\end{equation}
and
\begin{equation}
\label{eq:strain_rate_at_xp}
{E}_{ij}(\mathbf{x}_{p}) =  - \frac{1}{2} \int \left[ \frac{\partial}{\partial {x}_{f,i}} (\mathbf{v}_{s}(\mathbf{x}_{s})  \cdot \bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \mathbf{x}_{f}) \cdot \mathbf{\hat{n}} ) +  \frac{\partial}{\partial {x}_{f,j}}(\mathbf{v}_{s}(\mathbf{x}_{s})  \cdot \bm{\sigma}'^{(i)} (\mathbf{x}_{s}; \mathbf{x}_{f}) \cdot \mathbf{\hat{n}} ) \right]_{\mathbf{x}_{f} = \mathbf{x}_{p}} \, dS,
\end{equation}
where $i$, $j$, and $k$ are elements of $\{x,y,z\}$, $x_{f,i}$ is the $i$ component of $\mathbf{x}_{f}$,
and $\bm{\sigma}'^{(j)}(\mathbf{x}_{s}; \mathbf{x}_{f})$ is
given by Eq. (\ref{eq:sigma_all_modes}).

We have now completely specified how to calculate the motion of a ``point-like'' particle
advected by the flow in the volume of the solution. In turn, this flow is driven by
the surface flow $\mathbf{v}_{s}(\mathbf{x}_{s})$ on a planar wall. At this point, for
convenience, we introduce some new definitions and perform certain further reformulations.
Concerning the rotational velocity, we define $\bm{\Omega}^{ws} \equiv
\bm{\Omega}^{V} + \bm{\Omega}^{E}$, where we have distinguished the contributions from the
vorticity (V) and the fluid strain rate (E), respectively. Given the assumed
restriction of $\mathbf{\hat{d}}$ to the xy plane, we are interested only in the $\hat{z}$
component  $\Omega_{z}^{ws} = \Omega_{z}^{V} + \Omega^{E}_{z}$.
\MP{By expanding the vector and scalar products in the last term of Eq. (\ref{eq:strain_rate_at_xp}), one
arrives at}
\begin{equation}
\Omega^{E}_{z} =  \Gamma \,  \epsilon_{zjk} d_{j} E_{kl} d_{l},
\end{equation}
where
\begin{equation}
\begin{split}
 \epsilon_{zjk} d_{j} E_{kl} d_{l} &= \epsilon_{zxy} d_{x} E_{yl} d_{l} + \epsilon_{zyx} d_{y} E_{xl} d_{l} \\
 &=  d_{x} E_{yl} d_{l} - d_{y} E_{xl} d_{l} \\
 &= E_{yx} d_{x}^{2} + E_{yy} d_{x} d_{y} -  E_{xx} d_{x} d_{y} - E_{xy} d_{y}^{2}.
 \end{split}
\end{equation}
Since $E_{xy} = E_{yx}$, one has
\begin{equation}
\Omega^{E}_{z} = \Gamma E_{xy} (d_{x}^{2} - d_{y}^{2}) + \Gamma (E_{yy} -  E_{xx}) d_{x} d_{y}.
\end{equation}
It is convenient to introduce
\begin{equation}
\label{eq:decomp_Ez}
\Omega^{E}_{z} \equiv \Omega^{E,{cross}}_{z}  + \Omega^{E,{diag}}_{z},
\end{equation}
where we have distinguished the contributions from the diagonal and the cross
(i.e., off-diagonal) terms of the strain rate tensor.

\subsection{Point-particle approach for the solute field}

In order to be able to use Eqs. (\ref{eq:velocity_at_xp})-(\ref{eq:strain_rate_at_xp}),
one needs analytical expressions for $\mathbf{v}_{s} = -b(\mathbf{x}_{s}) \nabla_{||}
c(\mathbf{x})$. Here, we develop approximate, ``point-particle-like'' expressions for
the solute number density field $c(\mathbf{x})$.

The starting point is a multipole expansion for the solute number density field in
\textit{unbounded solution}, i.e., with no confining boundaries. We consider the centroid
of the particle (whether a sphere or a spherocylinder) to be located at the origin of a
Cartesian coordinate system which is co-moving with the particle. The axis of symmetry
of the particle is aligned with the $\hat{z}$ axis. The solute number density
field is obtained, as defined in the beginning of Sect. II, by solving the Laplace
equation subject to the boundary conditions noted there except for the one on the wall.

The \textit{f}ree \textit{s}pace distribution $c^{fs}(\mathbf{x})$ of solute
can be obtained numerically (e.g., by the BEM), or, for certain shapes of the
particle, analytically. We seek to extract multipole expansion coefficients
$\alpha_{i}$ from the result for $c^{fs}(\mathbf{x})$, where the coefficients
$\alpha_{i}$ are determined by its expansion in terms of Legendre
polynomials \cite{golestanian07}:
\begin{equation}
\label{eq:multipole_expansion}
c^{fs}(r, \theta) = c^{\infty} + \frac{R_{1}}{D} \sum_{l=0}^{\infty} \frac{\alpha_{l}}{l + 1}
\left(\frac{R_{1}}{r}\right)^{l+1} P_{l}(\cos(\theta)), \; r \geq R_{1}\,.
\end{equation}
This expansion, expressed in terms of spherical coordinates, is taken to be valid in
the fluid solution outside a sphere of radius $R_{1}$, where $2 R_{1}$ is the largest
dimension of the particle. For a \textit{spherical} particle of radius $R$, this expansion
is valid over the entire exterior region, i.e., $R_{1} = R$.
Furthermore, if the solute number density field around the sphere obeys the boundary
condition $[\mathbf{\hat{n}} \cdot \nabla c^{fs}(\mathbf{r})]|_{R,\theta}
= -\kappa/D$ over the lower hemisphere (i.e., the catalyst covered area) of the
sphere, and zero elsewhere on the surface of the sphere, the multipole coefficients
$\alpha_{i}$ are straightforwardly obtained by using the orthogonality of the Legendre
polynomials:
\begin{equation}
\label{eq:alphal_sphere}
\frac{\alpha_{l}}{\kappa} = \frac{2l + 1}{2} \int_{\pi/2}^{\pi} d\theta \sin(\theta) P_{l}
(\cos(\theta))\,.
\end{equation}
For instance, the monopole and dipole terms are $\alpha_{0} = \kappa/2$ and
$\alpha_{1} = -3 \kappa/4$, respectively.

For other particle geometries, such as a spherocylinder, the expression for the
coefficient $\alpha_{0}$ can be inferred from the following intuitive considerations.
In the multipole expansion of Eq. \ref{eq:multipole_expansion}, the monopole term
$\alpha_{0}$  encodes the net rate of solute production. For arbitrary geometries, we
can write
\begin{equation}
\label{eq:alf_0}
\alpha_{0} = \frac{A_{catalyst}}{4 \pi R_{1}^{2}} \kappa,
\end{equation}
i.e., $\alpha_{0}$ is proportional to the ratio of catalyst area to the area of a sphere
with diameter equal to $2 R_{1}$.  The higher order coefficients require detailed
calculations.  In Appendix A, we discuss how these are obtained for an arbitrary shape of
the particle from the numerically determined (e.g., via BEM) solute number density
field $c^{fs}(\mathbf{x})$. Particularizing the general results to our specific
spherocylinder geometry, we obtain (see Appendix A) $\alpha_0 = 0.2 \, \kappa$,
$\alpha_1 = -\, 0.2262 \, \kappa$, etc.

For a particle in the vicinity of the wall, the solute
number density field can be approximated from the multipole expansion as follows. We
restrict our consideration to the
first two terms in the multipole expansion, and describe the particle as a point
monopole plus a point dipole.\footnote{
\MP{We note that the strategy employed here is to keep the minimum number of terms in
the multipole expansion based on which the approximate dynamics is in qualitative agreement
with the numerical solution of the complete (exact) dynamics. In the sense of a perturbative
approach, the approximation based on just the first two terms in the multipole expansion, combined with a
point-particle solution of the hydrodynamics, is mathematically not justified when
$h/R$ is not much larger than 1, as studied here. Thus, the range of validity is determined
``a posteriori'', similar to many other cases in which such approximations turn out to be accurate
well beyond the mathematically rigorous limits of validity (see, e.g., Ref. \cite{spagnolie12}). For our system the approximate analysis seems to remain valid (at least
quasi-quantitatively) down to very small values of $h/R-1$.}
}
The direction of the dipole (defined to be from the
point source to the point sink) is $\mathbf{\hat{d}}$. We place an image monopole and an
image dipole below the wall at the position $(x_p, y_p, -h)$. The image monopole has the
same strength $\alpha_0$ as the actual monopole. Likewise, the image dipole has the
same strength $\alpha_1$ and orientation $\mathbf{\hat{d}}$ as the actual dipole
(recalling that the dipole is parallel to the wall surface, since we assumed
$\mathbf{\hat{d}} \cdot \hat{z} = 0$). Accordingly, one has
\begin{equation}
c(\mathbf{x}_{s}) \approx c^{mp}(\mathbf{x}_{s}) + c^{dp}(\mathbf{x}_{s}),
\end{equation}
where $c^{mp}(\mathbf{x}_{s})$ is the solute number density field due to the monopole
at the position $\mathbf{x}_{s}$ on the wall, plus its image, and $c^{dp}(\mathbf{x}_{s})$
is the field due to the dipole at the position $\mathbf{x}_{s}$ on the wall, plus its
image. For the monopole term, we obtain
\begin{equation}
\label{eq:c_mp}
c^{mp}(\mathbf{x}_{s}) = \frac{2 \alpha_{0} R_{1}^{2}}{D}
\frac{1}{|\mathbf{x}_{s} - \mathbf{x}_{p}|}\,,
\end{equation}
and for the dipole term,
\begin{equation}
\label{eq:c_dp}
c^{dp}(\mathbf{x}_{s}) = - \frac{|\alpha_{1}| R_{1}^{3}}{D}
\frac{\mathbf{\hat{d}} \cdot \mathbf{r}_{s}}{|\mathbf{x}_{s} - \mathbf{x}_{p}|^{3}},
\end{equation}
where $\mathbf{r}_{s} \equiv (x_s - x_p, y_s - y_p, 0)$. Consequently, the slip
velocity $\mathbf{v}_{s}(\mathbf{x}_{s}) = -
b(\mathbf{x}_{s}) \nabla_{||} c(\mathbf{x})|_{\mathbf{x}=\mathbf{x}_{s}}$ on the wall
is decomposed as $\mathbf{v}_{s}(\mathbf{x}_{s}) \approx \mathbf{v}_{s}^{mp}(\mathbf{x}_{s}) + \mathbf{v}_{s}^{dp}(\mathbf{x}_{s})$.  The contribution of the monopole to the wall slip velocity is
\begin{equation}
\label{eq:mp_vs}
\mathbf{v}_{s}^{mp} = \frac{2 \alpha_{0} R_{1}^{2}}{D} \frac{b_{w}(\mathbf{x}_{s}) \, \mathbf{r}_{s}}{|\mathbf{x}_{s} - \mathbf{x}_{p}|^{3}}\,,
\end{equation}
and the contribution of the dipole is
\begin{equation}
\label{eq:vs_dp}
\mathbf{v}_{s}^{dp} = \frac{b_w(\mathbf{x}_{s}) |\alpha_{1}| R_{1}^{3}}{D |\mathbf{x}_{s} - \mathbf{x}_{p}|^{3}} \,\; \mathbf{\hat{d}} \cdot  \left( \mathcal{I} - \frac{3 \mathbf{r}_{s} \mathbf{r}_{s}}{|\mathbf{x}_{s} - \mathbf{x}_{p}|^{2}} \right) .
\end{equation}

\subsection{Monopole contribution to the particle velocity}

The combination of Eqs. (\ref{eq:velocity_in_fluid}), (\ref{eq:sigma_all_modes}), and
(\ref{eq:mp_vs}) yields the monopole contributions to the fluid velocity at an arbitrary
point $\mathbf{x}_{f}$ above a chemically patterned wall:

\begin{equation}
{u}_{x}^{mp}(\mathbf{x}_{f}) =  \frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \int \frac{b_{w}(\mathbf{x}_{s})(x_{s}-x_{f})[(x_{s}-x_{f})(x_{s}-x_{p}) + (y_{s} - y_{f})(y_{s}-y_{p})]}{((x_{f}-x_{s})^2 + (y_{f} - y_{s})^{2} + z_{f}^{2})^{5/2} ((x_{p} - x_{s})^{2} + (y_{p} - y_{s})^{2} + h^{2}))^{3/2}} \, dS
\end{equation}
and
\begin{equation}
{u}_{y}^{mp}(\mathbf{x}_{f}) =  \frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \int \frac{b_{w}(\mathbf{x}_{s})(y_{s}-y_{f})[(x_{s}-x_{f})(x_{s}-x_{p}) + (y_{s} - y_{f})(y_{s}-y_{p})]}{((x_{f}-x_{s})^2 + (y_{f} - y_{s})^{2} + z_{f}^{2})^{5/2} ((x_{p} - x_{s})^{2} + (y_{p} - y_{s})^{2} + h^{2}))^{3/2}} \, dS.
\end{equation}

Now we consider the derivatives of the fluid velocity. In order to obtain $E_{xy}^{mp}(\mathbf{x}_{p})$, we calculate $\frac{1}{2} \left( {\partial u_{x}^{mp}}/{\partial y_{f}} + {\partial u_{y}^{mp}}/{\partial x_{f}} \right)\big|_{\mathbf{x}_{f} = \mathbf{x}_{p}}$. The two partial derivatives are identical, yielding
\begin{equation}
E_{xy}^{mp}(\mathbf{x}_{p}) =  \frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \int \frac{b_{w}(\mathbf{x}_{s}) (x_{p} - x_{s})(y_{p} - y_{s}) [4 \, ((x_{p} - x_{s})^2 + (y_{p} - y_{s})^2) - h^{2}]}{((x_{p} - x_{s})^{2} + (y_{p} - y_{s})^2 + h^{2})^{5}} \, dS.
\end{equation}

Analogously one obtains $E_{xx}^{mp}(\mathbf{x}_{p})$ and $E_{yy}^{mp}(\mathbf{x}_{p})$:
\begin{equation}
\begin{split}
& E_{xx}^{mp}(\mathbf{x}_{p}) = \frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \\ & \, \,
\times \int \frac{b_{w}(\mathbf{x}_{s}) [3 (x_p-x_s)^4 + 2(x_p-x_s)^2(y_p-y_s)^2 - (y_p-y_s)^4-(2(x_p-x_s)^2+(y_p-y_s)^2)h^2]}{((x_p - x_s)^2 + (y_p - y_s)^2 + h^2)^5} \, dS
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
& E_{yy}^{mp}(\mathbf{x}_{p}) = -\frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \\ & \;\;
\times  \int \frac{b_{w}(\mathbf{x}_{s}) [((x_p-x_s)^2 - 3(y_p-y_s)^2)((x_p-x_s)^2+(y_p-y_s)^2)+((x_p-x_s)^2+2(y_p-y_s)^2)h^2]}{((x_{p} - x_{s})^{2} + (y_{p} - y_{s})^2 + h^{2})^{5}} \, dS\,.
\end{split}
\end{equation}

The monopolar contribution to the vorticity component $\omega_{z}$ vanishes, because the two partial derivatives in $\omega_z = \left( {\partial u_{x}^{mp}}/{\partial y_{f}} - {\partial u_{y}^{mp}}/{\partial x_{f}} \right)\big|_{\mathbf{x}_{f} = \mathbf{x}_{p}}$ are identical so that
\begin{equation}
\omega^{mp}_{z} = 0.
\end{equation}

If the pattern exhibits translational symmetry in the $\hat{\mathbf{y}}$ direction,
such that $b(\mathbf{x}_{s}) = b(x_{s})$, the particle velocity does not depend on $y_p$.
Accordingly, in calculating the velocity, one can set $y_p = 0$ without loss of generality.
Concerning $E_{xy}^{mp}({x}_{p})$, the resultant expression involves an integral over an
odd function of $y_s$, and hence it is zero:
\begin{equation}
\label{eq:Exx_y_sym}
E_{xy}^{mp}({x}_{p}) = - \frac{3 h}{2 \pi} \frac{2 \alpha_{0} R^{2}}{D} \int dx_{s} \int dy_{s} \,  \frac{b_{w}({x}_{s}) (x_{p} - x_{s}) y_{s} [4 \, ((x_{p} - x_{s})^2 +  y_{s}^2) - h^{2}]}{((x_{p} - x_{s})^{2} + y_{s}^2 + h^{2})^{5}}  = 0.
\end{equation}

Now we consider the specific case of a chemical step by substituting
Eq. (\ref{eq:step_surface_mob}) for $b_{w}(\mathbf{x}_{s})$.  According to
Eq. (\ref{eq:Exx_y_sym}), $E_{xy}^{mp}$ vanishes. This leaves four piecewise integrals to
be performed: the contributions of the left ($x_s < 0$) and of the right ($x_s \,>\, 0$)
side to $E_{xx}^{mp}$ and $E_{yy}^{mp}$. After straightforward but cumbersome analysis, we
obtain the contribution of the monopole to particle rotation:
\begin{equation}
\label{eq:omega_z_mp}
\Omega_{z}^{mp} = -\frac{3 \alpha_{0} \Gamma R_{1}^{2}}{128 D} \, (b_{w}^{l} - b_{w}^{r}) \, \frac{h (32 x_{p}^{3} + 7 x_{p} h^{2})}{(x_{p}^{2} + h^{2})^{7/2}} \sin(\phi) \cos(\phi)\,.
\end{equation}

\subsection{Dipole contribution to the particle velocity}

By combining Eqs. (\ref{eq:velocity_in_fluid}), (\ref{eq:sigma_all_modes}), and
(\ref{eq:vs_dp}) one finds the dipolar contributions to the fluid velocity at
$\mathbf{x}_{f}$:
\begin{equation}
\begin{split}
\label{eq:ux_dp}
u_{x}^{dp}(\mathbf{x}_{f}) = \frac{3 h}{2 \pi} \frac{|\alpha_{1}| R_{1}^{3}}{D} \left[\int   \frac{  b_{w}(\mathbf{x}_{s})(x_s - x_f) ((x_s-x_f) d_x + (y_s - y_f) d_y)}{((x_f - x_s)^2 + (y_f - y_s)^2 + h^2)^{5/2} ((x_p - x_s)^2 + (y_p - y_s)^2  h^2)^{3/2}} \, dS \, + \, \right. \\ \left.
- \int \frac{3 b_{w}(\mathbf{x}_{s}) (x_s-x_f)[(x_s-x_p) d_x + (y_s - y_p) d_y] [(x_s - x_f)(x_s - x_p) + (y_s - y_f)(y_s - y_p)]}{((x_f - x_s)^2 + (y_f - y_s)^2 + h^2)^{5/2} ((x_p - x_s)^2 + (y_p - y_s)^2 + h^2)^{5/2}} \, dS \right]
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\label{eq:uy_dp}
u_{y}^{dp}(\mathbf{x}_{f}) = \frac{3 h}{2 \pi} \frac{|\alpha_{1}| R_{1}^{3}}{D} \left[\int   \frac{  b_{w}(\mathbf{x}_{s})(y_s - y_f) ((x_s-x_f) d_x + (y_s - y_f) d_y)}{((x_f - x_s)^2 + (y_f - y_s)^2 + h^2)^{5/2} ((x_p - x_s)^2 + (y_p - y_s)^2  h^2)^{3/2}} \, dS \, + \, \right. \\ \left.
- \int \frac{3 b_{w}(\mathbf{x}_{s}) (y_s-y_f)[(x_s-x_p) d_x + (y_s - y_p) d_y] [(x_s - x_f)(x_s - x_p) + (y_s - y_f)(y_s - y_p)]}{((x_f - x_s)^2 + (y_f - y_s)^2 + h^2)^{5/2} ((x_p - x_s)^2 + (y_p - y_s)^2 + h^2)^{5/2}} \, dS \right],
\end{split}
\end{equation}
where, in each equation, the two integrals correspond to the two terms in the parentheses in
Eq. (\ref{eq:vs_dp}). As with the monopole contributions,  one can construct surface
integrals for the vorticity and the strain rate tensor associated with the dipole by
taking partial derivatives of Eqs. (\ref{eq:ux_dp}) and (\ref{eq:uy_dp}). We omit these
lengthy intermediate expressions, and provide only the final results, which are, somewhat
surprisingly, rather compact expressions. The contribution of the dipole to rotation
via vorticity (see the text below Eq. (\ref{eq:strain_rate_at_xp})) is
\begin{equation}
\label{eq:omega_z_v_dp}
\Omega_{z}^{V,dp} = - \frac{3 |\alpha_{1}| R_{1}^{3} }{64 D} \, (b_{w}^{l} - b_{w}^{r}) \,
\frac{h}{(h^2 + x_p^{2})^{5/2}} \sin(\phi)\,.
\end{equation}
The dipolar contributions to rotation via the rate of strain (see Eq.
(\ref{eq:decomp_Ez}) and the text below Eq. (\ref{eq:strain_rate_at_xp})) are
\begin{equation}
\label{eq:omega_z_E_cross_dp}
\Omega_{z}^{E,{cross},dp}(\mathbf{x}_{p}) = \frac{3 |\alpha_{1}| R_{1}^{3} \Gamma }{512 D} (b_{w}^{l} - b_{w}^{r}) \,  \frac{h (22 x_p^2 + 7 h^{2})}{(x_p^2 + h^2)^{7/2}}  \sin(\phi) \, [\cos^{2}(\phi) - \sin^{2}(\phi)]
\end{equation}
and
\begin{equation}
\label{eq:omega_z_E_diag_dp}
\Omega_{z}^{E,{diag},dp}(\mathbf{x}_{p}) = - \frac{15 |\alpha_{1}|R_{1}^{3}  \Gamma }{512 D}  (b_{w}^{l} - b_{w}^{r}) \, \frac{h (20 x_p^4 + 2 x_{p}^{2} h^{2} + 3 h^{4})}{(x_p^2 + h^2)^{9/2}} \, \cos^{2}(\phi) \sin(\phi)\,.
\end{equation}
\begin{figure}[!htb]
\includegraphics[scale=0.9]{./Fig7.pdf}
\caption{\label{fig:sphere_spherocyl_omegaZ} (a) and (b) Angular velocity $\Omega_{z}^{ws}$ of
a half-covered sphere due to chemi-osmotic slip at the wall as a function of orientation
$\phi$ for several lateral distances $x_{p}$ from a chemical step. The step is characterized
by $b_{w}^{r}/b_{w}^{l} = 4$ and $b_{w}^{l} < 0$ and is located at $x_{p} = 0$. In (a),
the center of the sphere is at a height $h/R = 1.1$ above the surface. In (b), the height is $h/R = 3$.  (c) and (d) Angular velocity of a half-covered spherocylinder as a function of orientation $\phi$ for several lateral distances from the same chemical step. In (c), the centroid of the spherocylinder is at height $h/c = 0.6$ above the surface, while
in (d) it is at height $h/c = 3$, where $2 c$ is the tip-to-tip length of the
spherocylinder. In (a) the theoretical and numerical data for $x_p/R = \pm 1$ \textit{de facto} coincide. In (b) the theoretical and numerical data for $x_p/R = \pm 1$ \textit{de facto} coincide, as do the theoretical and numerical data for $\pm 2$. }
\end{figure}

\subsection{Comparison of analytical and numerical results}

Finally, by combining Eqs. (\ref{eq:omega_z_mp}), (\ref{eq:omega_z_v_dp}),
(\ref{eq:omega_z_E_cross_dp}), and (\ref{eq:omega_z_E_diag_dp}) one obtains an approximation
expression for the rotation of a particle in the vicinity of a patterned wall:
\begin{equation}
\label{eq:omega_Z_ws_all_terms}
\Omega_{z}^{ws}(\mathbf{x}_{p}) \approx \Omega_{z}^{mp}(\mathbf{x}_{p}) + \Omega_{z}^{V,dp}(\mathbf{x}_{p}) + \Omega_{z}^{E,{cross},dp}(\mathbf{x}_{p}) + \Omega_{z}^{E,{diag},dp}(\mathbf{x}_{p}).
\end{equation}
We first consider the rotation of a sphere near a chemical step. For this shape of the
particle, $\Omega_{z}^{mp}(\mathbf{x}_{p})$, $\Omega_{z}^{E,{cross},dp}(\mathbf{x}_{p})$,
and $\Omega_{z}^{E,{diag},dp}(\mathbf{x}_{p})$ all vanish, because they originate from the
strain rate tensor. The remaining term, $\Omega_{z}^{V,dp}(\mathbf{x}_{p})$, is a simple
sinusoid (Eq. \ref{eq:omega_z_v_dp}), and hence crosses zero only at $\phi = 0$,
$\phi = \pi$, and $\phi = 2 \pi$. Therefore, the only possible stable orientations of the
sphere are docking states (i.e., pointing towards or away from the step). In
Figs. \ref{fig:sphere_spherocyl_omegaZ}(a) and (b) we compare this expression with
\MP{the numerical solution of the exact dynamics, which is} obtained by BEM. In this figure, (a)
corresponds to $h/R = 1.1$ and (b) to $h/R = 3$, and $\Omega_{z}^{ws}$ is plotted as a
function of $\phi$ for various lateral distances $x_{p}/R$ from the step. We obtain excellent
agreement for $h/R = 3$, for which we expect the truncation of the multipole expansion to the
monopole and dipole terms to be a good approximation. Closer to the surface ($h/R = 1.1$), we
obtain qualitative, but only semi-quantitative agreement. In particular, we note that while
the numerical results show a stronger rotation by the surface than predicted by the
approximate expressions, they retain the theoretically predicted (see above) sinusoidal
dependence on $\phi$.

For the spherocylinder, all terms in Eq. (\ref{eq:omega_Z_ws_all_terms}) contribute.
The resultant expression for $\Omega_{z}^{ws}$ is complicated, and can cross $\Omega_{z}^{ws}
= 0$ between $\phi = 0$ and $\phi = \pi$. This more complicated dependence enables the
occurrence of the edge-following state. In Figs. \ref{fig:sphere_spherocyl_omegaZ}(c) and
(d), we show the theoretical and numerical results for a spherocylinder at $h/c = 0.6$ and
$h/c = 3$, respectively. For $h/c = 3$, we again obtain excellent agreement.
For $h/c = 0.6$, there is no \textit{a priori} reason to expect agreement, because our
mapping of the particle solute field to the multipole expansion was performed for distances
from the particle centroid larger than $R_{1} = c$. Nevertheless, the agreement remains
surprisingly good. The strongest deviation occurs for $x_p = 0$, for which the numerical
results exhibit a more complicated functional dependence on $\phi$ than the sinusoidal one
predicted theoretically. Note that for $x_p/c = 1$ (brown circles and brown curve) and
$x_p/c = -1$ (purple circles and purple curve), the disagreement between theory and
numerics can hardly be distinguished by eye.

\section{Conclusions}

We have studied the effect of particle shape on the motion of a catalytically active particle
near a chemically patterned surface. The active particle locally induces chemi-osmotic flows
on the surface, which, in turn, drive flows in the volume of the solution
that couple back to the particle. This chemi-osmotic contribution to particle motion is sensitive
to the material identity of the surface, and it competes with the inherent
self-diffusiophoretic motion of the particle. We have restricted our focus to
``inert-forward'' particles which tend to move away from their catalytic caps via
self-diffusiophoresis.  As first shown in previous work, near a chemical step
``inert-forward'' spherical particles exhibit a rich phenomenology \cite{uspal16}. A sphere
can pass over the step, or stably ``dock'' at the edge between the two materials, depending
on the system parameters and the initial position and orientation of the particle.
A docked particle remains motionless with its axis of symmetry aligned perpendicular to
the edge of the step.  A particle which passes over the step can exhibit
significant deviations of its orientation from the angle of approach.  If the particle is
non-spherical, such as a spherocylinder, this phenomenology is significantly enriched. In addition
to the docking behavior, a rod-like particle can stably follow the edge of the step, with its axis
of symmetry tilted slightly away from alignment with the edge.  This ``edge-following'' state can
coexist with the docking state for a broad range of parameters. Additionally, if the surface has a
narrow chemical stripe (i.e., two neighboring step-like interfaces), the edge-following state is
transformed into a stable ``stripe-following''
state in which the particle is located at the stripe center and is perfectly aligned
with the stripe.

These observations motivated the development of an approximate analytical approach in order to
understand the underlying physical mechanisms. A central insight of our approach is a distinction
between the orientational response of a particle to vorticity and to the fluid strain rate. All
particles are driven to rotate by fluid vorticity, but only elongated (i.e., non-spherical)
particles respond to the hydrodynamic strain rate. Our analytical expressions demonstrate that
this significantly more complicated orientational behavior is the basis of the enriched
phenomenology observed for spherocylinders. Therefore, our findings are not specific to
spherocylinders, but generalize to other non-spherical shapes (e.g., prolate
spheroids \cite{popescu10}), as detailed numerical calculations (omitted here) confirm. Our
numerical and analytical calculations have considered a self-diffusiophoretic particle with a
monopole term, i.e., a net producer of product molecules. However, our theoretical framework can
also be applied to confined self-electrophoretic particles\MP{, under the additional assumption of the presence of
a thin Debye layer required by a mapping to an osmotic slip velocity (which has been used in, e.g.,
Ref. \cite{liu16}),} with appropriate renaming of the variables: $c(\mathbf{x})$ as an electrical
potential $\psi(\mathbf{x})$, the surface mobility $b(\mathbf{x}_{s})$ as a quantity proportional
to the zeta potential $\zeta(\mathbf{x}_{s})$, etc. \cite{anderson89,golestanian07,chiang14,liu16}.
A self-electrophoretic particle would have no monopole term ($\alpha_{0} = 0$), but
otherwise our findings would carry over to this new setting.

\MP{While the focus of the current work is on the deterministic dynamics, the effects of
moderate thermal fluctuations can be assessed, to a certain extent, from the dynamical stability of
the corresponding fixed points. The linear stability analysis of the deterministic equations is
straightforward, requiring only evaluations of the Jacobians at the fixed points. Similarly with
the observations in our previous work \cite{uspal15}, the attractors of the dynamics are
``stiff'', which prevents large excursions from the vicinity of the attractor in the phase plane. For instance, following the procedure of Ref. \cite{uspal15}, we obtain a longest relaxation time of $3.7\;\textrm{s}$ for the docking state in Fig. \ref{fig:sphere_traj_portraits}(c), which compares favorably with the characteristic rotational diffusion time of $D_{r}^{-1} = 95\;\textrm{s}$. However, one often encounters, as in Fig. \ref{fig:stripe_figure}(a), a combination of a fast dynamics along one coordinate (the translation) with a (very) slow dynamics along the other coordinate (i..e, the rotation of the particle axis). This  implies that the small excursions in the phase plane may actually translate into extended periods
of time that a particle, subject to stochastic dynamics, spends outside the fixed point (see also,
e.g., Ref. \cite{mozaffari17}).
}

Our findings open new possibilities for programming the motion of catalytically active particles
in lab-on-a-chip devices. The edge-following and stripe-following states can be used to direct
rod-like particles from one location to another. If the ratio $U^{sd}/U_{0}$ (characterizing the
relative contributions of self-diffusiophoresis and chemi-osmosis to motility) is sensitive to
light or some other stimulus, a rod-like particle could be switched between edge-following and
docking states on demand. This would be the case, for instance, if light could be used to evoke
changes in the surface properties of either the particle or the wall. We note that we had
previously predicted stripe-following behavior for spheres, but only for the ``catalyst-forward''
direction of motion, which is much less commonly seen in experiments. Additionally, rod-like
particles quite naturally realize our assumption of quasi-2D motion, in which the particle is
restricted to have a constant height above the wall and an orientation vector that always remains
within a plane parallel to the wall. \WEU{Our findings concerning the role of the particle aspect ratio show that for $e < 0.85$ the edge-following states exist over a broad range of self-propulsion speeds.}

Further work could also investigate more complicated shapes. L-shaped particles which
are restricted to quasi-2D motion have a well-defined chirality \cite{kummel13}. The effect
of fore-aft asymmetry (e.g., for an asymmetric dumbbell with a large ``head'') is an open
question \cite{valadares10,uspal13}. Additionally, one could consider shapes with a more
complicated topology, such as ring or toroidal shapes \cite{schmieding17,singh13}.

\acknowledgments We thank C. Pozrikidis for making the \texttt{BEMLIB} library freely available
\cite{pozrikidis02} \WEU{and an anonymous referee for suggesting Fig. \ref{fig:phase_map}}. W.E.U., M.T., and M.N.P. acknowledge financial support from the German Science Foundation (DFG), grant no. TA 959/1-1. \WEU{M.T. acknowledges financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts no. IF/00322/2015.}

\section*{Appendix A: numerical calculation of multipole coefficients}

For many geometries of interest, such as the spherocylinder, there is no suitable system of
coordinates in terms of which the Laplace equation is separable. For these geometries,
instead a numerical method, such as BEM, can be used to obtain $c(\mathbf{r})$. We
briefly outline how to obtain the multipole coefficients from a known solute density
$c(\mathbf{r})$ surrounding an axisymmetric particle in unbounded solution.

Any solution $c(\mathbf{r})$ of the Laplace equation can be expressed as an integral
over the domain boundaries \cite{pozrikidis02}:
\begin{equation}
\label{eq:boundary_integral_c}
c(\mathbf{r}') = c_{\infty} -
\int_{\cal D} G(\mathbf{r}', \mathbf{r}) [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] dS + \int_{\cal D} c(\mathbf{r}) [ \mathbf{\hat{n}} \cdot \nabla_{\mathbf{r}} G( \mathbf{r}', \mathbf{r}) ] dS.
\end{equation}
Here, ${\cal D}$ is the particle surface, $\mathbf{r}'$ is a point of interest in the
fluid domain, and the local surface normal $\mathbf{\hat{n}}$ is defined to point from the particle surface into the fluid solution. The Green's function $G(\mathbf{r}, \mathbf{r}')$ is
\begin{equation}
\label{eq:fs_gf}
G(\mathbf{r}, \mathbf{r}') \equiv \frac{1}{4 \pi |\mathbf{r} - \mathbf{r}'|}.
\end{equation}
Equations (\ref{eq:boundary_integral_c}) and (\ref{eq:fs_gf}) indicate that one can
interpret the solution ${c}(\mathbf{r})$ as arising from a distribution of point
sources (monopoles) and point source - point sink pairs (dipoles) over the particle
surface which, respectively, correspond to the first and the second integrals
in Eq. (\ref{eq:boundary_integral_c}).

Now we replace the left hand side of Eq. (\ref{eq:boundary_integral_c}) by
Eq. (\ref{eq:multipole_expansion}):
\begin{equation}
\frac{R_{1}}{D} \sum_{l=0}^{\infty} \frac{\alpha_{l}}{l + 1}
\left(\frac{R_{1}}{r'}\right)^{l+1} P_{l}(\cos(\theta')) = - \int_{\cal D} G(\mathbf{r}',
\mathbf{r}) [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] dS + \int_{\cal D} c(\mathbf{r})
[ \mathbf{\hat{n}} \cdot \nabla_{\mathbf{r}} G( \mathbf{r}', \mathbf{r}) ] dS.
\end{equation}
We denote the spherical surface defined by $r' = R_{1}$ as ${{\cal D}_1}$.
We multiply both sides by $P_{n}(\cos(\theta'))$ and integrate them as
$\int_{{\cal D}_1} dS' $. Due to the orthogonality of the Legendre polynomials, one has
\begin{multline}
\label{eq:alpha_n_bie}
\frac{2 \pi R_{1}}{D}  \frac{\alpha_{n}}{n + 1} \frac{2}{2n + 1} =
- \int_{{\cal D}} dS \, [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] \int_{{\cal D}_1} dS' \, G(\mathbf{r}', \mathbf{r}) P_{n}(\cos(\theta')) \\
+ \int_{{\cal D}} dS \, c(\mathbf{r}) \, \mathbf{\hat{n}} \cdot \nabla_{\mathbf{r}} \int_{{\cal D}_1} dS' \, G( \mathbf{r}', \mathbf{r}) P_{n}(\cos(\theta'))\,.
\end{multline}
In the next step we expand the Green's function in terms of spherical harmonics:
\begin{equation}
G(\mathbf{r}', \mathbf{r}) = \sum_{l=0}^{\infty} \frac{1}{2l + 1} \sum_{m=-l}^{l} (-1)^{m}
\frac{r^{l}}{r'^{l+1}} Y_{l,-m}(\theta, \phi) Y_{l,m}(\theta', \phi')\,.
\end{equation}
This leads to
\begin{equation}
\label{eq:integr_G}
\begin{split}
\int_{{\cal D}_1} dS' \, G(\mathbf{r}',\mathbf{r}) P_{n}(\cos(\theta'))
& =
\sum_{l=0}^{\infty} \frac{1}{2l + 1} \sum_{m=-l}^{l} \frac{r^{l}}{R_{1}^{l+1}}
Y_{l,-m}(\theta, \phi) \int_{{\cal D}_1} dS'\, Y_{l,m}(\theta', \phi')
\, P_{n}(\cos(\theta'))  \\
& = \frac{1}{2n + 1} \frac{r^{n}}{R_{1}^{n+1}} Y_{n,0}(\theta, \phi)
\sqrt{\frac{4 \pi}{2n + 1}} \\
&= \frac{1}{2n + 1} \frac{r^{n}}{R_{1}^{n+1}} P_{n}(\cos(\theta))\,,
\end{split}
\end{equation}
where upon computing the integral on the right hand side above we have used
$P_{n}(\cos(\theta')) = \sqrt{\frac{2n+1}{4 \pi}} ~ Y_{n,0}(\theta',\phi')$ and the
orthogonality of the spherical harmonics.

Substituting Eq. (\ref{eq:integr_G}) into Eq. (\ref{eq:alpha_n_bie}) renders
\begin{equation}
\label{eq:alpha_n_bie_2}
\frac{4 \pi R_{1}}{D}  \frac{\alpha_{n}}{n + 1}  = - \int_{{\cal D}} dS
\, [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] \frac{r^{n}}{R_{1}^{n+1}} P_{n}(\cos(\theta))
+
\int_{{\cal D}} dS \, c(\mathbf{r}) \, \mathbf{\hat{n}} \cdot \nabla_{\mathbf{r}}
\left[ \frac{r^{n}}{R_{1}^{n+1}} P_{n}(\cos(\theta)) \right].
\end{equation}

For $\alpha_{0}$, only the first integral contributes:
\begin{equation}
\alpha_{0} = - \frac{D}{4 \pi R_{1}^{2}} \int_{D} dS \, [\mathbf{\hat{n}}
\cdot \nabla c(\mathbf{r})].
\end{equation}
Recalling that $[\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] = -\kappa/D$ over the
catalyst covered area $A_{catalyst}$ and zero elsewhere (see the first paragraph in
Sect. II), we have
\begin{equation}
\alpha_{0} = \frac{A_{catalyst}}{4 \pi R_{1}^{2}} \kappa,
\end{equation}
which we obtained earlier by simple arguments (see the text preceding
Eq.(\ref{eq:alf_0})).

The dipole term $\alpha_{1}$ has contributions from both integrals in
Eq. (\ref{eq:alpha_n_bie_2}):
\begin{equation}
\frac{2 \pi R_{1}}{D} \alpha_{1}  = - \int_{D} dS \, [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] \frac{r}{R_{1}^{2}} P_{1}(\cos(\theta))  + \int_{D} dS \, c(\mathbf{r}) \, \mathbf{\hat{n}} \cdot \nabla_{\mathbf{r}} \left[ \frac{r}{R_{1}^{2}} P_{1}(\cos(\theta)) \right].
\end{equation}
Using $r \, P_{1}(\cos(\theta)) = z$, one finds
\begin{equation}
\label{eq:alpha1_BIE}
\frac{2 \pi R_{1}^{3}}{D} \alpha_{1}  = - \int_{D} dS \, [\mathbf{\hat{n}} \cdot \nabla c(\mathbf{r})] z  + \int_{D} dS \, c(\mathbf{r}) \, \mathbf{\hat{n}} \cdot \hat{z}\,.
\end{equation}
One can readily verify that the solution for a sphere
(Eq. (\ref{eq:multipole_expansion})) satisfies Eq. (\ref{eq:alpha1_BIE}).

redpajama_set_name: RedPajamaArXiv


text:
define([
  'angular',
  'definitionsLoader',
  'uiRouter',
  'Sortable',
  'ngSortable',
  'controllers/index',
  'directives/index',
  'filters/index'], function (angular, definitionsLoader) {
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  return angular.module('QuickSale', [
    'QuickSale.controllers',
    'QuickSale.directives',
    'QuickSale.filters',
    'ng-sortable',
    'ui.router'
  ].concat(definitionsLoader.modulesToLoad));
});

redpajama_set_name: RedPajamaGithub


text:
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Check out the students that were members of South San Francisco Music. Visit the registration page if you would like to be listed.
Looking for South San Francisco alumni who participated in Music but are not listed? Classmates.com® has hundreds of more South San Francisco alumni listed.


Kitchen, dining room and living room in one 20 m2 walk-through space. Nearly impossible task requested a radical concept. Absolute minimalism. The solution lies in a single piece of furniture. A compact block contains all necessary features (complete kitchen, coffee machine, refridgerator, dining table, chairs, sofa, projector, footrests, coffee tables, lighting, storage and doors leading to the private part of the flat).


Here's a great deal on a 2018 BMW 320i. Smooth gearshifts are achieved thanks to the 2 liter 4 cylinder engine, and all wheel drive keeps this model firmly attached to the road surface. Well tuned suspension and stability control deliver a spirited, yet composed, ride and drive Turbocharger technology provides forced air induction, enhancing performance while preserving fuel economy. A wealth of standard features means that you no longer have to sacrifice. Like power windows, mirrors, and seats, variably intermittent wipers, adjustable headrests in all seating positions, automatic dimming door mirrors, heated seats, front and rear air conditioning, power moon roof, and seat memory. Everything is where it ought to be, from the dashboard controls to the door locks and window controls. Enjoy your favorite music via the stereo system, which includes a CD player with MP3 capability, steering wheel mounted audio controls, and 9 speakers, providing excellent sound throughout the cabin. BMW also prioritized safety and security with features such as: dual front impact airbags with occupant sensing airbag, front side impact airbags, traction control, brake assist, anti-whiplash front head restraints, a security system, an emergency communication system, and 4 wheel disc brakes with ABS. This vehicle has achieved Certified Pre-Owned status, by passing BMW's rigorous certification process. We have the vehicle you've been searching for at a price you can afford. Stop by our dealership or give us a call for more information.


Our new corporate site is coming soon!
TrekAce® Technologies enables people to THRIVE (surviving > coping > enjoying) at ANY situation or context …no matter how extensive it may be, and where other technologies fail!
Outdoors sports – Never look at a map again: Navigate with ease when running, cycling, motorcycling, hunting, skiing, kayaking. If it works well for the world's best Special Forces, it will work well for you!
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These lovely holiday apartments are situated in quiet surroundings of the small, medieval town Vodnjan. Furnished with comfortable furniture, they will surely provide you a pleasant stay. CIC954 has access via the terrace and the minimum ceiling height in this apartment is approx. 100 cm. Relax on the terrace with your loved ones. Vodnjan is known for its olive oil and for the highest bell tower in Istria. In Fazana you can reach the first beaches, from where you can make day trips to the Brijuni islands.


text:
About Chaunie
Your Lines
Chaunie Brusie
Mom | Writer | Nurse
August 22, 2018 By Chaunie
Jenna Dewan Thinks Becoming a Mom Made Her Sexier
I have to admit that I'm a little fascinated by Jenna Dewan and before you say that it's just because she was married to my main crush, Channing Tatum, I will say that I've always thought she was awesome. Way back to her first days on "Step Up," in fact. I mean, she had that inner confidence that I so admire in women, the kind of confidence that, fine, led her to eventually date and marry Channing, but also has led her to have her own career and arguably, way better dance moves than her ex.
Sorry, Chan, but it's true.
And now that Jenna Dewan is making a name for herself with a brand-new power career and making headlines on magazine covers and in articles on her own while Channing seems to have taken a bit of a back seat, it's fascinating to see how she has evolved as a mother and as a woman too. For instance, Dewan recently graced the cover of Women's Health, literally baring it all–and she had a lot to say on the topic of being sexy as a mom.
In her cover story for the mag, Dewan confessed that she was a bit frustrated by the newfound limitations to her sexuality that she encountered once she had a baby. "Apparently, when you become a mother, you're supposed to leave your sexuality at the door, and I never understood that," she protested in Women's Health's September issue.
To Dewan, however, who has famously described herself as a very sexual person, largely due to the fact that as a dancer, she is very sensual and in tune to her body, motherhood did not take away anything from her sexuality. In fact, she thinks that becoming a mom made her even sexier.
Photo by Jonathan Gallegos on Unsplash
"I think there's nothing sexier than becoming a mother. You give life. It's everything. And you don't change who you are inside just because you have a kid."
Read the rest on EverydayFamily.com
Couple Crush Alert: Channing and Jenna Dewan Tatum
What Is It With Dads and Babies?
Countdown To Baby
Filed Under: Freelance Writing Tagged With: channing tatum, Jenna Dewan, motherhood, Sexy Leave a Comment
Hi, I'm Chaunie, a coffee mug addict, a labor and delivery nurse turned freelance writer, and a young(ish) mom of four. I'm the author of Tiny Blue Lines: Preparing For Your Baby, Moving Forward In Faith, & Reclaiming Your Life In An Unplanned Pregnancy and The Moments That Made You A Mother. Read More…
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All Images & Content Copyright © 2020 Chaunie Brusie · Design by Stephanie Jones
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When the angel says, "When the darkness falls, please close your eyes." Close all eyes except the angels.
After the game, all the cards returned to the angel to re-shuffle; angel resign after reshuffle, everyone re-starting a new Council, re-draw.


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Game, David - consultant nephrologist
Dr David Game trained at Cambridge and Oxford Universities. He started as a consultant at Guy's and St Thomas' in November 2009.
Garagnani, Lorenzo – consultant orthopaedic hand and wrist surgeon
Dr Garagnani is a specialist hand surgeon. He joined the team at Guy's and St Thomas' in November 2015.
Gardezi, Syed - consultant cardiologist
Syed Gardezi is a consultant cardiologist at Guy's and St Thomas' NHS Foundation Trust
Gennatas, Spyros - consultant medical oncologist
Spyros Gennatas is a consultant medical oncologist and he specialises in the treatment of thoracic malignancies, including lung cancer, thymic epithelial tumours and mesothelioma
George, Mark – consultant general and colorectal surgeon
Mr Mark George was appointed as a consultant colorectal surgeon at Guy's and St Thomas' in 2003 and is lead clinician for colorectal cancer.
George, Rob - consultant physician palliative care and honorary professor of palliative care, King's College London
Professor Rob George is a consultant physician palliative care and honorary professor of palliative care, King's College London.
George, Simi - consultant pathologist
Dr Simi George is a children's and adult pathologist specialising in GI histopathology and post mortem examinations in babies, children and adults.
Mr Alexis Schizas was appointed as a consultant colorectal surgeon at Guy's and St Thomas' Hospital in 2011.
Gidwani, Sam – consultant hand and wrist surgeon
Sam Gidwani is a consultant hand and wrist surgeon.
Gill-Barman, Baljit - consultant histopathologist
Information about Dr Baljit Gill-Barman, consultant histopathologist


I don't like it when one of my students gets sick and misses multiple days of school. I cannot control the ravages of flu and other infectious, highly contagious diseases; I merely observe the ramifications of their rampage. The students who make life fun are especially missed. I miss my daily banter with them. A little bit of sparkle is missing from my everyday routine. My eyes stray to the assigned seat; I sigh to myself and enjoy the bright-eyes faces of the ones who are there with me. We sometimes deepen our rapport on those days.
On the other hand, I notice that the students who make life challenging sometimes leave me holding my breath and wondering about how the chemistry of the class will shift again to "normal." I perceive that I am more relaxed and open in my affect; my intuitive students pick up on it as well. When my challenging-to-me personality returns, I once again clamp down on procedures. My expectations never relax. I simply feel at ease to joke with my class again like I did before I unearthed the one or two persons who are too thick to understand my humor or take it in an improper vein. Some years I have multiple students who force me to earn another grey hair or two. Understand this: I never want any of them to be sick and to miss school. I just want the hearts of some of them to change for the better. I want them to make good decisions and to behave properly. I want them to concentrate on school and its priorities and not any other circumstances.
We had two days of state testing aka STAAR with our students. I let them listen to our novel we've been reading after the testing concluded today and yesterday. I had already made them finish reading on their own and thought that listening might benefit them. My colleagues took their classes outdoors. I wasn't dressed for the occasion with four-inch strappy sandals on my feet, so we relaxed indoors. I even began to redecorate my classroom during my conference period. We are required to remove any instructional writing or posters from our walls. Motivational posters may be all right, but I took no chances and only had the pledge of allegiance to the American and Texas flags visible along with the safety information that we're required to post for quick reference. I know the students are relieved to be done; we teachers who are proctors share their relief. I am also glad to have my classroom back to "normal" setting with its colorful, eclectic display of motivation and instructional works. Tomorrow will find me posting their haiku poems we created for National Haiku Day last week on April 17. Last year, I wore my yukata kimono with obi for Haiku Day; this year I'd already worn it for the day we reviewed foreign language vocabulary since that's one of the words on the school district's list. I've included a picture of me in my kimono below. My young charges like the fact that I don unique clothing/costumes to tie in to what we're studying that day. I come by it honestly since Mama matches her clothes and dresses thematically at times. She is a Red Hatter, so she's the proud owner of a few red hats and purple dresses to accompany them. It's part of the costume. I relish the day I get to join her group. I'm still not old enough yet. ¡Ciao!
Today, and I kid you not, I saw a student leaning his head forward his mouth (and perhaps tongue!) on the door and door frame of my classroom. He was the first in line to enter my classroom for the class period. I walked up to him and said,"You're being disgusting. Get out." This translated to him going to the office (short-term removal) for inappropriate behavior. My other students waiting for me saw me disinfect my door way where Mr. Lips had been. I've even seen him chewing on his hoodie string with gusto. I really didn't know what to say since I've never seen that before in my nineteen years of teaching. Students aren't allowed to touch cleaning products that could be potentially harmful, so he wasn't going to clean that mess off for me today.
I have a former colleague who didn't like to see kids sliding along the lockers. It was as though they were trying to blend in with the scenery; however, they were the only ones in the hallway. It makes sense to compact yourself in a huge group. When you are by yourself, wall-sliding isn't safe. I fear sneeze and/or cough residue. There could also be restroom residue because not everyone washes her hands after leaving the restroom. Gross!
This is a good poem to keep handy for the dark days and hours. I found it on the site www.psalm40.org.
If this poem was helpful to you, please copy and share it with my blessing. I wish I'd written it, but I remember a wise friend gave me a copy years ago, and I only recently recalled its existence. The written word is a powerful tool and a balm for a hurting soul. Psalm 8:4 says "In peace I will both lie down and sleep; for You alone, O Lord, make me to dwell in safety." I have to put my thoughts, worries, and prayers into His hands and know that He will hold on them for me through the night. I have to believe that He gives me enough rest for each day I face with my young charges. ¡Ciao!
My mother gave this poem to me when I first started teaching. I look at it at least once a year.
Well, Father, what do You think?
The room looks nice, doesn't it?
Breathe in. Breathe out. Relax. Relax.
like the first day of the rest of my life.
in addition to all the personalities in it.
than the sum of its parts, and all that.
I think that just about covers it, Lord!
Just one more thing, Lord.
You're not going anyplace are You?
More anon, dear ones. ¡Ciao!


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