text:
\section{Introduction}

Asymmetries in the proton's spin and flavor quark distributions
provide direct information on QCD dynamics of bound systems.
Difference between quark and antiquark distributions in the sea
almost universally signal the presence of phenomena which require
understanding of strongly coupled QCD, beyond the realm of
perturbation theory.
On the other hand, at extreme kinematics at $x$ near unity
the $x$-dependence of quark distributions is determined by
perturbative QCD, and can be tested by studying the asymptotic
behavior of valence quark distributions.

Most information on quark distributions in the proton to date
has come from inclusive scattering, which sums over all
charge-squared-weighted flavors.
Semi-inclusive scattering, on the other hand, in which a hadron is
detected in the final state in coincidence with the scattered lepton,
offers considerably more freedom to explore the individual quark
flavor content of the nucleon.

In the quark-parton model the cross section for producing a
hadron $h$ at a given $x$ and $z$ and large photon virtuality
$Q^2$ can be written (to leading order) as a product of a quark
distribution function, $q(x,Q^2)$, and a fragmentation function,
$D_q^h(z,Q^2)$, giving the probability of quark $q$ fragmenting
into hadron $h$ with a fraction $z$ of the quark's center of mass
energy:
\begin{eqnarray}
\label{Ndef}
N^h &\sim& \sum_q\ e_q^2\ q(x,Q^2)\ D_q^h(z,Q^2).
\end{eqnarray}
Here one assumes factorization of the scattering and hadronization
processes, which is generally true only at high energy.
Recent data from HERMES~\cite{HERMES_DU} suggests that the
fragmentation functions are, within experimental errors,
independent of $x$, and agree with previous measurements
by the EMC~\cite{EMCFRAG} at somewhat larger energies.

\section{Current Fragmentation Region}

The most direct way to study fragmentation is in the current
fragmentation region, in which the observed hadrons are produced
(in the photon--hadron center of mass frame) along the direction
of the current.
Here the photon transfers (hard) momentum to a single parton in
the hadron, which is then knocked out and fragments into other
hadrons by picking up $q\bar q$ pairs from the vacuum.
Because it requires only a single $q\bar q$ pair, the leading
hadrons in this region are predominantly mesons.
Furthermore, by studying the fragmentation process at large $z$,
where the knocked out quark is most likely to be contained in
the produced meson, one can obtain direct information on the
momentum distribution of the scattered quark in the proton.
(At small $z$ this information becomes diluted by additional
$q\bar q$ pairs from the vacuum which contribute to secondary
fragmentation.)

\subsection{Valence Quark Asymmetries}

The valence $d/u$ ratio contains important information about
the spin-flavor structure of the proton~\cite{ISGUR}.
In particular, its asymptotic behavior at large $x$ reflects
the mechanism(s) responsible for the breaking
of SU(2)$_{\rm spin} \times$ SU(2)$_{\rm flavor}$ symmetry.
Furthermore, there are firm predictions for this behavior from
perturbative QCD~\cite{FJ}, so that measurement of this ratio
would be an important indicator of the appropriate kinematics
at which QCD can be treated perturbatively.

So far, a direct measurement of $d/u$ has been rather difficult,
mainly because the cross sections decrease rapidly in the extreme
kinematics near $x \sim 1$.
Previous analyses have used inclusive deep-inelastic scattering
data on proton and deuteron targets to extract the $d/u$
ratio from the neutron to proton structure function ratio.
However, the neutron data suffer from the fact that nuclear effects,
even in the deuteron, become quite significant~\cite{SMEAR,NP}
at large $x$.
In particular, whether one corrects for Fermi motion only, or in
addition for binding and nucleon off-shell effects, the extracted
neutron structure functions for $x > 0.7$ can differ
dramatically~\cite{NP}.
The question is therefore how to avoid uncertainties in the
extraction procedure introduced by nuclear effects~\cite{W,SEMID}.

One possibility is to measure the relative yields of $\pi^+$
and $\pi^-$ mesons in semi-inclusive scattering in the current
fragmentation region.
At large $z$ the $u$ quark fragments primarily into a $\pi^+$,
while a $d$ fragments into a $\pi^-$, so that at large $x$ and
$z$ one has a direct measure of $d/u$.
(Although one should not be too close to $z=1$, as the fragmentation
process there may no longer be incoherent, or factorizable into a
partonic cross section and a target-independent fragmentation
function.)

The HERMES Collaboration has previously extracted the $d/u$ ratio
from the $\pi^+$--$\pi^-$ difference using both proton and bound
neutron targets~\cite{HERMES_DU}.
The advantage of using both $p$ and $n$ is that all dependence
on fragmentation functions cancels, removing any uncertainty
that might be introduced by incomplete knowledge of the
hadronization process.
For a proton target one has (dropping the explicit $Q^2$ dependence):
\begin{eqnarray}
N_p^{\pi^+} &\sim& 4 u(x)\ D(z)\ +\ d(x)\ \bar D(z),    \\
N_p^{\pi^-} &\sim& 4 u(x)\ \bar D(z)\ +\ d(x)\ D(z),
\end{eqnarray}
where $D(z) \equiv D_u^{\pi^+} = D_d^{\pi^-}$ is the
leading fragmentation function (assuming isospin symmetry),
and $\bar D(z) \equiv D_d^{\pi^+} = D_u^{\pi^-}$ is the
non-leading fragmentation function.
With the corresponding expression for a neutron, one can
construct a ratio:
\begin{eqnarray}
\label{Rnp}
R_{np}(x,z)\ \equiv\
{ N_n^{\pi^+} - N_n^{\pi^-} \over N_p^{\pi^+} - N_p^{\pi^-} }
&=& { 4 d(x)/u(x)  - 1 \over 4 - d(x)/u(x) }\ ,
\end{eqnarray}
which is then independent of the fragmentation function,
and is a function of $x$ only.

The disadvantage of extracting $d/u$ from this ratio is that
one must still account for nuclear effects in obtaining
$N_n^{\pi^\pm}$ from the proton and deuteron data.
The HERMES Collaboration~\cite{HERMES_DU} assumed that
$N_n^{\pi^\pm} = N_d^{\pi^\pm} - N_p^{\pi^\pm}$.
Beyond $x \sim 0.7$, however, the difference between the
ratios with $N_n^{\pi^\pm}$ corrected for nuclear effects
and that which is not is quite dramatic~\cite{MST_SEMI}.
Consequently a $d/u$ ratio obtained from such a measurement
without nuclear corrections could potentially give misleading
results.

\begin{figure}[h]
\hspace*{1.5cm}
\psfig{figure=epic1.ps,height=5.5cm}
\caption{$d/u$ ratio from semi-inclusive $\pi$ production,
	for two integration regions, $z > 0.3$ and $z > 0.5$,
	and for two asymptotic $x \rightarrow 1$ behaviors
	for $d/u$: $d/u \rightarrow 0$ (dashed) and
	$d/u \rightarrow 1/5$ (solid).}
\end{figure}

One can avoid the problem of nuclear corrections altogether
by comparing data for $\pi^+$ and $\pi^-$ production on protons
alone.
{}From the ratio of the $\pi^-$ to $\pi^+$ proton cross sections
one finds:
\begin{eqnarray}
\label{Rpi}
R^{\pi}(x,z)\ \equiv\
{ N_p^{\pi^-} \over N_p^{\pi^+} }
&=& { 4 \bar D(z)/D(z) + d(x)/u(x)
\over 4 + d(x)/u(x) \cdot \bar D(z)/D(z) }\ .
\end{eqnarray}
In the limit $z \rightarrow 1$, the leading fragmentation
function dominates, $D(z) \gg \bar D(z)$, and the ratio
$R^{\pi} \rightarrow (1/4) d/u$.
In the realistic case of smaller $z$, the $\bar D/D$ term in $R^\pi$
contaminates the yield of fast pions originating from struck primary
quarks, diluting the cross section with pions produced from secondary
fragmentation by picking up extra $q\bar q$ pairs from the vacuum.
Nevertheless, one can estimate the yields of pions using the
empirical fragmentation functions measured by HERMES~\cite{HERMES_DU}
and the EMC~\cite{EMCFRAG}.
Integrating the differential cross section over a range of $z$,
as is more practical experimentally, the resulting ratios for cuts
of $z > 0.3$ and $z > 0.5$ are shown in Fig.~1 for two different
asymptotic $x \rightarrow 1$ behaviors~\cite{ISGUR,FJ,NP}:
$d/u \rightarrow 0$ (dashed) and $d/u \rightarrow 1/5$ (solid).
Decreasing the lower limit for $z$ has the effect of raising the
cross section ratio, because of the larger integrated contribution
from non-leading fragmentation, which is more important at
smaller $z$.
Although the relative difference between the ratios for the two
asymptotic $d/u$ behaviors then becomes smaller, the absolute
difference between these remains relatively constant, and should
be measurable with the high luminosities proposed for EPIC.

\subsection{Light Sea Quark Asymmetries}

By tagging charged pions produced off protons and neutrons at
small $x$ one can also extract information on the flavor dependence
of {\em sea quarks} in the proton.
Interest in the proton's $\bar d$ and $\bar u$ distributions
has been renewed recently with the measurement by the E866
Collaboration~\cite{E866} at Fermilab of the $x$-dependence of
the $pd$ to $pp$ cross section ratio for Drell-Yan production,
which is sensitive to $\bar d/\bar u$ at small and medium $x$.
(Note that nuclear shadowing corrections, which are known to be
important in deep-inelastic scattering off the deuteron~\cite{SHAD}
at very small $x$, are relatively small in the region covered by
E866.)
Complementary information on this ratio can also be obtained in
semi-inclusive scattering by taking the ratio of the isovector
combination of cross sections for $\pi^+$ and $\pi^-$
production~\cite{LMS}:
\begin{eqnarray}
{ N_p^{\pi^+ + \pi^-} - N_n^{\pi^+ + \pi^-} \over
  N_p^{\pi^+ - \pi^-} - N_n^{\pi^+ - \pi^-} }
&=& {3 \over 5}
    \left( { u - d - \bar d + \bar u  \over  u - d  + \bar d - \bar u}
    \right)
    \left( { D + \bar D \over D - \bar D } \right) .
\end{eqnarray}
The HERMES Collaboration at DESY has in fact recently measured
this ratio~\cite{HERMES_DUBAR}, although the errors are currently
still relatively large compared with the Drell-Yan data~\cite{E866}.

As pointed out by Thomas~\cite{AWT83}, an excess of $\bar d$ quarks
in the proton arises naturally from the chiral structure of QCD,
in the form of a pion cloud.
If part of the proton's wave function can be approximated by a
$\pi^+ n$ component, a deep-inelastic probe scattering from the
virtual $\pi^+$, which contains a valence $\bar d$ quark, will
automatically lead to $\bar d > \bar u$ in the proton.
On the other hand, one can also expect the bare nucleon itself
(i.e. that which is not dressed by pions) to be asymmetric with
respect to its SU(2)$_{\rm flavor}$ sea.
As suggested long ago by Field and Feynman~\cite{FF}, the Pauli
exclusion principle can contribute to the asymmetry on the basis
of the $u$ and $d$ valence quarks already being unequally represented
in the proton.

\begin{figure}[h]
\hspace*{1.5cm}
\psfig{figure=epic2.ps,height=5.5cm}
\caption{$\bar d-\bar u$ difference arising from the pion cloud
	(dashed) and antisymmetrization (dotted) effects, and the
	combined effect (solid), compared with E866 Drell-Yan
	data~\protect\cite{E866}.}
\end{figure}

The $x$ dependence of $\bar d-\bar u$ provides much more stringent
constraints on models of the SU(2) flavor symmetry breaking in
the proton sea than earlier measurements~\cite{NMC} which only
extracted the first moment of the difference,
$\tau = \int_0^1 dx (\bar d-\bar u)$.
Analysis~\cite{DYN} of the E866 data suggests that it is rather
difficult to ascribe the whole asymmetry over the entire range
of $x$ to a single mechanism.
On the other hand, Fig.~2 illustrates that a hybrid model, where
effects of pions as well as antisymmetrization are taken into account,
gives quite a good fit to the data.

The pion cloud contributions are evaluated taking into account
both nucleon and $\Delta$ recoil states (the latter cancel
to some extent the $\bar d$ excess through coupling to the
$\Delta^{++} \pi^-$ channel).
The hadronic form factors, in the ${\cal M}$-dependent (light-cone)
dipole representation~\cite{DYN} (${\cal M}$ is the invariant mass
of the pion--baryon system), are taken to have cut-offs
$\Lambda = 1$~GeV and 1.3 GeV for the $\pi N$ and $\pi \Delta$
components, respectively.
These values, indicating a harder $\pi N \Delta$ contribution than
the $\pi N N$, are consistent with the measured values of the axial
elastic $N$ and $N \Delta$ transition form factors~\cite{AXIAL}.

The contribution from Pauli blocking, which can be parameterized as
$(\bar d~-~\bar u)^{\rm Pauli}
= \tau^{\rm Pauli} (\alpha+1) (1-x)^\alpha$,
where $\alpha$ is some large power, is based on MIT bag model
calculations~\cite{BAG} which indicate that the normalization,
$\tau^{\rm Pauli}$, can be as large as 25\%.
Phenomenologically, one finds a good fit with $\alpha \approx 14$
and a normalization $\tau^{\rm Pauli} \approx 7\%$, which is at
the lower end of the expected scale but consistent with the bag
model predictions~\cite{BAG}.
Together with the integrated asymmetry from pions,
$\tau^{\pi} \sim 0.05$, the combined value
$\tau = \tau^{\pi} + \tau^{\rm Pauli} \approx 0.12$ is in quite
reasonable agreement with the experimental result, $0.100 \pm 0.018$
from E866.

Although the presence of an asymmetry is now clearly established,
the trend of $\bar d/\bar u$ at large $x$, where the cross sections
are smaller and errors larger, needs to be confirmed in future
experiments.
This should provide even stronger constraints on the relative
contributions of the pion cloud and antisymmetrization
contributions.

\subsection{Strange Quark Asymmetries}

Asymmetries for light sea quarks are correlated with the valence
content of the proton, so that effects of $\pi$ clouds are
difficult to distinguish from those of antisymmetrization.
Asymmetries between strange and antistrange quarks, on the other
hand, do not suffer from antisymmetrization effects, and therefore
give more direct information on the non-perturbative structure of
the nucleon sea~\cite{JI}.

Evidence for non-perturbative strangeness is currently being
sought in a number of processes, ranging from semi-inclusive
neutrino induced deep-inelastic scattering to parity violating
electron--proton scattering.
Perturbative QCD alone generates identical $s$ and $\overline s$
distributions, so that any asymmetry would have to be
non-perturbative in origin.

\begin{figure}[h]
\hspace*{1.5cm}
\psfig{figure=epic3.ps,height=5.5cm}
\caption{$s-\bar s$ difference extracted from $\nu$
	deep-inelastic data~\protect\cite{CCFR} (shaded),
	with kaon cloud model predictions for ${\cal M}$-dependent
	(solid) and $t$-dependent (dashed) $KNY$ vertex functions
	with cut-off mass $\Lambda=1$ GeV.}
\end{figure}

In deep-inelastic scattering, the CCFR collaboration~\cite{CCFR}
analyzed charm production cross sections in $\nu$ and $\bar \nu$
reactions, which probe the $s$ and $\bar s$ distributions in the
nucleon, respectively.
The resulting difference $s-\bar s$, indicated in Fig.3 by the
shaded area, has been extracted from the $s/\bar s$ ratio and
absolute values of $s+\bar s$ from global data parameterizations.
The curves correspond to chiral cloud model predictions for the
asymmetry, in which the strangeness in the nucleon is assumed
to be carried by its $KY$ ($Y=\Lambda, \Sigma$) components,
so that the $s$ and $\bar s$ quarks have quite different
origins~\cite{ST,GI}.
Because the $s$ quark originates in the $\Lambda$, its distribution
is like that of the $u$ quark in the proton, $\sim (1-x)^3$ at
large $x$.
The $\overline s$ in the kaon, on the other hand, has a much harder
shape, $\sim (1-x)$.
In the ${\cal M}$-dependent parameterization of the $KNY$ form
factor on the light-cone~\cite{MM} the $K$ distribution in the
nucleon is softer than the $\Lambda$, thereby compensating somewhat
for the harder $\bar s$ quark in the $K$.
Nevertheless, for typical form factor cut-offs of $\Lambda \sim 1$ GeV,
the softer $K$ distribution in the nucleon is not sufficient to cancel
the harder $\bar s$ quark in the $K$, so that the resulting $\bar s$
distribution in the nucleon is still bigger than $s$ at large $x$,
as in Fig.3.
As a comparison, Fig.3 also shows the asymmetry calculated for
a $t$-dependent $KNY$ form factor~\cite{MM}, for which the $K$
distribution in the nucleon is much softer than the $\Lambda$,
and does in this case overcompensate for the harder $\bar s$
distribution in $K$.
However, unlike the light-cone (${\cal M}$-dependent) forms,
the $t$-dependent form factor introduces additional problems
with gauge invariance~\cite{MM}.

\begin{figure}[h]
\hspace*{1.5cm}
\psfig{figure=epic4.ps,height=5.5cm}
\caption{Strange form factor combination measured by the HAPPEX
	Collaboration~\protect\cite{HAPPEX}, with kaon cloud
	predictions for the electric ($E$) and magnetic ($M$)
	contributions, and the total (solid) indicated.
	For the HAPPEX kinematics $r \approx 0.39$.}
\end{figure}

Within the same formalism as used to discuss the quark distributions,
one can also calculate the strangeness form factors of the nucleon
at low $Q^2$, which are being measured at MIT-Bates~\cite{SAMPLE}
and Jefferson Lab~\cite{HAPPEX}.
The latest data on the strange electric and magnetic form factors
is shown in Fig.4, together with the kaon cloud predictions,
using the same parameters for the $KNY$ couplings and vertex
functions as for the $s-\bar s$ difference in Fig.3.
With a soft $KNY$ form factor the contributions to both $G_E^s$
and $G_M^s$ are small and slightly positive~\cite{MM}, in good
agreement with the data.

Although the experimental results on non-perturbative strangeness
in both structure functions and form factors are still consistent
with zero, they are nevertheless compatible with a soft kaon cloud
around the nucleon.
New data on $G_{E/M}^s$ from Jefferson Lab with smaller error
bars and over a large range of $Q^2$ will hopefully provide
conclusive evidence for the presence or otherwise of a tangible
non-perturbative strange component in the nucleon.

\subsection{Intrinsic Charm}

Extending the above discussion to heavier quarks, one finds that
similar relative asymmetries (albeit smaller in magnitude) between $c$
and $\bar c$ quarks can also be generated by various non-perturbative
mechanisms.
Based on some early indications that charm production cross sections
in hadronic collisions were larger than predicted by leading order
perturbative QCD, Brodsky et al.~\cite{BROD} suggested that the
discrepancy could be resolved by introducing an intrinsic,
non-perturbative, charm component in the nucleon, associated with a
higher order, five-quark configuration in the nucleon wave function.

More recently, alternative models for non-perturbative charm have
also been proposed~\cite{NAVARRA,BM,MT_CHARM}, in which the charm
sea is assumed to arise from the quantum fluctuation of the nucleon
to a virtual $D^- \Lambda_c$ configuration, along the lines of the
$\pi$ and $K$ cloud models discussed above.
A natural prediction of this model is non-symmetric $c$ and $\bar c$
distributions, unlike the five-quark model~\cite{BROD} which assumes
$c=\bar c$.
This difference could be tested by comparing charged current
events in $e^+ p$ and $e^- p$ scattering~\cite{MT_CHARM}.

\begin{figure}[h]
\hspace*{1.5cm}
\psfig{figure=epic5.ps,height=4.5cm}
\caption{Charm structure function calculated through the interpolating
	scheme~\protect\cite{SMT} with different amounts of intrinsic
	charm: 0\% (solid), 0.4\% (dashed) and 1\% (dotted).
	The data are from the EMC~\protect\cite{EMC}.}
\end{figure}

The most direct information on charm in the nucleon still comes
from charm production in deep-inelastic scattering, parameterized
by the charm structure function, $F_2^c$.
Since $F_2^c$ receives contributions from both perturbative and
non-perturbative charm, to unambiguously isolate the latter one
needs a reliable method of describing perturbative charm generation
over a large range of $Q^2$, including effects of heavy quark
masses and thresholds.
Such a scheme was developed recently by Steffens et al.~\cite{SMT}
to smoothly interpolate between the large-$Q^2$ limit in which
massless evolution is applicable, and the low-$Q^2$ limit where
the photon--gluon fusion process is appropriate, and at the same
time allow for the presence of non-perturbative charm.

In Fig.5 we show $F_2^c$ calculated using the interpolating
scheme~\cite{SMT} together with different amounts of intrinsic
charm from the quantum fluctuation model~\cite{MT_CHARM},
compared with data at large $x$ from the EMC~\cite{EMC}.
The EMC data appear to rule out intrinsic charm at the 1\%
level, although non-perturbative contributions of order 0.4\%
are still allowed.
More data on the semi-inclusive production of charm at large $x$
would be very valuable in settling the question of the level at
which intrinsic charm is present in the nucleon.

\section{Target Fragmentation Region}

While the hadrons produced in the current fragmentation region
are predominantly mesons, the baryon yield in the backward
hemisphere of the photon--target center of mass frame, or the
target fragmentation region, is observed to be higher by about
one order of magnitude.
After a collision with a high-energy photon, the remnants of the
incident hadron, when viewed in the target rest frame, usually
move with small momentum.
This makes identification of target fragments in fixed target
experiments problematic.
On the other hand, a collider set-up such as that envisioned
for EPIC, with a high luminosity beam and polarized hadrons,
would make the task of studying properties of target fragments
considerably easier.

Measurement of baryon fragments in semi-inclusive scattering from
hydrogen can provide clear indications of the mechanism underlying
the fragmentation process.
In fact, the nucleon's meson cloud gives rise to rather
characteristic fragmentation distributions in comparison
with the predictions of parton hadronization models, which
are significantly enhanced when initial and final state
polarization effects are included~\cite{TFR,ZPA}.

\subsection{Kinematics of Target Fragmentation}

Consider the semi-inclusive production of a polarized baryon $B$
(with momentum $P$) from a polarized proton (momentum $p$),
$e \vec p \rightarrow e' \vec B X$ (the electron can be unpolarized).
The four-momentum transfer squared between the initial and final
hadrons is\
$t = ( -p_T^2 - (1-\zeta) (M_B^2 - M^2 \zeta) )/ \zeta$,
where $\zeta = p \cdot q / P \cdot q$ is the light-cone fraction
of the proton's momentum carried by the produced baryon and $p_T$
its transverse momentum.
The requirement that the transverse momentum squared of the
baryon be positive leads to a kinematic upper limit on $t$,
namely $t_{max} = -(1-\zeta) (M_B^2 - M^2 \zeta) ) / \zeta$.

For polarized scattering, the proton polarization will be defined
to be parallel to the photon direction, and the spin of the
produced baryon quantized along its direction of motion.
We focus on production of polarized $\Delta^{++}$ baryons,
rather than for example nucleons, which will reduce backgrounds
due to $\Delta \rightarrow N$ decays.
Experimentally, the polarization of the produced $\vec\Delta^{++}$
can be reconstructed from the angular distribution of its decay
products.
For the case of $B = \Lambda$, which is relevant when studying
the distribution of strangeness in the proton, the polarization
of the $\Lambda$ hyperon will be easier to identify since the
$\Lambda$ is self-analyzing.

\subsection{Dynamics of Non-perturbative Sea Generation}

In the pion cloud model the differential cross section is determined
by the $\pi N \Delta$ vertex function, the pion structure function,
and by the amplitude, $T^{s_\Delta s_N}(t)$, for a nucleon of spin
$s_N$ to emit a pion with four-momentum squared $t$, leaving behind
a $\Delta$ with spin $s_\Delta$.
Because it is emitted collinearly (in the target rest frame) with
the pion, production of $\Delta$ baryons with helicity $\pm 3/2$
is forbidden.
The yield of spin projection $\pm 1/2$\ states, however,
is given by:
\begin{eqnarray}
{\cal T}^{ +{1 \over 2}\ \pm{1 \over 2} }(t)
&=& { 1 \over 12 M_{\Delta}^2 }
    \left[ (M - M_{\Delta})^2 - t \right]\
    \left[ (M + M_{\Delta})^2 - t \right]^2\
    (1 \pm \cos\alpha)
\end{eqnarray}
where $\alpha$ is the angle (in the target rest frame) between
the produced baryon and the direction of the photon.
With production of $\Delta$ baryons limited to forward angles,
the factor $(1 \pm \cos\alpha)$ associated with the final state
polarization significantly suppresses the $s_\Delta = -1/2$
yield relative to that of $s_\Delta = +1/2$ final states.
The spectrum should therefore show strong correlations between
the polarizations of the incident proton ($s_N=+1/2$) and the
$\Delta^{++}$ in the pion exchange model.

A competing process to one pion exchange will be uncorrelated
spectator (target remnant, or ``diquark'' $qq$) fragmentation.
One can estimate the importance of this within the parton model
framework~\cite{TFR}, in which the differential cross section
factorizes into a product of spin-dependent quark distributions
and spin-dependent ``diquark'' fragmentation functions.

\begin{figure}[h]
\centering{
\begin{picture}(-30,150)(180,0)
\psfig{figure=epic6a.ps,height=3.8cm}
\put(15,0){\psfig{figure=epic6b.ps,height=3.8cm}}
\end{picture}}
\caption{(a) Polarization asymmetry for $\Delta^{++}$ production in
	the $\pi$ exchange (solid) and $qq$ fragmentation (dashed)
	models.
	(b) Asymmetry for $\Lambda$ production in the $K$ exchange
	(solid) and $qq$ fragmentation (dashed) models.}
\end{figure}

In Fig.6 we show the difference $\sigma^{+3/2} - \sigma^{+1/2}$,
where $\sigma \equiv Q^2 d^3\sigma / dx dQ^2 d\zeta $,
as a fraction of the total spin-averageed cross section,
$\sigma_{\rm tot}$, for the two models at typical kinematics:
$x = 0.075$, $Q^2 = 4$ GeV$^2$ and $\sqrt s \approx 8$ GeV.
For the pion exchange model the form factor cut-off is
$\Lambda = 0.8$ GeV (giving an average number density of the
$\pi\Delta$ component\ $< n >_{\pi\Delta} \approx$ 2\%),
although the ratio is almost insensitive to the size of the
form factor cut-off.
For the $qq~\rightarrow~\Delta$ fragmentation functions we use
empirical information on unpolarized $qq$ fragmentation
together with SU(6) relations for the spin dependence~\cite{TFR}.
The resulting $\zeta$ distributions are almost flat, but
dramatically different for the two models.
In particular, the pion model predicts a large {\em negative}
asymmetry, while the asymmetry in the $qq$ fragmentation model
is small and {\em positive}.

Similarly, a kaon cloud model predicts the initial proton and
recoil $\Lambda$ polarizations to be very strongly correlated,
so that the asymmetry in Fig.6 for the cross sections is almost
unity.
The $K$ exchange ratios are very similar to the $\pi$ exchange
results, indicating the similar spin transfer dynamics inherent
to the meson cloud picture of the nucleon.
This is in strong contrast with the expectation from
$qq \rightarrow \Lambda$ diquark fragmentation, in which
the initial--final state spin correlation is much weaker.
In fact, since the probabilities to form a $\Lambda^{\uparrow}$
and $\Lambda^{\downarrow}$ from an uncorrelated $qq$ pair are
equal, in the leading fragmentation approximation the asymmetry
will be zero.
Of course, SU(6) symmetry breaking effects, as well as non-leading
fragmentation contributions, will modify this result,
as will contributions from the production and decay of
$\Sigma^{0 \uparrow\downarrow}$ hyperons.
However, the qualitative result that the asymmetry is small
should remain true.

Of course the model curves in Fig.6 represent extreme cases,
in which the $\Delta$ ($\Lambda$) baryons are produced entirely
via $\pi$ ($K$) emission or $qq$ fragmentation, with no interference
effects between them.
In reality one can expect experimental asymmetries to lie somewhere
between the two limits.
The amount of deviation from the parton model curve will indicate
the extent to which the meson-exchange process is relevant.
Unlike inclusive deep-inelastic scattering, which can only be
used to place upper bounds on the pion density~\cite{AWT83},
the semi-inclusive measurements could pin down the absolute
value of $<n>_{\pi\Delta}$ ($<n>_{K\Lambda}$).

\section{Conclusion}

In summary, semi-inclusive scattering offers unique opportunities
for probing the spin and flavor structure of the nucleon not
available with inclusive processes.
The polarized electron--hadron collider set-up proposed at EPIC
would allow new regions of kinematics to be accessed in which
the dynamics of strongly interacting QCD can be probed.

We have outlined a number of specific examples where measurement
of asymmetries in the nucleon's valence and sea quark distributions
can reveal hitherto hidden details of the non-perturbative structure
of the nucleon.
The ratio of the valence $d/u$ quarks at large $x$, which reflects
the mechanism of SU(2)$_{\rm spin}\times$SU(2)$_{\rm flavor}$
symmetry breaking, can be determined by tagging pions~\cite{MST_SEMI}
in the current fragmentation region at large $z$.
Existing measurements of the neutron/proton structure function ratio,
from which $d/u$ is usually extracted, are plagued by large nuclear
binding and Fermi motion effects~\cite{NP} in the deuteron in the
region $x \approx 1$.

Pion production can also be used to pin down the ratio of the
$\bar d/\bar u$ distributions at small and medium $x$.
This can provide constraints on models which attempt to describe
the non-perturbative generation of the sea, for example in terms
of a chiral cloud of mesons around the nucleon~\cite{AWT83,DYN}.
Measurements of the light antiquark asymmetry can also be correlated
with similar asymmetries for strange and even charm quarks and
antiquarks, where effects of antisymmetrization are no longer
present.

The major advantage in studying semi-inclusive processes with
a polarized collider will be in the target fragmentation region.
As well as allowing the problem of spectator fragmentation to
be tackled seriously for the first time, careful measurements
of polarized baryon remnants from polarized protons could enable
one to unambiguously establish the magnitude of a pion and kaon
cloud of the nucleon at high energies~\cite{TFR}.
While the experiments proposed are difficult, requiring all
the intensity and duty factor one can obtain at EPIC, it does
open the way for studying the non-perturbative structure of
the nucleon in previously enexplored terrain.

\newpage
\section*{Acknowledgments}
I would like to thank the organizers of the EPIC'99 Workshop and the
IUCF for their hospitality and support, and travel support from the
Forschungszentrum J\"ulich.
I am indebted to M. Malheiro, J. Speth, F.M. Steffens and A.W. Thomas
for their collaboration on much of the work presented here.

\section*{References}


text:
\section{Introduction}

In micro-organisms such as bacteria, self-propulsion helps them in the efficient exploration of their surroundings
to find nutrient rich areas or to swim away from toxic environments. A common feature among natural microswimmers is an
ability to adjust their self-propulsion in response to local stimuli such as chemical gradients, temperature gradients,
shear flows or gravitational fields. While the mechanism of some of these responses, such as chemotaxis, is active,
involving the ability to sense gradients and signals, others are passive, purely resulting from external forces and torques.
Gravitaxis in Paramecium, for example, is thought to arise solely due to the organism's fore-rear asymmetry \cite{roberts02}. Recently, there has been substantial effort to develop artificial micro-swimmers which mimic their natural counterparts in many ways \cite{Sanchez2015}. They transduce energy from their local surroundings to engage in self-propulsion and display persistent random-walk trajectories, reminiscent of bacterial run and tumble behaviour \cite{howse07}. They also respond to gravitational fields \cite{Ebbens2013,tenHagen14}, physical obstacles \cite{volpe11,spagnolie12,takagi2014,uspal15a,das15,simmchen16}, and local fuel gradients, seeking fuel-rich regions over depleted ones \cite{Hong2007,baraban13,Peng2015}. A number of applications have been envisioned for these bio-mimetic swimmers, ranging from targeted drug delivery \cite{Patra2013,Gao2014a} to environmental remediation \cite{Soler2014, Gao2014}. In many of these applications, it is inevitable that the artificial microswimmers will encounter flow fields \textcolor{black}{while operating in confined spaces}.  While their swimming behaviour in various complex, static environments has been studied in detail, little is known about their response when they are additionally exposed to flow fields \cite{tao10,zottl12,uspal15b,zottl16}.

In anticipation of this response, one can look to decades of previous work on biological micro-organisms in flow (\textcolor{black}{see the review papers \cite{pedley92,guasto12} and the references therein.}) The interplay of flow and swimming activity can lead to robust directional response of these microswimmers through a rich variety of mechanisms \cite{marcos12,chengala13,bukatin15,sokolov16}. For instance, in ``gyrotaxis,'' bottom-heavy micro-organisms like algae adopt a stable, steady orientation and swim against the direction of gravity \cite{Kessler85, pedley87, durham09, thorn10, durham11}. Elongated microswimmers such as sperms display ``rheotaxis'', the ability to orient and swim against the direction of flow \cite{bretherton61, hill07, kantsler14, figueroa15, bukatin15, omori16}. However, much of the work with biological microswimmers has focused on swimming in the bulk, and the detailed role of confining boundaries (e.g., in rheotaxis \cite{kantsler14,uspal15b, palacci14, bukatin15, omori16}) is an emerging area of research.

In this work, using spherical Janus particles as a model system, we demonstrate with experiments and theory
that spherical particles swimming near a surface can exhibit surprising and counterintuitive spontaneous transverse orientational order.  The particles align almost -- but not exactly -- in the vorticity direction. As we discuss in detail, this slight misalignment is a key experimental observation that allows us to identify and understand the physical mechanism behind the directional alignment. An inactive, bottom-heavy particle in flow
only rotates as it is carried downstream. However, near-surface swimming activity adds an effective ``rotational friction''
to the dynamics of the particle, since it tends to drive the particle orientation into the plane of the wall. The effective ``friction'' damps the particle rotation and stabilises the cross-stream orientation. Finally, the directional alignment, combined with self-propulsion, leads to the cross-stream migration of active particles. The analysis reveals that this mechanism can generically occur for spherical microswimmers in flow near a surface. Our findings exemplify the complex behaviour that can emerge for individual microswimmers from the interplay of confinement and external fields. \textcolor{black}{Moreover, they show that the qualitative character of the emergent behaviour is sensitive to the details of the interactions (e.g., hydrodynamic interactions) between individual microswimmers and bounding surfaces}.

\section{Results}

\subsubsection*{Active particles without external flow}

\begin {figure*}[htp]
\centering
\includegraphics[width = 14.5cm]{./schematic_v2.pdf}
\caption{\label{fig:schematic} (a) Schematic \textcolor{black}{illustration of the model system}. A particle (white sphere) with axisymmetric coverage by catalyst (black) is driven by an external shear flow (large gold arrows) in the $\hat{x}$ direction near a planar wall (grey). The particle has a height $h$ above the wall and an orientation vector $\mathbf{p}$, which can be specified by the angles $\theta$ and $\phi$. When the particle is active, the cap emits solute molecules (green spheres). (b) Schematic illustrating the self-diffusiophoretic and chemi-osmotic mechanisms that drive the motion of a chemical microswimmer. The self-generated solute gradient (green spheres) drives flows localized to thin boundary layers on the particle surface (magenta arrows) and on the nearby wall (blue arrows). The particle is shown in the ``primed'' frame (red arrows). This frame co-rotates with the particle around the $\hat{z}$ axis, so that the $\hat{z}' = \hat{z}$ and the particle orientation vector $\mathbf{p}$ is always in the plane spanned by $\hat{y}'$ and $\hat{z}'$.}
\end{figure*}

In our experiments, we use silica colloids (1 $\mu m$ radius, or 2.5 $\mu m$ radius where noted) half coated with a thin layer of Pt (10 $nm$) as active particles.  When the particles are suspended in an aqueous $H_2O_2$ solution, the Pt cap catalyses the degradation of $H_2O_2$ while the silica half remains inert. The asymmetric distribution of reaction products creates a concentration gradient along the surface of the particle which induces a phoretic slip velocity, resulting in its propulsion away from the Pt cap. \textcolor{black}{Additionally, particle-generated concentration gradients can induce chemi-osmotic slip on a nearby bounding surface, giving an additional contribution to particle motility. The details of this mechanism, shown schematically in Fig. \ref{fig:schematic}(b),} are comprehensively discussed elsewhere \cite{anderson89,golestanian07}, \textcolor{black}{and are the subject of ongoing research \cite{brown14,ebbens2014,brown2016,brown2017}.}

When the silica-Pt particles are suspended in water, they quickly sediment
to the bottom surface as they are density mismatched ($\rho_{SiO_2}$ =
2.196 $g/cm^3$). They orient with their caps down ($\theta = 0^\circ$,
\textcolor{black}{where $\theta$ is defined in Fig. \ref{fig:schematic}(a)})
due to the bottom heaviness induced by the Pt layer ($\rho_{Pt}$ = 21.45 $g/cm^3$). Addition of $H_2O_2$ introduces activity
into the system and changes the orientation distribution of the particles. The particles assume an orientation with
\textcolor{black}{the propulsion axis} parallel to the bottom surface ($\theta = 90^\circ$, see Fig. S1). We have previously
reported on the dynamics leading to this change in orientation of the active particles \cite{simmchen16}. Briefly, we could show that while the hydrodynamic interactions and the bottom heaviness of the particles tend to drive the particles towards the surface, the wall induced asymmetry of the distribution of the \textcolor{black}{chemical} product and the chemi-osmotic flow along the substrate (see Fig. \ref{fig:schematic}(b)) tend to have the opposite effect, leading to a stable orientation at $\theta \approx 90^{\circ}$. Once parallel to the surface, the particles are confined to a single plane of motion where they propel away from their Pt caps (Fig. \ref{fig:imposed_flow_expt}\textcolor{black}{(c) and (d))} \textcolor{black}{with a typical speed $V_p = 6\;{\mu m/s}$}. Due to the contrast between the dark Pt hemisphere and the transparent silica, we can measure the angular orientations of these particles (see methods and Fig. S2 for details of the tracking process). Within the 2D plane, the particles have no preferred directionality and are diffusive on long time scales.

\begin {figure*}[thp]
\includegraphics[width=16cm]{./fig2_cropped_v2.pdf}
\caption { \label{fig:imposed_flow_expt} Effect of imposed flow on passive and active colloidal particles.   (a) Passive silica-Pt particles in a flow \textcolor{blue}{($R = 1\;{\mu m}$)}. The plot shows the dependence of the rotation time on flow velocity. $V^*$ is the translational velocity of the passive particle and is used to characterize the flow rate.  \textcolor{black}{The red dashed line is a theoretical scaling derived in  Sect. S4 and fitted to the data.} Inset: Time-lapse images of a passive particle rolling in flow. (b) Tracked trajectories of passive particles in flow ($V^* = 14 \, \mu m/s$).  (c) Optical
microscopy image capturing the distribution of orientations of active particles \textcolor{black}{($V_{p} \approx 6\;{\mu m}/s$)} without an imposed flow.  (d) Tracked trajectories of active particles without flow. (e) The same system as (c) with an external flow imposed, $V^* = 14 \, \mu m/s$. (f) Tracked trajectories of active particles in flow ($V^* = 14 \, \mu m/s$). (g) The angular probability distributions of active colloidal particles in the absence of an imposed flow, $V^* = 0 \, \mu m/s$ (green), at $V^* = 14 \, \mu m/s$ (blue) and at $V^* = 24  \, \mu m/s$ (red). Inset shows the angular evolution of two different active colloids at imposed flow rates of $V^* = 14 \, \mu m/s$ (blue) and $V^* = 24 \, \mu m/s$ (red).}
\end{figure*}

\subsubsection*{Passive particles in external flow}

Now we seek to characterise the behaviour of these silica-Pt particles in an imposed flow. Initially a suspension of the particles in water is introduced in a square glass capillary (1 $mm$, Vitrocom) connected to a computer controlled microfluidic pump (MFCS-EZ, Fluigent). \textcolor{black}{We allow the particles to sediment to the bottom surface before we impose any external flows.} The desired flow rate in the capillary is maintained by using a flow rate monitor (Flowboard, Fluigent) which is in a feedback loop with the microfluidic pump. We begin by imposing a flow of water (no activity) in the $x$ direction. Close to the non-slipping capillary surface, the flow velocity varies linearly as $v_{flow} = \dot{\gamma} z$ and the particles which are sedimented near the surface experience a shear flow \textcolor{black}{(see Section S3 for a calculation of the flow profile) \cite{bruus08}}.  In terms of their translational behaviour, we observe that the particles act as ``tracers'' and translate in nearly straight lines along the direction of flow \textcolor{black}{(Fig. \ref{fig:imposed_flow_expt}(b))}. The translational velocity of these particles is proportional to the imposed flow rate. We use the translational velocity \textcolor{black}{$V^*$} of these inactive ``tracer''  particles to characterize the flow rate \cite{palacci14}. Before we start the flow, the particles are all in the cap down orientation ($\theta = 0^\circ$). The shear flow induces a torque on the particles and they rotate around \textcolor{black}{the axis of flow vorticity $\hat{y}$} as they translate in flow (See SM Video 1). The rotation speed of the particles is also dependent on the flow rate \textcolor{black}{(Fig. \ref{fig:imposed_flow_expt}(a))}, with higher flow rates leading to faster rotation. \textcolor{black}{Via a simple model  for particle rolling developed in Section S4 of the SM, we predict that the rotational period $\tau = 2 \pi/\sqrt{a V^{*,2} + b}$, where $a$ and $b$ are fitting parameters. This relation shows excellent fit to the data (Fig. 2a, red curve), and from the fitted $a$, we extract $\dot{\gamma} \approx 1.22 \, V^{*}/R$.}

\subsubsection*{Active particles in external flow}

We then start a flow of $H_2O_2$ to introduce activity into the system. We notice that the dynamics of particle behaviour in flow is drastically influenced in the presence of activity. Firstly, the particles stop rolling and reach a stable orientation parallel to the bottom surface ($\theta = 90^{\circ}$). This is similar to previously observed behaviour for the particles in the absence of flow \cite{simmchen16}. More surprisingly, the particles also evolve to a stable orientation  that is nearly perpendicular to the direction of imposed flow ($\phi \approx 90^{\circ}$ or $-90^{\circ}$) (Fig. \ref{fig:imposed_flow_expt}(e)). While the particles continue to translate in $x$ due to the imposed flow, they also have the self-propulsion velocity $V_p$ away from their Pt caps. A combination of these effects results in the cross-streamline migration of self-propelled particles, i.e,  \textcolor{black}{migration of particles in the y-direction, perpendicular to the flow along the x-direction.}
Typical trajectories of cross-stream migrating particles are presented in Fig. \ref{fig:imposed_flow_expt}(f). Further, we observe that the stability of cross-stream migration (due to the stability of the steady orientation angle \textcolor{black}{$\phi^{*}$} perpendicular to the direction of flow) is dependent on the flow rate, with higher flow rates resulting in a stronger alignment effect.  The inset of Fig. \ref{fig:imposed_flow_expt}(g) shows the angular evolution of two self-propelled particles in imposed flows of $V^* = 14 \, \mu m/s$ and $V^* = 24 \, \mu m/s$. The deviations away from the $\phi \approx  90^\circ$ positions occur more frequently and \textcolor{black}{at larger amplitude} than for particles in lower flow rates.

In order to study this effect at a population scale we flow a suspension of self-propelled particles with $H_2O_2$ and record the angular orientations and positions of every particle in each frame, which allows us to determine the probability distribution of $\phi$ in the system. These are plotted in Fig. \ref{fig:imposed_flow_expt}(g) for two different flow rates and compared to the system of self-propelled particles without any imposed flow. In the absence of flow, the distribution is nearly flat, indicating the lack of preference for any orientation $\phi$, and at long time scales, the particle behaviour is purely diffusive. However, in the case of imposed flow, we observe distinct peaks that appear at $\phi \approx  90^\circ$ and $-90^\circ$. These correspond to the particles exhibiting the cross-stream behaviour. These peaks also become sharper when we increase the flow rate, as can be seen in the distributions for $V^* = 14 \, \mu m/s$ and $V^* = 24 \, \mu m/s$. \textcolor{black}{In both cases,} closer observation of the angular probability distributions reveals a small bias of orientations in the direction of flow for particles migrating \textcolor{black}{across flow streamlines}.
Since the particles are subject to Brownian fluctuations, they can occasionally also orient with or against the flow ($\phi \approx 0^\circ$ or $180^\circ$). This leads to an intermittent state where the particles ``tumble'' in the direction of flow before recovering their $\phi \approx  \pm 90^\circ$ orientation and the cross-streamline behaviour. The recovered orientation of these particles can be different from their initial states, changing the direction of particle migration (e.g., from $+y$ to $-y$) (see \textcolor{black}{Fig. S5}).

\textcolor{black}{Apart from the imposed flow rate $V^{*}$, we find that the stability of the $\phi^{*}$ is also dependent on the particle radius. Using particles of larger radius ($2.5\;{\mu m}$), we show that the distribution of $\phi$ is narrower around $\phi = \pm 90^{\circ}$ as compared to the distribution for $R = 1\;{\mu m}$ particles for identical $V_{p}$ and $V^{*}$ (Fig. S6). Since the effect of Brownian noise is significantly lower on the larger particles, we also observe that they seldom switch their migration direction within the width of our capillary (Fig. S7).}

\begin{figure*}[htp]
\centering
\includegraphics[width=13.5cm]{./fig3_updated_v2.pdf}
\caption {\label{fig:alphas} The dependence of cross-stream migration on $V_p$. (a) The probability distributions of $\alpha$ for a particle in the absence of flow with $V_p = 6 \, \mu m/s$\textcolor{black}{, as well as} with an imposed flow corresponding to $V^* = 14 \, \mu m/s$ for particles of $V_p = 6 \, \mu m/s$ and $V_p = 3 \, \mu m/s$. \textcolor{black}{(b, c, and d) Tracked orientation vectors (line segments) and instantaneous velocities (arrows) for trajectories at different flow and self-propulsion speeds.}}
\end{figure*}

We can further control the behaviour of particle migration by tuning the self-propulsion velocity $V_p$ of the particles. \textcolor{black}{Firstly, we find that higher propulsion velocities dampen the fluctuations around $\phi^{*}$ due to higher activity (See Fig. S8). Secondly, the propulsion velocity also controls the ``slope'' of the cross-stream migration. In order to quantify this, we} define $\alpha$ to be the offset between the orientation vector $\mathbf{p}$ and the \textcolor{black}{tracked} velocity vector $\mathbf{v}$ of the particle. For a self-propelled particle in the absence of flow, the particles translate in the direction of the orientation vector $\mathbf{p}$ as the particles propel away from their Pt caps (\textcolor{black}{i.e.,} $\alpha = 0^\circ$), \textcolor{black}{as shown in} Fig. \ref{fig:alphas}(a and \textcolor{black}{b}). However, with an imposed flow, the translational direction differs from the orientation vector\textcolor{black}{,} and the offset $\alpha$, for a given flow rate, is determined by the $V_p$ of the particles. For particles with $V_p = 6 \, \mu m/s$ the probability distribution function of $\alpha$ has a peak around $-64^\circ$, whereas for a particle with $V_p = 3 \, \mu m/s$, $\alpha$ is peaked at $-83 ^\circ$ (Fig. \ref{fig:alphas}(a, c and d)).

\textcolor{black}{The offset $\alpha$ can be rationalized as a contribution of the transverse propulsion induced by the cross-stream orientation and the longitudinal advection by flow. The $\alpha$ should then simply be given by $\alpha = \arctan (V^{*}/V_{p})$. Substituting the experimental values for $V_{p}$ and $V^{*}$, we get $\alpha = 66.9^{\circ}$ and $\alpha = 78.3^{\circ}$, which agree reasonably well with the observed values.  This clear separation in peaks of $\alpha$ for different propulsion velocities could eventually be used for the separation of particles based on activity. }

\subsubsection*{Construction of theoretical model}

\textcolor{black}{The experimental observations can be qualitatively captured and understood within a generic mathematical model of swimming near a surface in external flow \cite{uspal15b}. In this model, we construct dynamical equations for the height and orientation of a heavy spherical microswimmer by exploiting physical symmetries and the mathematical linearity of Stokes flow. As detailed below, mathematical analysis of these equations reveals that qualitatively distinct steady state behaviors, including cross-stream migration, can emerge from the interplay of external shear flow, near-surface swimming, and gravity. We note that the analysis does \textit{not} depend on a particular mechanism of self-propulsion (e.g., self-diffusiophoresis, self-electrophoresis, or mechanical propulsion by motion of surface cilia); it is, in that sense, \textit{generic}. Accordingly, our major theoretical findings are, first, that there is a physical mechanism that can produce the surprising transverse orientational order observed in experiments, and, secondly, that this mechanism can \textit{generically} occur for a spherical, \textcolor{black}{axisymmetric} microswimmer exposed to flow near a bounding surface.}

\textcolor{black}{We now proceed to construction and analysis of the dynamical equations. As discussed in the Methods section, the flow in the suspending fluid is characterised by a low Reynolds number, and hence governed by the Stokes equations. Since these equations are linear,} the contributions of external \textit{\textbf{f}}low, \textit{\textbf{g}}ravity, and \textit{\textbf{s}}wimming \textcolor{black}{to the particle translational and angular velocities} can be calculated independently and superposed: $\mathbf{U} = \mathbf{U}^{(f)} + \mathbf{U}^{(s)} + \mathbf{U}^{(g)}$ and $\mathbf{\Omega} = \mathbf{\Omega}^{(f)} + \mathbf{\Omega}^{(s)} + \mathbf{\Omega}^{(g)}$. \textcolor{black}{The velocity of the orientation vector is determined by  $\dot{\mathbf{p}} = \mathbf{\Omega} \times \mathbf{p}$. We will generally describe $\mathbf{p}$ in Cartesian coordinates, but will sometimes find it useful to use the spherical coordinates $\mathbf{p} = (\sin(\theta) \cos(\phi), \sin(\theta) \sin(\phi), \cos(\theta))$. Note that $|\mathbf{p}| = 1$.}

\textcolor{black}{In order to calculate the contribution of the external flow to the particle velocity, we consider a neutrally buoyant and inactive sphere of radius $R$ in shear flow at a height $h$ above a uniform planar wall.  Shear spins the particle around the vorticity axis $\hat{y}$, as shown in the left panel of Fig. \ref{fig:decomposition}. This is because the flow is faster near the upper surface of the particle (i.e., the surface farther away from the wall) than near the bottom surface.  Accordingly, the particle has angular velocity $\mathbf{\Omega}^{(f)} = \frac{1}{2} \dot{\gamma} f(h/R) \hat{y}$. Additionally, the particle is carried downstream with velocity $\mathbf{U}^{(f)} =  \dot{\gamma} h g(h/R) \hat{x}$.  The functions $f(h/R)$ and $g(h/R)$ represent the influence of hydrodynamic friction from the wall \cite{goldman67}; they are provided in \textcolor{black}{SM Sect. S8}.  Due to the spinning motion of the particle, the tip of the vector $\mathbf{p}$ traces a circle in the shear plane:}
\begin{equation}
\label{eq:pf}
\dot{\mathbf{p}}^{(f)} \equiv \mathbf{\Omega}^{(f)} \times \mathbf{p} =  \frac{1}{2} \dot{\gamma} f(h/R) \left[ p_{z} \hat{x}  - p_{x} \hat{z} \right]
\end{equation}
The component $p_{y}$ is a constant determined by the initial orientation of the particle. The radius of the \textcolor{black}{circular orbit} is $\sqrt{1 - p_{y}^{2}}$, and the speed of the tip \textcolor{black}{$|\dot{\mathbf{p}}^{(f)}|$} is proportional to the circle radius.

\textcolor{black}{Now we consider a heavy, active particle moving in quiescent fluid (no external flow) near the same surface. The particle has an \textbf{\textit{ax}}isymmetric geometry and surface activity profile; accordingly, we define  $\mathbf{U}^{(ax)} \equiv \mathbf{U}^{(s)} + \mathbf{U}^{(g)}$ and $\mathbf{\Omega}^{(ax)} \equiv \mathbf{\Omega}^{(s)}  + \mathbf{\Omega}^{(g)}$. This system, depicted in the middle panel of Fig. \ref{fig:decomposition}, has a plane of mirror symmetry defined by $\hat{z}$ and $\mathbf{p}$. Accordingly, translation of the particle is restricted to this plane, and $\mathbf{p}$ is restricted to remain within this plane (i.e., $\dot{\phi} = 0$), although $\mathbf{p}$ can rotate towards or away from the wall ($\dot{\theta} \neq 0$). We write $\dot{\mathbf{p}}^{(ax)} = - \Omega_{x'}^{(ax)}(\theta, h/R) \, \hat{\theta}$, where we have defined a new, ``primed'' frame that has $\hat{y}'$ and $\hat{z}'$ in the plane of mirror symmetry, with $\hat{z}' = \hat{z}$. This frame is convenient for calculations because $\mathbf{\Omega}^{(ax)}$ is strictly in $\hat{x}'$  (see Fig. \ref{fig:schematic}(b)). The function $ \Omega_{x'}^{(ax)}(\theta, h/R)$ incorporates the effect of bottom-heaviness, as well as interactions with the wall (e.g., hydrodynamic interactions) that originate in swimming activity. Both $ \Omega_{x'}^{(ax)}(\theta, h/R)$  and $\mathbf{U}^{(ax)}(\theta, h/R)$ depend on the particle design and the model for the propulsion mechanism. In order to show that our analysis is generic in the sense discussed above, we will leave them unspecified until later in the text. Since the particle is axisymmetric and the \textcolor{black}{wall} is uniform, these functions have no $\phi$ dependence.}

\textcolor{black}{Now we superpose all contributions and obtain equations for $\dot{\mathbf{p}} = \mathbf{\Omega} \times \mathbf{p}$ and $\dot{h} = U_{z}$ in the following form}:
\begin{eqnarray}
\label{eq:dynamical_eqns}
\dot{p}_{x} &=&  \, \frac{1}{2} \dot{\gamma} p_{z}
f(h/R) -\frac{\Omega_{x'}^{\textcolor{black}{(ax)}}(p_{z},
h/R) p_{x} p_{z}}{\sqrt{1 - p_{z}^{2}}}\,  \,,
\label{dot_px} \\
\dot{p}_{y} &=& \,- \frac{\Omega_{x'}^{\textcolor{black}{(ax)}}(p_{z}, h/R) p_{y} p_{z}}{\sqrt{1 - p_{z}^{2}}} \label{dot_py} \\
\dot{p}_{z} &=&   -\frac{1}{2} \dot{\gamma} p_{x}
f(h/R) + \Omega_{x'}^{\textcolor{black}{(ax)}}(p_{z}, h/R) \sqrt{1 - p_{z}^2} \,, \label{dot_pz} \\
\dot{h}_{~} &=& U_z = \, U_{z'}^{\textcolor{black}{(ax)}}(p_{z}, h/R)\,\label{dot_h}.
\end{eqnarray}
\noindent \textcolor{black}{Note that the axisymmetric contributions are evaluated in the ``primed'' frame, which is defined to co-rotate with the particle, and we used $\hat{\theta} = \cos(\theta) \cos (\phi) \hat{x} + \cos(\theta) \sin(\phi) \hat{y} - \sin(\theta) \hat{z}$.} The components of the translational velocity $\mathbf{U}$ in the $x$ and $y$ directions are determined by $\mathbf{p}$ and $h$ as:
\begin{eqnarray}
 U_{x} &=& \dot{\gamma} \textcolor{black}{h} g(h/R) + \frac{U_{y'}^{\textcolor{black}{(ax)}}(p_{z}, h/R) p_{x}} {\sqrt{1 - p_{z}^{2}}} \\
\label{uy}
 U_{y} &=& \frac{U_{y'}^{\textcolor{black}{(ax)}}(p_{z}, h/R) p_{y}} {\sqrt{1 - p_{z}^{2}}}.
\end{eqnarray}

\subsubsection*{Steady state solutions}

\textcolor{black}{Eqs. \ref{dot_px}-\ref{uy} fully describe a deterministic active sphere in external shear flow near a planar surface.} Now we look for the fixed points $(\mathbf{p}^{*}, h^{*})$ of Eqs. \ref{dot_px}-\ref{dot_h}. A fixed point is a particle configuration in which the particle translates along the wall with a steady height and orientation, i.e., $\dot{\mathbf{p}} = 0$ and $\dot{h} = 0$ \cite{uspal15b}. We find that the system has three fixed points, shown schematically in \textcolor{black}{Fig. S10}. Of these three, \textit{planar alignment} shows excellent qualitative agreement with the experiment observations: the particle orientation is within the plane of the wall ($p_{z}^{*} = 0$), and has non-zero components in \textit{both} the flow and vorticity directions. These criteria are not satisfied by the other two fixed points. \textcolor{black}{Slight misalignment from the vorticity axis is therefore a key experimental observation that discriminates between the steady states predicted by theory.}  For planar alignment, the \textcolor{black}{streamwise} component \textcolor{black}{$p_{x}^{*}$ is
\begin{equation}
\label{eq:px_star}
p_{x}^{*} = \cos(\phi^{*}) = \frac{2 \Omega_{x'}^{(ax)}(p_{z}^{*} = 0, h^{*}/R)}{\dot{\gamma} f(h^{*}/R)},
\end{equation}
which can be} either downstream ($p_{x}^{*} > 0$), \textcolor{black}{in agreement with our experimental observations,} or upstream ($p_{x}^{*} < 0$), depending on the sign of $\Omega_{x'}^{\textcolor{black}{(ax)}}(p_{z}^{*} = 0, h^{*})$. Due to the mirror symmetry of Eqs. \ref{dot_px}-\ref{dot_h} with respect to $p_{y} = 0$, this fixed point always occurs in pairs $(p_{x}, \pm p_{y}^{*}, 0, h^{*})$. Further mathematical details are provided in the SM.

\begin{figure*}[thp]
\centering
\includegraphics[width = 0.9 \textwidth]{./decomposition-01.pdf}
\caption{\label{fig:decomposition} \textcolor{black}{(left) A neutrally buoyant, inactive sphere driven by an external shear flow (gold arrows) near a planar wall. Since the external flow spins the particle around the vorticity axis (i.e., the normal to the shear plane), the tip of the particle orientation vector $\mathbf{p}$ traces a circular path (magenta). (middle) An active, heavy sphere in quiescent fluid. Since the wall is uniform, and the geometry and activity profile of the particle are axisymmetric, the tip of the orientation vector can only rotate in the $\hat{\theta}$ direction, i.e., directly towards or away from the wall. (right) The \textit{planar alignment} steady state, where $p_{z}^{*} = 0$, but the particle orientation vector has non-zero components $p_{x}^{*}$ and $p_{y}^{*}$ in the flow and vorticity directions, respectively. For planar alignment, the component $p_{x}^{*}$ in the flow direction can be either upstream ($p_{x}^{*} > 0$) or downstream ($p_{x}^{*} < 0$), as determined by the function $\Omega_{x'}^{\textcolor{black}{(ax)}}(p_{z}, h/R)$; the downstream case is shown. All contributions to $\dot{\mathbf{p}}$ are in the $\hat{z}$ direction (see Eqs. \ref{dot_px}-\ref{dot_h} and the discussion in the main text). At a certain angle $\phi^{*}$, all contributions to $\dot{\mathbf{p}}$ balance, as shown by the arrows, so that $\dot{\mathbf{p}} = 0$. Note that this fixed point always occurs in pairs related by mirror symmetry across the shear plane; we show the state with $p_{y} > 0$.}}
\end{figure*}

\textcolor{black}{In order to understand why planar alignment is a steady state, we consider the conditions for the contributions of shear flow, activity, and gravity to the angular velocity to cancel out at some $\mathbf{p}^{*}$. We recall that the axisymmetric contributions to $\dot{\mathbf{p}}$ are in the $\hat{\theta}$ direction. For most orientations $\theta$ and $\phi$, $\hat{\theta}$ will have a component in the vorticity direction $\hat{y}$. However, from Eq. \ref{eq:pf}, we see that the shear contribution \textit{never} has a $\hat{y}$ component. As a way out of this dilemma, we see from the definition of $\hat{\theta}$ that when $\theta = 90^{\circ}$  (i.e., $p_{z} = 0$), the $\hat{x}$ and $\hat{y}$ components of $\hat{\theta}$ both vanish, and $\dot{\mathbf{p}}^{(ax)}$ is strictly in  the $\hat{z}$ direction. Likewise, when $\theta = 90^{\circ}$, $\dot{\mathbf{p}}^{(f)}$ is strictly in $\hat{z}$. This suggests the possibility that all contributions can cancel out for $\theta^{*} = 90^{\circ}$ and some unknown $\phi^{*}$.  \textcolor{black}{Now we consider the role of $\phi$}. The axisymmetric contributions \textcolor{black}{$\dot{\mathbf{p}}^{(ax)}$} have no dependence on $\phi$, as discussed previously. The contribution from shear does depend on $\phi$, \textcolor{black}{as can be seen by substituting $p_{z} = 0$ and $p_{x} = \cos(\phi)$ into Eq. \ref{eq:pf}}.  Therefore, the sign and magnitude of the shear contribution can be ``tuned'' via the angle $\phi$ to cancel out the axisymmetric contributions \textcolor{black}{to $\dot{\mathbf{p}}$}  (provided these are not too large).}   \textcolor{black}{Finally, we consider the height of the particle. Since shear does not contribute to vertical motion, the particle must obtain a steady height ($\dot{h} = 0$) through the combined effects of near-surface swimming activity and gravity.}

\textcolor{black}{As an additional note, if we assume that the steady height of the particle is not significantly affected by flow rate, Eq. \ref{eq:px_star} predicts that the steady state orientation approaches the vorticity axis as the flow rate is increased, i.e., $\cos(\phi^{*}) \sim 1/V^{*}$.  Accordingly, we perform further experiments with $5\;{\mu m}$ diameter particles, which allow for excellent resolution of their orientation (see Fig. S2). We performed the experiments at six different flow rates in the range accessible with the current experimental setup, $V^* = 5\;{\mu m/s}$ to $V^* = 75\;{\mu m/s}$, for particles with $V_p = 6\;{\mu m/s}$. In Fig. S11, we plot $\cos|\phi^{*}|$ against $V^*$ and find that the data recovers the predicted asymptotic behavior.}

\subsubsection*{Linear stability analysis}

\textcolor{black}{It is not enough to find a fixed point that corresponds to experimental observations; we must also consider its stability.} Since, experimentally, the active particles \textcolor{black}{spontaneously adopt a steady cross-stream orientation}, the fixed point should be a stable attractor for active particles. Secondly, since the inactive particles are  observed to  \textcolor{black}{continuously rotate} in the experiments, the fixed point should be associated with closed orbits for inactive particles. \textcolor{black}{In the following, we carry out a linear stability analysis and show that our model meets both criteria, provided a certain condition on the particle/wall interaction is satisfied.}

 We define the generalized configuration vector $\mathbf{f} \equiv (\mathbf{p}, h/R)$, and $\tilde{\mathbf{f}}$ as a small perturbation away from the fixed point: $\mathbf{f} = \mathbf{f}^{*} + \tilde{\mathbf{f}}$.  We obtain the linearized governing equations $\dot{\tilde{{\mathbf{f}}}} = \mathbf{J} \tilde{\mathbf{f}}$, where the Jacobian matrix $J_{ij} = \frac{\partial \dot{f}_{i}}{\partial f_{j}} \biggr\rvert_{\textcolor{black}{\mathbf{f}^{*}}}$ is given in detail in the SM.  As a simplifying approximation, we take the particle to have a constant height $h = H$, so that $\dot{\tilde{h}} = 0$. This approximation is motivated by the experimental observation that the particles never leave the microscope focal plane. Having made this approximation, we can find an intuitive analogy between our system and a damped harmonic oscillator (see SM):

\begin{figure*}[bhp]
\centering
\includegraphics[width=15.cm]{./traj_PDFs_combined_wall_slip_035-01.pdf}
\caption{\label{fig:traj_pdfs_combined} (a)  Numerically computed trajectory of a passive bottom-heavy particle near a wall (grey) and driven by an external shear flow (gold arrows) with dimensionless strength $\dot{\gamma} R/U_{0} = 0.1$ (concerning dimensionless parameters, see Methods). The initial condition of the particle is $\theta_{0} = 30^{\circ}$ and $\phi_{0} = 315^{\circ}$, and the particle height is fixed as $h/R = 1.2$. The particle rotates and is carried downstream by the external flow with no cross-streamline migration. (b) Numerically computed trajectory of an active Janus particle with the external flow strength, particle materials, and initial conditions as in (a). The particle rotates so that its inert face points largely in the $-\hat{y}$ direction, with a slight downstream orientation ($p_{x} > 0$). With this steady orientation, the particle swims across flow streamlines as it moves downstream.  (c) Probability distribution function for $\phi$, for a catalytic Janus particle in a shear flow with $\dot{\gamma}R/U_{0} = 0.1$. (d) Probability distribution function for the same particle with $\dot{\gamma}R/U_{0} = 0.5$.}
\end{figure*}

 \begin{figure*}[thp]
 \centering
\includegraphics[width=17cm]{./phase_portraits_wall_slip_035_new_v5-01.pdf}
\caption{\label{fig:phase_portraits} Phase portraits on the sphere $|\mathbf{p}| = 1$ for (a) an inactive, \textcolor{black}{bottom-heavy} particle in shear flow with $\dot{\gamma} R/U_{0} = 0.1$, and (b-c) for an active, \textcolor{black}{bottom-heavy} particle in a flow with (b) $\dot{\gamma} R/U_{0} = 0.1$ and (c) $\dot{\gamma} R/U_{0} = 0.5$.  The red dashed lines indicate the plane of mirror symmetry $p_{y} = 0$.  The cyan circle in (a) indicates the center of oscillatory motion. Green circles in (b) and (c) indicate stable fixed points (``attractors''). For (a) and (b), the magneta circles show the initial conditions for (a) the trajectory in Fig. \ref{fig:traj_pdfs_combined}(a) and (b) the trajectory in Fig. \ref{fig:traj_pdfs_combined}(b). The parameters characterizing the particle heaviness and activity are given in Methods. \textcolor{black}{(d) Vector field on the unit sphere representing the contribution of activity to the motion in (b) and (c). The green dashed line indicates $p_{z} = 0$.}  (e) Oscillation in $p_{z}$ with time for the inactive particle in Fig.  \ref{fig:traj_pdfs_combined}(a), also corresponding to the magenta circle in (a). The black dashed line gives the result obtained by numerical integration. The red dashed line gives the analytical solution for the linearized equations. The slight disagreement between them is due to the large amplitude of the oscillation, for which the effect of the nonlinearity in Eqs. \ref{eq:dynamical_eqns}-\ref{dot_pz} is important. (f) For the active particle in Fig. \ref{fig:traj_pdfs_combined}(b), with initial conditions corresponding to the magneta circle in (b), the oscillation in $p_{z}$ decays with time, and the orientation $\mathbf{p}$ eventually approaches the stable fixed point $\mathbf{p}^{*}$. The analytical solution is shown for the later part of the decay, for which $p_{z}$ has small amplitude and is well described by linear theory. }
\end{figure*}

\begin{equation}
\label{eq:effective_osc}
\ddot{\tilde{p}}_{z} = - \left(\frac{1}{2} \dot{\gamma} f(H/R)  p^{*}_{y} \right)^{2} \tilde{p}_{z} + \frac{\partial \Omega_{x'}^{\textcolor{black}{(ax)}}}{\partial p_{z}} \biggr\rvert_{p_{z}^{*} = 0, H} \dot{\tilde{p}}_{z}.
\end{equation}

\noindent \textcolor{black}{The motion of the orientation vector} resembles a damped harmonic oscillator with intrinsic frequency $\omega_{0} = \frac{1}{2} \dot{\gamma} f(H/R)  p^{*}_{x}$. \textcolor{black}{Recalling that inactive, bottom-heavy particle were observed to continuously rotate in the experiments, we consider the case in which $\Omega_{x'}^{\textcolor{black}{(ax)}} =  \Omega_{x'}^{(g)}$.}  We note that

\begin{equation}
\label{eq:PartialOmegaPartialPz_Justgrav}
 \frac{\partial \Omega_{x'}^{(g)}}{\partial p_{z}} \biggr\rvert_{p_{z}^{*} = 0, H} \textcolor{black}{=  \frac{\partial \dot{\theta}^{(g)}}{\partial \theta} \biggr\rvert_{\theta^{*} = 90^{\circ}, H}} = 0,
\end{equation}

\noindent \textcolor{black}{where we have used $\dot{\theta}^{(ax)} = -\Omega_{x'}^{(ax)}$  and $\dot{\theta}^{(g)} \sim \sin(\theta)$.} Therefore, the dissipative term in Eq. \ref{eq:effective_osc} is zero, and the orientation $\mathbf{p}$ has a continuous family of closed orbits centered on $\mathbf{p}^{*}$.
Now we consider active, bottom-heavy particles, for which  $\Omega_{x'}^{\textcolor{black}{(ax)}} = \Omega_{x'}^{(s)} + \Omega_{x'}^{(g)}$. The fixed point is a stable attractor if

\begin{equation}
\label{eq:stability_attractor}
 \frac{\partial \Omega_{x'}^{(s)}}{\partial p_{z}} \biggr\rvert_{p_{z}^{*} = 0, H}  \textcolor{black}{=  \frac{\partial \dot{\theta}^{(s)}}{\partial \theta} \biggr\rvert_{\theta^{*} = 90^{\circ}, H}} < 0.
\end{equation}

\noindent \textcolor{black}{This condition has the following interpretation: if the particle orientation is perturbed out of the plane of the wall, the active contribution to rotation responds (increases or decreases)  so as to oppose the perturbation.} \textcolor{black}{If this condition is satisfied,} activity \textcolor{black}{induces an effective ``friction''}  that damps oscillations of the orientation $\mathbf{p}$,  \textcolor{black}{driving attraction of $\mathbf{p}$ to $\mathbf{p}^{*}$. \textcolor{black}{Satisfaction of the condition depends on the details of the interactions between the particle and the wall that originate in swimming activity. In turn, these details depend on the character of the self-propulsion mechanism (see, e.g., the comparison of pushers and pullers in Ref. \cite{spagnolie12}.)}}

\subsubsection*{Application to a self-phoretic particle}
\textcolor{black}{Up to this point, we have proceeded without specifying a model for the particle composition and self-propulsion, and we derived and analyzed  Eqs. \ref{eq:dynamical_eqns}-\ref{dot_h} in general terms. Now we seek to obtain illustrative particle trajectories by numerical integration. Accordingly,  hereafter we calculate the various terms in Eqs. \ref{eq:dynamical_eqns}-\ref{dot_h} by using a simple, well-established model of neutral self-diffusiophoresis in confinement \cite{golestanian05, golestanian07, uspal15a, simmchen16, mozaffari16, ibrahim16}}. In the model, the particle has a hemispherical catalytic cap (Fig. \ref{fig:schematic}(b), black), and the orientation vector $\mathbf{p}$ points from its catalytic pole to its inert pole. The particle emits solute molecules (i.e., oxygen) at a constant, uniform rate from its cap, leading to self-generated solute gradients in the surrounding solution. These gradients drive surface flows on the particle (Fig. \ref{fig:schematic}(b), magenta arrows) and on the wall (blue arrows), leading to directed motion of the particle. \textcolor{black}{We note that our generic model does not account for effects (e.g., ionic or  electrokinetic) specific to the detailed self-phoretic mechanism \cite{brown14,ebbens2014,brown2016,brown2017}. Further details are provided in Methods}.

We consider some illustrative examples, using dimensionless parameters comparable to those in the experiments (see Methods for definition and estimation of parameters). We fix the particle height as $H/R = 1.2$.  In Fig. \ref{fig:traj_pdfs_combined}(a), we show a particle trajectory in a shear flow with strength $\dot{\gamma} R/U_{0} = 0.1$ and initial orientation $\theta_{0} = 30^{\circ}$ and $\phi_{0} = 315^{\circ}$. The particle rotates so that its inert face points largely in the $-\hat{y}$ direction, but with a small downstream orientation ($p_{x} > 0$). Notably, this slight downstream orientation agrees with experimental observations. If the particle is inactive (but still bottom-heavy), then from the same initial orientation, the particle simply translates in the flow direction (Fig. \ref{fig:traj_pdfs_combined}(b)), rotating as it does so.

\textcolor{black}{Since $h = H$ is taken to be constant}, \textcolor{black}{the instantaneous value of} $\mathbf{p}$ \textcolor{black}{completely} determines the instantaneous velocity of the particle. In Fig. \ref{fig:phase_portraits}(a)-(c), phase space trajectories are shown on the unit sphere $|\mathbf{p}| = 1$. The initial orientation in Fig. \ref{fig:traj_pdfs_combined}(a) and (b) is indicated by magenta circles in Fig. \ref{fig:phase_portraits}(a) and Fig. \ref{fig:phase_portraits}(b), respectively. We see that an inactive particle (Fig. \ref{fig:phase_portraits}(a)) has a continuous family of closed orbits in $\mathbf{p}$. When the particle is active, these oscillations are damped, and trajectories in phase space are attracted to the fixed point (Fig. \ref{fig:phase_portraits}(b)-(c)). In (c), we show the effect on the structure of trajectories as the flow strength is increased. For stronger flows, the approach to the fixed point is more oscillatory. In Fig. \ref{fig:phase_portraits}(e), we plot the component $p_{z}$ (black line) for the inactive particle in Fig. \ref{fig:traj_pdfs_combined}(a). In Fig. \ref{fig:phase_portraits}(f), we show $p_{z}$ for the active particle in Fig. \ref{fig:traj_pdfs_combined}(b). We can also obtain analytical solutions to the linearized equations, characterized by a few input parameters that are evaluated numerically (see SM). These solutions are shown as dashed red lines in Figs. \ref{fig:phase_portraits}(e) and (f), and agree well with the numerical data. \textcolor{black}{Additionally, we note that if the particle height is allowed to change, one can still obtain the phenomenology studied here of attraction to a planar alignment steady state; an example is given in \textcolor{black}{SI Sect. S8}.}

\textcolor{black}{The phase portraits also provide an intuitive way to understand the stability condition expressed in Eq. \ref{eq:stability_attractor}. Consider the bottom-heavy, inactive particle with the phase portrait shown in Fig. \ref{fig:phase_portraits}(a). How does adding activity transform this portrait into that shown in Fig. \ref{fig:phase_portraits}(b)? The contribution of activity to $\dot{\mathbf{p}}$ is shown as the vector field in Fig. \ref{fig:phase_portraits}(d). Since this vector field is small but non-zero on the equator $p_{z} = 0$, adding it to the portrait in Fig. \ref{fig:phase_portraits}(a) will shift the center of oscillation (cyan circle) slightly towards the axis $p_{x} = 0$, producing the green circle in Fig. \ref{fig:phase_portraits}(b). More significantly, the addition of this vector field will destabilize continuous oscillatory motion in the following way. Consider the closed trajectories in Fig.  \ref{fig:phase_portraits}(a) that are closest to the cyan circle.  In the neighborhood of the equator, activity always ``pushes'' the vector $\mathbf{p}$ towards the equator  (here neglecting the small value on the equator, the main effect of which is to shift the fixed point.)   Therefore, over each period of oscillation, the orientation vector will get slightly closer to the fixed point -- transforming the closed circular orbits into decaying spiral orbits.}

\subsubsection*{Effect of thermal noise}

All of the preceding analysis assumed deterministic motion. \textcolor{black}{As a further exploration, we consider} the effect of thermal noise on the particle orientation by performing Brownian dynamics simulations (see Methods). In Fig. \ref{fig:traj_pdfs_combined}(c), we show the probability distribution functions for the $\phi$ obtained for a Janus particle driven by shear flow at dimensionless inverse temperature $Pe_{p} = 500$ (see Methods for definition) and shear rate $\dot{\gamma}R/U_{0} = 0.1$. The distribution is symmetric, and has two peaks near the \textcolor{black}{steady angle} $\phi^{*} = \pm \textcolor{black}{80.9}^{\circ}$ \textcolor{black}{predicted by the deterministic model} (Fig. \ref{fig:traj_pdfs_combined}(c)). For a higher shear rate $\dot{\gamma}R/U_{0} = 0.5$, we find that the peaks in $\phi$ are sharper (Fig. \ref{fig:traj_pdfs_combined}(d)), in qualitative agreement with the experiments, \textcolor{black}{with the peaks shifted near the deterministic prediction for this shear rate, $\phi^{*} = \pm {88.2}^{\circ}$.} Additional probability distributions for the components of $\mathbf{p}$ are given in \textcolor{black}{Fig. S14}.

\section{Discussion}
In this paper, \textcolor{black}{we use catalytic Janus particles as a well-controlled model experimental system to study spherical active particles in confined flows.} We demonstrate that \textcolor{black}{spherical} active particles near surfaces, when exposed to external flows, \textcolor{black}{can} exhibit robust alignment and motion along the cross-stream direction. Our model reveals how this behaviour arises from the interplay of shear flow and swimming in confinement. \textcolor{black}{The steady orientation is determined by a balance of contributions to the angular velocity of the particle from shear flow, bottom-heaviness, and swimming near a planar substrate.
Near-surface swimming introduces an effective ``friction''} opposing rotation of the particle away from the preferred orientation. \textcolor{black}{The mechanism is generic in the sense that it can occur for any spherical microswimmer with axisymmetric actuation, and is not specific to a particular mechanism of propulsion (e.g., chemical or mechanical.)}  \textcolor{black}{As a consequence of the alignment, the particles migrate across the streamlines of the external flow as they are carried downstream.}

To the best of our knowledge, our results are the first to demonstrate that swimmer/surface interactions (e.g. hydrodynamic interactions) can drive a rich directional response of spherical particles to external flows. This is in contrast with previous works on natural microswimmers (e.g., bacteria) where complex body shapes and flagellar beat patterns were implicated in directional response \cite{marcos12, bukatin15}. Additionally, we have obtained semi-quantitative agreement between particle orientational statistics obtained from the experiments and from the theoretical model.

 For lab-on-a-chip devices that use continuous flows and artificial microswimmers, our findings imply that the microswimmers would have a tendency to migrate to the confining side walls of the device. Our findings additionally raise the possibility that in dense suspensions of microswimmers, for which fluid flows are self-generated, the collective behavior of the suspension may be sensitive to the detailed interactions between individual microswimmers and bounding surfaces.

\section{Materials and methods}
\subsubsection*{Sample preparation}
The Janus particles were obtained by electron-beam deposition of a Pt layer on a monolayer of silica colloids. The monolayers were prepared either by a Langmuir-Blodgett (LB) method or a drop casting method. For the LB method, silica colloids ($R = 1 \, \mu m$, Sigma Aldrich) were first surface treated with allyltrimethoxysilane to make them amphiphilic. A suspension of these particles in chloroform-ethanol mixture (80/20 $v/v$) was carefully dropped onto the LB trough and compressed to create a closely packed monolayer. The monolayer was then transferred onto a silicon wafer at a surface pressure of $20 \, mN/m$.  The silicon wafer was then shifted to a vacuum system for the electron beam deposition of a thin layer of Pt ($10 \, nm$) at $10^{-6} \, mmHg$. The Janus particles were released into deionized water using short ultrasound pulses. The suspension of Janus particles in water was stored at room temperature. The monolayers of $R = 2.5 \, \mu m$ silica particles (Sigma Aldrich) were prepared by drop casting of the suspension of colloids onto an oxygen plasma treated glass slide. The plasma treatment was used to make the glass slide hydrophilic and ensure uniform spreading of the particle suspension. The solvent was subsequently removed by slow evaporation. The Pt deposition step was identical to the one used for $R = 1 \, \mu m$ particles.

\subsubsection*{Tracking}
Particle tracking was performed using an automated tracking program developed in-house. The Python based program uses OpenCV library for image processing and numpy for data handling. In source videos filmed in grayscale, each frame is first cleaned of noise by using blurring techniques, which substitute each pixel with an average of its surroundings. The particles are then separated from the background by using either of the two segmentation methods: threshold and gradient. In the threshold method, given a grayscale image \textit{img(x,y)}, and a threshold value \textit{T}, this operation results in a binary image \textit{out(x,y)} given by:

\begin{equation}
\text{out}(x,y) =
\left\{
\begin{array}{cc}
1  & \text{if img}(x,y) \geq T \\
0 & \text{if img}(x,y) < T
\end{array}
\right.
\end{equation}

\noindent The gradient method is used for images with irregular brightness or when the particles are hard to distinguish from the background. In the first step of this method the gradient of the image \(\nabla img = (\frac{\partial img}{\partial x}, \frac{\partial img}{\partial y}) \), is approximated by convolving the original frame with a Sobel operator. This results in two images, one which is the derivative along the X axis and another along the Y axis. These images are then thresholded and joined together to obtain the segmented image. The final result has the edges of the detected particles.

The center of each particle is approximated as the center of mass of the contours obtained after segmentation. Particle trajectories are calculated using Bayesian decision-making, linking every particle center with the previous closest one. Intermediate missing positions, if any, are interpolated using cubic or linear splines. \\
In order to calculate the $\mathbf{p}$  vector, a line of predefined radius (approximately equal to the particle radius) is drawn through the center at a test angle. The standard deviation of the pixel values along the test-line is calculated and stored. The process is repeated multiple times and the standard deviations are compared. The test-angle with the least standard deviation is assumed to correspond to the separator between the silica and Pt halves and the vector orthogonal to it, the orientation vector (see Fig. S2).

\subsubsection*{Theoretical calculation of particle velocity}

Here, we present our model for calculation of the particle velocities $\mathbf{U}$ and $\mathbf{\Omega}$ as a function of $h$ and $\mathbf{p}$.  We take the instantaneous position of the particle to be $\mathbf{x}_{p} = (x_{p}, y_{p}, h)$ in a stationary reference frame.  The catalytic cap emits a product molecule at a constant and uniform rate $\kappa$. We take the solute number density $c(\mathbf{x})$ to be quasi-static, i.e., it obeys the equation $\nabla^{2} c = 0$ with boundary conditions $-D \hat{n} \cdot \nabla c = \kappa$ on the cap, $\hat{n} \cdot \nabla c  = 0$ on the inert region of the particle surface, and $\hat{n} \cdot \nabla c = 0$ on the wall. Here, $D$ is the diffusion coefficient of the solute molecule, $\mathbf{x}$ is a location in the fluid, $\hat{n}$ and is the normal vector pointing from a surface into the liquid. For each instantaneous configuration ($h$, $\mathbf{p})$, this set of equations can be solved for $c(\mathbf{x})$, e.g., numerically by using the boundary element method \cite{pozrikidis02,uspal15b}.

We take the velocity $\mathbf{u}(\mathbf{x})$ in the fluid solution to obey the Stokes equation $-\nabla P + \eta \nabla^{2} \mathbf{u} = 0$, where $\eta$ is the solution viscosity and $P(\mathbf{x})$ is the pressure in the solution. Additionally, the fluid is incompressible, so that $\nabla \cdot \mathbf{u} = 0$. The Stokes equation is a linear equation. Therefore, the contributions to $\mathbf{U}$ and $\mathbf{\Omega}$ from various boundary conditions for $\mathbf{u}$, as well as from various external forces and torques, can be calculated individually as the solution to separate subproblems and then superposed. Hence, we write $\mathbf{U} = \mathbf{U}^{(f)} + \mathbf{U}^{\textcolor{black}{(ax)}}$ and $\mathbf{\Omega} = \mathbf{\Omega}^{(f)} + \mathbf{\Omega}^{\textcolor{black}{(ax)}}$, where $(f)$ indicates the contributions of the external \textit{\textbf{f}}low, and the \textit{\textbf{a}}xi\textit{\textbf{s}}ymmetric contributions are from \textit{\textbf{g}}ravity and \textcolor{black}{\textbf{\textit{s}}}wimming activity: $\mathbf{U}^{\textcolor{black}{(ax)}} = \mathbf{U}^{(g)} + \mathbf{U}^{\textcolor{black}{(s)}}$ and $\mathbf{\Omega}^{\textcolor{black}{(ax)}} = \mathbf{\Omega}^{(g)} + \mathbf{\Omega}^{\textcolor{black}{(s)}}$. For each subproblem, the fluid is governed by the Stokes equation and incompressibility condition.

We first consider the subproblem for the contribution of the external flow. The fluid velocity is subject to no-slip boundary conditions $\mathbf{u} = 0$ on the planar wall and $\mathbf{u} = \mathbf{u}_{ext} + \mathbf{U}^{(f)} + \mathbf{\Omega}^{(f)} \times (\mathbf{x} - \mathbf{x}_{p})$ on the particle. Here, $\mathbf{u}_{ext}$ is the external flow velocity, $\mathbf{u}_{ext} = \dot{\gamma} z \hat{x}$. Additionally, the particle is free of external forces and torques, closing the system of equations for $\mathbf{U}^{(f)}$ and $\mathbf{\Omega}^{(f)}$. The solution of this subproblem is well-known; see, for instance, Goldman et al. \cite{goldman67}

The two axisymmetric subproblems are calculated in the ``primed'' frame co-rotating with the particle (Fig. \ref{fig:schematic}(b)). For the subproblem associated with particle activity, we employ the classical framework of neutral self-diffusiophoresis \cite{anderson89, golestanian05,golestanian07,uspal15a,simmchen16}. In this subproblem, the self-generated solute gradients drive surface flows on the wall and the particle surface, $\mathbf{v}_{s} = -b(\mathbf{x}_{s}') \nabla_{||} c(\mathbf{x}')$, where $\nabla_{||} \equiv (\mathbf{1} - \hat{n}\hat{n}) \cdot \nabla$ and $\mathbf{x}_{s}'$ is a location on a surface (in the primed frame). The ``surface mobility'' $b(\mathbf{x}_{s}')$ encapsulates the details of the molecular interaction between the solute and the bounding surfaces. We  write the boundary conditions $\mathbf{u} = \mathbf{U}^{\textcolor{black}{(s)}} + \mathbf{\Omega}^{\textcolor{black}{(s)}} \times (\mathbf{x}' - \mathbf{x}_{p}') + \mathbf{v}_{s}(\mathbf{x}_{s}')$ on the particle, and $\mathbf{u} = \mathbf{v}_{s}(\mathbf{x}_{s}')$ on the wall. Again, specifying that the particle is force and torque free closes the system of equations for $\mathbf{U}^{\textcolor{black}{(s)}}$ and $\mathbf{\Omega}^{\textcolor{black}{(s)}}$. This subproblem can be solved numerically using the boundary element method \cite{pozrikidis02,uspal15b}. For the subproblem associated with gravity, we use the ``eggshell'' model of Campbell and Ebbens for the shape of the cap, taking the cap thickness to vary smoothly from zero at the particle ``equator'' to a maximum thickness of $t$ at the active pole \cite{Ebbens2013}. Details concerning this subproblem, which is solved using standard methods \cite{simmchen16}, are provided in Supplementary Materials.

We now specify the parameters characterizing the system.  We choose to take the inert and catalytic regions
of the particle to have different surface mobilities, $b_{inert}$ and $b_{cap}$, with $b_{inert}/b_{cap} =
0.3$ and $b_{cap} < 0$. For this parameter, a neutrally buoyant Janus particle, when it is far away from bounding surfaces, moves in the $\mathbf{p}$ direction (i.e., away from its cap) with a velocity $U_{fs} = 13/80 \; U_{0}$, where $U_{0} = |b_{cap}| \kappa/D$ \cite{michelin14}. The wall is characterized by a surface mobility $b_{w}$. We choose $b_{w}/b_{cap} = -0.35$. These surface mobility ratios are chosen to be similar to those used in previous work (Ref. \cite{simmchen16}, where we had $b_{inert}/b_{cap} = 0.3$ and $b_{w}/b_{cap} = -0.2$), and to give a slightly downstream steady orientation. We non-dimensionalize length with $R$, velocity with $U_{0}$, and time with $T_{0} = R/U_{0}$. In order to non-dimensionalize the gravitational and shear contributions, we must estimate $U_{0}$ in real, dimensional units. Rather than calculate $U_{0}$ directly, which requires estimates for $\kappa$ and $|b_{cap}|$, we use the expression $U_{fs} = 13/80 \; U_{0}$ \cite{michelin14}. Knowing that, experimentally, the particle characteristically moves at $U_{fs} \approx 5\;{\mu m/s}$ when it is far from surfaces, we obtain $U_{0} \approx 30\;{\mu m/s}$. The parameters describing the heaviness of the particle are given in Supplementary Information.
Finally, we consider the shear rate $\dot{\gamma}$. This is not known experimentally, but can be roughly estimated by considering inactive particles to act as passive tracers. From Goldman et al., the velocity of a spherical particle driven by shear flow near a wall is $U^{(f)}_{x} = g(h/R) \dot{\gamma} h$ \cite{goldman67}. Experimentally, inactive particles in flow are observed to move with $V^{*} \sim 10 \, {\mu m/s}$. The particle height $h$ is difficult to observe experimentally, but we take it to be set by the balance of gravity and electrostatic forces. Hence, the particle/wall gap $\delta$ is on the order of a Debye length $\lambda_{D} \sim 0.1 R$, so that $h \sim 1.1R$. For $h/R \geq 1.05$, the factor $g(h/R) \approx 1$. We therefore estimate $\dot{\gamma} \approx 10\, {s^{-1}}$, and a typical dimensionless shear rate to be $\dot{\gamma} R/U_{0} \approx 0.8$.  As a reminder, our aim is to establish semi-quantitative agreement with experiments, and therefore we seek only order of magnitude accuracy in the dimensionless parameters.

In assuming the concentration field to be quasi-static, we neglected the advective effects on the solute field by the external shear flow and by the finite velocity of the particle. These approximations are valid for small Peclet numbers $Pe \equiv U_{fs} R/D$ and $Pe_{\dot{\gamma}} \equiv \dot{\gamma} R^{2}/D$. At room temperature, the diffusion coefficient of oxygen is $D \sim 4 \times 10^{-9} m^{2}/s$ \cite{popescu10}, so that $Pe \approx 0.003$ and $Pe_{\dot{\gamma}} \approx 0.015$. Furthermore, in taking the fluid velocity $\mathbf{u}$ to be governed by the Stokes equation, we neglected fluid inertia. This approximation is justified for low Reynolds number, $Re = \rho_{fluid} U_{0} R/ \eta$, where $\eta$ is the dynamic viscosity of the solution. Using $\eta \sim 10^{-3} Pa\,s$ for water, we obtain $Re \approx 10^{-4}$.

\subsubsection*{Effect of thermal noise}

The particle Peclet number $Pe_{p} = U_{fs} R/D_{0}^{t}$ characterizes the relative strengths of self-propulsion and translational diffusion of the particle. Here, $D_{0}^{t} = k_{B} T/6 \pi \eta R$ is the translational diffusion coefficient of the particle in free space. For a particle with $R = 2.5\;{\mu m}$ in water  at room temperature ($k_{B}T_{room} \sim 4 \times 10^{-21} J$) with $U_{fs} \approx 5\;{\mu m\textcolor{black}{/s}}$, we estimate $Pe_{p} \approx 880$.  In order to perform Brownian dynamics simulations, we adapt the Euler-Maruyama integration scheme introduced by Jones and Alavi \cite{Jones92}, and later presented by Lisicki \textit{et al.} \cite{Lisicki14}, which explicitly includes the effects of the wall on diffusion. Our principal modification to the method is inclusion of deterministic contributions to $\dot{\mathbf{p}}$ (Eqs. \ref{dot_px}-\ref{dot_pz}). Further details are provided in Supplementary Information and Ref. \cite{Lisicki14}.

\acknowledgments
The authors thank M. N. Popescu for  helpful discussions. W.E.U. acknowledges financial support from the DFG, grant no. TA
959/1-1. S.S., J.S. and J.K. acknowledge the DFG grant no. S.A 2525/1-1. S.S. acknowledges the Spanish MINECO for grant CTQ2015-68879-R (MICRODIA). This research also has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 311529.

redpajama_set_name: RedPajamaArXiv