∎

[$\star$]t1Contribution to the Topical Issue “Motile Active Matter”, edited by
Gerhard Gompper, Clemens Bechinger, Roland G. Winkler, Holger Stark e1e-mail:
<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>
11institutetext: Institute for Theoretical Physics, Leipzig University, 04103
Leipzig, Germany 22institutetext: Peter Debye Institute for Soft Matter
Physics, Leipzig University, 04103 Leipzig, Germany

# Thermotaxis of Janus Particlest1

Sven Auschrae1,addr1  Andreas Bregullae2,addr2  Klaus Kroye3,addr1  Frank
Cichose2,addr2

(Received: date / Accepted: date)

###### Abstract

The interactions of autonomous microswimmers play an important role for the
formation of collective states of motile active matter. We study them in
detail for the common microswimmer-design of two-faced Janus spheres with
hemispheres made from different materials. Their chemical and physical surface
properties may be tailored to fine-tune their mutual attractive, repulsive or
aligning behavior. To investigate these effects systematically, we monitor the
dynamics of a single gold-capped Janus particle in the external temperature
field created by an optically heated metal nanoparticle. We quantify the
orientation-dependent repulsion and alignment of the Janus particle and
explain it in terms of a simple theoretical model for the induced
thermoosmotic surface fluxes. The model reveals that the particle’s angular
velocity is solely determined by the temperature profile on the equator
between the Janus particle’s hemispehres and their phoretic mobility contrast.
The distortion of the external temperature field by their heterogeneous heat
conductivity is moreover shown to break the apparent symmetry of the problem.

††journal: Eur. Phys. J. E

##  1 Introduction

Ranging from flocks of birds via schools of fish to colonies of insects, a
distinctive trait displayed by the individual constituents of motile active
matter ramaswamy2010StatMechActMat ; menon2010ActMatt ;
cates2012MicroiolStatMech  is a unique capability to adapt to environmental
cues vicsek1995SelfDrivenParticles . Down to the microbial level where all
kinds of “animalcules” gest2004discoveryOfMicroorganisms  struggle to locomote
through liquid solvents Purcell77 ; Lauga2011LifeTheorem , interactions with
boundaries and neighbors and the sensing of chemical gradients
adler1966chemotaxis  are key features involved in the search of food, suitable
habitats or mating partners. Inspired by nature, scientist designed synthetic,
inanimate microswimmers that mimic the characteristics of biological swimmers
and are more amenable to a systematic investigation of their interactions. A
very popular design exploits self-phoresis Anderson1989  for which numerous
experimental and theoretical studies are available zoettl2016EmergBehav ;
romanczuk2012ABP ; poon2013Review . Such self-phoretic propulsion relies on
the interfacial stresses arising at the particle–fluid interface in the self-
generated gradient of an appropriate field (temperature, solute concentration,
electrostatic potential). On a coarse-grained hydrodynamic level, this effect
is captured by an effective tangential slip of the fluid along the particle
surface Anderson1989  that drives the self-propulsion of the swimmer.
Accordingly, thermophoresis Wurger2007ThermophoresisForces ;
Fayolle2008ThermophoresisParticles ; wuerger2010Review , diffusiophoresis
golestanian2005Diffusophoresis ; Julicher2009Colltrans ; Wurger2015Self-
DiffusiophoresisMixtures  and electrophoresis squires2004Electroosmosis ;
squires2006Electrophoresis  can deliberately be exploited for (or may
inadvertently contribute to) the swimming of Janus particles Jiang2010 ;
Bregulla2014 ; selmke2018PhotonNudgingI ; selmke2018PhotonNudgingII ;
popescu2016SelfDiffPhor ; golestanian2007DesigningSwimmers ;
bazant2010electrokonPhen ; gangwal2008Electrophoresis ;
moran2010electrophoresis . The classical Janus-particle design consists of a
spherical colloid with hemispheres of distinct physico-chemical properties,
which define a polar symmetry axis. Due to the broken symmetry, one expects
the axis to align with an external field gradient Bickel2014Polarization , the
direction being determined by the precise surface properties and the chosen
solvent saha2019PairingWaltzingScattering ; pohl2014ChemotactCollapse . The
reorientation of microswimmers in external fields is often referred to as
_taxis_ and has been studied for various phoretic propulsion mechanisms
boymelgreen2012electrophor ; Lozano2016Phototaxis ; Gomez2017Directionality ;
Jin2017 ; Geiseler2017Self-PolarizingWaves ; uspal2019LightActivatedParticles
.

What we call an external field can be understood as a template for the
influence of container walls or neighboring microswimmers
liebchen2019DomInteract ; popescu2019CommentWhichInteract ;
zoettl2016EmergBehav ; popescu2018Chemotaxis  that are at the core of the rich
collective phenomena emerging in active fluids golestanian2012Collbehav ;
saha2014ClustersAtsersOscill ; Husain2017emergStruct ; Liebchen2017DynClust ;
Liebchen2015Cluster ; Theurkauff2012Clustering ; Kaiser2012Capture ;
Pohl2015Collapse ; Cates2015MIPS . That microswimmers are constantly
exchanging linear and angular momentum with the ambient fluid generally
renders their apparent mutual interactions non-reciprocal. Next to the
thermodynamic field gradients also the hydrodynamic flow field generated by
one swimmer at the position of another one affects the swimmers’ interactions
elgeti2015Review ; zoettl2016EmergBehav . Generally, interactions mediated by
hydrodynamic flow fields takagi2014HydrOrbits ; varma2018Clustering ;
dunkel2010Scattering ; alexander2008Scattering ; martinez2018HydBoundStates ;
berke2008HydrAttr , optical shadowing uspal2019LightActivatedParticles ;
Cohen2014OpticallyDriven  and chemical or optical patterns
Volpe2011MicroswimmersEnvironments ; palagi2016lightstruct ;
Simmchen2016Topograph  may have to be considered, and which of these
contributions dominate the observed motion of microswimmers has recently been
under debate liebchen2019DomInteract ; popescu2019CommentWhichInteract ;
liebchen2019ResponseToComment .

In the present contribution, we report results from an experiment designed to
allow for direct measurements of the induced polarization and motion of a
single passive (i.e., not self-driven) Janus particle in the temperature field
emanating from a localized heat source in its vicinity. In other words, it
enables us to single out the passive phoretic response of a self-
thermophoretic swimmer to an external temperature field, without having to
bother with the autonomous motion that would result from a direct
(laser-)heating of the swimmer itself. Typically such measurements are
difficult to conduct since pairwise collisions are rare at low concentrations
and hard to discern among interfering the many-body effects, at high
concentrations. However, by adopting the technique of photon nudging Qian2013
; Bregulla2014  we can direct an individual Janus particle into the vicinity
of the local heat source selmke2018PhotonNudgingI ; selmke2018PhotonNudgingII
, without imposing potentially perturbing external fields. This allows us to
precisely record the polarization effects and thereby characterize the elusive
phoretic repelling and aligning interactions with good accuracy and good
statistics. Our experimental results are substantiated by a theoretical model
that addresses the thermophoretic origin of the interactions and complements
recent calculations of the phoretic interactions between two chemically active
particles nasouri2020PhoreticInteractions2Particles  and the axis-symmetric
interactions between two diffusiophoretic Janus particles
nasouri2020AxissymmInteract2JPs .

##  2 Experimental Setup

We experimentally explore the interaction of a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter Janus particle with a
$50\text{\,}\mathrm{nm}$ thin gold cap with the temperature field generated by
an immobilized $250\text{\,}\mathrm{nm}$ gold nano-particle optically heated
by a focused laser (wavelength $532\text{\,}\mathrm{nm}$). To confine the
Janus particle to the vicinity of the heat source, we employ the feedback
control technique of photon nudging Qian2013 ; Bregulla2014  that exploits its
autonomous motion to steer it to a chosen target. As illustrated in Fig. 1
(a), the steering is only activated when the Janus particle leaves an outer
radius $r_{\rm o}$ around the heat source until it has migrated back across an
inner radius $r_{\rm i}$, followed by a waiting time of $10$ rotational
diffusion times $\tau_{\mathrm{r}}$ to allow for the decay of orientational
biases and correlations Auschra2020ActivityFieldsPRE ;
Soeker2020ActivityFieldsPRL .

Figure 1: Schematic of the experimental setup and the phoretic motion. (a) A
passive Janus polystyrene (ps) bead with a thin gold (au) cap (thermophoretic
mobilities $\mu_{\rm ps}$, $\mu_{\rm au}$) is exposed to the temperature
gradient around a laser-heated immobilized gold nanoparticle. Particle
translation is restricted to the sample plane due to the thin liquid film
thickness ($1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The coordinate frame
attached to the particle’s geometric center has its $x$-axis aligned with the
particle’s symmetry axis and pointing towards its ps-side, while the $z$-axis
points into the paper plane. The in-plane angle $\phi$ and normal angle
$\theta$ are measured with respect to the $x$-axis and $z$-axis, respectively,
and technically (yet not with respect to the particle’s polarity) take the
role of what is conventionally called “azimuthal” and “polar” angles,
respectively. The orientation angle of the swimmer relative to the heat source
is $\gamma$. (b–d) Phoretic translational and rotational velocities $\bm{u}$,
$\bm{\Omega}$, arise from slip fluxes with velocities $\bm{v}_{\mathrm{s}}$
induced by the temperature gradient, chiefly near the particle equator. Arrow
lengths and orientations indicate the magnitude and direction of the
velocities.

All data recording and feedback is carried out in a custom-made dark field
microscopy setup with an inverse frame rate and exposure time of
$5\text{\,}\mathrm{ms}$. Further details regarding the sample preparation, the
experimental setup, and the position and orientation analysis are contained in
A–D. The temperature increment $\Delta T=$12\text{\,}\mathrm{K}$$ of the
heated gold nanoparticle relative to the ambient temperature
($T_{0}=$295\text{\,}\mathrm{K}$$) is known from a separate measurement using
the nematic/isotropic phase transition of a liquid crystal (see E). We account
for the direct influence of the heating laser on the Janus particle and the
phoretic velocities, as detailed in F.

##  3 Results and Discussion

###  3.1 Theory

On the hydrodynamic level of description, the temperature gradient
$\bm{\nabla}_{\parallel}T\equiv(\bm{I}-\bm{e}_{r}\bm{e}_{r})\bm{\nabla}T$
along the surface of the Janus particle induces a proportionate interfacial
creep flow derjaguin1987SurfaceForeces ; bregulla2016Thermoosmosis , where
$\bm{e}_{r}$ denotes the unit vector normal to the particle surface and
$\bm{I}$ the unit matrix. Since the interfacial flow is localized near the
particle surface, it is conveniently represented as a slip boundary condition
with slip velocity Anderson1989 ; golestanian2007DesigningSwimmers ;
Bickel2014Polarization

$\bm{v}_{\mathrm{s}}\left(\theta,\phi\right)=\mu\left(\theta,\phi\right)\bm{\nabla}_{\parallel}T\left(\theta,\phi\right).$
(1)

The particle surface is parametrized in terms of the in-plane and normal
angles $\phi$ and $\theta$, as sketched in Fig. 1 (a,b). They technically take
the role of "azimuthal" and "polar" angles, respectively, although these
notions are not associated with the particle’s polar symmetry, here. And
$\mu\left(\theta,\phi\right)$ is a phoretic mobility characterizing the
varying strength of the creep flow due to the distinct interfacial
interactions with the solvent Bickel2014Polarization . The resulting
translational propulsion velocity $\bm{u}$ and the angular velocity
$\bm{\Omega}$ of the Janus particle of radius $a$ are given by averages over
its surface $\mathcal{S}$: anderson1991DiffPhoresis ; Bickel2014Polarization

$\displaystyle\bm{u}$ $\displaystyle=-\frac{1}{4\pi
a^{2}}\oint_{\mathcal{S}}\mathrm{d}S\,\bm{v}_{\rm{s}},$ (2)
$\displaystyle\bm{\Omega}$ $\displaystyle=-\frac{1}{4\pi
a^{2}}\oint_{\mathcal{S}}\mathrm{d}\bm{S}\times\frac{3}{2a}\bm{v}_{\rm{s}}.$
(3)

Further analysis of Eqs. (2) and (3) becomes possible by the experimental
observation that the Janus particle is preferentially aligned with the sample
plane. This effect is presumably mostly due to the hydrodynamic flows induced
by the heterogeneous heating in the narrow fluid layers between the particle
and the glass cover slides uspal2015ParticlesNearWall ;
Simmchen2016TopographicalMicroswimmers ; Das2015BoundariesSpheres . For
simplicity, the following analysis assumes perfect in-plane alignment, thereby
neglecting weak perturbations due to rotational Brownian motion and the weak
bottom-heaviness of the Janus particle rashidi2020influCapWeight . For any
given temperature profile $T(\phi,\theta)$ at the surface of the Janus sphere,
the components of the translational and rotational velocity can then be
expressed as

$\displaystyle u_{x}$ $\displaystyle=-\frac{1}{\pi
a}\int_{0}^{2\pi}\mathrm{d}\phi\,\mu(\phi)\cos\phi\left\langle
T\sin\theta\right\rangle_{\theta}(\phi)$ $\displaystyle\hskip
11.38092pt+\frac{\mu_{\rm ps}-\mu_{\rm au}}{2\pi
a}\left[\left\langle\frac{T}{\sin\theta}\right\rangle_{\theta}\biggr{\rvert}_{\phi=\frac{\pi}{2}}+\left\langle\frac{T}{\sin\theta}\right\rangle_{\theta}\biggr{\rvert}_{\phi=\frac{3\pi}{2}}\right],$
(4) $\displaystyle u_{y}$ $\displaystyle=-\frac{1}{\pi
a}\int_{0}^{2\pi}\mathrm{d}\phi\,\mu(\phi)\sin\phi\left\langle
T\sin\theta\right\rangle_{\theta}(\phi),$ (5) $\displaystyle\Omega_{z}$
$\displaystyle=\frac{3}{4\pi a^{2}}(\mu_{\rm ps}-\mu_{\rm
au})\left[\left\langle
T\right\rangle_{\theta}|_{\phi=\frac{3\pi}{2}}-\left\langle
T\right\rangle_{\theta}|_{\phi=\frac{\pi}{2}}\right],$ (6)

where we have introduced the average over the normal angle
$\left\langle\bullet\right\rangle_{\theta}\equiv\frac{1}{2}\int_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta(\bullet)$. All other velocity components give zero contributions,
as the detailed derivation of Eqs. (4)–(6) in G shows.

Motivated by theoretical studies of chemotactic active colloids
saha2019PairingWaltzingScattering , we further employ the following model for
the angular velocity:

$\Omega_{z}(\gamma)=\Omega_{1}\sin\gamma+\Omega_{2}\sin(2\gamma),$ (7)

introducing the independent parameters $\Omega_{1,2}$. This is a natural
extension of $\Omega_{z}\propto(\mu_{\rm ps}-\mu_{\rm au})\sin\gamma$ for
particles with isotropic heat conductivity Bickel2014Polarization , to account
for the material heterogeneities of the Janus sphere. Similar (reflection)
methods as presented in saha2019PairingWaltzingScattering ;
boymelgreen2012electrophor ; boymelgreen2012acDielecJP  might be employed to
establish a connection between $\Omega_{1,2}$ and the material and interaction
parameters of the Janus particle and the ambient fluid, but we do not pursue
this further, here. The crucial feature is that the term
$\propto\sin(2\gamma)$ acknowledges the higher periodicity of the effect of
the two hemipsheres’ distinct heat conductivities onto the rotational motion.

The competition between the phoretic alignment of the Janus particle and its
orientational dispersion by rotational diffusion can be described by the
Fokker–Planck equation doi86 ; golestanian2012Collbehav

$\partial_{t}f=-\bm{\mathcal{R}}\cdot\left(\bm{\Omega}-{D_{\mathrm{r}}}\bm{\mathcal{R}}\right)f,$
(8)

for the dynamic probability density $f(t,\bm{n})$ to find the particle at time
$t$ with an orientation $\bm{n}$ (relative to the heat source). The rotational
operator $\bm{\mathcal{R}}\equiv\bm{n}\times\bm{\nabla}_{n}$ includes the
nabla operator $\bm{\nabla}_{n}$ with respect to the particle’s orientational
degrees of freedom, and $D_{\mathrm{r}}$ denotes the (effective
rings2012rotHBM ; falasco2016HBM ) rotational diffusion coefficient. With the
mentioned approximation of a strict in-plane orientation of the particle axis,
Eq. (8) greatly simplifies Bickel2014Polarization  to
$\partial_{t}f=-\partial_{\gamma}J,$ with the flux

$J(\gamma,t)=-\Omega_{z}(\gamma)f(\gamma,t)-{D_{\mathrm{r}}}\partial_{\gamma}f(\gamma,t).$
(9)

In the steady state, the flux $J$ is required to vanish identically, and for
an angular velocity $\Omega_{z}$ of the form (7) the orientational
distribution reads

$f(\gamma)=N^{-1}\mathrm{exp}\left\\{\frac{\Omega_{1}}{D_{\mathrm{r}}}\cos\gamma+\frac{\Omega_{2}}{D_{\mathrm{r}}}\cos^{2}\gamma\right\\},$
(10)

with a normalization factor111
$N=\mathrm{e}^{B-A}\left.\left[\mathrm{e}^{2A}\mathcal{D}\left(\frac{A+2B}{2\sqrt{B}}\right)-\mathcal{D}\left(\frac{A-2B}{2\sqrt{B}}\right)\right]\right/\sqrt{B}$
with $A\equiv\Omega_{1}/{D_{\mathrm{r}}}$,
$B\equiv\Omega_{2}/{D_{\mathrm{r}}}$, and
$\mathcal{D}(x)\equiv\mathrm{e}^{-x^{2}}\int_{0}^{x}\mathrm{d}y\leavevmode\nobreak\
\mathrm{e}^{y^{2}}.$ $N$.

###  3.2 Experimental Results

Figure 2: Distance- (a,b) and orientation- (c,d) dependence of translational
and (mean) angular swim speed. The Janus particle’s swim speed $u$ (a) and
mean angular speed $\bar{\Omega}_{z}$ (averaged over initial orientations
$\gamma$) (b) both decay like $r^{-2}$ (dash-dotted line) in the distance $r$
from the heat source, as expected from Fourier’s law of heat diffusion. Upper
and lower branch in panel (b) correspond to clockwise and counterclockwise
rotation, respectively. The orientational dependence of the swim speed in
panel (c), measured at a distance $r=$1.25\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ from the heat source, conforms with the theoretical fit
$\left(u_{x}^{2}+u_{y}^{2}\right)^{1/2}$ with $u_{x,y}$ obtained from Eqs.
(4), (5) using numerically determined temperature profiles (see Fig. 3). (d).
To resolve the orientation-dependence of the angular velocity $\Omega_{z}$,
data in the interval $r=1$-$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was
pooled. The theoretical fit (solid curve) was obtained from Eq. (6), again
using the numerically determined temperature profiles. The least-square fits
in c) and d) yield $\mu_{\rm
ps}=$2.88\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
and $\mu_{\rm
au}=$1.82\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
for the mobilities. The alternative fits shown in panel d) follow from Eq. (7)
with $\Omega_{1,2}$ as independent fit parameters (dashed) and
$\Omega_{z}=\Omega_{0}\sin\gamma$ Bickel2014Polarization , with $\Omega_{0}$
as fit parameter (dotted), and yield
$\Omega_{0}=$17.4\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$$,
$\Omega_{1}=$17.2\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$$,
$\Omega_{2}=$-3.25\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$$.

Figure 2 displays the experimental results for the magnitude of the phoretic
propulsion speed $u(\gamma,r)$ as a function of the distance $r$ from and
orientation $\gamma$ to the heat source. The speed $u$ decays with the squared
reciprocal distance, as expected for an external temperature gradient $\nabla
T\propto 1/r^{2}$ consistent with Fourier’s law. The maximum speed is
$u=$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\mathrm{/}\mathrm{s}$$ at a
distance of $r=$1.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Closer to the
heat source, tracking errors limit the acquisition of reliable data. The
experiments also provide direct evidence for a thermophoretic rotational
motion of the Janus particle. According to Eq. (6), the boundary temperatures
as well as the phoretic mobility coefficients must therefore differ between
the gold and polystyrene parts of the particle. Figure 2 (b) shows the mean
angular velocity $\bar{\Omega}$ for clockwise ($+$) and counter-clockwise
($-$) rotation, with the mean over positive and negative values of the initial
orientation $\gamma$ taken separately. The angular velocity also decays with
the squared reciprocal distance from the heat source from
$\bar{\Omega}=$100\text{\,}\mathrm{\SIUnitSymbolDegree}\mathrm{/}\mathrm{s}$$
at short distances.

The translational and rotational speeds depend on the orientation $\gamma$ to
the heat source, due to the Janus-faced particle surface and its heterogeneous
mobility coefficients $\mu$ and thermal conductivities $\kappa$. We have
therefore also analyzed the particle’s motion as a function of the initial
orientation $\gamma$. The experimental results are plotted in Figure 2 (c) and
(d). For the translational phoretic speed $u$ we observe a clear minimum
between $\gamma=$$$ and $\gamma=$$$. Local maxima are observed when the
polymer side is facing the heat source ($\gamma=$$$) or pointing away from it
($\gamma=$$$). That the latter orientation displays a smaller speed suggests
that the polymer side yields the major contribution.

Figure 3: Numerically determined azimuthal temperature variations. The
temperature increments $\langle T\rangle_{\theta}(\phi)-T_{0}$ (in Kelvin)
along the Janus particle circumference but averaged over the normal angle, are
depicted for 4 different orientations $\gamma$ and a fixed distance
$r=$1.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ between the Janus particle
and the source: (a) $\gamma=$$$, (b) $\gamma=$$$, (c) $\gamma=$$$, (d)
$\gamma=$$$.

In spite of averaging $\Omega_{z}$ over the measured distance range
($1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$–$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$) the $\gamma$-dependent angular
velocity exhibits some residual scatter. It is still seen to vanish for
$\gamma=$$$ and $\gamma=$$$ [Fig. 2 (d)], in line with the expected symmetry
of the temperature field around the axis of the Janus particle. At
$\gamma\approx$$$, we observe a maximum angular and minimum translational
speed.

To compare the experimental results to our theoretical expectations (4)–(7),
we require further information on the angular dependence of the temperature at
the surface of the Janus particle. For this purpose, we numerically solved the
complex heat conduction problem with a commercial PDE solver comsol  (H). The
obtained profiles of the mean temperature increment $\left\langle
T\right\rangle_{\theta}(\phi)-T_{0}$ along the circumference of the Janus
particle are displayed in Fig. 3. They reveal that the largest temperature
difference between the gold (au) and polystyrene (ps) side is attained when
the polymer is facing the heat source, confirming the experimental trend. They
also exhibit unequal mean boundary temperatures $\left\langle
T\right\rangle_{\theta}|_{\phi=3\pi/2}$ and $\left\langle
T\right\rangle_{\theta}|_{\phi=\pi/2}$ , as required by Eq. (6) for angular
motion.

The experimental results on the translational and the angular velocity as a
function of the orientation angle $\gamma$ can be compared to the theoretical
predictions (4)–(6) while using the numerically calculated surface temperature
profiles to obtain estimates for the phoretic mobility coefficients pertaining
to the different surface regions of the Janus particle. A least-square fit of
the theoretical prediction (6) for the angular velocity $\Omega_{z}$ yields
our best estimate for $\mu_{\rm ps}-\mu_{\rm au}$. Inserting it into Eqs. (4)
and (5) for the translational velocity components, another least-square fit
for the phoretic speed $(u_{x}^{2}+u_{y}^{2})^{1/2}$ eventually yields the
optimum values $\mu_{\rm
ps}=$2.88\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
and $\mu_{\rm
au}=$1.82\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
for the phoretic mobilities. The theoretical fits are shown in Fig. 2 (c,d) as
solid lines, while the dashed line is a fit of Eq. (7) with $\Omega_{1,2}$ as
independent fit parameters. It nicely reproduces the experimental data. In
contrast, assuming the rotational speed to be of the form
$\Omega_{z}\propto\sin\theta$ Bickel2014Polarization  (dotted), as for
homogeneous heat conductivity, misses the experimentally observed asymmetry.

Figure 4: Probability density to find the Janus sphere pointing at an angle
$\gamma$ to the heat source. The panels show data (symbols) measured for
various distances $r$ between particle and heat source: (a)
$1.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, (b)
$1.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$, (c)
$2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$, (d)
$2.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The solid lines are best fits by
Eq. (10), with free fit parameters $\Omega_{1,2}/D_{\mathrm{r}}$. The dashed
lines are fits by
$f\propto\mathrm{exp}[(\Omega_{0}/D_{\mathrm{r}})\cos\gamma]$
Bickel2014Polarization  for an angular speed profile
$\Omega_{z}=\Omega_{0}\sin\gamma$ with $\Omega_{0}/D_{\mathrm{r}}$ as a free
fit parameter. These fits yield for $\Omega_{0}/{D_{\mathrm{r}}}$,
$\Omega_{1}/{D_{\mathrm{r}}}$, $\Omega_{2}/{D_{\mathrm{r}}}$ the values (a)
0.596, 0.722, -0.284, (b) 0.493, 0.526, -0.091, (c) 0.355, 0.377, -0.0855, (d)
0.224, 0.228, -0.0248, respectively.

Besides these dynamical properties, we also assessed the stationary
distribution of the Janus particle’s orientation relative to the heat source,
at various distances. Figure 4 verifies that the particle aligns with the
external temperature gradient. In accordance with the positive angular
velocities observed for $0<\gamma<$$$ in Fig. 2 (d), we measure a
significantly higher probability to find the particle’s gold cap pointing
towards the heat source than away from it.

###  3.3 Discussion

The motion of a colloidal particle in an external temperature gradient is
determined by the thermo-osmotic surface flows bregulla2016Thermoosmosis
induced by the temperature gradients along the particle’s surface via its
physio-chemical interactions with the solvent. Knowing both the temperature
profile and interfacial interaction characteristics should thus allow the
behavior of our Janus particle in an external temperature field to be
explained. Note, however, that the heterogeneous material properties of the
Janus particle matter in two respects. First, if the hemispheres do not have
the same heat conductivities, this will distort the temperature profile in the
surrounding fluid in an unsymmetric, orientation-dependent manner. Secondly,
their generally unequal thermo-osmotic mobility coefficients $\mu$ will
translate the resulting surface temperature gradients differently into
phoretic motion. The numerically determined temperature profiles for our Janus
particle, shown Fig. 3, reveal that the presence of the Janus particle indeed
distorts the external field significantly, and that the difference between the
heat conductivities of the two hemispheres matters. The large thermal
conductivity of gold creates an almost isothermal temperature profile on the
gold cap (even if the thin film conductivity is somewhat lower than the bulk
thermal conductivity). The resulting temperature distribution is for some
orientations $\gamma$ reminiscent of the temperature distribution on the
surface of a self-propelled Janus particle. In the latter case, the metal cap
itself is the major light absorber and thus the heat source creating the
surrounding temperature gradient. In our case the gradient is primarily caused
by the external heat source, but modulated by the presence of the Janus
sphere. Unless the particle’s symmetry axis is perfectly aligned with the heat
source [Fig. 3 (a),(c)], the mean temperature profile is generally asymmetric
along the particle’s circumference. Such asymmetric distortions of the
temperature field were not considered in previous theoretical studies
Bickel2014Polarization  but matter for the proper interpretion of Eqs. (4)–(6)
for the particle’s linear and angular velocities.

Equation (5) yields the transverse thermophoretic velocity, $u_{y}$, of the
particle, i.e., the velocity perpendicular to its symmetry axis. Assuming a
constant temperature on the gold hemisphere, the only contribution for the
transverse velocity $u_{y}$ results from the temperature gradients along the
polystyrene side — due to the $\sin\phi$ term in Eq. (5) — and $u_{y}$ is
determined by the mobility coefficient $\mu_{\rm ps}$. The velocity component
$u_{x}$ along the particle’s symmetry axis contains two terms according to Eq.
(4). The first term yields a propulsion along the symmetry axis to which both
hemispheres contribute according to the $\cos\phi$ term. It tends to suppress
the details at the au–ps interface, where the temperature gradients are
typically most pronounced. Hence, the temperature profile in the vicinity of
the particle poles and the corresponding mobilities largely determine the
first term in Eq. (4). The second term, which only depends on the boundary
values of the (weighted) mean temperature at the au-ps interface and the
mobility step $\mu_{\rm ps}-\mu_{\rm au}$, is of opposite sign and thus
reduces the total propulsion velocity. (It disappears if $\mu_{\rm
ps}\approx\mu_{\rm au}$.)

Figure 5 illustrates the orientation dependence of the phoretic velocity
compoents $u_{x}$ and $u_{y}$ obtained from Eqs. (4) and (5). The longitudinal
component $u_{x}$ (along the particle’s symmetry axis) is positive or negative
depending on whether the ps-hemisphere faces away from or towards the heat
source. Its smooth sign change at $\gamma\approx$$$ simply reflects the fact
that the interaction is overall repulsive. Notice, however, that the higher
thermal conductivity of the gold cap creates a surface temperature
contribution mimicking that for an optically heated Janus swimmer. The ensuing
(self-) propulsion along the $x$ direction shifts the zero crossing slightly
from $None$. This thermophoretic "swimmer-contribution" to the propulsion is
not generally parallel to the direction of the external temperature gradient,
unless it is perfectly aligned to the heat source, thereby causing subtle
deviations from predictions for particles with isotropic heat conductivity
Bickel2014Polarization . The transverse velocity component $u_{y}$ naturally
vanishes if the particle axis is aligned or anti-aligned with the heat source
($\gamma=$$$ and $\gamma=$$$). It attains a maximum at $\gamma\approx$$$, when
the polystyrene hemisphere is oriented somewhat towards the heat source, which
allows for the maximum lateral surface temperature gradients. For the same
reason, the maximum propulsion speed $u$ is attained for $\gamma=$$$ (ps-side
facing the heat source) and only a lesser local maximum is seen at $\gamma=$$$
(au-side facing the heat source), in Fig. 2 (c).

Figure 5: Orientation dependence of the phoretic propulsion (a) Longitudinal
and transverse velocity components $u_{x}$ and $u_{y}$ ($u_{y}$ starting at
the origin) from Eqs. (4) and (5), respectively. For the phoretic mobilities,
three sets of values are considered: those obtained from the fits in Fig. 2
(b,d), namely $\mu_{\rm
ps}=$2.88\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
and $\mu_{\rm
au}=$1.82\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
(solid lines); $\mu_{\rm
ps}=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
and $\mu_{\rm
au}=$-0.55\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
(dashed); $\mu_{\rm
ps}=$-1.82\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
and $\mu_{\rm
au}=$-2.88\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}^{2}\mathrm{/}\mathrm{s}\mathrm{K}$$
(dotted); (b) the corresponding total propulsion speeds
$\left(u_{x}^{2}+u_{y}^{2}\right)^{-1/2}$.

From our fits in Fig. 2 we obtained $\mu_{\rm ps}>\mu_{\rm au}>0$. The first
condition ensures the correct sign for the angular velocity according to Eq.
(6) and Fig. 2(d), and is in agreement with previous findings for thermo-
osmotic interfacial flows bregulla2016Thermoosmosis . The step in the phoretic
mobility at the particle equator determines the magnitude and sign of the
angular velocity Bickel2014Polarization . While different absolute values can
lead to the same step height $\mu_{\rm ps}-\mu_{\rm au}$, the
$\gamma-$dependence of the translational velocity also constraints these
absolute values. This is illustrated by the dashed and dotted lines in Fig. 5,
representing other combinations of phoretic mobilities, including negative
signs ($\mu_{\rm ps}>0,\leavevmode\nobreak\ \mu_{\rm au}<0$ or $\mu_{\rm
au}<0<\mu_{\rm ps}$). Such choices would result in a quantitative and
qualitative mismatch between theory and data. They also serve to demonstrate
that the motion of the particle is very sensitive to these values, for a given
temperature profile.

According to Eq. (6), the angular velocity component $\Omega_{z}$ only depends
on the equatorial interfacial values at $\phi=\pi/2$ and $3\pi/2$ of the
average temperature $\left\langle T\right\rangle_{\theta}$ and the jump in the
mobility coefficients. In other words, the details of the temperature profile
on both sides of the particle are irrelevant for the rotational motion as long
as the two boundary temperatures and the two mobility coefficients differ
appreciably, but rotational motion will cease if either pair coincides. Hence,
irrespective of the negligible temperature gradient on the gold side, the
rotational velocity is sensitive to the thermo-osmotic mobility coefficient
$\mu_{\rm au}$, which can thus confidently be inferred from the measurement.
Compared to the substantial thermal-conductivity contrast, the role of mass
anisotropy, which can lead to similar polarization effects
Olarte_Plata_2018thermophorTorque ; Gittus2019thermalOrient ;
Olarte_Plata2020orientJPmassAnisotrop , plays presumably a negligible role in
our experiments, as the thin gold cap makes the Janus particle only slightly
bottom heavy.

##  4 Conclusions

To summarize, we have investigated the interaction of a single gold-capped
Janus particle with the inhomogeneous temperature field emanating from an
immobilized gold nanoparticle. The setup allows for a precise and well-
controlled study of thermophoretic interparticle interactions that dominate in
dilute suspensions of thermophoretic microswimmers. To our knowledege, this is
the first time, the repulsion of the Janus particle from the heat source and
its thermophoretically induced angular velocity have quantitatively been
measured. An interesting consequence of the induced angular motion is an
emerging polarization of the Janus particle in the thermal field, which should
generalize to any type of Janus swimmer in a motility gradient. In our case,
it means that the metal cap preferentially points towards the heat source.

In combination with numerically determined surface temperature profiles for
various particle-heat source orientations, the standard hydrodynamic model for
colloidal phoretic motion was found to nicely reproduce our experimental data.
Theory and observation corroborate that the rotational motion hinges on two
necessary conditions: (i) the phoretic mobilities of the Janus hemispheres
must be distinct and (ii) the values of the driving field (in our cases the
temperature) must differ accross the equator — irrespective of its behavior in
between. In return, we could therefore infer the phoretic mobilities from the
observed rotational and translational motion in an external field gradient. We
found them to be positive for both polystyrene and gold.

As an interesting detail, we found that the distinct heat conductivities
moreover break the naively expected symmetry of the particle’s translational
and rotational speeds as a function of the orientation, and, accordingly, of
the resulting polarization of the Janus sphere with respect to the heat
source. The observed asymmetries are quantitatively explained by the high heat
conductivity of gold, which renders the metal cap virtually isothermal. This
induces a robust translational motion that mimicks the self-propulsion of a
Janus swimmer in its self-generated temperature gradient, along it symmetry
axis. Since phoresis generally involves gradients in some (typically long-
ranged) thermodynamic fields, our principal results should also apply to
similar setups involving other types of phoretic mechanisms.

Acknowledgements:  We acknowledge financial support by the Deutsche
Forschungsgemeinschaft via SPP 1726 "Microswimmers" (KR 3381/6-1, KR 3381/6-2
and CI 33/16-1, CI 33/16-2).

## Author Contribution Statement

AB conducted the experiments and analyzed the data. FC carried out the
numerical calculations. SA carried out analytical calculations. KK, FC and SA
drafted the manuscript.

##  Appendix A Preparation of the Janus particles

The Janus particles have been prepared on standard microscopy glass cover
slips, which have been treated in an oxygen plasma. A solution of polystyrene
beads ($R=$0.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$; Microparticles GmbH)
is deposited on these cover slips in a spin coater at
$8000\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$. The particle concentration of
the bead solution has been adjusted such that the particles do not form a
closed packed monolayer but settle as rather isolated particles. This reduces
the number of aggregates formed during the gold layer depostion. The samples
have been further covered with a $5\text{\,}\mathrm{nm}$ chromium and a
$50\text{\,}\mathrm{nm}$ gold film by evaporation in a vacuum chamber. The
chromium layer has been added to make sure that the gold layer adheres to the
glass slide when removing the Janus particles from the glass substrate by
sonification. Fig. 6 displays a REM image of the prepared particles.

Figure 6: Raster electron microscopy image of the prepared Janus Particles.

##  Appendix B Sample preparation

The samples consist of two glass cover slips, which were rinsed with acetone,
ethanol and deionized water, and treated with an oxygen plasma. They have been
further coated with Pluronic F-127 (Sigma Aldrich) in a
$5\text{\,}\mathrm{\char 37\relax}$ aqueous solution for a few hours. The
Pluronic is adsorbed to the glass surface and residual Pluronic has been
removed by rinsing the coated slides with deionized water. A mixture of Janus
particles and $R=$125\text{\,}\mathrm{nm}$$ gold colloids (British Biocell)
was then deposited between the two slides and sealed with polydimethylsiloxane
(PDMS) to prevent evaporation of the solution. The typical thickness of the
liquid layer between the glass slides has been adjusted to be on the order of
the diameter of the Janus particle
($\approx$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) to prevent motion in
vertical direction.

##  Appendix C Experimental setup

The experimental setup consists of 2 parts: the heating and the illumination
part. For the heating part a common laser source at a wavelength of
$532\text{\,}\mathrm{nm}$ was used. This beam was first enlarged by a beam
expander to fully illuminate an acousto-optic-deflector (AOD). This AOD is
utilized to freely steer the focused beam within the sample. The optical path
is arranged such that the beam waist is approximately
$500\text{\,}\mathrm{nm}$. This beam is then focused by an oil immersion
objective lens (Olympus 100x NA 0.5-1.3) into the sample.
The illumination of the sample is realized by an oil immersion dark field
condenser (Olympus NA 1.2-1.4). The scattered white light is collected by the
objective and imaged on the CCD-camera (Andor iXon). For the spatial position
of the sample a piezo-scanner was used (Physik Instrumente, PI).

Figure 7: Experimental setup used for the experiments. See text for
explanations and additional details.

##  Appendix D Particle tracking

To determine the position of the Janus particles a binary image at a threshold
above the background noise was taken. The particle with the larger visual area
was identified as the Janus particle the smaller one as the gold heat source.
The geometric centers of the visual images was identified with the particle
position. For small distances ($<$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$)
beween the Janus particle and the gold particle, the determination of the
position of both particles fails. In this case the data is disregarded.
The image of he Janus particle is further analyzed with multiple binary images
that are obtained by limiting the maximum image intensity to thresholds
between $10\text{\,}\mathrm{\char 37\relax}$ and $80\text{\,}\mathrm{\char
37\relax}$. For each binary image the geometric center is determined. The $x-$
and $y-$coordinates of the geometric center are fitted with
$f_{\rm{x}}=a_{\rm{x}}\,x+b_{\rm{x}}$ and
$f_{\rm{y}}=a_{\rm{y}}\,x+b_{\rm{y}}$, respectively. From both fits, the in-
plane orientation can be determined by
$\gamma=\arctan\left(a_{\rm{x}}/a_{\rm{y}}\right)$.

##  Appendix E Measurering the temperature profile on the surface of the
heated gold nanoparticle

Figure 8: (a) The refractive index of 5CB in the nematic phase
($<$35\text{\,}\mathrm{\SIUnitSymbolCelsius}$$) and the isotropic phase
($>$35\text{\,}\mathrm{\SIUnitSymbolCelsius}$$)HornR.G.1978  (b) Example
images of the growing isotropic bubble with heating power.

For the estimate of the temperature increment $\Delta T_{\rm{Au}}$ at the
surface of the gold nanoparticle heat source an additional experiment has been
performed. In this experiment the solvent was replaced by a liquid crystal
(5CB). Its nematic-to-isotropic melting transition upon heating beyond
$T_{\rm{ph}}=$35\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ HornR.G.1978 ;
Marinelli1998  was employed as a temperature sensor. The colloidal heat source
generates a radial temperature field

$T\left(r\right)=T_{\rm{0}}+\frac{\Delta
T_{\rm{Au}}\left(r\right)R}{r}=T_{0}+\frac{P_{\rm{abs}}}{4\pi\kappa r}$ (11)

with $\kappa$ being the thermal conductivity of the medium, $P_{\rm{abs}}$ the
absorbed power, proportional to the incident light power $P_{\rm{inc}}$,
$T_{\rm{0}}=$22\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ the ambient
temperature, and $R=$125\text{\,}\mathrm{nm}$$ the radius of the gold colloid.
Whenever the temperature exceeds the phase transition temperature
$T\left(r\right)>T_{\rm{ph}}$ the nematic order melts. Since the molecular
temperature field $T\left(r\right)$ varies locally, the phase transition is
confined to the vicinity of the heat source. Due to the radially symmetric
shape of the temperature profile, an isotropic bubble forms around the gold
colloid if $\Delta T_{\rm{Au}}+T_{\rm{0}}>T_{\rm{ph}}$. The size $r_{\rm{ph}}$
of the bubble scales linearly with $\Delta T_{\rm{Au}}$ and therefore with the
heating power $P_{\rm{inc}}$:

$r_{\rm{ph}}=\frac{\Delta
T_{\rm{Au}}R}{\left(35^{\circ}-T_{\rm{0}}\right)}\sim\Delta T_{\rm{Au}}\sim
P_{\rm{inc}}$ (12)

Figure 9: Surface temperature increment on the gold colloid (relative to the
ambient temperature of the solvent) for a liquid-crystalline solvent and
water, respectively. Symbols represent the experiment. The dashed line is a
linear fit to the data to extract the temperature increment per heating power.
The values in water have been calculated from the known thermal conductivities
of the liquid crystal and water.

In the experiments, the size of the isotropic bubble $r_{\rm{ph}}$ as the
function of the incident power $P_{\rm{inc}}$ is of interest. Its observation
in the dark field setup, (see Fig. 8(a), HornR.G.1978 ) exploits the
refractive-index change upon melting. Similar to a colloidal particle with a
refractive index deviating from the surrounding material the molten bubble
scatters the incident white light and appears as a bright ring in the dark
field microscope. Example images are displayed in Fig. 8(b) for different
incident heating powers. The black circle indicates the estimated bubble size.
Its knowledge allows the surface temperature of the gold colloid in the liquid
crystal to be estimate by:

$\Delta
T_{\rm{Au}}=\frac{\,r_{\rm{ph}}\left(35^{\circ}-T_{\rm{0}}\right)}{R}.$ (13)

Since the heat equation is linear, the estimate for the temperature increment
in water is determined by its thermal conductivity
$\kappa_{\rm{H_{2}O}}=$0.6\text{\,}\mathrm{W}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{K}\mathrm{)}$$
relative to that of the isotropic liquid crystal,
$\kappa=$0.15\text{\,}\mathrm{W}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{K}\mathrm{)}$$.
The result is displayed in Fig. 9 where the approximate temperature increment
in water is displayed in addition to the estimated temperature increment in
5CB in dependence on the heating power. From the linear fit a temperature
increment per heating power of
$\approx$14\text{\,}\mathrm{K}\mathrm{/}\mathrm{(}\mathrm{m}\mathrm{W}\mathrm{)}$$
can be obtained.

##  Appendix F Influence of the laser heating on the Janus particle

The focused laser beam (beam-waist $\approx$500\text{\,}\mathrm{nm}$$) used
for the heating of the immobile gold colloid may also heat the Janus particle
directly. To quantify this effect, the experiment was repeated with and
without the immobile gold colloid with identical focus position. The influence
of the laser beam can be estimated by calculating the particle velocity $u$
and the radial velocity $v_{\rm{R}}$. The particle velocity is obtained by
projecting the translational step
$\Delta\mathbf{s}_{i}=\mathbf{r}_{i-1}-\mathbf{r}_{i}$ onto the particle
orientation $\mathbf{n}^{i}_{o}$ and then performing the ensemble average
$\mathbf{u}=\left\langle\Delta\mathbf{s}_{i}\cdot\mathbf{n}^{i}_{o}\right\rangle_{i}/\Delta
t$ divided by the experimental timescale $\Delta t$ being the exposure time of
the camera. Fig. 10 (a) displays the absolute value of $\mathbf{u}$ for 3
different orientations of the particle relative to the heat source $\phi$. The
particle velocity is always positive as a result of the direct laser
illumination, and quickly diminishes over a length scale comparable to the
laser beam width. The radial velocity
$v_{\rm{R}}=\left\langle\Delta\mathbf{s}_{i}\cdot\mathbf{e}^{i}_{\rm{R}}\right\rangle_{i}/\Delta
t$ being the ensemble average of the scalar product of $\Delta\mathbf{s}_{i}$
and the unit vector in radial direction $\mathbf{e}^{i}_{\rm{R}}$ divided by
the experimental timescale $\Delta t$ decays on similar length scales as the
particle velocity. Even though the influence of the direct laser heating on
the Janus particle diminishes rather quickly with increasing distance, its
influence is still noticeable, and was therefore subtracted for the velocities
presented in the main text.

Figure 10: (a) Particle speed $u$ and (b) radial speed $v_{\rm{R}}$ in
dependence on radial distance to the heat source for three orientations
$\gamma$ of the particle relative to the heat source.

##  Appendix G Derivation of the phoretic velocities

###  G.1 The setup

The considered setup of a Janus particle exposed to an external heat source,
and conventions used in the following derivations, are summarized in Fig. 1.
The induced slip velocity $\bm{v}_{\rm s}$ at the surface of a Janus sphere of
radius $a$ is given by [see Eq. (1)]

$\bm{v_{\mathrm{s}}}(\theta,\phi)=\mu(\theta,\phi)\bm{\nabla}_{\parallel}T(\theta,\phi),$
(14)

where the in-plane angle $\phi$ and normal angle $\theta$ are employed to
parametrize the particle surface (rather than conventional polar coordinates
adjusted to the particle symmetry). In the above equation, $\mu(\phi,\theta)$
is the thermophoretic mobility and

$\bm{\nabla}_{\parallel}T\equiv\frac{\partial_{\theta}T}{a}\hat{\bm{\theta}}+\frac{\partial_{\phi}T}{a\sin\theta}\hat{\bm{\phi}}$
(15)

denotes the tangential part of the temperature gradient at the particle
surface, expressed in terms of spherical coordinates. As they are constantly
used in the following derivations, we note the corresponding unit vectors:

$\displaystyle\hat{\bm{r}}$
$\displaystyle\equiv\begin{pmatrix}\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta\end{pmatrix}^{\top},$
(16) $\displaystyle\hat{\bm{\theta}}$
$\displaystyle\equiv\begin{pmatrix}\cos\phi\cos\theta,\sin\phi\cos\theta,-\sin\theta\end{pmatrix}^{\top},$
(17) $\displaystyle\hat{\bm{\phi}}$
$\displaystyle\equiv\begin{pmatrix}-\sin\phi,\cos\phi,0\end{pmatrix}^{\top}.$
(18)

The translational and rotational phoretic velocities, $\bm{u}$ and
$\bm{\Omega}$, follow from $\bm{v_{\mathrm{s}}}$ as [see Eqs. (2) and (3)]

$\displaystyle\bm{u}$
$\displaystyle=-\frac{1}{|\mathcal{S}|}\oint\limits_{\mathcal{S}}\mathrm{d}S\leavevmode\nobreak\
\bm{v_{\mathrm{s}}}=-\frac{1}{4\pi}\int\limits_{0}^{2\pi}\mathrm{d}\phi\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\leavevmode\nobreak\ \bm{v_{\mathrm{s}}},$ (19)
$\displaystyle\bm{\Omega}$
$\displaystyle=-\frac{3}{2a}\frac{1}{|\mathcal{S}|}\oint\limits_{\mathcal{S}}\mathrm{d}\bm{S}\times\bm{v_{\mathrm{s}}}$
$\displaystyle=-\frac{3}{8\pi
a}\int\limits_{0}^{2\pi}\mathrm{d}\phi\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\leavevmode\nobreak\ \hat{\bm{r}}\times\bm{v_{\mathrm{s}}},$ (20)

where $|\mathcal{S}|=4\pi a^{2}$ is the area of the particle surface
$\mathcal{S}$.

It is experimentally observed that the particle preferentially aligns
horizontally with the close-by cover slides. This observation enters our
theory through the assumption that the swimmer rotates only about the
$z$-axis, i.e., perpendicular to the observation plane. This implies that the
swimmer also translates only in the $x$-$y$ plane. Once the swimmer’s $z$-axis
remains invariant, the surface-temperature profile consequently always obeys
the (approximate) symmetry

$T(\phi,\pi/2-\alpha)=T(\phi,\pi/2+\alpha),\qquad\alpha\in[0,\pi/2]$ (21)

in the normal angle, in accord with the heterogeneous material composition of
the Janus sphere. The local phoretic mobility $\mu$ may likewise be expressed
as

$\mu=\mu(\phi)\equiv\begin{cases}\mu_{\rm ps}\quad\mathrm{for}\quad
0\leq\phi\leq\phi_{\mathrm{pa}}\\\ \mu_{\rm
au}\quad\mathrm{for}\quad\phi_{\mathrm{pa}}<\phi\leq\phi_{\mathrm{ap}}\\\
\mu_{\rm ps}\quad\mathrm{for}\quad\phi_{\mathrm{ap}}<\phi<2\pi\end{cases},$
(22)

where $\mu_{\rm ps}$ and $\mu_{\rm au}$ are the constant phoretic mobilities
corresponding to the polystyrene and gold part of the swimmer, respectively,
and $\phi_{\mathrm{pa}}$ and $\phi_{\mathrm{ap}}$ denote the angles pertaining
to the equator between the distinct surface materials.

###  G.2 Rotation

We start with the term $\hat{\bm{r}}\times\bm{v_{\mathrm{s}}}$ inside the
integral on the r.h.s. of Eq. (20). Plugging in Eqs. (14) and (15), and using
$\hat{\bm{r}}\times\hat{\bm{\theta}}=\hat{\bm{\phi}}$ and
$\bm{\hat{r}}\times\bm{\hat{\phi}}=-\hat{\bm{\theta}}$, one obtains

$\displaystyle\hat{\bm{r}}\times\bm{v_{\mathrm{s}}}$
$\displaystyle=\frac{\mu(\phi)}{a}\left[(\partial_{\theta}T)\hat{\bm{\phi}}-\frac{\partial_{\phi}T}{\sin\theta}\hat{\bm{\theta}}\right]$
(23)
$\displaystyle=\frac{\mu(\phi)}{a}(\partial_{\theta}T)\begin{pmatrix}-\sin\phi\\\
\cos\phi\\\ 0\end{pmatrix}$ (24)
$\displaystyle\quad+\frac{\mu(\phi)(\partial_{\phi}T)}{a\sin\theta}\begin{pmatrix}-\cos\phi\cos\theta\\\
-\sin\phi\cos\theta\\\ \sin\theta\end{pmatrix}.$ (25)

The symmetry relation (21) implies

$\displaystyle\partial_{\theta}T(\phi,\pi/2-\alpha)$
$\displaystyle=-\partial_{\theta}T(\phi,\pi/2+\alpha),$ (26)
$\displaystyle\partial_{\phi}T(\phi,\pi/2-\alpha)$
$\displaystyle=\partial_{\phi}T(\phi,\pi/2+\alpha),$ (27)

for $\alpha\in[0,\pi/2]$. Hence, when calculating the surface average of Eq.
(23), the $x$ and $y$-components vanish:

$\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\leavevmode\nobreak\
\partial_{\theta}T=\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\cos\theta\leavevmode\nobreak\ \partial_{\phi}T=0.$ (28)

Via Eq. (20), the remaining $z$-component is given by

$\displaystyle\Omega_{z}$ $\displaystyle=-\frac{3}{8\pi
a}\int_{0}^{2\pi}\mathrm{d}\phi\,\mu(\phi)\int_{0}^{\pi}\mathrm{d}\theta\,\frac{\sin\theta}{a\sin\theta}\partial_{\phi}T(\theta,\phi)\sin\theta$
$\displaystyle=-\frac{3}{4\pi
a^{2}}\int_{0}^{2\pi}\mathrm{d}\phi\,\mu(\phi)\partial_{\phi}\left\langle
T\right\rangle(\phi),$ (29)

where we introduced the mean ($\theta-$averaged) temperature $\left\langle
T\right\rangle_{\theta}(\phi)$ via

$\left\langle\bullet\right\rangle_{\theta}(\phi)\equiv\frac{\int_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\bullet(\phi,\theta)\sin\theta}{\int_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta}=\frac{1}{2}\int_{0}^{\pi}\\!\\!\mathrm{d}\theta\leavevmode\nobreak\
\bullet(\phi,\theta)\sin\theta.$ (30)

In contrast to Eqs. (4)–(6) in the main text, we omit the subscript $\theta$
in the averaging notation $\langle\bullet\rangle$ throughout the rest of this
section for the sake of brevity. Using the mobility profile (22) and
$2\pi$-periodicity of $\mu$ and $T$ in the angle $\phi$, the angular velocity
simplifies to

$\displaystyle\Omega_{z}$ $\displaystyle=-\frac{3}{4\pi a^{2}}\Bigg{(}\mu_{\rm
ps}\int\limits_{0}^{\phi_{\mathrm{pa}}}\mathrm{d}\phi\,\partial_{\phi}\left\langle
T\right\rangle(\phi)$ (31) $\displaystyle\qquad+\mu_{\rm
au}\int\limits_{\phi_{\mathrm{pa}}}^{\phi_{\mathrm{ap}}}\mathrm{d}\phi\,\partial_{\phi}\left\langle
T\right\rangle(\phi)+\mu_{\rm
ps}\int\limits_{\phi_{\mathrm{ap}}}^{2\pi}\mathrm{d}\phi\,\partial_{\phi}\left\langle
T\right\rangle(\phi)\Bigg{)}$ (32) $\displaystyle=-\frac{3}{4\pi
a^{2}}\big{\\{}\mu_{\rm ps}\left[\left\langle
T\right\rangle(\phi_{\mathrm{pa}})-\left\langle
T\right\rangle(\phi_{\mathrm{ap}})\right]$ (33)
$\displaystyle\qquad\qquad+\mu_{\rm au}\left[\left\langle
T\right\rangle(\phi_{\mathrm{ap}})-\left\langle
T\right\rangle(\phi_{\mathrm{pa}})\right]\big{\\}}$ (34)
$\displaystyle=\frac{3}{4\pi a^{2}}(\mu_{\rm ps}-\mu_{\rm
au})\left[\left\langle T\right\rangle(\phi_{\mathrm{ap}})-\left\langle
T\right\rangle(\phi_{\mathrm{pa}})\right].$ (35)

The final experession yields Eq. (6) upon identifying
$\phi_{\mathrm{pa}}=\pi/2$ and $\phi_{\mathrm{ap}}=3\pi/2$ for a half-coated
Janus sphere.

###  G.3 Translation

Plugging Eq. (15) in to Eq. (14), and using the expessions for the unit
vectors (16)–(18), the local slip velocity at the particle surface reads

$\bm{v_{\mathrm{s}}}=\frac{(\partial_{\theta}T)\mu(\phi)}{a}\begin{pmatrix}\cos\phi\cos\theta\\\
\sin\phi\cos\theta\\\
-\sin\theta\end{pmatrix}+\frac{(\partial_{\phi}T)\mu(\phi)}{a\sin\theta}\begin{pmatrix}-\sin\phi\\\
\cos\phi\\\ 0\end{pmatrix}.$ (36)

Calculating the surface average of Eq. (36), one finds that its $z$-component
vanishes, because

$\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin^{2}\theta\leavevmode\nobreak\ \partial_{\theta}T=0,$ (37)

by virtue of the symmetry relation (26). We now decompose the remaining $x$
and $y$ components of the translational velocity into
$\bm{u}^{(\theta)}+\bm{u}^{(\phi)}$, corresponding to the contributions
$\partial_{\theta}T$ and $\partial_{\phi}T$ of the temperature gradient (15),
respectively. We furthermore apply integration by parts to get rid of the
temperature gradients and deal with the bare temperature profiles instead.

####  $\bm{\partial_{\theta}}$-part

Using Eq. (19), the $\theta$-derivative of the temperature gradient (15) gives

$\displaystyle\bm{u}^{(\theta)}$
$\displaystyle\equiv-\frac{1}{|\mathcal{S}|}\oint\limits_{\mathcal{S}}\mathrm{d}S\leavevmode\nobreak\
\frac{\mu(\phi)}{a}(\partial_{\theta}T)\begin{pmatrix}\cos\phi\cos\theta\\\
\sin\phi\cos\theta\end{pmatrix}$ $\displaystyle=-\frac{1}{4\pi
a}\int\limits_{0}^{2\pi}\mathrm{d}\phi\leavevmode\nobreak\
\mu\begin{pmatrix}\cos\phi\\\
\sin\phi\end{pmatrix}\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\cos\theta\leavevmode\nobreak\ \partial_{\theta}T.$ (38)

Applying integration by parts and using

$\partial_{\theta}(\sin\theta\cos\theta)=\cos^{2}\theta-\sin^{2}\theta,$

the $\theta$-integral in Eq. (38) can be written as

$\displaystyle\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\cos\theta\leavevmode\nobreak\ \partial_{\theta}T$
$\displaystyle=\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
\sin\theta\leavevmode\nobreak\
\left(\sin\theta-\frac{\cos\theta}{\tan\theta}\right)T$
$\displaystyle=2\left\langle\left(\sin\theta-\frac{\cos\theta}{\tan\theta}\right)T\right\rangle,$
(39)

with the $\theta$-average as defined in Eq. (30). The $\theta$-contribution to
the velocity thus reads

$\displaystyle\bm{u}^{(\theta)}=-\int\limits_{0}^{2\pi}\frac{\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)}{2\pi
a}\left\langle\left(\sin\theta-\frac{\cos\theta}{\tan\theta}\right)T\right\rangle(\phi)\begin{pmatrix}\cos\phi\\\
\sin\phi\end{pmatrix}.$ (40)

####  $\bm{\partial_{\phi}}$-part

Analogously, the $\phi$-derivative of the temperature gradient (15)
contributes

$\bm{u}^{(\phi)}\equiv-\frac{1}{4\pi
a}\int\limits_{0}^{2\pi}\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)\begin{pmatrix}-\sin\phi\\\
\cos\phi\end{pmatrix}\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
(\partial_{\phi}T).$ (41)

The $\phi$-derivative appearing on the r.h.s. of the above equation can be
pulled out of the first integral. The remaining $\theta$-integration of the
bare temperature profile can be expressed as

$\int\limits_{0}^{\pi}\mathrm{d}\theta\leavevmode\nobreak\
T(\phi,\theta)=2\left\langle\frac{T}{\sin\theta}\right\rangle\equiv
2\widetilde{T}(\phi),$ (42)

with the $\theta$-average $\left\langle\bullet\right\rangle$ as defined in Eq.
(30). With the profile (22) of the local phoretic mobility $\mu(\phi)$, one
finds

$\displaystyle\bm{u}^{(\phi)}$
$\displaystyle=\int_{0}^{2\pi}\frac{\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)}{2\pi a}\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\ \partial_{\phi}\widetilde{T}(\phi)$
$\displaystyle=\frac{\mu_{\rm ps}}{2\pi
a}\int\limits_{0}^{\phi_{\mathrm{pa}}}\mathrm{d}\phi\,\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\ \partial_{\phi}\widetilde{T}$
$\displaystyle\quad+\frac{\mu_{\rm au}}{2\pi
a}\int\limits_{\phi_{\mathrm{pa}}}^{\phi_{\mathrm{ap}}}\mathrm{d}\phi\,\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\ \partial_{\phi}\widetilde{T}$
$\displaystyle\quad+\frac{\mu_{\rm ps}}{2\pi
a}\int\limits_{\phi_{\mathrm{ap}}}^{2\pi}\mathrm{d}\phi\,\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\ \partial_{\phi}\widetilde{T}$ (43)
$\displaystyle=\frac{\mu_{\rm ps}}{2\pi a}\left.\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\
\widetilde{T}\right|_{\phi_{\mathrm{ap}}}^{\phi_{\mathrm{pa}}}-\frac{\mu_{\rm
au}}{2\pi a}\left.\begin{pmatrix}\sin\phi\\\
-\cos\phi\end{pmatrix}\leavevmode\nobreak\
\widetilde{T}\right|_{\phi_{\mathrm{ap}}}^{\phi_{\mathrm{pa}}}$
$\displaystyle\quad-\int_{0}^{2\pi}\frac{\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)}{2\pi a}\begin{pmatrix}\cos\phi\\\
\sin\phi\end{pmatrix}\leavevmode\nobreak\ \widetilde{T}$ (44)
$\displaystyle=\frac{\mu_{\rm ps}-\mu_{\rm au}}{2\pi a}\times$
$\displaystyle\quad\left[\begin{pmatrix}\sin(\phi_{\mathrm{pa}})\\\
-\cos(\phi_{\mathrm{pa}})\end{pmatrix}\widetilde{T}(\phi_{\mathrm{pa}})-\begin{pmatrix}\sin(\phi_{\mathrm{ap}})\\\
-\cos(\phi_{\mathrm{ap}})\end{pmatrix}\widetilde{T}(\phi_{\mathrm{ap}})\right]$
$\displaystyle\quad-\int\limits_{0}^{2\pi}\frac{\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)}{2\pi a}\begin{pmatrix}\cos\phi\\\
\sin\phi\end{pmatrix}\widetilde{T},$ (45)

where we applied integration by parts from (43) to (44), and exploited
$2\pi$-symmetry.

#### Combining both contributions

Adding the results (40) and (45) for $\bm{u}^{(\theta)}$ and
$\bm{u}^{(\phi)}$, one finally arrives at

$\displaystyle\begin{pmatrix}u_{x}\\\ u_{y}\end{pmatrix}$
$\displaystyle=\frac{\mu_{\rm ps}-\mu_{\rm au}}{2\pi a}\times$ (46)
$\displaystyle\quad\Bigg{[}\begin{pmatrix}\sin(\phi_{\mathrm{pa}})\\\
-\cos(\phi_{\mathrm{pa}})\end{pmatrix}\left\langle\frac{T}{\sin\theta}\right\rangle(\phi_{\mathrm{pa}})$
$\displaystyle\qquad\qquad-\begin{pmatrix}\sin(\phi_{\mathrm{ap}})\\\
-\cos(\phi_{\mathrm{ap}})\end{pmatrix}\left\langle\frac{T}{\sin\theta}\right\rangle(\phi_{\mathrm{ap}})\Bigg{]}$
$\displaystyle\quad-2\int\limits_{0}^{2\pi}\frac{\mathrm{d}\phi\leavevmode\nobreak\
\mu(\phi)}{\pi a}\begin{pmatrix}\cos\phi\\\ \sin\phi\end{pmatrix}\left\langle
T\sin\theta\right\rangle(\phi),$

where we replaced $\widetilde{T}(\phi)$ by $\left\langle
T/\sin\theta\right\rangle$ [see Eq. (42)] and used the identity
$1+\sin^{2}\theta-\cos^{2}\theta=2\sin^{2}\theta$ to arrive at the term
$\left\langle T\sin\theta\right\rangle$ appearing inside the integral in the
last line of Eq. (46). Setting $\phi_{\mathrm{pa}}=\pi/2$ and
$\phi_{\mathrm{ap}}=3\pi/2$ renders Eqs. (4) and (5).

##  Appendix H Finite-element simulation of the temperature field

To calculate the surface temperature of the Janus particle at different
orientations, we use the COMSOL Multi-physics® software comsol  to employ a
finite-element solver for the considered heat conduction problem sketched in
Fig. 11.

Figure 11: Geometry of the setup used for the numerical temperature
calculations.

The Janus particle is realized as a polystyrene particle of
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter with a gold cap which is
tapered to the edges and has a maximum thickness of $50\text{\,}\mathrm{nm}$.
The heat source is a gold sphere of $250\text{\,}\mathrm{nm}$ diameter placed
at $1.25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ distance from the Janus
particle center. Both particles are placed in a box of an edge length of
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The gold nanoparticle is heated
with a heat source density of
$1\text{\times}{10}^{15}\text{\,}\mathrm{W}\,\mathrm{m}^{-3}$. Other
parameters used for the numerical calculations are listed in Table 1.

Medium | $\kappa\leavevmode\nobreak\ \left[$\text{\,}\mathrm{W}\mathrm{/}\mathrm{(}\mathrm{K}\leavevmode\nobreak\ \mathrm{m}\mathrm{)}$\right]$ | $\rho\leavevmode\nobreak\ \left[$\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$\right]$ | $c_{p}\leavevmode\nobreak\ \left[$\text{\,}\mathrm{J}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\leavevmode\nobreak\ \mathrm{K}\mathrm{)}$\right]$
---|---|---|---
polystyrene | 0.14 | 1000 | 1250
water | 0.6 | 1000 | 4185
gold | 318 | 19300 | 1860
Table 1: Heat conductivity $\kappa$, density $\rho$ and heat capacity $c_{p}$
used for the numerical calculations in COMSOL comsol .

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