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 [1,2,3] is a unique capability to adapt to environmental cues [4]. Down to the microbial level where all kinds of “animalcules” [5] struggle to locomote through liquid solvents [6, 7], interactions with boundaries and neighbors and the sensing of chemical gradients [8] 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 [9] for which numerous experimental and theoretical studies are available [10,11,12]. 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 [9] that drives the self-propulsion of the swimmer. Accordingly, thermophoresis [13,14,15], diffusiophoresis [16,17,18] and electrophoresis [19, 20] can deliberately be exploited for (or may inadvertently contribute to) the swimming of Janus particles [21,22,23,24,25,26,27,28,29]. The classical Janus-particle design consists of a spherical colloid with hemispheres of distinct physicochemical properties, which define a polar symmetry axis. Due to the broken symmetry, one expects the axis to align with an external field gradient [30,31,32,33], as experimentally confirmed, e.g., in [34] for diffusiophoretic Janus particles. The reorientation of microswimmers in external fields is often referred to as taxis and has been studied for various phoretic mechanisms [30, 31, 33,34,35,36,37,38,39]. Generally, the direction of alignment (parallel or anti-parallel) with respect to an imposed non-homogeneous field gradient is determined by the precise surface properties of the particle and the chosen solvent [30, 40, 41]. An active Janus sphere can thus, in principle, move toward or away from regions of high concentrations/temperatures/electrostatic potentials [30, 33, 34, 41, 42]. Fabricating microswimmers with appropriate physical and chemical properties may enable important developments, such as autonomous particles steering towards a target in a concentration gradient or along suitably patterned surfaces or motility landscapes [32, 43,44,45,46,47,48].

Beyond that, what we call an external field can be understood as a template for the influence of container walls or neighboring microswimmers [10, 49, 50] that are at the core of the rich collective phenomena emerging in active fluids [42, 51,52,53,54,55,56,57,58]. That microswimmers are constantly exchanging linear and angular momentum with the ambient fluid generally renders their apparent mutual interactions non-reciprocal [40, 59]. 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 [10, 60]. Generally, interactions mediated by hydrodynamic flow fields [61,62,63,64,65,66], optical shadowing [39, 67] and chemical or optical patterns [68,69,70] may have to be considered, and which of these contributions dominate the observed motion of microswimmers has recently been under debate [49, 50, 71].

The appreciation of the relevance of taxis for the two-body interaction of Janus particles has recently grown as it was shown to yield (or significantly contribute to) remarkable two-particle dynamics, e.g., phase-locked circular motion around a common center, bound or scattering states, or the existence of stable/unstable fixed points (see [40, 42, 72,73,74] and references therein). The goal of the present article is to quantify the contribution of thermotaxis to the two-body interactions of thermophoretic Janus spheres both theoretically and experimentally.

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 [22, 75], we can direct an individual Janus particle into the vicinity of the local heat source [23, 24], 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 an approximate theoretical analysis that addresses the thermophoretic origin of the interactions, improves previous literature results [30], and complements recent calculations of the phoretic interactions between two chemically active particles [73] and the axis-symmetric interactions between two diffusiophoretic Janus particles [74]. Our combination of idealized microscopic modeling and experimental measurements allows us to determine the sign and order of magnitude of the surface mobility coefficients. These tend to be poorly known and are treated as free parameters in many active colloid studies. For instance, in the context of self-diffusiophoresis, it is difficult to sustain the chemical gradients that could, in principle, be used to measure surface mobility. Here, we exploit the photon nudging technique and laser heating to obtain good particle tracking statistics for a single Janus particle in a controlled temperature gradient.

2 Experimental setup

We experimentally explore the interaction of a \({1}\,{\upmu }\hbox {m}\) diameter Janus particle with a 50-nm-thin gold cap with the temperature field generated by an immobilized 250-nm gold nanoparticle optically heated by a focused laser (wavelength 532nm). To confine the Janus particle to the vicinity of the heat source, we employ the feedback control technique of photon nudging [22, 75] that exploits its autonomous motion to steer it to a chosen target. As shown in Fig. 1a, the steering is only activated when the Janus particle leaves an outer radius \(r_{\mathrm{o}}\) around the heat source until it has migrated back across an inner radius \(r_{\mathrm{i}}\), followed by a waiting time of 10 rotational diffusion times \(\tau _{\mathrm {r}}\) to allow for the decay of orientational biases and correlations [45, 76]. During the photon nudging periods, the central gold particle heating was turned off.

Fig. 1
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 \(\mu _{\mathrm{ps}}\), \(\mu _{\mathrm{au}}\)) 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 (\({1.5}\,{\upmu }\hbox {m}\)). The coordinate frame attached to the particle’s geometric center has its x-axis aligned with the particle’s symmetry axis and pointing toward 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 \({\varvec{u}}\), \({\varvec{\varOmega }}\), arise from slip fluxes with velocities \({\varvec{v}}_\mathrm {s}\) 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 \({5}\hbox {ms}\). Further details regarding the sample preparation, the experimental setup, and the position and orientation analysis are contained in “Appendix A”–“Appendix D.” The temperature increment \(\Delta T = {12}\,\hbox {K}\) of the heated gold nanoparticle relative to the ambient temperature (\(T_0 = {295}\,\hbox {K}\)) is known from a separate measurement using the nematic/isotropic phase transition of a liquid crystal (see “Appendix E”). We account for the direct influence of the heating laser on the Janus particle and the phoretic velocities, as detailed in “Appendix F.”

Below, we offer an approximate theoretical analysis that fits the experimental data reasonably well and point out some of its technical limitations.

3 Results and discussion

3.1 Theory

On the hydrodynamic level of description, the temperature gradient \( {\varvec{\nabla }}_{\parallel }T \equiv ({\varvec{I}} - {\varvec{e}}_r {\varvec{e}}_r) {\varvec{\nabla }}T \) along the surface of the Janus particle induces a proportionate interfacial creeping flow [77, 78], where \({\varvec{e}}_r\) denotes the unit vector normal to the particle surface and \({\varvec{I}}\) 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 [9, 26, 30]

$$\begin{aligned} {\varvec{v}}_{\mathrm {s}}\left( \theta ,\phi \right) = \mu \left( \theta ,\phi \right) {\varvec{\nabla }}_{\parallel }T\left( \theta ,\phi \right) . \end{aligned}$$

The particle surface is parameterized in terms of the in-plane and normal angles \(\phi \) and \(\theta \), as sketched in Fig. 1a, 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 [30]. The resulting translational propulsion velocity \({\varvec{u}}\) and the angular velocity \({\varvec{\varOmega }}\) of the Janus particle of radius a under three-dimensional bulk conditions are given by averages over its surface \({\mathcal {S}}\): [30, 79]

$$\begin{aligned} {\varvec{u}}&= - \frac{1}{4 \pi a^2} \oint _{{\mathcal {S}}} \mathrm {d}S \, {\varvec{v}}_{\mathrm{{s}}}, \end{aligned}$$
$$\begin{aligned} {\varvec{\varOmega }}&= -\frac{1}{4 \pi a^2} \oint _{{\mathcal {S}}} \mathrm {d}{\varvec{S}} \times \frac{3}{2a} {\varvec{v}}_{\mathrm{{s}}}. \end{aligned}$$

We note that Eqs. (2) and (3) correspond to a free-space approximation of the particle’s phoretic motion where the influence of the cover slides (walls) is disregarded. In reality, the Janus sphere “senses” and responds to the presence of the boundary via the hydrodynamic fields it creates. The cover slides generally alter the coefficients of the hydrodynamic stress tensor and the temperature field around the particle [80]. We point out that no signatures of attractive thermoosmotic contributions [78] to the particle motion stemming from the cover slides were detected, indicating that the film thickness between the Janus particle and its confining walls is sufficiently large to neglect this effect. The free-space approximation is moreover appears reasonable, as our aim is primarily to acquire a qualitative understanding of the particle’s alignment and repulsion, to obtain the right signs and symmetries of the observed effects, and to see to how well a simple analytical model can quantify the experimental observations.

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 [80,81,82]. 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 [83]. Within the free-space approximation, (2) and (3), the in-plane alignment is maintained by assuming that both the heat source and the Janus particle are perfectly centered between the cover slides. Within this idealization of the experimental setup, their center-to-center distance r is then the same as in the in-plane projection (cf. Fig. 1a). 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

$$\begin{aligned} u_x&= - \frac{1}{\pi a} \int _0^{2\pi } \mathrm {d}\phi \, \mu (\phi ) \cos \phi \left\langle T \sin \theta \right\rangle _{\theta }(\phi ) \nonumber \\&+ \frac{\mu _{\mathrm{ps}}- \mu _{\mathrm{au}}}{2 \pi a} \left[ \left\langle \frac{T}{\sin \theta } \right\rangle _{\theta } \biggr |_{\phi =\frac{\pi }{2}} + \left\langle \frac{T}{\sin \theta } \right\rangle _{\theta } \biggr |_{\phi =\frac{3\pi }{2}} \right] , \end{aligned}$$
$$\begin{aligned} u_y&= -\frac{1}{\pi a} \int _0^{2\pi } \mathrm {d}\phi \, \mu (\phi ) \sin \phi \left\langle T \sin \theta \right\rangle _{\theta }(\phi ), \end{aligned}$$
$$\begin{aligned} \varOmega _z&= \frac{3}{4 \pi a^2} (\mu _{\mathrm{ps}}- \mu _{\mathrm{au}}) \left[ \left\langle T \right\rangle _{\theta }|_{\phi =\frac{3\pi }{2}} - \left\langle T \right\rangle _{\theta }|_{\phi =\frac{\pi }{2}} \right] , \end{aligned}$$

where we have introduced the average over the normal angle \( \left\langle \bullet \right\rangle _{\theta } \equiv \frac{1}{2} \int _0^\pi \mathrm {d}\theta ~ \sin \theta (\bullet ) \). All other velocity components give zero contributions, as the detailed derivation of Eqs. (4)–(6) in “Appendix G” shows.

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 [51, 84]

$$\begin{aligned} \partial _t f = - {\varvec{{\mathcal {R}}}} \cdot \left( {\varvec{\varOmega }}- {D_\mathrm {r}}{\varvec{{\mathcal {R}}}} \right) f, \end{aligned}$$

for the dynamic probability density \(f(t,{\varvec{n}})\) to find the particle at time t with an orientation \({\varvec{n}}\) (relative to the heat source). The rotational operator \( {\varvec{{\mathcal {R}}}} \equiv {\varvec{n}} \times {\varvec{\nabla }}_n \) includes the nabla operator \({\varvec{\nabla }}_n\) with respect to the particle’s orientational degrees of freedom, and \(D_\mathrm {r}\) denotes the (effective [85, 86]) rotational diffusion coefficient. With the mentioned approximation of a strict in-plane orientation of the particle axis, Eq. (7) greatly simplifies [30] to \( \partial _t f = -\partial _\gamma J, \) with the flux

$$\begin{aligned} J(\gamma ,t) = - \varOmega _z(\gamma ) f(\gamma ,t) - {D_\mathrm {r}}\partial _\gamma f(\gamma ,t). \end{aligned}$$

To obtain a tractable steady-state solution (\(J=0\)) to the above equation, we further employ the following heuristic model for the angular velocity:

$$\begin{aligned} \varOmega _z(\gamma ) = \varOmega _1 \sin \gamma + \varOmega _2 \sin (2\gamma ), \end{aligned}$$

introducing the independent parameters \(\varOmega _{1,2}\). Including the term \(\varOmega _2 \sin (2\gamma )\) is a natural extension of \(\varOmega _z \propto (\mu _{\mathrm{ps}}- \mu _{\mathrm{au}})\sin \gamma \) for particles with isotropic heat conductivity [30], to account for the material heterogeneities of the Janus sphere. A similar expression for the angular velocity (9) was recently found in theoretical studies of interactions between two chemotactic active colloids [40] using perturbation series in powers of a/r, where r is the distance between the two particles. Similar (reflection) methods as presented in [31, 40, 87] might be employed to justify our heuristic model (9), and to establish a connection between \(\varOmega _{1,2}\) 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 broken symmetry of the particle’s rotational motion due to the two hemispheres’ distinct heat conductivities.

For an angular velocity \(\varOmega _z\) of the form (9) the stationary orientational distribution (solving Eq. (8) for \(J=0\)) reads

$$\begin{aligned} f(\gamma ) = N^{-1} \mathrm {exp} \left\{ \frac{\varOmega _1}{D_\mathrm {r}} \cos \gamma + \frac{\varOmega _2}{D_\mathrm {r}} \cos ^2\gamma \right\} , \end{aligned}$$

with a normalization factorFootnote 1N.

3.2 Experimental results

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 \( u = {2}\,{\upmu }\hbox {m}/\hbox {s} \) at a distance of \(r = {1.25}\,{\upmu }\hbox {m}\). 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 2b shows the mean angular velocity \({{\bar{\varOmega }}}\) 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 \( {{\bar{\varOmega }}} = {100}^{\circ }/\hbox {s} \) at short distances.

Fig. 2
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 \({{\bar{\varOmega }}}_z\) (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 \(r = {1.25}\,{\upmu }\hbox {m}\) 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 \(\varOmega _z\), data in the interval \(r = 1\)\({4}\,{\upmu }\hbox {m}\) 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, d) yield \(\mu _{\mathrm{ps}}= {2.88}\,{\upmu }\hbox {m}^{2}/\hbox {sK}\) and \(\mu _{\mathrm{au}}= {1.82}\,{\upmu }\hbox {m}^{2}/\hbox {sK}\) for the mobilities. The alternative fits shown in panel d) follow from Eq. (9) with \(\varOmega _{1,2}\) as independent fit parameters (dashed) and \(\varOmega _z = \varOmega _0 \sin \gamma \) [30], with \(\varOmega _0\) as fit parameter (dotted), and yield \(\varOmega _0 = {17.4}^{\circ }/\hbox {s}\), \(\varOmega _1= {17.2}^{\circ }/\hbox {s}\), \(\varOmega _2= {-3.25}^{\circ }/\hbox {s}\). The error bars in all panels have been estimated from the standard deviation of the mean values

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 Fig. 2c, d. For the translational phoretic speed u, we observe a clear minimum between \(\gamma ={50}^{\circ }\) and \(\gamma ={135}^{\circ }\). Local maxima are observed when the polymer side is facing the heat source (\(\gamma ={180}^{\circ }\)) or pointing away from it (\(\gamma ={0}^{\circ }\)). That the latter orientation displays a smaller speed suggests that the polymer side yields the major contribution.

Fig. 3
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 \(r={1.25}\,{\upmu }\hbox {m}\) between the Janus particle and the source: (a) \(\gamma ={0}^{\circ }\), (b) \(\gamma ={90}^{\circ }\), (c) \(\gamma ={180}^{\circ }\), (d) \(\gamma ={270}^{\circ }\)

In spite of averaging \(\varOmega _z\) over the measured distance range (\({1}\,{\upmu }\hbox {m}\)\({4}\,{\upmu }\hbox {m}\)) the \(\gamma \)-dependent angular velocity exhibits some residual scatter. It is still seen to vanish for \(\gamma ={0}^{\circ }\) and \(\gamma ={18}^{\circ }\) (Fig. 2d), in line with the expected symmetry of the temperature field around the axis of the Janus particle. At \(\gamma \approx {90}^{\circ }\), we observe a maximum angular and minimum translational speed.

To compare the experimental results to our theoretical expectations (4)–(9), 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 [88] (“Appendix 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 \(\varOmega _z\) yields our best estimate for \(\mu _{\mathrm{ps}}- \mu _{\mathrm{au}}\). 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 \( \mu _{\mathrm{ps}}={2.88}\,{\upmu }\hbox {m}^{2}/\hbox {sK} \) and \( \mu _{\mathrm{au}}= {1.82}\,{\upmu }\hbox {m}^{2}/\hbox {sK} \) for the phoretic mobilities. The theoretical fits are shown in Fig. 2c, d as solid lines, while the dashed line is a fit of Eq. (9) with \(\varOmega _{1,2}\) as independent fit parameters. It nicely reproduces the experimental data. In contrast, assuming the rotational speed to be of the form \(\varOmega _z \propto \sin \gamma \) [30] (dotted), as for homogeneous heat conductivity, misses the experimentally observed asymmetry. With respect to various neglected non-ideal effects that would seriously complicate the analysis (e.g., the vertically non-central position of the heat source and its effect onto the particle orientation, diverse wall effects, etc.), the obtained optimum values for the mobility coefficients should be regarded as “effective for the setup” rather than numerically precise for the molecular composition of the particle and solvent.

Fig. 4
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) \(1.1\,{\upmu }\hbox {m}\), (b) \(1.7\,{\upmu }\hbox {m}\), (c) \(2.3\,{\upmu }\hbox {m}\), (d) \(2.8\,{\upmu }\hbox {m}\). The solid lines are best fits by Eq. (10), with free fit parameters \(\varOmega _{1,2}/D_\mathrm {r}\). The dashed lines are fits by \(f \propto \mathrm {exp}[(\varOmega _0/D_\mathrm {r})\cos \gamma ]\) [30] for an angular speed profile \(\varOmega _z = \varOmega _0 \sin \gamma \) with \(\varOmega _0/D_\mathrm {r}\) as a free fit parameter. These fits yield for \(\varOmega _0/{D_\mathrm {r}}\), \(\varOmega _1/{D_\mathrm {r}}\), \(\varOmega _2/{D_\mathrm {r}}\) 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 \(0< \gamma < {180}^{\circ }\) in Fig. 2d, we measure a significantly higher probability to find the particle’s gold cap pointing toward 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 [78] induced by the temperature gradients along the particle’s surface via its physicochemical 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 unsymmetrical, orientation-dependent manner. Secondly, their generally unequal thermoosmotic 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. 3a, 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 [30] but matter for the proper interpretation 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 \(\mu _{\mathrm{ps}}\). 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 \( \mu _{\mathrm{ps}}- \mu _{\mathrm{au}}\), is of opposite sign and thus reduces the total propulsion velocity. (It disappears if \(\mu _{\mathrm{ps}}\approx \mu _{\mathrm{au}}\).)

Figure 5 illustrates the orientation dependence of the phoretic velocity components \(u_x\) 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 toward the heat source. Its smooth sign change at \(\gamma \approx {90}^\circ \) 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 \({90}^\circ \). 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 [30]. The transverse velocity component \(u_y\) naturally vanishes if the particle axis is aligned or anti-aligned with the heat source (\(\gamma = {0}^\circ \) and \( \gamma = {180}^\circ \)). It attains a maximum at \(\gamma \approx {120}^\circ \), 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 \(\gamma = {180}^\circ \) (ps-side facing the heat source) and only a lesser local maximum is seen at \(\gamma = {0}^\circ \) (au-side facing the heat source), in Fig. 2c.

Fig. 5
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. 2b, d, namely \(\mu _{\mathrm{ps}}={2.88}\,{\upmu }\hbox {m}/\hbox {sK}\) and \(\mu _{\mathrm{au}}={1.82}\,{\upmu }\hbox {m}/\hbox {sK}\) (solid lines); \(\mu _{\mathrm{ps}}={0.5}\,{\upmu }\hbox {m}/\hbox {sK}\) and \(\mu _{\mathrm{au}}={-0.55}\,{\upmu }\hbox {m}/\hbox {sK}\) (dashed); \(\mu _{\mathrm{ps}}={-1.82}\,{\upmu }\hbox {m}/\hbox {sK}\) and \(\mu _{\mathrm{au}}={-2.88}\,{\upmu }\hbox {m}/\hbox {sK}\) (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 \(\mu _{\mathrm{ps}}> \mu _{\mathrm{au}}> 0\). The first condition ensures the correct sign for the angular velocity according to Eq. (6) and Fig. 2d, and is in agreement with previous findings for thermo-osmotic interfacial flows [78]. The step in the phoretic mobility at the particle equator determines the magnitude and sign of the angular velocity [30]. While different absolute values can lead to the same step height \(\mu _{\mathrm{ps}}-\mu _{\mathrm{au}}\), 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 (\(\mu _{\mathrm{ps}}>0,~\mu _{\mathrm{au}}<0\) or \(\mu _{\mathrm{au}}<0<\mu _{\mathrm{ps}}\)). 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 \(\varOmega _z\) 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 \(\mu _{\mathrm{au}}\), 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 [89,90,91], 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 inter-particle interactions that dominate in dilute suspensions of thermophoretic microswimmers. To our knowledge, 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 toward 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. Our idealized 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 across 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 mimics the self-propulsion of a Janus swimmer in its self-generated temperature gradient, along its 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.