Active colloidal propulsion over a crystalline surface

First we describe the experimental system and how the activity of the particles may be tuned by changing the concentration of a chemical propellant, as shown in several previous reports [10, 11, 51]. An HCP monolayer consisting of spherical silica (${\mathrm{SiO}}_{2}$) microbeads (average diameter $d=2.07\,\mu {\rm{m}}$ with a coefficient of variation of 10%–15%, Bangs Laboratories) forms the periodic surface upon which the active colloids move; the lattice constant of the crystal is set by the particle diameter. The HCP monolayer was prepared with a Langmuir–Blodgett (LB) deposition technique [52] and covered an entire silicon wafer. A scanning electron microscope (SEM) image of the monolayer can be seen in figure 1(b), and the actual topography of the surface is inferred from the atomic force microscope (AFM) image in figure 1(d).

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The silica microspheres were first functionalized with allyltrimethoxysilane then dispersed in chloroform. This colloidal suspension was then distributed over the air–water interface of an LB trough. A cleaned silicon wafer is dipped into the trough and, upon slowly pulling out the wafer, the monolayer is compressed to form a close-packed assembly. This process transfers the monolayer from the air–water interface to the silicon wafer. The wafer is then dried and treated with air plasma to remove any organic impurities before the experiments. While the LB technique yields large area HCP monolayers of silica beads, microscopic line defects can result from the lattice mismatch between adjacent self-assembled colloidal crystals [53]. In order to ensure consistency of the underlying substrate topography, the lattice experiments were carried out on the same piece of wafer by varying the peroxide concentration for the same batch of particles.

The active colloids were fabricated by evaporating a 2 nm Cr adhesion layer followed by 5 nm of Pt onto microbeads of the same type as used for the monolayer; see figure 1(a) for an SEM image. The thus formed Janus spheres were then suspended into ${{\rm{H}}}_{2}{\rm{O}}$ and subsequently pipetted onto an HCP lattice surface (figure 1(c)). The Pt on the Janus particle catalyzes the decomposition of hydrogen peroxide (${{\rm{H}}}_{2}{{\rm{O}}}_{2}$) and thus gives rise to self-propulsion [11]. The strength of the propulsion was altered by adding different concentrations of aqueous ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ to the colloidal suspension, covering concentrations between 0% and 6% (v/v). For each concentration, trajectories from 10 randomly chosen Janus particles were recorded for 100 s at a frame rate of 10 fps with a Zeiss AxioPhot microscope in reflection mode with a 20× objective coupled to a CCD camera (pixel size $5.5\,\mu {\rm{m}}\times 5.5\,\mu {\rm{m}}$, resolution 2048 × 1088).

We computed time-averaged mean-square displacements (MSDs) of 10 trajectories for each ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration, and by averaging the MSDs at each lag time we obtain the 'averaged MSD' and its standard error. Data fitting was performed with the software OriginLab (OriginLab Corp., Northampton, MA) using a Levenberg–Marquadt iteration algorithm. Due to the linearly spaced time grid, the data points accumulate in the double-logarithmic representation at large times. To account for the different density of data points at short and long lag times on logarithmic scales, we used $1/t$ as a weighting factor. The fits to equations (2) and (10), respectively, were then performed simultaneously for all 10 data sets of each concentration such that the different scatter of the data points enters the error estimate of the fit parameters. The free diffusivity D₀ was fixed initially to its value for the passive particle moving over a smooth surface and was slightly adjusted afterwards for each ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration to obtain the best match with the averaged MSD curves.

Under gravity the Janus particles settle onto the substrate; once settled, Brownian motion leads to effectively two-dimensional diffusion in the gravitational potential imposed by the surface. The potential exhibits a periodic, hexagonal structure of potential wells with adjacent energy minima separated by a distance $d/\sqrt{3}$. A series of 'hops' are observed between adjacent energy minima, or in analogy to surface diffusion, adjacent 'adsorption sites.' Figure 2(a) schematically demonstrates a single hop from one minimum to an adjacent one. A successful hop requires the Janus particle to overcome an energy barrier of height ${E}_{{\rm{a}}}$, as depicted in figure 2(b).

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The gravitational potential $U({\boldsymbol{x}})={\rm{\Delta }}mgz({\boldsymbol{x}})$ is given by the buoyant mass ${\rm{\Delta }}m$ of the Janus particle, the acceleration g due to gravity, and the height profile $z({\boldsymbol{x}})$ at the two-dimensional position ${\boldsymbol{x}}$. The energy barrier between adjacent potential minima is thus ${E}_{{\rm{a}}}={\rm{\Delta }}mg{\rm{\Delta }}z$, where ${\rm{\Delta }}z$ follows from elementary geometry as shown in figure 2(c) for the configurations of maximal and minimal height. At the barrier maximum (left column of figure 2(c)), the centers of two substrate particles and the mobile one form an equilateral triangle. The Janus particle is thus elevated by ${z}_{{\rm{\max }}}=\sqrt{3}d/2$ above the centers of the substrate particles. If the Janus particle is found in a potential minimum, the centers of three substrate particles and the mobile one form a regular tetrahedron, thus ${z}_{{\rm{\min }}}=\sqrt{6}d/3$. For a successful hop, the particles must overcome a geometric barrier of height ${\rm{\Delta }}z={z}_{{\rm{\max }}}-{z}_{{\rm{\min }}}\approx d/20$, which evaluates to ${\rm{\Delta }}z\approx 100\,\mathrm{nm}$ for $d=2\,\mu {\rm{m}}$. This is consistent with the surface's height profile obtained from AFM, see figure 1(d).

The second ingredient to the energy barrier E_a is the total force on the Janus particle, which results from the competition of gravitation and buoyancy, i.e. the energy barrier is also a function of the material from which the Janus particle is made. Let us first consider a silica sphere which has no metal coating. Then the buoyant mass is ${\rm{\Delta }}m={\rm{\Delta }}\varrho {V}_{{\mathrm{SiO}}_{2}}$, where ${\rm{\Delta }}\varrho ={\varrho }_{{\mathrm{SiO}}_{2}}-{\varrho }_{{{\rm{H}}}_{2}{\rm{O}}}$ is the difference in density between ${\mathrm{SiO}}_{2}$ and ${{\rm{H}}}_{2}{\rm{O}}$, and ${V}_{{\mathrm{SiO}}_{2}}=(4/3)\pi {a}^{3}$ is the volume of the fluid displaced by the particle of radius $a=d/2$. For a bare ${\mathrm{SiO}}_{2}$ bead of 2 μm diameter, this yields ${E}_{{\rm{a}}}=1.7\,{k}_{{\rm{B}}}T$ in terms of thermal energy ${k}_{{\rm{B}}}T$.

In the case of the Janus particle, the asymmetric distribution of the metallic coating needs to be taken into account. Even though the volume of the cap is small in comparison to that of the bead, it has a significant effect on E_a due to the higher density of $\mathrm{Pt}$. Following [54], we model the hemispherical cap as ellipsoidal in shape; the thickness ${\rm{\Delta }}a$ of the deposited metal is largest at the top of the sphere and tapers to zero at the equator. This assumption is justified from the deposition process, which delivers the atoms in the vapor plume ballistically to the surface of the sphere. The volume of the cap then reads ${V}_{\mathrm{Pt}}=[(4/3)\pi {a}^{2}(a+{\rm{\Delta }}a)-(4/3)\pi {a}^{3}]/2=(2/3)\pi {a}^{2}{\rm{\Delta }}a$, and with this, the buoyant mass of the Janus particle is ${\rm{\Delta }}m=({\varrho }_{{\mathrm{SiO}}_{2}}-{\varrho }_{{{\rm{H}}}_{2}{\rm{O}}}){V}_{{\mathrm{SiO}}_{2}}+({\varrho }_{\mathrm{Pt}}-{\varrho }_{{{\rm{H}}}_{2}{\rm{O}}}){V}_{\mathrm{Pt}}$. Adopting a value of ${\rm{\Delta }}a=5\,\mathrm{nm}$ as the maximal thickness of the $\mathrm{Pt}$ cap and using ${\varrho }_{\mathrm{Pt}}=21.4\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, we estimate ${E}_{{\rm{a}}}={\rm{\Delta }}mg{\rm{\Delta }}z\approx 2.1{k}_{{\rm{B}}}T$ for the energy barrier of the Janus particle.

For the Janus particle moving passively over a smooth plane, we have measured for the translational diffusion constant ${D}_{0}=0.13\,\mu$m² s⁻¹, which implies a translational (hereafter indicated by the subscript 'T') hydrodynamic friction of ${\zeta }_{{\rm{T}}}={k}_{{\rm{B}}}T/{D}_{0}=3.2\times {10}^{-8}\,\mathrm{Pa}\,{\rm{s}}\,{\rm{m}}$ at $T=298\,{\rm{K}}$. As expected, the presence of a surface increases the hydrodynamic friction compared to unbounded motion: comparing with the Stokes friction ${\zeta }_{{\rm{T}}}^{{\rm{St}}}=6\pi \eta a\approx 1.74\times {10}^{-8}\,\mathrm{Pa}\,{\rm{s}}\,{\rm{m}}$ in ${{\rm{H}}}_{2}{\rm{O}}$ ($\eta =0.89\,\mathrm{mPa}\,{\rm{s}}$), we find ${\zeta }_{{\rm{T}}}\approx 1.8\,{\zeta }_{{\rm{T}}}^{{\rm{St}}}$. For a planar surface, the friction coefficient ${\zeta }_{{\rm{T}}}$ of a sphere of radius a dragged parallel to the surface at a distance h from the sphere center obeys Faxén's famous result [15, 55]:

$$\begin{eqnarray}&&\displaystyle \frac{{\zeta }_{{\rm{T}}}^{{\rm{St}}}}{{\zeta }_{{\rm{T}}}}\simeq 1-\displaystyle \frac{9}{16}\displaystyle \frac{a}{h}+\displaystyle \frac{1}{8}{\left(\displaystyle \frac{a}{h}\right)}^{3}-\displaystyle \frac{45}{256}{\left(\displaystyle \frac{a}{h}\right)}^{4}-\displaystyle \frac{1}{16}{\left(\displaystyle \frac{a}{h}\right)}^{5},\quad a\ll h.\end{eqnarray} \tag{ 1 }$$

Inserting the above experimental value for ${\zeta }_{{\rm{T}}}$ and solving for h with $a=1\,\mu {\rm{m}}$, we obtain $h\approx 1.3\,\mu {\rm{m}}$ leaving a gap of $h-a\approx 0.3\,\mu {\rm{m}}$ between the two surfaces of the Janus particle and the planar substrate.

Note that Faxén's calculation relies on a far-field expansion of the flow field and is justified only for small ratios a/h. As can be seen a posteriori we have $a/h\approx 0.8$, implying slow convergence. Indeed, truncating after the 3rd order in a/h yields an unphysical $h\lt a$, which is fixed by the 4th order term. The 5th order term contributes merely a relative correction of 2%, which suggests convergence of the series. Furthermore, we have compared this far-field estimate with the predictions from lubrication theory [55] and a rigorous series expansion [56] accurate at all separations of the particle and the surface, including the singular limit of close approach. This cross-check shows that our present experimental situation is still outside the lubrication regime and equation (1) provides very good estimates of the friction coefficient and the elevation h. In the following, we anticipate that the translational friction ${\zeta }_{{\rm{T}}}$ does not change appreciably for the range of ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentrations used, although it may be modified for active motion due to altered boundary conditions at the colloid's surface.

The behavior of the active colloids moving across the crystalline surface is significantly different from the planar case. The catalyzed chemical reaction on the $\mathrm{Pt}$ side leads to two effects described first qualitatively: (i) similar to the planar case, the Janus particles are actively and directionally propelled over the surface away from the catalyst coating [57]. (ii) In contrast to the planar case, motion over the crystalline surface is hindered by the particle becoming transiently trapped within potential wells.

According to the above reasoning, the active Brownian motion in the periodic potential $U({\boldsymbol{x}})$ may be modeled in a simplified way by an over-damped Langevin equation:

$$\begin{eqnarray}&&\dot{{\boldsymbol{x}}}(t)={\boldsymbol{v}}(t)-{\zeta }_{{\rm{T}}}^{-1}{\rm{\nabla }}U({\boldsymbol{x}}(t))+\sqrt{2{D}_{0}}\,{{\boldsymbol{\xi }}}_{{\rm{T}}}(t).\end{eqnarray} \tag{ 3 }$$

Here, ${\boldsymbol{x}}(t)$ and ${\boldsymbol{v}}(t)$ are, respectively, the vectors of the particle position and of the propulsion velocity projected onto the plane of the crystalline surface. The latter is incorporated via the second term on the right hand side of equation (3). Further, ${{\boldsymbol{\xi }}}_{{\rm{T}}}(t)$ is a two-dimensional Gaussian white noise of zero mean and unit (co-)variance, $\langle {{\boldsymbol{\xi }}}_{{\rm{T}}}(t)\otimes {{\boldsymbol{\xi }}}_{{\rm{T}}}(s)\rangle ={\bf{I}}\,\delta (t-s)$ with ${\bf{I}}$ the identity tensor, to describe passive Brownian motion over a planar surface with diffusion coefficient ${D}_{0}={k}_{{\rm{B}}}T/{\zeta }_{{\rm{T}}}$.

The partial case of passive Brownian diffusion taking place in a periodic surface potential is described by equation (3) with vanishing self-propulsion, ${\boldsymbol{v}}=0$:

$$\begin{eqnarray}&&{\dot{{\boldsymbol{x}}}}_{{\rm{c}}}(t)=-{\zeta }_{{\rm{T}}}^{-1}{\rm{\nabla }}U({{\boldsymbol{x}}}_{{\rm{c}}}(t))+\sqrt{2{D}_{0}}\,{{\boldsymbol{\xi }}}_{{\rm{T}}}(t),\end{eqnarray} \tag{ 4 }$$

where the subscript 'c' stands for crystalline surface. The corresponding MSD, ${\rm{\Delta }}{R}_{{\rm{0,c}}}^{2}(t)=\langle | {{\boldsymbol{x}}}_{{\rm{c}}}(t)-{{\boldsymbol{x}}}_{{\rm{c}}}(0){| }^{2}\rangle$, is well approximated by a simple exponential memory [58, 59], which manifests itself in the velocity autocorrelation function (VACF) as

$$\begin{eqnarray}&&{Z}_{{\rm{0,c}}}(t):= \displaystyle \frac{1}{4}\displaystyle \frac{{{\rm{d}}}^{2}}{{{\rm{d}}{t}}^{2}}{\rm{\Delta }}{R}_{{\rm{0,c}}}^{2}(t)={D}_{0}\,\delta (t-{0}^{+})-\displaystyle \frac{{\rm{\Delta }}{D}_{{\rm{c}}}}{{\tau }_{{\rm{c}}}}{{\rm{e}}}^{-t/{\tau }_{{\rm{c}}}}.\end{eqnarray} \tag{ 5 }$$

Here, ${\tau }_{{\rm{c}}}$ is the longest relaxation time of the process and ${\rm{\Delta }}{D}_{{\rm{c}}}\gt 0$ describes the reduction of the long-time diffusivity due to the presence of the periodic surface relative to the planar case. This ansatz for the VACF corresponds to keeping only the largest non-zero eigenvalue of the Smoluchowski operator [60]. For the MSD, one readily calculates:

$$\begin{eqnarray}&&{\rm{\Delta }}{R}_{{\rm{0,c}}}^{2}(t)=4{\int }_{0}^{t}(t-s)\,{Z}_{{\rm{0,c}}}(s)\,{\rm{d}}s=4({D}_{0}-{\rm{\Delta }}{D}_{{\rm{c}}})t-4{\rm{\Delta }}{D}_{{\rm{c}}}{\tau }_{{\rm{c}}}({{\rm{e}}}^{-t/{\tau }_{{\rm{c}}}}-1).\end{eqnarray} \tag{ 6 }$$

It describes a simple crossover from free, unconfined diffusion with ${D}_{0}={k}_{{\rm{B}}}T/{\zeta }_{{\rm{T}}}$ at short times ($t\ll {\tau }_{{\rm{c}}}$) to diffusion at long times with a reduced diffusion constant ${D}_{{\rm{0,c}}}={D}_{0}-{\rm{\Delta }}{D}_{{\rm{c}}}\leqslant {D}_{0}$ for $t\gg {\tau }_{{\rm{c}}}$. The crossover timescale ${\tau }_{{\rm{c}}}$ describes the time after which the particle has explored a single potential minimum. Its inverse, ${\tau }_{{\rm{c}}}^{-1}$, may be interpreted as the attempt rate for escaping from the potential well [60, 61].

Next, we combine this result for passive motion in a periodic potential with active motion under the approximation that the diffusive motion in the potential be independent of the direction of the propulsion velocity. More precisely, we neglect terms of the form $\langle {\rm{\nabla }}U({\boldsymbol{x}}(t))\cdot {\boldsymbol{v}}(0)\rangle ;$ merely the increment ${{\boldsymbol{\xi }}}_{{\rm{T}}}(t)$ is strictly independent of ${\boldsymbol{v}}(t)$. Then, the autocorrelation of the propulsion velocity is simply added to the VACF for passive diffusion,

$$\begin{eqnarray}&&{Z}_{{\rm{a,c}}}(t)={Z}_{{\rm{0,c}}}(t)+\tfrac{1}{2}\langle {\boldsymbol{v}}(t)\cdot {\boldsymbol{v}}(0)\rangle .\end{eqnarray} \tag{ 7 }$$

Rotational diffusion of the cap orientation determines the propulsion velocity vector ${\boldsymbol{v}}(t)$ projected onto the surface plane. Neglecting the small gravitational torque on the present Janus particles (see also section 4.3), it follows that

$$\begin{eqnarray}&&\langle {\boldsymbol{v}}(t)\rangle =0,\quad \langle {\boldsymbol{v}}(t)\cdot {\boldsymbol{v}}(0)\rangle ={v}^{2}{{\rm{e}}}^{-t/{\tau }_{{\rm{rot}}}},\end{eqnarray} \tag{ 8 }$$

with ${\tau }_{{\rm{rot}}}$ the persistence time of the orientation; for free three-dimensional rotation ${\tau }_{{\rm{rot}}}={(2{D}_{{\rm{rot}}})}^{-1}$. With this, the MSD of an active particle moving atop a crystalline surface follows from equations (5) and (7), again by integration:

$$\begin{eqnarray}&&{\rm{\Delta }}{R}_{{\rm{a,c}}}^{2}(t)=4\left({D}_{0}-{\rm{\Delta }}{D}_{{\rm{c}}}+\tfrac{1}{2}{v}^{2}{\tau }_{{\rm{rot}}}\right)t-4{\rm{\Delta }}{D}_{{\rm{c}}}\,{\tau }_{{\rm{c}}}({{\rm{e}}}^{-t/{\tau }_{{\rm{c}}}}-1)+2{v}^{2}{\tau }_{{\rm{rot}}}^{2}({{\rm{e}}}^{-t/{\tau }_{{\rm{rot}}}}-1).\end{eqnarray} \tag{ 9 }$$

In section 4.3, this prediction will be checked against simulations.

Figure 4 shows the averaged MSD data of Janus particles being actively propelled atop the crystalline surface for six ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentrations. As expected, increasing the ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration leads to higher observed propulsion speeds and higher long-time diffusion. The latter can be directly inferred from the rectification ${\rm{\Delta }}{R}_{{\rm{a,c}}}^{2}(t)/t$ displayed in the inset of figure 4. The data suggest further a monotonic dependence on time, either decreasing or increasing depending on the concentration of fuel, which we interpret as a competition of the suppression of diffusivity due to the potential landscape with the enhancement due to active motion.

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Fitting equation (9) to the experimental MSD data would, in principle, provide an estimate for the parameters ${\rm{\Delta }}{D}_{{\rm{c}}}$, v, ${\tau }_{{\rm{c}}}$, ${\tau }_{{\rm{rot}}}$. Following this approach, it turned out all of the parameters depend on the ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration. Specifically, fixing the values of v and ${\tau }_{{\rm{rot}}}$ to those from the experiments with the planar surface does not produce satisfying fits. Further, the four-parameter fits suggest similar values for ${\tau }_{{\rm{c}}}$ and ${\tau }_{{\rm{rot}}}$, which motivated us to merge both timescales into a single parameter, τ. This is consistent with the absence of any minimum or maximum at intermediate lag times in the data for ${\rm{\Delta }}{R}_{{\rm{a,c}}}^{2}(t)/t$, which would be supported by equation (9). However, ${\tau }_{{\rm{c}}}\approx {\tau }_{{\rm{rot}}}$ implies that the parameters ${\rm{\Delta }}{D}_{{\rm{c}}}$ and v are no longer independent, merely the combination ${\rm{\Delta }}D={\rm{\Delta }}{D}_{{\rm{c}}}-{v}^{2}\tau /2$ can be obtained. Thus, equation (9) reduces to a simplistic, effective model of the MSD of a self-propelled particle atop a periodic surface,

$$\begin{eqnarray}&&{\rm{\Delta }}{R}_{{\rm{a,c}}}^{2}(t)\approx 4{D}_{{\rm{a,c}}}t+4({D}_{{\rm{a,c}}}-{D}_{0})\tau \,({{\rm{e}}}^{-t/\tau }-1).\end{eqnarray} \tag{ 10 }$$

By construction, this result has the same form as equations (2) and (6), but the interpretation of the parameters is different in each case. For long times ($t\gg {\tau }_{{\rm{c}}},{\tau }_{{\rm{rot}}}$), the MSD increases linearly, and the combination ${D}_{{\rm{a,c}}}:= {D}_{0}-{\rm{\Delta }}{D}_{{\rm{c}}}+{v}^{2}\tau /2$ is the long-time diffusion coefficient on the crystalline surface.

We used equation (10) to fit the MSD data with D₀, ${D}_{{\rm{a,c}}}$, and τ as free parameters. The results obtained for each concentration are given in table 2 and the fits shown as solid curves in figure 4 provide a consistent description of the data. We again observe a slight variability of D₀ and a strong dependence of τ on the ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration. The enhancement of the long-time diffusivity ${D}_{{\rm{a,c}}}$ with increasing ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentration is much less pronounced compared to the case of a planar surface (${D}_{{\rm{a,p}}}$ in table 1), which is a direct consequence of the trapping potential. The experimental values for the ratio ${D}_{{\rm{a,c}}}(c)/{D}_{{\rm{a,p}}}(c)$ vary between 0.5 and 0.1 and suggest a decrease towards higher concentrations c. We note that the MSD for $c=0.1 \%$ seems to deviate from the overall trend, and we exclude this data set from the remaining discussion.

As an independent check of the above approximate predictions and to gain further insight, we finally proceed to a refined theoretical model that explicitly includes both the translational and rotational degrees of freedom. The over-damped dynamics of an active bottom-heavy microswimmer [62] sedimenting due to gravity onto an HCP monolayer can effectively be described by Langevin equations for the projected position ${\boldsymbol{x}}=(x,y)$ and the orientation ${\boldsymbol{u}}=({u}_{x},{u}_{y},{u}_{z})$ [63, 64]:

$$\begin{eqnarray}&&\dot{{\boldsymbol{x}}}(t)={v}_{0}{{\boldsymbol{u}}}_{\parallel }(t)-{\zeta }_{{\rm{T}}}^{-1}{\rm{\nabla }}U({\boldsymbol{x}}(t))+\sqrt{2{D}_{{\rm{T}}}}\,{{\boldsymbol{\xi }}}_{{\rm{T}}}(t),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\dot{{\boldsymbol{u}}}(t)=[{\zeta }_{{\rm{R}}}^{-1}{\boldsymbol{T}}+\sqrt{2{D}_{{\rm{R}}}}\,{{\boldsymbol{\xi }}}_{{\rm{R}}}(t)]\times {\boldsymbol{u}}(t)-2{D}_{{\rm{R}}}{\boldsymbol{u}}(t),\end{eqnarray} \tag{ 12 }$$

where Itō's interpretation of the white noise is utilized for the second equation. The first equation describes translational motion parallel to the surface and simply reproduces equation (3), where we specify the propulsion term as ${\boldsymbol{v}}={v}_{0}{{\boldsymbol{u}}}_{\parallel }$. Here, v₀ is the propulsion strength and ${{\boldsymbol{u}}}_{\parallel }:= ({u}_{x},{u}_{y})$ is the orthogonal projection of the three-dimensional unit vector ${\boldsymbol{u}}$ onto the xy-plane. The second equation governs the cap orientation ${\boldsymbol{u}}$ of the Janus particle, in which ${\boldsymbol{T}}={{mr}}_{0}{\boldsymbol{u}}\times {\boldsymbol{g}}$ is the gravitational torque with $m={V}_{{\mathrm{SiO}}_{2}}{\varrho }_{{\mathrm{SiO}}_{2}}+{V}_{\mathrm{Pt}}{\varrho }_{\mathrm{Pt}}$ the mass of the particle and r₀ the displacement of the center of mass from the center of the sphere due to the heavy cap; for the Janus particle used in the present experiments, ${r}_{0}\approx 0.02\,a$. The strength of thermal fluctuations is determined by ${D}_{{\rm{T}}}={k}_{{\rm{B}}}T/{\zeta }_{{\rm{T}}}$ and ${D}_{{\rm{R}}}={k}_{{\rm{B}}}T/{\zeta }_{{\rm{R}}}$, the diffusivities of the translational and rotational motions, respectively, with ${{\boldsymbol{\xi }}}_{{\rm{T}}}$ and ${{\boldsymbol{\xi }}}_{{\rm{R}}}$ being independent Gaussian white noises having zero mean and unit covariance. The effective substrate potential $U(x,y)={\rm{\Delta }}mgz(x,y)$ arises from the buoyancy-corrected gravitational force ${\rm{\Delta }}mg$ on the Janus particle with the height of its center given by the landscape $z(x,y)$. The latter is created by the HCP colloidal monolayer and is composed of the upper non-intersecting segments of spheres,

$$\begin{eqnarray}&&z(x,y)=\sqrt{{(h+a)}^{2}-{(x-{x}_{i})}^{2}-{(y-{y}_{i})}^{2}},\end{eqnarray} \tag{ 13 }$$

located at the centers $({x}_{i},{y}_{i})$ of the monolayer particles, which form a hexagonal lattice of lattice constant $d=2a=2\,\mu {\rm{m}}$, see figure 2(a). For the sphere radius, we use $h+a\approx 2.3\,\mu {\rm{m}}$ as estimated in section 2.4, see also figure 2(d). Such an effectively two-dimensional representation requires that the particle height adjusts rapidly to the changes of the landscape, which is the case if the propulsion velocity v₀ is sufficiently smaller than the sedimentation velocity; ${v}_{{\rm{sed}}}={\rm{\Delta }}mg/{\zeta }_{{\rm{T}}}^{\perp }\approx 2.8\,\mu {\rm{m}}\,{{\rm{s}}}^{-1}$ for the current experiment.

The presence of the surface also modifies the rotational hydrodynamic friction. With the above value of the elevation h, we estimate for the rotational diffusion constants, which are distinct for rotations parallel and perpendicular to the surface [15, 65]:

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{{\rm{R}}}^{\parallel }}{{D}_{{\rm{R}}}^{{\rm{St}}}}\approx 1-\displaystyle \frac{5}{16}{\left(\displaystyle \frac{a}{h}\right)}^{3},\qquad \displaystyle \frac{{D}_{{\rm{R}}}^{\perp }}{{D}_{{\rm{R}}}^{{\rm{St}}}}\approx 1-\displaystyle \frac{1}{8}{\left(\displaystyle \frac{a}{h}\right)}^{3},\end{eqnarray} \tag{ 14 }$$

where ${D}_{{\rm{R}}}^{{\rm{St}}}={k}_{{\rm{B}}}T/8\eta \pi {a}^{3}$. Again, these far-field expressions for the rotational friction are sufficiently accurate for the given experimental conditions, which we have checked by comparing to [56]. For colloidal motion in close proximity to a surface, the translational and rotational degrees of freedom are, in general, coupled through hydrodynamic fields of the solvent. This coupling is quantified by off-diagonal terms $\pm {\zeta }_{{\rm{TR}}}$ in the grand resistance matrix [15]. Adopting the far-field result, ${\zeta }_{{\rm{TR}}}/6\pi \eta {a}^{2}\simeq {(a/h)}^{4}/8$ [55], one can show that the eigenvalues of the dimensionless grand resistance matrix are perturbed by a change of ${(a/h)}^{7}/36$ relative to 1. Thus, not only by the smallness of this term, but also by asymptotic consistency, we conclude that the translation–rotation coupling can safely be neglected here. Further, for the sake of simplicity and since we are only interested in the correlation $\langle {\boldsymbol{u}}(t)\cdot {\boldsymbol{u}}(0)\rangle$, we use the average rotational diffusion constant

$$\begin{eqnarray}&&{D}_{{\rm{R}}}:= \displaystyle \frac{1}{3}(2{D}_{{\rm{R}}}^{\parallel }+{D}_{{\rm{R}}}^{\perp })=\left[1-\displaystyle \frac{1}{4}{\left(\displaystyle \frac{a}{h}\right)}^{3}\right]{D}_{{\rm{R}}}^{{\rm{St}}},\end{eqnarray} \tag{ 15 }$$

such that ${D}_{{\rm{R}}}\approx 0.15\,{{\rm{s}}}^{-1}$ and ${\tau }_{{\rm{rot}}}={(2{D}_{{\rm{R}}})}^{-1}\approx 3.3\,{\rm{s}}$. With this, the model (equations (11) and (12)) is fully specified with v₀ as a control parameter and is solved numerically with the standard Euler–Maruyama scheme. We have validated the numerical scheme for the planar case (U = 0) by comparing simulated MSDs with the exact solution, equation (2), for this case.

First of all we note that equation (12) for the orientational dynamics does not include the position ${\boldsymbol{x}}(t)$ and thus can be solved independently of equation (11). Yet, it remains analytically challenging due to the gravitational torque term, ${\boldsymbol{T}}\ne 0$. The numerical results suggest that the dynamics of the orientation is well approximated by the free solution $\langle {\boldsymbol{u}}(t)\rangle =0$ and $\langle {\boldsymbol{u}}(t)\cdot {\boldsymbol{u}}(0)\rangle ={{\rm{e}}}^{-2{D}_{{\rm{R}}}t}$ and therefore ${\tau }_{{\rm{rot}}}={(2{D}_{{\rm{R}}})}^{-1}$. Thus, the problem described by equations (11) and (12) becomes equivalent to the model defined by equation (3) with the velocity correlation prescribed via equation (8). Note that although the orientation of the particle is three-dimensional, the translational motion of the particle is essentially restricted to the horizontal plane. As a result, for the mean-square velocity entering equation (8) we have ${v}^{2}:= \langle {({v}_{0}{{\boldsymbol{u}}}_{\parallel })}^{2}\rangle =2{v}_{0}^{2}/3.$ The latter step tacitly assumes that all orientations of ${\boldsymbol{v}}$ are equally probable, which is fulfilled under the approximation of free rotational diffusion.

Simulations of the active motion of the Janus particle in the hexagonal landscape, equations (11) and (12), were performed for different values of v₀ with all other parameters fixed to their values as mentioned above. The obtained MSDs shown in figure 5 capture the trends of the experimental data (figure 4) semi-quantitatively and reproduce the long-time diffusion coefficients for propulsion velocities $v=\sqrt{2/3}{v}_{0}$ similar to the experimental values.

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Finally, the numerical solutions permit a number of insights into the analytical predictions for the MSD. First, simulations addressing the passive motion above the crystalline surface show that equation (6) is a very good approximation for the experimental regime investigated here. With ${D}_{{\rm{T}}}=0.13\,\mu {{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$ and ${D}_{{\rm{R}}}=0.15\,{{\rm{s}}}^{-1}$, excellent quantitative agreement between the simulated MSD and the theory is found for the parameters ${\tau }_{{\rm{c}}}\approx 0.18\,{\rm{s}}$ and ${\rm{\Delta }}{D}_{{\rm{c}}}\approx 0.055\,\mu {{\rm{m}}}^{2}\,{{\rm{s}}}^{-1}$. Second, simulations of active motion above the crystalline surface indicate that the approximation of equation (9) does not work universally. For the considered range of propulsion velocities v₀, the simulated MSDs are significantly overestimated by equation (9) if ${\rm{\Delta }}{D}_{{\rm{c}}}$ and ${\tau }_{{\rm{c}}}$ are fixed to their values for passive motion (${v}_{0}=0$). For the cases where ${\rm{\Delta }}{R}_{{\rm{a,c}}}^{2}(t)/t$ is monotonic in t, it is, however, possible to obtain good descriptions if we allow ${\rm{\Delta }}{D}_{{\rm{c}}}$ and ${\tau }_{{\rm{c}}}$ to depend on the activity, v₀, and estimate them by the fit for each value of v₀. Third, for these monotonic cases we have found that equation (10) is a good approximation to the MSD, see figure 5 and table 3. On one hand, it provides less flexibility because it contains fewer parameters, in particular it allows for only a single crossover. On the other hand, this is sufficient to capture the full time-dependence in these cases and the formula is simpler to handle than equation (9). Note that for high ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentrations, when ${D}_{{\rm{a,c}}}$ is larger than D₀, the crossover time saturates, $\tau \approx {\tau }_{{\rm{rot}}}$. For low ${{\rm{H}}}_{2}{{\rm{O}}}_{2}$ concentrations, when ${D}_{{\rm{a,c}}}\simeq {D}_{0}$, τ may differ from ${\tau }_{{\rm{R}}}$ quite significantly.