Supplementary Information Supplementary Figures

Supplementary Figure 1 |Definition of the angles of the microswimmer symmetry axis relative to the substrate and the side step.

, including the effect of gravity. Except for the size of the particle, all parameters are the same as in Fig. 2d in the main text.

Supplementary Figure 3 | 2.5
Janus microswimmer trajectories following well features for 227s; scale bar corresponds to 20 . Inset: 2.5 Janus microswimmer circling around a step for more than 89s; both tracks were recorded in 5 vol% H 2 O 2 , scale bar corresponds to 10 .
Supplementary Figure 4 | Wall-induced change of the particle velocity (a), for a particle with its cap oriented towards the wall ( = 0 ∘ ). Chemi-osmotic and gravitational effects are not included. (b), Same plot as in (a), but with a log-log scale. The red line shows a (ℎ/ ) scaling.
Supplementary Figure 5 | Lithography based creation of patterns a, Schematic representing the lithography based method to create submicron step and well structures for particle guidance. b, Optical images of the resulting patterns; the scale bar in the inset correspond to 10 , while the scale bars in the main images correspond to 500 .

SUPPLEMENTARY TABLES
Supplementary . "Bwds" indicates steady motion towards the cap, which is not observed experimentally. "Limit cycle" indicates one case with a sustained oscillation of small amplitude in ℎ/ and . "Unstable" indicates the presence of an unstable fixed point. In one case, there are clear signs of a "hovering" attractor with ℎ / below the numerical cut-off. Although we listed the amplitudes of the first three squirming modes, we note that the effective squirmer can also have ≠ 0 for > 3.

Supplementary Note 1: Calculation of Gravitational Contribution to Particle Motion
The particle experiences a gravitational force from the weight of the spherical silica core and platinum cap, as well as a gravitational torque from the bottom-heaviness imparted by the platinum cap. For a given height and orientation of the particle, we use the BEM to calculate the hydrodynamic resistance tensor of the particle. We then obtain the gravitational contributions and to the particle velocity as the product of the inverse of this tensor (i.e. the hydrodynamic mobility tensor) and the vector containing the six components of gravitational force and torque. (Our calculation of the gravitational contribution to velocity therefore includes the effect of hydrodynamic interaction with the planar wall.) The linearity of the Stokes equation allows one to sum the separate contributions of activity and gravity to determine the complete translational and angular velocities as = + and = + . However, the two sets of velocities must be expressed in the same units. The contributions from particle activity are obtained in terms of U ≝ |b |κ/D and Ω ≝ U /R, i.e., as /U and /Ω . We estimate U and Ω by taking U = 6 μm s -1 as a typical particle velocity. Within the neutral self-diffusiophoretic framework, U /U can be calculated analytically or numerically as a function of the material parameters of the particle, i.e. the extent of catalyst coverage and the spatial variation of surface mobility b ( ). 1,2 For instance, U /U = 0.25 for half coverage and uniform surface mobility. 3 For a given set of material parameters, we can therefore calculate the characteristic velocity U in dimensional units from U /U and U = 6 μm s -1 .

Supplementary Note 2: Isolation of contributions to particle motion
We seek to isolate and quantify the various physical contributions to the motion of a particle. We focus on the −component of the angular velocity Ω . We recall that the linearity of the Stokes equations permits us to solve for active, gravitational, and wall slip (chemi-osmotic) contributions separately and superpose them to obtain the full angular velocity: Ω = Ω + Ω + Ω . To obtain the active contribution, we use the Lorentz reciprocal theorem. This theorem allows the problem of determination of ( , ) to be related to six "primed" problems with the same geometry but different boundary conditions. We obtain six coupled equations: Where and are the force and torque, respectively, exerted by quiescent fluid on a particle in steady translation (or rotation) with a no-slip boundary condition on its surface. The index denotes steady translation in the , or ̂ direction for = 1, 2, 3, respectively, or steady rotation in , or ̂ for = 4, 5,6. Likewise, is the fluid stress tensor for the steadily translating or rotating particle. Further details concerning the derivation of Eq. 1 are provided in our previous work . We recall that = − ( )∇ || ( ). From Eq. 1, we find the component Here, the forces and torques = F , F , F and = τ , τ , τ have been compactly The contribution in Eq.
(2) to the angular velocity of the particle is shown in Fig. 3a of the main text.
We now write the activity-induced solute concentration ( ) = ( ) + ( ) with a free space component ( ) and the wall correction ( ). Similarly, we write = + and = + . Using these representations in the integral in Eq. 2, we may estimate various contributions to the particle rotation.
gives the contribution to Ω strictly from hydrodynamic interactions with the wall. It is plotted in Fig. 3b of the main text.
gives the contribution to Ω , shown in Fig.   3(c) of the main text strictly from wall-induced solute modifications. In other words, Ω , represents phoretic rotation of the particle from wall-induced concentration gradients. Note that this term is nonzero only when ≠ ; in the case considered here, Finally, the term Ω , = ∑ ( ) ( )∇ || ( ) • • is due to higher order coupling between the chemical and hydrodynamic effects of the wall. It is depicted in Fig. 3d of the main text. Interestingly, it is not necessarily small when the particle is close to the wall.

Supplementary Note 3: Hydrodynamics-only models
We turn to the wider question of whether the "squirmer" model, in which the interaction with the wall is purely hydrodynamic, can reproduce the experimentally observation for some set of parameters. We recall that the slip velocity of an axisymmetric squirmer can be written as follows: where is an angle defined with respect to the axis of symmetry, ( ) = √ ( ) ( ), ( ) is the Legendre polynomial of order , and is the amplitude of squirming mode . 5 An exhaustive search through the squirming mode amplitudes is beyond the scope of this work. Nevertheless, we can gain some insight by restricting our consideration to the first two squirming modes. The amplitude of the first mode is set by the free space swimming velocity . . = 6 -1 , since . . = 2/3 . We vary the ratio / , taking = 0 for > 2, and determine whether a sliding state emerges (i) in the absence of gravity and (ii) in the presence of gravity, which respectively represent (i) swimming near a side wall, and (ii) swimming above a substrate. The results are shown in Table 1 for = 2.5 . For situation (i), our results show good agreement with Gaffney and Ishimoto, 5 including the finding that sliding states emerge only for / ≥ 3.
We find that, for the parameters considered, the squirmer model cannot reproduce the experimental observation that a particle swims at ≈ 90 ∘ in both situation (i) and situation (ii). Gravity shifts the sliding states that occur for / ≥ 3 to ≈ 45 ∘ . This steady angle would be detectable experimentally as a large apparent coverage of the particles by catalyst. We conclude that, for the parameters considered in Supplementary Table 1, the effect of the force dipole (the strength of which is proportional to the amplitude of the second squirming mode) is too weak to balance the effect of gravity at ≈ 90 ∘ .
In constructing Supplementary Table 1, we made two simplifying and physically plausible assumptions: 1.) The contributions of the higher order modes ( > 2) to the disturbance velocity decay rapidly with distance from the swimmer, and hence contribute negligibly to interaction with the wall; and 2.) the ratio / ∼ (1). We can relax both assumptions by considering an "effective squirmer" obtained within our model for a self-diffusiophoretic swimmer. The effective squirmer is obtained for a given = / by neglecting the effect of the wall on the concentration field of a self-diffusiophoretic particle, i.e., by using . . as described in Supplementary Note 2. Additionally, chemi-osmotic effects are neglected. An effective squirmer could have, in principle, non-zero amplitude for all . Secondly, as shown in Supplementary Table 2, the ratio / ⟶ ∞ as ⟶ −1 from above. This is because, for ≈ −1, one face of the particle is attracted by solute, and the other repelled by solute; furthermore, the strengths of attraction and repulsion are approximately equal.
is therefore nearly zero, since it is proportional to the velocity of the particle.
Our results are in Supplementary Table 2. We find that, for the parameters studied, the effective squirmer cannot reproduce the experimental observations in both situation (i) and situation (ii

Supplementary Note 4: Range of interaction with wall
The interaction of a chemically active particle with a planar wall has a long-ranged character. However, we show that the amplitude (strength) of this interaction is very small except when the particle is close to the wall. This makes it difficult to detect its effects experimentally (at least with the equipment and techniques currently available to us) In Supplementary Figure 4, we plot the wall-induced change of the wall normal component of the selfdiffusiophoretic velocity of a particle that has its cap oriented towards the wall (θ = 0 • ). The subtraction of the free space self-diffusiophoretic velocity , = 0.1625 from isolates the effect of the wall. (Note that chemi-osmotic and gravitational contributions are not included in the figure.) At h/R = 5, the contribution to from the wall has already decayed to approximately three percent of the free space self-diffusiophoretic velocity. This change in speed is too small to be apparent when viewing an optical microscopy video of a particle near a step or a side wall. The effect of the wall is even weaker for orientations θ > 0 • .
Our numerical calculations recover the long-ranged character of the interaction. In Supplementary  Figure 4(b), we show the same plot as in (a), but with a log-log scale. Far away from the wall, Δ follows a (ℎ/ ) power law, which is shown as a red line. The effect of the wall on the solute field decays as 1/ , i.e., the leading order term representing the wall is an image point source. Since Δ is proportional to the wall-induced concentration gradient, it decays as 1/ .

Supplementary Note 5: Robustness of the sliding state
Finally, we comment on the robustness of the sliding states against thermal fluctuations, which were not included in our model. To this end we perform a standard linear stability analysis of our dynamical system, which may be written as {ℎ = (ℎ, ), = (ℎ, )}, at a fixed point ℎ , at which { ℎ , = 0, ℎ , = 0}. This amounts to determination of the eigenvalues , of the Jacobian matrix evaluated at (ℎ , ), i.e.