Diffusion-based height analysis reveals robust microswimmer-wall separation

Microswimmers typically move near walls and can be strongly influenced by them. However, direct experimental measurements of swimmer-wall separation remain elusive to date. Here, we demonstrate that this separation can be obtained from the height dependence of the passive component of the swimmer's mean-squared displacement. We apply our approach to catalytic microswimmers and find that they exhibit"ypsotaxis", a tendency to assume a fixed height above the wall for a range of salt concentrations, swimmer surface charges, and swimmer sizes. Our results show that nearby walls play an important role in controlling their motion, and provide new insights for modelling their propulsion mechanism.

Confining surfaces, such as planar walls, have a farreaching impact in the microswimmer world, often ensuring microswimmer function and survival [1]. Encounters with surfaces give rise to accumulation, as seen for sperm [2], algae [3] and bacteria [4], and enable the formation of bacterial biofilms that facilitate their spreading, cooperation, and capture of nutrients [5][6][7]. Moreover, surfaces can significantly modify swimming trajectories; e.g., bacteria often exhibit circular motion with direction controlled by the boundary condition [8][9][10][11], in stark contrast to their run-and-tumble motion in bulk.
Striking surface effects are not only found in biological systems, but are also present for synthetic microswimmers [12][13][14][15][16][17]. Model catalytic colloidal swimmers exhibit autonomous directed motion due to self-generated chemical gradients [18]. Recently, neighboring walls were shown to significantly alter the magnitude of their swim speeds [15][16][17]. This revealed that walls play a far greater than previously expected role on self-propulsion, providing a path towards resolving seemingly conflicting experimental observations. For example, speed differences under similar conditions may stem from the phoretic interplay between the hydrodynamic wall boundary condition and the out-of-equilibrium self-generated chemical species [17]. Turning to theory, current models predict a wide range of behaviors close to walls, including hovering, sliding, forward and/or backward propulsion [12,[19][20][21][22][23][24][25][26][27][28][29][30][31]. This diversity is partly due to the complexity of and uncertainties in the propulsion mechanism, and partly due to the large number of hydrodynamic and phoretic couplings that wall proximity can introduce. Thus, quantitative insight into swimmer-wall separation is pivotal to pinpointing missing details of the propulsion mechanism, and in turn tailoring swimming behaviors, e.g., for guiding microswimmers in complex environments.
To date, no reported experiment has directly measured swimmer-wall separations. However, based on qualitative observations, separations are anticipated to be smaller than the swimmer size [13,32], even as small as a few tens of nm [33,34]. Such separations cannot be directly resolved by standard optical microscopy [33], which is why holographic microscopy has been proposed [16]. Digital in-line holographic microscopy (DIHM) measurements fitted with Mie scattering theory have been shown to yield three-dimensional positions of spherical particles with high precision [35]. However, fitting holograms of spheres half-coated with a metal is computationally expensive, especially when studying dynamics, since discrete dipole approximations have to be employed in the numerical calculations to obtain their positions [36]. Furthermore, coatings, being inhomogeneous in thickness, shape, and refractive index, introduce additional fit parameters. With all parameters being correlated, uncertainties in determining the coating's orientation lead to uncertainties in determining particle positions along the optical axis. An alternative technique to measure small particle-wall separations is Total Internal Reflection Microscopy (TIRM), which yields separations through the scattering of evanescent waves from the particles [37]. Here too, the asymmetry of the swimmer surface and the reflection from its coating interferes with interpreting the TIRM result and obtaining accurate measurements. Hence, a novel measurement approach is needed.
In this Letter, we present a facile and straightforward method for obtaining microswimmer-wall separations in situ. We determine the translational diffusion coefficient of the swimmer from mean-squared displacement curves, and obtain the height from its theoretically predicted dependence on swimmer-wall separation. As such, our method can be applied to any microswimmer, be it biological or synthetic. Here, we demonstrate its potential by applying it to catalytically propelled model microswimmers. We systematically varied parameters previously known to affect swim speeds as well as particlewall separations in passive systems, e.g., the salt concentration in solution, swimmer size, and swimmer zeta potential, and were able to gain unprecedented insight into the effect thereof and the presence of a wall on the c NaCl . Lines show theoretical predictions based on balancing electrostatics and gravity [49,50]. C) Diffusion coefficient (circles) and separation (squares) in the active state in aqueous 10% where A is 0.35 ± 0.09 µm/s the remaining speed in high salt, B a prefactor, and C is 0.09 ± 0.07 mM the ion concentration already present in solution, following from ionic diffusioosmosis along the wall.
swimming behavior.
We obtained swimmer-wall separations from experimental measurements of the separation-dependent translational diffusion coefficient, D, of the swimmers. D as well as propulsion speeds V were extracted from mean square displacements (MSDs) following Ref. [38]. That is, we fitted the short-time regime of the MSDs with ∆r 2 = 4D∆t + V 2 ∆t 2 [38]. The first term corresponds to the passive diffusion contribution that is usually obscured by the activity-induced, short-time ballistic behavior, but may be obtained with sufficient statistics. See the Supplemental Information (SI) [39] for details on tracking and MSD calculation [40]. Subsequently, we converted the measured D with respect to free bulk diffusion, D bulk = k B T 6πηR with R the radius, η the viscosity, k B the Boltzmann constant, and T the absolute temperature, into a separation using the theoretical expression for the dependence of the D/D bulk on height, h, see Figure 1A for illustration. The expression is based on numerical calculations [41,42] and analytic theory [43][44][45] and is provided in the SI Section II-A [39]. It covers the entire separation range from the well-known far-field prediction by Faxén [43] -predominantly used in analysing experiments [46,47] -to the h R regime captured by lubrication theory [45].
In all experiments, we used 3-(trimethoxysilyl)propyl methacrylate (TPM) monodisperse colloids [48] halfcoated with a thin Pt layer at dilute concentration (≈ 10 −7 v/v). In water, colloids exhibited passive Brownian motion, while dispersion in 10% H 2 O 2 rendered them active through a catalytic process. Colloids quickly reached the lower glass wall and continued to move adjacent to it, while their motion was recorded with an inverted Nikon Eclipse Ti microscope through a 60x oil objective (NA=1.4). Swimming experiments were performed at 18.92 fps for 30 s, see the SI Section I-C [39].
To demonstrate the effectiveness of our method, we first performed control experiments in deionized water and in water at pH 3.3, equivalent to the pH in the swimming experiments, at 5 fps. In these cases D was acquired from fitting MSDs with ∆r 2 = 4D∆t. Figure 1B shows that we recovered the expected decrease in separation with increasing salt concentration due to a decrease in the Debye length. Even more so, the extracted separations are in good agreement with a theoretical prediction based on a balance of electrostatic repulsion and gravity [49,50] that uses no fit parameters, see the SI Section II-B [39]. To verify our method further, we compared separations resulting from our diffusion coefficient-based method to those directly measured with DIHM, for uncoated silica spheres with well-known size and refractive index [51]. Despite using a computed value of D bulk in the analysis, we found good agreement between the two methods confirming again that we are indeed recovering colloid-wall separations.
Having established the validity of our method, we employed it to our catalytic microswimmers. First, we studied the effect of salt concentration in solution. For these experiments, we used TPM spheres of 2.77 ± 0.08 µm diameter half-coated with 4.4 ± 0.2 nm Pt. Surprisingly, in the active system we found a behavior that is completely unlike that of the passive system in Figure 1B. For the same particles and salt concentration range, the diffusion coefficient and separation remain constant within measurement precision, see Figure 1C. Particles propel themselves parallel to the wall at constant separations of 0.25 ± 0.06 µm.
At the same time, we found a decrease in speed with increasing salt concentration, see Figure 1D, where the line represents the least-squares fit with V = A+(B/(C +c)). This expression follows from a salt-gradient based contribution to the observed speed [32], with A the remaining speed in the limit of high salt, B a prefactor, and C the ion concentration already present in the medium. From the fit we find the reasonable numbers 0.35 ± 0.09 µm/s and 0.09 ± 0.07 mM, for A and C, respectively. We will return to how the salt gradient impacts the speed once we have put forward additional pieces of experimental evidence to substantiate our claim.
Second, we explored the effect of colloid zeta potential on self-propulsion from the swimmer-wall separation perspective. We used 2.70 ± 0.06 and 2.77 ± 0.08 µm diameter colloids with different surface functionalizations [52] and thus zeta potentials. The reported zeta potentials correspond to those of the parent colloids, see SI Sections I-A and I-B [39] for characterization, before adding the Pt-coating. We therefore use the term "base" zeta potential ζ b , to indicate that we know only the zeta potential of the uncoated colloid and not that of the swimmer.
Unexpectedly, in the swimming experiments we found that the wall separation remained unaffected for the wide range of ζ b under study, see Figure 2A. In all cases, particles moved at 0.24 ± 0.04 µm from the wall. This value agrees furthermore well with the separations measured for different salt concentrations, see Figure 1C. We note that the constant separation with ζ b in the swimming experiments sharply contrasts the passive behavior in water at the same pH, and thus ζ b . In the latter, separation distance is well-known to be affected by ζ b , and in our experiments colloids with ζ b > −12 mV were typically stuck on the wall, see also SI Sections II-B and I-D [39]. In the active state, colloids self-propelled not only at similar separations from the wall, but also at quantitatively comparable speeds, see Figure 2B. When plotting speed as function of wall separation in Figure 2C, we indeed see the collapse of the data, further demonstrating that ζ b does not affect the swimming behavior. We note that the direction of motion was away from the Pt cap both for positive and negative ζ b .
Third, we focused on swimmer size, another parameter that is known to significantly affect propulsion speeds [27]. We performed experiments using TPM spheres with a wide range of radii, but with similar Pt coating thicknesses and zeta potentials, see SI Sections I-A and I-B for characterization [39]. In Figure 3A we show the obtained swim speeds, together with a fit with V = a/R [27] (a = 2.4 ± 0.4 µm 2 /s). In addition, we found that the diffusion coefficient also scales inversely with the swimmer size, see Figure 3B, where the solid line represents the a/R fit (a = 0.120 ± 0.005 µm 3 /s). Strikingly, however, swimmer-wall separation remained relatively unaffected with size in Figure 3C; the dashed line is a guide to the eye, showing the mean separation of 0.33 ± 0.08 µm. The inset further shows that indeed the relative swimmer-wall distance, h/R, follows an 1/R dependence, with the solid line the a/R fit (a = 0.35 ± 0.05 µm).
The above experiments reveal that swimmers exhibit "ypsotaxis": a tendency to assume a specific height, irrespective of salt concentration, base zeta potential, and even size, see Figures 1C, 2A, and 3C, respectively. For our Pt-coated TPM particles, this robust separation distance was found to be on average 0.27 ± 0.11 µm, in line with the observation that micron-sized catalytic swimmers do not self-propel over steps with heights of few hundred nanometers [13]. Such a height is further consistent with wall-dependent speeds [15][16][17], for which the swimmer-wall distance must not substantially exceed the swimmer size to ensure strong osmotic coupling [12,19,24]. Interestingly, previous studies have commented on a strange constancy of the diffusion coefficient as a function of H 2 O 2 concentration [14]. However, they were unable to relate this to a height, presumably due to the larger distance between the wall and their swimmers, which led to observed values of D ≈ D bulk [14,32]. The radical departure of the active particle behavior from the passive behavior, see Figure 1B, suggests that ypsotaxis is reaction dominated. Phoretically and osmotically driven fluid flows thus seem prime contributors to the observed behavior. A significant buoyancy component seems unlikely in view of the range of swimmer sizes employed in Figure 3. Upon an inversion of our sample holders, we observed swimmers moving along the top wall for a period of time, which further indicated a significant activity-based component. Catalytic swimmers are furthermore known to align their direction of propulsion along surfaces [12,13,53], which we also observed here. Ypsotaxis and angular alignment likely share a common origin, as we explore in SI Section II-D [39].
Next, interpreting the effect of salt in the context of the literature, we draw a number of interesting conclusions. Simple salt such as NaCl is known to greatly decrease propulsion speeds [32,54], suggesting that the originally proposed self-diffusiophoretic mechanism [38] is insufficient in capturing details of the motion. This has led to debates on the propulsion mechanism, bringing forward alternative mechanisms such as self-electrophoresis [32,55]. Yet, these arguments are all based on theory for catalytic swimmers in bulk, while experiments are almost exclusively conducted in the vicinity of a wall. The lack of speed variation with ζ b , following our near-wall experiments in Figure 2B, however, is not typical of self-electrophoretic mechanisms, wherein the zeta potential strongly affects the speed, e.g., see the theoretical overview in Brown et al. [56], or the experimental result for self-electrophoretic bimetalic-nanorods [57]. Similarly, other ion-involving self-propulsion mechanisms are also sensitive to ζ b variation [56]. We therefore conclude that what happens at the Pt cap dominates the swimmer's behavior, and that this is either largely unaffected by the presence of NaCl, or that potentially the selfpropulsion in bulk is not governed by ionic species.
Still, in Figure 1D we find that swim speed is sensitive to increasing NaCl. To explain this, we draw upon the conclusion of our previous work that osmotic flows near the wall coupled to modification of the hydrodynamic slip can affect the observed speed [17]. Thus, we speculate that whilst the bulk speed of the swimmer remains unaffected when adding salt, the effective speed of the swimmer near the wall is modified by an osmotic counterflow coming from the wall; we provide details in the SI Section II-C [39]. Our fit in Figure 1D reveals that this (wall-based) osmosis bears the hallmarks of ionic diffusioosmosis [58]. This implies that a dissociated salt gradient must be present around the swimmer, and by extension, the surface reactions on the Pt cap must generate equal amounts of positive and negative ions with unequal mobilities [56]. The generation of speed at the Pt surface may thus be dominated by momentum-transfer mecha-nisms [59], while the ionic components act further away along the walls, where the gradients can act over a larger area, ultimately generating comparable osmotic backflow speeds. The above leads to an alternative interpretation of the swimming direction reversals observed by Brown and Poon [32]. These could be a consequence of the wallterm opposing the bulk motion of the swimmer and for a sufficiently large CTAB concentration dominating the speed generated at the Pt surface. Ions such as CTAB used in their experiments are known to affect zeta potential and contact angle, both of which can substantially alter the osmotic flow generated by the wall [58,60].
In summary, we established a novel method for measuring microswimmer-wall separations utilizing the height dependence of the diffusive component of their meansquared displacement. Our method can be applied to any spherical swimmer, biological or synthetic, and could also be adapted for asymmetric swimmers. We used it here to investigate the behavior of model catalytic microswimmers near a wall. We found that swimmers propel at roughly fixed heights of few hundred nanometers from the wall. Our work further showed that nearby walls are dominant factors in controlling observed variations of swim speed and that phoretic mechanisms may only play a role at the wall, rather than at the swimmer surface. This would necessitate a paradigm shift in modeling experimental observations and in identifying the still missing details of their propulsion mechanism. We are confident that further application of our method to other types of microswimmers will provide novel insights on the impact of confining surfaces in the microswimmer world, and in turn facilitate predicting swimming behaviors in complex environments.
We gratefully acknowledge Rachel Doherty for providing TPM colloids and for discussions on colloid functionalizations. We thank Ruben Verweij and Nikos Oikonomeas for useful discussions on holographic microscopy and Aidan Brown for discussions on the propulsion mechanism and for pointing out a relevant passage in the literature. J.d.G. thanks NWO for funding through Start-Up Grant 740.018.013 and through association with the EU-FET project NANOPHLOW (766972) within Horizon 2020. D.J.K. gratefully acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement no. 758383).