Slip length dependent propulsion speed of catalytic colloidal swimmers near walls

TL;DR

This study experimentally demonstrates that the propulsion speed of catalytic colloidal swimmers near surfaces depends on the substrate's slip length, which influences their velocity through osmotic coupling effects.

Contribution

It reveals a quantitative relationship between swimmer velocity and substrate slip length, highlighting the role of hydrodynamic slip in microswimmer propulsion.

Findings

Swimmer velocity scales with contact angle and slip length.

Velocity depends on osmotic coupling between swimmer and substrate.

Hydrodynamic slip significantly affects catalytic swimmer propulsion.

Abstract

Catalytic colloidal swimmers that propel due to self-generated fluid flows exhibit strong affinity for surfaces. We here report experimental measurements of significantly different velocities of such microswimmers in the vicinity of substrates made from different materials. We find that velocities scale with the solution contact angle $\theta$ on the substrate, which in turn relates to the associated hydrodynamic substrate slip length, as $V\propto(\cos\theta+1)^{-3/2}$. We show that such dependence can be attributed to osmotic coupling between swimmers and substrate. Our work points out that hydrodynamic slip at the wall, though often unconsidered, can significantly impact the self-propulsion of catalytic swimmers.

Figures

Equations

$${\xi }_w$$

$$V\propto (1+\cos \theta {)}^{-3/2},$$

Full text

Colloidal swimmers constitute a new class of non-equilibrium model systems, that also hold great promise for applications owing to their fast directed motion in liquid environments. A simple experimental realization of such microswimmers are spherical colloids half coated with Pt Ebbens and Gregory (2018). These colloids move autonomously in H2O2 solutions due to asymmetric catalytic reactions taking place on their surfaces Golestanian et al. (2005) and are typically found self-propelling parallel to a substrate Brown and Poon (2014); Ke et al. (2010); Das et al. (2015); Simmchen et al. (2016). This substrate-affinity leads to accumulation Brown and Poon (2014) and retention Das et al. (2015); Simmchen et al. (2016); Brown et al. (2016) of swimmers at surfaces, such as walls and obstacles, and can be exploited as a means to guide their motion Das et al. (2015); Uspal et al. (2016).

Strikingly, upon approaching a surface, numerical and theoretical models predict both an increase or decrease in swimming velocity depending on the considered propulsion mechanism and the physico-chemical properties of the swimmer and wall Popescu et al. (2009); Crowdy (2013); Chiang and Velegol (2014); Uspal et al. (2015); Ibrahim and Liverpool (2015); Mozaffari et al. (2016); Shen et al. (2018). Even more so, experimental observations also hint at non-negligible substrate effects for synthetic swimmers M. et al. (2018); Palacci et al. (2014). The perhaps most puzzling observation for catalytic swimmers is the inconsistency in swimming velocities under comparable experimental conditions. For example, velocities as disparate as 3 $\mu$m/s Howse et al. (2007) and 18 $\mu$m/s Brown and Poon (2014) were found for polystyrene spheres with 5 nm Pt coating in 10% H2O2. This discrepancy is even more surprising when one considers that the slower speeds were observed for the smaller species, whereas the velocity of Pt-coated swimmers should scale inversely with size Ebbens et al. (2012). Resolving these discrepancies will facilitate the development of a quantitative framework for these microswimmers.

Recent experiments on bimetallic microswimmers moving along glass surfaces showed a velocity decrease upon functionalizing the glass with polyelectrolytes M. et al. (2018). Surprisingly, the decrease on the velocity was observed for both positively and negatively charged polyelectrolytes, indicating that wall zeta potential does not have a dominant effect on the velocity of electrophoretic swimmers. At the same time, the effect of two different substrates, glass and Au-coated glass, on photo-activated TiO2/SiO2 swimmers was considered Holterhoff et al. (2018). Interestingly, 4 $\mu$m/s average velocities were found on Au, as opposed to 3 $\mu$m/s on glass. It was proposed, based on zeta potential values for Au and glass at neutral pH conditions, that the increase in the velocity stemmed from the lower zeta potential of the Au surface. However, neutral conditions are likely not met in H2O2 solutions. Results obtained using Au-coated substrates are hard to interpret, because Au could in principle catalyze H2O2 decomposition and thus interfere with the propulsion reaction. To corroborate the intriguing prospect that differences in swimming speeds originate from the substrate, surfaces other than Au ought to be examined. Furthermore, to pinpoint the origin of potential velocity differences a quantitative approach is required. Understanding potential surface effects on colloid self-propulsion is essential not only for their use as model systems, but also for future applications that may require motion in complex environments comprising obstacles or confining walls Bechinger et al. (2016).

In this Letter, we quantitatively examine the effect of various substrates, namely glass, glass coated with the organosilicon compound polydimethylsiloxane (PDMS), and plastic substrates made of a polyethylene (PE) or polystyrene (PS) derivative, on the velocity of catalytically self-propelled colloidal swimmers. Under otherwise fixed conditions, we observe significant differences in the velocities, which cannot be fully accounted for by the substrate zeta potential. Instead, we find that velocities upon different substrates fall on a single curve as a function of the solution-substrate contact angle which relates to the substrate slip length Huang et al. (2008). This finding indicates a velocity dependence on substrate slip for catalytic swimmers. After careful examination of the observed dependence in view of qualitative and scaling arguments, and accounting for possible couplings between swimmers and the substrate, we show that substrate-dependent velocities may result from osmotic coupling.

For all experiments, we used TPM colloids van der Wel et al. (2017) of diameter 2.7 $\mu$m half-coated with 4.9 nm of Pt by sputter-coating, see inset in Figure 1A. Colloids were prepared in one batch, and thus any inhomogeneities arising from their preparation, including Pt thickness that affects H2O2 decomposition, should be universal. Measurements were taken with a 60x ELWD air objective (NA 0.7) on an inverted Nikon TI-E microscope at 18.92 fps in the dark typically within the hour after dispersing the colloids at dilute particle concentration ($\approx$ 10*-7* w/v) in deionized water containing 10% H2O2. The colloids quickly reached the lower surface and continued to self-propel parallel to it, as illustrated in Figure 1A. Figure 1B shows 35 representative xy trajectories on glass, PE, and PDMS substrates acquired over a time interval of 8 s. We find that the colloids cover significantly greater distances on PDMS than on glass and PE, clearly demonstrating that the substrate affects colloid motion. To quantify the differences in the observed behavior, we first obtain the velocity of each individual colloid from its short-term mean squared displacement following Ref. Howse et al. (2007). We fit the corresponding probability density function (PDF) of the velocity with a log-normal distribution following Ref. Lisicki et al. to obtain the velocity distribution parameters on each substrate. Details on the determination of colloid velocities can be found in SI Section I D. The most frequently encountered velocities, as obtained from the fitted peak position of each PDF, are 1.05 $\pm$ 0.09, 1 $\pm$ 0.2, and 2.8 $\pm$ 0.3 $\mu$m/s, above glass, PE, and PDMS, respectively. Interestingly, though all three substrates are chemically different, the colloids show similar velocities for two of the substrates and a notably different velocity for the third. In the absence of H2O2, however, the translational diffusion coefficients are similar, namely 0.099 $\pm$ 0.005, 0.098 $\pm$ 0.008, and 0.105 $\pm$ 0.005 $\mu$m2/s, for glass, PE and PDMS, respectively. Therefore, substrate-dependent differences arise only in the active state.

While velocities may be influenced by colloid properties, such as size Ebbens et al. (2012), roughness Longbottom and Bon (2018) and slip Ajdari and Bocquet (2006); Manjare et al. (2014), these effects are negligible here since the same colloid batch was used in all experiments. Thus, the observed velocity differences arise from differences in the substrate properties. To quantitatively unravel the origin of our observations, we consider substrate properties that may influence colloid motion. The fluid flow generated by the anisotropic catalytic reaction on the swimmer surface Campbell et al. (2018), and hence the swimming velocity Anderson (1989), has been predicted to be affected by the swimmer-wall distance Popescu et al. (2009); Crowdy (2013); Chiang and Velegol (2014); Uspal et al. (2015); Ibrahim and Liverpool (2015); Mozaffari et al. (2016); Shen et al. (2018), wall zeta potential Chiang and Velegol (2014); Holterhoff et al. (2018) and wall surface inhomogeneities Holterhoff et al. (2018). Surprisingly, little consideration has been given until now on whether slip on the substrate impacts swimming velocities, even though slip on the colloid has already been shown to do so Manjare et al. (2014). Considering that hydrodynamic attraction in the active state pulls the colloids close to surfaces, to the extent that they even propel along the top of their container Brown and Poon (2014), colloid-substrate distances are expected to be small. Pt-coated swimmers of 2.5 $\mu$m radius have been found to not swim over 200 nm steps Simmchen et al. (2016), and other experiments pointed out that distances may even be of the order of tens of nm Takagi et al. (2013, 2014). Since wall slip lengths ranging from tens Craig et al. (2001); Baudry et al. (2001); Bhushan et al. (2009); Bocquet and Barrat (2007) to hundreds Pit et al. (2000); Cottin-Bizonne et al. (2002) of nanometers and even micrometers Zhu and Granick (2001) have been reported, boundary conditions could strongly affect the velocity. Following Ref. Ajdari and Bocquet (2006), we hypothesize that deviations from the no-slip condition on the substrate may enhance the nearby swimmer velocities.

Surface slip relates to liquid-solid interactions, and thus surface wetting properties, and generally, though not always, increases with increasing hydrophobicity and thus contact angle $\theta$ Bocquet and Barrat (2007); Huang et al. (2008); Bhushan et al. (2009); Siboulet et al. (2013). Since a larger slip length $b$ on hydrophobic surfaces leads to a larger slip fluid velocity Kezirian (1992), we hypothesize that this leads to a larger swimming velocity. Conversely, the no-slip approximation on hydrophilic surfaces would lead to a lower velocity, see Figure 2A. Indeed, the measured contact angles for the H2O2 solution agree with this hypothesis: $\theta$

is 46 $\pm$ 9$\degree$, 51 $\pm$ 3$\degree$, and 100 $\pm$ 3$\degree$, for glass, PE, and PDMS, respectively. PE is normally hydrophobic, thus a modification had been performed by the supplier.

To further test this hypothesis, we modulated the hydrophilicity of the employed substrates and repeated the experiments. We increased the hydrophilicity of glass by either a cleaning procedure ($\theta=29.5\pm 3\degree$) or treatment with HCl ($\theta=13\pm 3\degree$) and observed a concomitant velocity decrease by 30% and 45% (0.78 $\pm$ 0.03 and 0.6 $\pm$ 0.05 $\mu$m/s), respectively. Conversely, when we rendered the glass more hydrophobic ($\theta=80\pm 2$°), we found that the velocity increased by 28% (1.35 $\pm$ 0.13 $\mu$m/s) compared to untreated glass. Similar behavior was seen on PDMS that was rendered hydrophilic through UV-ozone treatment ($\theta=37\pm 7\degree$): colloidal particles propelled 4 times slower (0.7 $\pm$ 0.05 $\mu$m/s) than on the hydrophobic PDMS. Finally, we employed commercially available PS substrates that were rendered hydrophilic ($\theta=46\pm 6\degree$) and found 0.8 $\pm$ 0.45 $\mu$m/s. We summarize these findings by plotting velocity as a function of $\theta$ in Figure 2B. The collapse of the data onto a single curve suggests that $\theta$, and thus slip length, is the most relevant parameter and that other differences among substrates, besides their effect on $\theta$, are of lesser importance.

Next, we develop a quantitative framework for the slip-dependent swimming velocities. For our analysis we consider that the height above the substrate remains relatively unaffected by the change of substrate, as supported by our experimental measurements of the diffusion coefficient, see SI Section I D. When the height is left unperturbed by varying $\cos\theta$ — possibly due to electrostatic or even hydrodynamic coupling — the dominant source of change to the swimming velocity comes from solute gradients near the substrate. As mentioned earlier, these are generated by reactions taking place on the swimmer surface and, similar to the way they cause self-propulsion, lead to an effective surface fluid velocity along the wall Anderson (1989). This is often referred to as ’slip’ velocity, but we do not use this term to avoid confusion with the concept of hydrodynamic surface slip. This effective surface fluid velocity couples back to the swimmer, modifying its net velocity Chiang and Velegol (2014); Das et al. (2015); Uspal et al. (2016); Popescu et al. (2018). In SI Section II, we show that neither purely hydrodynamic coupling Spagnolie and Lauga (2012); Lintuvuori et al. (2016); Shen et al. (2018), solute confinement Popescu et al. (2009); Crowdy (2013); Uspal et al. (2015); Ibrahim and Liverpool (2015); Mozaffari et al. (2016), nor reaction-based coupling Ebbens et al. (2012) can account for the significant wall effect. Instead, as we will show, our observation can be attributed to osmotic coupling Keh and Chen (1988); Anderson (1989); Chiang and Velegol (2014); Das et al. (2015); Uspal et al. (2016); Popescu et al. (2018).

The osmotic coupling scales linearly with the slip-velocity parameter $\xi_{w}$, i.e., the prefactor that converts solute gradients into effective hydrodynamic surface velocities Anderson (1989). Ajdari and Bocquet Ajdari and Bocquet (2006) have shown that for a partial-slip wall the result by Anderson Anderson (1989) can be generalized to

[TABLE]

where slippage is expressed by the slip length $b_{w}$; $b_{w}=0$ for a no-slip surface and $b_{w}\rightarrow\infty$ for a full-slip surface, respectively. Here, we have introduced $k_{\mathrm{B}}$ Boltzmann’s constant, $T$ the temperature, $\mu$ the dynamic viscosity, $\lambda_{w}$ a length scale for the solute-surface interactions, $\gamma_{w}$ a length measuring the solute excess Ajdari and Bocquet (2006). For smooth surfaces, as we consider here, the value $\lambda_{w}$ is left relatively unaffected by changes in $\theta$, but $b_{w}\propto(1+\cos\theta)^{-2}$ and $\gamma_{w}\propto\sqrt{1+\cos\theta}$ Huang et al. (2008). This leads to the following leading-order proportionality of the measured velocity with $\theta$:

[TABLE]

which requires that $b_{w}/\lambda_{w}\gg 1$, see also SI Section II for a more in depth discussion on the osmotic coupling based mechanism. We use this quantitative relationship between velocity and contact angle to fit the experimental data presented in Figure 2B. The proportionality factor A, which contains all other contributions to the velocity that are slip-independent, is 1.84 $\mu$m/s. The excellent agreement between data and model further corroborates in a quantitative manner the influence of slip.

To provide additional support to our hypothesis, we test whether the above dependence persists in the presence of salt. Previous experiments employing 2 $\mu$m PS spheres showed that even 1 mM salt considerably decreases colloid velocities Ebbens et al. (2014); Brown and Poon (2014). Although velocities for similar H2O2 concentration without salt were different above glass, namely around 4 $\mu$m/s Ebbens et al. (2014) and 18 $\mu$m/s Brown and Poon (2014), they reduced to 0.45 and 1 $\mu$m/s, respectively, in 1 mM salt. In agreement with these experiments, we find that velocities above different substrates decrease with added salt, see Figure 3. More importantly, we observe that the velocities still follow the same slip dependence. In salt solution, the proportionality factor A is 1.2 $\mu$m/s, i.e. it shows a 33% decrease compared to the salt-free case. Considering that the influence of salt is complex, potentially affecting zeta potentials, separation, higher-order hydrodynamic moments or possibly more properties including bulk velocities, this decrease is not surprising. However, that the same dependence persists strongly supports the importance of slip, and may provide additional insights into the propulsion mechanism Brown and Poon (2014).

We emphasize that other substrate properties besides slip may affect the swimming velocity. As mentioned earlier, lowering the substrate zeta potential has been proposed to increase velocity Holterhoff et al. (2018). For completeness thus we measured substrate zeta potentials using a Surface Zeta Potential Cell from Malvern by laser Doppler electrophoresis following Ref. Corbett et al. (2012), see SI Section I E. We find zeta potentials of -38.3 $\pm$ 1.1 mV and -22 $\pm$ 0.9 mV for glass and PDMS, respectively, which is in line with this proposal. However, we find an even lower zeta potential, -11 $\pm$ 5 mV, for hydrophilic PDMS. Based on the low velocity on hydrophilic PDMS, we conclude that the substrate zeta potential is, surprisingly, not the dominant effect. Secondly, an increase in the substrate roughness was shown to increase the velocity Holterhoff et al. (2018). We thus performed Atomic Force Microscopy (AFM) measurements, see SI Section I F. The average substrate roughness Ra, is 1.5 and 5 nm, for glass and PDMS, respectively; with Ra denoting the arithmetic mean of the deviations in height from the roughness mean value. Again, this is in line with previous observations. However, hydrophilic PDMS, with a roughness equal to untreated PDMS, featured significantly different velocities. We thus conclude that substrate roughness is also not the dominant effect.

Finally, we note that the velocity increase on Au-coated surfaces Holterhoff et al. (2018) may be due to increased surface slip, since contact angles on Au are typically higher than on glass Erb (1968); Abdelsalam et al. (2005), while the velocity decrease in  M. et al. (2018) may be due to the hydrophilic polyelectrolyte coatings employed on the glass. Besides, our findings may shed new light on discrepancies in the velocities between previous experiments despite the use of a similar material, coating and solution, as mentioned in the introduction. Even though glass substrates were used, glass can differ in composition, homogeneity and hydrophilicity due to different preparation, coatings, treatment and cleaning methods from the supplier or the researchers themselves, as we have also demonstrated. During contact angle measurements we have noticed that discrepancies of about 10$\degree$ can be found between substrates even within the same batch and/or different parts of the same substrate. AFM indicated that this is likely due to inhomogeneous application or even local absence of coatings applied by the supplier. Thus, similarly to inhomogeneities in colloid properties arising from preparation, inhomogeneities in substrate properties may arise as well. If the coating or treatment that is applied to render the material hydrophilic is inhomogeneous or unstable, locally enhanced substrate slip would persist especially when the underlying material is in principle hydrophobic. Finally, we note that it is possible for substrate slip to be influenced in time by chemically reacting with H2O2; for example, we found a 15o increase in contact angle for the PS substrate before and after being exposed to H2O2 for several hours.

In conclusion, we show that the velocity of catalytic colloidal swimmers near a wall is influenced by the wall slip boundary condition. This quantitatively follows from theoretical predictions on the basis of an osmotic coupling mechanism, indicating further control and understanding of the behavior of self-propelled particles. Our work points out that wall properties, though often unconsidered, can significantly impact self-propulsion of catalytic swimmers. We expect that the hydrodynamic slip at nearby walls is also relevant for other types of microswimmers.