Diffusion-based height analysis reveals robust microswimmer-wall separation
DDiffusion-based height analysis reveals robust microswimmer-wall separation
Stefania Ketzetzi, Joost de Graaf, and Daniela J. Kraft Soft Matter Physics, Huygens-Kamerlingh Onnes Laboratory,Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena,Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Dated: June 12, 2020)Microswimmers typically move near walls and can be strongly influenced by them. However, directexperimental measurements of swimmer-wall separation remain elusive to date. Here, we demon-strate that this separation can be obtained from the height dependence of the passive component ofthe swimmer’s mean-squared displacement. We apply our approach to catalytic microswimmers andfind that they exhibit “ypsotaxis”, a tendency to assume a fixed height above the wall for a rangeof salt concentrations, swimmer surface charges, and swimmer sizes. Our results show that nearbywalls play an important role in controlling their motion, and provide new insights for modellingtheir propulsion mechanism.
Confining surfaces, such as planar walls, have a far-reaching impact in the microswimmer world, often en-suring microswimmer function and survival [1]. Encoun-ters with surfaces give rise to accumulation, as seen forsperm [2], algae [3] and bacteria [4], and enable the for-mation of bacterial biofilms that facilitate their spread-ing, cooperation, and capture of nutrients [5–7]. More-over, surfaces can significantly modify swimming trajec-tories; e.g. , bacteria often exhibit circular motion withdirection controlled by the boundary condition [8–11], instark contrast to their run-and-tumble motion in bulk.Striking surface effects are not only found in biologicalsystems, but are also present for synthetic microswim-mers [12–17]. Model catalytic colloidal swimmers ex-hibit autonomous directed motion due to self-generatedchemical gradients [18]. Recently, neighboring walls wereshown to significantly alter the magnitude of their swimspeeds [15–17]. This revealed that walls play a far greaterthan previously expected role on self-propulsion, provid-ing a path towards resolving seemingly conflicting ex-perimental observations. For example, speed differencesunder similar conditions may stem from the phoretic in-terplay between the hydrodynamic wall boundary con-dition and the out-of-equilibrium self-generated chemicalspecies [17]. Turning to theory, current models predict awide range of behaviors close to walls, including hovering,sliding, forward and/or backward propulsion [12, 19–31].This diversity is partly due to the complexity of and un-certainties in the propulsion mechanism, and partly dueto the large number of hydrodynamic and phoretic cou-plings that wall proximity can introduce. Thus, quan-titative insight into swimmer-wall separation is pivotalto pinpointing missing details of the propulsion mecha-nism, and in turn tailoring swimming behaviors, e.g. , forguiding microswimmers in complex environments.To date, no reported experiment has directly measuredswimmer-wall separations. However, based on qualitativeobservations, separations are anticipated to be smallerthan the swimmer size [13, 32], even as small as a few tens of nm [33, 34]. Such separations cannot be directly re-solved by standard optical microscopy [33], which is whyholographic microscopy has been proposed [16]. Digitalin-line holographic microscopy (DIHM) measurementsfitted with Mie scattering theory have been shown toyield three-dimensional positions of spherical particleswith high precision [35]. However, fitting holograms ofspheres half-coated with a metal is computationally ex-pensive, especially when studying dynamics, since dis-crete dipole approximations have to be employed in thenumerical calculations to obtain their positions [36]. Fur-thermore, coatings, being inhomogeneous in thickness,shape, and refractive index, introduce additional fit pa-rameters. With all parameters being correlated, uncer-tainties in determining the coating’s orientation lead touncertainties in determining particle positions along theoptical axis. An alternative technique to measure smallparticle-wall separations is Total Internal Reflection Mi-croscopy (TIRM), which yields separations through thescattering of evanescent waves from the particles [37].Here too, the asymmetry of the swimmer surface andthe reflection from its coating interferes with interpretingthe TIRM result and obtaining accurate measurements.Hence, a novel measurement approach is needed.In this Letter, we present a facile and straightforwardmethod for obtaining microswimmer-wall separations insitu . We determine the translational diffusion coefficientof the swimmer from mean-squared displacement curves,and obtain the height from its theoretically predicteddependence on swimmer-wall separation. As such, ourmethod can be applied to any microswimmer, be it bi-ological or synthetic. Here, we demonstrate its poten-tial by applying it to catalytically propelled model mi-croswimmers. We systematically varied parameters pre-viously known to affect swim speeds as well as particle-wall separations in passive systems, e.g. , the salt con-centration in solution, swimmer size, and swimmer zetapotential, and were able to gain unprecedented insightinto the effect thereof and the presence of a wall on the a r X i v : . [ c ond - m a t . s o f t ] J un FIG. 1:
Salt-dependent motion above a planar wall:
A) Schematic of the experiment. The swimmer-wallseparation, h , is obtained from the measured translational diffusion coefficient of the swimmer and its theoreticallypredicted dependence on wall separation. B), C), and D): Effect of salt concentration c NaCl on the motion of2.77 ± µ m colloids with 4.4 ± st quartiles. B)Diffusion coefficient (orange diamonds) and separation (purple hexagons) in the Brownian state in water at different c NaCl . Lines show theoretical predictions based on balancing electrostatics and gravity [49, 50]. C) Diffusioncoefficient (circles) and separation (squares) in the active state in aqueous 10% H O with c NaCl . Dotted linesindicate mean values. D) Speed decrease in 10% H O with c NaCl . Solid line is a fit with V = A + ( B/ ( C + c NaCl )),where A is 0.35 ± µ m/s the remaining speed in high salt, B a prefactor, and C is 0.09 ± D , of the swimmers. D aswell as propulsion speeds V were extracted from meansquare displacements (MSDs) following Ref. [38]. Thatis, we fitted the short-time regime of the MSDs with∆ r = 4 D ∆ t + V ∆ t [38]. The first term correspondsto the passive diffusion contribution that is usually ob-scured by the activity-induced, short-time ballistic be-havior, but may be obtained with sufficient statistics.See the Supplemental Information (SI) [39] for detailson tracking and MSD calculation [40]. Subsequently, weconverted the measured D with respect to free bulk dif-fusion, D bulk = k B T πηR with R the radius, η the viscosity, k B the Boltzmann constant, and T the absolute temper-ature, into a separation using the theoretical expressionfor the dependence of the D/D bulk on height, h , see Fig-ure 1A for illustration. The expression is based on nu-merical calculations [41, 42] and analytic theory [43–45]and is provided in the SI Section II-A [39]. It covers theentire separation range from the well-known far-field pre-diction by Fax´en [43] — predominantly used in analysingexperiments [46, 47] — to the h (cid:28) R regime captured bylubrication theory [45].In all experiments, we used 3-(trimethoxysilyl)propylmethacrylate (TPM) monodisperse colloids [48] half-coated with a thin Pt layer at dilute concentration( ≈ − v/v). In water, colloids exhibited passive Brow-nian motion, while dispersion in 10% H O rendered them active through a catalytic process. Colloids quicklyreached the lower glass wall and continued to move ad-jacent to it, while their motion was recorded with aninverted Nikon Eclipse Ti microscope through a 60x oilobjective (NA=1.4). Swimming experiments were per-formed at 18.92 fps for 30 s, see the SI Section I-C [39].To demonstrate the effectiveness of our method, wefirst performed control experiments in deionized waterand in water at pH 3.3, equivalent to the pH in the swim-ming experiments, at 5 fps. In these cases D was acquiredfrom fitting MSDs with ∆ r = 4 D ∆ t . Figure 1B showsthat we recovered the expected decrease in separationwith increasing salt concentration due to a decrease inthe Debye length. Even more so, the extracted separa-tions are in good agreement with a theoretical predictionbased on a balance of electrostatic repulsion and grav-ity [49, 50] that uses no fit parameters, see the SI SectionII-B [39]. To verify our method further, we comparedseparations resulting from our diffusion coefficient-basedmethod to those directly measured with DIHM, for un-coated silica spheres with well-known size and refractiveindex [51]. Despite using a computed value of D bulk inthe analysis, we found good agreement between the twomethods confirming again that we are indeed recoveringcolloid-wall separations.Having established the validity of our method, we em-ployed it to our catalytic microswimmers. First, we stud-ied the effect of salt concentration in solution. For theseexperiments, we used TPM spheres of 2.77 ± µ m di-ameter half-coated with 4.4 ± Swimmer base zeta potential dependence of propulsion above a planar wall:
The base zetapotential ζ b of TPM colloids with diameters between 2.70 ± ± µm was varied through surfacefunctionalization. A) Diffusion coefficient (circles) and separation (squares) in the active state in aqueous 10% H O with ζ b . B) Speed for the same solvent and ζ b range as in (A). C) Speed as function of separation (purple circles).The white circle marks the intersection of speed and separation mean values; the rectangle denotes standarddeviations. All reported values are medians. Error bars denote 1 st quartiles. Dotted lines represent mean values.the active system we found a behavior that is completelyunlike that of the passive system in Figure 1B. For thesame particles and salt concentration range, the diffu-sion coefficient and separation remain constant withinmeasurement precision, see Figure 1C. Particles propelthemselves parallel to the wall at constant separations of0.25 ± µ m.At the same time, we found a decrease in speed with in-creasing salt concentration, see Figure 1D, where the linerepresents the least-squares fit with V = A +( B/ ( C + c )).This expression follows from a salt-gradient based contri-bution to the observed speed [32], with A the remainingspeed in the limit of high salt, B a prefactor, and C theion concentration already present in the medium. Fromthe fit we find the reasonable numbers 0.35 ± µ m/sand 0.09 ± A and C , respectively. We willreturn to how the salt gradient impacts the speed oncewe have put forward additional pieces of experimentalevidence to substantiate our claim.Second, we explored the effect of colloid zeta potentialon self-propulsion from the swimmer-wall separation per-spective. We used 2.70 ± ± µ m diam-eter colloids with different surface functionalizations [52]and thus zeta potentials. The reported zeta potentialscorrespond to those of the parent colloids, see SI SectionsI-A and I-B [39] for characterization, before adding thePt-coating. We therefore use the term “base” zeta poten-tial ζ b , to indicate that we know only the zeta potentialof the uncoated colloid and not that of the swimmer.Unexpectedly, in the swimming experiments we foundthat the wall separation remained unaffected for the widerange of ζ b under study, see Figure 2A. In all cases, par-ticles moved at 0.24 ± µ m from the wall. This valueagrees furthermore well with the separations measured for different salt concentrations, see Figure 1C. We notethat the constant separation with ζ b in the swimming ex-periments sharply contrasts the passive behavior in wa-ter at the same pH, and thus ζ b . In the latter, separationdistance is well-known to be affected by ζ b , and in ourexperiments colloids with ζ b > −
12 mV were typicallystuck on the wall, see also SI Sections II-B and I-D [39].In the active state, colloids self-propelled not only at sim-ilar separations from the wall, but also at quantitativelycomparable speeds, see Figure 2B. When plotting speedas function of wall separation in Figure 2C, we indeedsee the collapse of the data, further demonstrating that ζ b does not affect the swimming behavior. We note thatthe direction of motion was away from the Pt cap bothfor positive and negative ζ b .Third, we focused on swimmer size, another param-eter that is known to significantly affect propulsionspeeds [27]. We performed experiments using TPMspheres with a wide range of radii, but with similar Ptcoating thicknesses and zeta potentials, see SI SectionsI-A and I-B for characterization [39]. In Figure 3A weshow the obtained swim speeds, together with a fit with V = a/R [27] ( a = 2.4 ± µ m /s). In addition, wefound that the diffusion coefficient also scales inverselywith the swimmer size, see Figure 3B, where the solidline represents the a/R fit ( a = 0 . ± . µ m /s).Strikingly, however, swimmer-wall separation remainedrelatively unaffected with size in Figure 3C; the dashedline is a guide to the eye, showing the mean separationof 0.33 ± µ m. The inset further shows that in-deed the relative swimmer-wall distance, h/R , followsan 1 /R dependence, with the solid line the a/R fit( a = 0.35 ± µ m).The above experiments reveal that swimmers exhibitFIG. 3: Size-dependent propulsion above a planar wall:
A) Swim speed in 10% H O with swimmer radius, R . B) Diffusion coefficient in 10% H O with R . C) Separation with R . Dotted line shows the mean separation,0.33 ± µ m. The inset shows the relative distance h/R with R . All TPM colloids have similar ζ b . All reportedvalues are medians. Error bars denote 1 st quartiles. Solid lines are fits with a/R ; a is 2.4 ± µ m /s (A),0.120 ± µ m /s (B), and 0.35 ± µ m (C). Insets show the data on a linear (A, B) and log-log scale (C).“ypsotaxis”: a tendency to assume a specific height, ir-respective of salt concentration, base zeta potential, andeven size, see Figures 1C, 2A, and 3C, respectively. Forour Pt-coated TPM particles, this robust separation dis-tance was found to be on average 0.27 ± µ m, in linewith the observation that micron-sized catalytic swim-mers do not self-propel over steps with heights of fewhundred nanometers [13]. Such a height is further con-sistent with wall-dependent speeds [15–17], for whichthe swimmer-wall distance must not substantially ex-ceed the swimmer size to ensure strong osmotic cou-pling [12, 19, 24]. Interestingly, previous studies havecommented on a strange constancy of the diffusion co-efficient as a function of H O concentration [14]. How-ever, they were unable to relate this to a height, pre-sumably due to the larger distance between the walland their swimmers, which led to observed values of D ≈ D bulk [14, 32]. The radical departure of the ac-tive particle behavior from the passive behavior, see Fig-ure 1B, suggests that ypsotaxis is reaction dominated.Phoretically and osmotically driven fluid flows thus seemprime contributors to the observed behavior. A signifi-cant buoyancy component seems unlikely in view of therange of swimmer sizes employed in Figure 3. Upon aninversion of our sample holders, we observed swimmersmoving along the top wall for a period of time, whichfurther indicated a significant activity-based component.Catalytic swimmers are furthermore known to align theirdirection of propulsion along surfaces [12, 13, 53], whichwe also observed here. Ypsotaxis and angular alignmentlikely share a common origin, as we explore in SI SectionII-D [39].Next, interpreting the effect of salt in the context of theliterature, we draw a number of interesting conclusions.Simple salt such as NaCl is known to greatly decreasepropulsion speeds [32, 54], suggesting that the originally proposed self-diffusiophoretic mechanism [38] is insuffi-cient in capturing details of the motion. This has led todebates on the propulsion mechanism, bringing forwardalternative mechanisms such as self-electrophoresis [32,55]. Yet, these arguments are all based on theory forcatalytic swimmers in bulk, while experiments are al-most exclusively conducted in the vicinity of a wall. Thelack of speed variation with ζ b , following our near-wallexperiments in Figure 2B, however, is not typical ofself-electrophoretic mechanisms, wherein the zeta poten-tial strongly affects the speed, e.g. , see the theoreticaloverview in Brown et al. [56], or the experimental re-sult for self-electrophoretic bimetalic-nanorods [57]. Sim-ilarly, other ion-involving self-propulsion mechanisms arealso sensitive to ζ b variation [56]. We therefore concludethat what happens at the Pt cap dominates the swim-mer’s behavior, and that this is either largely unaffectedby the presence of NaCl, or that potentially the self-propulsion in bulk is not governed by ionic species.Still, in Figure 1D we find that swim speed is sensitiveto increasing NaCl. To explain this, we draw upon theconclusion of our previous work that osmotic flows nearthe wall coupled to modification of the hydrodynamicslip can affect the observed speed [17]. Thus, we specu-late that whilst the bulk speed of the swimmer remainsunaffected when adding salt, the effective speed of theswimmer near the wall is modified by an osmotic coun-terflow coming from the wall; we provide details in the SISection II-C [39]. Our fit in Figure 1D reveals that this(wall-based) osmosis bears the hallmarks of ionic diffusio-osmosis [58]. This implies that a dissociated salt gradientmust be present around the swimmer, and by extension,the surface reactions on the Pt cap must generate equalamounts of positive and negative ions with unequal mo-bilities [56]. The generation of speed at the Pt surfacemay thus be dominated by momentum-transfer mecha-nisms [59], while the ionic components act further awayalong the walls, where the gradients can act over a largerarea, ultimately generating comparable osmotic backflowspeeds. The above leads to an alternative interpretationof the swimming direction reversals observed by Brownand Poon [32]. These could be a consequence of the wall-term opposing the bulk motion of the swimmer and fora sufficiently large CTAB concentration dominating thespeed generated at the Pt surface. Ions such as CTABused in their experiments are known to affect zeta poten-tial and contact angle, both of which can substantiallyalter the osmotic flow generated by the wall [58, 60].In summary, we established a novel method for measur-ing microswimmer-wall separations utilizing the heightdependence of the diffusive component of their mean-squared displacement. Our method can be applied to anyspherical swimmer, biological or synthetic, and could alsobe adapted for asymmetric swimmers. We used it here toinvestigate the behavior of model catalytic microswim-mers near a wall. We found that swimmers propel atroughly fixed heights of few hundred nanometers fromthe wall. Our work further showed that nearby wallsare dominant factors in controlling observed variationsof swim speed and that phoretic mechanisms may onlyplay a role at the wall, rather than at the swimmer sur-face. This would necessitate a paradigm shift in model-ing experimental observations and in identifying the stillmissing details of their propulsion mechanism. We areconfident that further application of our method to othertypes of microswimmers will provide novel insights on theimpact of confining surfaces in the microswimmer world,and in turn facilitate predicting swimming behaviors incomplex environments.We gratefully acknowledge Rachel Doherty for provid-ing TPM colloids and for discussions on colloid func-tionalizations. We thank Ruben Verweij and NikosOikonomeas for useful discussions on holographic mi-croscopy and Aidan Brown for discussions on the propul-sion mechanism and for pointing out a relevant pas-sage in the literature. J.d.G. thanks NWO for fundingthrough Start-Up Grant 740.018.013 and through associ-ation with the EU-FET project NANOPHLOW (766972)within Horizon 2020. D.J.K. gratefully acknowledgesfunding from the European Research Council (ERC) un-der the European Union’s Horizon 2020 research and in-novation program (grant agreement no. 758383). [1] Jens Elgeti, Roland G. 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