Relating microswimmer synthesis to rheotactic tunability
Relating microswimmer synthesis to rheotactic tunability
Brosseau, Quentin;Usabiaga, Florencio Balboa;Lushi, Enkeleida;Wu, Yang;Ristroph, Leif;Zhang, Jun;Ward, Michael;Shelley, Michael J.
2019-06-11 00:00:00
Relating microswimmer synthesis to hydrodynamic actuation and rheotactic tunability 1 2 3 Quentin Brosseau , Florencio Balboa Usabiaga , Enkeleida Lushi , Yang 4 1 1 4 1;2 Wu , Leif Ristroph , Jun Zhang , Michael Ward and Michael J. Shelley Courant Institute, New York University, New York, NY 10012, USA, Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Chemistry, New York University, New York, NY 10012, USA (Dated: June 13, 2019) We explore the behavior of micron-scale autophoretic Janus (Au/Pt) rods, having various Au/Pt length ratios, swimming near a wall in an imposed background
ow. We nd that their ability to robustly orient and move upstream, i.e. to rheotax, depends strongly on the Au/Pt ratio, which is easily tunable in synthesis. Numerical simulations of swimming rods actuated by a surface slip show a similar rheotactic tunability when varying the location of the surface slip versus surface drag. Slip location determines whether swimmers are Pushers (rear-actuated), Pullers (front-actuated), or in between. Our simulations and modeling show that Pullers rheotax most robustly due to their larger tilt angle to the wall, which makes them responsive to
ow gradients. Thus, rheotactic response infers the nature of dicult to measure
ow- elds of an active particle, establishes its dependence on swimmer type, and shows how Janus rods can be tuned for
ow responsiveness. We demonstrate the eectiveness of a simple geometric sieve for rheotactic ability. Swimming microorganisms must contend with bound- aries and obstacles in their natural environments [1{ 3]. Microbial habitats have ample surfaces, and swim- mer concentrations near them promote attachment and bio lms [4, 5]. Motile bacteria and spermatozoa accu- mulate near boundaries, move along them [6, 7], and self-organize under con nement [8{11]. Microswimmers also exhibit rheotaxis, i.e. the ability to actively reorient and swim against an imposed
ow [12]. Surfaces are key for rheotactic response:
uid shear near boundaries re- sults in hydrodynamic interactions which favor swimmer alignment against the oncoming
ow and prevent swim- FIG. 1: The dierent bimetallic swimmers. The ratio of the mer displacements across streamlines [13{17]. Swimmers metallic segments varies from (a) 1:1 for symmetric, to (b) 3:1 with dierent propulsion mechanisms { front-actuated for long-gold and to (c) 1:3 for long-platinum. Scale bar 1m. like puller micro-algae, or rear-actuated like pusher bac- (d) Each swimmer type is tested in a rectangular micro
uidic teria { exhibit associated dipolar
ow elds [18{20] which channel where it is gravitationally con ned near the bottom. result in dissimilar collective motions [21{23] and behav- Under shear
ow the metallic particles swim upstream. ior near boundaries or in
ows [24{29]. In this Letter, we address this question with experi- Recent advances in the manufacture and design of arti- ments using chemically powered gold-platinum (Au/Pt) cial swimmers have triggered an acute interest in devel- microswimmers combined and compared with numerical oping synthetic mimetic systems [3, 30{34]. Like their bi- simulations. In experiments we vary the position of the ological counterparts, arti cial swimmers can accumulate Au/Pt join along the swimmer length, postulating that near boundaries [35, 36], navigate along them [37, 38], be this varies the location of the
ow actuation region, and guided by geometric or chemical patterns [39{42] or ex- that observed dierences in rheotaxis can be related to ternal forces [43, 44], and can display rheotaxis near pla- having dierent pusher- or puller-like swimmers. In sim- nar surfaces [45{47]. While models have been developed ulation, we study the rheotactic responses of rod-like mi- to study their locomotion and behavior [24, 35, 48, 49], croswimmers that move through an active surface slip. the relevance of the swimmers' actuation mechanism and Dierent placements of the slip region allow us to cre- the resulting hydrodynamic contributions to their rheo- ate pullers, symmetric, and pusher microswimmers. We tactic motion remains an open question. In large part nd measurably dierent rheotactic responses in simula- this is due to the diculty in directly assessing swim- tion which show quantitative agreement with our experi- mers'
ow- elds, particularly near walls, and relating ex- ments with Au/Pt active particles conducted in micro
u- perimental observations to our theoretical understand- idic channels. Lastly, we show that mixed swimmer pop- ing of swimmer geometry, hydrodynamics and type (i.e., ulations can be sorted through a micro
uidic sieve that pusher or puller). exploits the swimmers' dierent rheotactic behaviors. arXiv:1906.04814v1 [cond-mat.soft] 11 Jun 2019 2 Experimental setup and measurements. Our Janus microswimmers are elongated Au/Pt rods, 2m in length and 0:3m in diameter, which propel them- selves through self-electrophoresis in aqueous H O so- 2 2 lutions [32, 34]. The swimmers are synthesized by elec- trodeposition [30, 50] to a prescribed ratio of the two metallic segments: symmetric with Au:Pt (1:1), asym- metric long-gold with Au:Pt (3:1) and asymmetric long- platinum with Au:Pt (1:3); see Fig. 1a-c, details in [51]. FIG. 2: Computed velocity elds around simulated self- The swimmers' rheotactic abilities are tested in a propelled rods with a surface slip region (shown in red) (a) at the center, (b) at the front, and (c) at the rear, corresponding rectangular PDMS micro
uidic channel of width W = to, symmetric, puller and pusher swimmers, respectively. 300m built following classical soft-lithography tech- niques [52]. We control the background unidirectional ow down the channel (the x-direction) using an o-stage Eq. (1) represents the balance of the geometric con- hydrostatic column. Suspended glass beads of radius straint forces with the external force F and torque r 2:5m serve as markers to measure the
ow pro- generated by steric interactions with the substrate and le close to the bottom of the channel where the rods gravity. Eq. (2) gives the balance of
uid, propulsive, move. We record the trajectories of swimmers and beads and thermal forces, with ue the active slip velocity, u (r ) i 0 i 1=2 over 1 minute and extract the instantaneous velocities of the background
ow velocity, and 2k T=t(M W ) B i swimmers V and of beads U , along the x-axis. See Fig. the Brownian noise, with k the Boltzmann constant, T x b 1d and videos in [51]. the temperature, t the time step, W a vector of white 1=2 Thermal
uctuations are important at this scale and noises, and M representing the square root of the mo- the swimmers mean square displacement for U = 0 at bility tensor [59]. Half the blobs along the rod are \pas- a xed H O concentration are used to estimate their sive" with ue = 0, while the other half have an active 2 2 translational and rotational diusivities, D and D , and slip of constant magnitude jue j = u parallel to the rod's i s t r deterministic base-line swimming speeds V [31, 51]. At main axis. We can set the active slip at the rear, middle, xed H O concentration, swimming speeds are smaller or front; See Fig. 2a-c. After solving Eqs. (1,2), we up- 2 2 for asymmetric rods than for symmetric ones, therefore date the con guration with a stochastic integrator [60]. H O concentration is adjusted to maintain a comparable Here, the background
ow is linear shear: u (x) =
_zx. 2 2 velocity V between experiments. 0 X X = F; (r q) = ; (1) The background
ow pro le close to the wall U (z) is i i i measured by the drift velocity U of the suspended glass i2(1;N ) i2(1;N ) b b beads. As the beads move close to the wall, we nd M = u + ! (r q) u (r ) + ue + (2) ij j i 0 i i it important to account for the lubrication forces that j2(1;N ) act upon them [53]. The
ow velocity is estimated to be 1=2 U (r +h ) 2:5U for a thermal height h 30nm[51]. 2k T=t M W for i 2 (1; N ): B b 0 b th b th Model and Simulations. Resolving the chemi- cal and electro-hydrodynamics near a wall is challeng- Fig. 2b & c show that asymmetrically placed slip re- ing. The electro-osmotic
ow near an self-diusiophoretic sults in a contractile (or puller) dipolar
ow for front-slip swimmer is the result of charge gradients localized on a particles, and an extensile (or pusher) dipolar
ow for small surface region near the junction of the two metal- rear-slip particles. The former corresponds to physical lic segments [30]. We make the simplifying assumption long-gold particles, and the latter to short-gold. Plac- that this results in a surface slip velocity yielding the ing the slip region in the middle (symmetric swimmers) rod propulsion with the Pt segment leading. As we do { see Fig. 2a { yields a higher-order Stokes quadrupole not know the extent of the slip region, we simply assume
ow as its leading order contribution. This corresponds that it covers half the rod length. The propulsion speed to a symmetric Au/Pt particle. depends on the slip coverage. Simulation results To build intuition, we rst ex- We model the swimmer as a rigid, axisymmetric rod plore the simulations' predictions, which motivate a yet immersed in a Stokes
ow and sedimented near an in - simpler dynamical model of rheotactic response. nite substrate. The rod is discretized using N \blobs" Fig. 3a illustrates the basic rheotactic response evinced at positions (r q) with respect to the rod center q by our microswimmer model for all swimmer types [54, 55]. Linear and angular velocities u and ! satisfy (pusher, symmetric, puller). Here, Brownian
uctua- the linear system Eqs. (1,2) where are unknown con- tions are neglected, and all swimmers are initially set straint forces enforcing rigid body motion and M is the to swim downstream in a linear shear
ow. In reaction Rotne-Prager mobility tensor [56] corrected to include to the background shear each swimmer turns to swim the hydrodynamic eect of the substrate [57, 58]. upstream, with the pusher being the least responsive. 3 For symmetric swimmers, Fig. 3b shows the competition between rheotaxis induced by
ow with thermal
uctua- tions whose eect is to de-correlate the swimming direc- tion. In the absence of background
ow (
_ = 0), swim- mers diuse isotropically over long times. This yields a symmetric bimodal distribution P (V ) for the x-velocity V . As the shear-rate becomes increasingly positive, the distribution becomes asymmetric and increasingly biased towards upstream swimming (negative V ). The distri- bution curves also shift right, yielding smaller peak up- stream velocities and larger peak downstream velocities. Simulations show that active rods swim with a down- ward tilt towards the substrate, i.e. with their Pt head segment closer to the wall [35, 47]. The tilt angle de- pends weakly on the shear rate but is dierent for puller, pusher and symmetric swimmers, see Fig. 3c. It is this tilt that allows the microswimmer to respond to the shear FIG. 3: (a) Trajectories of deterministic swimmers with ini- ow near the wall, and is the origin of rheotaxis. tial orientation = =16, seen from above for simulations The fact of a nonzero tilt angle has been explored most (solid lines) and theory (dashed lines). (b) Particle velocity thoroughly by Spagnolie & Lauga [24] who, in seeking distribution in the
ow direction (V ) for hydrodynamic simu- to understand capture of active particles by spherical lations with brownian noise in a shear
ow with
_ =0s (), 1 1 4s () and 8s (), and weathervane model (lines). (c) obstacles [35], numerically studied idealized ellipsoidal From simulations, equilibrium tilt angle relative to the sub- \squirmers" moving near a spherical surface. For our nu- strate as a function of the center of the slip layer x (with merical model we associate the tilt with the appearance x = +0:5; 0; 0:5 representing front/middle/aft slip). of high (and low) pressure regions between the swimming rod and the substrate that tilt the swimmer. These re- along the
ow is hV i =
_ (h V sin =2D ) which sets x 0 r gions appear where surface velocities, both from slip and the critical speed V = 2D h= sin where the role of the 0c r no-slip regions, are converging (and diverging). Moving tilt angle is evident. the slip/no-slip boundary moves the high pressure region, From Eqs. (3)-(4) we derive the distribution P (V ) of and thus changes the tilt angle (see [51]). the swimmer velocities down the channel [51], see Fig. A weather-vane model From these observations we 3b. Although the weathervane model neglects hydrody- build an intuitive model displaying a behavior akin to namics interactions with the substrate, it agrees with the that of a weathervane. Due to its downward tilt, the full numerical simulations for the range of shear rates shear
ow imposes a larger drag on the tail of a swimming and also underlines the in
uence of parameters in
u- rod. The drag dierential promotes upstream orientation encing rheotaxis. Note that the absence of lubrication by producing a torque that depends on the tilt angle forces results in an overestimated swimmer velocity in . The rod's planar position x = (x; y) and orientation the x-direction, a discrepancy re
ected in the distribu- angle evolve as: tion peaks shifted to larger absolute values of V . p Experimental validation of the theory. In exper- x_ = V n() +
_he + 2D W ; (3) 0 x t x iments the velocity distribution P (V ) follows the same phenomenology described for the numerical simulations =
_ sin sin + 2D W : (4) and the reduced model; see Fig.4a. Under weak shear Eq. (3) describes a swimming rod that moves with in- ow we observe that passive particles (i.e. no H O ) are 2 2 trinsic speed V at an angle [n = (cos(); sin())], while washed downstream whereas all three types of active rods is advected by a shear
ow with speed
_ h at a character- orient themselves against the
ow and swim upstream. istic height h along the x-axis. Eq. 4 imposes that the As suggested by Fig. 3a, both experiments and sim- rod angle orients against the shear
ow. The particle's ulations reveal that pushers are the least robust rheo- translational and angular diusion are D and D . W tactors. Upstream swimming bias is measured by hV i t r x x and W are uncorrelated white noise processes. as a function of the shear rate, shown in Fig. 4b-d. This model is sucient to reproduce the deterministic Upstream rheotaxis is found for moderate shear rates, trajectories of symmetric, puller and pusher swimmers,
_ < 20 30s , with the characteristic non-monotonic Fig. 3a. The tilt angle controls how fast a rod re- trends previously described [46, 47]. The swimmers' orients against the
ow and it explains why pushers are ability to move against the
ow reaches a maximum at less responsive to shear
ows. The model also predicts
_ 10s . When the viscous drag overcomes the propul- a critical swimming speed to observe positive rheotaxis sive forces, i.e.
_ > 20s , the rods enter a drifting (upstream swimming). As
_ ! 0, the average velocity regime characterized by a rectilinear downstream motion 4 FIG. 4: (a) From experiments: velocity distribution P (V ) of symmetric swimmers in the absence of background
ow ( ), 1 1 with background
ow
_ = 8:7s ( ) and for immotile particles in
ow
_ = 9:5s ( ). Mean velocity vs. shear rates for (b) symmetric, (c) long-gold puller, and (d) long platinum pusher swimmers respectively in experiments (+) and simulations () and compared to experiment with immotile particles (+). Region of upstream swimming bias is shaded in gray. (hV i > 0). For large shear rates the reduced model pre- dicts a linear average velocity hV i V + h
_ . This x 0 trend is consistent with numerical and experimental re- sults of Fig. 4b-d beyond the minimum of hV i, though with slightly dierent slopes. Even in the drifting regime the average speed hV i, of active particles is smaller than immotiles ones because they are directed and swimming upstream. Both the symmetric and asymmetric swimmers' rheo- tactic behavior agrees with the numerical predictions. This result corroborates the partial slip model used in FIG. 5: A rheotactor sieve: (a) The local mean swimmer the numerical model to describe asymmetric Au/Pt dis- velocity for critical () and low ()
ow rates in a micro
u- tributions. Qualitatively, simulations indicate that the idic sieve geometry (inset). At the critical
ow rate, particles swim upstream in the wide part of the channel and down- maximum velocity upstream should be larger for puller stream in the narrow part. (b) The time-integrated swimmer and symmetric swimmers than for pushers. Experiments density pro le normalised by the average swimmer density found roughly a factor of two dierence between the max- = reveals a concentration of swimmers where the mean l tot imum upstream velocities between pushers and pullers swimmer velocity changes sign (indicated by an arrow). at comparable shear values, implying that the reorient- ing torque is strongest for pullers. This observation fur- swimmers changes little in this range of induced shear. ther agrees with the deterministic trajectories presented In a critical regime (red), the swimmers drift downstream in Fig. 3a. There, the parameter that dierentiates those from the narrow part of the channel but swim upstream swimmers' dynamics is their tilt angle , identifying it as in the wider sections. The change of sign in the local aver- a crucial parameter to engineer ecient rheotactors. age swimmer velocity corresponds roughly with a peak of A rheotactor sieve. Fig. 5a presents the concept of swimmer density, showing the accumulation of the swim- a \micro
uidic sieve" consisting of a diverging channel. mers in this region Fig. 5b (arrow). This geometry al- A constant
uid in
ux yields a decreasing shear gradient lows the sorting of motile swimmers based on their speed downstream. We show that in limited windows of
ow- or tilt angle. From the examples presented above, this rates, the rheotactors travel upstream in the nozzle until method could conveniently separate mixed populations facing a \shear wall" that prevents them from traveling of asymmetric swimmers in the same channel. further, thus concentrating them in those locations. Discussion. Through experiment, simulation, and Fig. 5b compares the symmetric swimmers' local den- modeling, we demonstrate how to modify rheotactic re- sity integrated over a period of three minutes for two sponse by changing swimmer type, which for Au/Pt dierent shear regimes. The local swimmer density, Janus rods amounts to changing the location of the along the x-direction is normalized by the average den- Au/Pt join. Rheotactic tunability is determined primar- sity within the channel, , to compare experiments ily by the tilt angle of the swimmer to the wall, which tot with dierent total numbers of microswimmers. For is controlled by the distribution of the surface slip. The small
ow-rates (blue), rheotactors swim upstream at any quantitative agreement between experiment and simu- point of the nozzle. The population density is evenly dis- lation demonstrates that we can infer \by proxy" the tributed within the whole channel as the velocity of the pusher and puller nature of arti cial microswimmers for 5 which direct
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Relating microswimmer synthesis to rheotactic tunability