Harnessing Thermal Fluctuations for Purposeful Activities:

The Manipulation of Single Micro-swimmers by Adaptive

Photon Nudging†

Bian Qian,a‡ Daniel Montiel,a Andreas Bregulla,b Frank Cichos,b and Haw Yang∗a

Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX

First published on the web Xth XXXXXXXXXX 20XX

DOI: 10.1039/b000000x

A simple scheme is presented for remotely maneuvering individual microscopic swimmers by means of on-demandphoto-induced actuation, where a laser gently and intermittently pushed the swimmer along its body axis (photonnudging) through a combination of radiation-pressure force and photophoretic pull. The proposed strategy utilizedrotational random walks to reorient the micro-swimmer and turned on its propulsion only when the swimmer wasaligned with the target location (adaptive control). A Langevin-type equation of motion was formulated, integratingthese two ideas to describe the dynamics of the stochastically controlled swimmer. The strategy was examined usingcomputer simulations and illustrated in a proof-of-principle experiment steering a gold-coated Janus micro-spheremoving in three dimensions. The physical parameters relevant to the two actuating forces under the experimentalconditions were investigated theoretically and experimentally, revealing that a ∼7 ◦C temperature differential onthe micro-swimmer surface could generate a propelling photophoretic strength of ∼0.1 pN. The controllability andpositioning error were discussed using both experimental data and Langevin dynamics simulations, where the latterwas further used to identify two key unitless control parameters for manipulation accuracy and efficiency; they were thenumber of random-walk turns the swimmer experienced on the experimental timescale (the revolution number) andthe photon-nudge distance within the rotational diffusion time (the propulsion number). A comparison of simulationand experiment indicated that a near-optimal micron-precision motion control was achieved.

Introduction

The capacity to manipulate a single microscopic object,navigating it toward its destination and/or along a pro-grammed path in a liquid environment could have wide-ranging implications; one might envision such applica-tions as local surgery of individual living cells, preciselytargeted transport and release of chemicals in cellularmilieu, and bottom-up assembly of complex micro- andnano-structures, to name a few.

A major challenge to realizing this overarching vision

† Supplementary Information (SI) available: the movie for the traceshown in Fig. 5; computation simulation details; calculation of theradiation-pressure force; reconstruction of the absolute orientationof the particle. See DOI: 10.1039/b000000x/a 225A Frick Chemistry Laboratory, Princeton University, Prince-ton, New Jersey 08544, USA. Fax: +1 609 258 3708; Tel: +1 609258 3578; E-mail: hawyang@princeton.edub Molecular Nanophotonics, University of Leipzig, 04103 Leipzig,Germany.‡ Current address: Mechanical Engineering, Massachusetts Insti-tute of Technology, 77 Massachusetts Avenue, Building3-264, Cam-bridge, Massachusetts 02139

is managing the omnipresent thermal fluctuations. Thesmaller the particle is, the more significant thermal fluc-tuations are to the particle’s movements; the time it takesfor the fluctuating forces to move a particle over a dis-tance of its size scales as its size squared (t ∝ a2). Ageneral strategy to overcome the influence of thermal fluc-tuations is to introduce external confining potentials. Acelebrated example of this is the laser tweezers, whichuse a very strong laser field to create a trapping poten-tial around the particle to restrict the particle’s Brown-ian motion; this way, the position of the trapped micro-particle can be changed by translating the laser beam.1

Using laser tweezers to manipulate small objects, how-ever, has several drawbacks. For example, laser tweezerslack specificity in that any object with sufficient dielectriccontrast against the medium will be trapped. In addition,the high laser power tends to result in photo-damage andheating in the sample.2 Limitations like these seriouslyconfine the scope of laser tweezers for applications in com-plex environments and living systems.

Another example is the anti-Brownian electrokinetic

1–12 | 1

Fig. 1 The adaptive photon-nudging idea. (a) Illustrationof the algorithm for adaptively steering individual half-metalcoated swimmers via laser nudging. The position and dis-placements (solid lines), as well as orientations (magenta ar-rows) of the swimmer are tracked in real time. When theswimmer orientation aligns with the target (red cross), a laserbeam (green cone) is turned on to propel the swimmer. (b)Normalized swimmer-target distance, |r|/a, as a function oftime calculated from the trajectory shown in inset (numericalsimulation). (c) Using a sequence of target points, it is pos-sible to steer a micro-swimmer along prescribed paths usingour adaptive steering strategy (numerical simulation).

trap,3 which not only confines the Brownian motion toa fixed potential, but also introduces a feedback controlto actively reduce the translational Brownian motions.The trapping and steering of nano-objects in this schemeis, however, restricted to a small volume related to theelectrode structure on the substrate, and is not readilyamenable to manipulation in complex environments.

Here, we introduce a conceptually new method to steerindividual micro-objects. This concept is based on re-cent efforts in the realization of artificial micro-swimmersutilizing a variety of propulsion mechanisms.4–8 Theseminiaturized man-made swimmers can propel themselvesalong their body axis so that, at short times, they ex-hibit directed movements in the lab frame. Due to ro-tational Brownian motion, however, the propelled swim-mers eventually lose track of their traveling direction andbehave diffusively at long times. The inescapable ther-mal fluctuations make it extremely challenging to gener-ate persistently directed motions for self-propelled micro-swimmers. Whereas naturally occurring molecular mo-tors overcome this difficulty by confining the motion toa track,9 many artificial systems use again an externalfields to restrict the rotational motion in a controlledway to maneuver the swimmers to perform complex mo-tions.10–14 In all of these cases the external field influ-

ences all swimmers in the sample at the same time, mak-ing it impossible to steer individual swimmers in an in-dependent fashion.

The new concept here is not to forcefully trounce thethermal fluctuations by an external potential but to uti-lize them to reorient and move the swimmer. We imple-ment an adaptive control strategy that is based on thefact that the swimmer propulsion can be regulated so thatit is turned on only when the swimmer has the correctorientation towards its target. This enhances the over-all movement in the desired direction relative to thosein other directions. This idea is illustrated in Fig. 1a.Even though a swimmer turns randomly, the probabil-ity for it to be in the correct alignment per unit time isfinite. Therefore, as long as the forced directional move-ments overcome the random diffusive movements withinthe time period that the swimmer is aligned, a sequence ofcontrolled on-and-off actuations will navigate the swim-mer to its destination (cf. Fig. 1b). Consequently, there isno need to impose an additional mechanism to steer theswimmer rotation; in stead, we let the omnipresent ther-mal fluctuations do the work and realign the swimmertowards the target, albeit in a stochastic manner.

Photon Nudging

To realize our control strategy, a ‘power engine’ thatcan be quickly switched on and off is crucial. We rea-son that a photo-induced propulsion system could meetthis requirement. Light-induced particle migration hasbeen known for more than a century. This general phe-nomenon can be attributed to two primary mechanisms;they are radiation pressure15–17 and photophoresis.18,19

The radiation-pressure force, caused by changes in themomentum of photons upon impinging on the particle, iswell understood and is the mechanism for laser trapping.The less-well-understood photophoretic effect is believedto be caused by the temperature gradient around a parti-cle due to the uneven light-induced heating of the parti-cle. Recently, the latter phenomenon has gained renewedinterests not only for a better understanding of its mech-anism but also for exploring possible applications,20,21

though often under the designation of thermophoresis.Historically, however, thermophoresis is not the same asphotophoresis. Photophoresis refers to the particle lo-comotion arising from a temperature asymmetry at theparticle created by light whereas thermophoresis refers toparticle or molecule population migration along a macro-scopic temperature gradient, e. g., two parallel platesmaintained at two different temperatures.22–24

Photophoresis in general can be attributed to differ-ent mechanisms, all related to temperature-dependent

2 | 1–12

interfacial properties such as interfacial tension (theMarangoni forces), the nature of the solute, or the soluteconcentration. In the case of colloids suspended in a solu-tion, the major contribution of photophoresis is thoughtto come from a distortion of the local counter ion cloudaround the colloidal particle, which has to meet the hy-drodynamic boundary conditions at the particle surfaceand also at infinity. As a result, an interfacial flow at theparticle interface is generated that drives the particle. In-dependent of the detailed mechanism, all phoretic mech-anisms reveal a particle velocity directly proportional tothe gradient of a physical quantity, which in our case isthe temperature,25–27

Uth = −DT∇T,

where DT is the thermal diffusion coefficient, which couldbe temperature dependent, and ∇T the temperature gra-dient.

In the current proof-of-principle demonstration, we em-ploy ∼1-µm gold-polystyrene Janus particles as a modelto test the proposed control scheme. A laser is focusedat the particle to switch on light-induced propulsion thatincludes both the radiation-pressure and photophoresismechanisms. The scattering and absorption of the laserphotons by the particle contributes to the radiation-pressure force that drives the particle along the laser-propagation direction. In addition, the temperature ofthe gold-coated hemisphere will rise rapidly upon laser ir-radiation to build up a local temperature gradient aroundthe Janus particle. The resulting photophoretic pull willact on the particle along the symmetry axis, with thepolystyrene hemisphere as the forward-facing end. Sinceboth types of actuation are directional and arise almostinstantaneously with the laser actuation (relative to therelevant diffusional time scales), they both contribute tothe photo-induced propulsion of the micro-swimmer.

Our actuation scheme is distinctly different from earlierparticle-manipulation works such as the laser tweezers28

because in our case, relatively much weaker driving forces(low photon fluxes) are applied intermittently rather thancontinuously. We therefore term our actuation methodusing low-light intermittent illumination ‘photon nudg-ing.’

The Light-Particle Interactions

We next investigate, theoretically, the physical param-eters under the experimental conditions. In particular,we are interested in the fractional reflectance and ab-sorbance by the particle and in the manner by whichthey may differ as the photon-nudging light interacts with

the sphere with the polystyrene first, or with the goldcoating first. These pieces of information allow us tounderstand the manner in which radiation-pressure ac-tuation is applied to the Janus particle and to estimatethe forces it generates. The photon energy absorbed bythe gold cap of the Janus particle in turn heats up theparticle but in an asymmetric way where the gold caphemispherical shell is expected to exhibit a higher tem-perature than the polystydene particle body does. Theresulting steady-state temperature distributions are alsocomputed to evaluate the temperature gradient at theparticle. The discussion hereafter assumes that the par-ticle is located at the center of a high numerical aperturemicroscope objective at all times. This assumption isvalid for the current experimental setup (vide infra)—areal-time 3D single-particle tracking spectrometer with a∼2.3 µs feedback loop time. A ∼1-µm microsphere in60% glcyerol (η = 10.9 mPa s)29 moves ∼4.3 A in 2.3µs by diffusion, well within the tracking capability of theinstrument. Indeed, when appropriate, experimental pa-rameters are used to obtain numerical values to provideadditional physical insights.

The relative reflectance and absorbance at theJanus particle

The interaction between a diffraction-limited focusedlaser beam and the gold-capped Janus particle can bewell approximated using simple ray-tracing optics be-cause the incident wavelength (532 nm) is less than thediameter of the sphere.30 As the particle is kept at thefocus of the microscope objective at all times via real-time 3D tracking, all the light rays will travel at the nor-mal angle across the medium-polystyrene, polystyrene-gold, and gold-medium interfaces (cf. inset of Fig. 2).This greatly simplifies the calculation to that of multiple-layer parallel plates with normal incidence. The fi-nite element method, implemented in COMSOL Multi-physics version 4.2a, was used to carry out the calcula-tions which took into account the multiple reflection andabsorption across all interfaces. The index of refractionfor polystyrene beads was calculated using the formula,ns = 1.5725 + 0.0031080/λ2 + 0.00034779/λ4, where thewavelength λ was in µm.31 This gives ns = 1.5878 at532 nm. The index of refraction for 60% (w/w) glyc-erol/water mixture was nm = 1.413,32 and that of wa-ter nw = 1.333.29 And finally the wavelength-dependentelectric permittivity values for gold are interpolated frompublished values for bulk materials.33 The results are dis-played in Fig. 2.

The transmittance results at zero gold-cap thicknessindicate that the ∼1-µm polystyrene sphere is practi-

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Fig. 2 Calculated coating-thickness dependence of thefractional transmittance (green), reflectance (blue), and ab-sorbance (red) of a hemispherically gold-coated polystyrenesphere in 60% glycerol. The diameter of the sphere is1 µm and the wavelength for the incident light is 532nm. The experimental gold cap thickness (50 nm) is in-dicated by a vertical bar at the bottom axis. The in-set shows a possible Janus particle orientation in a photon-nudging experiment. A photon-nudging light emanating fromthe microscope objective can interact with the Janus par-ticle through the medium→polystyrene→gold→medium in-terfaces (the right-hand line tracing the focusing cone), orthrough the medium→gold→polystyrene→medium interfaces(the left-hand line tracing the focusing cone). The solid linesare the results for the first scenario and the dashed lines thesecond scenario.

cally transparent under the experimental condition. Thisindicates that the radiation-pressure forces, will comeprimarily from the hemispherical gold coating. Fur-thermore, due to the symmetry of the Janus particle,the radiation pressure exerted on the micro-swimmerwill always be along the laser-propagation direction re-gardless of the micro-swimmer’s orientation. On theother hand, the fractional absorbance, which is responsi-ble for creating the local temperature gradient to gen-erate a photophoretic pull, is primarily from the goldcoating and is seen to quickly level out at around 40-nm thickness. The calculated total absorbance, Asg,and reflectance, Rsg, at the experimental 50-nm thick-ness for the medium→polystyrene→gold→medium rayare 0.3788 and 0.4676, respectively. On the other hand,the total absorbance, Ags, and reflectance, Rgs, for themedium→gold→polystyrene→medium ray are calculatedto be 0.4065 and 0.4285, respectively. Using these results,we next discuss the expected radiation-pressure forces.

The radiation-pressure forces

Following Roosen,34,35 Wright et al.,36 and Ashkin,30 theradiation-pressure force on a surface exerted by a ray oflight impinging on the surface at a normal angle is,

FR = (A+ 2R)I0(c

nm)−1k

A and R are the total absorbance and reflectance of thesystem, I0 the light intensity in Watts, c the light speedin vacuum, k the propagation direction of the light. Thisexpression is valid for rays passing through the center ofa sphere as in our current case. The net force is thencalculated by integrating over the cone angle swept bythe focusing laser, which is assumed to exhibit a TEM00

laser mode with a Gaussian intensity profile.The 532-nm steering laser used in the experiment was

adjusted to a beam diameter of 5 mm. With the effectivefocal length of the microscope objective estimated to be1.19 mm (based on the microscope objective back aper-ture, ∼7 mm in diameter, and the full numerical apertureof 1.4), we calculate the net radiation-pressure force to be4.06 pN/mW and 4.23 pN/mW for the cases where thelaser beam first impinged on the gold or the polystyrenesurfaces, respectively. These numbers serve as an order-of-magnitude consistency reference for comparison withexperimental measurements. A more accurate and quan-titative description should include a full electromagneticmodeling of the interaction of a Gaussian beam with theJanus particle, which is, however, beyond the scope ofthis report.

This simple ray-tracing model is expected to capturemost of the physics in radiation-pressure induced forcesunder the current experimental conditions. As the mea-surement precision and accuracy improves in future ex-periments, however, additional subtle features will haveto be explicitly taken into account. They may include theaccurate accounting of the multiple-material nature of aJanus particle, an optics model beyond the simple ray-tracing picture, and potential particle-to-particle varia-tions. We next discuss in more detail the temperaturegradient under the experimental condition.

The near-field steady-state temperature gradient

Generation of a net photophoretic pull from laser heatingrelies on the production of an asymmetric temperaturefield at and around the Janus particle.37 To gain insightsabout the temperature distribution and to evaluate thetemperature change upon laser actuation under the ex-perimental conditions, we used the COMSOL package tocarry out finite-element computer simulations. The cal-culation involves two parts: First the energy absorbed

4 | 1–12

by laser illumination is calculated as outlined previouslybut using a linearly polarized focused Gaussian beamthat is consistent with the experiment. Then the steady-state temperature distribution is calculated by solvingthe heat equation. Literature values for the density,heat capacity, and thermal conductivity for gold,29,38,39

polystyrene,40–42 and glycerol/water mixture43 are usedfor the second part of the calculation.

Fig. 3 Finite-element simulations of the steady-state tem-perature distribution of a Janus particle reveals the near-fieldnature of the temperature asymmetry. (a) Cartoon of particleorientation with respect to propagation vector of the incidentelectric field, k. (b–f) 2D cross-sections of the near-field tem-perature contours in the steady state limit. The orientationangles, θ, are 0◦, 45◦, 90◦, 135◦, and 180◦ for panels b–f,respectively.

We investigate whether the relative orientation of theJanus particle with respect to the laser propagation di-rection has an impact on the steady-state temperatureprofile. The results, displayed in Fig. 3, immediatelyshow that the temperature gradient is a near-field phe-

nomenon, consistent with the picture that phoretic forcesare an interfacial phenomenon.44 Moreover, the resultsindicate that the temperature gradient around the parti-cle in the near field is nearly symmetric with respect tothe particle axis for all orientations examined. These re-sults indicate that the net photophoretic pull should actalong the particle axis, independent of particle orienta-tions.

Fig. 4 Finite-element calculation of the average temperaturedifferences between the gold and exposed polystyrene surfacesof a 1-µm Janus particle. The results indicate that the tem-perature difference is proportional to the laser power but in-dependent of the particle orientation with respect to the laserpropagation direction. The numerical aperture in each simu-lation is 0.8. Powers are normalized against the lowest powerused in the simulations, 6.63 × 108 W/m2. The dashed linerepresents a linear fit to the θ = 0◦ data set where the slopecorresponds to 3.748× 10−8 K/(W/m2) .

The relationship between the excitation power and theparticle orientation is also examined. In the literature,the average temperature difference between the gold andexposed polystyrene sides of the particle is used to char-acterize the near-field temperature gradient.21 In the cur-rent computation, the average temperature differences forthe two surfaces are calculated and plotted as a functionof normalized excitation power. As shown in Fig. 4, themagnitude and scaling of the temperature differences forall particle orientations are linear with respect to laserexcitation power, independent of particle orientations.The maximum calculated temperature in any of the sim-ulations is ∼600 ◦C, below the melting temperature of1064.18 ◦C for gold.45 In additional to providing a the-oretical estimate of the temperature gradient generatedby the actuating laser, the results presented in this sec-tion re-enforce the idea that the gold cap in the Janusparticle can serve as the ‘engine’ compartment under the

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experimental conditions.

The Governing Equation of StochasticPropulsion

Following the above discussion, the governing equationfor a stochastic over-damped photon-nudged Janus par-ticle can be written as,

X(t) = µ[−gV∆ρz + FR(t)] + Uth(t) + ξ(t), (1)

where X represents the particle position, g the gravita-tional acceleration, V volume of the particle, ∆ρ the den-sity difference between the particle and the medium. µis the particle mobility and is related to the viscosity ofthe medium, η, by

µ = (6πηa)−1,

where a is the radius of the particle. Under the non-slipboundary condition. FR(t) is the force due to radiationpressure and Uth(t) the effective velocity due to thermalgradient around the particle. Eq. (1) is general and isexpected to be applicable to a variety of swimmer designand steering implementation.

For the current implementation, following the discus-sion in previous sections, these two propulsion mecha-nisms can be expressed as FR(t)k and Uth(t)n, in which

FR(t) and Uth(t) represent the magnitude, k and n(t) areunit vectors denoting the laser propagation direction andthe particle orientation. Both FR(t)k and Uth(t)n(t) areintermittently turned on and off depending on the ori-entation n(t) of the particle. The coordinate system ofthis equation is the laboratory frame with +z pointingup where ‘ˆ’ indicates a unit vector. ξ is the stochas-tic thermal fluctuation and has the following properties:〈ξi(t)〉 = 0 and 〈ξi(t)ξj(t′)〉 = 2Dtδ(t − t′) for i = j and〈ξi(t)ξj(t′)〉 = 0 otherwise, where i and j denotes theCartesian coordinates x, y, or z. The translational diffu-sion coefficient Dt is related to the particle mobility, µ,via the fluctuation-dissipation theory,

Dt = µkBT,

where kB is the Boltzmann constant and T temperaturein Kelvin. Here we assume that the temperature risearound the particle is much smaller than the ambienttemperature. In the experiment (vide infra), the powerdensity of the actuating laser at the swimmer is estimatedto be ∼1.9×108 W/m2. Using the finite-element simu-lations, the temperature differential is estimated to be∼7.1 K, much less than the ambient room temperature.Therefore, the locally altered viscosity and temperature

around the particle only weakly modify the translationaland rotational diffusion. In the case of strong heating, thecorresponding generalized expressions for the fluctuationdissipation theory—the so called Hot Brownian motion—have to be considered.20

The propulsion-rotation coupling is through the n(t)term, which follows the rotational diffusion equation,

dn(t)/dt = (I− n(t)n(t))ζ(t)− 2Drn(t). (2)

Here I is a 3×3 identity matrix, ζ(t) the random ro-tational motion, characterized by 〈ζi(t)〉 = 0, and〈ζi(t)ζj(t′)〉 = 2Drδijδ(t − t′) with the rotational diffu-sion coefficient Dt = 4a2/3Dr. Eq. (1) and (2) togetherfully describe the controlled motions of a photon-nudgedparticle and will be the working equation of motion forthe remainder of the article.

Since experiments have finite time resolution (say ∆t),we re-cast Eq. (1) in a finite-difference form so that it canbe directly compared with experiments. This leads to,

∆X(t)

∆t= −µgV∆ρz + µFR(t)k + Uth(t)¯n(t) +

w(∆t)

∆t,

(3)where the over-bar denotes finite-time averaging, e. g.,

FR(t) ≡ 1∆t

∫ t+∆t

tFR(t)dt, ∆X(t) is the particle displace-

ment at time t, defined by ∆X(t) ≡ X(t+∆t)−X(t), andw(∆t) is a stochastic variable for zero-mean Wiener pro-

cesses, given by w(∆t) =∫ t+∆t

tξ(t)dt with 〈w(t)〉 = 0.

Formally, the mean value of the displacement 〈∆r(t)〉can be related to the radiation-pressure force and thephotophoretic speed by taking the ensemble average ofEq. (3),

〈∆X(t)〉∆t

= −µgV∆ρz + µFR(t)k + Uth(t)〈¯n(t)〉, (4)

where 〈· · · 〉 denotes ensemble average, which can be car-ried out within a sub-ensemble of a specific orientation.Eq. (4) shows that the intermittent photon nudging canproduce deterministic displacements while the details willdepend on the experimental implementation.

To gain additional insight about the relative contribu-tions of the two propulsion mechanisms and to place thephoton-nudging strength in the context of the conven-tional laser tweezers, one may define the photophoreticstrength as,

Fth ≡ Uth/µ,

mirroring the definition of the apparent speed due to theradiation-pressure force,

UR = µFR.

It is important to note, however, that the photophoreticstrength is not a Newtonian body-centered force exerted

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externally. In the present case, there is a force actingagainst the liquid (due to the temperature gradient) sothat the micro-swimmer appears to be pulled forwardrelative to the flowing fluid that surrounding it. Thus, thephotophoretic pull strength, Fth, should be understoodas the equivalent body force that would be required tobring a particle of the same size under Stokes friction tothe phoretic speed Uth.

Computer Simulation Studies of the Steer-ing Algorithm

Through the simulation studies (see Supplementary In-formation for simulation details), we are able to identifyand define two dimensionless parameters (scaling lengthby a and time by τr = 1/Dr) that are important for thesteering control. They are the revolution number,

λ = ∆t/τr, (5)

which quantifies the number of turns the swimmer expe-riences during the experimental time scale ∆t, and thepropulsion number,

κp = Uthτr/a, (6)

which quantifies the photon-nudge distance within therotational time in terms of the particle radius. It is in-teresting to note that κp here shares a similar spirit as,but not identical to, the dimensionless Knudsen numberin fluid mechanics, which is defined as the mean-free pathdivided by the characteristic length scale of the system.

A typical simulated micro-swimmer trajectory is shownin the inset of Fig. 1(b). As expected, while the micro-swimmer does not swoop to its target, it eventually ar-rives at the destination. In the distance plot, |r| decreasessteadily with time and then fluctuates about zero whenthe micro-swimmer arrives at its destination. For eachsimulated trajectory, the arrival time ta required for nav-igating the micro-swimmer to its destination and the po-sitioning error σ in maintaining the micro-swimmer at thetarget can be characterized. Here, ta is defined as the firstpassage time for |r(t)| ≤ σ with σ2 ≡ 〈r(t > ta)2〉. Dueto the interrelated definition of ta and σ, they need tobe calculated iteratively until they converge. Note thatσ represents the motion control accuracy and sets theeffectiveness of resolving two distinguishable targets. Bysequentially navigating between a series of prescribed tar-get points, the proposed control scheme enables a micro-swimmer to move along complex paths (Fig. 1(c)).

Fig. 5 (a) Schematic of the experimental configuration. Theinset is a typical dark-field image of a 1-µm Janus particleused in the experiment. The bright hemisphere correspondsto the gold coating. The scale bar in the inset is 1µm. (b)Typical experimental trajectory of a single Janus-like micro-swimmer. The yellow dot at the top-left corner representsthe beginning of the trajectory and the yellow cross at thebottom-right corner the target. The trajectory is color codedto denote segments with (red) and without motion control(blue). (c) Normalized swimmer-target distance |r|/a as afunction of time. The vertical green lines represent the timeinstances when the nudging laser is on.

Experimental Demonstration

Encouraged by the simulation results, we proceed tocarry out an experimental validation for the proposedstrategy. The micro-swimmers we use are made of 1-µm diameter polystyrene beads hemispherically cappedwith gold coatings (∼50-nm thick). They move within a∼10-µm thick liquid chamber enclosed by two microscopecover slips and filled with 60% (w/w) glycerin/water mix-ture. The experiments are carried out using our real-time 3D single-particle tracking microscope (Fig. 5(a))46

with an all-digital feedback loop47 and synchronized con-current imaging (Cascade 650, Roper Scientific / Photo-metrics).48 To actuate the micro-swimmer, a low-power532-nm continuous-wave laser beam (∼20.9µW at thesample with a diffraction-limited spot, too weak for anydetectable trapping) is introduced to the microscope andfocused on the tracked particle through a microscope ob-jective (Leica, numerical aperture set to 0.7). The powerdensity of the actuating laser at the swimmer is estimatedto be ∼1.9×108 W/m2 by which the finite-element simu-lation predicts to cause a temperature differential of ∼7.1◦C. A computer-controlled electro-optic light modulator

1–12 | 7

is used to control the laser on-off state with an updaterate of 11 Hz. The maximum update rate of laser stateis limited by the rate of image capturing and processingfor determining particle orientations. For the purpose ofdemonstrating the steering strategy with minimum com-plication, the micro-swimmers are only steered on thetwo-dimensional image plane while they are allowed tomove freely in a three dimensional space to avoid possi-ble complications at the liquid-container interface.49–51

An experimental trajectory of the micro-swimmermovements is shown in Fig. 5(b) (See SupplementaryInformation for a representative movie). At the begin-ning of the trajectory, the control was not activatedand the micro-swimmer underwent random walks (re-gion I). Switching on the control, the micro-swimmerwas actuated intermittently and directed toward the tar-get (II). Once the micro-swimmer arrived at the destina-tion, it hovered about the destination. In region III, weturned off the photon-nudging control and lightly tappedthe sample chamber to create a disturbance in orderto move the micro-swimmer away from the target area.When the control was engaged again in region IV, themicro-swimmer trekked back to the original destination.Fig. 5(c) shows the experimentally measured swimmer-target distance |r| corresponding to this trajectory. With-out the photon-nudging control, |r| either changes slightly(I) or increases in the presence of external disturbance(III). In contrast, under our 2D photon-nudging control,|r| falls off rapidly and then maintains around a smallvalue, |r| ≈ 1.2 – 1.4µm or σ2D

expt ≈ 1.4 – 1.7µm. Aswill be discussed shortly, this is very close to an optimalphoton-nudging control.

Discussion

Experimental evaluation of radiation-pressureforce and the photophoretic strength

Since the photon-nudging laser in our experimental setuptravels along the +z direction, the equation of motionappropriate for this experiment is (cf. Eq. (4)),

〈∆X(t)〉∆t

= µ[−gV∆ρ+ FR(t)]z + Uth(t)〈n(t)〉. (7)

Here, ∆t is taken to be the exposure time of the camera(50 ms). Experimentally, the absolute orientation n ofthe micro-swimmer can be determined for each frame si-multaneously with the three-dimensional position of themicro-swimmer position X (see Supplementary Informa-tion). It allows us to average the experimentally mea-sured displacement, ∆X(t), over a certain orientation ofthe micro-swimmer to separate the photophoretic pulland the radiation-pressure force for fixed laser inputs.

We define two orientation sub-ensembles: One sub-ensemble is the collection of all the frames with n · z > 0while the photon-nudging laser is on. The average of thisensemble is denoted by 〈· · · 〉↑. The other sub-ensembleis the collection of the frames with n · z < 0 also withthe photon-nudging laser on. The average of this latterensemble is denoted by 〈· · · 〉↓. Apply Eq. (4) on thesetwo sub-ensembles and extract the z-component by mul-tiplying the equations by z,

〈∆X(t)〉↓∆t = µ(−gV∆ρ+ FR) + Uth〈n(t)〉↓

〈∆X(t)〉↑∆t = µ(−gV∆ρ+ FR) + Uth〈n(t)〉↑. (8)

Note that the Eq. (4) which is derived from Eq. (3) isvalid for both sub-ensembles since the rotational motionis independent of the translational diffusion process. Bysubtracting the above two equations, the magnitude ofthe thermophoretic force is given as,

Uth =1

∆t

[〈∆X〉↓ − 〈∆X〉↑][〈n〉↓ − 〈n〉↑]

, (9)

Analyzing the experimental data gives Uth/µ ∼ 0.103 pN,corresponding to κp ∼ 16.9 with a theoretically estimatedDr = 0.12 s−1.

During the experiment, the very weak tracking illumi-nation is always on, inevitably exerting a finite radiation-pressure forces on the swimmer along the −z direction.This weak force together with the buoyancy/gravitationterm can be experimentally characterized in situ by fol-lowing the swimmer’s drift along the −z direction whenthe laser is off. Thus, the magnitude of FR for a givenphoton-nudging illumination can be obtained by replac-ing the Uth in Eq. (7) with Eq. (9) and subtracting out thecontributions from the tracking illumination and buoy-ancy/gravitation pull. This gives an experimental valueof FR ∼ 0.051 pN, consistent with the prediction of theray-tracing model discussed earlier. We note that theseparation of the radiation-pressure force and the pho-tophoretic pull is made possible through the constructionof the stochastic equation of motion and the experimental3D tracking capability.

The evaluations of absolute values of photo-actuatedforces are of importance for the swimmer design andcontrol-parameter selection. For the swimmer design,one may wish to minimize the radiation-pressure actu-ated motion because its unidirectional nature could de-grade the control performance. To achieve that, for ex-ample, the radiation-pressure force can be balanced outusing buoyancy force −∆ρgV by adjusting the thicknessof metal coating since the absolute value of radiation-pressure force is now separable from the photophoretic

8 | 1–12

Fig. 6 (a) Experimentally measured probability density func-tion (PDF) of the apparent swimmer speed Uth = |∆r|/∆t,where measurements are made with ∆t = 90 ms. (b) Experi-mentally measured probability distribution of the angles φ be-tween consecutive displacements ∆r and the swimmer-targetvector r. The gray and white bars represent the swimmerwith and without laser actuation, respectively. The solid anddashed lines are obtained from numerical simulations using anexperimentally evaluated κp = 16.9 and κp = 0 for cases withand without the actuating laser, respectively.

effect. For motion control, it will be seen in later discus-sion that to achieve optimal control the time duration ofnudging pulse is determined by the photophoretic driftspeed.

The controllability of the adaptive strategy

To evaluate the controllability of our steering strategy,we first calculate the dimensionless apparent speed of themicro-swimmer, defined as |∆r|τr/(a∆t) at ∆t = 90 ms.The results, displayed in Fig. 6(a), show that the micro-swimmer is more energetic when actuated, exhibiting agreater average speed and a broader speed distribution.This enhanced motility is necessary in order to overcomethe Brownian motion to achieve motion control. Motilityamplification alone, however, does not guarantee that themicro-swimmer will move faithfully to the target becauseits orientation changes during ∆t due to rotational ran-

dom walk. Thus, another important parameter is the di-rectionality of the actuated micro-swimmer, which can bequantified by measuring the directional angles φ betweenits consecutive movements ∆r(∆t) and position vectorsr (Fig. 6(b)). It is clear that without actuation, φ dis-tributes evenly whereas upon actuation, φ peaks at zerowith a narrow distribution. The accentuated alignment of∆r with r in steered micro-swimmers confirms that pho-ton nudging indeed navigates the micro-swimmer towardthe destination. These measurements are further corrob-orated by numerical simulations (solid lines in Fig. 6).

The conditions for optimal control

We next turn to conditions for optimal control, whichwill have important implications in swimmer design andexperimental implementation. We focus on the most in-tuitive merit function—the positioning error. The di-mensionless positioning error is defined by σ/a, which isa function of control parameters λ and κp. For fixed λ, asshown in Fig. 7, there exists an optimal κ◦p at which thepositioning error is minimum. For κp < κ◦p, translationalrandom walks become dominant so that the swimmer mo-tion is weakly controlled, i.e., Uth∆t�

√2Dtτr. On the

other hand, for κp > κ◦p, the swimmer is over corrected

by the forced movements, i.e., Uth∆t�√

2Dtτr, result-ing in large positioning error when it is near the target.Therefore, optimally, the propulsion must be tuned tostrike a fine balance between the forced motion and theBrownian motion. Note that the optimal κ◦p decreaseswith λ, approximately following a straight line on thedouble logarithmic axes. The line has a slope of -2 forthe 2D case and -4 for the 3D case, which means that un-der optimal control, the propulsion speed must inverselyscale with the selected pulse duration, Uth ∝ ∆t−

14 . For

fixed κp, the control accuracy decreases with λ becauseduring an actuation action, the rotational diffusion causesn to deviate from r and this deviation is proportional to∆t1/2.

We note that the dimensionless positioning error σ/ais not explicitly dependent on the swimmer size a. Infact, the estimated minimum 2D and 3D positioning er-rors from simulations are σ2D

min/a ≈ 2.6 and σ3Dmin/a ≈ 3.8,

respectively, applicable to swimmers of all sizes with cor-rectly chosen λ and κp. This implies that the smallerthe swimmer, the more precisely its motion can be con-trolled. To understand this observation, we note that thepositioning error arising from the diffusive movements be-tween two control pulses scales linearly with the particlesize,

√2Dtτr ∼ a, because the waiting time between de-

sired swimmer orientations is approximately proportionalto the particle’s rotational time, τr = 1/Dr. To attain

1–12 | 9

Fig. 7 Computer-simulation studies of control conditions forthe two-dimensional (a) and three-dimensional (b) cases. Thedimensionless position errors σ/a are plotted as a functionof the control parameters, the revolution number λ = ∆t/τrand the propulsion number κp = Uthτr/a. The dotted linesapproximate the minimum σmin/a. The control condition forthe proof-of-principle experiment discussed in this work is in-dicated by the red dot in (a). The plots are generated fromover 100 simulated swimmer trajectories.

good control accuracy for smaller swimmers, however,one has to reduce the actuation duration while raising theinput power, since ∆t ∼ τr ∼ a3 and Uth ∼ τ−1

r a ∼ a−2.We further note that as the swimmer becomes smaller, sois the timescale (∝ a−2) for the buildup or dissipation oftemperature fields around the swimmer, suggesting thateven very small particles can be effectuated by a pho-tophoretic control mechanism.

Concluding Remarks

We have put forward an original strategy for controllingthe motion of individual micro-swimmers. The two new

ideas embodied in it are adaptively regulating the actu-ation according to the swimmer’s orientation, and pho-ton nudging where a weak light is applied to the micro-swimmer intermittently to generate propulsion.

Using a Janus particle as an illustrating example, wenot only have shown the first proof-of-principle exper-iment but also have sketched the key mechanistic ele-ments that are anticipated to be important for future de-velopments. We have calculated the expected radiation-pressure force and have found that, with a very goodapproximation, the radiation-pressure force follows thelaser propagation direction. For the less well-understoodphotophoretic effect, we have used finite-element simula-tions to study the temperature differential between thegold and the polystyrene surfaces of the Janus particle,and have found that the near-field temperature gradientfollows the Janus particle symmetry axis regardless ofthe particle’s relative orientation to the actuating laser.These pieces of information have allowed us to constructthe governing equations of the stochastically controlledmotions, providing the physical underpinning for the pro-posed control scheme (Eq. (1) and Eq. (2)). Through thestudy of the governing equations, we have identified twodimensionless control parameters: they are the propul-sion number κp which quantifies the photon-nudged dis-tance within the rotational time in terms of the parti-cle radius, and the revolution number λ which quantifiesthe number of turns the swimmer experiences during thecharacteristic experimental time.

The steering scheme has been validated via numeri-cal simulations, and implemented in a proof-of-principleexperiment. Under the experimental conditions, theradiation-pressure force has been found to be ∼0.05 pN,consistent with the theoretical prediction of a simple ray-tracing model. On the other hand, the photophoretic pullstrength has been found to be ∼ 0.1 pN, as a result of ∼7◦C temperature differential between the surfaces of thegold cap and the polystyrene body according to finite-element simulations. We point out that the experimentalevaluation of the radiation-pressure force and the pho-tophoretic pull strength has been made possible by theformulation of the governing equations together with theuse of real-time 3D single-particle tracking technology.

Though demonstrated using a model swimmer and aparticular control implementation, the concept of adap-tive steering should be useful for a broad range of ap-plications since it is not necessary to know the explicitdetails of the swimmer nor the environment in which itnavigates, as long as the swimmer speed can be rapidlymodulated. For example, it is now conceivable that thetechnology can be further developed to drive a micro-swimmer in such a complicated environment as cellular

10 | 1–12

milieu. Also become realizable is the simultaneous con-trol of multiple micro-swimmers, which not only couldprovide new ways for micro-fabrication but also couldserve as a new platform for the basic research on the in-terfacial fluctuations at the mesoscopic scales.

Extending to the nanometer scale, the orientation ofthe nano-swimmer would have to be determined usingthe now-mature single-molecule or single-particle tech-niques. The relative contributions between the radiation-pressure force and the photophoretic pull would dependon the configuration of the nano-swimmer. For exam-ple, for Janus nanoparticles, the phorophoretic velocity isnot expected to depend on the particle size because thescaling of temperature gradient and gold film absorptioncross section will cancel out. Nonetheless, our controlscheme makes use of rotational random walks to align theswimmer, which could prove advantageous in nanoscalemotion control because of the significantly reduced reori-entation time for nano-swimmers. More generally, thiswork serves as an example showing that it is possible toexploit thermal fluctuations, in this case rotational ran-dom walk, to do purposeful works—a feat that has beenroutinely achieved by biological systems.

Acknowledgments

This work was supported by the National Institute ofStandards and Technology, the Camille and Henry Drey-fus Foundation, Princeton University, and the Eric andWendy Schmidt Transformative Technology Fund (toHY), as well by the Deutsche Forschungsgemeinschaftwithin the Research Unit FG 877 (to FC).

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