Colloidal Microworms Propelling via a Cooperative Hydrodynamic Conveyor Belt

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by  Fernando Martinez-Pedrero

Colloidal Microworms Propelling via a Cooperative Hydrodynamic Conveyor-Belt
Fernando Martinez-Pedrero1, Antonio Ortiz-Ambriz1, Ignacio Pagonabarraga 2,3, and Pietro Tierno1,3∗
1Estructura i Constituents de la Mat`eria, Universitat de Barcelona, 08028 Barcelona, Spain.
2Departament de F´ısica Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain.
3Institut de Nanoci`encia i Nanotecnologia, IN2UB, Universitat de Barcelona, Barcelona, Spain.
(Dated: January 6, 2016)
We study propulsion arising from microscopic colloidal rotors dynamically assembled and driven
in a viscous fluid upon application of an elliptically polarized rotating magnetic field. Close to a
confiningplate,themotionofthisself-assembledmicroscopicwormresultsfromthecooperativeflow
generatedbythespinningparticleswhichactasahydrodynamic”conveyor-belt”. Chainsofrotors
propel faster than individual ones, until reaching a saturation speed at distances where induced-
flow additivity vanishes. By combining experiments and theoretical arguments, we elucidate the
6 mechanism of motion and fully characterize the propulsion speed in terms of the field parameters.
1
0
PACSnumbers: 82.70.Dd,87.85.gj
2

n
a Propulsion in viscous fluids plays a key role in many fect generated by the cooperative flow of the spinning
J different contexts of biology, physics, and chemistry. particlesclosetoasurface. Recenttheoreticalworks[28–

5 From a fundamental perspective, the transport of mi- 30] have addressed the rich dynamics of active rotors ly-
croscopicobjectsinaliquidmediumposestheappealing ing and spinning in the same plane. Here we rotate our
]
t challenge to find an adequate swimming strategy due to particles perpendicular to the bounding plane and use
f
o the negligible role of inertial force compared to viscous real time/space experiments to elucidate the mechanism
s one. At low Reynolds number the Navier-Stokes equa- underlying the collective propulsion.
.
t tions become time reversible [1], and any strategy based Weusemonodisperseparamagneticcolloidsofradiusa=
a
on reciprocal motion, i.e. a motion composed by sym- 1.4µm,composedbyapolymermatrixevenlydopedwith
m
metric backward and forward displacements, will fail to nanoscale superparamagnetic iron oxide grains (Dyn-
-
d produce net propulsion [2]. abeads M-270, Invitrogen). The particles are diluted
n Facing this challenge, the last few years have witnessed in Millipore water and deposited above a glass sub-
o the theoretical propositions of several suitable geome- strate where they sediment due to density mismatch.
c
tries and procedures to propel micromachines in viscous After sedimentation, each particle floats at a distance
[
fluids [3–9]. Parallel advances in miniaturization have h from the plate due to the balance between gravity

1 led to the generation of new classes of chemically pow- and electrostatic interactions with the negative charged
v
ered [10, 11] or externally actuated [12–15] prototypes substrate. The particles display small Brownian motion
6
with exciting applications in emerging fields such as mi- above the substrate ((x,y) plane), with a diffusion co-
0
8 crosurgery [16, 17] or lab-on-a-chip technology [18, 19]. efficient D = 0.14µm2s−1, and negligible out-of-plane
0 In contrast to reaction driven microswimmers, actuated fluctuations.
0 magneticmicropropellersdonotpresenttheautonomous As shown in the schematic of Fig.1(a), we have assem-
.
1 behaviourwhichdistinguishesforce-andtorque-freemo- bled the magnetic particles into translating chains by
0 tion of biological organisms [20], but instead avoid effi- applying a rotating magnetic field elliptically polarized
6 ciency reduction due to fuel shortage or directional ran- in the (x,z) plane with angular frequency ω, H(t) ≡
1
domization. To date, three main approaches have been H cos(ωt)e −H sin(ωt)e . The amplitude H and el-
: x x z z 0
v developed to transport microscopic particles in a viscous lipticity β ∈ [−1,1] of the applied field are defined as
Xi fluid using uniform magnetic fields [21], namely by actu- H0 = (cid:112)(Hx2+Hz2)/2 and β = (Hx2 −Hz2)/(Hx2 +Hz2),
ating flexible magnetic tails [12, 22], by rotating helical respectively. The applied modulation induces a finite
r
a shaped structures [23, 24], or by using the close proxim- magnetic torque on the particles, T = µ (cid:104)m × H(cid:105),
m 0
ity to a bounding wall [25, 26]. In the latter case, it is which sets them in rotation at an angular speed Ω close
well established that in the Stokes regime the rotational to the plane. Here µ = 4π · 10−7Hm−1, m is the
0
motion of a body close to a surface can be rectified into particle magnetic moment and (cid:104)...(cid:105) denotes a time av-
nettranslationduetothehydrodynamicinteractionwith erage. For high driving frequencies, ω (cid:38) 50rads−1,
the boundary [27]. Surface rotors are optimal to work in the magnetic torque arises due to a finite internal re-
confined geometries such as microfluidic channels or bio- laxation time of the particle magnetization [31–33], t ,
r
logical networks characterized by narrow pores. which in our case is t ∼ 10−4s. Upon balancing T
r m
InthisLetterweshowthatanensembleofrotorsdynam- with the viscous torque arising from the rotation in the
ically self-assembled via attractive dipolar interactions medium T = −8πηΩa3 the average rotational speed
v
(cid:112)
can be propelled by a hydrodynamic ”conveyor-belt” ef- reads (cid:104)Ω(cid:105) = H2 1−β2χt ω/6η(1 + t2ω2) [34]. Here
0 r r
2
In order to identify the range of field parameters where
propelling worms are observed, we perform a series of
experiments by varying the two amplitudes of the ap-
plied field, (H ,H ) which correspond to values of β ∈
x z
[−0.73,0.97] and H ∈ [600,2400]Am−1, and at a fixed
0
angular frequency of ω = 62.8rads−1. The graph in
Fig.1(c) shows the existence of two distinct types of dy-
namics: for low ellipticity (β <0.2), the chains rotate as
a whole, performing a 3D ”walking”-like motion as ob-
served in previous works [35, 36]. In contrast, for larger
values of β >0.45, where the out of plane component of
the field H (cid:28) H , we observe translating worms. Note
z x
that in the intermediate region both types of dynamic
states can be found. At fixed frequency (data with dif-
ferentω areincludedin[34])walkerscanbetransformed
into worms by increasing the field ellipticity. This ef-
FIG. 1. (Color online) (a) Schematic of an ensemble of para-
fect is caused by the decrease in the azimuthal part of
magneticcolloidsrotatingclosetoaglassplateandsubjected
to an elliptically polarized rotating field. (b) Microscope im- the dipolar force Fz ∼ Hz∂∂Hzz due to the corresponding
ages showing the motion of a chain (blue line) and of an in- decrease in H , and hence the relative vertical displace-
z
dividual rotor (red line) subjected to a rotating field with ment of the colloids, in favor of the in-plane part (H )
x
H0 =1500Am−1 , β =0.49, and ω =62.8rads−1 at t=0 s of the rotating field during one oscillation period. As
(top) and 8.3 s later (bottom). (c) Regions in the (H ,H )
x z a consequence, increasing β at constant ω decreases the
plane where colloidal ”walkers” [35, 36] (green), translating
oscillations of the phase-lag angle between the main axis
worms(white)orbothtypesofdynamics(orangepoints)are
observed. (ω = 62.8rads−1) (d) Average speed of the col- of the chain and the rotating field [34]. In the limit case
loidal worms (cid:104)v(cid:105) vs number of particles N. of β = 1 worms still form but the chains do not show a
net motion. As shown in Fig.1(d), at a fixed ellipticity
of β =0.49, and amplitude of H =1500Am−1, the av-
0
erage speed (cid:104)v(cid:105) initially increases with the worm length,
and later saturates to a maximum value of 5.3 µms−1.
χ = 0.4 is the magnetic susceptibility under static field Such a trend of the velocity in terms of the number of
and η =10−3Pa·s denotes the dynamic viscosity of wa- rotors is observed both by varying the frequency, or the
ter. Individualrotorstranslateataspeed(cid:104)v0(cid:105)∼(cid:104)Ω(cid:105)aex field amplitude H0.
due to the close proximity of the surface which breaks The series of experiments displayed in Fig.2 serveto elu-
the symmetry. A pair of particles, (i,j), a distance rij cidate the mechanism underlying the motion of the col-
away,experienceamagneticdipolarinteractionofenergy, loidal worms. Fig.2(a) demonstrates that the rotation
Ud = (µ0/4π)[(mi·mj)/ri3j −3(mi·rij)(mj ·rij)/ri5j], of the individual particles within the chain is an essen-
which is maximally attractive (repulsive) for particles tial ingredient for its net displacement. At the top part
withmagneticmomentsparallel(normal)tor. Perform- of Fig.2(a) a chain of N = 7 particles is propelled to-
ing a time average of the dipolar energy between two wards the right at an average speed of (cid:104)v(cid:105) = 2.0µms−1.
colloids for a rotating magnetic field in the (x,z) plane The pair of images at the bottom of Fig.2(a) shows a
(β = 0), reveals an effective attractive potential in this chain composed by the same number of particles, but
plane(cid:104)U (cid:105)=− µ0m2 ,leadingtochainingalongthex- where some of them have been previously permanently
d 8π(x+z)3
direction,whilebeingrepulsiveintheperpendicularone, linked by screening the electrostatic interactions (orange
(cid:104)U (cid:105) = µ0m2 (similar results holds for β (cid:54)= 0). Thus the segments) [37]. We observe no net translational motion
d 4πy3
actuatingmagneticfieldforcestheparamagneticcolloids when subjecting the linked particles to the same field
to assemble into elongated chains or ”worms”, while it conditions, as shown in VideoS2 in [34].
alsoensurestherotationalmotionoftheindividualunits. Next, in a separate experiment, we analyze the effect of
Thedynamicsofacolloidalwormisillustratedbythetwo the boundary on the propulsion direction and speed of
microscopic images in Fig.1(b) (VideoS1 in [34]). Under the colloidal worms. In Fig.2(b) we compare two chains
theconditionsofthisexperiment(captionFig.1),achain composed by N =5 particles and subjected to the same
of N = 11 rotors covers a distance of ∼ 25µm at an av- field conditions, but propelling close to a glass/water in-
erage speed of (cid:104)v(cid:105) = 3.0µms−1. In the same pair of terface (images at the top), or to a water/air interface
images, belowthetravelingchain, oneindividualrotoris (bottom). The water/air interface was realized by sus-
shown covering a smaller distance of ∼5.6µm at a lower pending ∼ 0.2g of surfactant-free water in a specially
speedof(cid:104)v (cid:105)=0.6µms−1,makingevidenttheenhanced prepared circular cavity of volume of 0.18cm3 and pro-
0
propulsive behaviour of the colloidal worm. vided with a small hole (2mm diameter) at the bottom.
3
FIG. 2. (Color online)(a) Sequence of images showing a worm composed by N = 7 free rotors (top) or rotors with pairs of
particles permanently linked (orange segments, bottom). (b) Propelling chain of N = 5 rotors at glass/water interface (top)
andatair/waterinterface(bottom). Fieldparametersfor(a)and(b)areH =1500Am−1,β =0.49andω=125rad s−1. (c)
0
Schematicsillustratingtherotorslocationswithrespecttotheinterfacein(b). (d)Position(x−x )versustimeforthechains
0
of particles. Black (blue) line refers to (a), grey (pink) to (b).
Surfacetensionisabletoholdthedepositeddropletwhile a solid interface additional terms must be included, as
evaporation effects are negligible during the experimen- described in [34], and T∗ =−T. Thus for a liquid/solid
tal time (see [34] for details). The paramagnetic colloids interface the torque acting on the image has the same
sediment approaching the water/air interface. However magnitude that the one actuating on the deposited par-
as observed previously [38, 39], these particles remain in ticle, but for a liquid/air interface the image torque in-
the water phase rather than protruding in the air phase creasesbyafactorη/η [42]. Hydrodynamicrectification
g
due to their corresponding small contact angle with this emergesfromtheflowinducedbytheimagesingularities
interface. As shown in VideoS2 in [34], close to the wa- because the flow will generally have a component paral-
ter/air interface the chain propels in opposite direction. lel to the interface. As a result, a particle close to an
Although the interfaces in both cases bind the motion of actuated one will be displaced by such induced flow. We
thecolloidalparticlesandinducearectificationofthero- canquantifytheeffectofthisdisplacingflowonanarray
tatingcolloids,thelocalflowinducedbythestickbound- on N equi-spaced colloids placed parallel to the wall by
ary condition characteristic of the solid wall differs from computingthevelocityatthecenterofsuchanassembly
the one generated by the slip boundary condition im- v . If we consider that each colloid moves with the total
c
posed by the liquid/air interface, leading to an opposite flowvelocityinducedbytherestofrotatingcolloidsatits
displacement of the worms. We note that similar effect position (hence assuming it experiences no friction force
hasbeenusedin[40]toreversetherotationofE.coli. We with respect to the local flow), we can arrive at [34]:
also find that the average speed of the colloidal worm at
thewater/airinterface((cid:104)v(cid:105)=−1.8µms−1)is25%higher v =v e + |v0|(cid:88)N (cid:88)(cid:20)1−ξ+ξ(cid:16)a(cid:17)−2(cid:21)×
thanthespeedclosetothesolidsurface,(cid:104)v(cid:105)=1.5µms−1. c 0 x N h
j=0i(cid:54)=j
In order to explain the hydrodynamic origin of the ob-
(cid:26) 1−ξ 3ξ(cid:15)2(i−j)2 (cid:27)
served rectification effect, we propose a model where the + e (1)
effectofaplanarinterfaceonthemotionofasolidsphere [1+(cid:15)2(i−j)2]3/2 [1+(cid:15)2(i−j)2]5/2 x
located at a distance h can be accounted for, to lowest
which shows that the instantaneous velocity depends on
order, through an appropriate set of hydrodynamic sin-
the worm length, N, and the geometrical parameter,
gularities at the same distance h below the position of
(cid:15) ≡ δ/2h, where δ is the center-to-center separation be-
the interface. For a liquid/air interface, the image re-
tween consecutive colloids in the array, while ξ = 1(0)
duces to a particle rotating in the opposite sense that
for a solid (liquid) substrate. Here v stands for the rec-
0
the actuated colloid, while for a solid surface an addi-
tifyingvelocityofanisolatedcolloidplacedatadistance
tional stresslet and source doublet must be added [41]. h above the liquid/solid interface v = T a2/(32πηh4),
0 0
A colloid subject to a torque T will rotate and gener- while v = −T /(32πη h2) for a liquid/gas interface.
0 0 g
ate a flow field, v, around it that will decay asymptot-
Therefore, one can understand both the increase in the
ically as v = −T × r/(8πηr3) where r stands for the
magnitude of the speed of a worm in the presence of
position vector from the colloid center. In the pres-
a liquid/gas interface and the change in direction with
ence of a liquid interface, the induced flow decays as
respect to the solid/liquid interface. Moreover, Eq. (1)
v = −T × r/(8πηr3) − T∗ × r∗/(8πηr3) where r∗ is
illustrates that the difference between interfaces enters
the position vector from the center of the image and
through the overall magnitude v and that the remain-
0
T∗ =−ηT/η ,beingη thegas(air)shearviscosity. For
g g ing dependence in the worm size and elevation from the
4
FIG. 4. (Color online) (a) Normalized average flow speed
(cid:104)v (cid:105)/(cid:104)v (cid:105) vs elevation from the surface z/h and calculated
f 0
following the model in [34]. (b) Colloidal worm dragging a
2µmsilicaparticle(redtrajectory),VideoS3in[34]. (c)Plots
of the distance versus time for the worm (empty circles) and
silica particle (filled squares).
FIG. 3. (Color online) Normalized average speed (cid:104)v(cid:105)/(cid:104)v (cid:105) of
0
colloidal worms as a function of the number of rotors N for
different ellipticity β (ω = 62.8 rad s−1, H = 2300Am−1).
0
Pointsdenoteexperimentaldatawhilethecontinuouslineisa age flow velocity (cid:104)v (cid:105)/(cid:104)v (cid:105) produced by a moving worm
f 0
fit following Eq.(1). Inset shows the average surface distance as a function of the rescaled elevation z/h (details of
(cid:104)d(cid:105) vs β.
the calculation are given in [34]). In the small region
between the worm and the bounding plate the flow ve-
locityisnegative,whileitchangessignabovethemoving
chain. The flow field can be visualized by tracking the
interface are purely geometrical. position of non-magnetic silica particles used as tracers
To test the model, we measure the time average velocity and floating above the plate similar to particle image
(cid:104)v(cid:105) for a series of worms characterized by different num- velocimetry, Fig.4(b). Once close to the moving chain,
ber of particles and at different values of β. Fig.3 shows the tracers are captured by the hydrodynamic belt and
the comparison between the experiment and theory by rapidly dragged along the worm till they are expelled at
fitting the rescaled velocity (cid:104)v(cid:105)/(cid:104)v (cid:105) with Eq.(1). First, the front, as shown in VideoS3 in [34]. We confirm this
0
the data show that the propulsion speed is almost inde- mechanismbymeasuringtheaveragespeedoftwodiffer-
pendent on β at parity of actuating amplitude of the ap- entcolloidal”cargos”havingsizes1µmand2µmshuffled
pliedfield. Moreover,weuseonlyoneadjustablevariable bytheconveyorbeltatanaveragespeedof8.3µms−1and
as a fitting parameter, finding (cid:15) = 0.92. From particle 7.4µms−1 resp. (here v =1.1µms−1).
0
tracking we measure the average distance between two In conclusion, we have demonstrated and theoreti-
particles (cid:104)δ(cid:105) = (cid:104)d(cid:105)+2a, being (cid:104)d(cid:105) the distance between cally supported a concept of directed propulsion at low
their surfaces, and confirm that this quantity is indeed Reynolds number based on the use of magnetically as-
independent on β, and given by (cid:104)d(cid:105) = 120nm, as shown sembled colloidal rotors. Propulsion arises due to the
intheinsetofFig.3. Closecontact((cid:104)d(cid:105)=0)betweenthe non-linearcooperativerectificationofflowsgeneratedby
particle is prevented by the electrostatic double layer in- the spinning particles close to the bounding wall. The
teractions between the particles and steric repulsion due hydrodynamicflowgeneratedbyourmagneticprototype
tothepolymercoating. Thevalueof(cid:104)d(cid:105)allowsustofind can be used as an efficient mechanism to transport bi-
an elevation of the worm from the substrate h=1.58µm ological or colloidal cargos entrapped and translated by
(surface distance from the glass plane ∼ 180nm), sim- theconveyorbelt. Ontheotherhand,trappingthelead-
ilar to the one found by evanescent light scattering of ing rotor of the worm by optical tweezers could convert
individualparticles[43]. Thequantitativeagreementbe- the magnetic propeller into a fluid pump which can be
tweenexperimentsandtheory,despitethesimplifications readily employed in micro- and nanofluidc systems.
used in the model [34], demonstrates that the latter well F.M.P., A.O.A. and P.T. acknowledge support from
captures all the basic features observed, and provides a the ERC starting Grant ”DynaMO”. P.T. acknowl-
clear description of the physical mechanism underlying edges support from the ”Ramon y Cajal” Program
the worm propulsion. No.RYC-2011-07605, from MINECO (FIS2013-41144-P)
A moving worm generates a cooperative flow field which and DURSI (2014SGR878). I.P. acknowledges support
continuously advects the colloidal chain in a similar way from MINECO (Spain), Project FIS2011-22603, DURSI
to a hydrodynamic ”conveyor-belt” [44]. This effect is Project2014SGR-922,andGeneralitatdeCatalunyaun-
illustrated in Fig.4, where we plot the normalized aver- der Program ”ICREA Acad`emia”.
5
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