P opulsion E iciency o a Dynamic Sel -Assembled Helical Ribbon
Nebojsa Casic,
1
Niu ka Quin e o,
2
Rena o Al a ez-Noda se,
2
F anz G. Me ens,
1
Le an Jibu i,
1
Wal e Zimme mann,
1
and Thomas M. Fische
1,
*
1
Ins i u e o Physics, Uni e si a
¨ Bay eu h, 95440 Bay eu h, Ge many
2
Ins i u e o Ma hema ics, Uni e si y o Se ille, E-41012 Se ille, Spain
(Recei ed 8 Oc obe 2012; published 15 Ap il 2013)
We s udy he dynamic sel -assembly and p opulsion o a ibbon o med om pa amagne ic colloids in a
dynamic magne ic ield. The sedimen ed ibbon assembles due o ime a e aged dipola in e ac ions
be ween he beads. The ime dependence o he dipola in e ac ions oge he wi h hyd odynamic
in e ac ions cause a wis ed ibbon con o ma ion. Domain walls o high wis connec domains o nea ly
cons an o ien a ion and negligible wis and a el h ough he ibbon. The pa icula o m o he domain
walls can be con olled ia he equency and he eccen ici y o he modula ion. The lux o wis walls—
a ue ibbon p ope y absen in slende bodies—p o ides he h us on o he su ounding liquid ha
p opels his biomime ic lagellum in o he opposi e di ec ion. The p opulsion e iciency inc eases wi h
equency and ceases ab up ly a a c i ical equency whe e he con o ma ion changes discon inuously o a
la s anding ibbon con o ma ion.
DOI: 10.1103/PhysRe Le .110.168302 PACS numbe s: 82.70.Dd, 87.15.hm
Na u e dynamically sel -assembles a ich a ie y o
swimme s o di e en size and geome y [1,2]. La ge
swimme s mo e a a high Reynolds numbe . They gene a e
eddies, he eby e icien ly p oducing ine ial h us .
Mic on sized small swimme s lack hese possibili ies and
mus use non ecip ocal less e icien con o ma ional
dynamics [3–6]. Swimming s a egies o low-Reynolds-
numbe swimme s a y. ‘‘Squi me s’’ a e s a iona y
swimme s o la ge olume- o-su ace a io ha main ain
hei shape. P opulsion is achie ed by a s eady lux o he
su ace, om a sou ce a he on o he sink a he ea .
O he swimme s change geome y cycling h ough a se ies
o con o ma ions o hei shapes. Slende swimme s [7]o
small olume- o-su ace a io use non ecip ocal bending
bea s o mo e [3,4,8]. Swimme s o mode a e olume- o-
su ace a io pe o m mo e complex shape changes o
p opel.
Technology [9] usually ies o mimic his a ie y [10]
and ebuild [11,12] such swimme s using op-down
app oaches. Bibe e e al. [12] buil a i icial lagella by
connec ing pa amagne ic colloidal pa icles wi h DNA
links o o m a semi lexible chain a ached o a la ge
pa icle a he ea o he swimme . Bending wa es induced
ia magne ic ields p opel his biomime ic swimme .
Slende body hyd odynamics, i.e., ea ing he swimme
as a one-dimensional semi lexible objec , could explain he
p opulsion mechanism. I s e iciency depends on he spe m
numbe , i.e., he a io o elas ic bending o ques o iscous
o ques o he luid.
The use o bo om-up me hods o build a i icial
swimme s is a a e excep ion success ully used only o
high-Reynolds-numbe swimme s. P ominen examples o
swimme s dynamically sel -assembled om hei compo-
nen s a e magne ic snakes [13] and ings [14]. He e,
we dynamically sel -assemble an a i icial low-Reynolds-
numbe swimme om he same pa amagne ic pa icles
used by Bibe e. A complex magne ic ield ins ead o DNA
links o ces he pa icles in o wo pa icle chains ha bind
side-by-side and o m a ibbon. In con as o Bibe e’s
swimme , ou swimme p opels due o wis o he aniso-
opic c oss sec ion o he ibbon. A heo e ical desc ip ion
beyond slende body hyd odynamics is needed o explain
his ibbon-speci ic p opulsion mechanism.
Ma hema ically, a ibbon di e s om a cu e since i
addi ionally has a one-dimensional c oss sec ion. While a
cu e can bend and wind, a ibbon can addi ionally wis .
The con o ma ion o a ibbon can be desc ibed by i s wis
and i s w i he. The sum o he wis (a local ibbon p op-
e y) and he w i he (a global con o ma ion p ope y o he
neu al line and hus a cu e p ope y) adds up o he link
numbe . Bea ing cilia o lagella can be desc ibed by
cu es and p opel by changes in w i he. In ou ibbons,
he neu al line emains a s aigh line and he w i he
anishes. The ibbon does no p opel in he la bu in he
wis ed con o ma ion. We can de ine he wis densi y as
well as a lux o wis . Open ends o he ibbons allow a
s eady lux o wis h ough he ibbon. A pa ame ically
modula ed magne ic o que ac ing on he colloidal ibbon
se es as a con ol pa ame e o he dynamically sel -
assembled shape. Shape ansi ions occu in he o m o
o =2walls ha a el along he ibbon. The numbe
de e mining he p opulsion is he a io o wis and iscous
o ques, no he spe m numbe , and he assembly is p o-
pelled by a ibbon-speci ic mechanism.
The ibbon (Fig. 1) is o med in wa e om nega i ely
cha ged (COOH) pa amagne ic Dynabeads M-270 o
adius a¼1:4m. The polys y ene beads ha e a co e
illed wi h supe pa amagne ic nanog ains ha ende he
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bead pa amagne ic. The beads we e dilu ed in Millipo e
wa e (5106bead=ml). Because o g a i y along he z
di ec ion, he colloids sedimen ed on op o a glass su ace
ha was p e ea ed wi h a solu ion o polysodium 4-s y ene
sul ona e o p e en adhesion. Wi hou a magne ic ield,
he la e al dis ibu ion o beads is andom.
The magne ic ield Hð Þinduces magne ic momen s
mð Þ¼0VHð Þ. He e, 0deno es he pe meabili y o
acuum, V he olume o he ensemble, and  he e ec i e
suscep ibili y. The magne ic momen s o he beads hence
in e ac ia dipola in e ac ions. The dimensionless Mason
numbe M¼=02H2cha ac e izes he a io o
iscous e sus magne ic in e ac ions, whe e ¼
103Nsm
2deno es he wa e iscosi y and  he modu-
la ion equency a which he di ec ion o he magne ic
ield changes. A he condi ions used he e, he Mason
numbe is la ge M>1and he mo ion o he beads is
wi h a lowe a e !< han ha o he magne ic ield
because iscous o ces a e oo s ong o allow o a syn-
ch onous (!¼) mo ion. F equencies =2>11 Hz
we e necessa y o p e en disin eg a ion o he ibbon.
Ou sys em is d i en by a magne ic ield
Hð Þ¼ ^
H½cosex exþsinex ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1Þ
peycosð Þþ
sinex ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ð1þÞ
pezsinð Þ o he a e age p ecession
angle ex and eccen ici y  ha mo es a ound he di ec o
along he xaxes. In his ex e nal ield, we conside a pai o
pa amagne ic beads sepa a ed by he bond ec o b
enclosing a pola angle bwi h he di ec o and an
azimu hal angle bwi h he yaxis (Fig. 2). The dipola
ene gy o his pai is hen gi en by Wð Þ¼
02
beadV2H2ð Þ
4 3
b
P2ðcosð ÞÞ, whe e ð Þdeno es he angle
be ween he magne ic ield and he bond ec o . The ime
a e aging can be done by measu ing bo h he o ien a ion o
he magne ic ield and he bond ec o wi h espec o he
di ec o . The ime a e aged dipola in e ac ion be ween
wo beads eads 
W¼02
beadV2^
H2
4 3½P2ðcosex ÞP2ðcosbÞ
P2
2ðcosex ÞP2
2ðcosbÞcosð2bÞ, whe e P2and P2
2a e
Legend e polynomials o deg ee 2 and associa ed
Legend e polynomials o deg ee 2 and o de 2, espec-
i ely. We assembled ou ibbons wi h a magne ic ield o
^
H¼2200 A=m, a p ecession angle o ex ¼=6, and
nega i e eccen ici ies 0:05, o which he ime
a e aged dipole in e ac ions a e a ac i e o a pai o
beads sepa a ed along he xdi ec ion (b¼0), weakly
a ac i e o indi e en along he ydi ec ion (b¼=2,
b¼0), and epulsi e along he zdi ec ion (b¼=2,
b¼=2). Since he in e ac ion is weak along he y
di ec ion, collec i e demagne iza ion e ec s in ol ing
h ee o mo e beads play a ole along his di ec ion. Yan
e al. [15] used hose collec i e e ec s o o m hollow
ubes. He e, collec i e e ec s lead o weak a ac ion in
he ydi ec ion be ween single chains bu o epulsion
be ween a ibbon and a hi d chain o an addi ional bead.
We can c ea e ibbons o ypical leng h o up o 50 beads
pe chain. De ec s in he o m o acancies on he ibbon o
adso bed beads s a ing a hi d chain can be elimina ed by
using an annealing p ocedu e [16]. The op pa in Fig. 1
shows an annealed ibbon.
FIG. 1 (colo online). Top: Mic oscopy image ( op iew) o a
healed colloidal ibbon on a glass su ace. The ibbon p epa ed
a a equency o =2<18 Hz and 0:05 lies in an
un wis ed con o ma ion. =2¼11–40 Hz: Mic oscopy im-
ages ( op iew) o he con o ma ion o he ibbon o di e en
equencies and an eccen ici y o þ0:05. A he highes
equency =2¼40 Hz, he ibbon is s anding on he glass
su ace. A low equencies, lying domains (ske ched in ed in
he scheme a he bo om) a e sepa a ed by walls (ske ched in
g ay) ha , when app oaching =2<40 Hz, spli in o =2
walls sepa a ing lying om s anding (cyan in he ske ch) do-
mains. Mo ies o he mo ion a e shown in Re . [16].
FIG. 2 (colo online). The le scheme shows he de ini ions o
angles de ined be ween he di ec o (g een a ow), he magne ic
ield, and he bond ec o . The magne ic ield p ecesses on a
do ed pu ple ellipse ha de ia es om a do ed blue ci cle bu
on a e age encloses he same angle ex as he a e age blue ield
ec o wi h he di ec o . The igh image shows he colo coded
a e aged dipole in e ac ions o a pai o dipoles in a ious
di ec ions o nega i e eccen ici y. Dipoles o m bonds in he
a ac i e (pu ple) di ec ion and a oid bonds in he epulsi e
( ed) di ec ion. Along he cyan di ec ion, he a e aged pai
in e ac ion is indi e en , and he o ma ion o bonds o no bonds
is decided by collec i e highe o de e ec s.
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Once he ibbon is healed o de ec s, we swi ch o a
posi i e eccen ici y. This u ns he ydi ec ion epulsi e
and he zdi ec ion a ac i e, a o ing an up igh o ien a-
ion. Figu e 1(40 Hz) shows he con o ma ion o 0:05
a =2¼40 Hz and a magne ic ield o ^
H¼2200 A=m
s ong enough o o ce he ibbon agains g a i y in o an
up igh con o ma ion. The sepa a ion o he ield modula-
ion equency om he a e o o a ion is s ong enough o
wipe ou all dynamic e ec s o he modula ion on o he
ibbon. This is no longe he case i we dec ease he
modula ion equency. Dynamic o ques may now dis o
he con o ma ion. The neu al line emains along he x
di ec ion, and he con o ma ion is en i ely desc ibed by
he angle bðxÞ ha he no mal ec o o he ibbon plane
encloses wi h he zdi ec ion. Fo equencies below
=2<40 Hz, wis walls be ween subsequen quasis a-
ble o ien a ions a el in he o m o soli ons wi h a speed
wall h ough he ibbon. These walls o m ia spon aneous
symme y b eaking. The ibbon always o a es wi h he
same sense as he magne ic ield. Twis walls o le and
igh chi ali y nuclea e wi h equal p obabili y and a el
in o opposi e di ec ions on he ibbon. Once a s eady s a e
is eached, only walls o one chi ali y a el on one ibbon.
Fo low equencies 11 Hz <=2<18 Hz, such walls
a e walls connec ing a lying ibbon sec ion wi h ano he
lying ibbon sec ion. Fo equencies 18 Hz <=2<
28 Hz, hese walls spli in o wo =2walls, he i s
connec ing a lying sec ion wi h a s anding sec ion and he
second connec ing a s anding sec ion wi h a lying sec ion
wis ed by wi h espec o he i s lying sec ion. Fo
equencies 28 Hz <=2<40 Hz, he =2walls
me ge again o walls ha connec wo s anding seg-
men s. Finally, abo e =2>40 Hz, a s anding la ib-
bon emains.
In Fig. 3, we show space ime plo s o he angle bðx; Þ
ex ac ed om he ideos o h ee equencies. These plo s
show he beha io o he wis walls a eling a ela i ely
la ge eloci ies and he o wa d p opulsion wi h a much
smalle p opulsion eloci y opposing he mo ion o he
walls. A spa ially mo e esol ed e sion o he p opulsion
is shown in he inse o he op igu e. The space ime
plo s span he ange xbð Þ<x<x
eð Þ, whe e xb;eð Þ¼
x0
b;e þ p op deno e he p opelling beginning and end o
he ibbon. The colo coding o he plo encodes he angle
b, ed colo s co espond o a lying (b¼0) sec ion, and
cyan colo s co espond o a s anding (b¼=2) sec ion.
The domina ing colo shows whe he he ibbon is lying o
s anding. Ab up changes in colo occu wi hin he wis
walls. walls connec egions o simila colo , while he
colo changes om ed o cyan when passing a =2wall.
In Fig. 4, we plo he domain wall and he p opulsion
speed as a unc ion o he modula ion equency. Domain
walls and p opulsion a e obse ed in he equency band
11 Hz <=2<40 Hz. We de ine a geome ic p opul-
sion e iciency e¼ p op= wall analogous o ha in
Re . [17] plo ed in he hi d g aph in Fig. 4 e sus
=2. I measu es he dis ance a ibbon p opels du ing
he mo ion o one domain wall by he wa eleng h.
The e iciency inc eases wi h equency. We exp ess i in
e ms o he equency o o a ion o he ibbon and
he densi y o walls n¼L= as e¼n p op=L , whe e L
is he leng h o he ibbon and  he wa eleng h. Since
bo h he p opulsion eloci y and o a ion equency o he
ibbon a e ai ly independen o he modula ion equency
, he e iciency inc eases as he densi y o domain walls
inc eases. I equi es so ibbons o achie e high densi ies
o walls.
We can unde s and he beha io by a damped elaxa ion
equa ion @2b=@ 2þ@b=@ ¼F=bðxÞ, whe e
F¼Rdx Ug a ðbÞþUmagnðb; Þþð@b=@xÞ2=2g
is a escaled unc ional o he ield bðx; Þwi h
Ug a ðbÞ¼jsinbj he g a i a ional po en ial and
Umagnðb; Þ¼h2
þcosð2b2 Þþ2hþhcosð2bÞþ
h2
cosð2bþ2 Þ he magne ic po en ial a ising due o
he magne ic o que densi y ac ing on o he aniso opic
c oss sec ion o he wo chains in a ibbon. hþand h
a e escaled le and igh ci cula ly pola ized magne ic
ield ampli udes. The p e ac o deno es he aniso opy
o he e ec i e suscep ibili y o he ibbon c oss sec ion.
The las e m in he unc ional deno es he o sional igid-
i y. Following [18–20], we decompose he local o ien a ion
in o a as and a slow componen b¼sþ , expand
FIG. 3 (colo online). Th ee space ime plo s o he o ien a ion
angle bðx; Þ o di e en equencies =2. The inse mag-
ni ies he egion o one end o he ibbon.
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in e ms o he as componen , and equa e he esul ing
e ms o he as componen s and he ime a e aged slow
componen s. This esul s in a ime a e aged equa ion o he
slow componen o he o m @2s=@ 2þ@s=@ 
@2s=@x2¼ e dUe =dswi h an e ec i e po en ial
o he o m Ue ¼jsinsjþ2hþhcosð2sÞ
2h2
þh2
cosð4sÞ=22and an e ec i e o ce e ¼
2ðh4
þh4
Þ=23. The slow componen ends o s ay
wi hin he minima o he e ec i e po en ial, while he as
componen will algeb aically anish wi h inc easing e-
quency, educing luc ua ions a ound he a e age o ien a-
ion a la ge equencies. Fluc ua ions ende he ime
a e aged c oss sec ion mo e iso opic. The aniso opy o
he suscep ibili y also esul s sel -consis en ly om depo-
la iza ion ields o neighbo ing c oss sec ions. When
applying an ellip ical ex e nal magne ic ield wi h he
majo axis along he zaxis, hen he eccen ici y o he
magne ic momen will be enhanced o an up igh o ien a-
ion and educed o a ho izon al o ien a ion. This beha io
can be modeled by a equency dependen inc easing
ðÞ, which shi s he global minima om b¼0 o
b¼=2wi h equency. Whe he he e ec i e po en ial
exhibi s minima a one o a bo h loca ions decides whe he
he domain walls a e walls o =2walls, espec i ely.
Fo e y high equencies, he e ec i e o ce and he
luc ua ions a ound he minima a e oo weak o o e come
he ba ie be ween wo minima sepa a ed by and he
ibbon is o ced in o he ully up igh con o ma ion, whe e
no p opulsion is possible.
In conclusion, dipola in e ac ions and g a i y o ce an
ensemble o pa amagne ic beads in o a ibbon. The ibbon
changes om a la lying ibbon ia a eling wis walls
owa d a s anding ibbon. Fo equencies whe e domain
walls a e o med, he ibbon is p opelled wi h an e iciency
ha scales wi h he domain-wall densi y.
This wo k was suppo ed by he Ge man Science
Founda ion ia he p io i y p og am 1164; he SFB 840;
FEDER-MINECO FIS2011-24540, MTM2009-12740-
C03-02, and PR2011-0123; FEDER-JA P09-FQM-4643;
and by he Humbold Founda ion SPA 1146358 STP.
*[email p o ec ed]
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FIG. 4. Domain-wall eloci y, p opaga ion eloci y, and he
p opulsion e iciency e e sus modula ion equency =2.
The e o ba s ep esen he s anda d de ia ion o e 3–7 mea-
su emen s o each poin .
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