Propulsion efficiency of a dynamic self-assembled helical ribbon

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 PRL 110, 168302 (2013) PHYSICAL REVIEW LETTERS week ending 19 APRIL 2013 0031-9007=13=110(16)=168302(4) 168302-1 Ó2013 Ame ican Physical Socie y 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. PRL 110, 168302 (2013) PHYSICAL REVIEW LETTERS week ending 19 APRIL 2013 168302-2 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. PRL 110, 168302 (2013) PHYSICAL REVIEW LETTERS week ending 19 APRIL 2013 168302-3 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. 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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 . PRL 110, 168302 (2013) PHYSICAL REVIEW LETTERS week ending 19 APRIL 2013 168302-4