Organization of Paramagnetic and Nonmagnetic Colloidal Particles in Ferrofluid

O ganiza ion o Pa amagne ic and Nonmagne ic Colloidal Pa icles in Fe o luid Von de Uni e si ä Bay eu h zu E langung des G ades eines Dok o s de Na u wissenscha en (D . e . na .) genehmig e Abhandlung on Ayan Ray gebo en am 27. No . 1983 in Kalku a/Indien 1. Gu ach e : P o . D . Th .M. Fische 2. Gu ach e : P o . D . W. Köhle Tag de Ein eichung: 17.04.2012 Tag des Kolloquiums: 21.06.2012 Dedica ed o my belo ed a he P o . K. K. Ray (baba) and mo he M s. S. Ray (maa) Acknowledgemen I would like o hank and exp ess my since e g a i ude o P o esso D . Thomas Fische o his kind and ca ing guidance, and cons an suppo du ing my doc o al s udy a he Uni e si y o Bay eu h, Ge many. I would also like o hank him specially o in oducing me o he new wo ld o "Dynamics o So Ma e " in physics. This hesis wo k would no ha e been possible wi hou his scien i ic ad ice, pe sonal guidance and unde s anding. P o esso Thomas is mo e like a iend han a supe iso o me. I ake his oppo uni y o hank all my g oup membe s Uli Lange , Nebojsa Casic, Tobias Geh ing, Saeedeh Aliaska isohi and Ch is iane Jungnickel who helped me wi h hei scien i ic and echnical knowledge apa om hei pe - sonal help whene e equi ed. Ge ing such a galaxy o iendly colleagues in one place is ha d o ind now-a-days. Al oge he i was a small amily wi h p ecious swee memo ies. I would also like o hank all he membe s o Expe imen al Physics V o hei cons an suppo whe he echnical o scien i ic discussions. A his poin I would like o hank specially M s. Ca men Ke ling o helping me in he IT sec o , M . Klaus Oe e o his assis ance in manu ac u ing o di e en machined componen s a Bay eu h and ou g oup sec e a y M s. Ch is ine Linse helping me wi h he o icial, adminis a i e wo ks and he i aluable sugges ions owa ds socie al ela ions. A e all, a cup o co ee is an unequalled medium o es o e and keep up spi i s. Taking his oppo uni y I would like o hank he Welcome Cen e o he Uni e si y o Bay eu h and pe sonally D . Co nelia Nicodemus o he help and guidance h oughou my s ay a Bay eu h. I ake his oppo uni y o exp ess my hea el hanks o my con empo a y esea ch colleagues and iends - Sonal Di, Swas ik Da and boudi (Adi i), D . Himad i da and boudi (Dolon), Im an, Somna h, P a ap, Andy, Mo i z, Ch is ian and Ma ion and many o he s o he Uni e si y o Bay eu h o hei kind help a equi ed momen s and o making my s ay a Bay eu h a happy and memo able one. All along his wo k my belo ed iend Sayan i has emained as a sou ce o inspi a ion, and assu ance; I exp ess my hea el deep g a i ude o Sayan i o he lo e, pa ience and men al company a all momen s o make his achie e- men a eali y. Finally, I would like o hank my pa en s, o always being wi h me h ough- ou my s udies and o hei endless lo e and suppo . Thei lo e and mo i a- ion was one o he key o he success o his disse a ion. Ayan Ray Con en s 1 In oduc ion 1 1.1 In oduc ion............................. 1 1.2 Colloidal lowe ........................... 5 1.3 T ansi ions eng h......................... 6 1.4 Colloidalphases........................... 9 2 Ma e ials and Me hod 13 2.1 Ma e ials .............................. 13 2.1.1 Fe o luid .......................... 13 2.1.2 Magne icField ....................... 15 2.1.3 Op ical Mic oscopy . . . . . . . . . . . . . . . . . . . . . 15 2.2 Me hod ............................... 16 2.2.1 Dynamics o sel -assembly o lowe -shaped magne ic col- loidalclus e s........................ 16 2.2.2 The ansi ion s eng h om solid o liquid colloidal dipola clus e s in p ecessing magne ic ields . . . . . . . 17 2.2.3 Magne ic ield con olled composi e pa amagne ic-diamagne ic colloidalphases....................... 20 iii 3 Colloidal lowe 23 4 T ansi ion s eng h 31 5 Colloidal phases 41 6 Summa y 65 Lis o Figu es 1.1 a) schema ic ep esen a ion o colloidal lowe -shaped clus e s [7] o med in a pe pendicula ield Hz. The cen e pa icle is a pa amagne ic pa icle, which is he co e o he lowe and he pa icles a ound he co e a e he diamagne ic pa icles ha a e e e ed as pe als o he lowe . b) ep esen s X, Y and Z a e he coo dina e axis wi h Hx, Hyand Hza e he ex e nal magne ic ield espec i ely. .......................... 6 1.2 a)Schema ic ep esen a ion o he beha iou o supe -pa amagne ic and nonmagne ic pa icles in an applied magne ic ield when im- me sed in a hin ilm o e o luid; Figu e 2a indica es ha he di ec ion o he applied magne ic ield is in he z di ec ion. Fig- u e 2b shows a mix u e o nonmagne ic and supe -pa amagne ic pa icles imme sed in hin ilm o e o luid be ween wo glass co e slips unde he in luence o he magne ic ield. Figu e 2c e eals he alignmen o dipole momen o e o liud and Figu e 2d shows he e ec i e magne ic momen o he magne ic and nonmagne ic pa icles unde an ex e nal applied magne ic ield. I can be no ed ha Figu e 2c and Figu e 2d can combine o o m Figu e 2b. He e ep esen χ he suscep ibili y ac o . . . . 7 1.3 a) schema ic ep esen a ion o diamagne ic clus e o med in a o a ing magne ic ield H|| in x-y plane. A co e o pe al size a io is chosen o o m he colloidal clus e . b) ep esen s X, Y and Z a e he coo dina e axis wi h Hx, Hyand Hza e he ex e - nal magne ic ield wi h H|| being he in plane o a ing e ec i e magne ic ield and being he p ecession angle. Ωis he ex e nal applied equency........................... 9 1.4 Schema ic ep esen a ion o magic angle. . . . . . . . . . . . . . 10 1.5 Schema ic p esen a ion o he beha iou o nonmagne ic pa i- cles unde a o a ing magne ic ield when imme sed in a hin ilm o e o luid. Figu e 5a indica es he di ec ion o he ap- plied magne ic ield in he x-y plane. Figu e 5b shows wo di - e en sizes o nonmagne ic pa icles imme sed in hin ilm o e o luid be ween wo glass co e slips unde he in luence o he o a ing magne ic ield. Figu e 5c e eals he alignmen o dipole momen o e o liud and Figu e 4d shows he e ec i e magne ic momen o nonmagne ic pa icles unde an ex e nal applied o a ing magne ic ield, assuming he e ec o e o luid o be negligible. I can be obse ed ha Figu e 5c and Figu e 5d can combine o o m igu e 5b. . . . . . . . . . . . . . . . . . 11 1.6 Schema ic ep esen a ion o he ex e nal magne ic ield applied ~ H( ) = ˆ Hcos ϑex ~ez+ˆ Hsin ϑex (~exsin Ω +~eysin 2Ω ). H ( ) is he o al ex e nal magne ic ield s eng h applied o he sample. Whe e x,y and z a e he coo dina e axises. . . . . . . . . . . . . 12 2.1 a) schema ic ep esen a ion o sample on op o solenoid and b) Hz being he ex e nal s a ic magne ic ield in he z di ec ion. He e x, y and z a e he coo dina e axises. . . . . . . . . . . . . . 17 2.2 a) Schema ic ep esen a ion o a angemen s o i e se s o solenoid coils and b) he combined o a ing magne ic ield H|| and he pe pendicula ield H wi h being he angula equency and he p ecessionangle........................... 19 2.3 a) schema ic ep esen a ion o he a angemen o solenoid coils and b) ime dependen magne ic ield p oduced by he i e se s o solenoid coils simila o Lissajou cu e. . . . . . . . . . . . . . 21 1.1. INTRODUCTION CHAPTER 1. INTRODUCTION si ua ion, all pa amagne ic beads ha e magne ic momen s ha poin in o he same di ec ion. We can en ich he s uc u e o he assembly [5] by inco po a - ing diamagne ic pa icles. Such diamagne s eac o an ex e nal ield wi h a magne ic momen an i-pa allel o he ex e nal ield. Since diamagne ic suscep- ibili ies o mos ma e ials a oom empe a u e a e small, we mus use a ick o ob ain e ec i e diamagne s. This ick consis s o imme sing nonmagne ic colloids in o a e o luid. When using a e o luid wi h suscep ibili y be ween he ze o suscep ibili y o he non-magne ic colloids and he suscep ibili y o he pa amagne ic colloids he pa amagne ic colloids s ill ac as pa amagne s while he non-magne ic beads ac e ec i ely as diamagne ic pa icles in he backg ound o he e o luid. Such e ec i e diamagne s un unde he name magne ic holes. In chap e 5, we expose a mix u e o pa amagne s and mag- ne ic holes [1] [3] o ime dependen ex e nal ields o sel assemble he mix u e in o a ious s uc u es. The ques ion add essed in his chap e is which ype o aniso opic s uc u es o he mixed sys em may be assembled when using a ious o ms o ex e nal magne ic ield modula ions. To answe hese ques ions I ha e a anged he hesis in o he ollowing s uc u e: chap e 1 includes a b ie in oduc ion o he hesis wi h mo i a ion as a subsec ion. Expe imen al de ails ha e been p o ided in he subsec ion i led me hodology wi h he ma e ials pa ame e o chap e 2. Chap e 3, chap e 4, and chap e 5 a e he a ached published manusc ip s wi h he esul s and conclusion. Finally, chap e 6 includes he summa y. 4 CHAPTER 1. INTRODUCTION 1.2. COLLOIDAL FLOWER 1.2 Dynamics o sel -assembly o lowe -shaped magne ic colloidal clus e s In chap e 3, we we e in e es ed o s udy he e ec s o dynamic in e ac ions o pa amagne ic and nonmagne ic pa icles in a 1-dimensional sys em. We obse ed single ile di usion p esen in ou sys em. Single ile di usion e e s o he 1-dimensional mo ion o in e ac ing pa icles in po es, which a e so na ow ha he mu ual passage o such pa icles is excluded. Since he sequence o pa icles in such a si ua ion emains una ec ed o e ime , leads o de ia ion om no mal di usion. Such a single ile di usion o colloids in 1-dimensional ha e al eady been epo ed [C. Lu z e al, 2004]. Whe e he colloidal pa icles we e apped by a scanning lase beam o a ci cula op ical ap. Ou sys em consis s o pa amagne ic and nonmagne ic pa icles imme sed in e o luid unde s a ic magne ic ield (magne ic ield s eng h ~ H( ) = ˆ H~ez, z-di ec ion), sandwiched be ween wo glass co e slips. Unde such condi ions lowe shaped magne ic colloids a e o med, whe e he pa amagne ic pa icle is a he cen e i.e. he co e o he lowe and he nonmagne ic pa icles a e a he equa o , he pe als, shown in Figu e(1.1). Ex e nal magne ic ield in- duces magne ic momen s in he pa icles ha in e ac s ia he dipole dipole in e ac ion. Due o he p esence o s a ic magne ic ield in he sys em o magne ic and nonmagne ic pa icles imme sed in e o luid (chap e 3), he e ec i e dipoles i.e. he magne ic dipole minus he e o luid backg ound o he wo so s o pa icles poin in o opposi e di ec ions Figu e(1.2). Hence, in p esence o s a ic magne ic ield he nonmagne ic pa icles imme sed in e o luid beha es as diamagne s and he pa amagne s beha es s ill as pa a- magne s. These diamagne s a e a ac ed owa ds he co e (pa amagne ) o 5 1.3. TRANSITION STRENGTH CHAPTER 1. INTRODUCTION Figu e 1.1: a) schema ic ep esen a ion o colloidal lowe -shaped clus e s [7] o med in a pe pendicula ield Hz. The cen e pa icle is a pa amagne ic pa icle, which is he co e o he lowe and he pa icles a ound he co e a e he diamagne ic pa icles ha a e e e ed as pe als o he lowe . b) ep esen s X, Y and Z a e he coo dina e axis wi h Hx, Hyand Hza e he ex e nal magne ic ield espec i ely. o m a ci cula channel. A ound he co e he diamagne s ha e a epulsi e o ce be ween each o he and in e ac by so -co e in e ac ions. The mo ions o hese in e ac ing pa icles (diamagne s) in he ci cula a ay made us mo- i a ed o s udy and cha ac e ize he single ile di usion in he sel -assembled lowe -shaped magne ic colloidal clus e s. 1.3 The ansi ion s eng h om solid o liquid colloidal dipola clus e in p ecessing mag- ne ic ields Due o he p esence o ha d-co e and dipola in e ac ions p esen in he mag- ne ic colloidal lowe sys em we can s udy he in luence o long ange in e - ac ions on o he single ile di usion chap e 3. Long- ange in e ac ions also 6 CHAPTER 1. INTRODUCTION 1.3. TRANSITION STRENGTH Figu e 1.2: a)Schema ic ep esen a ion o he beha iou o supe -pa amagne ic and nonmagne ic pa icles in an applied magne ic ield when imme sed in a hin ilm o e o luid; Figu e 2a indica es ha he di ec ion o he applied magne ic ield is in he z di ec ion. Figu e 2b shows a mix u e o nonmagne ic and supe -pa amagne ic pa icles imme sed in hin ilm o e o luid be ween wo glass co e slips unde he in luence o he magne ic ield. Figu e 2c e eals he alignmen o dipole momen o e o liud and Figu e 2d shows he e ec i e magne ic momen o he magne ic and nonmagne ic pa icles unde an ex e nal applied magne ic ield. I can be no ed ha Figu e 2c and Figu e 2d can combine o o m Figu e 2b. He e ep esen χ he suscep ibili y ac o . play an essen ial ole o phase ansi ions be ween di e en ly o de ed phases. A i s -o de phase ansi ions exhibi a discon inuous change in he o de pa- ame e . The change o one phase o o he occu s ia a coexis ence o he wo phases. The a ea o he hys e esis measu es he dissipa ed ene gy when a e sing he coexis ence egion back and o h. Whe eas a second o de phase ansi ion is a ansi ion whe e he o de pa ame e changes con inuously a he ansi ion. Second o de ansi ions a e associa ed wi h c i ical beha io o esponse unc ions as a unc ion o he con ol pa ame e while i s o de ansi ions exhibi no c i ical beha io . The o ma ion and up u e o lowe -shaped magne ic colloidal clus e s can be conside ed as a ini e size phase ansi ions. I was a ques ion o in e es , is he change in he hys e esis could e eal he s eng h and o de o he phase ansi ions in he sys em. The o ma ion and up u e o he lowe - 7 1.3. TRANSITION STRENGTH CHAPTER 1. INTRODUCTION shaped magne ic colloidal clus e s and diamagne ic clus e s akes place wi h he change in he p ecession angle, he con ol pa ame e . A hys e esis loop is obse ed when uned he p ecession angle, om low o high and ice e sa. Whe he he s udy o he wid h o he hys e esis could e eal he o de o he sys em? Besides, is i possible o de ine he o de and s eng h by he measu ing he esponse unc ion, he angula eloci y o he pa icles as a unc ion o change in he p ecession angle? These lowe -shaped magne ic colloidal clus e s we e o med om pa amag- ne ic and nonmagne ic pa icles imme sed in dilu ed e o luid unde a s a ic magne ic ield in he z-di ec ion and sandwiched be ween wo glass co e slips (Figu e (1.1)). Whe eas he diamagne ic colloidal clus e s we e o med om nonmagne ic pa icles imme sed in concen a ed e o luid unde a o a ing ield Figu e(1.3) and sandwiched be ween wo glass co e slips. The magne ic ield s eng h being ~ H( ) = ˆ H(~exsin Ω +~eycos Ω ), wi h Ωbeing he angula equency in x-y plane. The lowe -shaped magne ic colloidal we e s able a low angles and nea by he magic angle hese s uc u es we e uns able whe eas, he diamagne ic clus e s we e s able a high angles and hei s abili y dec eased eaching owa ds he magic angle. He e he magic angle ϑmagic is he de- ined as a unique angle, which is app oxima ely 54.73◦. I is he oo o a second-o de Legend e polynomial P2(cos θ)=0and in e ac ions depending on his second-o de Legend e polynomial anishes a his angle. Ma hema - ically ϑmagic =θm= a c an √2≈54.73◦,Figu e(1.4). As ex e nal magne ic ield induces magne ic momen s in he pa icles and hey in e ac ia dipole dipole in e ac ion. The e ec i e dipoles (diamagne ic clus e o ma ion) i.e. he magne ic dipole minus he e o luid backg ound o he wo so s o pa i- cles poin in o same di ec ions (x-y plane), shown in Figu e(1.5). Simila ly, in 8 CHAPTER 1. INTRODUCTION 1.4. COLLOIDAL PHASES Figu e 1.3: a) schema ic ep esen a ion o diamagne ic clus e o med in a o a ing magne ic ield H|| in x-y plane. A co e o pe al size a io is chosen o o m he colloidal clus e . b) ep esen s X, Y and Z a e he coo dina e axis wi h Hx, Hyand Hza e he ex e nal magne ic ield wi h H|| being he in plane o a ing e ec i e magne ic ield and being he p ecession angle. Ωis he ex e nal applied equency. bo h he sys ems o colloidal lowe and diamagne ic clus e due o he p esence o ex e nal magne ic ield he pa icles in e ac ia dipole dipole in e ac ion. 1.4 Magne ic ield con olled composi e pa amag- ne ic-diamagne ic colloidal phases Neu aliza ion o opposi e cha ge is one o he majo concep s in o dina y ma e whe e wo opposi e cha ges cancel each o he . The in e ac ions aking place be ween hese opposi e cha ges is iso opic and is independen o di ec- ion. This cha ge neu aliza ion is he key owa ds he o ganiza ions o ma e on he a omic and molecula scale leading o sel -assembly. I is spon aneous b eaking o o a ional symme y [4] and he quan iza ion o angula momen um ha p oduces c ys alline s uc u es wi h o ming di ec bonds in a oms and molecules. Whe eas, neu aliza ion p ocess is di e en in case o mesoscopic 9 1.4. COLLOIDAL PHASES CHAPTER 1. INTRODUCTION Figu e 1.4: Schema ic ep esen a ion o magic angle. sized pa icles due o he absence o he quan um phenomena and angula momen um being a con inuous quan i y. In a colloidal sys em he di ec bond o ma ion does no wo k. S e ic in e ac ions a e he means o spon aneously b eak he o a ion symme y o o m colloidal c ys al o iso opic s uc u es. Di ec bond in colloidal sys em a e only possible using in insically aniso opic colloidal pa icles e.g. Janus o ellipsoid pa icles. One o he o he possibili ies o use he magne ic o elec ic dipole momen using an ex e nal ield. In case o a mix u e o pa amagne ic and nonmagne ic pa icles imme sed in a magne ic luid unde magne ic ield. The e ec i e dipole momen induced due o he same ex e nal magne ic ield esul s in poin ing he dipoles in o opposi e di ec ion o pa amagne ic and nonmagne ic pa icles. The induced magne ic momen neu alizes each o he simila ly like he cha ge neu aliza ion, o ming ich a ie y o aniso opic sel -assembled s uc u es. An a emp has been made o s udy his cha ge neu aliza ion o magne ic momen s in an ex e nal magne ic ield esul ing in o ming di e en aniso opic s uc u es. 10 CHAPTER 1. INTRODUCTION 1.4. COLLOIDAL PHASES Figu e 1.5: Schema ic p esen a ion o he beha iou o nonmagne ic pa icles unde a o a ing magne ic ield when imme sed in a hin ilm o e o luid. Figu e 5a indica es he di ec ion o he applied magne ic ield in he x-y plane. Figu e 5b shows wo di e en sizes o nonmagne ic pa icles imme sed in hin ilm o e o luid be ween wo glass co e slips unde he in luence o he o- a ing magne ic ield. Figu e 5c e eals he alignmen o dipole momen o e o liud and Figu e 4d shows he e ec i e magne ic momen o nonmagne ic pa icles unde an ex e nal applied o a ing magne ic ield, assuming he e ec o e o luid o be negligible. I can be obse ed ha Figu e 5c and Figu e 5d can combine o o m igu e 5b. Ou sys em consis s o pa amagne ic and nonmagne ic pa icles imme sed in e o luid unde a magne ic ield ~ H( ) = ˆ Hcos ϑex ~ez+ˆ Hsin ϑex (~exsin Ω + ~eysin 2Ω )as shown in Figu e (1.5), sandwiched be ween wo glass co e slips. We use magne ic ield wi h h ee di e en equencies wi h ze o- equency,Ω and 2 -Ω equency along di e en axes. This magne ic ield was applied o he pa icles such ha he e is no o que. Dipola in e ac ions a e aniso opic and di e in sign o in e ac ions be- ween simila (pa amagne ic o diamagne ic ) pa icles and opposi e (pa am- agne ic and diamagne ic ) pa icles. The composi e s uc u e o a mix u e o diamagne ic s and pa amagne s is he e o e expec ed o exhibi a ich a ie y o s uc u es. These s uc u es will be explo ed in chap e 5. 11 1.4. COLLOIDAL PHASES CHAPTER 1. INTRODUCTION Figu e 1.6: Schema ic ep esen a ion o he ex e nal magne ic ield applied ~ H( ) = ˆ Hcos ϑex ~ez+ˆ Hsin ϑex (~exsin Ω +~eysin 2Ω ). H ( ) is he o al ex- e nal magne ic ield s eng h applied o he sample. Whe e x,y and z a e he coo dina e axises. 12 Chap e 2 Ma e ials and Me hod 2.1 Ma e ials 2.1.1 Fe o luid Fe o luid is a complex luid, which has magne ic p ope ies like solid while being a luid in i s physical s a e. The e o luids con ain iny magne ic ma- e ials o he o de 10 −12 nm in size in a liquid medium. These nanome e - sized pa icles a e coa ed wi h a s abilizing dispe sing agen , which p e en s pa icle agglome a ion e en unde an applied s ong magne ic ield g adien . Depending on he medium, hese e o luids can be classi ied ei he as (a) oil based o (b) wa e based. Fo he cu en expe imen s, wa e based e o lu- ids we e p ocu ed om Fe o ec Fe osound. Fe o luid EMG 705 and EMG 707 we e wo wa e -based e o luids used o he p esen expe imen s. The EMG 705 has a sa u a ion magne iza ion a 22 mT wi h magne ic suscep ibil- i y o 4.04 (SI uni s) whe eas he EMG 707 has 11 mT wi h suscep ibili y o 1.51 (SI Uni s) [Fe o ec Fe osound USA]. Supe -pa amagne ic beads Sphe i- 13 2.2. METHOD CHAPTER 2. MATERIALS AND METHOD u ed using a Leica high-speed came a (Leica DFC 360 FX). The dynamics o he colloidal lowe s and clus e s o ma ion we e analyzed by using image- p ocessing echniques in wi h he help o a comme cially a ailable so wa e package (MATLAB) and open sou ce packages such as ImageJ and Vi ual Dub. 2.2.3 Magne ic ield con olled composi e pa amagne ic- diamagne ic colloidal phases Sample P epa a ion: A mix u e o pa amagne ic pa icles (diame e 2a = 2.8µm) wi h nonmagne ic luo escen ( ed) polys y ene pa icles (diame e 2a =1.0µm) imme sed in concen a ed e o luid EMG 707.was p epa ed in con- olled p opo ions ( pa amagne ic 2 : nonmagne ic 4 by olume). This mix- u e was igo ously shaken o o m a homogenous mix u e. Using a pipe e a small amoun 0.5µlo his mix u e was placed a he cen e be ween wo p e-cleaned glass co e slips. Ex e nal Field and Op ical Mic oscopy: The sample was placed on op o a solenoid, shown in Figu e(2.3). A combina ion o s a ic magna ic ield in he z-di ec ion was applied wi h an in plane ime dependen magne ic ield . This sample was obse ed unde luo escence mic o- scope in a e lec ing mode. Red luo escence il e was used o obse e he ed luo escence pa icles whe eas he Pola iza ion il e was used o obse e he non- luo escence pa amagne ic pa icles. Obse a ions and eco ding: Chang- ing he s a ic magne ic ield aniso opic s uc u es e ol ed in 2-dimension and 3-dimension. A high s a ic magne ic ield H 26.5 mT colloidal lowe s a e ob- se ed whe e as dec easing his magne ic ield esul s in o ming 3-dimensional anis opic sandwiched s uc u e. Whe e he pa amagne s a e a he middle 20 CHAPTER 2. MATERIALS AND METHOD 2.2. METHOD Figu e 2.3: a) schema ic ep esen a ion o he a angemen o solenoid coils and b) ime dependen magne ic ield p oduced by he i e se s o solenoid coils simila o Lissajou cu e. laye and he diamagne s a e on he ei he sides o he pa amagne s. Mo ies o hese colloidal lowe s, sandwiched s uc u es, deco a ed s ings we e cap- u ed using a Leica came a (Leica DFC 360 FX). 21 2.2. METHOD CHAPTER 2. MATERIALS AND METHOD 22 Chap e 3 Dynamics o sel -assembly o lowe -shaped magne ic colloidal clus e s 23 CHAPTER 3. COLLOIDAL FLOWER Dynamics o sel -assembly o lowe -shaped magne ic colloidal clus e s A. Ray, S. Aliaska isohi, and T. M. Fische , Phys. Re . E 82, 031406 (2010) Copy igh by The Ame ican Physical Socie y 2010 DOI: 10.1140/epje/i2008-10421-5 24 Dynamics o sel -assembly o lowe -shaped magne ic colloidal clus e s A. Ray, S. Aliaska isohi, and T. M. Fische * Ins i u e o Physics, Uni e si ä Bay eu h, Bay eu h 95440, Ge many 共Recei ed 11 May 2010; published 24 Sep embe 2010兲 In a s a ic magne ic ield pa amagne ic and nonmagne ic colloids imme sed in a e o luid sel -assemble in o luc ua ing colloidal lowe s. Adso p ion and deso p ion o nonmagne ic pe als o la ge pa amagne ic co es and changes in he pe al con o ma ion a ound he pa amagne ic co e induce a luc ua ing dynamics. We ack he mo ion o colloidal pe als on he pa amagne ic co e. Adso p ion and deso p ion o pe als occu on a la ge ime scale han he o a ional di usion o he pe als. Magne ic dipole in e ac ions spli he mo ion o he pe als in o di e en modes o o a ional di usion. Modes o o a ional di usion ha change he pe al con o ma ion a e supp essed compa ed o he con o ma ion in a ian o a ional di usion o all pe als. The supp ession o highe modes o o a ional di usion esul s in a subdi usi e dynamics o he indi idual pe als. DOI: 10.1103/PhysRe E.82.031406 PACS numbe 共s兲: 82.70.Dd I. INTRODUCTION Colloidal assemblies a e mesoscopic sys ems in he mo- dynamic equilib ium. Unde s anding he complex s uc u es o hese assemblies, he so in e ac ions be ween he indi- idual pa icles, and he esul an dynamics in eal space is o cu en in e es ; because colloidal assemblies a e being used as models o a omic c ys als 关1兴 o glasses 关2兴, o an de Waals c ys als 关3兴, and as sys ems o he s udy o dynamic sel -assembly 关4,5兴. The so ness o he in e ac ions gi es ise o luc ua ions a ound he equilib ium ha allows ob- se ing di ec ly he anspo p ocesses 关6–8兴which lead o he dynamic sel -assembly o he sys em. Di usion is con- side ed as one o hese basic passi e means o i e e sible anspo in o equilib ium. I a ises om luc ua ions o he pa icle eloci y due o s ochas ic o ces. These o ces ac on he di using pa icles due o collisions wi h o he pa icles om a ese oi a a ce ain empe a u e. In he p esence o s ochas ic and de e minis ic mic oscopic o ces, mac oscopic di usion can be exp essed as he ze o h momen o he pa - icle eloci y au oco ela ion and/o c oss-co ela ion unc- ions 关9兴. Kubo 关9兴ex ended a gene alized concep o di u- sion ha allows de ining and measu ing he di usion o in e ac ing pa icles. I has been shown by E b e al. 关5兴 ha pa amagne ic and nonmagne ic colloidal pa icles imme sed in a e o luid can sel -assemble in o colloidal lowe s in a s a ic magne ic ield. The colloidal lowe s esul om he e ec i e dipola a ac ion o he pa amagne ic colloids in which nonmagne ic pa icles beha e as magne ic holes in he e o luidic backg ound. The dipole in e ac ion is a enso ial aceless in e ac ion ha depends on he angle be ween he magne ic momen s and he pa icle sepa a ion. Fo holes si - ing a he pole posi ions abo e o below he pa amagne ic bead he dipole in e ac ion wi h he pa amagne ic bead is epulsi e. In he equa o ial plane on he o he hand i is a - ac i e. The dipole in e ac ion be ween wo magne ic holes on he o he hand is epulsi e in he plane no mal o he magne ic momen s and a ac i e along he di ec ion o he magne ic momen s. The plana s uc u e o he colloidal lowe s is a esul o he complex angula dependency o he dipola in e ac ions. He e, an a emp has been made o measu e he no mal modes o di usion, as well as he adso p ion and deso p ion kine ics o he pe als in colloidal lowe s using he concep p oposed by Kubo 关9兴. Kubo gene alized he concep o di - usions o si ua ions whe e he pa icle kine ics is a supe - posi ion o andom mo ion and di ec ed in e ac ions ha o ce he pa icles in o de e minis ic di ec ions. The in e ac- ions co ela e he mo ion o he pa icles ha would o he - wise show a degene a e indi idual di usion. The co ela- ions spli he indi idual di usion in o s a is ically independen no mal modes o di usion. I is demons a ed ha he adso p ion and deso p ion kine ics as well as he mode dependence o he no mal modes o pe al di usion can be unde s ood by he compe i ion o dipola o ces wi h he luc ua ing o ces om he iscous ca ie luid. II. EXPERIMENT We s udy he supe pa amagne ic Dynabeads M-270 ca - boxylic acid, 2.8 ␮ m in diame e 共Ca . No. 143.05 D兲ob- ained om In i ogen Dynal 共Oslo, No way兲, and Flu o- Max ed luo escen polyme mic osphe e beads wi h 1.0 ␮ m diame e 共Ca . No. R0100兲ob ained om Duke Scien i ic 共Palo Al o, CA兲. The pa icles om Dynal a e supplied in concen a ions o app oxima ely 2⫻109beads ml−1 共10–30 mg ml−1兲and om Flu o-Max supplied wi h concen a ion o app oxima ely 1% olume ac ion suspended in wa e and espec i e su ac an . Pa a- magne ic pa icles a e mixed wi h nonmagne ic pa icles and dilu ed e o luid EMG 705 Fe oTec Fe osound 共Fe oTec GmbH, Ge many兲wi h con olled p opo ions depending on he expe imen . Elec ic cu en o 0.43 A was supplied o he wa e -cooled coils o p oduce a magne ic ield o 10.0 mT, machined a Uni e si y o Bay eu h. The mix u e o he beads wi h e o luids was aken on a p ecleaned glass slide wi h a co e slip o educe he ai d i . S a ic magne ic ield om he zdi ec ion was applied o he sample and was ob- se ed unde he LEICA DM4000B 共Leica Mic osys ems We zla GmbH, Ge many兲 luo escence mic oscope h ough 63⫻pola iza ion lens in e lec ing mode. Videos we e cap- *[email p o ec ed] PHYSICAL REVIEW E 82, 031406 共2010兲 1539-3755/2010/82共3兲/031406共6兲©2010 The Ame ican Physical Socie y031406-1 u ed using a colo cha ge-coupled de ice Basle came a 共Basle A311 c兲high ame a e om Basle AG, Ge many. III. ADSORPTION AND DESORPTION Nonmagne ic beads o adius a=0.5 ␮ m in a dilu ed aqueous e o luid 共EMG 705 Fe o ec Fe osound/wa e =1:4兲adso b a and deso b om he pa amagne ic beads o adius R=1.4 ␮ m. When hey adso b hey o m a colloidal lowe wi h one pa amagne ic bead a he co e o he lowe su ounded by se e al nonmagne ic beads o ming he pe als. A ypical colloidal lowe is depic ed in Fig. 1. The assembly is a dynamic s uc u e and he numbe o pe als N共 兲 luc u- a es as a unc ion o ime because nonmagne ic beads adso b a and deso b om he pa amagne ic co e. I we assume a Bol zmann dis ibu ion o he numbe o pe als we may ex- ac he po en ial ene gy o adso p ion o Nbeads U共N兲as U共N兲−U共N e 兲=−kBTln 冉 共N兲 共N e 兲 冊 ,共1兲 whe e 共N兲deno es he o al ime when one inds he colloi- dal lowe wi h Npe als, N e deno es a e e ence numbe o pe als, and Tis he empe a u e. In Fig. 2we plo he adso p- ion po en ial as a unc ion o he numbe o pe als ob ained ia Eq. 共1兲by measu ing N共 兲o e a ime du a ion o 4000 ideo ames. The adso p ion po en ial shows a p onounced minimum nea six pe als. Assuming he po en ial o a ise ia dipola a ac ion o he nonmagne ic beads o he pa amag- ne ic co e and due o dipola epulsion be ween he equally spaced nonmagne ic pe als, we p edic a po en ial o U共N兲=4 ␲ ␮ 0 ␹ F 2H2a3 9共R/a+1兲3N 冋 − 冉 ␹ p ␹ F −1 冊 R3 a3 +1 2兺 j=1 N−1 1 8 sin3共j ␲ /N兲 册 .共2兲 In Eq. 共2兲 ␮ 0deno es he acuum pe meabili y, ␹ Fand ␹ pa e he e ec i e suscep ibili ies o he e o luid and o he pa a- magne ic pa icle, and His he ex e nal magne ic ield. The po en ial has a minimum o an equilib ium numbe o pa - icles gi en app oxima ely by Neq =2 ␲ 冑3冑 ␹ p ␹ F −1R3/2 a3/2.共3兲 The dashed line in Fig. 2shows a i o he expe imen al da a 共solid line兲ob ained om Eq. 共1兲 o he heo e ical p edic ion in Eq. 共2兲using ␹ P=0.082 and ␹ F=0.063. No e ha he he- o e ical i exhibi s a minimum a ound N=7 ins ead o he alue N=6 in he expe imen . The 2N-dimensional con o ma ional space o he pe als is spanned by he posi ions 共 j, ␸ j,j=1,...,N兲o he pe als. In an N- old colloidal lowe he equilib ium con igu- a ion is de e mined by he con o ma ion j=R+aand ␸ j=2 ␲ j/N共j=1,...,N兲. A ansi ion o a 共N−1兲- old lowe happens when, o example, he N h pe al sepa a es om he lowe 共 N→⬁兲and he emaining N−1 pe als ea ange hei angula posi ions ␸ j共j=1,...,N−1兲. We desc ibe he eac ion pa hway o such a con o ma ional change by he eac ion coo dina e ⌬ . The posi ion o he N h pe al is N=R+a+⌬ N, ␸ N=0 and he o he beads adap he posi ions j=R+a, ␸ j= ␣ 共⌬ N兲+2关 ␲ − ␣ 共⌬ N兲兴共j−1兲/共N−2兲. The angle 2 ␣ 共⌬ N兲desc ibes he angle be ween he i s and he 共N−1兲 h pe als ha eadjus 关 om ␣ =2 ␲ /N o ␣ = ␲ /共N−1兲兴, while he N h pe al lea es he lowe 共see op in Fig. 3兲. We compu e he eac ion pa hway such ha he emaining pe als j=1,...,N−1 adjus hei posi ions o he ene gy minimum o he dipola ene gy o he Npe al sys em while he N h pe al is ixed a he posi ion N=R+a+⌬ N. Usually no signi ican changes in ene gy a e compu ed when he sepa a ion ⌬ No he lea ing pe al has exceeded ⌬ N⬎4 ␮ m. Hence, sepa a ions la ge han 4 ␮ m can be conside ed as quasi-in ini e sepa a ions. In Fig. 3we plo he dipola ene gy e sus he eac ion coo dina es ⌬ N共N=3,...,11兲 o a cascade o ansi ions om an 11- old colloidal lowe owa d a lowe wi h wo pe als. The cascade om he 11- olded lowe o he heo e ical mini- mum lowe wi h se en pe als is plo ed on he le side. The emaining cascade om he minimum se en old lowe o- wa d a wo-pe al lowe is plo ed a he igh . The eac ion coo dina es al e na e be ween he lowe 共e en N兲and uppe FIG. 1. 共Colo online兲共a兲Fluo escence mic oscope image o a six-pe aled colloidal lowe and 共b兲scheme o a colloidal lowe . The pa amagne ic co e pa icle is non luo escen and hence no is- ible in he luo escence image. The nonmagne ic luo escence pe al pa icles a e isualized as b igh spo s in he luo escence mic o- scope image. FIG. 2. Adso p ion po en ial o he colloidal pe als. The solid line is ob ained om he expe imen al da a by using Eq. 共1兲. This po en ial le els o nea 5kBTdue o lack o e en s. The dashed line is a i acco ding o Eq. 共2兲. RAY, ALIASKARISOHI, AND FISCHER PHYSICAL REVIEW E 82, 031406 共2010兲 031406-2 共odd N兲axes. Numbe s indica e equilib ium lowe s o he co esponding numbe o pe als. The po en ial hus changes om he Npe al lowe ene gy EN o he 共N−1兲pe al lowe ene gy EN−1. The po en ial o a Npe al lowe wi h he N h pe al a a dis ance ⌬ =5 ␮ m is indis inguishable om he po en ial ene gy o a 共N−1兲-pe aled lowe . This con i ms ha a pe al a a dis ance ⌬ ⬎5 ␮ m can be conside ed as ully sepa a ed om he lowe . Fo he deso p ion o he se en h pe al he ene gy exhibi s a maximum EAalong he eac ion pa hway. This maximum co esponds o a ansi ion s a e, i.e., a saddle poin in con o ma ional space loca ed a a dis ance ⌬ 7,max⬇0.7 ␮ m om he minimum posi ion o he se en h pe al wi h an ac i a ion ba ie o he deso p ion o 共EA−E7兲⬇0.7kBT. The ac i a ion ene gy o he adso p ion is 共EA−E6兲⬇0.5kBT. A quali a i ely simila ansi ion s a e is compu ed be ween he se en- and eigh -pe aled lowe s. All o he ansi ions in he numbe o pe als show no ansi- ion s a e. Hence, all lowe s wi h N⬍6 and N⬎8 a e un- s able. The six- and eigh -pe aled lowe s a e me as able E6,E8⬎0, and he se en old lowe is he s able con o ma- ion E7=0 o he gi en pa ame e se . Assuming an A hen- ius beha io o he a e cons an k6→7o he adso p ion p o- cess o he se en h pe al one would expec a a e cons an o he o de k6→7=kBT 6 ␲ ␩ a共⌬ max兲−2exp关−共EA−E6兲/kBT兴,共4兲 whe e ␩ =10−3 Nsm −2 is he e o luid iscosi y. Inse ing he alues ⌬ max⬇0.7 ␮ m and 共EA−E6兲⬇0.5kBT om Fig. 3in o Eq. 共4兲we ob ain k6→7⬇0.3 s−1. In Fig. 4we plo he au oco ela ion unc ion o he pe al numbe , 具 ␦ N共 兲 ␦ N共 + ␶ 兲典,共5兲 whe e ␦ N共 兲=N共 兲−Neq deno es he pe al numbe luc ua ion. The au oco ela ion unc ion decays wi h a ypical a e o kex⬇0.3 s−1 in good ag eemen wi h he es ima e gi en by Eq. 共4兲. Fo la ge imes ␶ ⬎10 s he expe imen al au oco - ela ion unc ion becomes s a is ically un eliable since he numbe o e en s 共⬀ ␶ meas- ␶ 兲d ops o 1 as he ime sepa a- ion ␶ app oaches he ime ␶ meas o he measu emen . IV. PETAL CONFORMATION AND DYNAMICS Once he pe als adso b o he pa amagne ic co e he e is some eedom o con o ma ion, and one obse es lowe s wi h pe als equally spaced a ound he co e as well as con o - ma ions whe e he pe als a e c owded a one side o he co e. We de ine he one-dimensional densi y o pa icles as ␳ =N/⌬ ␾ ,共6兲 whe e ⌬ ␾ deno es he minimum angula ange o e which he Npe als a e dis ibu ed and 2 ␲ −⌬ ␾ is he la ges gap FIG. 3. 共Colo online兲共Top兲Scheme o a N-pe aled lowe los- ing he N h pe al along he eac ion coo dina e ⌬ N, while he an- gula posi ions o he emaining pe als adjus . 共Bo om兲The po en ial-ene gy cascade om a 11-pe aled lowe ia he s able VII pe al lowe 共le 兲 owa d a wo-le el lowe 共 igh 兲. The lowe loses he N h pe al along he eac ion coo dina e ⌬ N; black cu es co espond o he deso p ion o a N=e en pe al 共lowe abscissa兲, and g een 共g ay兲cu es co espond o he deso p ion o a N=odd pe al 共uppe abscissa兲. The ene gy o a pe al sepa a ed by ⌬ N=5 ␮ m is indis inguishable om an in ini ely sepa a ed pe al and hence equals o he ene gy o a 共N−1兲-pe aled lowe . The numbe s labeling he ends o he cu es co espond o he numbe o he pe als in he lowe . The ansi ion s a e be ween six old and se en old pe al lowe s 关 ed 共black兲a ow兴is a a dis ance o ⌬ =0.7 ␮ m om he equilib ium posi ion o he se en h pe al and has an ac i a ion ene gy o EA=0.7kBT. FIG. 4. 共Colo online兲The au oco ela ion unc ion 具 ␦ N共 兲 ␦ N共 + ␶ 兲典 e sus ime as ob ained om he expe imen al da a 共solid line兲. The numbe o pe als changes on a ime scale o 3 s. The dashed line co esponds o an exponen ial decay wi h a e con- s an 0.3 s−1. The s a is ical e o 共e o ba s兲o he co ela ion unc ion inc eases when he ime lag ␶ app oaches he ime o mea- su emen ␶ meas=70 s. DYNAMICS OF SELF-ASSEMBLY OF FLOWER-SHAPED …PHYSICAL REVIEW E 82, 031406 共2010兲 031406-3 be ween he pe als. We compu e he po en ial ene gy o a con o ma ion U共 ␳ 兲as U共 ␳ 兲−U共 ␳ e 兲=−kBTln 冉 g共 ␳ e 兲⌬ ␳ e 共 ␳ ,⌬ ␳ 兲 g共 ␳ 兲⌬ ␳ 共 ␳ e ,⌬ ␳ e 兲 冊 ,共7兲 whe e 共 ␳ ,⌬ ␳ 兲is he o al ime when he pe als in he lowe show a densi y in he in e al 关 ␳ , ␳ +⌬ ␳ 兴and whe e g共 ␳ 兲⬀ 冉 N ␳ −N ␳ hc 冊 N−2 共8兲 is he leading-o de app oxima ion o he con igu a ional space densi y 关10兴a ailable o con o ma ions o densi y ␳ , whe eas ␳ hc=共R/a+1兲/2 is he maximum 共ha d-co e兲pack- ing densi y o he pe als a ound he co e. Figu e 5shows he po en ial U共 ␳ 兲compu ed ia Eq. 共7兲 o lowe s consis ing o an a bi a y numbe o pe als. The esolu ion ⌬ ␳ a ies wi h ␳ and is chosen in a way so as o ensu e ha 共 ␳ ,⌬ ␳ 兲⬎0 o all ␳ . Since he da a a highe po en ial a e spa se he eso- lu ion 1/⌬ ␳ is bes a he minimum and dec eases when mo ing owa d highe po en ial. We ind he lowes po en ial o densi ies ␳ ⬇1 co esponding o a hexagonal a angemen o he pe als wi h equal spacing o ␲ /3 be ween he pe als. The pe al con o ma ion esul s om he simul aneous mini- miza ion o he pe al numbe and he minimiza ion o he dipola epulsion be ween he pe als. The dipola epulsion be ween he pe als, howe e , is weak and allows o signi i- can luc ua ions a ound a con o ma ion. We he e o e acked he angula posi ion ␾ j共 兲关j=1,2,3,...,N共 兲兴 o he adso bed pe als as a unc ion o ime. The accu acy o he acking o ␾ j共 兲was be e han 2°. The angula equency ␻ j共 兲= ␾ ˙j共 兲o each indi idual pe al is a luc ua ing unc ion o ime. We measu e he angula equency using ini e di - e ences o he angula posi ions o consecu i e ames. The ame a e o he came a was 30 ames pe second. We de ine he au oco ela ion unc ion o he angula equency o wo pe als o a colloidal lowe wi h Npe als as CN共 ␴ , ␶ 兲=具 ␻ j共 兲 ␻ j⫾ ␴ 共 + ␶ 兲 ␦ „N共 兲−N… ␦ „N共 + ␶ 兲−N…典. 共9兲 He e, ␴ deno es he neighbo numbe 共 ␴ =0 is he same pa icle, ␴ =1 is he nea es neighbo , e c.兲. Bo h del a unc- ions ␦ (N共 兲−N)and ␦ (N共 + ␶ 兲−N)disca d all imes whe e he pe al numbe de ia es om he ixed pe al numbe N om he co ela ion. In Fig. 6we plo C6共 ␴ , ␶ 兲 e sus ␶ o ␴ =0,1,2,3. The angula equencies a e co ela ed o ze o ime delay 共i.e., ␶ =0兲, showing ha pa o he pe al di usion can be conside ed as a Ma ko ian p ocess on he ime scale ␶ ⬎0.03 s o he measu emen . The mos p ominen obse - a ion is ha neighbo ing pe als a e no s a is ically indepen- den . As does he pe al au oco ela ion unc ion C6共0, ␶ 兲, he pe al c oss-co ela ion unc ions C6共 ␴ ⫽0, ␶ 兲also show he same albei weake ins an aneous posi i e co ela ion. This is a dynamic p oo o he de e minis ic in e ac ion o he pe als. Apa om his posi i e co ela ion a weak an ico ela ed decay is obse ed o he au oco ela ion C6共0, ␶ 兲and he c oss co ela ion C6共 ␴ ⫽0, ␶ 兲 o ␶ ⬎0.05 s 共see he inse in Fig. 6兲. I is a measu e o he e a da ion o he in e ac ion. In single ile di usion 关11–13兴, whe e pa icles in e ac only ia ha d-co e epulsion, a s ong algeb aic an ico ela ion signi ican ly al e s he di usion o he pa icles. Neighbo ing pa icles in single ile di usion emain unco ela ed a sho imes and become an ico ela ed only a imes ypical o he indi idual di usion ime needed o encoun e each o he . The e a da ion o such a ha d-co e in e ac ion is signi ican . Single ile di usion becomes mos p ominen in he he mo- dynamic limi N→⬁, whe e he ime scale o he simul a- neous co ela ed di usion o he igid lowe sepa a es om he indi idual di usion o he pe als. Ou sys em di e s om a sys em exhibi ing single ile di usion. I has a small numbe o pe als, and he pe als in e ac ins an aneously ia he so dipola in e ac ions; e- a da ion e ec s a e weak. In no ime a e he pe als allowed o di use indi idually. Hence, he ela i ely weak delayed an ico ela ion ollows he ins an aneous del a co ela ion wi h a ela i e sho delay. The di usion cons an o he pe - als is gi en by hal he a ea unde he au oco ela ion unc- FIG. 5. E ec i e pe al po en ial as a unc ion o he pe al den- si y ␳ as ob ained om he expe imen al da a ia Eq. 共7兲. The dashed line is a linea i . FIG. 6. 共Colo online兲Angula equency au oco ela ion and c oss-co ela ion unc ions o a colloidal lowe wi h six pe als. The black line co esponds o he au oco ela ion, while he ed, blue, and g een lines co espond o c oss co ela ions be ween nea - es 共 ␴ =1兲, second-nea es 共 ␴ =2兲, and hi d-nea es 共 ␴ =3兲neigh- bo s, espec i ely. RAY, ALIASKARISOHI, AND FISCHER PHYSICAL REVIEW E 82, 031406 共2010兲 031406-4 ion. While he ini e ame a e o he came a b oadens he expe imen al co ela ion unc ion, he a ea unde he co e- la ion unc ion is no a ec ed by he con olu ion o he da a wi h he ime esolu ion unc ion o he came a. Hence, he di usion cons an s ha e no signi ican dependence on he ame a e o eco ding, DN共 ␴ 兲= 冕 0 ⬁ d ␶ CN共 ␴ , ␶ 兲.共10兲 Equa ion 共10兲is Kubo’s 关9兴gene aliza ion o he concep o di usion o pa icles ha in e ac . The in e ac ion o he pa - icles causes he mo ion o one pa icle o s a is ically depend on he mo ion o ano he . The s a is ically dependen mo ion o he pa icles can be decomposed in o s a is ically indepen- den no mal modes o mo ion. In Fig. 7we plo he di usion cons an D6共 ␴ 兲 e sus ␴ . The pe als beha e like being coupled by so sp ings, wi h pe als no di using indepen- den ly, bu wi h neighbo s pe o ming a co ela ed di usion. The co ela ion dec eases when mo ing away owa d u he dis an neighbo s. We may decompose he co ela ed mo ion o he pe als in o unco ela ed no mal modes o di usion ia ␾ 共m, 兲=1 冑N兺 j=1 N e2 ␲ imj/N ␾ j共 兲.共11兲 The co esponding s a is ically independen di usion con- s an s o he no mal modes, DN共m兲=1 N兺 ␴ =1 N e2 ␲ im ␴ /NDN共 ␴ 兲,共12兲 a e plo ed in Fig. 8. The mode m=0 has he highes di u- sion cons an , and he di usion cons an dec eases wi h he mode numbe m. The mode m=0 co esponds o a igid o- a ion o all pe als by he same amoun . I he e o e co e- sponds o he o a ional di usion o he en i e lowe ha lea es he con o ma ion o he lowe unchanged. The highe modes m⬎0 in ol e ela i e mo ion o pe als ha change he con o ma ion. Such modes a e supp essed o di use by he dipola epulsion be ween he pe als. The highe is m, he sho e is he dis ance 2 ␲ /mbe ween pe als ha a e mo ing in opposi e di ec ions. The mos likely con o ma ion is an equilib ium con o ma ion such ha an m⫽0 mode usually aises he dipola ene gy o he sys em. This explains why he di usion o highe modes 兩m兩⬎0 is supp essed by he dipole-dipole in e ac ion. Con a y o single ile di usion he di usion mode o he pe als a ises om mos ly ins an aneous esponse o he lowe o con o ma ional changes. In single ile di usion he supp ession o highe modes a ises om a e a ded esponse o con o ma ional changes ha only se s in when one pe al di uses o i s neighbo and encoun e s i s ha d-co e epul- sion. In conclusion we ha e cha ac e ized he dynamic luc ua- ions o magne ic colloidal lowe s. These luc ua ions can be unde s ood as a esul o de e minis ic o ces a ising due o dipola in e ac ions and s a is ical o ces a ising om he collisions o he embedding luid. The so cha ac e o he dipola in e ac ions places his sys em be ween ha o a ee sys em and a sys em in e ac ing ia ha d-co e in e ac ions. The so con inemen o he pa icles leads o a mode- dependen di usion ha di e s om single ile di usion. The deso p ion and adso p ion o he pe als can be unde - s ood as ac i a ed p ocesses. The colloidal lowe s a e hus a wo-dimensional model sys em o he dynamics o mo e complex h ee-dimensional colloidal assemblies such as Picke ing emulsions 关14兴and colloidosomes 关15兴. FIG. 7. Di usion cons an D6共 ␴ 兲 e sus ␴ . FIG. 8. No mal-mode di usion cons an s D6共m兲 e sus he mode numbe m. DYNAMICS OF SELF-ASSEMBLY OF FLOWER-SHAPED …PHYSICAL REVIEW E 82, 031406 共2010兲 031406-5 Eu . Phys. J. E (2012) 35:17 Page 3 o 6 Fig. 1. a)-b) Reflec ion pola iza ion — espec i ely, fluo escence— mic oscope image o a colloidal flowe consis ing o a pa amagne ic (non-fluo escen ) co e o diame e 2a1=2.8µm in an aqueous dilu ed e ofluid (EMG707 : H2O = 20 : 80) su ounded by an iso opic ing o diamagne s o diame e a) 2a2=3.1µm and b) 2a2=1.0µm. The images we e ob ained in a no mal field o ˆ H⊥=7mT. c)-e) Fluo escence mic oscope images o iso opic clus e s o diamagne s o diame e c) 2a1=3.1µmand2a2=3.1µm, d) 2a1=3.1µmand2a2=2.0µmande)2a1=9.9µmand 2a2=3.1µm, imme sed in o an undilu ed e ofluid (EMG 707). The clus e s we e assembled in an in-plane o a ing field o ˆ H=1.62 mT a a p ecession angula equency o Ω= 188 s−1. The scale ba in all images co esponds o 3 µm. The mo ie in he suppo ing in o ma ion shows he o a ing clus e s unde he in-plane field o ˆ H=1.62 mT and wo diffe en no mal fields wi h a p ecession angle close and a om he magic angle. fluo escence o eflec ion mic oscopy, LEICA DM5000 (Leica Mic osys ems We zla GmbH, Ge many). Fo colloidal clus e s non-magne ic pa icles wi h di - e en size diame e s we e imme sed in undilu ed e ofluid EMG 707 sandwiched be ween wo co e slips. The co e was hen subjec ed o a o a ing magne ic field whe e iso opic colloidal clus e s a e o med. Then a s a ic mag- ne ic field no mal o he film was supe posed o he o- a ing in-plane field and he dynamics o he clus e s we e obse ed unde he fluo escence mic oscope. The field di ec ion o he magne ic field changes om he ai in o he e ofluid film acco ding o ˆ H e o luid ⊥= ˆ Hai ⊥/(1 + χF)and ˆ H e o luid =ˆ Hai ⊥, and he p eces- sion angle in he e ofluid and in he ai a e ela ed ia an ϑ e o luid =(1+χF) an ϑai .χFdeno es he magne ic suscep ibili y o he e ofluid. All ex e nal fields and ex- e nal p ecession angles a e gi en in e ms o hei alues inside he e ofluid. 3Resul s Iso opic colloidal flowe s we e assembled in a s a ic mag- ne ic field no mal o he sample consis ing o a mix u e o pa amagne ic and non-magne ic pa icles dispe sed in a Fig. 2. (Colou on-line) Hys e esis loops o he o ma ion and up u e o colloidal flowe s and o diamagne ic clus e s as a unc ion o he s a ic no mal field ˆ H⊥. The colloidal flowe s consis ed o a pa amagne ic (non-fluo escen ) co e o diame- e 2a1=2.8µm in an aqueous dilu ed e ofluid (EMG 707 : H2O = 20 : 80) su ounded by an iso opic ing o diamagne s o diame e 2a2=1µm in a o a ing field o ˆ H=1.62 mT a a p ecession angula equency o Ω= 188 s−1. Blue up- wa d iangles co espond o inc easing he no mal field and pink downwa d iangles o dec easing no mal field. The dia- magne ic clus e s consis ed o co e pa icles o diame e 2a1= 3.1µm and pe als o diame e 2a2=3.1µm imme sed in an aqueous undilu ed e ofluid (EMG 707). The o a ing in-plane field s eng h and equency we e he same as o he colloidal flowe s. Red ci cles a e measu ed upon inc easing and he g een squa es upon dec easing he no mal field. The inse shows he same hys e esis loops in e ms o he p ecession angle. dilu ed e ofluid. I has been shown [11,12] ha wi h he p ope dilu ion he magne ic suscep ibili y can be uned o p e e a numbe o diamagne ic pe als abso bing a he magne ic co e co esponding o a ull monolaye o pe als a ound he co e. Such kinds o iso opic colloidal flowe s a e displayed in fig. 1a)-b). Clus e s o a bidispe se ( adii a1and a2) mix u e o effec i e diamagne s in a e ofluid we e o med in an in-plane o a ing magne ic field. The diamagne ic clus e s o med a e plana clus e s lying in he mid plane o he e ofluid sample ha ing a ich a i- e y o con o ma ions wi h diffe en numbe s o diamagne s o ming one clus e s. Amongs his a ie y we picked ou clus e s ha ing a co e o med by a bead o adius a1su - ounded by a comple e monolaye o beads wi h adius a2. Examples o such iso opic diamagne ic clus e s a e shown in fig. 1c)-e). Bo h ypes o clus e s we e exposed o a p ecessing magne ic field being a supe posi ion o a o a ing mag- ne ic field H( )= ˆ H[excos Ω +eysin Ω ] in he plane o he e ofluid film and a s a ic field H⊥( )= ˆ H⊥ez. The p ecession angle is defined by he a io o his wo componen s o he field ia an ϑ=ˆ H⊥/ˆ H. The ex e - nal fields epo ed a e hose in he e ofluid film a away om he clus e s. In fig. 2 we show he s abili y o such Page 4 o 6 Eu . Phys. J. E (2012) 35:17 Fig. 3. (Colou on-line) Dependence o he wid h o he hys- e esis loop o he o ma ion and up u e o colloidal flowe s ( ed) and diamagne ic clus e s (blue) on he a io o he co e adius and he pe al adius. The ed and blue lines a e fi s acco ding o eq. (8). clus e s as we sweep he no mal componen ˆ H⊥o he p ecessing field. Colloidal flowe s a e s able o low p e- cession angles (la ge no mal field ˆ H⊥) while clus e s o holes a e s able a la ge p ecession angles (small no mal field ˆ H⊥). Dec easing he no mal componen o he field des abilizes he colloidal flowe s and hey all apa a a c i ical field ˆ H⊥c1. I we s a he expe imen a ˆ H⊥=0 one obse es a mix u e o magne ic hole clus e s and pa a- magne ic beads. Colloidal flowe s o m om his mix u e upon su moun ing a second h eshold ˆ H⊥c2>ˆ H⊥c1.We cha ac e ize he wid h o his hys e sis by he diffe ence o he wo c i ical fields ∆ˆ H⊥=ˆ H⊥c2−ˆ H⊥c1. The wid h o he hys e sis ∆ˆ H⊥is a measu e o how s ongly he ansi ion is o fi s o de . A simila hys e esis is measu ed when diassembling diamagne ic clus e s by inc easing he no mal componen ˆ H⊥o he p ecessing field and eassembling a clus e o a gene ically diffe en shape and size when dec easing he field. The s eng h o he ansi ion bo h o he colloidal flowe s as well as o he magne ic hole clus e s depends on he size a io a1/a2o he colloids o he co e and o he pe als. In fig. 3 we plo he wid h o he hys e sis ∆ˆ H⊥ e sus he size a io a1/a2. The wid h o he hys e sis inc eases wi h he size a io. The o a ing pa allel componen and he con as o he imagina y pa o he magne ic suscep ibili y ∆χ′′ o he clus e o he su ounding e ofluid esul in a o que τ=4πµ0∆χ′′Vˆ H2sin2ϑ. He e µ0is he acuum pe meabili y, Vdeno es he olume o he clus e , and ˆ His he absolu e alue o he magne ic field. This o que causes he clus e s o o a e a ound hei co e wi h an angula equency ω<Ω. The a io Fig. 4. The angula eloci y o diffe en colloidal flowe s and diamagne ic clus e s as a unc ion o he p ecession angle eco ded a a cons an in-plane o a ing field o ˆ H=1.62 mT a a cons an p ecession angula equency o Ω= 188 s−1. Fig. 5. The angula eloci y a io o diffe en colloidal flowe s and diamagne ic clus e s nea and a om he magic angle as a unc ion o he a io o he co e o he pe al adii. ω/ ˆ H2sin2ϑmeasu es how efficien he magne ic field o- a ion is con e ed in o a o a ion o he clus e . In fig. 4 we plo he angula equency ωa fixed in-plane field s eng h and equency as a unc ion o he p ecession angle ϑ o diffe en clus e s. Some o he clus e s show a speeding up when one app oaches he magic angle [13], whe e he clus e s all apa . O he clus e s do no change hei angula equency when changing he no mal com- ponen o he field. We cha ac e ize he clus e speed up by he a io ω as /ωslow, whe e ω as deno es he angula equency jus be o e up u e and ωslow is he angula e- quency a low (high) p ecession angle whe e he colloidal flowe (magne ic hole clus e ) is s able. Eu . Phys. J. E (2012) 35:17 Page 5 o 6 Fig. 6. The angula eloci y a io o diffe en colloidal flowe s and diamagne ic clus e s nea and a om he magic angle as a unc ion o wid h o he hys e esis. In fig. 5 we plo he clus e speed up e sus he size a io a1/a2o he colloids o he co e and o he pe als. The speed up dec eases wi h he size a io o bo h he colloidal flowe s and o he magne ic hole clus e s. Figu es 3 and 5 show ha bo h he wid h o he hys- e eses and he speed up o he o a ion co ela e wi h he a io o he co e- o- he-pe al adius aco e/ape al.Wemay combine figs. 3 and 5 o measu e he speed up as a unc- ion o he s eng h o he fi s o de ansi ion. Hence, in fig. 6 we plo he clus e speed up e sus he wid h o he hys e esis. A la ge speed up is obse ed o small hys e esis while no speed up occu s a la ge hys e esis. 4 Discussion I we conside he co e pa icle o be la ge han he pa i- cles in he ing i is a good app oxima ion o desc ibe he local magne ic field as ha in he absence o he pe al pa - icles. Toussain e al. [14] ha e shown ha image dipoles due o he p esence o he e ofluid glass walls can cause a fi s -o de ansi ion wi h wo s able dis ances be ween he diamagne s. He e hose effec s a e neglec ed since he sample hickness is much la ge han he sepa a ion o he pe als om he co e. Neglec ing he image dipoles, he field om he co e is desc ibed by H=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ I+a3 1(χc−χF) 1+χc+2(1+χF) 3 − 2I 5·Hex , o >a 1, 3(1+χF) 1+χc+2(1+χF)Hex , o <a 1, (1) whe e Ideno es he uni enso , and χcdeno es he sus- cep ibili y o he co e pa icle o adius a1. The effec- i e magne ic momen o he co e and pe al pa icles mc=Vc(χc−χF)H( =0)andmP=VP(χP−χF)H( = a1+ 2) a e hus de e mined by he local field a he pa - icle posi ions =0and =a1+ 2, he olumes Vcand VPo he pa icles and he suscep ibili y con as s o he e ofluid. The dipola in e ac ion hence eads W=−µ0 4πmc·3 − 2I 5·mP(2) =−γHex ·3 − 2I 5+a3 1(χc−χF) 1+χc+ 2(1 + χF) ×3 − 2I 52·Hex ,(3) whe e γ=µ0 4πVPVc 3(1 + χF) 1+χc+ 2(1 + χF)(χc−χF)(χP−χF) (4) and is he sepa a ion ec o be ween he co e and pe al pa icle. The fi s e m in (3) co esponds o he in e - ac ion o he pe al pa icle in he unpe u bed ex e - nal field and he second e m is he pe u ba ion o he magne ic momen o he pe al pa icle due o he p es- ence o he co e pa icle. Fo an ex e nal field Hex = Hex (sin ϑex [excos Ω +eysin Ω ]+cos ϑex ez)andape al pa icle si ing in he equa o ial plane a a dis ance = a1+ 2 he ime-a e aged dipole in e ac ion ene gy eads W=1 2 γH2 ex (a1+ 2)31+ β (1 + 2/a1)3 ×P2(cos ϑex )−4β β+(1+ 2/a1)3,(5) whe e β=(χc−χF) 1+χc+ 2(1 + χF).(6) The fi s e m in (5) co esponds o a eno malized long- ange dipole in e ac ion ha scales wi h second Leg- end e polynomial P2(cos ϑex ) o he p ecession angle ϑex and swi ches sign when passing he magic angle. This pa o he in e ac ion is a ac i e i (χc−χF)(χP− χF)P2(cos ϑex )<0 and explains he s abili y o he col- loidal flowe s (χc−χF>0,χ P−χF<0,P 2(cos ϑex )>0) o small p ecession angles ϑex <ϑ magic and he s abili y o he diamagne ic clus e s (χc−χF<0,χ P−χF< 0,P 2(cos ϑex )<0) o la ge p ecession angles ϑex > ϑmagic. The second e m is independen o he p ecession angle. I s sign does no depend on he sign o he suscep- ibili y con as sign(χc−χF) o he co e pa icle o he e ofluid. The second e m is epulsi e o pe al pa icles ha a e magne ic holes (χP−χF<0), while o pa amag- ne ic pa icles i is a ac i e. The des abilizing co ec ion e m is sho ange. This esul s in an equilib ium dis ance o he pe al om he co e gi en by 2,min =a13 4β P2(cos ϑex )−β−1,(7) ha mo es om infini y a he magic angle ϑex =ϑmagic owa d he ha d-co e dis ance a2as one mo es away om Page 6 o 6 Eu . Phys. J. E (2012) 35:17 he magic angle. The pic u e changes when he dipole in- e ac ions be ween he pe als a e aken in o accoun as well. He e he diffe en ange o bo h in e ac ions becomes impo an when summing up he in e ac ion o all pe al pa icles. We expec ha in a clus e o Npa icles ha he dipole in e ac ion inc eases wi h he numbe o pai s o pa icles ha scales as N2, while he sho - ange co - ec ion inc eases wi h he numbe o nea es neighbo pa - icles ha scales like N. This explains he hys e eses since once a clus e is o med i can be s abilized by he long- ange dipole in e ac ions e en when a single pai o pa - icles is no ye s able. The minimum adius o Npe als will hence be diffe en om ha o one pe al desc ibed by eq. (7). We expec he hys e eses o oughly scale wi h he a io o he long- ange o sho - ange in e ac ions such ha ∆H ∝1/(β+(1+a2/a1)3).(8) In fig. 3 we ha e inco po a ed cu es acco ding o eq. (8) wi h he p e ac o o eq. (8) fi ed o he da a. The fi ag ees well o he colloidal clus e bu is less accu a e o he colloidal flowe s. This is no oo sup ising since he diffe en suscep ibili y o he co e o he flowe adds o he complexi y o he phenomenon. The wid h o he hys e eses is a measu e o he s eng h o he fi s -o de ansi ions. I he ansi ion is weakly fi s o de , some o he second-o de c i ical phenomena a e likely o pe sis . This is wha we obse e in he c i ical speeding up. Fo a second-o de ansi ion we would expec he o a ion speed o he clus e o di e ge. Fo a weakly fi s -o de ansi ion he e is significan inc ease when app oaching he ansi ion, while no significan inc ease is obse ed when he ansi ion is s ongly fi s o de . The in e ac- ion be ween pa icles a he magic angle in an iso opic en i onmen anishes. Some in e ac ion will pe sis i he la ge size o he co e ende s he en i onmen aniso opic. The sel -consis en de ia ion o he sys em om iso opic is wha s abilizes o des abilizes he pa icula con o ma- ion and ende s he ansi ion om second o fi s o - de . I is he e o e concei able ha he p esence o a la ge co e pa icle is esponsible o he s ong fi s -o de ype o ansi ions in he clus e s wi h a la ge co e. The co e- sponding second-o de speeding up o he o a ion o he clus e is des oyed by he la ge co e and pa ially pe sis s o smalle co e sizes. 5 Conclusions The o a ion o colloidal clus e s o non-magne ic holes and o mix u es o pa amagne ic beads wi h non-magne ic holes in a e ofluid in a p ecessing ex e nal magne ic field depends on he p ecession angle o he ex e nal field ha se es as a con ol pa ame e o he s abili y o he clus- e s. Nea he magic angle clus e -shape–dependen de- pola iza ion fields cause an o ien a ion o he local field de ia ing om he ex e nal field and ende clus e an- si ions weakly o s ongly fi s o de . I he ansi ion is weakly fi s o de a c i ical speeding up o he clus e o a- ion is obse ed. No speeding up occu s o s ongly fi s - o de clus e ansi ions wi h hys e esis. The s eng h o he fi s -o de ansi ion is la ge he la ge he size o he co e as compa ed o he pe al pa icles o he clus e . This wo k is suppo ed by he Ge man Science Founda ion wi hin he clus e o excellence SFB840. Open Access This is an open access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons.o g/licenses/by/2.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. Re e ences 1. P. Pie anski, Con emp. Phys. 24, 25 (1983). 2. A. anBlaade en, R. Ruel, P. Wil zius, Na u e 385, 321 (1997). 3. James E. Ma in, Eugene Ven u ini, Ge ald L. Gulley, Jona han Williamson, Phys. Re . E 69, 021508 (2004). 4. N. Casic, S. Sch eibe , P. Tie no, W. Zimme mann, Th.M. Fische , EPL 90, 58001 (2010). 5. A.P. Gas , C.F. Zukoski, Ad . Colloid In e ace Sci. 30, 153 (1989). 6. N. Os e man, I. Pobe aj, J. Dobnika , D. F enkel, P. Zihe l, D. Babic, Phys. Re . Le . 103, 228301 (2009). 7. G. Helgesen, P.O. Pie anski, A.T. Skjel o p, Phys. Re . A 42, 7271 (1990). 8. E.R. And ew, A. B adbu y, R.G. Eades, Na u e 182, 1659 (1958). 9. J. Ce nak, G. Helgesen, A.T. Skjel o p, Phys. Re . E 70, 031504 (2004). 10. R.M. E b, H.S. Son, B. Saman a, V.M. Ro ello, B.B. Yellen, Na u e 457, 999 (2009). 11. K.H. Li, B.B. Yellen, Appl. Phys. Le . 97, 083105 (2010). 12. A. Ray, S. Aliaska isohi, Th.M. Fische , Phys. Re . E 82, 031406 (2010). 13. P. Tie no, R.M. Mu ugana han, Th.M. Fische , Phys. Re . Le . 98, 028301 (2007). 14. R. Toussain , J. Aksel oll, G. Helgesen, A.T. Skjel o p, Phys. Re . E 69, 011407 (2004). CHAPTER 4. TRANSITION STRENGTH 40 Chap e 5 Magne ic ield con olled composi e pa amagne ic-diamagne ic colloidal phases 41 CHAPTER 5. COLLOIDAL PHASES Magne ic ield con olled composi e pa amagne ic-diamagne ic colloidal phases A. Ray, and Th. M. Fische , Submi ed o The Jou nal o Physical Chemis y B 42 Magne ic ield con olled composi e pa amagne ic-diamagne ic colloidal phases A. Ray, and Th. M. Fische ( Ins i u ü Expe imen alphysik, Uni e si ä Bay eu h, 95440 Bay eu h, Ge many. Abs ac We epo on di e en ly o de ed colloidal phases o a mix u e o pa amagne ic and diamagne ic colloids subjec o a quickly a ying ime dependen magne ic ield. E ec i ely pa amagne ic and e ec i ely diamagne ic colloids a e c ea ed om pa amagne ic and nonmagne ic colloids imme sed in o a hin ilm o aqueous e o luid. The ime a e aged dyadic p oduc o he magne ic ield wi h i sel se es as a con ol pa ame e o a sequence o ansi ions be ween di e en ly co ela ed o ien a ion o de be ween he pa amagne ic and diamagne ic colloids. We obse e an i- and equimagne ic o de along di ec ions ha a e o hogonal o each o he . A he magic angle equimagne ic and an imagne ic di ec ions o he colloidal o de change ia an in e ening biaxial o de ed phase o a phase we e he equimagne ic o de ed di ec ion is eplaced by an an imagne ic o de ing and ice e sa. email: homas. ische @uni-bay eu h.de 1. In oduc ion Neu aliza ion o opposi e cha ges is one o he d i ing concep s leading o he o ganiza ion o ma e on he a omic and molecula scale. The in e ac ion be ween poin cha ges is iso opic and does no depend on di ec ion. I is spon aneous b eaking o o a ional symme y and he quan iza ion o angula momen um ha ne e heless p oduces c ys alline s uc u es wi h di ec ed bonds in a oms and molecules. Colloidal pa icles ha e been used as a model o a oms on a la ge scale1,2. They howe e a e o mesoscopic size, whe e quan um phenomena a e absen , and angula momen um is a con inuous quan i y. The p inciples leading o di ec ed bonds in mic oscopic sys ems he e o e do no wo k on he colloidal scale. In iso opically in e ac ing colloids s e ic in e ac ions a e he means o spon aneously b eak o a ion symme y and o m a colloidal c ys al3. The only possibili y o ob aining di ec ed bonds in colloidal sys ems is by using colloidal pa icles ha a e in insically aniso opic. Fo his eason chemis s ha e syn hesized Janus pa icles4,5 and pa chy colloids6,7 wi h su ace unc ionali ies ha a y as a unc ion o he loca ion on he pa icle su ace. O he possibili ies a e he use o ellipsoidal pa icles8-10 he shape o which is di e en in di e en di ec ions. A hi d possibili y is o use a magne ic11 o elec ic12 dipole momen using an ex e nal magne ic o elec ic ield. Induced pa amagne ic dipoles do no neu alize in an ex e nal ield bu build up an induced magne iza ion wi h a mac oscopic magne ic momen gi en by he magne iza ion o he sample imes i s olume. The si ua ion changes when conside ing a mix u e o pa amagne ic and diamagne ic colloids13. Diamagne s and pa amagne s poin in o opposi e di ec ions in he same ield. They a e able o neu alize each o he on a mac oscopic scale. In his sense, mix u es o pa amagne s and diamagne s in an ex e nal magne ic ield a e a model sys em o neu alizing, aniso opically in e ac ing pa icles ha posses a a ie y o mesoscopic a angemen s ha is iche han ha o iso opic colloids and han ha o non-neu alizing aniso opic colloids. In he cu en manusc ip , we show a ew o he mos ob ious colloidal phases ha o m in such a sys em, when we apply a ield a ying on ime-scales as e han he in e pa icle dynamics. 2. Expe imen Figu e 1 a) Scheme o he expe imen al se up. b) Scheme o he ield modula ion The ield a ies in ime acco ding o [ ] HH yxex zex Ω+Ω+= 2sinsinsin ˆ cos ˆ )( eeeH ϑϑ wi h he ip o he magne ic ield ec o ollowing he black Lissajou igu e. This modula ion p oduces he same ime a e aged dipole in e ac ions as a ield p ecessing a a p ecession angle o ex ϑ a ound he z-axis (g een cone) bu causes no ne o que on he colloidal assembly. γ ( ) deno es he angle be ween he pa icle sepa a ion ec o ij and he ield. Only he p ojec ion angles o he ield ex ϑ and he bond b ϑ en e in he angula dependence o he ime a e aged in e ac ion. Whe he he in e ac ion be ween he induced momen s mi and mj is a ac i e o epulsi e depends on whe he ex ϑ and b ϑ a e smalle and la ge han he magic angle. The in e ac ion also is p opo ional o he Ω 2Ω H ϑ ex a b c ϑ b γ i j x y z mi m j glass e o luid Hai H e o luid diamagne s while pola iza ion e lec ion mic oscopy images isualize he pa amagne s. The igh igu e shows he co esponding angula dependence o he dipole in e ac ions as explained in igu e 2. The images in igu e a) show he andom a angemen in he absence o a magne ic ield. Figu es b-d a e eco ded in a magne ic ield o ( ) mTHHH yx 82.12/ ˆˆ 22 || =+= and a equency Ω =120s‐1. The e ical ield (p ecession angle) in he images we e b) Hzai =26.5mT ( ° = 10 ex ϑ ) c) Hzai =4.0mT ( °= 49 ex ϑ ) d) Hzai =2.22mT ( °= 64 ex ϑ ) e) Hzai =1.27mT ( ° = 75 ex ϑ ). We applied a ield o he o m equa ion 5 wi h an eccen ici y o less han 5%. This ensu es ha he dyadic p oduc o he magne ic ield a wo di e en imes is a symme ic enso ( 0HHHH = ′ − ′)()()()( ), whe e he ba deno es he ime a e age. As a consequence he e is no ne ime a e aged o que on o he colloidal s uc u e14. In wha ollows we desc ibe he assemblies o pa amagne ic and diamagne ic pa icles as we inc ease he angle ϑex . Colloidal lowe s In a s a ic ield = H ˆ21200 A/m, ϑex =0 no mal o he e o luid ilm we obse e he o ma ion o colloidal lowe s. Such lowe s o m due o he dipola a ac ion o diamagne ic pa icles in he equa o ial plane 2/ π ϑ = bo he pa amagne s. They ha e been i s disco e ed by E b e al.13 They a e highly dynamic s uc u es whe e he pe als o he lowe s may di use15 and hey can be easily se in o o a ion wi h ime dependen magne ic ields ha ing an asymme ic pa in he dyadic p oduc 16. Figu e 3b shows a luo escence mic oscope image o such colloidal lowe s wi h 2ap=2.8μm pa amagne ic co es and 2ad=1.0μm pe als. An ensemble o lowe s can be seen ia he luo escen pe als o he lowe su ounding he non luo escen pa amagne ic co es. The lowe s a e loca ed in he middle o he sample indica ing ha g a i a ion and image dipoles p e en he binding o pa amagne ic beads in o one dimensional s ings wi h a diamagne ic man le. Deco a ed s ings Upon inc easing he p ecession angle o ϑex =49° we obse e he o ma ion o pa amagne ic s ings undula ing a ound he middle plane o he ilm wi h a pe iod o h ee o i e beads ( igu e 3c). The en i e s uc u e is deco a ed wi h a collec ion o diamagne s ha ho izon ally adso b o he undula ing s ing a he sides o he s ing. The bonds be ween diamagne s and pa amagne s in his s uc u e a e also in he ho izon al plane bu pe pendicula o he bonds be ween he pa amagne s in he s ing. A scheme o he deco a ed s ings is shown o he igh o igu e 3c These s ings co espond o he biaxial angula dependence o he dipola in e ac ions. Sandwiched memb anes A p ecession angles o he o de ϑex =64° he pa amagne ic beads o m memb anes ins ead o s ings. These pa amagne ic memb anes a e sandwiched be ween wo laye s o diamagne s ha adso bed o he memb ane on ei he side. A he ansi ion angle ϑ=51° he o ien a ion o he memb ane no mal is in he plane o he e o luid making he sandwich s uc u e clea ly isible in he luo escence mic oscope image. The wo diamagne ic adso p ion laye s appea as b igh ly luo escing lines o diamagne ic beads sandwiching he non luo escen pa amagne s. Upon inc easing he p ecession angle he memb ane bends ( igu e 3d) such ha pa o he memb ane no mal emains in he ho izon al di ec ion while he no mal o he lowe pa o he memb ane now aligns wi h he ilm no mal. E en ually upon u he inc easing he p ecession angle he memb ane la ens and en i ely lies in he ilm plane ( igu e 3e), allowing a close inspec ion o he diamagne ic o de o he abso bed laye s. Fo all sys ems s udied he e he pa amagne ic memb ane is a close-packed wo dimensional s uc u e wi h a hexagonal uni cell wi h uni ec o s ha ing he leng h o a pa amagne ic bead diame e 2ap. The o de o he diamagne ic adso ba e on he con a y a ies a lo and sensi i ely depends on he size o he diamagne ic beads, on he concen a ion a io o diamagne s e sus pa amagne s and on he suscep ibili y o he dilu ed backg ound e o luid. In wha ollows we desc ibe he o de o he diamagne ic adso ba e unde a ious condi ions. Pa amagne ic c ys al ensla ed diamagne ic gas phase Figu e 3b shows a supe posi ion o a e lec ion mic oscopy image o he sample wi h a luo escence mic oscope image o he same sample aken immedia ely one a e ano he o a il angle o ϑex =π/2. The pa amagne ic pa icles o de in o a se ies o plana clus e s su ounded by egions ha a e comple ely deple ed o pa amagne ic colloids. Wi hin he clus e s a c ys alline hexagonal a angemen o he pa amagne ic beads is obse ed. The a angemen o he diamagne ic colloids is no comple ely 0,1 1 10 10-3 10-2 10-1 100 101 beyond uni cell diamagne ic gas diamagne ic ensla ed c ys al Δ 2/A ime [sec] wi hin uni cell  Figu e 4:Mean squa e displacemen o he diamagne ic beads upon a clus e o a diamagne ic gas Hz=0.8mT, Ω =120s-1, H||=1.82mT, 2ad=1 μ m (black) and o an ensla ed c ys al Hz=1.01mT, Ω = 120s-1, H||=1.82mT, 2ad=2 μ m (o ange). The shaded egion co esponds o mean squa e displacemen s smalle han he pa amagne ic uni cell size. unco ela ed o he pa amagne s. Diamagne ic pa icles om he pa amagne ic deple ed egions adso b on op and below he pa amagne ic c ys alline clus e s. As a esul he densi y o diamagne ic pa icles on op and below he clus e s is la ge han he densi y in he pa amagne ic deple ed egions. The pa amagne ic c ys al is sandwiched be ween wo laye s o diamagne ic gas. The diamagne ic pa icles pe o m B ownian mo ion, and he mean squa e displacemen o he diamagne s inc eases linea ly ( igu e 4) wi h a slope de ining he gaseous di usion cons an o he diamagne s. The inc ease o he mean squa e displacemen beyond he a ea o he uni cell o he pa amagne ic c ys al shows ha he diamagne s emain mobile in his phase. Fo his eason we call his phase he pa amagne ic c ys al ensla ed diamagne ic gas phase. This does no mean ha he diamagne ic gas possesses no o de . In igu e 5 we plo he adial co ela ion unc ions ∑∫−−=Δ Δ+ ji jdid d dd d N g , 2)( 2 1 )( δ and ∑∫−−=Δ Δ+ ji jpip p pp d N g , 2)( 2 1 )( δ o he pa amagne s and diamagne s, whe e Np and Nd a e he numbe o pa amagne s and diamagne s in a pa icula clus e and he ip and jd a e he posi ions o he i h pa amagne and he j h diamagne . While he long ange beha io o bo h co ela ion unc ions is go e ened by he shape o he clus e , he sho ange beha io shows ha despi e o he mobili y o he diamagne ic gas, he c ys al o de o he pa amagne is imp in ed upon he gas ia he magne ic ield modula ions om he pa amagne . 110 0,0 0,5 1,0 1,5 pa amagne ic shee diamagne ic ensla ed gas (μm) g[ ]  Figu e 5 adial co ela ion unc ion o pa amagne ic pa icles (blue) and diamagne ic pa icles ( ed) in a diamagne ic gaseous phase clus e . Al hough he diamagne ic gas is mobile he c ys al s uc u e o he pa amagne s is imp in ed upon he diamagne s.  The au o-co ela ion- unc ion o he diamagne s sha e he peaks occu ing in he au oco ela ion unc ion o he pa amagne s. Since he diame e o he diamagne s is much smalle han ha o he pa amagne s mo e han one diamagne can eside on op and below one pa amagne . We obse e a diso de in he occupancy numbe o he diamagne s o he si es abo e and below he pa amagne ic c ys als. A si e can be acan , o ha e one, wo, h ee o ou diamagne s on op o a pa amagne . This diso de is exp essed by he subs uc u e in he c oss co ela ion unc ion occu ing in he ha d co e egion o he au o co ela ion unc ion. Pa amagne ic c ys al ensla ed diamagne ic c ys al phase Figu e 6 op le ) Pola iza ion e lec ion mic oscope image o an ensla ed c ys alline phase o he diamagne s eco ded a Hz=1.01mT, Ω = 120s-1,H||=1.82mT, 2ad=2.0 μ m. The magne ic holes a e si ing on op o he pa amagne s as ske ched in he scheme o he op igh . The scheme a he bo om shows a side iew wi h wo o he us a ed bonds be ween he diamagne s shown in yellow. Upon inc easing he adii o he diamagne s and upon dilu ing he e o luid we obse e a slowing down o he la ge scale di usion ha e en ually s ops comple ely. Fo a bead diame e ad=2.0μm he diamagne s emain on op and below he pa amagne ic pa icle hey eside. In he plo o he meansqua e displacemen o he diamagne ic beads in igu e 5, we obse e a much weake inc ease o he mean squa e displacemen wi h ime ha e en ually se les a oughly 1 pe cen o he a ea o a pa amagne ic uni cell. Acco ding o he Lindemann c i e ion a c ys al should mel when he oo mean squa e displacemen s o i s elemen s amoun s o one en hs o he la ice spacing. We would hence expec a diamagne ic c ys al o immedia ely mel unde he cu en condi ions. I is, howe e , no he in e ac ions be ween he diamagne s bu he in e ac ion wi h he c ys al po en ial o he pa amagne s ha causes he c ys alline o de o he diamagne s. The diamagne s a e hence ensla ed by he pa amagne ic c ys al and o m wo c ys al laye s g owing epi axial wi h he same uni cell on he pa amagne ic c ys als. Pa amagne ic c ys al incommensu a e diamagne ic c ys al phase Fo la ge densi ies o he diamagne s and when using concen a ed e o luids he a ac ion be ween he diamagne s o e comes he pa amagne ic c ys al po en ial and he diamagne s o m close-packed hexagonal c ys als on op and below he pa amagne ic close-packed hexagonal c ys al ha has i s own uni cell u ned by 30 degees wi h espec o he pa amagne ic uni cell. The close-packed cells o he pa amagne ic and diamagne ic c ys al laye s ha e pe iodici ies de ined by he diame e s o he pa amagne ic and diamagne ic beads ha gene ically a e incommensu a e. Figu e 7 shows such an incommensu a e c ys al s uc u e. Figu e 7: luo escence- ( op le ), and pola iza ion e lec ion mic oscope image ( op middle) o an incommensu a e c ys alline phase Hz=0mT, Ω =120s‐1,H||=1.82mT, 2ad=2.0 μ m(black). The op igh pic u e shows a scheme o he packing o he pa amagne s ( ed) and diamagne s (g een). On he bo om we ha e a side iew scheme o he incommensu a e s uc u e, whe e wo pa amagne ic diamagne ic bonds ha a e pa ially us a ed a e shown in yellow. We also obse e he o ma ion o diso de ed s uc u es when nei he he in e diamagne ic in e ac ion no he in e ac ion o he diamagne s wi h he pa amagne s domina es. Unde such ci cums ances diamagne s may o m small close packed incommensu a e clus e s on op o he pe ec ly o de ed pa amagne ha ollow he pe iodici y o he pa amagne ic la ice on a la ge scale. 4 Discussion The s uc u e o he phases obse ed can all be unde s ood by conside ing he ime a e aged dipola in e ac ions be ween he cons i uen s (equa ion 7). Depending on he p ecession angle o he ex e nal ield we expec pa amagne s o bind o la ge s uc u es in bond angle di ec ions ha a e a ac i e ( iole in igu e 2). In his way we ob ain an assembly o pa amagne s in he a ac i e bond di ec ions equi b equi b ϑϕ , ha a e all poin ing wi h hei magne ic momen s in he same di ec ion pa allel o he ex e nal ield. The o de esembles a e omagne ic o de ing, howe e , he magne ic momen s he e a e no pe manen bu a e induced by he ex e nal ield. We hence named he o de ing an equimagne ic o de ing. The ime a e aged dipole in e ac ion be ween diamagne s beha es he same way c ea ing a diamagne ic equimagne ic o de wi h he diamagne ic momen s all poin ing an ipa allel o he magne ic ield. Bonds be ween diamagne s and pa amagne s a e a ac i e in bond di ec ions an i b an i b ϑϕ , pe pendicula o he equimagne ic bonddi ec ions. In hose o hogonal di ec ions (o ange bond di ec ions in igu e 2) we ob ain an an imagne ic o de o al e na ing pa a- and diamagne s ha esembles a e imagne , howe e , he al e na ing momen s a e induced momen s no pe manen momen s. The en i e o de hence consis s o opposi e magne ic pa icles ha assemble in an al e na ing induced an imagne ic sequence in one o wo di ec ions while he a angemen is equimagne ic in he emaining di ec ions. Whe he he an imagne ic o de ing is in plane and he equimagne ic is no mal o he ilm o he o he way ound is con olled by he p ecession angle ex ϑ o he ex e nal magne ic ield. An imagne ic equa o ial o de ing magic an i b ϑϑ >and equimagne ic pola magic equi b ϑϑ <o de ing is suppo ed by p ecession angles magicex ϑ ϑ < below he magic angle, while equimagne ic equa o ial magic equi b ϑϑ >o de ing and an imagne ic pola o de ing magic an i b ϑϑ < is suppo ed by angles magicex ϑ ϑ >. I is o his eason colloidal lowe s o m a magicex ϑ ϑ < while sandwich s uc u es a e s able o magicex ϑ ϑ >. When he p ecession angle o he magne ic ield is nea magic magicex ϑ ϑ ≈we a e in he egime we e biaxial o de ing p e ails wi h equimagne ic o de ing along one equa o ial di ec ion magic equi b equi b ϑϑϕ >= ,0 and an imagne ic o de ing along magic an i b an i b ϑϑπϕ >= ,2/ he o he equa o ial di ec ion. A la ge p ecession angles we obse e he equa o ial equimagne ic o de ing wi h c ys alline packing o he pa amagne s and di e en ypes o packing o he diamagne s. The gaseous and di e en c ys alline diamagne ic s uc u es a e con olled by he s eng h o he mal luc ua ions and he dipole in e ac ions. Whe he he dipole in e ac ion be ween pa amagne s o diamagne s o be ween diamagne s and pa amagne s domina es can be con olled ia he suscep ibili y con as s ha can be changed by dilu ing he e o luid, he size o he pa icles, and he olume ac ion o bo h ypes o pa icles. Small pa icles a e mobile and p e e gaseous phases, la ge pa icles a e immobile. A low olume ac ions o diamagne s d φ in a dilu ed e o luid (1<< F χ ) hei in e ac ion wi h he pa amagne s is s onge ( 11 pF χχ ∝) han he in e ac ion be ween hem ( 2 F χ ∝). Each pa amagne binds one diamagne o i s no hpole lea ing diamagne ic bonds us a ed because he diamagne s a e sepa a ed mo e han hei close-packed dis ance. I is o such condi ions whe e we obse e he ensla ed c ys al phase. In concen a ed e o luid a a high ac ion o diamagne s each pa amagne in he memb ane can bind mo e han one diamagne , he diamagne ic dipole in e ac ion becomes s onge , such ha diamagne s o m a close packed memb ane abo e he pa amagne s as well. As a d aw back some o he diamagne s eside a posi ions wi h bond angles o he pa amagne ha a e subop imal ( igu e 7 bo om). In his limi pa amagne ic diamagne ic bonds a e pa ially us a ed and incommensu a e phases a e obse ed. We can es ima e he amoun o neu aliza ion be ween he pa amagne s and diamagne s by he excess suscep ibili y [] ddpp ex excess e H M φχφχχ Δ+Δ==Δ (8) , whe e p φ , and d φ a e he olume ac ions o pa a- and diamagne s. Fo ou samples we had 0>Δ e χ such ha in e ac ions be ween pa amagne s domina e all o he dipole in e ac ions. They hence o med s uc u es hey also would ha e o med wi hou he p esence o he diamagne s. The diamagne s, howe e , had o accep he dis o ed s uc u e o he magne ic ield, gene a ed by he pa amagne s and a ange hemsel es acco dingly. P esumably when using uly neu alizing mix u es 0≈Δ e χ , he hen mo e symme ic si ua ion be ween pa a- and diamagne s would p oduce e en mo e in e es ing supe s uc u es. A p esen we do no ha e e o luids o su icien magne ic suscep ibili y o es such ully dipola neu alized supe s uc u es. Howe e , e en wi hou ha ing explo ed he ull pa ame e space o possible s uc u es i is clea ha he con ol o he di e en pa ame e s in he dipole in e ac ion o di e en pa icles as well as he con ol o he olume ac ion o pa icles allows he cons uc ion a ich a ie y o phases in a mixed diamagne ic and pa amagne ic sys em. 5 Conclusions An imagne ically o de ed colloidal phases wi h al e na ing a angemen s o e ec i ely diamagne ic and pa amagne ic pa icles a e o med in mix u es o pa amagne ic and diamagne ic colloids imme sed in o a e o luid and subjec o a quickly a ying ime dependen magne ic ield. Depending on he mean o ien a ion o he ime a e aged dyadic p oduc o he ex e nal magne ic ield he al e na ing o de is obse ed in he plane o he ilm in o m o colloidal lowe s o no mal o he ilm in he o m o 2D pa amagne ic c ys als sandwiched be ween a diamagne ic gas o c ys al. Nea he magic angle eccen ici y o he modula ion c ea es also biaxial s uc u es. The o de o he diamagne ic sandwich laye depends on a sub le balance o pa ame e s en e ing in o he dipole in e ac ions a wo k be ween he di e en pa icles 6 Acknowledgemen We hank Thomas F ied ich o helping wi h suscep ibili y measu emen s o he e o luids. This wo k is suppo ed by he Ge man Science Founda ion wi hin he clus e o excellence SFB840. BIBLIOGRAPHY BIBLIOGRAPHY [6] Xia, Y. N. ; Ga es, B. ; Li, Z. Y.: Sel -assembly app oaches o h ee-dimensional pho onic c ys als. In: Ad . Ma e . 13 (2001), 409–413. h p://dx.doi.o g/10.1002/1521-4095(200103)13:6<409:: AID-ADMA409>3.0.CO;2-C [7] Ze ouki, D. ; Baud y, J. ; Pine, D. ; Chaikin, P. ; Bibe e, J.: Chi al colloidal clus e s. In: Na u e 455 (2008), 380–382. h p://dx.doi. o g/10.1038/na u e07237 68