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
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68