Shapes and Textures of Ferromagneti Liquid Droplets
Shubho Banerjee
and M. Widom
Department ofPhysis,CarnegieMellonUniversity, Pittsburgh,Pennsylvania 15213
Reeivedon21May,2001
Theoretialalulations, omputersimulationsandexperimentsindiatethepossibleexistene of
a ferromagneti liquid state. Should suh a state exist, demagnetization eets would fore a
nontrivial magnetizationtexturegovernedbytheshapeoftheliquiddroplet.Sine liquiddroplets
are deformable, the droplet shape ouples to the magnetization texture. This paper solves the
jointshape/textureproblemsubjettotheassumptionofylindrialdropletsymmetry.Theshape
undergoes a hange in topology from spherial to toroidal as exhange energy grows or surfae
tensiondereases.
I Introdution
Inaspontaneouslymagnetizedliquidstate,longrange
magneti order would exist in the liquid without
ap-pliationof anyexternal eld. The existeneof a
fer-romagneti liquid state in dipolar uids has been
in-diated bymean eldalulations [1-6℄and omputer
simulations [7-10℄. Experiments to observeliquid
fer-romagnetisminferrouids[11℄arehallengingbeause
the uids often freeze [12, 13℄ or phase separate [14℄
well abovethe preditedlow temperaturesfor the
on-setof spontaneousmagnetization. Reentexperiments
onstronglyinteratingFe
3
N ferrouids[15℄ doshowa
hint of a possible ferromagneti transition. Similarly,
experimentsonsuper-ooledCo-Pdalloys[16℄indiate
the possibility of ferromagnetism in superooled
liq-uid metals. In this ase it is the strong exhange
in-teration, and not the dipole interation, that would
ausethespontaneousmagnetization. The
experimen-tal evidene in bothferrouidand superooledCo-Pd
isinonlusive,sinethebulkofevideneaddressesthe
temperaturedependeneofparamagnetisuseptibility
above the Curie temperature [15-18℄. Denitive proof
offerromagnetismbelowtheCurietemperatureremains
elusive.
Although the existene of a ferromagneti liquid
state is yet to be proven experimentally, spontaneous
polarization oupled with other order parameters has
alreadybeenobserved. Someeletriallypolarized
liq-uid rystals [19℄ show ahelial ordering of the dipole
momentsintheliquid. Insuperuid 3
Hethemagneti
momentouplestothesuperondutingorder
parame-ter[20℄. Manysuperuid 3
Hephasesarethereforealso
magnetiallyordered.
It isinteresting to onsider themagnetization
tex-ture(spatialvariationoftheorientationof
magnetiza-tion)insideadropletofsuhaferromagnetiliquid[21℄.
The magnetization texture likesto avoidpoles [22℄ to
minimizeits energy. However,this leadsto defets
in-sidethetexture. Forexample,therotating
magnetiza-tiontexturewithylindrialsymmetryinside asphere
M(r;;)=M ^
; (1)
where ^
istheunitvetorforthevariable,avoidsall
poles but has a vortex line running through the
en-ter. Near the vortex of suh a texture the
magneti-zation is topologially unstable [23℄ and mightesape
intothethirddimension[24℄withanonzeroomponent
along the vortex line. Whether this happens depends
on thebalane betweendemagnetizing and vortex
en-ergies. Simulatedannealingofthemagnetizationinside
aubiboxsuggeststhat replaingvortieswithpoint
defets may be favorable [6℄. However,for suÆiently
largedropletsthedemagnetizingenergywill dominate
andonlytextureswithvanishingdemagnetizingenergy
(perunit volume)willour.
Any defet is likely to have a
system-shape-dependent energy ost ausing a deformable liquid
droplettodeviatefromaspherialshape. Theomplete
alulationoftheshapeofanunonnedferromagneti
liquid droplet in three dimensions, oupled with the
alulationofitsmagnetizationtexture,remainsan
in-terestingandhallengingunsolvedproblem.
This problem has a simple solutionin two
sions in zero eld. The magnetization texture inside
anysoft(zeroanisotropy)ferromagnetisolidthin lm
is givenbyvandenBerg'salgorithm[25℄whihavoids
all poles, and thus all magnetostati energy, at the
expense of a domain wall through the lm. A liquid
droplet, whih an hange itsshape,prefersa irular
shapeto minimize its surfaeenergy. The
magnetiza-tionlinesinside airleformonentriirles
aord-ing to van den Berg'salgorithm. For airular shape
the domain wall energyis also minimized beause the
domain wallshrinksto apointvortex. Theirle thus
solvesthe oupled texture and shape problem in zero
eld.
In a previous paper [26℄ addressing the thin lm
limit,wedesribedtheevolutionofmagnetization
tex-tureand dropletshapeundertheappliation ofan
in-plane magneti eld. Thevortexstrethes to beome
adomainwall,whihdisplaestowardstheedgeofthe
droplet. Ifexhange energy istaken into aount, the
dropletexhibitsreetionsymmetry-breaking,
immedi-atelyrevealingitsmagnetizedstate.
Our present goal is to analyze the shape and
tex-tureofanunonnedferromagnetiliquiddroplet. The
problem ishighlynontrivialevenin theabseneof
ap-plied eld. To simplify the study, we restrit our
at-tentiontoshapesandtextureswithanaxisof
ontinu-ousrotational symmetry. Wendthetextureexhibits
a planar harater with a vortex line along the
sym-metry axis. When surfaetensionis high or
magneti-zation weak, the droplet shape remains nearly
spher-ial, with slight \bulging at the waist" and dimpling
where the vortex line meets the surfae (Fig. 1). As
surfaetensiondropsormagnetizationgrows,the
dim-pling inreases, shorteningthevortexline. Eventually
thedropletundergoesahangeintopologytoatoroidal
shape(Fig. 2).
II Shape, texture and energy
We onsider droplets of volume V and shape with
an internal magnetization texture M(r). Four terms
ontributeto theenergy,
E
tot =E
surf +E
exh +E
ore +E
demag
: (2)
Thersttermistheenergyofthedropletsurfae,,
E
surf =
Z
d
2
r=A (3)
with A the total droplet surfae area and the
sur-faetensionwhihwetaketobeisotropi. Thesurfae
Figure1. Appleshapeobtainedbyminimizingeq.(13)for
M 2
=0:17.
Figure2. Donutshapeobtainedbyminimizingeq.(13)for
M=7:0.
Theseondtermistheexhangeenergy,whih,for
anisotropimedium,anbewrittenastheintegralover
thedropletvolumeof
U
exh =
1
2
M
k
x
i M
k
x
i
; (4)
where is the exhange onstant and thesummation
onvention is employed [27℄. The exhange onstant
ferromag-order a 2
, where a is aharateristi atomi size and
10 4
istheratioofexhange eldto magnetostati
eld [28℄. Fora ferrouid, we an derive an eetive
exhangeonstantreetingtheenergyostofplaing
nitesizepartilesintoarotatingmagnetitexture. By
alulatingtheorretionto themeaneldin aavity
ofdiameteraausedbyrotationofthemagnetization,
we nd=4a 2
=15. Again,is oforder a 2
, where
now=1beausetheenergyostismagnetostatiin
origin.
The third term is the energy assoiated with
un-magnetizedregionswithin theuid. Wepresume that
throughoutmostofthedropletthemagnetizationM(r)
has onstant magnitude, M, that minimizes the free
energy density f(M). Inspeting eq. (4) we see that
exhange energy density may diverge at the ore of a
vortex, where U
exh
=r
2
. Within a distane r
of
the enter of the vortex the exhange energy density
U
exh
mathes the free energy density ost U
ore
f(0) f(M). The ore radius so-obtained does not
dependondropletshapeormagnetizationtexture.
The nal term in (2) is the demagnetizing energy
thatarisesasaonsequeneofthelongrange1=r 3
har-aterofthedipoleinteration. Thisweakfall-oofthe
interation makes the total energy of the system
de-pendent on global magnetization texture and system
shapein general,hallengingournotionsof
thermody-namilimits[29℄. Theshapedependentdemagnetizing
energy for dipolar systemsan be best understood in
termsofthedemagnetizingeld
H D (r)= Z S d 2 r 0 (M(r 0
)^n(r 0 )) r r 0 jr r 0 j 3 Z V d 3 r 0
(rM(r 0 )) r r 0 jr r 0 j 3 : (5) d
Heren^isthenormaltothesurfaeofthesystem. The
rst term on the right hand side reets the surfae
poles that are reatedwhere the magnetization has a
omponentnormalto thesurfae. Theseond termis
the ontribution from the bulk hargedensity that is
reatedbyanonzerodivergeneofthemagnetization.
H
D
is alledthe demagnetizingeld beauseit
in-hibits the magnetization, as shown by the
magneto-statienergy E D = 1 2 Z V d 3 rH D
(r)M(r); (6)
whihanberewritten
E D = 1 8 Z allspae d 3 rjH D (r)j 2 : (7)
This energy is manifestly positive denite and hene
inhibitsthemagnetization.
Toavoidademagnetizingenergybyeliminatingits
demagnetizingeld,magnetizationtexturesin rystals
breakupintodomainsseparatedbydomainwalls. For
suitable domain struture the demagnetizing energy
an be removed entirely. The domain wall width is
set by balaning the ost in exhange energy for
ro-tatingtextures againstthe ostin magneto-rystalline
anisotropyenergy when athe magnetization does not
pointalong aneasyaxis. Beause aferromagneti
liq-uidshould lakmagneto-rystallineanisotropy,the
do-main wall width diverges [21℄ unless it is limited by
Thefourtermsin eq.(2)growatdieringratesas
dropletvolume inreaseswhiledroplet shapeand
tex-tureareheldxed. Wend
E
surf
R h (8)
E
exh
M
2
hlogR =r
E ore U ore h E demag DM 2 R 2 h:
Intheabove,wehavetakenmagnetizationtexture(1)
inaylinderofheighthandradiusRforouralulation
ofE
exh
,andwehavetakenM(r)onstantinsidean
el-lipsoid ofdemagnetizationfatorDforouralulation
ofE
demag
. ForsuÆientlylargedroplets,E
demag
dom-inates unlessatextureis foundto redueit orremove
it altogether. Antiipating that textures with
vanish-ingE
demag
willemerge, thedominantenergybeomes
E
surf
,solargedropletswillfavorompatshapes. For
large R , we nd E
exh
dominates E
ore
. At xed R ,
the exhange energy mimis the ore energy, with an
energyostproportionaltothevortexlength.
III Energy minimization
We now onfront the problem of simultaneously
ahievethelowest totalenergy. Tosimplify ourwork,
wehoosetoworkwithinalimitedsubsetofthefamily
of possible textures and shapes. We impose
ylindri-al symmetry, motivated both by our suspiion that
the trueenergy minimummayexhibit suh a
symme-tryandbytheonsiderablealulationalsimpliations
that itallows.
Hene weassumethe shape and the
magnetiza-tiontextureM(r)aresymmetriunderrotationsabout
thez^axis. Themagnetizationvetoreldobeys
M(r;;z)= 0
os sin 0
sin os 0
0 0 1
1
A
M(r;0;z) (9)
The droplet shape is dened by its boundary
whih we parameterize by the funtion z(r). This
family ofshapesexhibits reetionsymmetry through
the z = 0 plane in addition to rotational symmetry
aboutthez^axis.
Forlargedropletsthedemagnetizingenergyis
dom-inant, as weshowed in eq. (9). Consider theproblem
of minimizing E
D
within a symmetri shape. Sine
E
D
0,weansolvethis minimizationwith any
tex-ture that ahievesE
D
=0,whih, byequation(7)
re-quires H
D
=0through allspae. Inspeting (5), we
an ahieve H
D
= 0 by eliminating all surfae poles
andvolumedivergene.
The simplest texture with H
D
= 0 is given in
eq.(1). Thistextureispurelyplanar,satisesH
D =0,
butexhibitsavortexlinerunningalongthez-axis. The
exhangeenergydensityvariesas
U EX = M 2 2r 2 (10)
and diverges at r =0. Therefore, we impose ashort
lengthut-o ofr
0
. Insidethe vortex (r <r
0
) we
as-sumeauniformenergydensityU
ore
. Thevortexradius
r
0
anbehosenbybalaningM 2 =r 2 0 againstU ore at r 0 yieldingr 0 (M 2 =U ore ) 1=2
. Multiplyingtheore
energydensitybytheross-setionalareayields a
vor-texenergyostperunitlength,U
V =r 2 0 U ore .
Thistexture isprobablynottheabsolutelowestin
energy. We expet that themagnetization will rotate
out-of-planeloseto thevortexore. Suhadistortion
omes witha small ost in demagnetizing energy, but
bringsalargesavingsinexhangeenergy.Sinethis
dis-tortionislimitedtotheregionloseto thevortex[24℄,
wemayonsider itaspartoftheorestruture,andit
is notrelevant forthe bulk dropletshape and energy.
Largesaleout-of-planerotationsauselarge
demagne-tizingenergyostsunlesstheyarespatiallynonuniform.
Spatialnonuniformitywillraiseexhangeenergyosts.
Hene,weexpetthetexture(1)isoptimalsuÆiently
farfrom thevortex.
Theexhange energyanberedued,and the
vor-tex ore energy eliminated altogether, by a hange in
topologyfrom spherialto toroidal. Consider atorus
withylindrialsymmetryaboutthez^axis,with
ross-setionz(r)for R
i
rR
o
. Assumingtexture(1),
andnotingE
ore =E
demag
=0,thetotalenergyisthe
sumofsurfaeandexhangeenergy
E tot =2 Z 2 0 d Z Ro Ri rdr p
1+(dz=dr) 2 + M 2 2 Z 2 0 d Z Ro Ri rdr Z z(r) z(r) dz r 2 : (11)
The urvatureenergyompeteswith thesurfaeenergyto determine theshapeof thedropletin this model. For
shapeswith spherialtopology, we must add anadditional termto eq. (11)of the form U
V L
V
, where L
V is the
lengthofthevortex. However,sinetheexhangeenergymimistheeetofthevortexoreenergy,wesetU
V =0
in thefollowing.
The relativestrength of eah term is governedby a singledimensionless magnetization parameter dened so
that M 2 M 2 L (12)
whereLisameasureofthelineardropletdimension(L(volume) 1=3
). SalinglengthsbyLandenergyby4L 2
,
weintrodueadimensionless formofthetotalenergy
E tot [z(r)℄= Z R o R i rdr p
wheresriptquantities aredimensionless.
To alulate the shape of the droplet we used the
programSurfaeEvolverbyKennethBrakke[30℄. The
program approximates a surfae using a triangular
mesh, andthevertiesofthemesh evolvetominimize
thetotal energyof thesurfaesubjetto various
on-straintsoftheproblem (xedvolumein ourase).
WendthatforsmallvaluesofMthedroplettakes
adimpledandslightlyoblate\apple"shape(seeFig.1).
This ours beause the droplet redues its urvature
energybyreduingthelengthofthevortex(henethe
dimpling where the vortex meets the surfae) and by
plaingthebulkoftheuidasfaraspossiblefrom
vor-tex(henetheoblateshape). AsMinreases,the
dim-pling growsandthevortexshortens. This ourseven
in the absene of a vortex ore energy and is driven
bythedivergingexhangeenergydensitynearthe
vor-tex. ForsuÆientlylargeM,thevortexshrinkstozero
length and the droplet hanges topology, aquiring a
\donut"shape(seeFig.2).
ForlargeM theholeat the enter ofthe donut is
large. AsMdereases,theholeshrinks,andthewalls
oftheholebeomenearlyvertialformingaylinderof
radiusR
i
. Weanestimatethisradiusbybalaningthe
R
i
dependene of the ylinder surfae energy against
theR
i
dependene of themagnetienergyoutsidethe
ylinder(andthus insidethetorus). Weestimate
dE
surf
dR
i
2L (14)
and
dE
exh
dR
i
2R
i L
M 2
2R 2
i
(15)
from whih it follows that R
i
M
2
=2 (in
dimen-sionless variables, R
i
1
2 M
2
). Thus the donut hole
remainsfor allvaluesofM, vanishingasM!0.
In-triguingly,thestrutureofthedonutholeforsmallM
resemblesthedimple onthe applesurfaeat thesame
valueofM. Thedimple isaylindrialholeofradius
R
i
, but the hole extends only part way through the
appleore,withthevortexoupyingtheremainder.
The energy of the apple is lower than the energy
ofthedonutfor smallMand remainsloweruntil just
beforetheappleoreshrinkstozero. Forlargervalues
ofM,the appleshapedoesnotexist andthedonutis
thestableshape. Foradropletwithvolumesuh that
L=12and vortexradiusr
0
=0:1,thepinh-opoint
islosetoM=0:2.
IV Conlusion
In this paper, we x a simple magnetization texture
to optimization of shape. To expressthe total energy
of the droplet asa funtion of its shape, we required
that themagnetizationbeylindriallysymmetriand
onned to aplane. Thevortex orewas assumed to
bestraight,andwenegletedtheenergyofthevortex
ore.
For a more rigorous study, a simultaneous
alu-lation of the magnetization texture and the shape is
needed. Sine the magnetization may rotate out of
planenearthevortex[23℄,andpossiblybreakthe
ylin-drialsymmetry, theanalysis will requirebreaking up
thevolumeof thedropletintonite elements[31℄and
evolvingtheshapetominimizethesumof
demagnetiz-ing, urvature, vortexand surfaeenergies. However,
oursimple analysis illustrates thenontrivialnature of
the problem and gives an idea of shapes that might
ourforferromagnetiliquiddroplets.
Aknowledgements
Weaknowledgeusefuldisussionsand
ommunia-tionswithR. B.GriÆths,A. A.ThieleandL.Berger.
This workwassupported inpartbyNSF grant
DMR-9732567 at Carnegie Mellon University and by NSF
grantCHE-9981772atUniversityofMaryland.
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