Magneti Walls in Nemati Liquid Crystals
M.Sim~oes 1
and A.J. Palangana 2
1
Departamento deFsia,UniversidadeEstadual deLondrina,
CampusUniversitario,86051-970, Londrina(PR),Brazil
2
Departamento deFsia,UniversidadeEstadual deMaringa,
87020-900, Maringa(PR),Brazil
Reeivedon23November,2001
Inreentpublishedpapersithasbeenshownthatthetheoryabouttheformationofmagnetiwalls
intheneighborhoodsoftheFreederikszthreshold isinprofound disagreementwithexperiments.
This ndingleads to the development of a new theory for the onset of these strutures inthis
region ofmagnetields. Inthispaperwe presentareviewof thesedevelopments. Theprevious
theorydesribingtheseunstablestrutureslaimsthatthemodewiththefastestinitialgrowthwill
determine the observed properties of thesepatterns. But, justabove the Freederiksz threshold,
thereis aregion wherethis leadingmodevanishesand, therefore,a homogeneousbendingof the
diretor ouldbedeteted. Thispreditionwas notonrmedbytheexperiment, andwallswith
verywell denedwavelength werefound. Toexplain these experimentalfats ithas beenshown
thatthefastestgrowingmodeannotbedenedaroundtheFreederikszthresholdand,therefore,
anewwaytoomputetheobservedperiodiitymustbeformulated. Theobservedwallresultsfrom
asumofaontinuumandnon-sharpdistributionofmodesinwhihthenullmodeisattheenter.
Thisworkiswritteninsuhawaythatthemainoneptualdevelopmentsanbeeasilygeneralized
tosystemspresentingsimilarbehavior.
I Introdution
Oneofthemostoutstandingphenomenainoutof
equi-libriumphysisisthat,evenundertransientonditions,
the onset of veryregular and symmetri strutures in
uid systemsis frequentlyobserved[?℄. These kindof
phenomena,thathavetheRayleigh-Bernardonvetion
as a orn-stone, abound in many dierent areas and,
onlytoquotefewexamples,theyareregularlyfoundin
solidstatephysis,nonlinearoptis, hemistry,and
bi-ology. Asarule,itistheinternalmotionofthematter
omposing these systems that indues the prodution
ofthese ongurationsand,of ourse,to withoutsuh
dynamialbehavioritwouldbeimpossibleobserveone.
Takingthisobservationasevidentbyitselfoneanask
for the behavior of these patterns when their internal
uidowbeomessmallerandsmaller,approahingthe
limitinwhihtheuidveloityanbenegleted. What
will be the behaviorof the mehanism that produes
these strutures when this limit is onsidered? Will
ontinuepatternsalwaysappearingorarethereritial
points in their arising? The study of a physial
sys-tem where these kind of questions anbeformulated,
andanswered,is theaimofthis work. Theemergene
ofmagnetiwalls[2-5℄innematiliquidrystals(NLC)
willbeourlaboratoryforthestudyoftheseproblems.
magneti eld ~
H is applied perpendiularly to a
ho-mogeneously pre-oriented NLC. A ompetition our
betweenthe ationof the magnetield and the
elas-tiresistaneof the medium; the magnetield tends
to alignthediretoralong itsdiretionand theelasti
interation, due to the ohesion of the NLC with the
edgesofthesample,tendstoretainauniform
orienta-tionofthediretor. Forvaluesoftheappliedeldbelow
aritialvalue, ~
h
,nodistortion arises. Forthose
val-uesoftheeld whih arelargerthan ~
h
sometextures
appear,indiatingthatthemagnetiouplingbetween
thediretor~n,andtheeld ~
H,isbiggerthanthe
elas-tiinteration insidethenemati material. Whenthis
happenswehavethewellknownFreederiksztransition
that,fromthestatipointofview,hassome
harater-istisofaseondorder phasetransition.
But,thediretordoesnotrotatein ahomogeneous
way. When observed through rossed polarizers, the
sample exhibits very regular strutures; a set of
one-dimensional,periodiandparallellinesextendingalong
the diretion of the external magneti eld, the
mag-neti wallsof theNLC. Afundamental elementto
un-derstandthepreseneoftheselinesis the-symmetry
Therefore, abovetheFreederiksztransition the
dire-tor an bend lokwise or anti-lokwise. This
dou-ble hoie is typialof systemsexhibiting asymmetry
breakingand,asitisusualintheseonditions,two
dif-ferently orientated portions of the sample an be
re-ated. Awallisthe ontinuousbendingof thediretor
eld onneting these two dierent ongurations: a
kink. AtypialperiodistrutureisexhibitedinFig. 1.
Despiteitsimportanefortheexplanationofthis
sym-metrybreaking,the-symmetryannotexplainthe
as-tonishingsymmetriesobservedinthisgure. Thewalls
extended asrightlinesalongthediretionofthe
exter-nal magneti eld and, furthermore,they are
periodi-ally distributed along the~e
x
diretion. They appear
in thesampleasaone-dimensionalandperiodi
stru-ture. There is not an a priori reason leading to this
experimental nding. All that the -symmetry teah
us is thatthere maybe regionswithdierentbending
of the diretor, but itan notexplainthe symmetries
observedinthisgure. Therefore,somebasi
informa-tionabouttheproessthatbuildwallsaremissing.
Figure 1. Lyotropi nemati phase in 0.2 mm thik
mi-roslidesbetweenrossedpolarizers. Magneti eld(2kG)
alongthey-axisat0 o
ofthelightpolarizingdiretion. The
measuredlengthoftheperiodiityofthiswallwas=700
mandonlybeomepereptibleafter36hoursofonstant
expositiontothemagnetield.
It was onjetured by Guyon et al. [6℄ and
Lon-bergetal. [7℄thattheelastipropertiesofthenemati
mediumarenotenoughtoexplaintheobserved
geome-trywalls. Thenematimaterialisananisotropiliquid
and, at the moment of the walls reation,its internal
rearrangement must be taken in to aount. The
di-retor rotation stimulates the motion of the nemati
material and it is this internal motion that gives rise
to theone-dimensionaland periodioutstanding
har-ater of the walls. As aonsequene of these
investi-gations,itbeomeswellestablishedthatforhigh
mag-netieldstheinternalmotionofthenematimaterial,
whihstartsatthemomentatwhihtheexternal
mag-netieldisturnedon,hasadeisiverolefortheir
on-one-dimensionaland periodi harater. In fat, with
theuseoftheanisotropipropertiesofNLC,Guyonet
al. [6℄andLonbergetal. [7℄haveshownthat the
on-strution ofthe wallsis possible beause the oherent
motion of the nemati material driving them, has an
eetive visosity whih is lower than the one
result-ingfrom themattermovementforminganyotherkind
of pattern. That is, the observed periodiity results
from a seletion mehanism that amplies to
maro-sopisale somewell dened utuations. If fat,all
modes are amplied. But, theentral onept of this
theory aÆrms that the nal observed prole of these
patternsisdetermined,atthebeginningoftheproess,
by the modes having the fastest initial ampliation.
Fromnowonthismehanismwillbereferredtoasthe
leadingmodepriniple.
Furthermore, aording to the alulations
result-ing from these ideas, when the Freederiksz
thresh-old is approahed, from the upper side, the
oher-ent internal motion of the nemati material beomes
smaller and smaller and there is a point ~
h
w
; greater
thantheFreederikszritial point ~
h
; belowwhih it
disappears. Between this point and the Freederiksz
threshold the walls would be absent and the
dire-torwouldhaveahomogeneousalignment. Theregion
~
h
<H <
~
h
w
isknown astheforbiddenregion. So,
aordingto theusualinterpretation, in theforbidden
regionthe torque of the externaleld on thenemati
moleuleswouldprodueauniformalignment,andthe
periodiwallswould notbedeteted.
Nevertheless,therearesometheoretialand
experi-mentalevidenesindiatingthat, wheneveragood
im-ageof the physial proess behind the originof these
struturesisprovided,thisprinipleannotgiveafull
explanation to the pattern observed on the magneti
walls [?, ?℄. In some reent works it has been found
that at the neighborhoods of the Freederikszritial
point the omparison between the preditions of the
leadingmodeprinipleandexperimentalfats areina
irreoniliabledisagreement[?, ?℄. Inanexperimental
investigation, using the twist-bend geometry [?, ?, ?℄,
wefound that, whenthe Freederikszthresholdis
ap-proahed, the emergeneof these strutures ontinues
[?℄ and, no matter how lose the Freederiksz
riti-al point is approahed, a homogeneous alignment of
the diretor has never been found. Moreover, it was
observed that the time spent in the onstrution of
these periodi walls diverges as the ritial point is
approahed. Therefore, the preditions of the theory
basedontheleadingmodepriniplewerenotonrmed
andaforbiddenregionwasnotdetetedbythe
experi-ment.
Inthesequene,wehaveproposedatheoretial
ex-planation for this unexpeted behavior [?℄. A
Freederiksz threshold has been undertaken. It has
beenshownthat,atthisregion,auniqueleadingmode
an not be learly dened. In fat, there is a set
of neighboring modes, ontinuously distributed, that
growspratiallyat thesame rate. The resultof this
olletivegrowthisnotahomogeneousbendingofthe
diretor, as predited by the leading mode priniple,
butasetofverywelldened periodiwalls,whose
for-mation tends to spend an innite time interval when
theFreederikszthresholdisapproahed.
The aim of this paper is twofold. We will review
the theory of the formation of the magneti walls in
the neighborhoods of the Freederiksztransition and,
at the same time, we will examine the question
pro-posedabove, bystudying anexampleof patterns'
for-mation in the limiting situation where the uid ow,
that determine the appearing of these out of
equilib-riumstrutures,anbemadearbitrarilysmall.
Thework is divided in three setions. Inthe rst
setion,thegeneraltheoryofthewallsformationis
pre-sented. Inthenext one,theleadingmodeprinipleis
disussed and our experimental results, showing that
attheneighborhoodsoftheFreederikszthresholdthis
theory fail, are presented. Finally, at thenal setion
ourtheoryshowinghowthesestruturesarereatedat
thisregion,ispresented.
II Fundamentals
Inordertostudythedynamialproessleadingtothe
appearaneofthemagnetiwalls,aNLCsampleinside
amiroslideglasswithdimensions(a;b;d)that satisfy
therelationabdwillbeonsidered. Thediretor
isinitiallyuniformlyalignedalongthe~e
x
diretionand
anexternalontrolledmagnetield ~
H isappliedalong
the~e
y
diretion. Under these onditions we an
sup-pose that the diretoromponentswill alwaysremain
in the plane dened by the diretion of the magneti
eldandtheinitialorientationofthediretor[? ℄,that
is
n
x
=os(x;y;z); n
y
=sin(x;y;z); n
z
=0: (1)
Inorderto studythe NLCdynamis, the so-alled
Eriksen-Leslie-Parodi (ELP) approah [14-16℄ is used.
The time evolution of the diretor diretion and the
motionofthenematimaterialisgivenbyasetof
dif-ferentialequationsomposedbytheanisotropiversion
oftheNavier-Stokesequation[?℄,thebalaneoftorques
equation,andtheequationofontinuity[?,?℄. Thefull
formandthewaybywhihtheseequationsarehandled
in the formulation of this problem anbe found
else-where [?, ?, ?℄. As our aim is the obtainment of a
non-linearformulationofthisproblem,someattention
will begiventotheapproximationsdoneonthe
equa-tionsoftheELPapproah. Theusual approximations
makethe following assumptions: a) the uid veloity
doesnothasomponentsalongthe~e
z
diretion,b)the
visositydominatesthemotionofthenematimaterial
andtheinertialomponentanbenegleted,)the
on-ditionsprevailingatthebordersofthesamplehavenot
anydeisiveinuenein theshapeof thesestrutures,
d)theomponentoftheveloityalongthediretionof
themagnetield,V
y
,isthedominantoneandonlyit
needsto beonsidered [?℄,e) savefortheedgesof the
sample[?℄ thebending ofthediretorwill beonstant
alongthediretionoftheexternalmagnetield.
Normally,onlythelinearparts oftheequations
re-sultingoftheseapproximationsareanalytiallystudied
(there are numerialstudies of these non-linear
equa-tions [?, ?℄). Consequently, the walls' amplitude will
growwithout limit and this approximationleadsto a
desription that is valid at the initial moments of the
wallsarising. Inordertodesribethelongtime
behav-iorofthesestrutures,theirsaturatedproleisneeded.
So,thehoieofthenon-linearomponentsthatwillbe
retainedinourequationswillbeguidedbythefollowing
riterion: only those that are deisive for a saturated
proleofthewalls'amplitudewill bemaintained.
TheNavier-Stokesequationisgivenby
V
t +V
V
x
=
x
( pÆ
+
); (2)
beingthedensityofthesystem,V
theomponent
of the veloity, p the pressureand
the assoiated
anisotropistresstensor,whihdependsontheveloity
~
V oftheuid,onthebendingofthediretor,andon
itstimevariationrate _
[? ,?, ?℄.
Theapproximationassumedaboveallowsusto
on-siderthe problemastwo-dimensional. So,the
ompo-nentsoftheNavier-Stokesdesribingthemotionofthe
nemati uid alongthe ~e
x and ~e
y
diretions are
suÆ-ienttodesribethewalls'phenomenology. Hene,the
pressure pan be eliminated from these equations by
subtrating one of the omponents of these equations
from another [?℄. Furthermore, we will onsider that
theveloityofthematterinthesampleissuhthatwe
annegletthenon-lineartermV
V
. Thus,
d
dt (
x V
y
y V
x )=
2
x
xy
2
y
yx +
x
y (
yy
xx )+
z (
x
zy
y
zx
): (3)
therefore, any inertialterm an benegleted. Furthermore, using theexpressions for the stress tensor given, for
example,atthereferenes[?,?℄theaboveequationbeomes
1 () 2 x ( x V y )+ 2 () x V y + 3 () + 4 () 2 x (
)=0 (4)
where 1 () = 3 +( 2 3 )sin 2 +( 1 3 +2 3 sin 2 )os 2 ; 2 () = 1 2 d 2 ( 2 + 3 +( 3 2
)os2);
3 () = 1 2 x sin2; 4 () = 1 2 1
(1+os2); (5)
and 1 ; 2 and 3
aretheMiesowiz'soeÆients,
1
isthevisosity'soeÆientsrelatedtorotationofthediretor
(itwasassumedthat
2
=
1
)and, alongthediretion~e
z
,itwasassumedthat 2
z
= (=d) 2
:
Moreover, wealso onsider that the walls' periodiity along the ~e
x
diretion allows us to Fourierdeompose
x V
y
and. Thatis,
x V y = X k a k
os(kx); = X k b k os(kx); t = X k _ b k os(kx) 2 x ( x V y ) = X k a k k 2 os(kx); and 2 x ( t )= X k _ b k k 2 os(kx): (6) where _ b k = t b k
:So,Eq. (?? )beomes
X k n 2 () k 2 1 () a k + 3 () k 2 4 () _ b k o
os(kx)=0:
Consequently, a k = 3 () k 2 4 () 2 () k 2 1 () _ b k : (7)
Fromthisequation,weseehoweahFourieromponentof
t
induesashearmotionofthenemati material,
induinganonnullvaluetotheFourieromponentsof
x V
y
. Itisimportanttoobservethat,throughthefuntions
1 , 2 , 3 and 4
, given by Eq. (??), Eq. (?? ) is strongly dependent on . At this pointthe usual approah [?℄
restrited theanalysis ofthe wallsformationto the independent termof this equation. This proedure is valid
onlywhenthebendingofthediretorisverysmalland,onsequently,intherstmoments,whentheleadingmode
prinipleissupposedto at. But,whenthenextordertermsareonsidered,wehave
a k =R 0 (1 2 ' 2 o +O( 3 )) _ b k ; (8) where R 0 = 1 ~ k 2 3 + ~ k 2 1 and ' 2 o = 3 + ~ k 2 1 ( 2 +2 3 )+ ~ k 2 (2 1 + 2 +2 3 ) ; (9) and ~ k 2 =(kd=) 2
isthereduedwave-vetor. Thisequationshowsthat,foreahk,thereisanangle,'
o
,abovewhih
the Fourieromponentof the bending of thediretor, _
b
k
, nomore induesa non-nullvalueto the orresponding
Fourieromponent,a
k
,oftheshearing,
x V
y :
Usingthesameapproximationsused above,andmakingthehanges
= a H 2 1 t; h 2 = H 2 H 2 F ; ~ K= K 33 K 22 ; a H 2 F =K 22 ( d ) 2 +K 33 ( b ) 2 'K 22 ( d ) 2 ; (10) n x
= os()'1 2 2 ; n x n y
=os()sin ()'(1 2
3
2
); (11)
thebalaneoftorquesequation,
1 t = 1 n 2 ( x V y )+K
beomes _ b k 1 a H 2 F n 2 x a k + ~ K ~ k 2
+1 h 2 1 2 3 2 b k
=0 (13)
UsingtheresultofEq. (??)wearriveat
_ b k = 1 o 1 2 2 max +O( 3 ) b k ; (14) where o ( ~ k)= (1 R 0 ) h 2 1 ~ K ~ k 2 = (1 1 ~ k 2 3+ ~ k 2 1 ) h 2 1 ~ K ~ k 2 ; (15) and 2 max ( ~ k)= h 2 1 ~ K ~ k 2 (h 2 1 ~ K ~ k 2 ) R 0 1 R0 (1+ 1 ' 2 o )+ 2 3 h 2 : (16) d
Now, the term 2
of Eq. (?? ) is written in
terms of its Fourier omponents, given by Eq. (?? ),
and theresulting expression integratedin theinterval
(0;L);giving _ b k = 1 o b k 1 b 2 k 2 max +I nt (b k1 b k2 b k2 ) ; (17) where I nt (b k 1 b k 2 b k 2
) represents the terms of order
higher than three on b
k
. The argument of I
nt has
beenwritten as b
k 1 b k 2 b k 2
to expliitly indiate that it
is only at this term that the interation between the
dierent Fourier omponents is found. Consequently,
inthedynamialequationgivingthetimeevolutionof
b
k
, Eq. (?? ), the interation between dierent modes
isof fourthorder,orhigher. So, uptothird orderthe
equationgivingthetimeevolutionofb
k
isgivenby
_ b k = 1 o b k 1 b 2 k 2 max : (18)
This equationisintegrable,havingassolution
b k ()= max (k) v u u t A o e 2 o (k ) 2 max (k)+A
o e
2
o(k )
; (19)
whih gives the time independent evolution of eah
mode k. A
o
is a onstantof integration that may be
xedat =0.
III The leading mode and its
breakdown
Inorder tounderstand theleadingmodepriniple, let
usonsiderthelowestorderofEq. (??),thatgivesthe
timeevolutionoftheFourieromponentb :
_ b k = 1 o b k (20)
In this approximation, we see that b
k
would have
an exponential growing, and the rate of this growing
is determined by the value of
o
. That is, the fastest
growingrateofb
k
willhappenforthesmallest
o . F
ur-thermore, observe that aording to Eq. (??),
o is
k dependent. Consequently, eah Fourier omponent
b
k
willhaveadierentgrowingrate. Theleadingmode
priniplestatesthatthepatternobservedinthesample
orrespond totheFouriermode withthefastest
grow-ing rate; theonewith the lowest
o
, whih is solution
oftheequation
o
k
=0: (21)
Thisresultsinaseletionmehanismforthe
param-eterk 2
that,throughtheminimizationof,isgivenby
k=0;orbyoneoftherootsoftheequation
K( 1 k 2 + 3 ) 2 1 (K 1 k 4 +(h 2 1) 3
)=0 (22)
Thisseletionmehanismhasgivengoodresultsfor
the regions far abovethe Frederiks threshold, but as
an be easily veried [?, ?℄ this equation predits the
existeneofaregionh 2 h 2 h 2 w ;where h 2 w =1+
K
3
1
; (23)
in whih theseletedmode would beharaterized by
k 2
=0;that is, thefastest modewould bethe
homo-geneous bending of the diretor. Furthermore, it an
alsobeshownthatfor thismodetheoherentveloity
ofthenematimaterialwouldbezero.
In order to verify the validity of this approah in
the outputsof theparametersk 2
and
o
in the
neigh-borhoodsoftheFreederiksztransition. Anemati
ly-otropimixtureof potassiumlaurate(KL),potassium
hloride (KCl) and water, in the alamiti nemati
phase,withtherespetiveonentrationsinweight
per-entage: 34.5, 3.0, 62.5 was used. Nemati samples
were enapsulated in at glassmiroslide (length a =
20mm, width b =2:5 mm, and thiknessd=0:2mm)
from Vitrodynamis. Fig. 1 showsa periodi
distor-tion of~nwith wallsformed in thediretion of ~
H in a
polarizingmirosope. Duringtheexperimentthe
tem-peraturewasontrolled at 251 0
C. Aordingto the
theorypresentedabovewewouldexpet,fromEq.(?? ),
that intheneighborhoodsofh 2
'h 2
w
;itisobtained
k 2
=
1
2K
1 (h
2
h 2
w
); (24)
where h 2
w
wasgiveninEq. (??). Atthesametime,as
k 2
!0,thetime wouldonvergetothexedvalue
w =
1
K
3
: (25)
Summarizing, the aboveresults lead us to believe
thatask 2
!0itwouldbeexpetedthatagraphofk 2
vs. h 2
wouldbeastraightlineonvergingtothepoint
h 2
w
. Below thispoint ahomogeneousdiretorbending
wouldbefound. InFig. 2the resultsofour
measure-ments are shown. Surprisingly, the wallsalwaysexist
and a region with homogeneous alignment was never
found. Inthis gurethedotted line givesapitureof
this supposedresult[?℄.
Furthermore, agraph of vs. h 2
would onverge,
ask 2
!0, to thepoint
w
alulated above, whihis
learly a nite time interval. But, aording to our
experimental results the time spent with the
forma-tionofthesestruturesdivergesasthepointforwhih
k 2
!0isapproahed. Wehaveobtaineddataso lose
to thisritialpointthattheorrespondingwallsonly
appearedaftertwoorthreedaysofontinuum
expo-sitiontothemagnetield. Observethattherstpoint
only appearsin theurveof k 2
vs. h 2
. Atthis point
wedidnotndtheformationof anykindofstruture
in thesample, even after aweek of ontinuous
expo-sure to the magneti eld. Consequently, we are not
approahing the point h 2
= h 2
w
: Furthermore, no
sig-nal of any kind of homogeneous alignment was found
Figure2. Measuredpointsof and(2d=) 2
versush 2
. The
squaresgivethemeasuredpointsfor thatarereadinleft
side. The triangles give (1=) 2
and are read at the right.
Theontinuousline aompanyingthesquares andthe
tri-anglesis only for eyesguiding. The dotted line gives the
urvealongwhiharesupposedbetheexperimentalpoints
of (2d=) 2
versus h 2
. This urve would arrive at zero at
h 2
2:5 and remains zero until the point h 2
= 1. No
suhbehavior wasfoundinthe experiment. Themeasured
pointsgoesdiretlytothepointh 2
=1,andnoregionwith
(1=) 2
= 0 was found. Observe that the rst point only
appearsintheurveofk 2
vs. h 2
:Forthispointwedonot
foundtheformationofanykindofstrutureinthesample
evenafteraweekofontinuousexposureto themagneti
eld. Aswasdemonstratedalongourpaperwhenthepoint
h 2
w
is approahed the bending of the diretor should
be-omeshomogeneous-the walls would disappear -and the
timespentwiththeonstrutionofthishomogeneous
bend-ingmustbenite. Weneversawahomogeneousbending
of the diretor and, furthermore, the time lasted for the
wallsonstrutionbeomesinniteastheritialpointwas
approximated.
Therefore, it is experimentally evident that the
Freederikszthreshold must be above this point, and
weare foredto onludethat the twourves arenot
approahing the point h 2
= h 2
w
; but the Freederiksz
ritialpoint h 2
= 1. That is, the formation of walls
in the upper neighborhoods of the Freederiksz
tran-sition has been deteted in aregion where, aording
to the existing theory, these strutures would not
ex-ist. We have never seen a homogeneous bending of
thediretorand,furthermore,thetimetakenwiththe
walls'onstrutionbeameinniteastheritialpoint
wasapproximated. Furthermore, thenal wave
num-ber presentedby theobserved patterns is nottheone
obtainedbythefastest modeof Eq. (??). Indeed,the
fastest mode is not even an approximation of the
ob-served result, simply beause it predits the absene
of patterns. Ourmeasurement of thewavelength and
thetimespentwiththeformationoftheseobjets
indi-atethatthesewallsarebuiltbythesystemassoonas
theFreederikszthreshold is exeeded. Moreover,the
Lonbergmodelannotexplainthelongtimedemanded
on-three daysin avisous medium where the only
exter-nal fore is due to a onstant magneti eld. In the
neighborhoods of the Freederikszthreshold the time
spent with the onstrutions of the walls beomes so
largethatthevisosityofthesystemwoulddampany
oherentmotionofmatter. Forthisreason,weassume
thatthebigrandomutuationsexistingin the
neigh-borhoodsoftheritialpoint[?℄ aredrivingthebirth,
andseletion,oftheobservedstrutures.
IV The olletive growth
Astheexperimentalresultsreportedaboveareinlear
ontraditionwiththepreditionsoftheleadingmode
priniple, itbeomes neessaryto understand the
for-mation of these walls from another point of view. In
ordertodoit,ourrstattitudewill betoonsiderthe
next order approximation of Eq. (?? ). So, the
for-mationof magneti wallsat the neighborhoods ofthe
Freederikszthresholdwill beexamined losely andit
willbeshownthatinthisregionthestatementthatsays
that there is a unique isolated mode determining the
physial properties of the observed patterns is rather
ambiguousandannotbeonsidered- evenasan
ap-proximation-fortherealphysialsituation. Thatis,a
largesetof statesontributesequallytotheformation
ofthemagnetiwallsinthisregion.
Inthis order,themode that hasthe leadingmode
prinipleanberewrittenas
k
_
b
k
=0; at =0; (26)
that beomes Eq. (??) when the third order term is
abandoned.
Whenthisequation is applied toEq. (?? )it leads
to
k
o
o
1
2
o
2
max
+2
2
o
3
max
k
max
=0; (27)
where
o
standsfortheinitialdistributionofthemode
b
k
that,asusual[?,?℄,anbefoundusingthe
equipar-titiontheorem. Thebehaviorof thesolutionobtained
withtheEq. (??),whenthemagnetieldapproahes
the Freederiksz threshold (h 2
' 1) is now going to
be disussed. An analytial omputation shows that
aroundk 2
0therealsolutionsofEq. (?? )aregiven
by
k 2
0; or
k 2
2
3 3
3
1 (1 h
2
)+ ~
K
33
3
+ 2
o
1 3
2 (h
2
1)+
3 11h
2
9
P(
1 ;
2 ;
3 ;
1 ;h
2
)
; (28)
whereP(
1 ;
2 ;
3 ;
1 ;h
2
)isanawkwardandnonnullfuntionthathasnoinueneontheresultspresentedbelow.
Fromthisequationitiseasyto seethatin thewhole interval
1h 2
h 2
lm
; (29)
where
h 2
lm =(
H
lm
H 2
F )
2
=1+
3
3 ~
K
33
3 +2
2
o
1
1 (3
3
2
o (3
2 +11
3 ))
; (30)
theuniquerealsolutionofEq. (??)isgivenbyk 2
0:So,itseemsthatalsointhisorderwehavethepreseneofa
forbiddenregion. Nevertheless,letusonsider theseondderivative, 2
k
_
b
k
,at =0. Foritwehavefoundthat
2
k
_
b
k
2
2
3
3
1 (h
2
1)(
3
2
o
2 ) 3
~
K
33
2
3 +
1
3
2
o
(9 11h 2
)
6
4
3 ~
K
33 (h
2
h 2
lm )
(1 h 2
lm )
: (31)
Fromthisresult, itiseasy tosee that,atthepointh=h
lm
; theseondderivativeis nulland, furthermore,it
hangessignwhenthispointisrossed. Moreover,inthewholeinterval1h 2
h 2
lm
;theseondderivativeissmall
(itisproportionalto 4
3
)andnegative. Hene,astherangeofthemodesontributingtothewallsformationisgiven
by the inverse of 2
k
_
b
k
, the usual interpretation that followsfrom Eq. (?? ), whih says that in the forbidden
regiontheuniquemodeontributingtothewall'sformationisgivenby ~
k 2
=0,annotbetrue.
Likewise,astraightforwardalulationshowsthat
3
_
b
k
=0 at =0; for 1h 2
h 2
Finally,wealsohavefoundthat
4
k
_
b
k
= 24 ~
K
1
3 +O(
2
o
)<0 at =0; for h=h
lm
; (33)
d
whereO( 2
o
)representsthetermsoftheorderof 2
o ,or
higher, that are insigniant. So, exatly at h=h
lm ,
onlythefourthderivativeofthegrowingspeedofthe
di-retorwithrelationtok,isnon-nullanditisthisvalue
that ontrolsthewidthofthedistributionofmodesat
this point.
TheresultsexposedintheEq. (??)toEq. (??)are
graphiallyexhibitedinFig. 3,whereanumerial
om-putation, using the known parameters of the MBBA,
for thebehaviorof _
b
~
k
;as afuntion of ~
k;asthepoint
h
lm
is approahed, is shown. From these gures, we
learlyseethat,aroundh
lm
;theoneptofanisolated
mode ontributing to the observed periodiity of the
walls is meaningless, and a new way to ompute the
ontribution to the nal periodiity of thewallsmust
befound.
Inordertoproposeanewwaytoalulatethewalls'
periodiity,intheforbiddenregion,weobservethatthe
periodiity of thewallsannot be understood asa
re-sultoftheontributionofauniqueandisolatedmode.
It mustbesupposedthat thenalwalls'periodiityis
determined by the long time olletive growth of the
modes neighboring themode ~
k =0. In this ase, the
natural andidate to x the partiipation of eah of
these modes in the nal prole is just the maximum
amplitude thateahof themanattain. So, itanbe
assumed that, around ~
k ' 0, the nal prole (x) =
(x;t!1)ofthewallsouldbeapproximatedby
(x) = lim
t!1 X
~
k b
~
k (t)os(
~
kx)
Z
L=2
0
max (
~
k)os( ~
kx) (34)
where
max (
~
k)isgivenbyEq. (??). However,withthe
form for
max
(k)given by Eq. (?? ), thisintegrationis
probably impossible. As themaximum of
max (k)
o-urs at k = 0, it an be expanded around this point
and
max (k)'
p
a bk 2
; (35)
isobtained,where
a= 3
2h 2
(h 2
1) and b= 3
2 ~
K
h 2
: (36)
It hasbeenassumedthat (h 2
1) 2
issmall,so
(x) = p
b Z
p
a
b
p
a
b r
a
b k
2
os(kx)dk=
=
p
a
J
1 (
r
a
x); (37)
whereJ
1 (
p
a=bx) is arst kindBesselfuntion of
or-der 1. As, due to our non-linear approximations, our
resultsannotbeextendedtothewallsnodesand
on-sequently they are only valid for small x, we an use
theapproximationJ
1
(x)'(x=2)os(x=2)toobtain
(x) '
2 a
p
b os
1
2 r
a
b x
=
2 r
3
2 ~
Kh 2
(h 2
1)os 1
2 s
(h 2
1)
~
K
x:(38)
With thisequation fortheproleofthewallalong
the~e
x
diretionweangetitswavevetor,thatisgiven
by
~
k 2
= 1
4 ~
K (h
2
1): (39)
whih is just theexperimentally foundresult; alinear
relationshipbetweenh 2
and ~
k 2
[?℄. Furthermore, this
expression for ~
k 2
an be substituted in the Eq. (??)
toobtainthetimespentinthewalls'formation,inthe
forbiddenregion,asafuntion ofthemagnetieldh.
Theresultisgivenby
o (h
2
)= 4
3(h 2
1) (
h 2
1
(
1
1 )+4
~
K
3
(h 2
1)
1 +4
~
K
3 )
;
(40)
that, as it was experimentally found, diverges when
h 2
!1.
V Conlusion
In this work we have studied the behavior of an out
ofequilibriumdynamialsystemthatpresentspattern
formationduetotheinternalmotionofitsmatter,even
whentheveloityofitsuidowapproaheszero.
Con-trarily to theusual theory that assumes theexistene
ofaleading mode, hasbeenshown that the resultsof
theleadingmodepriniple,whihisthetheorythatup
to now is used to desribe their formation [?, ?℄, an
notexplainits arisingin this region. It isknownthat
in the interval 1 h 2
h 2
lm
; where h 2
lm
is given in
Eq. (??),the leadingmodepriniplepreditsthat the
fastest mode would orrespond to k = 0. This result
orrespondsto ahomogeneousbending ofthediretor
and, therefore, to theabsene of any texture. As this
2.5
3
3.5
4
4.5
/
t
q
k
0.2
0.4
0.6
0.8
1
1.1
1.2
1.3
1.4
/
t
q
k
0.2
0.4
0.6
0.8
1
1.1
1.2
1.3
1.4
/
t
q
k
0.1
0.2
0.3
0.4
0.5
/
t
q
k
Figure3.Setofpituresshowingthefuntion;at =0;
when ~
kishanged. Eahgurewasomputedforadierent
valueofh.Thesegures,obtainedusingtheknown
param-etersoftheMBBA,showntheevolutionoftheleadingmode
asthe pointhlm,denedatEq. (??),is approahed. The
(a)was omputedat h=2h
lm
, the point ~
k for whihthe
funtionhasthemaximumgrowingislearlyobserved,
this isthe fastest mode that will determinethe nal
peri-odiityofthewall. The(b)wasomputedath=1:35hlm,
theleadingmodeexistbutislesspronouned.The()was
omputed at h=1:1h
lm
, the leadingis not learly
reog-nizable,butweanseethatitexist. The(d)wasomputed
ath=h
lm
,theleadingmodeollapsedtothepoint ~
k=0.
Fromthesegureswehave alearevidene thatas h
lm is
approahedthewidthofthemodesaroundtheleadingmode
hasbeomesolargethatanisolatedleadingmodeannot
astrongevidenethat,atleastaroundtheFreederiksz
threshold,theleadingmodeprinipledeservessome
re-formulation. The main result presented by this work
statesthat, in thisregion, the modek =0isthe
en-terofalargedistributionofmodeswhereeahofthem
gives almost thesameontribution to the nal prole
of the observed walls. Inorder to obtaina model for
the magnetiwallsformation, wehave supposed that,
due to the long time involved in this proess, around
theFreederikszthreshold,eah modeattainsits
max-imum amplitude. These isolated ontributions have
beenaddedandthenalproleoftheperiodiwallshas
beenobtained. Wehavealsoomputedthetimespent
withtheformationofthesestruturesandwehave
ex-perimentallyfound that,astheFreederikszthreshold
isapproahed,thistimeapproahestheinnite.
Aknowledgments
ThisworkwassupportedbytheBrazilianagenies,
CNPqandArauaria.
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