B.Sadovaya1/4, Mosow 103787,Russia
Reeivedon30January,2002
Inthe last deadeithas beenexperimentally foundaperiodidomainpatternarisinginsmeti
C
liquid rystalsinsurfaestabilizedbookshelf geometry. Suhaperioditextureappearsafter
swithing-oanexternaleletrield,eveninstronganhoringonditions. Ithasastatiharater
and anbebidimensional, being dependenton bothdiretions normal to thesmetiplanes and
normaltotheellplates. Inthepresentworkanexplanationtothisphenomenonisproposed.
A-ordingtoourmodelintheantiferroeletriphasethebiperioditextureisathresholdphenomenon,
appearingfor valuesof thespontaneouspolarizationgreater thanaritial value, whereasinthe
ferroeletriphasethistypeofbidimensionalinstabilityishindered.
I Introdution
TheimportaneofFerroeletriLiquidCrystals(FLC)
is stritly related to the nonlinear eletro-opti eet
determined by theoupling of the spontaneous
polar-ization !
P
s
withanexternalappliedeletrield !
E
ext
[1℄. The eet anbe haraterizedby bistabilityand
optialmemoryinthesurfaestabilizedbookshelf[2,3℄
orquasibookshelf [4℄onguration. Inthisase,eah
single layer exhibits a uniform spontaneous
polariza-tion, oriented in the same way in the whole ell (see
Fig.1a).
But some smeti C
materials, like MHPOBC,
due to their moleular struture [5℄ an present also
a dual arrangement, in whih onseutive layers
nat-urally have alternative sense of !
P
s
- antiferroeletri
phase(AFLC).Inthisstatethewholeellspontaneous
polarizationvanishes. AnAFLCunderappliedeletri
eldanmovetowardsothertwostablestates,up-and
down- viaanazimuthal rotationÆ= ofalternate
layers(seeFig.1b),allowingtoobtainatristability
be-haviour[3℄. Inthelastdeade,theappearaneofstati
modulatedpattern wasobserved,bothin
ferroeletri-andinantiferroeletriphase[6℄-[12℄,forinstaneafter
swithingoaDCeldappliedtoauniformsurfae
sta-asstati stripes,oriented either parallel [8-11℄ or
per-pendiular[6,7,12℄tothesmetilayersnormal,whih
liesin aplaneparalleltotheellwalls(see Fig.2).
Up to now several qualitative models desribe the
possibleroleplayedbythepreseneofaperiodi
disli-nationarray[13℄,byexoeletriityandharge
ondu-tion[12℄.
Thepurpose of thepresentpaper is to investigate
anewmehanism,explainingthebiperiodiinstability
astheresultofaompetitionbetweenthe
polarization-eld ouplingand theoulombian interation between
the polarization harges in the smeti layers. This
ompetition is driven by the anhoring, mediated by
thebulkelastiity[14℄, biasedbythedieletri
ontri-butionand orretedbytheexoeletriity[15℄.
II Theory
Let us onsider a bookshelf arrangement of a
mate-rialexhibitingbothferroeletri-and
antiferroeletri-phases,likeMHOPBC.Theellrefereneframe[x;y;z℄
hasx-axisnormaltotheellwalls,y-axisparalleltothe
Figure 1. Surfae stabilized C
liquid rystal ell with
stronganhoringinbookshelfgeometry(a). Theellvolume
is D
x D
y D
z
,d is the smetilayerthikness. Thesmeti
oneis alsoreportedin(b),withazimuth,polarangle
andspontaneouspolarization ~
P
S
eitherparallel or
antipar-allel tothe unitvetorpb=b b
k,aording to thetypeof
moleularhirality.
Figure2. Periodidistortionmodeswithwave-vetoralong
thex-axis, normal to the ellplates (a), andalong the
z-axis,normaltothesmetiplanes(b). Intheaseofstrong
anhoring, the rstmode has wavelength x equalto the
doubleoftheellthiknessDx.
diretor,andb isthetiltdiretor. is thepolar angle
haraterizing thetilt one, whereas is the azimuth
of the n-diretorb orientation (see Fig. 1b). Any
vari-ationÆ is onnetedwith aharddistortion, hindered
at onstant temperature; instead, any variation Æ is
allowed, desribing a soft distortion. The LC
sponta-neous polarization !
P
s = P
s b
p, where pb= b b
k ( b
k is
paralleltothez-axis),liesinthesmetilayerparallel
to xy-plane. The initial onguration of b is parallel
toy-axis(
0
==2). Byapplyinganexternaleletri
eld !
E
ext
alongzthelinearouplingwiththe
polariza-tion !
P
s
produesaertainazimuthalrotation(
0 ).
Afterswithing o the eld, theb-distributionrelaxes
toaongurationdierentwithrespettotheprevious
one.Thenewongurationisbiperiodiallydistributed
in zx-plane,beinginvariant alongy-axis. Todesribe
themodulatedpattern,itisonvenienttodene
=
0
+'(x;z) (1)
where the wave amplitude j'j
0
. Moreover, the
whered
i
areexoeletrimoduli. !
P
f
(x;z)writes
!
P
f ' d
3 os
0 b '
x +[d
4 '
z +d
6 sin
0 '
x ℄pb d
9 os
0 '
x b
k; (5)
turningoutto bebidimensionallymodulatedaswell, thenreatinganinternal eld !
E(x;z),haraterizedby the
potential (x;z):
!
E(x;z)= grad (x;z)= [
x b
i+
z b
k℄; (6)
beingj (x;z)jP
s D
x
,whereD
x
istheellthikness. Intheloalframe[b; b
k;p℄b theinternaleld !
E(x;z)is:
!
E = E
b+E
z b
k+E
p b p=sin
0 x b
z b
k os
0 x b
p; (7)
andoupleswith !
P
f
,giving
f
flexo =
!
P
f
!
E =
= 1
2 (d
3 +d
6 )sin 2
0 '
x x +d
4 os
0 '
z x d
9 os
0 '
x z
: (8)
d
Dieletriterm
The dieletri ontribution to the ell free energy
densitywrites
f
diel =
1
8 !
D !
E; (9)
where
!
D =" !
E: (10)
The rank 2 dieletri tensor "("
k ;"
p ;"
t
) has only its
diagonal omponents dierent from zero [16, 17℄,
ex-pressed in the intrinsi frame [bn;bp; b
t℄ where b
t is the
transverseunit vetor b
tnbp.b It takesinto aount
thebondedhargesseparationinthehiralliquid
rys-tal. Duetothefatthatthewaveamplitudej'jofthe
periodidistortionismuhsmallerthan
0
,thediretor
b nreads
b
n = sinsin b
i+ossin b
j+os b
k=
' sin(sin
0
+'os
0 )
b
i+
+sin(os
0
'sin
0 )
b
j+os b
k: (11)
Thus,from Eqs. (9- 11)thefreeenergytermis
f
diel =
"
k
8 (
!
E n)b 2
"
p
8 (
!
Ep)b 2
"
t
8 (
!
E b
t) 2
(12)
andeventuallybeomes
f
diel =
"
k
8
(sinsin
0 x os
z )
2 "
p
8 (os
0 x )
2
+
"
t
(ossin
0 x +sin
z )
2
Eletrostatiinterationtermamongbondedharges
The Coulomb interation free energy density
be-tween spontaneous polarization harges Q, Q 0
sepa-rated at the same surfae in orrespondene of two
smeti layersl,l 0
isgivenby
f C = 1 2 jQjjQ 0 j j ! r 0 ! rj 1 V (14)
whereV istherelevantvolume, !
r; !
r 0
arethepositions
of Q, Q 0
, and the sign to be hosen is either (+) or
( ) respetivelyfor ferroeletri- and antiferroeletri
-phases, aording to the fat that interating dipole
harges separated at the same ell surfae are of the
same or dierent signs. This means that a hiral LC
inferroeletristatehasarepulsiveoulombian
poten-tial,sinethehargesseparatedatthesamesurfaein
orrespondene of adjaent layers have the same sign
(f C = 2 Q 2
=d),whereasinantiferroeletristateithas
attrativeoulombianpotential,sinetheharges
sepa-ratedhavealternativelyoppositesign(f
C = 2 Q 2 =d)
-notethatisaonvenientonstantanddisthe
sme-tilayerthikness. Morepreiselyf
C writes f C = 1 2 Z Z dA 0 dA 0 jz 0 zjD x D y D z ; (15) where , 0
are the surfae harges densities at the
walls, dA, dA 0
are the relevant surfae elements and
D
i
is the ell size along the i-diretion. Putting P
s ,
P 0
s
asthe involvedspontaneous polarizations, the
or-respondinghargesare
dA = ! P s d ! A = ! P s D y dz b i 0 dA 0 = ! P 0 s d ! A = ! P 0 s D y dz 0 b i: (16)
ApproximatingDiradeltafuntionbyÆ(z)=1=jz z 0
j,
theintegrationofEq. (15)overthewhole ellgives
f C = 1 2 (div ! P 0 s ) 2 D x D y : (17)
Theloal spontaneouspolarizationis
! P s = P s b p=P
s [os
b
i+sin b j℄= ' P s [(os 0 'sin 0 ) b
i+(sin
0
+'os
0 )
b
j℄; (18)
d
andthedivergeneis
div ! P s = P sx x ' P s sin 0 ' x : (19)
TheCoulombtermeventuallyreads
f C = 1 2 P 2 s sin 2 0 D x D y ' 2 x ; (20)
where(+)and( )arerelevanttotheferroeletri-and
antiferroeletri- states, respetively. The total bulk
free energy density is the sumof all ontributions
al-ready mentioned:
f =f
el +f flexo +f diel +f C : (21)
Inthehypothesis thatthe anhoringis strong,the
to-talfreeenergyoinideswiththetotalbulkfreeenergy.
Byminimizing itwith theusualproedure,the
Euler-LagrangeEqs. 8 < : z f ' z x f ' x + f ' =0 z f z x f x + f =0 (22)
anbewritten as
8 > > > > < > > > > : [B 1 sin 2 0 +B 2 os 2 0 1 2 P 2 s sin 2 0 D x D y ℄' xy + +B 3 ' zz 2B 13 sin 0 ' xz
+dsin2
0 xx +d os 0 xz =0 [sin 2 0 (" q sin 2 +" t os 2
)+"
p os 2 0 ℄ xx + +(" q os 2 +" t sin 2 ) zz (" q " t
)sin2sin
0 xz
+
2dsin2
+ 4[ 1 2 dsin2 0 +d os 0 r℄ 2 sin 0 (" k sin 2 +" t os 2 ) (" k " t )sin2r+(" k os 2 +" t sin 2 )r 2 =0 d
wherethewave-vetorratioisdened as
rq
z =q
x
: (26)
Theminimumvalue ofr, whih satisesthe ondition
(25),determinesthethresholdoftheappearaneofthe
biperiodiinstability. In theaseof stronganhoring,
itturnsouttobe
q x = D x '1m 1 ; (27)
sine at theellwallsthe diretoris keptin its
unde-formed position: '(x = 0;z)= 0, '(x = D
x
;z) = 0,
andtheminimumenergydistortionhasthewavelength
x
equalto thedoubleellthikness,
x =2D x ,where D x 2m.
IftheelastionstantsforbendB
1
andforsplayB
2
areequalandifthedieletripermittivityisisotropi,
" k =" p =" t
",thedispersionrelationanbewritten
moresimplyas
B 3 r 2 B 13 sin 0 r+B
2 P 2 s sin 0 D x D y +4 [dsin2 0 +d os 0 r℄ 2 "[sin 0 +r 2 ℄
=0; (28)
d
resultingina4 th
degreealgebraiequation. Letus
re-mind the orrespondene betweenthe smeti C
and
thenematielastionstants
8 > > < > > : B 1 = 1 4 K 22 sin 2
2+K
33 sin 4 ; B 2 =K 11 sin 2 ; B 3 =K 22 sin 4 + 1 4 K 33 sin 2 2; B 13 = 1 2 (K 33 K 22 )sin2sin 2 ; (29)
where the smeti tilt =(T) is only dependent on
temperatureT. AtroomtemperatureT =25 Æ
C ithas
thetypialvalue20 Æ
. Theusualvaluesoftheother
relevant parameters are B
1
B
2
B
3
0:10:5,
K ii 10 6 dyn, B 13 B 1 10 6 dyn, d 10 5 dyn, d 10 5 dyn, P s
(0:0550)nC/m 2 , 10 Æ 90 Æ
,"=1. HenethelasttermofEq. (28)
turns out to be negligible. If the elasti oupling no
ommarbend-twist,B
13
isnegligible(forrigidsmeti
planes,B
13
=0)andifthesplaydoesnotinuenethe
phenomenon(B
2
=0), thus(28)reduesto
B 3 r 2 1 2 P 2 s sin 0 D x D y
'0; (30)
givingsimply: r 2 = D x D y 2B 3 sin 0 P 2 s : (31)
Wepointoutthatonlyinantiferroeletristatethis
modelpreditstheexisteneofthebiperiodidistortion
forsmetiC
0
5
10
15
20
25
30
0
20
40
60
80
100
120
r=
λ
x
/
λ
z
φ
0
=85°
φ
0
=75°
φ
0
=15°
φ
0
=5°
(a)
0
0.5
1
1.5
2
2.5
3
0
2
4
6
8
10
polarization [ues]
r=
λ
x
/
λ
z
φ
0
=85°
φ
0
=75°
φ
0
=15°
φ
0
=5°
(b)
Figure 3. Ratio r =x=z between thewavelengthsof the twodistortion modes haraterizing thebiperiodiinstability
as afuntionof thespontaneous polarization PS. (a)Theonguration are desribedbythe pre-azimuth0 =90 Æ
with
=5 Æ
-15 Æ
(aroundmaximumpre-tilt),and with0 =
(aroundunidiretionalplanarity). (b)Zoomofthe same
diagramaroundthethreshold,showingthatintherstasethetransitionis2ndorder,whereasintheseondaseitis1st
order.
0
5
10
15
20
25
30
0
20
40
60
80
100
r=
λ
x
/
λ
z
φ
0
=65°
φ
0
=45°
φ
0
=25°
(a)
0
0.5
1
1.5
2
2.5
3
0
2
4
6
8
10
polarization [ues]
r=
λ
x
/
λ
z
φ
0
=65°
φ
0
=45°
φ
0
=25°
(b)
Figure 4. ThesameasinFig. 3,but(a)with
=25 Æ
-45 Æ
. (b)Zoomofthe samediagramaroundthe threshold. When
045 Æ
plates;when
0
=90 ,thetiltdiretorbliesintheell
plates, and the bn-orientation is unidiretional planar.
Aordingtothepresentmodel,numerialalulations
performed onthe dispersionrelation Eq. (28)
demon-strate that the biperiodi pattern is a threshold
phe-nomenon, with the wavelength
x
bonded to the ell
thikness in the ase of strong anhoring. Above the
threshold, the wavelength
z
is almost inversely
pro-portionaltothesmetilayerspontaneouspolarization
P
s
,andinreaseswiththepre-azimuthaswell. InFig.
3band4bazoomoftheoriginzoneofFig. 3aand4a
isreported,showingthatforpre-azimuth
0 0
Æ
,
im-plyingpre-tiltlosetothesmeti tilt,thetransition
isoftherstorder,whereasforpre-azimuth
0 90
Æ
,
implyingaquasi-planaronguration,thetransitionis
oftheseondorder. Therstorderharateryieldsin
therange
0 =0
Æ
-45 Æ
,whereastheseondorder
har-ater yields in the range
0 = 45
Æ
-90 Æ
. In any ase
theP
s
-threshold diminishesas the pre-azimuth
0
in-reasesaswell,onvergingtoaminimumritialvalue
0:2ues.
III Conlusions
Thearising ofperiodialinstability in smeti C
has
been analysed and a new model in the frame of
on-tinuumtheory hasbeenestablished. Aordingto our
model, a softmixed bend- and twist- distortion
spon-taneouslyarisesinhomogeneoussurfaestabilisedells
orderedin bookshelfgeometry,whentheexternaleld
isswithedo. Suhdistortionisbiperiodial,sinethe
twowave-vetorsinsteadofwavevetoranddiretions
insteadofdiretion. Inantiferroeletriongurationit
anbeexplainedwithoutinvokingsmetiplanes-and
layersdeformation, asaompetition betweenthe
ou-pling ofspontaneouspolarisation-internaleletrield
and the Coulombian interation among harges
sepa-ratedinsmetilayersattheellwalls. Suh
ompeti-tionisdrivenbyanhoringandismediatedbyelastiity.
Thedieletri-andexoeletri-ontributionsjustgive
ComplexFluidsandTheirAppliation",whihallowed
them to attend the Workshop. We are grateful to L.
Kramerforveryusefuldisussion.
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