Bistable director alignments of nematic liquid crystals confined in frustrated substrates

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by  Takeaki Araki

Bistable director alignments of nematic liquid crystals confined in frustrated
substrates
Takeaki Araki1,2 and Jumpei Nagura1
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2CREST, Japan Science and Technology Agency, Japan
(Dated: January 27, 2017)
We studied in-plane bistable alignments of nematic liquid crystals confined by two frustrated
surfaces by means of Monte Carlo simulations of the Lebwohl-Lasher spin model. The surfaces are
preparedwithorientationalcheckerboardpatterns,onwhichthedirectorfieldislocallyanchoredto
7
be planar yet orthogonal between the neighboring blocks. We found the director field in the bulk
1
tendstobealignedalongthediagonalaxesofthecheckerboardpattern,asreportedexperimentally
0
[J.-H. Kim et al., Appl. Phys. Lett. 78, 3055 (2001)]. The energy barrier between the two stable
2
orientationsisincreased,whenthesystemisbroughttotheisotropic-nematictransitiontemperature.

n Based on an elastic theory, we found that the bistability is attributed to the spatial modulation of
a the director field near the frustrated surfaces. As the block size is increased and/or the elastic
J modulus is reduced, the degree of the director inhomogeneity is increased, enlarging the energy

6 barrier. Wealsofoundthattheswitchingratebetweenthestablestatesisdecreasedwhentheblock
2 size is comparable to thecell thickness.

] PACSnumbers: 64.70.mf,61.30.Hn,64.60.De,42.79.Kr
t
f
o
s I. INTRODUCTION than the wavelength of visible light. With them, many
.
t types of structured surfaces for the liquid crystals and
a
m Liquid crystals have been utilized in many applica- the resulting director alignments have been reported in
the past few decades [9, 11, 23, 28–31]. For example,
- tions. In particular, they are widely used in optical de-
d vices such as flat panel displays [1, 2]. Because of the a striped surface, in which the homeotropic and planar
n anchorings appear alternatively, was used to control the
softnessoftheliquidcrystal,itsdirectorfieldisdeformed
o polar angle of the director field in the bulk [32–34].
by relatively weak external fields [3]. To sustain the de-
c Kim et al. demonstrated in-plane bistable alignments
[ formed state, the external field has to be constantly ap-
plied to the liquid crystal substance. In order to reduce by using a nano-rubbing technique with an atomic force
1 microscope [16, 18]. They prepared surfaces of orienta-
power consumption, a variety of liquid crystal systems
v tional checkerboard patterns. The director field in con-
showingmultistabledirectorconfigurationsorstorageef-
5
tact to the surfaces is imposed to be parallel to the sur-
8 fects have been developed [4–23]. In such systems, a
face yet orthogonal between the neighboring domains.
5 pulsed external field can induce permanent changes of
They found that the director field far from the surface
7 thedirectorconfigurations. Liquidcrystalsoflowersym-
0 metries,suchascholesteric,ferroelectricandflexoelectric tends to be alignedalongeither of the two diagonalaxes
. of the checkerboardpattern. More complicated patterns
1 phases, are known to show the storage effects [4–6].
are also possible to prepare [17].
0 Anematicliquidcrystalinasimplegeometry,e.g. that
7 sandwichedbetweentwoparallelplateswithhomeotropic In this paper, we consider the mechanism of the
1 bistable orientations of the nematic liquid crystals con-
anchoring, shows a unique stable director configuration
: fined in two flat surfaces of the checkerboard anchoring
v if externalfields are not imposed. By introducing elastic
patterns. We carriedoutMonte Carlosimulations ofthe
i frustrations, the nematic liquid crystals can have differ-
X Lebwohl-Lasherspin model [35] and argued their results
entdirectorconfigurationsofequalornearlyequalelastic
r withacoarse-grainedelastictheory. Inparticular,thede-
a energy [8–11, 21–26]. For instance, either of horizontal
pendences of the stability ofthe director patterns onthe
or vertical director orientation is possibly formed in ne-
temperature, and the domain size of the checkerboard
maticliquidcrystalsconfinedbetweentwoflatsurfacesof
patterns are studied. Switching dynamics between the
uniformly tilted but oppositely directed anchoringalign-
stable configurations are also considered.
ments [8]. Also, it was shown that the nematic liquid
crystal confined in porous media shows a memory effect
[27]. Thedisclinationlinesofthedirectorfieldcanadopt
alargenumberoftrajectoriesrunningthroughthechan- II. SIMULATION MODEL
nels of the porous medium [21, 22]. The prohibition of
spontaneouschangesofthedefectpatternamongthepos- We carryoutlattice-basedMonte Carlosimulationsof
sible trajectories leads to the memory effect. nematic liquid crystals confined by two parallel plates
Recent evolutions of micro- and nano-technologies en- [21, 35–41]. The confined space is composed of three-
able us to tailor substrates of inhomogeneous anchoring dimensionallatticesites(L L H)anditisdenotedby
× ×
conditions, the length scale of which can be tuned less . Eachlatticesiteihasaunitspinvectoru (u =1),
i i
B | |
2
(a)1.0 (b)3 The nematic order on the surface is larger than that in
0.8 the bulk S and is decreased continuously with T. Even
2 b
0.6 atandabove T , S does not vanishto zero. When the
IN w
0.4 1 w temperature is far below TIN, on the other hand, it is
0.2 close to that in the bulk S .
b
0.0 0 In this study, we prepare two types of anchoring cells.
0 1 2 3 0 0.5 1
In type I cells, we set hybrid substrates. At the bottom
surface (z = 0), the preferred direction d is heteroge-
j
neously patterned like a checkerboardas given by
FIG. 1. (a) Plots of the scalar nematic order parameter in
the bulk Sb (black circles) and on the surface Sw (red open (0,1,0) if ([x/D]+[y/D])iseven
squares) with respect to the temperature. (b) Plot of the dj(x,y)= (1,0,0) if ([x/D]+[y/D])isodd , (2)
elastic modulus K (black circles) of the nematic phase with (cid:26)
T. S2/K is also plotted with red open squares.
w where [X] stands for the largest integer smaller than a
real number X. D is the unit block size of the checker-
board pattern. At the top surface (z = H +1), on the
and the spins are mutually interacting with those at the
other hand, the preferreddirection is homogeneouslyset
adjacent sites. At z = 0 and z = H +1, we place sub- to d = d (cosφ sinθ ,sinφ sinθ ,cosθ ). θ and φ
j t t t t t t t t
strates, composed of two-dimensional lattices. We put ≡
arethepolarandazimuthalanglesofthepreferreddirec-
unit vectors d on the site j on , where represents
j tion at the top surface. In type II cells, both substrates
S S
the ensemble of the substrate lattice sites. We employ
arepatternedlikethecheckerboard,accordingtoEq.(2).
the following Hamiltonian for u ,
i We perform Monte Carlo simulations with heat bath
samplings. A trial rotation of the i-th spin is accepted,
= ε P (u u )
H − 2 i· j consideringthe localconfigurationsofneighboringspins,
hi,jXi(∈B) with the probability p(∆ ) = 1/(1+e∆H/kBT), where
H
P (u e) w P (u d ), (1) ∆ is the difference of the Hamiltonian between before
− 2 i· − 2 i· j anHd after the trial rotation. The physical meaning of
i∈B hi,ji,i∈B,j∈S
X X the temporal evolution of Monte Carlo simulations is
where P (x)=3(x2 1/3)/2 is the second-order Legen- sometimes a matter of debate. However, we note that
2
−
dre function and means the summation over the the method is known to be very powerful and useful for
hi,ji
nearest neighbor site pairs. We have employed the same studying glassy systems with slow relaxations, such as a
HamiltoniantostuPdythenematicliquidcrystalconfined spin glass [21, 42], the dynamics of which is dominated
in porous media [21]. by activation processes overcoming an energy barrier.
The first term of the right hand side of Eq. (1) is the In this study we fix the anchoring strengths at both
Lebwohl-Lasher potential, which describes the isotropic- the surfaces to w =ε, for simplicity. The lateral system
nematictransition[35–37]. InFig.1,weplotthetemper- size is L = 512 and the thickness H is changed. For
aturedependencesof(a)thescalarnematicorderparam- the lateral x- and y- directions, the periodic boundary
eter S and (b) the elastic modulus K in a bulk system. conditions are employed.
b
Thenumericalschemesformeasuringthemaredescribed
inAppendixA.Wenotethatacubiclatticewithperiodic
boundaryconditions(L3 =1283)isusedforobtainingS III. RESULTS AND DISCUSSIONS
b
and K in Fig. 1. As the temperature is increased, both
the scalar order parameter and the elastic modulus are A. Bistable alignments
decreased and show abrupt drops at the transition tem-
perature T =T , which is estimated as k T /ε=1.12 First, we consider nematic liquid crystals confined in
IN B IN ∼
[41]. cells with the hybrid surfaces (type I). Figure 2(a) plots
The second term of Eq. (1) is the coupling between the energies stored in the cell with respect to the az-
the spins in and an in-plane external field e. The last imuthal anchoring angle φ . Here the polar anchoring
t
B
term represents the interactions between the bulk spins angle is fixed to θ = π/2. The energy per unit area
t
E
and the surface directors, that is, the Rapini-Papoular is calculated as (θ ,φ ) = (θ ,φ ) /L2 , where
t t t t min
E hH i −E
type anchoring effect [1, 3]. w is the strength of the X means the spatial average of a variable X.
min
h i E
anchoring interaction. If d is parallel to the substrates is the lowest energy defined as = min /L2
j Emin θt,φthHi
and w >0, the planar anchoring conditions are imposed at each temperature (see below). is obtained after
E
tothespinsatthe -sitescontactingto . Thistermnot 5 104 Monte Carlo steps (MCS) in the absence of ex-
B S ×
onlygivestheangledependenceoftheanchoringeffectin ternal fields. The cell thickness is H =16 and the block
the nematic phase, but also enhances the nematic order size is D =8. The temperature is changed.
near the surface. In Fig. 1(a), we also plot the scalar Figures 2(a) indicates the energy has two minima at
nematic order parameter on a homogenuous surface of φ = π/4, while it is maximized at φ = 0 and π/2.
t t
± ±
w =ǫ. The definition of S is described in Appendix A. To see the dependence on the polar angle, we plot
w
E
3
(a) (b)
against θ with fixing φ = π/4 in Fig. 2(b). It is
t t 0.3
shown that is minimized at θ = π/2 for φ = π/4.
E t t 10-1
Hence we conclude that the stored energy is globally 0.2
lowest at (θ ,φ ) = (π/2, π/4), so that we set =
t t ± Emin 0.1
(θ =π/2,φ =π/4) /L2 in Fig. 2. This global min-
t t
ihmHum indicates that thie parallel, yet bistable configu- 0
-2
10
rations of the director field are energetically preferred 0.2 0.4 0.6 0.8 1 100 101
in this cell. This simulated bistability is in accordance
with the experimental observations reported by Kim et
al. [16]. When a semi-infinite cell is used, the bistable FIG. 3. (a) Plot of the energy difference ∆E per unit area
alignments of the director field would be realized. Here- againstthetemperature. ThecellthicknessisH =8(typeI)
and the block size D is changed. (b) Plot of the energy dif-
after,weexpressthesetwostabledirectionswithnˆ and
+
ferenceperunitareaagainsttheblocksize. Thetemperature
nˆ . That is, nˆ =(1/√2, 1/√2,0).
− ± ± is T/TIN =0.89 and the cell thicknessis changed.
(a) (b) liquid crystal is confined in the thicker cell.
0.8
0.2
Inordertoclarifythemechanismsofthebistablealign-
0.6
ments, we calculate the spatial distribution of the ne-
0.4
0.1
matic order parameter. In Fig. 4, we show snapshots of
0.2
xx-andxy componentsofatensorialorderparameterat
0 0
severalplanesparalleltothesubstrates. Usingu (t′),the
0 0 i
tensorialorderparameterQ iscalculatedbyaveraging
µν
3/2(u u /2 δ /3) in a certain period δt as,
µ ν µν
FIG.2. Dependencesofthestoredenergyperunitareawith −
respecttoφt atθt =π/2in(a),andtoθt atφt =π/4in(b). 1 t+δt−13 1
The liquid crystal is confined in a type I cell of D = 8 and Q (t)= u (t′)u (t′) δ , (3)
i,µν i,µ i,ν µν
H =16. The temperature is changed. δt 2 − 3
t′=t (cid:26) (cid:27)
X
wheret′ meanstheMonteCarlocycle,andµandν stand
Figure 2 also indicates the temperature dependencies
for x, y and z. In this study, we set δt = 102, which is
of the stored energies. When the temperature is much
chosensothatthe systemis wellthermalized. The block
lower than the transition temperature T , the curves of
IN
size is D = 8 in (a) and D = 64 in (b), and the cell
are are rather flat. As the temperature is increased,
E thickness is fixed to H = 8. The temperature is set to
the dependence becomes more remarkable. Figure 3(a)
T/T =0.89. Theanchoringdirectionatthetopsurface
plots the energy difference between the maximum and IN
isalongnˆ ,andwestartedthesimulationwithaninitial
minimum of for fixed θ = π/2 as functions of T. It +
E t condition,inwhichthedirectorfieldisalongnˆ ,sothat
is defined by the in-plane rotation of d as ∆ = (θ = +
t t
E E thedirectorfieldislikelytobeparalleltothesurfaceand
π/2,φ = π/4) (π/2,0). We plot them for several
t
− E along the azimuthal angle φ=π/4 in average.
blocksizesD,while the cellthicknessis fixedto H =16.
Q near the bottom surface shows the checkerboard
In Fig. 3(a), we observe non-monotonic dependences xx
pattern as like as that of the imposed anchoring direc-
of the energy difference on the temperature. ∆ is al-
E tions d . Q inside the block domains is small and it
most independent of T when T/T < 0.6. In the range j xy
IN
of 0.6 < T/T < 0.9, it is increased with increasing T. is enlarged at the edges between the blocks. With de-
IN
When∼T/T >0.9,itdecreaseswithT anditalmostdis- parting from the bottom surface, the inhomogeneity is
IN
reduced and the director pattern becomes homogeneous
appears if T >∼T . When T >T , the system is in the
IN IN alongnˆ . TheinhomogeneitiesinQ andQ aremore
isotropicstate,anditdoesnothavethelong-rangeorder. + xx xy
remarkableforthelargerD thanthoseforthesmallerD.
Thus, it is reasonable that ∆ vanishes when T > T .
IN
E In Fig. 5, we plot the corresponding profiles of the
When T < T , on the other hand, it is rather striking
IN
spatial modulations of the order parameter with respect
that the energy difference shows the non-monotonic de-
to z. The degree of the inhomogeneity of Q is defined
pendencesonT,inspiteofthatthelong-rangeorderand µν
by
the resultant elasticity are reduced monotonically with
increasing T (see Fig. 1).
1
Weplottheenergydifference∆E asafunctionofD in I(z)= L2S2 dxdy{Qµν(x,y,z)−Q¯µν(z)}2, (4)
Fig. 3(b), where the temperature is T/T = 0.89. The b Z
IN
cellthicknessischanged. Itisshownthattheenergydif- whereQ¯ (z)isthespatialaverageofQ inthez-plane
µν µν
ference ∆ is increased proportionally to the block size
and it is given by
E
D when D is small. When the cell thickness is large,
on the other hand, the energy difference is almost sat-
Q¯ (z)=L−2 dxdyQ (x,y,z). (5)
urated. The saturated value becomes smaller when the µν µν
Z
4
(a) (b)
2 2
1 1
0 0
0 10 20 30 0 10 20 30
FIG. 5. Profiles of the inhomogeneity of the nematic order
parameter I(z) along thecell thickness z. (a) The cell thick-
nessisH =32(typeI)andthetemperatureisT/TIN =0.89.
The block size is increased from D = 1 to D = 64. (b) The
cellthicknessisH =32andtheblocksizeisfixedtoD=16.
The temperatureT is changed.
instead of the Rapini-Papoular anchoring energy,
Wcos2φ /2. Here φ is the average azimuthal angle
0 0
−
of the director field on the patterned surface. K is the
elastic modulus of the director field in the one-constant
approximationoftheelastictheory,andW representsthe
anchoring strength in the continuum description. c is a
FIG. 4. Snapshots of the xx- and xy components of the
tensorialorderparameterQµν inthetypeIcellsofthethick- numerical factor, which is estimated as c ∼= 0.085 when
ness H = 8. The anchoring direction on the bottom surface H/D is large. g(φ0) has a fourfoldsymmetry and is low-
(z =0) is patterned like the checkerboard, while that on the ered at φ = π/4 and 3π/4. The resulting energy
0
topsurface(z=9)ishomogeneously alongnˆ+. Thetemper- difference per u±nit area is±given by
ature is T/TIN = 0.89. The block size is D = 8 in (a) and
D=64 in(b). Onlythesnapshotsin asmall area (1282) are π2K
∆ = . (7)
shown. Eth 32H 1+K2/(8cW2DH)
{ }
First we discuss the dependence of the energy differ-
Sb is the scalar nematic parameter obtained in the bulk ence on the block size D. Equation (7) indicates that
[asreeesFcaigle.d1b(ay)S].b2SininEceq.Q(4µν),∝inSorbdienrttohesebeutlhke,pthuerepdreogfirleees wthheicehneisrginycdrieffaesreednlcienebaerhlyavweisthasD∆,Ewthhe≈nπD2ciWssu2Dffi/c(ie4nKtl)y,
of the inhomogeneity of the director field. In Fig. 5(a), small. IfDislargeenough,ontheotherhand,theenergy
wechangedtheblocksizeDandthetemperatureisfixed difference converges to ∆ π2K/(32H). The latter
th
toT/TIN =0.89. Itisshownthatthe degreeofthe inho- energy difference agrees wEith≈the deformation energy of
mogeneity decays with z, and it is larger for larger D as the directorfield, whichtwistsalongthe z axisby π/4.
showninFig.4. Figure5(a)alsoshowsthatthedecaying It is independent of D, but is proportional to H−1±. The
length is also increased with the block size D. Roughly asymptotebehaviorsforsmallandlargeD areconsistent
it agrees with D. In Fig.5(b), we plot the profiles of with the numerical results shown in Fig. 3(b).
I(z) for different temperatures with fixing D = 16. It Nextweconsiderthedependenceof∆ onthetemper-
is shown that the spatial modulation is increased as the ature. Equation(7)alsosuggests∆ isEproportionalto
th
temperature is increased. This is because the nematic W2D/K whenW2DH/K2 issmall.EWe havespeculated
phasebecomessofterasthetemperatureisincreased(see the anchoring strength is simply proportional to the ne-
Fig. 1(b)). When the elastic modulus is small, the di- maticorderasW S . Ifso,theenergydifferenceisex-
b
rector field is distorted by the anchoring surface more pected to be indep∝endent of S as ∆ W2/K S0,
largely. since K is roughly proportionabl to S2E.thT∝his expect∝atiobn
b
Based on these numerical results, we consider the is inconsistent with the dependence of the numerical re-
bistable alignments with a continuum elasticity theory. sultsof∆ inFig.3(a). Apossiblecandidatemechanism
The details of the continuum theory is described in Ap- in explainEing this discrepancy is that we should use the
pendix B. In our theoretical argument, the spatial mod- nematic order on the surface S , instead of S , for es-
w b
ulation of the director field due to the heterogeneous timating W. Since S is dependent on T more weakly
w
anchoring plays a crucial role in inducing the bistable than S near the transition temperature [see Fig. 1(a)],
b
alignmentsalongthediagonaldirections. Aftersomecal- W2/K can be increased with T. The curve of S2/K
w
culations, we obtained an effective anchoring energy for is drawn in Fig. 1(b). Thus, the director field is more
D H as largely deformed near T as shown in Fig. 5(b), so that
≪ IN
cW2D theresultingenergydifferenceshowstheincreasewithT.
g(φ0)= sin22φ0, (6) Also, Fig. 3(a) shows ∆ turns to decrease to zero,
− K E
5
when we approach to T more closely. In the vicinity (a) 0.5
IN
of T , K is so small that W2/K becomes large. Then 0.4
IN
Eq. (7) behaves as ∆ th K/H. It is decreased to zero 0.3
E ∝
as K with approaching to T . In Fig. 3(a), we draw
IN 0.2
the theoretical curve of Eq. (7) with taking into account
0.1
the dependences of W and K on the temperature. Here
0
we assume W = W0Sw with W0 being a constant. The 10-2 10-1 100 101
theoreticalcurvereproducesthenon-monotonicbehavior
(b)
of the energy difference qualitatively. After the plateau
of ∆ in the lower temperature region, it is increased
th
E
with T. Then it turns to decrease to zero when the tem-
perature is close to the transition temperature. Here we
use W = 0.3, which is chosen to adjust the theoretical
0
FIG. 6. (a) Dependences of the averaged order parameter
curve to the numerical result.
on the block size. The cell thickness is H = 16 (type II).
The temperature is changed. (b) Schematic pictures of the
directorfieldnearthemid-planeinthetypeIIcellofD≫H.
B. Switching dynamics
Next we confine the nematic liquid crystals in the 0.5
0.01
type II cells, both the surfaces of which are patterned 0.02
as checkerboard. As indicated by Eq. (6), each checker- 0 0.03
0.033
board surface gives rise to the effective anchoring effect 0.035
with the fourfold symmetry. Hence, the director field is -0.5 0.04
expected to show the in-plane bistable alignments along
0
n or n also in the type II cells.
+ −
Figure 6(a) plots the spatial average of the xy com-
ponent of Qµν at equilibrium with respect to the block FIG.7. Timesequencesoftheaveragednematicorderalong
size. The equilibrium value of Qxy is estimated as nˆ+ inthein-planeswitchingprocesses. Att=0,thedirector
Qna∞xlyfi=eldh.QAxysi|tth=e5×in10it4iailnctohnedistiimonu,hlawtieoniems pwliotyhtnhoe edxirteecr-- 1fi0e4ld≤istc<om2×ple1t0e4lyanadlig3n×ed10a4lo≤ngt<n+4.×I1n04t,imtheeienxtteerrvnaallsfi1el×d
tor field homogeneously aligned along nˆ+, so that Q∞xy isappliedalongnˆ− andnˆ+,respectively. Thestrengthofthe
is likely to be positive. In Fig. 6(a), we also draw a externalfieldeischanged. ThetemperatureisT/TIN =0.89,
and thetypeII cell of D=16 and H =16 is employed.
line of 3S /4, which corresponds to the bulk nematic
b
order when the director field is along n . It is shown
+
that Q∞ is roughly constant and is close to 3S /4 for
xy b itive value, which agrees with Q∞ in Fig. 6(a). From
D H. It is reasonable since the inhomogeneity of xy
the≪director field is localized within D fromthe surfaces. t1 =104,wethenimposeanin-planeexternalfieldalong
When D >H, on the other hand, Q∞xy is decreased with nˆ−, and turn it off at t2 = 2×104. After the system is
D. When D H, the type II cell can be considered as thermalized during t=2 104 and t3 =3 104 with no
a collectionof≫squaredomainseachcarryingthe uniform external field, we apply t×he second extern×al field along
anchoring direction. Thus, the director field tends to be nˆ+ from t3 =3 104 until t4 =4 104. We change the
× ×
parallel to the local anchoring direction d , and then, strength of the external field e.
j
Q inside each unit block becomes small locally. Only When the external field is weak (e2 0.03), the av-
xy ≤
on the edges of the block domains, the director fields eraged orientational order is slightly reduced by the ex-
are distorted and adopts either of the distorted states as ternal field, but it recovers the original state after the
schematically shown in Fig. 6(b). With scaling D by H, field is removed. After a strong field (e2 0.04) is ap-
≥
the plots of Q∞ collapse onto a single curve. plied and is removed off, on the other hand, Q is
xy h xyi
Thenwe considerthe switchingdynamics ofthe direc- relaxed to another steady state value, which is close to
torfieldbetweenthetwostablealignmentswithimposing −Q∞xy. This new state ofthe negativehQxyicorresponds
in-planeexternalfieldseinthetypeIIcells. InFig.7,we to the other bistable alignment along nˆ−. After the sec-
plotthespatialaverageofthexy componentoftheorder ond field along nˆ+ is applied, the averagedorientational
pinagr.amTehteercheQllxsyiizeinisthHe p=ro1c6e,sstehseobfltohcekdsiirzeectiosrDsw=itc1h6- o+rQde∞r.hQxyi comes back to the positive original value,
xy
and the temperature is T/T =0.89. At t=0, we start InFig.8(a),weshowthedetailedrelaxationbehaviors
IN
the Monte Carlo simulation with the same initial condi- of Q in the first switching after t . ∆t means the
xy 1
h i
tion,inwhichthedirectorfieldishomogeneouslyaligned elapsed time in the first switching, that is ∆t = t t .
1
−
along nˆ , in the absence of the external field. As shown Here we change the block size D, while we fix the ex-
+
in Fig. 7, the nematic order is relaxed to a certain pos- ternal field at e2 = 0.03 and the cell thickness H = 16
6
(a) (b)105 withDisconsideredtobeattributedtotheenhancement
0.4
of the energy barrier. Here we note that a critical field
0.2 104 strengthfor the thermally activated switching cannotbe
0.0 defined unambiguously. Since the new alignment is en-
-0.2 103 ergetically preferred over the original one even under a
-0.4 weak field, the director configuration will change its ori-
100 101 102 103 104 102 100 101 102 entation if the system is annealed for a sufficiently long
period. Whenthefieldstrengthismoderate(e2 =0.035),
∼
theaveragedordergoestoanintermediatevalue,neither
FIG. 8. (a) Time sequences of the averaged nematic order of Q∞ or Q∞ in Fig. 7. Such intermediate values of
xy − xy
parameterhQxyiintheswitchingprocess. Before∆t=0,the Qxy reflect large scale inhomogeneities of the bistable
directorfieldisalignedalongnˆ+inaverage. After∆t=0,the ahligniments (see Fig. 9). At each block, the director field
in-plane externalfield is applied along nˆ−. The temperature adoptseitherofthe twostableorientations. Thepattern
is T/TIN = 0.89 and the cell thickness is H = 16 (type II). of the intermediate Q depends not only on the field
xy
The strength of the external field e is changed. (b) Plots of h i
strength, but also on the annealed time. Under large
thecharacteristictimeτ oftheswitchingprocesswithrespect
externalfields, on the other hand, the energy barrierbe-
to the block size D. τ is defined as hQxy(∆t=τ)i=0. The tween the two states can be easily overcome,so that the
temperatureisT/TIN=0.89 andthethicknessof typeIIcell
switching occurs without arrested at the initial orienta-
is changed.
tion (not shown here).
Regarding the local director field, which adopts either
(type II). We note that Qxy at ∆t = 0 depends on D of the two stable orientations (nˆ+ and nˆ−), as a bina-
as indicated in Fig. 6(a)h. IniFig. 8(a), it is shown that rized spin at the corresponding block unit, we found a
the switching rate depends alsoonthe block size D. No- similarity of the domain growth in our system and that
tably, the dependence of the switching behavior is not in a two-dimensional Ising model subject to an exter-
monotonic against D. nal magnetic field. If the switching of the director field
occurs locally only at each block unit, there is no corre-
In Fig. 8(b), we plot the characteristic switching time
lations between the director fields in the adjacent block
τ with respect to the block size D in the cells of H =8,
units. Therefore, the nucleation and growth switching
16 and 32. The temperature and the field strength are
behaviorimplies the director field ata block unit prefers
the samethoseforFig.8(a). The characteristictime τ is
to be aligned along the same orientation as those at the
definedsuchthattheaverageorientationalorderisequal
adjacent block units.
to zero at τ, Q (∆t = τ) = 0. Figure 8(b) shows
xy
h i
the characteristictime is maximized when the block size We observed string-like patterns as shown at ∆t =
is comparable to the cell thickness. When D < H, the 4000 for D = 1 in Fig. 9. Here we note that they are
switching process is slowed down as the block size is in- notdisclinationsofthedirectorfield. Theyrepresentdo-
creased. Ontheotherhand,itisspeededupwithDwhen main walls perpendicular to the substrates. In the type
D >H. InFig.8(b),itissuggestedthatthedependence II cells, we have not observed any topological defects,
of τ on D becomes less significant as H is increased. although topological defects are sometimes stabilized in
Figure 9 depicts snapshots of Q (t) at the midplane the frustrated cell [21]. The string-like patterns remain
xy
(z = H/2) during the first switching process. The pa- ratherstabletransiently. Ontheotherhand,suchstring-
rameters are the same as those in Fig. 8(a), so that likepatternsarenotobservedintheswitchingprocessin
the pattern evolutions correspondto the curves of Q the Ising model. This indicates the binarized spin de-
xy
in Fig. 8(a). Figure 9 shows that the switchinhg bei- scription of the bistable director alignments may be not
havior is slowed down when D is comparable to H, in adequate. Undertheexternalfieldalongnˆ−,thedirector
accordance with Fig. 8(b). When D < H, the snap- field rotates to the new orientationclockwiseor counter-
shots implies the switching proceeds via nucleation and clockwise. New domains,whichappearviathe clockwise
growth mechanism. From the sea of the positive Q , rotations, have some mismatches against those through
xy
where the director is aligned along nˆ , the droplets of the counter-clockwise rotations. The resulting bound-
+
the negative Q are nucleated. They grow with time aries between the incommensurate domains are formed
xy
and cover the whole area eventually. Under the exter- and tend to suppress the coagulationsof them more and
nal field along nˆ , the alignment of the director field less, although the corresponding energy barriers are not
−
along nˆ is more preferred than that along nˆ . Be- so large.
− +
cause of the energy barrier between these bistable align- When D >H, the switching occurs in a different way.
ments, the director field cannot change its orientation The director rotations are localized around the edges of
to nˆ smoothly under a weak external field. From the blocks as indicated in Fig. 6(b). As D is increased,
−
Eq. (6), the energy barrier for the local swiching of the the amount of the director field that reacts to the field
director field between the two stable states is given by is reduced. Although the director fields around the cen-
∆ = 8D2 g(φ = 0) g(π/4) = 8cW2D3/K, when ters of the blocks do not show any switching behaviors
0
F { − }
D <H. Thus,theslowingdownoftheswitchingprocess before and after the field application, they are distorted
7
tor configuration with imposing in-plane external fields.
Usually,theswitchingisconsideredtobeassociatedwith
the actual breaking of the anchoring condition. Thus,
the energy barrier for the switching is expected to be
proportional to W [16]. In this article, we propose an-
other possible mechanism of the switching, in which the
anchoring condition is not necessarily broken. Since the
energybarrierisincreasedwiththeblocksize,theswitch-
ingdynamicsnotablybecomesslowerwhentheblocksize
is comparable to the cell thickness.
By solving ∆ = ∆ǫE2/2 (4D2H), we obtain a
F ×
characteristic strength of the electric field E as E =
c ∼
8cW2D/(∆ǫKH) 1/2, where ∆ǫ is the anisotropy of
{ }
the dielectric constant [see Eq. (B1)]. If we apply an
in-plane external field larger than E , the switching oc-
c
curs rather homogeneously without showing the nucle-
ation and growth processes. This characteristic strength
is decreased with decreasing D, so that the checker-
FIG.9. SnapshotsofthenematicorderparameterQxy(x,y)
board pattern of smaller D is preferred to reduce the
at z = 8 in the type II cell of H = 16. The director field in
yellow regions is along nˆ+ and that in blue regions is along field strength. With smaller D, however, the stability of
nˆ−. The temperature is T/TIN = 0.89 and the block size D the two preferredorientations is reduced. If the effective
is changed. anchoring energy is lower than the thermal energy, the
bistable alignmentwillbe destroyedby the thermalfluc-
tuation. In this sense, the block size D should be larger
to orient slightly toward the field. It is considered that thanD (Kk T/8cW2)1/3,whereweassumedthatthe
c B
≈
the distortion of the director field inside the blocks ef- switchingoccurslocallyineachblock,thatis∆ k T.
B
F ≈
fectively reduces the energy barrier against the external For a typical nematic liquid crystal with K = 1pN and
field. We have not succeeded in explaining the mech- W = 10−5J/m2 at room temperature T = 300K, it is
anism of the reduction of the switching time with D. estimated as D =34nm.
c ∼
When D >H, the inhomogeneousdirectorfieldcontains In our theoretical argument, we assumed the one-
higher Fourier modes of the distortion. The energy bar- constant approximation of the elastic modulus. How-
rier for each Fourier mode becomes lower for the higher ever, the director field cannot be described by a single
Fourier modes [see Eq. (B8). Thus, such higher Fourier deformation mode in the above cells. The in-plane splay
modesaremoreactiveagainsttheexternalfieldandthey and bend deformations are localized within the layer of
would behave as a trigger of the switching process. D near the surface. On the other hand, the twist de-
formation is induced by the external field along the cell
thickness direction. If the elastic moduli for the three
IV. CONCLUSION deformationmodes are largely different from each other,
our theoretical argument would be invalid. We need to
In this article, we studied nematic liquid crystals con- improve both the theoretical and numerical schemes to
fined by two parallel checkerboard substrates by means consider such dependences more correctly. Also, we con-
ofMonte Carlosimulationofthe Lebwohl-Lashermodel. sideredonlythecheckerboardsubstrates. But,itisinter-
As observed experimentally by Kim et al., we found the esting and important to design other types of patterned
director field in the bulk shows the bistable alignments, surfaces[17]toappendmorepreferredfunctions, suchas
which are along either of the two diagonal axes. We faster responsesagainstthe externalfield, to liquidcrys-
attribute the bistability of the alignments to the spa- tal devices. We hope to reporta series of such studies in
tial modulation of the director field near the substrates. the near future.
Based on the elastic theory, we derived an effective an-
choringenergywith thefourfoldsymmetry(Eq.(6)). Its
anchoring strength is expected to behave as W2D/K,
ACKNOWLEDGEMENTS
when the block size D is smaller than the cell thickness.
As the temperature is increasedto the isotropic-nematic
transitiontemperature,the elastic modulus K ofthe ne- We acknowledge valuable discussions with J. Ya-
matic phase is reduced so that the director field is de- mamoto, H. Kikuchi, I. Nishiyama, K. Minoura and T.
formednearthe substratesmorelargely. Withthis effec- Shimada. This work was supported by KAKENHI (No.
tive anchoring effect, we can explain the non-monotonic 25000002 and No. 24540433). Computation was done
dependenceoftheenergystoredinthiscellqualitatively. using the facilities of the Supercomputer Center, the In-
We also studied the switching dynamics of the direc- stitute for Solid State Physics, the University of Tokyo.
8
Appendix A: Estimations of the nematic order and squaress. Notably, S remains a finite value even when
w
the elastic moduls T >T . In Fig. 1(a), we cannot see any drastic change
IN
of S , which is continuously decreased with T.
w
In this appendix, we estimate the scalar nematic or-
der parameter and the elastic modulus in Fig. 1 from
the Monte Carlo simulations with Eq. (1) [21, 35–41].
Appendix B: Analysis with Frank elasticity theory
First we consider the bulk behaviors of nematic liquid
crystals, which are described by the Lebwohl-Lasher po-
Here, we consider the nematic liquid crystal confined
tential. Here we remove the surface sites and employ
S in the checkerboard substrate on the basis of the Frank
the periodic boundary conditions for all the axes (x,y,
elasticity theory. The checkerboard substrate is placed
and z). We use the initial condition along u = (1,0,0)
i
atz =0,while wefixthe directorfieldatthe topsurface
and thermalize the system with the heat bath sampling.
The simulation box size is L3 with L=128. likethetypeIcellsemployedinthesimulations. Thefree
energy of the nematic liquid crystal is given by
It is well known that this Lebwohl-Lasher spin model
describes the first-ordertransitionbetweenisotropic and
K ∆ǫ
nematic phases. In Fig. 1, we plot the xx component = dr( n)2 dr(n E)2
F 2 ∇ − 2 ·
of the tensorialorderparameter after the thermalization
Z Z
(t ≤5×104) as a function of T. Since the initial condi- W dxdy(n d)2, (B1)
tion is along the x axis, the director field is likely to be − ·
Zz=0
aligned along the x axis. We here regard Q as the
xx
scalar nematic order parameter S . We sehe aniabrupt where n is the director field. The first term in the right
b
change of S around k T 1.12ǫ, which is consistent hand side of Eq. (B1) is the elastic energy. Here we
b B IN
≈
withpreviousstudies [41]. AboveT ,the nematic order employ the one-constant approximation with the elas-
IN
almost vanishes, while it is increased with decreasing T tic modulus K. E and ∆ǫ are external electric field and
when T <T . the anisotropyofthe dielectric constant. Here wedo not
IN
In the nematic phase (T < T ), the director field n considerthe effectofthe electric field. The thirdtermin
IN
can be defined. Because of the thermal noise, the local Eq. (B1) represents the anchoring energy in the Rapini-
director field is fluctuating around the average director Papoular form. W is the anchoring strength and d is
field, reflectingthe elastic modulus. The elasticmodulus the preferred direction on the surface at z = 0. For the
of the director field is obtained by calculating the scat- checkerboardsubstrates, we set d according to Eq. (2).
tering function of the tensorial order parameter as [37], At the top surface, we fix the director field as n(z =
H) = d (cosφ ,sinφ ,0), and the bottom surface also
k T t t t
Q˜ (q)2 = B , (A1) prefersthe planaranchoring. Fromthesymmetry,there-
h| xµ | iT A+4L sin2 q a/2
Q | | fore, we assume that the director field in the bulk lies
forµ=y andz. Q˜ (q)istheFouriercomponentofQ paralleltothesubstrateseverywhere. Then,wecanwrite
xµ xµ
at a wave vector q. a is the lattice constant, and it only with the azimuthal angle φ as
T
h···i
meansthethermalaverage. AandL arethecoefficients
Q
appearing in the free energy functional for Q . n=(cosφ,sinφ,0). (B2)
µν
In the case of T < T , the scattering function goes
IN
to zero for q a = 0. Then, we obtain the coefficient L Also, we assume that the director field is periodic for x
byfitting Q|˜| ∼2 −1 with4(k T)−1L sin2 q a/2. L Qis andydirections,sothatweonlyhavetoconsiderthefree
h| xµ| iT B Q | | Q energy in the unit block (0 x, y 2D). With these
proportionaltotheelasticmodulusK ofthedirectorfield ≤ ≤
n as L = KS2. In Fig. 1(b), the elastic modulus K is assumptions, the free energy per unit area is written as
Q b
plottedwithrespecttoT. Itisdecreasedwithincreasing
K 2D 2D H
T,ifT >TIN. Thisindicatesthesofteningofthenematic = dx dy dz( φ)2
phase near the transition temperature. E 8D2 ∇
Z0 Z0 Z0
Next we consider the effect of the surface term. The W D D 2D
surface effect not only induces the angle dependence of dx dysin2φ+ dycos2φ .
the anchoring effect in the nematic phase, but also leads −2D2 Z0 (Z0 ZD )(cid:12)(cid:12)z=0
to the wetting effect of the nematic phase to the surface (cid:12) (B3)
(cid:12)
inthe isotropicphase[43]. We sethomogeneoussurfaces (cid:12)
ofw =ǫatz =0andz =H+1asinthemaintext. The In the equilibrium state, the free energy is minimized
anchoringdirectionisdj =(1,0,0). Theperiodicbound- with respect to φ(x,y,z). Inside the cell (0 < z < H),
aryconditionsareimposedforthex-andy-directionsand the functional derivative of gives the Laplace equation
the initial condition is along ui = (1,0,0). The profile of φ as E
of Q¯ (not shown here)indicates Q¯ at z =1 becomes
xx xx
largerthanthatinthebulk,S . Thisvalueisthesurface δ
b E = K 2φ=0. (B4)
orderSw,whichisalsoplottedinFig.1(a)withredopen δφ − ∇
9
From the symmetry argument, we have its solution as In the limit ofH D, c convergesto c 0.085,while it
≫ ≈
behavesasc 0.5H/DifH D. Sincecispositive,the
φ(x,y,z)=φ +(φ φ )z/H+∆(x,y,z), (B5) ≈ ≪
0 t 0 last term in the right-hand side of Eq. (B10) represents
−
∞ (2m+1)πx (2n+1)πy an effective anchoring condition [Eq. (6)] in the main
∆(x,y,z)= ∆ sin sin
mn text. It indicates the director field tends to be along the
D D
mX,n=0 diagonal axes of the checkerboard surface, φ0 = π/4.
±
×sinh(πγmn(H −z)/D), Then, we minimize E with respect to φ0 and obtain
γ = (2m+1)2+(2n+1)2 (B6)
mn
where φ0 andp∆mn are determined later. W cW2D K (φ π/4)2
t
It is not easy to calculate the second term in Eq.(B3) = + ∓ (.B12)
E − 2 − K 2H 1+K2/(8cW2DH)
analytically. Assuming ∆ 1, we approximate sin2φ
| | ≪
as
sin2(φ0+∆) sin2φ0+∆sin2φ0+∆2cos2φ0,(B7) ItcorrespondstotheplotsinFig.2(a). Hereweassumed
≈
φ π/4 1, so that sin22φ =1 4(φ π/4)2. The
Then, we obtain the free energy per unit area as | 0∓ |≪ 0 ∼ − 0±
resulting energy difference is obtained as
K W
= (φ φ )2
t 0
E 2H − − 2
πK∆2 sinh(2πγ H/D) π2K
+ mn mn ∆ = . (B13)
16D E 32H 1+K2/(8cW2DH)
m,n(cid:20) { }
X
4W∆ sin2φ sinh(πγ H/D)
mn 0 mn
. (B8)
− (2m+1)(2n+1)π2
(cid:21)
In the strong anchoring limit, we can obtain φ rigor-
Firstweminimizethefreeenergywithrespectto∆
mn ously as
by solving ∂ /∂∆ =0. Then, we have
mn
E
16WDsech(πγ H/D)sin2φ
mn 0
∆ = , (B9)
mn (2m+1)(2n+1)γmnπ3K φ(x,y,z)=(φt∓π/4)z/H±π/4
4 1 (2m+1)πx (2n+1)πy
and + sin sin
π (2m+1)(2n+1) D D
K W W2D Xm,n
E ≈ 2H(φt−φ0)2− 2 −c K sin22φ0, (B10) I×tssiennhe(rπgγymdniff(Here−ncze)/isDt)hen given by ∆ =π2K/(3(B2H14)).
32tanh(πγmnH/D) E
c= . (B11) It is consistent with Eq. (B13) in the limit of W .
(2m+1)2(2n+1)2π5γ →∞
mn
mn
X
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