Point-defect haloing in curved nematic films

0.25 MB·English

by  Isaku Hasegawa

Point-defect haloing in curved nematic films
∗
Isaku Hasegawa
Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan
Hiroyuki Shima
Department of Applied Physics, Graduate School of Engineering,
Hokkaido University, Sapporo 060-8628, Japan and
Department of Applied Mathematics 3, LaCa`N, Universitat Polite`cnica de Catalunya (UPC), Barcelona 08034, Spain
(Dated: January 6, 2010)
We investigate the correlation between the point disclination energies and the surface curvature
0 modulation of nematic liquid crystal membranes with a Gaussian bump geometry. Due to the
1 correlation, disclinations feel an attractive force that confines them to an annulus region, resulting
0
in a halo distribution around the top of the bump. The halo formation is a direct consequence of
2
the nonzero Gaussian curvatureof the bump that affects preferable configurations of liquid crystal

n molecules around thedisclination core.
a
J PACSnumbers: 61.30.Jf,61.30.Hn,81.40.Lm,02.40.-k

6

I. INTRODUCTION tures. In actual systems, however, a slight physical (and
]
t chemical)disturbancecausesexcessdisclinations[19–22].
f
o These disclinations can travel over a surface via thermal
It is generally assumed that topological defects in liq-
s excitation,and they flow almostfreely if the disclination
. uid crystal membranes are excluded in the ground state
t densityissufficientlylowtoignoretheirinteraction. The
a becausetheyincreasetheelasticfreeenergyofthesystem
m to more than that in the defect-free equilibrium state. only possible force that restricts the free motion of the
excess disclinations arises from a coupling between the
- The situation is quite different when the liquid crystal
d surface curvature and the disclination energy. The sub-
phases are embedded in a curved surface. In a curved
n stratecurvaturebreaksthetranslationalsymmetryofthe
membranewithnonzeroGaussiancurvature,theappear-
o molecularconfigurationinthevicinityofthedisclination
ance of topological defects can be favorable or even in-
c core, which may cause a shift in the disclination energy.
[ evitable because of geometrical and/or topological con-
As a result,the energeticallypreferentialpositions ofex-
straints. A typical example is a disclination-containing
3 cess disclinations are believedto be tunable by imposing
equilibrium state in a nematic liquid crystal that is con-
v an appropriate curvature modulation.
strained in a spherical surface [1]. The defect structure
8
In this article, we study the energies of disclinations
2 insuchanematicsphericalshellhasrecentlybeenrecon-
in a nematic thin film assigned on a Gaussian bump. It
2 sideredboththeoretically[2–5]andexperimentally[6,7],
4 and these investigations revealed richer physical proper- is found that disclinations with a +1 charge prefer posi-
2. ties thanthose thathaveinitially been suggested. Inad- tioningatinflectionpointsofthebump,implyingthefor-
mation of a “halo,”i.e.,an annulus distribution of excess
1 dition to the nematic cases, the smectic phases of liquid
disclinations around the top of the bump. In addition,
9 crystalfilms[8,9]andcolumnarphasesofablockcopoly-
0 mer assembly [10] have also exhibited an interplay be- disclinationswitha 1chargearefoundtoexhibitdouble
−
: concentric halo structures, provided the bump height is
v tweendisclinationstructuresandthe geometry/topology
relatively large. These halo formations are consequences
i of the constraint surface. In particular, smectic ordering
X ofthe non-trivialcouplingbetweenthe disclinationener-
on a sinusoidal substrate has indicated a direct coupling
r between the substrate curvature and the preferential lo- gies and the surface curvature modulation.
a
cationsofdisclinationsinequilibriumstates[9]. Thecou-
pling is purely a geometric curvature effect in the sense
that the system involves no topological requirement for
II. MODEL
the existence of disclination. The geometric curvature
alsostronglyinfluencestheorientationalorderofthespin
lattice models defined on curved surfaces [11–18]. We consider nematic liquid crystals confined in a
thin curved membrane with thickness h, in which the
Many intriguing results have been obtained with re-
moleculesalignparalleltothe tangentialdirectionsinde-
gardtothegloballowest-energystateonacurvedsurface
pendently of the depth in the membrane. In line with
with a minimum number of disclinations, which should
the continuum theory [19, 23], the elastic free energy of
be relevantto liquid crystalphases at very low tempera-
themolecularensemblewithinthevolumeAhisgivenby
F =h dA K1(D n)2+ K3(D n)2 , (1)
∗Electronicaddress: [email protected] d Z (cid:26) 2 · 2 × (cid:27)
A
2
where the unit vector n (called the director field) de- shape when it is placed on a curved surface, whereas it
scribes the local molecular alignment. D is a vector op- is an exact circle when placed on a flat plane.
eratorwhosecomponentsD arethecovariantderivative Forlateruse,weintroduceanewpolarcoordinatesys-
µ
[24]onthecurvedsurface. K1andK3aretheelasticcon- tem(l,θ)ontheflatplanesuchthatthecoordinateorigin
stants associated with splay and bend distortions of the (i.e., l =0) locates at the point of interaction of the flat
molecular configuration, respectively [25]. plane and the vertical line penetrating the disclination
Supposethatthedirectorlieswithinathincurvedshell core (see Fig.1). Then, the basis vectors a1 ∂R/∂l
≡
having a Gaussian bump geometry. Points on the bump and a2 ∂R/∂θ span the curved surface and allow us
are represented by a three-dimensional vector R(r,φ) to write≡n = niai (i = 1,2), where ni and ai depend
given by on l and θ (See Appendix B). The director orientation
around the core is given by the angle ψ between n and
r2 a1. It is natural to assume that
R(r,φ)=rcosφi+rsinφj+vexp k, (2)
(cid:18)−2r2(cid:19)
0
ψ = (s 1)θ+ψ0, (3)
−
where i, j, and k are basis vectors of the Cartesian co-
where s denotes the disclination charge that quantifies
ordinate system, and r and φ are the polar coordinates
how many times n rotates about the core. When s =
assignedontheunderlyingflatplane(seeFig.1). Thege-
+1, for instance, all n contained within the disclination
ometric deviation of a bump from a flat plane is charac-
terized by a dimensionless parameter α v/r0 that rep- area A make the same angle ψ0 with respect to a1 (see
≡ Fig.3(a)); on the other hand, when s = 1, the angle
resents the ratio of the height to the width of the bump.
−
increases proportionally to θ (see Fig.4(a)).
Note that the constant r0 in the last term in Eq.(2) de-
In the calculation, we consider disclinations with two
termines the points of inflection of the Gaussian bump,
i.e., across which the sign of (∂2R/∂r2) k changes. We charges, s = 1. The disclination area A and the mem-
· brane thickn±ess h are set to be A = 400π µm2 and
demonstrate below that r0 plays a prominent role in de-
h=4µmbyreferringtothematerialconstantsofactual
scribing the correlation between the disclination energy
liquid crystal membranes [26]. The values of the elastic
and the substrate curvature.
Let us locate a disclination core on a point R = constantsK1andK3 arethoseof4-methoxybenzylidene-
′
4-butylaniline (MBBA), p-azoxyanisole (PAA), and 4-
R(r ,φ ), where φ is anarbitraryconstantwithout loss
c c c ′
pentyl-4-cyanobiphenyl(5CB) presented in Table I [25].
of generality. The disclination size is sufficiently smaller
than the bump width, and the perimeter of the discli- Theconstantψ0 inEq.(3)isfixedtobezerowithoutloss
of generality,bearing in mind the fact that other choices
nation is defined by the locus of points whose distances
fromthe coreare equalin the sense ofgeodesic distance; ofψ0 leadtonoessentialchangeinthe conclusionofthis
work.
therefore, the perimeter slightly deviates from a circular
III. RESULT
Figure 2 shows plots of the disclination energy F of
d
MBBAasafunctionofthecorepositionr . Thevertical
c
axis shows the difference in F between the bump and
d
the flat plane, i.e., F (α = 0) F (α = 0). The discli-
d d
6 −
nation charge s is s = +1 in (a) and 1 in (b), and the
−
parameter α ranges from α = 0 to 1.2 in both (a) and
(b). Fors=+1,Fd takesthe minimumatrc =r0 forall
values ofα, indicating thata disclinationis stable atthe
inflection point of the bump r =r0 independently of the
bump height. On the other hand, the plot for s = 1
−
exhibits a transitionfroma single downwardpeak struc-
ture(α<0.7)intoadouble-wellstructure(α>0.7)such
TABLE I: Elastic constants of MBBA, PAA, and 5CB re-
ported in Ref.[25]
Elastic constants MBBA PAA 5CB
FIG.1: SchematicofaGaussianbumpandmathematicalno- ◦ ◦ ◦
(pN) (25 C) (122 C) (26 C)
tationsusedinthepaper. Thethickarcdepictedonthebump
K1 6 6.9 6.2
isaprojectionofthethicklinesegmentontheunderlyingflat
K3 7.5 11.9 8.2
plane.
3
FIG. 2: Disclination energy in deformed MBBA films as a function of the core position rc. The parameter α characterizing
thebumpheight ranges from 0to1.2 (asindicated). (a) Energy of apositively charged disclination (s=+1). Themagnitude
of a downward peak at rc = r0 monotonically increases with α. (b) Energy of a negatively charged disclination (s = −1).
Transition from a single-well to a double-well structureis observed at α=0.7.
that r0 becomes unstable. The contrasting behaviors of Figures 3(a) and 3(b) illustrate how the configuration
F fors=+1and 1areattributedtosurfacecurvature varies with a shift in the core position r from r = 0
d c c
−
effects on the director configuration close to the core, as (a) to rc = r0 (b). In Fig.3(a), n aligns radially around
will be showninSec.IV. For PAAand5CB,we haveob- thetopofthebump,exhibitingrotationalsymmetrysuch
tained similar profiles of Fd exhibiting a downwardpeak thatn a1andn a2 =0ateverypoint. Thissymmetry
k ·
at rc = r0 for s = +1 and a double-well structure for breaks down for rc > 0. By increasing rc, the perimeter
s = 1, whereas the peak heights differ slightly from ofthedisclinationdistortsslightlyandbecomeselliptical,
−
those shown in Fig.2 because of quantitative differences andthen, the relationshipn a2 =0nolongerholds(see
·
in K1 and K3. We have also confirmed that the peak Fig.3(b)), except along the four specific line segments
positions (both downward and upward) are independent (θ = 0, π, π, 3π). The most important observation in
2 2
of our choice of ψ0 defined by Eq.(3). Fig.3(b) is a suppression of the splay deformation in the
We emphasize that the energy scales of the potential shadowed region. The area of the corresponding region
wellsshowninFigs.3(a)and3(b)exceedthe thermalen-
ergy scale at ambient temperature. For instance, the
depth of the single well for s =+1 and α=1.0 given in
Fig.3(a) is approximately 5.7 10−18J; this is of the or-
der of 103 k T at T =25◦C. ×This fact suggeststhat the
B
potential well at r = r0 overwhelms the thermal excita-
tionthatcausesarandommovementofdisclinationsover
thesurface. Asaresult,anensembleoftrappeddisclina-
tions forms a halo with radius r around the top of the
c
Gaussianbump, providedthatthedisclinationdensity is
sufficiently low to ignore their repulsive interaction. The
same scenario applies to disclinations with a charge of
1, implying the formation of doubly consentric halos
−
observed for α>1.0.
IV. DISCUSSIONS FIG. 3: Configurations of the director n around the core of
a disclination with a charge of +1. With a shift in the core
The physical origin of a downwardpeak in the plot of position from rc =0 (a) to rc =r0 (b), the radial symmetry
F for s=+1 (see Fig.2(a)) is a curvature-induced shift ofthedirectorconfiguration breaksdownsuchthatthesplay
d
deformation in the shadowed region in (b) is suppressed.
inthedirectorconfigurationaroundthedisclinationcore.
4
FIG. 4: Configurations of n in a disclination with a charge of −1. The core positions rc are rc =0 in (a), 0<rc <r0 in (b),
and rc =r0 in (c). In thelight- and dark-shadowed region, thesplay deformation is suppressed and enhanced,respectively.
ismaximizedatrc =r0,asaresultofwhichadownward ceeds the enhancement, and it increases in the opposite
peak Fd arises at rc =r0 at the expense of a slightsplay situation. The degree of splay enhancement in the dark
enhancement within a limited region around θ = π2. regions is maximized at rc = r0; this explains why r0 is
±
Parallelargumentstotheprecedingparagraphaccount the most unstable point for s= 1 disclinations.
−
for the double-well structure of F for s = 1 that is
d
− We have numerically confirmed that the splay term
shown in Fig.2(b). Figures 4(a)–(c) show the variation
of the n configuration under s = 1 and α > 0.7, each K1(D n)2 in Eq.(1)playsa dominantrolein describing
·
− the r - (and α-) dependences of F . Figures 5(a)–(b)
figure corresponding to the core position: r = 0 (a), c d
0 < rc < r0 (b), and rc = r0 (c). The dircector con- show contour plots of the splay term K1(D·n)2 within
thedisclinationareaofs= 1. Weobservethatwithan
figurationin(a)exhibitsafour-foldrotationalsymmetry
−
increase in r , the local splay energy decreases at points
thatbreaksdownasr increases. Symmetrybreakingin- c
c
near θ =π/2 or 3π/2 but increases near θ =0 or π; this
ducesasuppressionofthesplaydeformationinthelight-
is in agreement with our previous discussions. A more
shadowedregionsshowninFig.4(b)and4(c)inamanner
detailed version of the study that covers the variations
similar to the case of s = +1. In contrast, the splay de-
formationinthedark-shadowedregionsisenhancedwith of the disclination size A and the initial angle ψ0 will be
presented elsewhere.
an increase in r . These two competing effects result in
c
the double-well structure of F for s = 1 cases. That
d
−
is, F decreases with r when the suppression effect ex-
d c
V. SUMMARY
In conclusion, we investigated the elastic energy of
disclinations that arise in a curved nematic membrane
with a Gaussian bump geometry. When α < 0.7, the
disclination energy F exhibits a downward peak at r =
d
r0 independently of the sign of the disclination charge,
suggestinga curvature-drivenhalo formationof disclina-
tions around the top of the bump. Furthermore, when
α > 0.7, F for s = 1 exhibits a double-well structure
d
−
FIG.5: Contourplotsof thesplay termK1(D·n)2 of discli- in which the point r = r0 becomes unstable in contrast
nations with s = −1 placed at rc = 0 (a) and at rc = r0 to the case of s=+1; this implies doubly concentric ha-
(b). Light (dark) gray indicates a small (large) value of the los peculiar to negatively charged disclinations. In both
term at the position. With an increase in rc, the splay term cases of s = 1, the well depth overcomes the thermal
±
decreases around the line segments of θ =π/2 and 3π/2 but energy scale at room temperature, suggesting the exper-
increases around those of θ=0 and π. imental feasibility of our theoretical predictions.
5
Acknowledgements have
1 0
haWraefowrotuhlde flriukeitftuol dthisacnukssfioonrsK. .IHYaiksutbhoanaknfdulHfo.rOthrie- gµν =(cid:18)0 l2 (cid:19), (A4)
financial support from the 21st Century COE Program
“Topological Science and Technology”and support from as expected.
theJapanSocietyforthePromotionofScienceforYoung
scientists. HSacknowledgesM.Arroyoforassistingwith
the work carriedout of UPC. This work is supported by Appendix B: Contravariant components ni
the Kazima foundation and a Grant-in-Aid for Scientific
ResearchfromtheJapanMinistryofEducation,Science,
We presentthe derivationof the contravariantcompo-
Sports and Culture.
nent ni of the vector n=nia in terms of a . First, the
i i
relative angle between n and a1 is denoted as ψ. Sim-
Appendix A: Metric tensor gµν =aµ·aν ilarly, the angle between a1 and a2 is denoted as Ψ. It
follows from Eq.(A3) that
In this Appendix, we derive the metric tensor gµν = a1 a2 g12
aµ·aν intermsofthecovariantbasisvectorsa1 =∂R/∂l cosΨ = a1·a2 = √g11√g22, (B1)
anda2 =∂R/∂θintroducedinSec. II. Westartwiththe | || |
basis vectors e1 = ∂R(r,φ)/∂r and e2 = ∂R(r,φ)/∂φ sinΨ = |a1×a2| = √g , (B2)
associatedwiththe polarcoordinatesystem(r,φ)onthe a1 a2 √g11√g22
| || |
flatplane. Intermsofe1 ande2,themetrictensorg˜µν =
eµ eν becomes where g =det[gµν]. We thus have
·
f(r) 0 α2r2 r2 1
g˜µν =(cid:18) 0 r2 (cid:19), f(r)=1+ r02 exp(cid:18)−r02(cid:19).(A1) cos(Ψ−ψ) = √g11g22(g12cosψ+√gsinψ),(B3)
We use a coordinate transformation from (r,φ) to (l,θ) whichleadsustotheexplicitformsofthecovariantcom-
by shifting the coordinate origin from r = 0 to r = rc. ponents ni =n ai such as
This is accomplishedby multiplying both sidesofg˜ by ·
µν
the Jacobian β defined by n1 = n a1 cosψ =√g11cosψ, (B4)
| || |
∂r/∂l ∂φ/∂l
β = , (A2) n2 = n a2 cos(Ψ ψ)
(cid:18)∂r/∂θ ∂φ/∂θ (cid:19) | 1|| | −
= (g12cosψ+√gsinψ). (B5)
√g11
whose elements can be computed by using the follow-
ing relationships: r = r2+2r lcosθ+l2 and φ =
−1 c c Finally, the formulas nj = n gij with gij = [g ]−1 is
tan lsinθ/(rc+lcosθ)p. Finally we obtain i ij
{ } employed to obtain
g = a a =βg˜ (β)−1
µν µ ν µν
· 1 1 g12
n = (cosψ sinψ), (B6)
= r12 (cid:18)lpfq((r1)p2+f(qr2)) l2lp(fq((1r)−q2f+(rp)2))(cid:19),(A3) √g11 − √g
− 2 g11
n = sinψ. (B7)
r g
where p = r cosθ + l and q = r sinθ. In particular,
c c
when α = 0 (i.e., a flat plane), f(r) 1, and thus, we
≡
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