Presented at Faraday Discussions 144, Groningen, The Netherlands

1. Q u a d r u p o la r d e f e c t s t r u c t u r e s g e n e r a t e d b y c h ir a l
islands in freely suspended smectic C films
NM Silvestre1,2, P Patrício2,3, MM Telo da Gama1,2, A Pattanaporkratana4,5 , CS Park4, JE Maclennan4,
and NA Clark4
1
Departamento de Física da Faculdade de Ciências and 2 Centro de Física Teórica e Computacional, Universidade de Lisboa,
Avenida Professor Gama Pinto 2, P-1649-003 Lisboa, Portugal.
3
Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em ídio Navarro 1, P-1949-014 Lisboa, Portugal.
4
Department of Physics and Liquid Crystal Materials Research Center, University of Colorado, Boulder, CO, USA.
5
Physics Department, Kasetsart University, Bangkok 10900, Thailand.
Motivation Aim of this work
Textures of heterochiral islands interacting Provide a quantitative description of recent observations of the interaction
on a film of 25% chirally doped MX8068. (a )
(a) between topologically chiral islands in freely suspended Smectic C films, in which
The quadrupolar structure is in equilibrium the existence of an important class of inclusion interactions not studied previously
when the islands almost touch. (b ) The
(b) in either 2d or 3d was revealed, namely interactions that depend on inclusions
equilibrium separation between −1 defects chirality.
increases as the islands are separated using
optical tweezers. (c) When the separation is
(c)
The model: a theory of the Landau class
sufficiently large, the quadrupolar symmetry
is broken. (d ) When the islands are forced
(d) Smectic C order parameter:
even further apart, the quadrupole evolves
into two separate dipoles. Bulk free energy (in reduced units):
Energy density:
The system
Smectic C film: 2d structured fluid layers with LC molecules aligning an average
Ratio between bend and elastic constants:
direction tilted from the layer normal.
We assume that the effect of adding chiral component to the system results in
Projection of the molecular tilt onto the layers defines a 2d vector: c-director.
an effective bend elastic constant due to the spontaneous polarization.
♦
Islands: disk-like inclusions of extra layer.
Order parameter is fixed at the inclusions boundaries (clockwise or
Boundary is assumed to be sharp: neglecting layer modulation.
counterclockwise) and at far field (uniform alignment) with .
♦
c-director at the inclusions boundary is planar and points either clockwise or
anticlockwise.
Horizontal position of defect vs island separation
By adding chiral material elastic bend energy is modified: polarization of the
chiral material contributes to an effective bend elastic constant. (a ) Computed horizontal position of
(a)
the right defect xd as a function of
Equilibrium configurations: ƙ = 1 island center-to-center separation D
for ƙ=1.8, 3.0, and 4.0. Reference
(a ) D=2.2 R: the defects are equally
(a)
system has origin at the mid-point
shared by the islands, in a symmetric,
between islands.
quadrupolar configuration.
(b ) Th e q u a d ru po la r symme try o f
(b) rup lar
(b ) D=3.0 R: the symmetry is broken;
(b)
th e isla nd -d e fect p a irs is ge ne ra lly
isla -de
lan rally
here the defects are found in a
b ro ke n at critica l isla n d sep a ra tio n Dc, shown here as a function of ƙ. The noisiness
rok ritic
ica lan ration
metastable configuration where they
of the data for ƙ<1 reflects the numerical difficulties encountered when modelling for
are bothe closer to one of the islands.
this elasticity regime.
(c) D=3.5 R: each defect follows a
(c)
different island.
(d ) D=4.0R: even at this separation,
(d) Vertical position of defect vs island separation
the positions of the defects are still strongly influenced by the presence of the
(a ) Computed vertical position yd
(a)
other island-defect pair, with configurations typical of isolated dipoles occurring at
of the upper -1 defect as a
larger separations.
function of island separation D,
Vertical separation S vs islands separation D for ƙ=1.4, 2.4, and 4.5. For small
D (close to equilibrium position),
S is the vertical separation of the -1 th e d isp la ce me nt o f th e d e fe ct
isp
defects; D is the island center-to- in cre a se s lin ea rly with a we ll
increa line
center separation. d e fin e d slo pe m shown in (b ) as
ine lop (b)
The straight lines are linear fits of the a function of the elastic
experimental data. anisotropy ƙ.
The dashed lines are fits to the
numerical data to guide the eye. Conclusions
The value of ƙ=2.4 was chosen to
Quadrupolar symmetry brake was observed both experimentally and numerically at
reproduce the slope of the linear fit for the chiral mixture. well-defined island separation that depended on effective elastic anisotropy.
Flu ctua tio n s a re e xp e cted to incre ase th e a ve ra g e d efe ct se p a ra tio n , S, a n d
increa rag ration For small separations, computed vertical displacement of defects was found to
a cco u n t, a t le ast in pa rt, fo r th is discre pan cy.
lea rt, iscrep increase linearly, in general agreement with experiment.
Numerics suggests that the initial slope of S vs D curve may be used to estimate
Supported by FCT through grants: POCI/FIS/58140/2004, POCTI/ISFL/2/618, and SFRH/BPD/40327/2007
the effective elastic anisotropy.