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Tine Porenta
Mentor: prof. dr. Slobodan Žumer
Januar 2010
Seminar
 Introduction in liquid crystals
 Basics of flexoelectricity
 Theory
 Numerical method
 Radial nematic-filled sphere
 Radial nematic-filled sphere with point-like defect
 Radial nematic-filled sphere with hedgehog defect
 Conclusion
Introduction in liquid crystals
 Materials with properties most useful for different
applications in the modern world
 Liquid oily materials made of rigid organic molecules
 In proper temperature region they can self orginise and
form a mesophases between the liquid and solid state
 Mesophases are characterized by orientational and
positional order of the molecules
 Nematic phase is the least ordered phase influenced only
by long-range order but no positional order.
 Long-range order is the phenomenon that makes liquid
crystal unique
 They are typically highly responsive to external fields
 In confined geometries opposing
orientational ordering of different surfaces
can lead to formation of regions, where
orientation is undefined -> defects.
 Defects can be either point-like or lines.
 The average of the molecules are described as
a director n. Director is aporal, meaning the
orientation n an –n are equivalent.
 Degree of order: Orientational fluctuations of
the molecules are defined as an ensemble
average of the second Legendre polynomials
S = <P2(cos )>
 The director and the nematic degree of order can be joint
together in a single tensorial order parameter defined as
S
Qij  (3ni n j   ij )
2
 By definition Qij is symmetric and traceless. Its largest
eigenvalue is nematic degree of order S and the
corresponding eigenvector is the director n
 Phenomenological Landau – de Gennes (LdG) total free
energy is used to incorporate liquid crystal elasticity and
possible formation of defects:
1  Qij  Qij 

 dV
F   L
2  xk  xk 
LC
1
1
1
2


  AQijQ ji  BQijQ jk Qki  C QijQ ji dV
2
3
4
LC
Elastic deformation modes:
(a) splay, (b) twist and (c) bend
 Electric field couples with nematic through a dielectric
interaction with induced dipoles of the nematic molecules.
Within the LdG framework, the electric coupling is
introduced as an additional free energy density
contribution
1
Fd     0 ij Ei E j dV
2 LC
where ij is defined as
1
2
 ij  2    ||  ij   ||    Qij
3
3S
|| and  are dielectric constant measured parallel and
perpendicular to the nematic director
From piezoelectricity to flexoelectricity
 Piezoelectricity is the ability of some materials to generate
an electric field or electric potential in response to applied
mechanical stress.
 The effect is closely related to a change of polarization
density within the material's volume.
 The internal stress in this materials is proportional to
electric field inside.
 Stress tensor is defined as
 F 

 ij  


u
ij

T , E
1  ui u j
uij 


2  x j xi



T , E
 Electric displacement field is then
Di  Di   ij E j   i , jk ij
0
where i,jk tensor rank three with symmetry i,jk = i,kj .
If tensor is known, piezoelectric properties are entirely
determined
 In liquid crystals exist phenomenon similar to
piezoelectricity that occurs from the deformation of
director filed


 
Di   ij Ei  e1n(  n )  e3 ((  n )  n )
wiht coefficients e1, e3  10-11 As/m
Polarisation induced by splay and bent deformation
 The total macroscopic polarisation induced by
deformation of liquid crystal is introduced by using a
nematic degree of order
Qij
Pi  Gijkl
xl
where Gijkl is a general fourth rank coupling tensor,
which incorporates flexoelectric coefficients e1 and e3
 For simplicity -> one constant approximation Gijkl=G
 The corresponding free energy:
 Qij 
 E j dV
F    G
xi 
LC 
Numerical method
 Numeric relaxation method was developed to calculate
the effect of flexoelectricity
 Electric potential and the profile of the nematic order
parameter tensor are alternatively computed, until
converged to the stable or metastable solutions
 Cubic mesh with resolution of 10 nm
 Strong anchoring on boundaries is assumed
 The total free energy is
 Electric potential is
miminized by using EulerLagrange algorithm
calculated from Maxwell’s
equations in an anisotropic
medium
1  Qij  Qij 

 dV
F   L
2  xk  xk 
LC

1
1
1
2


AQ
Q

BQ
Q
Q

C
Q
Q
dV
ij
ji
ij
jk
ki
ij
ji
LC 2
3
4
    
1
 dV

   0 ij 


2 LC
 xi  x j 
 Qij   
 dV

  G

xi  x j 
LC 
2



Qij
 
 
 0 ij
 G
0


xi 
x j 
xi x j
1
2
 ij  2    ||  ij   ||    Qij
3
3S
 Scheme:
Nematic and dielectric constants A, B, C, L, || and 
are taken.
Radial nematic-filled sphere
 Effect of the flexoelectricity are typically small in the
absence of the external fields (Fflex < 1% Ftotal), but in
some geometries like nematic filled sphere can
become substantial importance.
Radial nematic-filled sphere with point-like defect
 existance of analytical solution of flexoelectric quantities
for isotropic medium
 only splay deformation of a director field
 director field can be represented in spherical coordinate
system as n=(1,0,0)



Pflex  e1n (  n )
1 r 2
 e1 (1,0,0) 2
r r
 2e1

  ,0,0 
 r



(0 E )    Pflex
 
0 E  Pflex  C
  2e1

E  
,0,0 
 r0

 Flexoelectric contribution to the total free energy
16e12 R
F flex    Pflex  EdV  
0
V
Radial nematic-filled sphere with hedgehog defect
 Electric potential induced by flexoelectricity affects
the nematic profile, primarily in the core region of the
defects.
 (a)Electric field induced by flexoelectricity and spatial
distribution of elastic (b), dielectric (c) and
flexoelectric (d) contribution to the
total free energy
Conclusion
 Coupled numerical method was developed for the study of
flexoelectricity in nematic liquid crystals to show us that
flexoelectricity induces substantial electric potential in the
regions surrounding the defects
 Flexoelectricity affects defect cores and changes their size
 Flexoelectricity could change stability of defect
configurations in confined geometries
 Flexoelectric contribution to the total free energy has
quadratic dependence on flexoelectric coefficient and
could become important factor for materials with high
flexoelectric coefficients