Transcript prenta-predstavitev
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 =
The director and the nematic degree of order can be joint together in a single tensorial order parameter defined as
Q ij
S
2 ( 3
n i n j
ij
) By definition Q ij 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:
F
LC
1 2
L
Q x k ij
Q ij
x k
d
V
LC
1 2
AQ ij Q ji
1 3
BQ ij Q jk Q ki
1 4
C
Q ij Q ji
2 d
V
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
F d
1 2
LC
0
ij E i E
j d
V
where ij is defined as
ij
1 3 2 ||
ij
2 3
S
||
Q ij
|| 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
ij
F u ij T
,
E u ij
1 2
u i
x j
u j
x i T
,
E
Electric displacement field is then
D i
D i
0
ij E j
i
,
jk
ij
where i,jk tensor rank three with symmetry determined i,jk = i,kj . If tensor is known, piezoelectric properties are entirely In liquid crystals exist phenomenon similar to piezoelectricity that occurs from the deformation of director filed
D i
ij E i
e
1
n
(
n
)
e
3 ( (
n
)
n
) wiht coefficients e 1 , e 3 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
P i
G ijkl
Q ij
x l
where G ijkl is a general fourth rank coupling tensor, which incorporates flexoelectric coefficients e 1 and e 3 For simplicity -> one constant approximation G ijkl =G The corresponding free energy:
F
LC G
Q ij
x i
E
j d
V
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 miminized by using Euler Lagrange algorithm
F
LC
1 2
L
Q ij
x k
Q ij
x k
d
V
LC
1 2
AQ ij Q ji
1 3
BQ ij Q jk Q ki
1 4
C
Q ij Q ji
2 d
V
1 2
LC
0
ij
x i
x j
d
V
LC G
Q ij
x i
x j
d
V
ij
Electric potential is calculated from Maxwell’s equations in an anisotropic medium
x i
0
ij
x j
G
2
Q ij
x i
x j
1 3 2 ||
ij
2 3
S
0 ||
Q ij
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 (F flex < 1% F total ), 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
P flex
e
1
n
(
n
) 1
e
1 ( 1 , 0 , 0 )
r
2 2
r e
1 , 0 , 0
r
2
r
( 0 0
E
E
)
P flex
P flex
C
E
r
2
e
1 0 , 0 , 0
Flexoelectric contribution to the total free energy
F flex
V
P flex
EdV
16
e
1 2 0
R
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