FlexoelectricityTine Porenta

Mentor: prof. dr. Slobodan ŽumerJanuar 2010

SeminarIntroduction in liquid crystalsBasics of flexoelectricityTheoryNumerical methodRadial nematic-filled sphere

Radial nematic-filled sphere with point-like defect

Radial nematic-filled sphere with hedgehog defect

Conclusion

Introduction in liquid crystalsMaterials with properties most useful for

different applications in the modern worldLiquid oily materials made of rigid organic

moleculesIn 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

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:

)3(2 ijjiij nnS

Q

LC

jiijkijkijjiij

LC k

ij

k

ij

VQQCQQBQQAQ

Vx

Q

x

QLF

d 4

1

3

1

2

1

d 2

1

2

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

where ij is defined as

|| and are dielectric constant measured parallel and perpendicular to the nematic director

LC

iijd VEEF d 2

1j0

ijijij QS |||| 3

22

3

1

From piezoelectricity to flexoelectricityPiezoelectricity 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

ETijij u

F

,

ETi

j

j

iij x

u

x

uu

,2

1

Electric displacement field is then

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

wiht coefficients e1, e3 10-11 As/m

ijjkijiji EDDi

,0

))(()( 31 nnenneED iiji

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

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:

l

ijijkli x

QGP

LC i

ij VEx

QGF d j

Numerical methodNumeric relaxation method was developed to

calculate the effect of flexoelectricityElectric 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 nmStrong anchoring on boundaries is assumed

The total free energy is miminized by using Euler-Lagrange algorithm

Electric potential is calculated from Maxwell’s equations in an anisotropic medium

LC ji

ij

LC jiij

LC

jiijkijkijjiij

LC k

ij

k

ij

Vxx

QG

Vxx

VQQCQQBQQAQ

Vx

Q

x

QLF

d

d 2

1

d 4

1

3

1

2

1

d 2

1

0

2

02

0

ji

ij

jij

i xx

QG

xx

ijijij QS |||| 3

22

3

1

Scheme:

Nematic and dielectric constants A, B, C, L, ||

and are taken.

Radial nematic-filled sphereEffect 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 defectexistance 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)

0,0,2

1)0,0,1(

)(

1

2

21

1

r

er

r

re

nnePflex

0,0,2

)(

0

1

0

0

r

eE

CPE

PE

flex

flex

Flexoelectric contribution to the total free energy

0

2116

Re

EdVPFV

flexflex

Radial nematic-filled sphere with hedgehog defectElectric 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

ConclusionCoupled 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