Lech Longa (Kraków, Poland)

Lebwohl-Lasher models of liquid crystals: from quadrupolar to spontaneously induced chiral order

Karol Trojanowski Grzegorz Pająk

2

The Lebwohl- Lasher model (Lattice Liquid Crystals)

Single-site soft potentials:<i,j>

= .

∙ ≝ … …

1. Mayer-Saupe uniaxial quadrupoles are assumed to occupy the sites of a three dimensional cubic lattice.

2. Model preserves nematic symmetry ( ).3. No dependence on the orientation of the radius vector connecting

molecular centers.4. Orientational structures identified not only characterize nematic-like

states but can also coexist with a long-range positional order, characteristic of smectic, columnar or crystalline phases.

Vij

i3 j3

3

<i,j>

Lebwohl- Lasher model (results, 3d cubic lattice)

<i,j>

∑ ∙ + const<i,j>

4

Generalizations of the Lebwohl- Lasher (LL) model (Lattice Liquid Crystals)

The model has proved to correctly account for the essential symmetry ofliquid crystalline orientational order and has been generalized to morecomplex situations.

(a) (PL PM ) models: (L,M) = (2,4), (2,1) (b) investigation of the nematic ordering in confined geometries, subject to different

surface anchoring fields (grating, patterned surfaces, …), (c) effect of an external field on the isotropic - nematic phase transition(s), (d) simulations of electro-optical devices, (e) simulation of chiral liquid crystal phases, (f) orientational properties of polymers and elastomers and (g) physics of two-dimensional systems (no topological phase transitions in O(3)

symmetric two dimensional (LL-molecules can reorient in three dimensions)

1.Lattice models emphasize symmetry aspects of orientational order2.Discretized versions of Landau-deGennes theories => LL models3.The same form of „effective” interaction for hard bodies ( Onsager)

and dispersion interactions ( t <-> 1/ρ )

5

1. Biaxial nematic phase: interacting biaxialquadrupoles of D2h symmetry (and beyond).

2. Spontaneously generated chiral states in thepresence of quadrupolar and octupolar interactions.

3. Possibility of phases with splay- twist– bentdeformations.

4. Summary

Outline

6

Interacting biaxial quadrupoles of D2h symmetry:

b2

b3

l1

l2

l3

,

=

6 parameters! (too many)

Only 3 parameters are independent

symmetry interchangeof l and b vectors

7

=

Interacting biaxial quadrupoles of D2h symmetry

b1 b2

b3

,

8

Interacting biaxial quadrupoles of D2h symmetry:(3d cubic lattice, nearest-neighbour interactions)

b2

b3

l1

l2

l3

+ const

; Z = ,…,

9

Duality transformation

V =

L.L. , P. Grzybowski, S. Romano, and E. Virga PRE 71, 051714 (2005)

, v 1/t

; Z = ,…,

10L.L. , P. Grzybowski, S. Romano, and E. Virga PRE 71, 051714 (2005)

Rod-like states

Disc-like states Disc-like states

No stable nematic

11

Dispersion model

12

wèè2,00 = - e

wèè2,20 =

2 e l

wèè2,22 = - 2 e l2

)()(),( jijiV QQ

QU=

QB=

Geometric mean rule for interactions(Berthelot rule)

DISPERSION MODEL

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1. Biaxial nematic phase: interacting biaxialquadrupoles of D2h symmetry and beyond.

2. Spontaneously generated chiral states in thepresence of quadrupolar and octupolar interactions.

3. Possibility of splay – bent deformations.

4. Summary

Outline

14

handedness, from the Greek

Chirality

15

Chiral interactions: Measure of chiralityMeasure of chirality

16

Chirality breaking from non-chiral molecules

Reviews:

H. Takezoe and Y. Takanishi, Japanese Journal of Applied PhysicsVol. 45, No. 2A, 2006, pp. 597–625

R. A. Reddy and C. Tschierske, J. Mater. Chem., 16, 2006, pp. 907–961

• Spontaneous chiral deracemization at microscopic andmacroscopic levels occurs and is controllable (banana smectic ph.)

• achiral bent-core molecules enhance system chirality

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• polarizers under decrossed conditions - method of confirming chiral segregated domains

(From H. Takezoe and Y. Takanishi, Japanese Journal of Applied PhysicsVol. 45, No. 2A, 2006, pp. 597–625; smectic B4 phase)

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• Selfassembly of twisted conformers• Conical twist-bend deformations

analyser rotated clockwise and anticlokwise by the same angle

R. Memmer Liquid Crystals, (2002) 29, 483-496

F. Yan, C. A. Hixson, and D. J. Earl, PRL 101, 157801 (2008)

22

2TrTrTr21

QQ

QQQ 232

V

aF

Dimensionless deGennes - Ginzburg - Landau:

1nn

31 Q SOseen-Zocher-Frank:

V

KKKF 23

22

212

1 nnnnn

S2 + T2 + B2 S2 + B2

20

Chirality: phenomenological level (orientational order only)(Dimensionless) minimal coupling

Landau-deGennes free energy

Splay, twist, bend … deformations areweighted by these terms

CALCULATIONS IN FOURIER SPACE :

, ≡

22

2TrTrTr21

QQ

QQQ 232

V

aF

-2

Cholesteric phase

h

F2

t=a= a0(T-T*)

22

Chirality: phenomenological level (orientational order only)(Dimensionless) Landau-deGennes free energy

-for ordinary chiral materials:

Where does the totally antisymmetrictensor come from?

Osipov, M. A., B. T. Pickup, and D. A. Dunmur, 1995, Mol. Phys. 84, 1193.B. Harris, R. D. Kamien, and T. C. Lubensky, RMP, Vol. 71, 1999

tetrahedrally coordinated molecule

Chirality: phenomenological level (orientational order only)(Dimensionless) Landau-deGennes free energy

For nonchiral molecules:

where does the totally antisymmetrictensor come from?

25

From symmetry point of view there are 2 possibilities that can be readfrom the previous Landau-deGennes formula:

should be replaced by:

and/or

octupolar tensor Polar vector

and lowest order bulk chiral invs.

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Partial discussion of these issues can be found in (work is still in progress):

Lech Longa and Hans-Rainer Trebin

Mass distribution:

The second mass moment

The third mass moment

The ‘steric dipole’ mass moment

ˆ kaˆ

kb

ˆkc

Dipole, quadrupole and octupole order parameters are relevant for bent-core systems

These 2 tensors are needed to getchiral order

L.G. Fel, Phys. Rev. E 52 (1995) 702.

28

Chirality is a combined effect of biaxial-quadrupolarand octupolar (local or global) order

Generic Lebwohl Lasher lattice model: -quadrupoles and octupoles occupy lattice sites,- dipoles are disregarded- cross-coupling terms involve Q and T and r- two-fold axes of the tensors coincide

)()()()(),( jijijiV TTQQ

,

i

,

j

( 2 ) U BQ Q Q

Degrees of freedom: proper rotations + :

+ c.c.t (Q,T,r)

Order parameters

.

.

.

( ) 3 1ˆ ˆ( )2 3

2U 0 k kQ T c c 1

( ) 1 ˆ ˆˆ ˆ( )2B 2

2 k k k kQ T a a b b

( )

ˆˆ ˆˆ ˆ ˆ( , , ) ( , , )

1 ˆ ˆ ˆ6

k k k

32

x y z a b c

T T x y z

.

.

.

< >< > =

Averaged values ofsymmetrized Wigner functions

< p >

L.Longa, G.Pająk, and T. Wydro PRE 79, 040701R 2009K.Trojanowski, G.Pająk, L.Longa, and T. Wydro, PRE 86, 011704 (2012)L.Longa and G. Pająk, MCLC Vol. 541, 390, (2011)

32

33

34

35

. , ,

36

l

. ,

37

. , ,

t=0.2

38

. , ,

t=0.55

39

. , ,

t=0.55

40

. , ,

t=0.7

41

. , ,

t=1.1

K. Trojanowski and L. Longa (submitted for publication)

42

1. Biaxial nematic phase: interacting biaxialquadrupoles of D2h symmetry and beyond.

2. Spontaneously generated chiral states in thepresence of quadrupolar and octupolar interactions.

3. Possibility of splay – bent deformations.

4. Summary

Outline

43

A discussed previously:

should be replaced by:

and/or

octupolar tensor Polar vector

Ground state deformations should contain all possible modes as discussedon p. 22. In addition: (a) presence of T or P fields renormalizes elasticconstants of nematics; (b) coupling of Q with P can stabilize helical polar structures; (c) presence of T tensor can give interesting optical propertiesof the system

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Summary:

•The simplest model with quadrupolar and octupolar interactions gives 3 new structures; one ofthem is chiral

•Cross-coupling terms impose spatially modulated order

•Different scenarios are possible when twofold axes of Qand T tensors do not coincide

•Polar order (dipole-dipole int.) and various scenarios of Q, T and P couplings are subject to present studies

.3 strsymmetricD h

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