Memory effects from topologicalconnectivity of nematic liquid crystals
confined in porous materials
Takeaki ARAKIa,b), Marco BUSCAGLIAc),
Tommaso BELLINIc) and Hajime TANAKAa)a)b)c)

Institute of Industrial Science, University of Tokyo, Japan
Department of Physics, Kyoto University, Japan
Department of Chemistry, Biochemistry and Biotecnology,
University of Milano, ItalyLiquid crystal
n-4'-MethoxyBenzylidene-n-ButylAnilin (MBBA)

Liquid crystals are substances thatexhibit a phase of matter that hasproperties between those of aconventional liquid, and those of a solidcrystal.

smectic

nematic

isotropic

orientational orderand layeringorientational orderdisorder(Liquid phase)temperature

Elasticity of nematic phasespray (K1)

twist (K2)

bend (K3)Frank elastic energy


1
22F   dr  {K1 (  nˆ )  K 2 (nˆ  (  nˆ ))  K3 (nˆ  (  nˆ )) 2 2 n (r ) local director field(| n | 1)

Defects in liquid crystals

When incommensurate domains contact, defects are formed at thedomain boundary

Schlieren texture (r )  s (r )  0
defect core energy

21Ed   dr    s 2K

 : angular coordinate
: angle of director fields: the strength of the defectDefects in liquid crystals
A defect of s interacts with that of s’.
Roughly, the strength of theinteraction is proportional to ss’.

(+,+) and (-,-)(+,-): repulsive: attractive

A line defect has a tension in 3D.
It tend to be shrunken.

Annihilation of defects
In bulk, defects of nematic
phase are not long-lived
A defect of the topologicalcharge s is annihilated1s21s
2with other one of ss  1s0

Minomura et al. (1997)

Re-organizations of defects
1s
2s

12s

12s

1
2

Energy barrier for a reorganization

E ~ K 3 / 2  1/ 2

K:elastic modulus
: chemical potential difference

Chuang et al. (1993)

Simulations of nematic ordering

Characteristic length(Typical separation between defects)

Rt

1/ 2

Anchoring effectnematic liquid crystal

nDirector field is aligned atsolid surface with tilt angle solid wall 0  90

planarhomeotropicLiquid crystal and solid objects
The inclusion of solid objects imposes the formation of defectin liquid crystals

nematic liquid crystal

particles (colloids)

porous media

elastic energy F (a.u.)

A particle and defect in NLC

anchoring strength W (a.u.)

H. Stark, Phys. Rep. 351, 387 (2001).

KWc  Ca(C  3.5)

Topology of defect structure
Araki and Tanaka (2006)

A Saturn-ring defect<Topologically arrested structure

Aggregation of colloids

Ravnik et al. (2007)

Musevic et al. (2006)

Nematic liquid crystal in porous
media

Nematic liquid crystal

porous media(PM)They show interesting behaviors due to the topological constraints of thedefects. And they provide promising properties for optical devices.
(Crawford and Zumer (1996)),
Bellini et al. (1996,2000,2002).

Nematic liquid crystal in a simple cell

EOffOnOff

The director field can be tuned by external field.
Soon after the field is switched off, it will recover the original pattern.

Memory effect of confined nematic
liquid crystal

Rotunno et al. (2005). Buscaglia (2006)
A nematic liquid crystal in a porous medium can efficiently keep its
orientational order, even after the field is switched off.
The nematic liquid crystal memorizes the orientational field.
The aim of this study is to explore the mechanism of the memory effects.

Preparation of random porous
media
Cahn-Hilliard-Cook equation  2     3   2  t  4.6

  11.8

We can obtain isotropic random porous media
with controlled sizes.

 : mean pore size  30.7

Spatial distributions of Q
Red regions represent the porous medium

W=0

W=1

TINbulk  0.703

The nematic order grows from the surfaces in a porous medium with finite W.

Defect structure of nematic liquid
crystal in porous media
Since all the channels do not necessarily havedisclination lines running through them, manymetastable configurations can be found.
The defect configurations are long-lived since theenergy barriers connecting them are associated tosimultaneous liquid crystal rotation in entire
channels, and hence larger than thermalfluctuations.
defect structures for different
simulations of the same condition

This results in non-ergodic glassy behaviors,
analogous to a spin glass.

The number of the possible configurations is estimated as, ( p  1)1  / 2p

average number of arms at nodes, typicallyp  36Euler characteristic (topological invariant)~101000000 for 1 mm cell of 1m pores !

1dS K
2 Transformation of defect structureby an external field
E=0.3

E=1.0
after the quench

under an electric field after the application

A strong field melts the defect structure andthe topology of defect structure can be changed.
The new topology is conserved even after the field is switched off!

Induced and remnant nematic orders
Orientational order along the fieldsimulations

Q* 131 nz nz  
Q0 (T ) 2 3QE

QM

experiments (5CB in millipore filter (3m) and silica gel (0.8m))

E=1.25V/mNLC in regular porous materials
Bicontinuous cubic (BC) Simple cubic packing (SC) Cylindrical channels (Cyl)The defect structureis also regular inordered porous
media.

Relaxation of memorized order
Bicontinuous cubic (BC)

Q(t)Q(t)Random porous medium (PPM)In BC, only a single relaxation mode is observed.
After the fast mode, the second slow relaxation appears in RPM.
BC: single stretched exponential decayQ(t )  QM  QS exp( (t /  S ) )RPM: stretched exponential decay and logarithmic decayQ(t )  QM  QS exp( (t /  S ) ) QL
1  log(1  t /  L )Logarithmic decay forsuperconductor:
Anderson (1962)
Kim et al. (1962)

Fast relaxation mode
In the both type of porous media, the fast relaxation mode represents therecovery of the elasticity of the nematic phase that is distorted by theexternal field.
In this fast process, the topology of the defect structure does not change.
30

10.8

sRPM20

 0.4

10BC
00

10

BC

0.6

RPM

0.2
2030

4050



00

10

20

30

4050



The elastic theory predicts the relaxation time is proportional
to l2 in the limit of the strong anchoring.
 1
2
 F n  K  dV n  W  dS n  s 22 1 Fnt nS  2K

T-Dependence of the relaxations
RPMBC
T=0.1

Q(t)

Q(t)

T=0.01

T=1.0

T<TNIbulk

T=1.0t (MCC)

T>TNIbulkt (MCC)In BC, the remnant order is almost proportional to the bulk nematic order.
Above TNIbulk , it decays to zero without the slow mode.

In RPM, the second mode becomes slower with decreasing temperature.

What is the mechanism of the second mode in RPM?

The topological change of thedefect structure in RPM

At T=0.1 for   43.9

The second slow mode stems from the reorganizations of topology ofthe defect structure

The second slow relaxation in RPMQ(t )  QM  QS exp( (t /  S ) ) QL1  log(1  t /  L )Pore size  -dependence at T=0.1

Temperature dependence for   12.7

105104104

L 10

L

103

102

3

 L   L 0 ' exp(b / T ) L   L 0 exp( a)

01020

3040

1020ln  L   / T

102030401/T

In BC, the defect structure reaches the more stable configuration
after the fast mode. Then, the second mode is absent.

5060

The topological change of thedefect structure in RPM
In a subunit of the volume  3, the elastic
energy density is estimated ase
11KK (n ) 2 ~22 2Thus, the stored elastic energy in thissmall volume isE  e3  KThe energy barrier against the topologicalchange is also scaled as K L  exp(cK / T )T and l - dependences of theremnant orders

1

1

BC:QM

0.8

RPM: QM  QL

0.6

Q0.6

Q
0.4

0.4

0.2
0102030



RPM:QM L400 -2
1010-1T
BC: Q(t )  QM  QS exp( (t /  S ) )RPM: Q(t )  QM  QS exp( (t /  S ) ) 

RPM: QM  QL

0.2

RPM:QM L

0BC: QM

0.8

QL
1  log(1  t /  L )In BC, the remnant order appears to be independent of the pore size.
This is consistent with the scaling argument for the strong anchoring case.
In RPM, the remnant depends on the mean pore size because of its nonergodic glassy behavior.

100Memory effect of nematic liquidcrystal in porous media
bicontinuous cubic

SC sphere array

inverted FCC sphere arrayrandom pores

FCC sphere array

Summary
We numerically studied the glassy behaviors of nematic liquid crystals
confined in porous materials.
We found the nematic liquid crystal in porous media shows the tworelaxation processes after the external field is removed.
The fast mode represents the viscoelastic relaxation of the director fieldwith keeping the topology of the defect structure.
The second slow one comes from the topological changes of the defectstructure toward a more stable configuration.

We also found the remnant order depends upon the pattern of the porous
media. By utilizing regular porous media, the second relaxation mode canbe controlled.