Soft bend elastic constant and transition to a modulated nematic phase

Alenka Mertelj,1 Martin Čopič,1,* Geoffrey R. Luckhurst2, R. P. Tuffin3, and Owain Parri3

1Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia

2School of Chemistry, University of Southampton, Southampton SO17 1BJ, UK 3Merck Chemicals Ltd, Chilworth Technical Centre, University Parkway, Southampton SO16

7QD, UK

Outline

• Observation of modulated phase in nematic phase of flexible dimers

• Nematic fluctuations and dynamic light scattering

• Temperature and order parameter dependence of elastic constants

• Conclusions

Modulated nematic phase

• Observed in flexible dimers of biphenyls like 1,7-bis(4-cyanobiphenyl-4-yl)heptane (CB7CB) and CB9CB [1], or CB11CB[2]

• I.Dozov [3] proposed that a softening of the bend elastic constant could lead to a modulated nematic phase with nematic director forming a twist-bend helix

• In [2] it was suggested that in CB11CB the modulation is due to soft splay elastic constant

• Numerical modeling of A. Ferrarini indicates that bend elastic constant in dimers can become negative

[1] M. Cestari et al., Phys. Rev. E 84, 031704 (2011)

[2] V. P. Panov et al., Phys. Rev. Lett.105, 167801 (2010).

[3] I. Dozov, Europhys. Lett. 56, 24 (2001).

Observed modulation under polarized microscope

Structures proposed by Dozov

Dozov’s model

2

2

2

2

3

2

2

2

2

2

2

2

12

32

22

1 4

1

2

1zjzji n

dz

dCnn

dz

dCnn

dz

dCKKKF bts

1

3332

1

320 27

,3

,3

4

CK

KF

C

Kk

K

Ksb

Splay-bend phase:

2

3332

2

320 54

,3

,2 CK

KF

C

Kk

K

Ksb

Twist-bend phase:

If 21 2KK

then twist-bend is the stable phase

Microscope observation - thin cell (8 m)

n

Microscope observation – thick cell (20 m)

n

Light scattering

• Elastic constants can be measured by observation of thermal director flucutations

• Relaxation rates give ratios Ki/ηi

• Scattering intensity gives (ε)2/Ki

• As ε is proportional to S, we get Ki /S2

• Ki /S2 are lowest order “bare” elastic coefficients in Landau-deGennes free energy

Bend

Splay

Twist

n/ ti ee rqn0

nqn

n q0( )nB

n

2

2k T

K q

Nematic fluctuations

Relaxation rates

2,1,1 2

32

iqKqK

i

zi

i

czzb

z

qqqq

qq

45431

224

22

23

2

11 )(

)(

cza

z

qq

q

222

2

12

Two modes: bend-splay and bend-twist, for q along n – pure bendRelaxation rates:

Effective viscosities:

Usually α2 >> α3, so that bend viscosity is smaller due to backflow

• The direction and polarization of the incoming and scattered light and n determine which mode is observed

Samples

NC (CH2)x CN X=7,9 - CB7CB and CB9CB

• CB7CB : TNI = 116 oC, TNX = 103 oC

• CB9CB : TNI = 124 oC, TNX = 109 oC

• Planar orientation

CB7CB diffusivities (K/η)

102 104 106 108 110 112 114 1160.0

2.0x10-11

4.0x10-11

6.0x10-11

8.0x10-11

1.0x10-10

Bend Twist Splay

Ki/

i(m2 /s

)

Temperature

CB7CB

Note increase in the splay diffusivity below TNX

CB9CB diffusivities (K/η)

106 108 110 112 114 116 118 120 122 124 1260.0

2.0x10-11

4.0x10-11

6.0x10-11

8.0x10-11

1.0x10-10

1.2x10-10 Bend Splay Twist

File: DLS_CB9CBcorT.org, 11-Aug-12 Window: Diff

Ki/

i(m2 /s

)

Temperature (oC)

Note increase in the splay diffusivity below TNX

Normalized “bare” elastic constants

102 104 106 108 110 112 114 116

1

Bend Bend Twist Splay

Ki/S

2

Temperature

CB7CB

T=116oC:K

1>K

2>K

3,

K1/K

2 2.8

K2/K

3 2.3

106 108 110 112 114 116 118 120 122 124 1260.1

1

10 Twist Bend Splay

Ki/S

2

Temperature

CB7CB CB9CB

• Absolute scattering cross-sections is difficult to measure, so we obtain only T dependence of Ki relative to the value at TNI

• The bend constant softens, but increases just above TNX

• The splay constant increases below TNX , also seen in diffusivity

“Bare” elastic constants – linear scale

102 104 106 108 110 112 114 1160.0

0.5

1.0

1.5

2.0

Bend Bend Twist Splay

Ki/S

2

Temperature

CB7CB

106 108 110 112 114 116 118 120 122 124 126

0.5

1.0

1.5

2.0 Twist Bend Splay

Ki/S

2

Temperature

CB7CB CB9CB

Ki are normalized to 1 at TNI.

CB9CB: True K3

• The increase close to TNI is due to S2

104 106 108 110 112 114 116 118 120 122 1240.12

0.15

0.18

n

T

108 110 112 114 116 118 120 122 124 126

0.4

0.6

0.8

1.0

1.2

K3/K

3(TN

I)

Temperature• Δn measured by

polarization interference• Δn is proportional to S

True elastic constants of CB9CB.

-16 -14 -12 -10 -8 -6 -4 -2 00

1

2

3

4

5

Ki

T-Tc [K]

Values are relative to the values at TNI. Black squares - splay, green triangles – twist, red circles – bend.

Mixture of dimers

The phase diagram for a mixture of KA and the liquid crystal dimer, CBF9CBF

Elastic constants of mixture

Elastic constants of mixture by Frederiks transition

Minimum K3 =0.63 pN – by light scattering 0.3 pN

Relation to cubic invariants

jkliklij QQQC ,,1 ljlkikij QQQC ,,2 ljklikij QQQC ,,3

)22(3/1 321)3(

1 CCCK

)2(3/1 31)3(

2 CCK

)4(3/1 321)3(

3 CCCK

•To quadratic order in gradient of Q splay and bend constants are equal.

•Cubic invariants that contribute to the elastic constants are

Values of third order coefficients

0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.160.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

Ki/

n2

n

K3 K2 K1

0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.140.0

0.5

1.0

1.5

2.0

Ki /

n2

n

Ki/S2 as functions of Δn for CB7CB (left) and CB9CB (right).

C1 negative, C2 and C3 about 0

•Transition seems to be driven by increase in S

Problems

• Bend constant increases just before the transition to Nx phase – Bent core molecules also have small bend constant,

but go to Sm phase – perhaps the increase of K3 due to competition with smectic order

• Standard methods based on Frederiks transition give

smaller decrease of the bend constant

Conclusions

• Bend elastic constant in the nematic phase of flexible dimers dramatically decreases with T and is probably the cause of an instability resulting in the modulated phase

• Just above the transition K3 slightly increases – effect of pretransitional fluctuations?

• Below the transition light scattering corresponding to splay fluctuations strongly decreases – analogy with SmA phase?

F

O

F

O

FF

F

N

FN