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
2
2








1
1
d
d
d

2
2
2
2
F  K1s  K 2t  K 3b  C1  2 ni n j   C2  2 nz n j   C3  2 nz  
2
4   dz

 dz

 dz
 



 

Splay-bend phase:
4K
0   3 ,
3K1
2
K
k  3 ,
3C
2
3
K3
Fsb 
27CK1
then twist-bend is
the stable phase
Twist-bend phase:
K
0   3 ,
2K 2
2
If K1  2K2
K
k2   3 ,
3C
3
K3
Fsb 
54CK 2
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
Nematic fluctuations
Bend
n0 qn e
iqn r t / n
e
Splay
n0 (qn )
n
Twist
n0
q
2

kBT
2
K qn
Relaxation rates
Two modes: bend-splay and bend-twist, for q along n – pure bend
Relaxation rates:
1
i
Ki q  K3qz
2

i
2
, i  1,2
Effective viscosities:
(q 3  qz  2 ) 2
1   1  4
qb  q2 qz2 (1  3   4  5 )  qz4c
2
qz  2
Usually α2 >> α3, so that bend viscosity
2   1  2
2
qa  qzc
is smaller due to backflow
2
•
2
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)
-10
1.0x10
Bend
Twist
Splay
CB7CB
-11
Ki/i(m /s)
8.0x10
-11
2
6.0x10
-11
4.0x10
-11
2.0x10
0.0
102
104
106
108
110
112
114
116
Temperature
Note increase in the splay diffusivity below TNX
CB9CB diffusivities (K/η)
File: DLS_CB9CBcorT.org, 11-Aug-12
Window: Diff
-10
1.2x10
Bend
Splay
Twist
-10
Ki/i(m /s)
1.0x10
-11
2
8.0x10
-11
6.0x10
-11
4.0x10
-11
2.0x10
0.0
106 108 110 112 114 116 118 120 122 124 126
o
Temperature ( C)
Note increase in the splay diffusivity below TNX
Normalized “bare” elastic constants
CB7CB
CB7CB
o
T=116 C:
K1>K2>K3,
K1/K2  2.8
K2/K3  2.3
CB9CB
Bend
Bend
Twist
Splay
Twist
Bend
Splay
2
Ki/S
2
10
Ki/S
1
102
104
106
108
110
112
Temperature
114
116
1
0.1
106 108 110 112 114 116 118 120 122 124 126
Temperature
• 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
CB9CB
CB7CB
2.0
2.0
CB7CB
Twist
Bend
Splay
Bend
Bend
Twist
Splay
1.5
Ki/S
2
Ki/S
2
1.5
1.0
1.0
0.5
0.0
0.5
102
104
106
108
110
112
114
116
Temperature
Ki are normalized to 1 at TNI.
106 108 110 112 114 116 118 120 122 124 126
Temperature
CB9CB: True K3
1.2
0.18
n
K3/K3(TNI)
1.0
0.8
0.15
0.6
0.12
104
106
108
110
112
114
116
118
120
122
124
T
• Δn measured by
polarization interference
• Δn is proportional to S
0.4
108 110 112 114 116 118 120 122 124 126
Temperature
• The increase close to TNI is
due to S2
True elastic constants of CB9CB.
5
4
Ki
3
2
1
0
-16
-14
-12
-10
-8
T-Tc
-6
-4
-2
0
[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
•To quadratic order in gradient of Q splay and bend constants are equal.
•Cubic invariants that contribute to the elastic constants are
C1QijQkl,iQkl, j
C2 QijQik , k Qjl,l
K1(3)  1/ 3(2 C1  2 C2  C3 )
K2(3)  1/ 3(2C1  C3 )
K3(3)  1/ 3(4C1  C2  C3 )
C3 QijQik,l Qjk,l
Values of third order coefficients
2.0
K3
K2
K1
1.5
1.4
1.3
1.5
1.2
1.1
1.0
2
Ki /n
Ki/n
2
0.9
0.8
0.7
1.0
0.6
0.5
0.5
0.4
0.3
0.2
0.1
0.0
0.07
0.0
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.07
0.08
0.09
n
0.10
0.11
0.12
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
0.13
0.14
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
F
F
F
O
O
F
N
F
N