Appendix
I.
The Total Free Energy of Half-V Mode Ferroelectric Liquid Crystal
Fig. A-1. The coordinate system of a HV-FLC cell.
The molecular director of FLC can be expressed as:
nˆ  (sin  cos  , sin  sin  , cos  )
(1)
where  and  are the cone angle and azimuthal angle, respectively. When the molecular
director of uniformly aligned FLC is parallel to the z-axis, the one-dimension total free
energy per unit area of FLC cell without applied external voltage can be expressed as:
F 
d /2
d / 2
Wd dy  Fs
(2)
where Wd is the elastic free energy density; Fs is the surface energy per unit area; and d is the
cell gap. The elastic free energy density, Wd of a SmC* material, can be written in
Oseen-Frank form as:
Wd  ( K1 / 2)(div nˆ )2  ( K 2 / 2)(nˆ rot nˆ  qt ) 2
 ( K3 / 2)(nˆ  rot nˆ  qˆb ) 2
(3)
where K1, K2, and K3 are the elastic constants. The wave vectors of spontaneous twist, qt, and
spontaneous bend, q̂b , are given by
qt  q sin 2 
(4)
qˆb  q cos  (nˆ  zˆ)
(5)
where q  2 / p . p is the pitch length of FLC, and the sign of q specifies the handedness
of the FLC helicoids. If the variation of  is within confines of the y axis at constant cone
angle in a HV-FLC cell, the elastic free energy density can be expressed as:
1
Wd   y 2 sin 2  [ K1 cos 2   ( K 2 cos 2   K 3 sin 2  ) sin 2  ]
2
  y q ( K 2  K 3 ) sin 3  cos  sin 
(6)
1
 q 2 sin 2  ( K 2 sin 2   K 3 cos 2  )
2
where  y is the partial derivative of  with respect to the y axis. FLC molecules are
assumed for their well alignments without variation at y axis, i.e.  y =0 or  =constant, in a
uniformly aligned SSFLC cell. The elastic free energy density of this uniform state can be
simplified as:
Wd 
2 2
sin 2  ( K 2 sin 2   K3 cos 2  )
2
p
(7)
The surface energy per unit area, Fs, in the Eq. (2) can be given as:
Fs   [  1(i ) (nˆ  sˆ) 2   2(i ) ( pˆ  sˆ) ]
i
 sin  (
2
(t )
1
sin t  
2
(b )
1
sin b ) ( 2 cos t   2
2
(t )
(b )
cos b )
(8)
where the surface energy per unit area is the energy summation of the top (t) and the bottom
(b) substrates;  1 and  2 are the non-polar and polar surface interaction coefficients,
respectively. p̂ and ŝ are the unit vectors of polarization and surface normal. The negative
and positive values correspond to Ps up and Ps down domains in the horizon chevron defects.
When FLC molecules are well oriented by rubbing direction in the SSFLC cell ( t  b   ),
the surface free energies can be rewritten as:
Fs  sin 2  ( 1( t ) sin 2    1(b ) sin 2  ) ( 2(t )   2(b ) ) cos 
(9)
Thus, the total free energy of FLC can be expressed as:
F
2 2
sin 2  ( K 2 sin 2   K3 cos 2  )
p2
(10)
 sin 2  ( 1(t ) sin 2    1(b ) sin 2  ) ( 2(t )   2(b ) ) cos 
II. Electrooptical Property of Half-V Mode Ferroelectric Liquid Crystal-R3206
Low frequency triangular driving waveform confirmed that both pure R3206 and its mixtures
were half-V switching mode FLC materials as indicated in the Fig. A-2.
Transmittance (arb. units)
2.0
1.5
R3206-70
R3206
1.0
0.5
0.0
-8.0 -6.0 -4.0 -2.0 0.0
2.0
4.0
6.0
8.0
Applied voltage (V)
Fig. A-2. Electro-optical properties of R3206 and R3206-70 driven by 30Hz triangular wave.
The pitches of R3206/R3206H at different weight percentages were listed in the Fig. A-3.
10
Pitch Length (um)
8
6
4
2
0
0
20
40
60
80
100
R3206 wt.%
Fig. A-3. The pitches of R3206/R3206H at different weight percentages.