ferroelectric chiral smectic liquid crystal confined between two aligning plates is studied in the case of thick samples. i-e- samples whose thickness is higher than.
Phys.
J.
France
II
Chiral
(1996)
6
Smectic
M.
Brunet
(~) G-D-P-C(~) Physique
1687-1725
Liquid Crystal,
C
and
(~>*)
Montpellier II,
UniversitA des
(Received11
Solides,
December1995,
revised
31
PAGE
1687
Textures
(~)
Montpellier. France Paris-Sud, 91405
34095
510,
Bitiment
Sample
Thick
~lartinot-Lagarde
Ph.
1996,
DECEMBER
UniversitA
Jul,v1996,
accepted
9
Orsay,
France
September1996)
Liquid crystals
PACS.61.30.-v
PACS.61.72.Lk
dislocations,
defects:
Linear
disclinations
Abstract. The of ferroelectric chiral smectic liquid crystal confined between texture two aligning plates is studied in the case of thick samples i-e- samples whose thickness is higher than the helical pitch. The helical with periodic unwinding discfination lines in these texture appears samples. A theoretical elastic calculation is made assuming in the defect lines the melting of the smectic C phase into the smectic A phase. It allows the calculation of the line positions and the local helical the sample pitch thickness. of the In the bookshelfgeometry and in that versus case of the che~ron observations well explained by the model. The pitch allows measurement our are estimation of the defect and confirms the A melting. We show the existence of an core energy the chevron plane in thick samples, plane which plays the role of an unwinding surface, for the helical
structure.
R4sum4. confin4
Atudions
Nous entre
deux
lames
de
la
du
texture
traitAes,
verre
liquide
cristal
dans
le
cas
d'un
smectique
C
Achantillon
4pais,
chiral
ferroAlectrique, c'est-I-dire
tel
que
l'4paisseur soit sup4rieure au pas de l'hAlice. On observe accompagnAe de enrou14e texture une lignes de disinclinaison dites fignes de ddroulement. de Nous prAsentons un calcul du minimum l'4nergie 41astique en supposant, dans les lignes de d4faut, fusion de la phase smectique C une dans la phase smectique A. Cette hypothAse permet de calculer la position des lignes et le pas hAlicoidal local en fonction de l'Apaisseur de l'4chantillon. Nos rAsultats expArimentaux sont bien expliquAs par le modAle dans le cas de la g40m4trie dite en bookshelf et dans le cas du chevron. La du pas permet d'4valuer l'4nergie du cceur du d4faut et confirme la fusion en phase A. mesure Nous l'existence du plan du chevron dans les 4chantillons 4pais, plan qui joue pour la montrons hAlicoidale
structure
le r61e
de
surface
de
dAroulement.
Introduction
phase [1] (C* phase) is made of layers. In the layers the molecules are with respect to the layer normal. The layers pile up while turning with a perpendicular spontaneous pitch Zo, it is a liquid (2D) in the layers, a solid (1D) in the direction only a helical symmetry and a twofold axis. This lack of symmetry to the layers. It possesses allows a spontaneous electrical polarization P. It also gives rise to difficult adjustment between the helical and the homogeneous often imposed by sample walls. spontaneous texture texture The
chiral
tilted
(*)
©
with
Author
Les
smectic
an
for
(ditions
C
angle 90
correspondence
de
Physique
(e-mail:
1996
brunet©gdpc.univ-mont2.fr)
PHYSIQUE
DE
JOURNAL
1688
N°12
II
phase has been studied in detail for thin samples because of the possible applications connected with the surface interactions than with displays. In this case the is more texture as the bulk properties. study the texture of thick samples. We give an elastic model of the defect In this we paper the lines that between the helical sample bulk and the homogeneous sample part near appear This
walls.
This
compare
predictions
Thick
Sample
The
1.
In
samples,
thick chiral
a
stripes, the
and
focal
unwound
is
untreated
with more
and
appearance
the
position
of
these
lines.
We
observations.
the and
have
bulk
and
lines"
been
arrays
interpreted unwound
the we
glass plates,
regular
less
or
They
conics. in
sample
the
Textures
"dechiralization
called
structure
with
shows
structure
the
calculate
to
us
prepared C
smectic
helical
First
allows
model
these
now
is
well-known
of lines
(Photo 1,
as
structure
it
the
the
because,
the
texture
1713),
of the
result
imposed by
prefer unwinding lines
that p.
first
of
called
connection
between
bounding
surfaces.
on
the
surfaces,
the
dechiralized.
not
surfaces but in the bulk. They go in pairs, one line of to the surface. line the other surface, the other close The relative the pair is to to one positions of the two lines of a pair change in relation to the relative anchoring direction on surfaces. The two lines of a pair can be superposed when the anchoring directions the two of the background parallel. Figure shows these superposed lines with between extinction are polarizers, indicating parallel anchoring. The made crossed observations mixture were on a A polarization. with no smectic A phase, and having a very small strong planar spontaneous anchoring is obtained by SiO evaporation at 60° incidence. shifted forward a half pitch. Figure 2 shows these lines with The lines can also be shifted ~/2 that about 290, of the background. between polarizers 50° is The angle is extinction of the polarizer, of the analyzer, of which indicates symmetrical anchoring. The direction direction and the anchoring deduced from the the aligning directions both the glass plates on indicated chiral C extinction in Figure The observations made in smectic 2. are were a pure compound presenting a sniectic A phase and having a medium polarization; the spontaneous surface is the in Figure 1. treatment same as The first situation is similar to a uniform molecular parallel to the plates in Surorientation Stabilized Ferroelectric face Liquid Crystal (SSFLC) [2] geometry (thin samples), the electrical polarization lying along the normal to the plates. The second one is similar to a rotation of for the half turn the surface. This molecules going from the lo~v-er to the upper on cone a called a splayed The made the situation is often interpretation assuming structure [3]. was "bookshelf' called The smectic layers perpendicular to the glass plates, a structure geometry. topology of the director field for superposed lines described by Brunet et Williams [4] is redescribed by Glogarova and Pavel [5] recalled in called in Figure 3a and that for shifted lines Figure 3b. observe lines on the plates as shown in Figure 21 the corresponding Sometimes texture we separated on the surface by half a pitch. This configuration is drawn in Figure 22. They are surface surface anchoring [6], a quasi degenerated planar anchoring or some indicates a bistable lines
These
are
located
located
close
close
hysteresis. Most
and
made for compounds studies during the last years were experimental conditions (surface treatment, spontaneous symmetrical anchoring where the thickness exceeds about
of the
under
rise
to
two
effects
a
polarization
to
explain the interaction
frequent with
the
with
the N* -SA
polarization, 1.5
~m.
symmetrical anchoring. occurrence surface polarization gives a polar surface of the
We The energy
transition
etc.) giving can
invoke
spontaneous Also term.
Sl
N°12
Fig.
Regular polarizers (A
I.
array
THICK
SAMPLES
superposed
of
pairs
P)
shows
that
bottom
of the
of
lines.
1689
The
extinction
background
the
of
between
anchoring directions on both surfaces are parallel to the aligning direction (D). The distance between neighbouring lines is about IS ~tm. Compound: a 5% two of cholesteryl mixture cinnamate with bis-(4'-n-decyloxybenzal) 2-chloro-1-4-phenylenediamine. The cell geometry is cylindrical: a cylindrycal lens ill thickness is 102 mm) is put on a glass plate. The crossed
and
the
=
growing
the
from
the
bulk
the
to
chirality
molecular
the
top
elastic
creates
bend.
spontaneous
a
because
energy
photo.
it
spdntaneous
This
integrated,
be
can
bend
does
not
integration gives
this
but
a
appear
polar
in
surface
[7].
term
2.
Calculations
2. I.
HYPOTHESIS
geometrical and
.
On
these
the
.
The
.
The the
.
The
axis
yJ is
and is
in
tilt
the
only
has
texture
obtain
the
to
the
the
simplify
To
plates
parallel
lie
equilibrium
are
of the
dimensions
smectic to
normal
molecular
angle with
azimuth
normal
the
define
we
the
energies,
elastic
the
use
plates,
the
the
plates.
following
anchoring
is
model:
strong.
very
The
is d.
angle
two
to
involved
the
equations.
Euler we
sample
limited
a
calculate
study
our
of
texture
defect
molecules
the
9 is the
layer.
plates x
the
layers
smectic
the
the
Conditions.
between
(Fig. 4a). to
and
surfaces
distance
To
model; we then energies by solving numerically the
Geometrical
2.1.1.
TeXtures
EQUATIONS.
AND
conditions
minimize
Sample
Limited
of
and
to
director
respect does
to
the
normal
to
vary
along
the y
layers. the
plates
perpendicular
z
respect
to X,
not
The
with
to
y
and
z.
their
axis
is
z,
the
the
local
normal. normal
plate.
direction,
parallel
to
JOURNAL
1690
PHYSIQUE
DE
N°12
II
a)
Top direction)
D(aligning Polarizer Bottom
Analyzer
b)
Regular array of pairs of shifted lines. The polarizer (P) and the analyzer (A) make 2. an angle of 50° (that is about ~/2 20), when is obtained the extinction of the background; it shows that surfaces (Bottom: B and Top: T) are symmetrical with respect to the directions both the anchoring on (D). The distance between neighbouring which is parallel to the aligning direction layer normal two p-decyloxybenzylidene p'-amino 2-methyl Compound: a 50% mixture of chiral lines is about lo ~tm. (D.O.B.A.M.B.C.) with the racemic. Cell thickness: about 30 ~tm. butyl cinnamate
Fig.
.
The
layer deformation position.
defined
is
by
u
its
displacement
along
z
with
to
a
respect
to
equi-
its
librium
.
The
molecular
shown
in
normal
local
perpendicular
2.1.2.
9 is
Defect
kept
Line
constant.
to to
4b. the
9 is
the
layer.
tilt
yJ is
in the
angle the
layer
of the
azimuth
with
respect
molecular
angle
with
director
respect
local
frame:
xiyz~,
with
respect
to
to
xi,
the
in
zi
as
the
layer and
y.
Model. To
defined
is
orientation
Figure
obtain
In the a
defect
continuous
line, topological variation
of the
argument molecular
indicates
a
~
orientation,
jump of ~ if imagine we
S[
N°12
SAMPLES
THICK
1691
z
z
~ X
X
a)
b)
Fig.
Topology of the director in two b) symmetrical anchoring. anchoring
3.
allel
cases
of
planar
anchoring
molecular
distance
The
between
b
the
two
the
on
families
a)
surfaces: of
defects
paris
lines
,
0 in
a
Zo/2
and
3a
of the
core
defect
of ~ jump discontinuity. the A phase
The ~
3b.
in
~ is
molecules
where
the
then
transformed
corresponds
This
to
a
normal
are
in
the
to
continuous
a
second
layer (9 0, ~ undefined). -90 to +90 without from the C* phase in the bulk
smectic
order
=
of 9 from
variation transition
The A melting Figure 5 shows a drawing of the defect. core. induces a displacement ~1 of the layers. corresponds to an increase of the layer thickness. This This displacement propagates along the normal to the layer near the plane parallel to the plates and going through the defect, because a smectic layer is like a sheet of paper it is easy to bend
to
difficult
and
the
in
to
defect
[8].
compress
thickis made of The infinite sample equilibrium Energies. constant texture 90. They parallel and tilted with turn parallel plane layers. In the layer the molecules are ness from one layer to the other making a helix of pitch Zo. The uniIn the central part of a sufficiently thick sample the helical equilibrium is reached. corresponding the distortion, form boundary by the plates conditions imposed create energy a and compression tilt smectic smectic density can be separated into nematic energy energy, Involved
2.1.3.
energy:
F=Fn+FS@+Fsc We is ~
call
nematic
function 9
and
of the
layer,
~~~
K
§
the
in
in
9,
approximation we
jd~ ox
orientation
orientation write
~
~~
direction [9]. It corresponding to a variation of the director energy of u. of The derivatives derivative of ~ and 9 and of the second
molecular
molecular
terms
energy
the
derivative the
measure
the
measures
energy
of the
~~
~
the
d~ 0z
due
variation
of only nematic 2~
Zo
in
variation
elastic
one
off ~
ox
K
constant
density Fn
energy ~
the
to
layer. The layer bending.
the
~
off ~
0z
and if
second For we
derivative low
neglect
of
fourth
order
as:
~~lj
~
~
0x2
~
2d~~ d(9 0x2
u,
displacement
cos
ox
~)
JOURNAL
1692
DE
PHYSIQUE
II
N°12
Plate
'
z
w
x
a)
Plate
z
w
~l
b) Fig.
a)
4.
molecular
angle
~a,
In
equilibrium
orientation
with
respect
in to
the
layer
the the
are
normal
layer with respect to a local frame, layer, zi, the azimuthal angle ~a, in
The
smectic
equilibrium
tilt energy tilt 90 (10]
layers are normal to the plates. The angles which define the tilt angle 0, with azimuthal respect to the layer normal z, the plates, x. bj The molecular orientation is defined in the to the the tilt angle 0, in with the local relation normal to the xi, yi, zii relation with xi, in the layer and perpendicular to y.
smectic
the
density Fs@
corresponds
Fs@
The This
length energy
~
gives
the
appears
in
ratio
the
between defect
line
"
the when
to
the
keeps the
molecules
at
tilt
constant.
the
9( )~
~~ (0~ 4~ nematic
9 goes
that
torque
constant to
0.
K
and
the
smectic
Si
N°12
i,
i
/ ,
,'
domain the
2r, it
=
defect
in the
only stays this
In
parallel
the
distortion radius.
Figure 8 is drawn the layer anchoring (~i ~2
In
Variation.
for 90 " 0A radian and z created by the layer thickness
is
distance
gives a sample
defect
and
displacement We
plotted
~~
of the
where
90
zero.
uers~s
~1
analytical
the
indicates
the
Smectic
2.2.2.
p.
~
~'
~~~~
~°~~~~~
given by the
r
with
calculation near
to
9
4@
I
sample
~
proportional
~~ 0~
oo
The
P
~°P~
solution:
~ The
0~9
°P
equation is:
the
p,
off
and gives 9
°P
=
P
For
9 ~
defect
in
a
the core
corresponds
domain maximum
radius to
a
due
core
to
displacement "
~/2).
the 9
This
decrease.
parallel to the plates of displacement is reached
(z
re
uniform
2r).
The
calculation
compression of the
JOURNAL
1696
DE
PHYSIQUE
Tilt
angle 0o/rad
N°12
II
~ w _~
~
b
o.oi
teristic
log-log plot length of the
1=
~tm.
Fig.
7.
20
One
can
of the that
see
line
C
smectic
r
radius
core
energy.
is
r
The
inversely
the
~ersus
calculation
proportional
angle
tilt
is
made
to
00.
9.
is
r
for Zo
"
d
by
divided =
2
~tm,
~t
p, =
the 30
charac~tm
and
/ 3
x
Fig. the
Sketch
8.
increase
layer
in
the until
of
the
layer
the
defect half
core
displacement
induces
pitch of the
a
t1
~ersus
z
layer displacement texture.
and u
~,
for 00
which
=
al
propagates
radian. in
the
The
layer
direction
thickness
normal
to
S)
N°12
~ i
8 E
~
~ .~
~'
~ ~
f
E j
~ .oi Tilt
THICK
SAMPLES
1697
JOURNAL
1698
o
PHYSIQUE
DE
II
N°12
z/2
z
z/2
z
z /2
a
o
o
x
~
3a /2 3a /2 a
d
Fig.
Fig.
Azimuthal
Fig.
line
corresponds
Azimuthal
11.
angle to
molecules,
of the
jump
a
angle
/2
~a
Fig.
10
10.
defect
~ ~
~i
/2
~a
/2
~a,
molecules,
of the
~ersus
angle.
of 2~ of this
~a,
x
The
~ersus
x
and
11
z
the
in
drawing
and
z
parallel
of
case
corresponds
in
the
one
anchoring. The helical pitch.
symmetrical
of
case
to
anchoring.
~
~
o
Fig.
The
azimuthal
angle
The
continuous
lines
12.
defect constant
.
line. 0
Zo/4
z
~a
~ersus
are
for
z
the
result
different of
distances
computer
the
z
from
the
calculation,
the
plate
upper
dashed
lines
near
are
the the
approximation.
for the
parallel anchoring,
~
) =
4£ m
+
p=o
~i
=
cos(2p
§J2
+
"
~/2:
1)~sin(2p (2p
+
+
1)~sinh(2p
1)sinh(2p
+
)
~
+
1)
1)
j lz
~ ~
(1)
Si
N°12
THICK
SAMPLES
1699
Eos/K Eo~/K
)
~°~~
E
~uS/K
8
] ~°~~
Eu~/K
I
'
iy3 ~j I ~
io-4
io-5
angle 00/rad
Tilt
Fig.
13.
and
reported
d
2
=
sgN
"
.
Energies ~m,
~
nematic
for the
v7
=
=
of the
K,
to
30
tilt
the
and
pm
energy,
j j
calculated
texture
elastic
1
~m. smectic
"
anchoring,
for
suN
tilt
~i
=
layer
"
~/2,
~2
~~"~ ~~
~"
computer bend
c~
energy,
pitch along
one
The
constant.
20
=
sgs
symmetrical
+
line
nematic
i
o-i
o.oi
=
"
energy,
total
-~/2
for
the
surface
line
one
unit
is
made
z
sus
nematic
"
layer azimuthal
length along for
Z
=
compression angle energy.
y,
Zo
#
energy,
3~/2:
or
"~~~/j
~ z~~~~~~~
~
~~ 12)
+ 2 ~
.
and
calculation
case,
~
~i
"
§J2
"
2d
+~/2:
sin(2p+1)isinh(2p+1)I 2fl m
~
~g
=
+
~ p=o
d
(2p
+
1)sinh(2p
1)I
4d
j~)
+
origin of x and z is on the first plate in the middle between the two defects near this plate. the pitch of the helix in the middle of the sample. We will see in a following paragraph that Z can be slightly larger than Zo, the equilibrium helical pitch of the compound.
The
Z is
JOURNAL
1700
The
the
defect
calculation
core
allows
to
us
defect
of the
outside
Energy
2.2.4.
calculation, sample. We
the
of ~
values
obtained
with
surprising if we not established in the paragraph 2.2.1. 9 the approximation constant use
numerical
the it
between
agreement
PHYSIQUE
DE
take
is
N°12
II
the
This
approximation
the 9
account
agreement
interpret
to
9
constant
into
is
very
sample
the
stability
and outside
with
of
interesting because everywhere texture
core.
Considerations
Defect
of the
Existence
Texture.
Line
the
From
computer
energies of a case we can calculate them for one pitch of the and one unit length direction in the z texture direction. in the y These energies divided by the elastic l~ are plotted in Figure 13 constant the angle. tilt The numerical calculation for sample made with parallel anchoring. is uers~s a The sample thickness is 2 ~m and the helical pitch is 2 ~m. We can see that the different energies are not at all equal. For a given 90, the layer displaceorder of magnitude lower than the molecular tilt angle energy s~ which ment energy su is one order of magnitude lower than the is also one azimuth angle energy s~. suN, the layer bend increases smectic compression energy is the lowest one; it very quickly with 90. sus, the energy stable. The 9 and ~ energies vary roughly as 9(. We can see that the ~ energy is not is more exactly proportional to 9(. An estimation of the 9 and ~ energies can be made: for the bulk that, out energy we saw of the core, the ~ configuration is very close to the one at 9 constant, which follows a Laplace equation. The electrical analogy allows us to write the term of the out-of-core energy of a ~ sample pitch as: of the
the
in
line
E~b
the
logarithmic the
For
thickness
other
two
proximate
radius,
core
sample
the
to
defect
the
is
~
terms
related
helical to
energies core expression of
it
the
~
origin is the
The
defect
lines in
two
center.
pitch
one
~° ~~~~ p
and
one
and
~
unit
the
are
different
terms
of this
c~~ c~c is
numerically
tions,
the
We The
can
tilt
two
tilt
compare nematic
-~
we
assume
~
with 90. expression from the apthe cylindrical symmetry
~
cylindrical coordinates. The by the integral: given are
(())~
+
P
E~C
The
an
of s~
approximate
core
energies
for
in y
P
o
variation
~(
length
+
be
when
obtained
~
l(~(~))~
£c=2/~
and R is a given length close independent of 90. In the bulk, to pitch Zo and are proportional to 9(. The
important calculate
distortion
2.2.1
has
spontaneous
the
different
ml
Section
R
Z.
the
Aoi
2~
=
from
pitch
slightly more is possible to configuration
defect:
the
in1
K91
the
induces
term
2~
expression
its
or
the
or
are
=
obtain
texture,
~(9~-9(~~) ~
2~pdp
E@s
E@N
integral give
o.i6~A~oj
coN
=
cos
~iioj ~ ~( =
-~
o.59~A~oj
calculated, coN and cos are analytically integrated. With these approximaenergies c~N and c~s are equal. these calculation analytic results with the computer given in Figure 13. exactly the The tilt is smectic energy coN same. energy c~s given by the
Si
N°12
THICK
SAMPLES
1701
E/K
)
E'q/K
j m
i
.Io «
W
z
O.
~
e
i ~
1 ~ n
W
angle oo/tad
Tilt
Fig. 14. reported
Total K
to
The
state.
~ersus
is 30%
questionable
higher
is
the
than
for
energy
of the
s)
made
depends
it
for
s
angle 00.
condition
the
with
energy tilt
calculation
because
the
find
energy
the
computer
computer To
distortion
the
is
analytic
=
value
of
the
defect
which
state
area
~m, ~t
=
30
which
line
and
~tm
analytic
The of R
is
not
texture,
1
defect
20
=
(y)
line
the
unwound
~tm.
approximation of c~ really defined. have
we
to
compare
is
its
exactly:
is
j
2~~A'9]
=
length along the of the sample in
unit
one same
2
=
the
on
existence
cj
d
=
approximation.
much
unwound
Zo
and
z
of the
energy
for Z
too
the
pitch along
one
o
cj
Figure 14,
In
We
can
defect
see
line
and
total
the
that, for this is
texture
energy
of the line
thickness,
unstable
the
if 90 >
texture,
c,
energies
two
are
reported for are equal for 90 *
d
=
Z
0.08
Zo
=
radian,
"
2
and
pm.
the
radian.
0.08
The Near each plate a family of lines exists. plates is mainly due to the bulk yJ distortion. We find the relative positions of the lines by minimizing the ~ energy only, using the constant close of defect the tilt angle 9 model. Indeed that outside the is to the core very we saw equilibrium tilt angle 90. ~Ve first write the angle ~ with a Fourier series in the general case calculate the ~ energy c~ for one pitch in when the defect line position is arbitrary. Then we families of defect lines b and the of the and The relative length position unit in two one z y. calculated by minimizing c~ uemw distance a between a family and its neighbouring plate are POSITION
THE
2.3.
and
a
OF
between
interaction
DEFECT
THE
these
LINES.
families
two
and
the
b.
The
of the
determination
calculations.
Indeed
defect
core
2.3.1.
General
be
obtained
the
two
energy
we as
well
of lines
of the the
to
the
bulk
energy
~ in
the
Constant
Fourier in
pitch
minimize
as
Expression of
using the
families
real
have
the
transformation, z
direction
difficult
texture
is
energy
density
have
9
to
be
more
Model.
in the
(Fig. 3)
is
uers~s
taken
general
For case
arbitrary.
into
~,
than
the
pitch;
two
the
following expression
can
in
this
preceding case
the
account.
the
where
the
distance
between
JOURNAL
1702
PHYSIQUE
DE
N°12
II
b-Z/2 0. The
increase
l do
>
~.
The
shear
with
vary
for ~2 last shear
very weak. For thick linear law, and the a
is
by
because
~ ~~~ 6
b.
vanishes
shear §J1
not
the
approximated
i~~
~~
"
shears this
Z/2
=
One §J1
is
"
effect
the
is ~,
to
»
1.5
Z),
equation
j
z
=
anchoring on the anchoring, and tends to the layer, it tends to decrease stay in the same layer.
of the
different
symmetrical
the
related
samples (d equilibrium
the
tilt of
pushed by the layer tilt. They try to When the layers are perpendicular to the plates (6 0), two particular cases are very For the symmetrical anchoring Z/2, all the three shears vanish for common. ~2 §J1 ~ and the two line families alternate, as drawn in Figure 3b; this alternate position is better are defined in thick samples. For the parallel anchoring ~i §J2, the anchoring tends to lower b. The equilibrium is obtained for b families 0 and the two line in phase as drawn in Figure are 3a; this position is better defined for thin samples. When the layers are tilted (6 # 0) in the case of parallel anchoring, the pairs of lines from the two families the of in smectic layer (at the foot of Fig. 30, 6 > 0). In the are same case symmetrical anchoring, (at the top of Fig. 30, 6 < 0), the tilt of the layer and the difference between the two anchoring angles can put the lines of the two families in front of each other. defects
are
=
=
"
"
=
Position
2.3.3. we
can
evaluate
of the the
Lines with Respect to the Plates. position of the defect lines with
Using respect
the to
previous the
energy
expression,
neighbouring plate.
We
JOURNAL
1704
DE
PHYSIQUE
N°12
II
fl
~
~2
"
f2
Z
£ g 41 ~ ©©
i
= Q
f2
~
I
Q
i
(
"
f~2
j o
~
0
Uneplmedistanc«a)
Fig.
16.
and
the
Force, f2, plates, in the
the
line
length
due
on
the
due to
a
length on a defect line, due samples id > xZ). For neighbouring plate; the to its image by the layer made by the images of the continuous per
case
unit
of
to
thick
the
interrupted lines
with
interaction
comparison of its
the
thin
all
other
lines
line, f( is the
force
line, Ii is family. Ii
the
the is
force
the
force
per
due
unit to
helix.
/2
derive c~
with
respect
to
a
to find
the
force in the
la
direction
x
on
one
length of
unit
one
defect.
~~ ~ ~~
with
~~2 J~@2
fl
~
20
f~
6
CDS
~~
~°~~~~ l~"~
-~~Koi)
-
~
)~~~j ()~"~
~
fi
~~2d
~~~
~
2d
force due to the helix. This force tends to repel whole perfect helix throughout the sample. The series a defects. This force lines by the plates and the other term f2 is the force applied on the defect for repels the line from the plate. In Figure 16 is plotted f2 thick sample a very uemw a (d oo). In comparison we have also plotted f( the force applied on one line by its image due neighbouring plate and fl', the force applied on one line by a continuous layer made by to the the image lines of its family. The
term
the
defects
can
be
toward
the
interpreted plates to
as
a
constant
obtain
r~
f~
.
If on
a
is one
smaller line is
"
~~~°
than Z/4~, f2 * f(, only the force due to
fi
this
means
its
image.
~~ "
that This
)~° the
line is close
force is
to
the
proportional
plate, the force to
1la.
Sl
N°12
THICK
SAMPLES
1705
4
4
t~
ti %
W
j
W
£
li z S
Z &
#
~
8
~
.E
f
fi
°
2
~
e
#
Sb
©©
fi
Q "
z
#
E
d>nZ
~
(
8U i
g
k
d>AZ
~
d 0
plme ~stancela)
Une
ii
Fig. plate
17.
are
in
are
I Red
p the helical
texture
=
In are
0.9Zo if 90
Figure reported
Zo/2:
if d < and
density of the structure stable. This gives an d~
other
paragraph
In
"
19 the uers~s
0.08
the can
upper
rad
gives
this we
limit
for the
thickness.
"
2
helical
Z
=
this
=
is
close
(Sect. 3.1.2) "
45°
~m
lt
Fig. tilt
pitch
radian,
in
the
calculation defect
spontaneous
two
Grandjean-Cano
the
sample.
For
the
to
method
about
the
of the
elastic
smectic
defect
thus
and
energy
core
the
Zo
to we
12
"
If
pm.
that
hypothesis
the
molecular
rectangles correspond error only adjustable used parameter
say
can
the
~ersus
The
pm.
curves
constants,
pitch Zo
spontaneous
is 15
theoretical
the
pitch and the best fit corresponds
Experimental LINE
3.I.
often
surface
to
we
an
into
experiment
the
of
take
A
C*
account
confirms
transition
in
to
is
the our
the
the
analyser
different
is
side of
each
on
between
difference
Figure
line.
a
an
background
This
60°.
when they exist, they are very unwinding volume line becoming
but,
shows
located
area
part of the
about
Textures
uncommon
(p. 1713)
2
the
21
other
very
are
Photo
Figure the
while
lines
line.
Sample
Limited
in
Surface bulk
a
In the
lines.
polarizers
polarizer and the glasses are this
Situation
POSITION.
connected
crossed
the
shows 22
that
shows
lines is black
surface
two
black
is
when
the
angle anchoring the
distribution
the
between directions
of the
with the on
director
case.
Generally unwinding lines are distance between the glass plate report
the
3.I.I.
Line-Plate
of
Angle 6
line.
3.
two
reported pitches.
de
thickness
for
the
in
spontaneous
uncertainty
in
by
measured
dispersion
the
sample
Critical
20.
angle 90
i
o-i
o.oi
results
measurement
of
not
surface
and
the of this
measurements
Distance
requires
edge
uersw a
lines
line is
Sample
wedge-shaped
and
about
distance
half
the
relation
in
Thickness:
geometry
reference
in
Measurement for
the
been
[4] it has
said
that
spontaneous pitch Zo. Here to the spontaneous pitch.
sample.
Method.
Straight
The
sharp possible to
the we
method ink
lines
glass plates. In this way it is to are on measure by focusing ~v.ith a large magnification on the place, of sample in given thickness the the a The slow displacement of the stage, which gives fine mark and then on the lower one. upper difference be measured. The between the two distances graduated, vertical focusing, is can so
perpendicular
the
drawn
the
S)
N°12
i"~~~
viii~,1,>~~
i
~~
~'l 1k
.%,p.,~~,=.,
j~~~iis".~$~[$.
i~
'÷7.....
~'
.~'
,i,i
1709
-cl >iz=liii~~li Zo. We call this results of these measured Z with The region we have respect to d. of d/Zo. Here the rectangles in Figure 19 giving Z/Zo as a function measurements are experimental results are close to the results given by the calculation (Sect. 2.3.4): too our the best fit corresponds to a pitch of12 ~lm and the by the measurements spontaneous Grandjean~cano method give Zo 15 ~lm.
Zo ~ d
where
.
pitch
Z.
