Anchoring energy and easy direction of non uniform surfaces

Transcription

Anchoring energy and easy direction of non uniform surfaces
Anchoring energy and easy direction of non uniform
surfaces
G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan
To cite this version:
G. Barbero, T. Beica, A. Alexe-Ionescu, R. Moldovan. Anchoring energy and easy direction
of non uniform surfaces. Journal de Physique II, EDP Sciences, 1992, 2 (11), pp.2011-2024.
<10.1051/jp2:1992248>. <jpa-00247785>
HAL Id: jpa-00247785
https://hal.archives-ouvertes.fr/jpa-00247785
Submitted on 1 Jan 1992
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Pllys.
J.
ii
France
2
(1992)
NOVEMBER1992,
2011-2024
2011
PAGE
Classification
Physics
Abstracts
61.30
Anchoring
(~),
G.
Barbero
(~)
Dipartimento
Italy
(~)
(~)
Institute
Physics
of
(Received
Abstract
The
of
energy
of the
allows
to
Inti~oductiou.
The
orientation
periodic
a
samples
sample
contains
Materials,
of
an
interference
an
(~)
Moldovan
degli
Duca
24,
Abruzzi
Bucharest,
MG 7
CP
Department
Rumania
Physics,Splaiul
of
by
Independentei,
distribution
of
directions
easy
the
on
substrate
on
liquid crystal
nematic
a
Alany
aspects.
Our
papers
devoted
are
to
is
this
consider
anchoring strength
(the
nematic
distortion.
very
is of the
literature
nematic
very
the
fs(n ar)
ar
is
is the
defined
the
important
substrate
understand
to
on
the
description
phenomenological
nematic.
>vhich
Of
mechanism
course
large.
very
expressions used
is
the
the
llowever
in
is
orientation
the
literature
and
ar
orientation
average
n
energy fs, in the absence of bulk
of fs around its
minimum.
A
curvature
[iii,
by
Rapini
and
Papoular
which
ago
as
responsible
number
both
surface
proposed long
one
interaction
fs(1)
=
researchers.
nematic
the
This
directions
phenomenological
w(n ar)~. The phenomenological approach
+
2
[12-16], because it gives the possibility to describe, by means of few
liquid crystal-substrate
interaction.
The microscopic approach is, on
kind:
microscopic
the
easy
direction
director) on the surface minimizing
The anchoring strength is defined as
popular expression of is
easy
surface
axis.
easy
the
The
w.
elective
important both for
subject [1-4]. The
problem relevant to the
nematic-substrate
The
from pl~enomenological [5-7] and microscopic [8-10] points of view.
description of the surface
is made by using the concepts of easy
interaction
of
313,
macroscopic
surface
analysis shows that the
interference
coming from the two diferent
term
uniform
direction, depending on the two
easy
term
too, as suggested a few years
ago by
some
solid
a
fundamental
analyses
Torino,
10129
August 1992)
4
considered.
are
elective
an
induced
technological and
of
erect
introduce
1
Technology
Bucharest,
Corso
surfaces
Rumania
nematic
containing
of
uniform
non
(~) and R.
Torino,
di
1992, accepted in final form
June
9
properties
published
and
of
Alexe-Ionescu
A-L-
Politecnico
Institute
Bucharest,
77216
(2),
Fisica,
di
direction
easy
Beica
T.
Politechnical
and
and
energy
of
for
parameters
microscopic
are
the
correct.
parameters,
the
necessary
also
papers
in
the
hand,
imposed by
complete
a
other
orientation
approach
The
widely used
is
for
that
confirms
relevant
to
the
JOURNAL
2012
PHYSIQUE
DE
II
N°11
phenomenological approach try to connect the physical properties of the substrate
induced on a given liquid crystal. The effect of the surface
orientation
geometry has
taken
into
account
with
also
the
been
[17-20].
published
the surface is considered
uniform and
characterized
by a well
papers
anchoring
strength.
is
the
But,
well
known,
solid
substrate
easy
energy
as
is never
homogeneous over a large scale, even if the chemical composition of the surface does
interesting to consider the effect of a spatial distribution of easy
Hence it
not change.
seems
experimentally
direction.
This analysis could be important also
directions
the
detected
easy
on
In
most
of the
defined
the
in
direction
which
in
case
evaporated
and
a
surface
is
by
characterized
than
more
axis,
easy
one
as
in
the
case
of
[21-23].
surfaces
organized
section 2 a
semi-infinite
periodic structure,
with
In
analysis, which generalizes the one proposed long ago by
Berreman
[17] and de Gennes [18] and recently by Durand [19] et al., gives the opportunity
of a periodic
introduce
the equivalent
surface
In
section 3 the finite
structure.
to
energy
anchoring energy, in a semi-infinite sample, is taken into account.
Our analysis, very
to
near
performed
by
the
results
obtained
section
will
show
the
Faetti
generalizes
in
2.
We
[20],
one
different
in the surface
there
that the
directions give rise to a kind of
interference
easy
energy.
characterized
by an imposed deformation in the bulk is considered.
In section 4 a finite sample
results with the
In this way >ve can define an equivalent easy axis. This is done comparing our
relevant to a uniform sample. It gives, furthermore, the possibility to treat a
well known
one,
homogeneous surface. In section 5 the main conclusions of
surface
with periodic
structure
as
a
given.
our
paper
are
Our
strong
paper
is
anchoring,
sample
Semi-infinite
2.
Let
us
consider
z-direction-
a
Les
solid
a
follows.
as
considered.
is
The
with
fl
be
alichoi~ing
by
characterized
surface
and
sti~ong
the
easy
axes
a
energy,
periodic
for 0 <
z
<
a
of
structure
and
<
a
z
<
wavelength ~ along
(see Fig. I):
~
~~
T
a
li
O
Fig.
i.
Semi-infinite
~
liquid crystal sample ~vith periodic
~
distribution
x
of easy
directions.
the
N°il
DIRECTION
EASY
NON
OF
UNIFORM
SURFACES
2013
aforo<z<a,
8(z)
(I)
=
fl for
total
The
elastic
of
energy
period, of the
one
<
a
~.
<
z
liquid crystal (unbounded along y) is
nematic
given by:
F
~iv~)~d~dY
lk it°'
=
12)
approximation, and by supposing that n is everywhere parallel to the
Equation (2) holds in the strong anchoring case, I-e- in the event in which R,
the angle made by the
director n with y, at y = 0 is imposed by the surface
In our
treatment.
the
in
constant
one
(z,y)-plane.
case:
~(z, 0)
8(z)
where
is
given by (I). By minimizing (2)
[email protected]
the
function
boundary
have
we
be
can
minimizing (2)
(I).
condition
determine
to
solved
expand R(z, y)
shown
as
Fourier
in
the
equation
~
fiy2
~~~
harmonic
a
known
well
£
~>
"
°
=
R(z, y)
By putting
series.
R(z, y)
for
function.
This
function
has to satisfy the
problem, called Dirichlet's problem, in which
function knowing its value on the boundary.
Equation (4)
Appendix I, but for further generalization it is better to
is
is the
harmonic
a
directly,
in
This
(3)
obtain
we
[email protected]
~
4
i-e-
8(z)
"
+
~
[D[cos(nqz)
E(~sin(nqz)]
+
(5)
e~~~Y
n=1
imposing boundary
and
(3),
condition
~l
~
0
D[
~'~
(2)
q
one
=
~
obtains
is the
vector
wave
+
~"
)
~
/q
found
are
to
be
~)fl
~
~
~si~l'~q~)
~~~
~°~~~~~~~'
periodic
of the
~
~
~
l~
coefficients
~
~
"
~~
where
expansion
the
~
~
~~~
By substituting (5) and (6) into
structure.
j~ ~~~~~~~
~~~
n=
This
is
the
equivalent
is
zero.
total
energy
surface
Furthermore
per
energy
it is
unit
of the
zero
length along the
sample [17, 18].
fl. These
for a
=
z-directionNote
results
that
are
It
for
a
can
=
obvious.
0
considered
be
or
a
=
Keeping
as
this
in
the
energy
mind the
JOURNAL
2014
definition
due
energy
of q, it is easy to show that F/~ is
distortion
imposed
to the periodic
D[
I
is the
of the
average
axis,
easy
y
Be
angle imposed by
tilt
considered
be
can
as
Appendix 2). The results
[24] for the particular case
Let
value
easy
this
In
w.
F
total
)k /~
=
previous
the
torded
for
second
term,
for
small
0 <
z
y
-
In
the
~
<
from
and
0 < y <
condition
boundary
the
c<,
I-e-
the
to
elastic
that:
(8)
Be,
silicon
in which
+
surface.
under
follows
It
analysis presented
the
that
(see
consideration
in
reference
surfaces.
energy~
on 0 < z <
energy of the
~(VR)~dzdy
the
structure
generalise
anchoi~ing
situation
from
far
very
periodic
the
vicinal
on
finite
the
easy
and
a
a
nematic
still the ones of equation
~,
and they have the
< z <
same
sample is given by
axes
are
8(z)]~dz.
jw /~[S(z, 0)
(9)
o
o
considered
the
contribution.
surface
deviation
c<,
section
weak
the sample is
equation (9)
case
c<.
-
elastic
o
As in
~~~
=
to
this
in
with
the
event
i-e-
Note
treatment.
~~
(
~
~"
due
anchoring
general
now
a
more
anchoring energies are
the
but
fl)2/~,
(a
to
=
at y
axis
consider
us
(I),
surface
obtained
sample
Semi-infinite
3~
a)fl
(~
N°11
c<
the
of
proportional
by the surface
~
R(z, y)
-
the
+
au
II
where
lim
is the
PHYSIQUE
DE
easy
R(z, y)
first
The
along
unbounded
the
represents
term
surface
is
term
y-direction and undiscontribution, and the
in the parabolic form, valid
the
bulk
written
calculations
By minimizing (9) trivial
still given by equation (4), but
it has
now
that
for
satisfy
the
axis.
show
is
to
())
81~)1"
)lsl~> 0)
+
(lo)
°>
y=o
at y
=
0. In
(10)
L
=
Neuman-Dirichlet's
knowing
a
relation
k/w is the extrapolation length. The
problem (mixed), in which we have
betwen
Appendix I this problem
putting expansion (5) into
derivative
normal
its
equation (10)
Do,,
"
2
~~
l
aa
+
we
fl(~
+~qL
~n
l
instead
substrate
of
(6).
S(z, y)
Note
tends
that
to
also
iii
its
obtain
now
determine
value
on
the
means
of
a
for the
border.
As
called
now
function
shown
expansion.
Fourier
expansion
is
harmonic
a
in
By
coefficents
a)
~
~iqL (ii
~~
and
easily only by
solved
be
can
problem
mathematical
to
this
case
~°
~~~~~~~~~~
~°
~~~~
~°~~'~~~~~
equation (8) holds,
Be (See Appendix 2). By assuming 8~
~~~~
I-eas
the
very
easy
far
from
direction
the
solid
imposed
N°ii
by
EASY
the
uniform
surface
sample
defined
non
z-uniform
DIRECTION
8~
treatment,
UNIFORM
NON
be
can
SURFACES
2015
by minimizing
deduced
a
surface
of
energy
a
as:
Is
a)~
wo(S
=
~wp(R
+
fl)~
a)/~ and
assumed
z-independent.
w(~
is
By putting
[aa
a)]/~,
previously
reported.
that if for
fl(~
Note
+
one
as
results
have to be
0 < z < a and a < z < ~ the anchoring energies are different, previous
modified.
in which the
chemical
of the surface is everywhere the
However in our
nature
case
w(0
a)
w(a
~)
be
justified.
to
to
<
<
< z <
same,
assume
z
seems
condition (10), equation (9) becomes:
Using the boundary
where
wa
fifs/fiR
=
w(a/~)
OF
=
and
wp
obtains
0
=
=
8~
=
=
( jk1/~ ~(VR)~dzdy
+ L
=
o
By substituting
into
o
~
2
and
II)
~
L
Equation (13) shows
the
elastic
energy
is
that
no>v
the
localised
total
energy
shown
L. As
over
f
~~~~~
in
this
in
Appendix
(~f~
~~
~=i
l
~
)2
(1+
n2q2
n=1
8
~)
proportional
is
case
~( ~
1(~~)
~~f~
~"~~
(12)
y=o
obtain
we
(°~fl)~f
~
~
Y
o
(12) equations (5)
~
/~ ())~ dz).
(a
to
fl)~/L,
I-e-
1.
~)
(~ ~)
Consequently
~~~~
~
i
and
(13)
be
can
rewritten
(nq)2
(1~
~~
l~~
interference
4~
Sample
Finally let
surfaces
works
between
of
us
of the
finite
consider
kind
~)
~
~~~
~~~~
l'
~~~~
8~~~
as
~'
Equation (16)
~~~
well
the
if qL
two
thickness
a
real
described
»
I, I-e-
surfaces
and
problem
above.
~
~
of
Let
for L
a
us
~~
~
Equation (16)
~.
-~
characterized
weak
~~L
by
different
anchoi~ing
sample
suppose
of
first
can
be
interpreted
as
an
directions.
easy
energy,
thickness
that
d
the
=
2e
surfaces
limited
are
by two
arranged
solid
in
an
JOURNAL
2016
PHYSIQUE II
DE
N°11
y
e
-fl
-a
0
~
a
x
fl
~
Fig.
2.-
liquid
Finite
distribution
of
antisymmetric
crystal
directions
easy
and
way,
sample
arranged
the
hence
thickness
of
in
angle
tilt
is
odd
an
by
limited
2e
antisymmetric
an
solid
two
surfaces
periodic
with
way.
with
function
respect
the
to
sample, as shown in figure 2.
middle
frame having the origin of the y-axis in the
By using the reference
half
thickness
is
given
by
total
period
and
(see Fig. 2), the
energy of one
middle
of
the
F
=
k
2
~(VR)~dzdy
/~
+
0
_e
8(z)
where
is
odd
function
still
w-r-t-
given by (I). Simple
-R(z, -y).
y : R(z, y)
=
~°
S(z, y)
The
y
=
expansion
Simple
-e.
=
coefficients
y +
2
now
are
calculations
£
determined
2
~"
~
~
~
"
=
ch(nqe)
and
Sn
8(z)]~dz
/~[S(z, -e)
(17)
0
considerations
Its
sample
Fourier
+
by the
show
R(z,y)
that
expansion
is
is
now
a
harmonic
written
Ensin(nqz)]sh(nqy).
boundary
condition
(18)
(10)
rewritten
for
give
e
Cn
w
[Dncos(nqz)
~°
where
2
of the
=
sh(iiqe).
aa
+
fl(~
~
+ L
a)
'
sin(nqa)
(flqLcn
+ Sn
flq
cos(flqa)
i
fl
(flqLcn
+ Sn)
flq
fl
a
~
a
~
~jg~
EASY
N°ii
By operating
length along the
in
as
previous
the
is found
z-axis
I
DIRECTION
lk Ii
=
NON
OF
UNIFORM
relevant
case,
anchoring,
weak
to
2017
total
the
energy
for
unit
be
to
~~[l~)lT'~j
I) l~ +1 1°1~)~
+
SURFACES
(2°)
nq
where Tn
Sn/Cn
=
and
°~
os
(~
+
j
~~
=
is the
average
surface
(Six, -e)1,
121)
angle.
tilt
y
=
~
~
fl
a
0
~
a
a
x
fl
e
Fig.
3.
Finite
distribution
of
In
is
an
the
even
event
liquid
in
crystal
directions
easy
which
with
function
the
sample
arranged
surfaces
respect
to
thickness
of
in
arranged
are
the
limited
2e
symmetric
a
middle
by
solid
two
surfaces
with
periodic
way.
in
of the
symmetric
sample, as
in
the tilt angle
figure 3, R(z,y) is
Elsin(nqz)I ch(flqy),
(22)
way,
hence
and
shown
given by
R(z, y)
~~
~
«
+
=
lDlcos(nqz)
+
n=i
where
the
expansion
coefficients
are
~*
given by
now
~"~
+
0
~"~
~
~~
fl(~
~
~
a
~
'
fl
sin(iiqa)
flq(flqLsn+Cn)~
~
q(nq~~~~~~~nl'
~
~~
~)
~~~~
JOURNAL
2018
By
substituting
the
total
expansion (22)
sample
of half
energy
PHYSIQUE II
DE
expression (17)
length along
into
unit
per
N°11
and
taking
the
z-axis
into
equations (23),
account
is
section, it is supposed that the two regions with easy axes a and fl have to be in exact
practical sample, one can expect a random arrangement of
as in figures 2 and 3. Iii a
calculations
the two regions on the two plates. In this case the above reported
be easily
can
generalised. Of course the results depend on the actual arrangement of the considered
surfaces.
(19-21)
different
considered
This
that
equations
cliange
for
from
the
arrangement
means
an
one
above.
However
the aim of our analysis is to show the expected results for two very simple,
and completely different,
surface
arrangements.
direction of a uniform
surface we choose as
criterion
In order to deduce the "equivalent"
easy
the equality of measurable
quantities. From the experimental point of view we can only measure
birefringence or other physical anisotropies of the sample. These quantities are proportional,
in the limit of small angles to the average
tilt angle, (R~). Consequently it is possible,
square
characterized
according to our point of view, to define two samples as equivalent if they are
bj
under
the
defining,
for
the
periodic
consideration,
by
of
By
(R~).
structure
(R~)
same
means
In this
register
(R~)
~ /~ /~ S~(z, y)dzdy,
=
e
and for the
z-uniform
0
by
characterized
structure
)/
of
(Sll
(25)
-e
~
=
a
R(y) tilt angle
the
same
quantity by
means
(26)
Sl(y)dy>
-e
"equivalent"
the
easy
axis is
by putting
obtained
(S()
In the
sion
event
(18)
which
in
and
the
sample is in
into (25)
the
equations (19)
one
(S~).
=
arrangement, by substituting
obtains, in the limit of e » ~,
antisymmetric
where
m
j~(~
'
To
the
contrary
for
an
uniform
~~~~2
~)
~
(27)
(iiq)3(1
'
antisymmetric
expan-
[email protected])
(~g)
2
+
sample,
nqL)2'
for
which
Su(y)
=
-(yle)Ssu,
[25]
equation (26) gives
~~u)
It
follows
that
in this
case
the
condition
~~~
=
3
(27) gives:
su.
(~~)
EASY
N°11
Consequently
DIRECTION
"equivalent"
the
(~
~~
~
Since in this
case
equivalent
the
axis is
easy
both
In
~
~
a,
(32)
L).
and
R(z, y)
hence
is
given by
(~ ~ ~)~ R(q, a, L).
+
homogeneous sample, (RI
in the
present
~j
(~
(°~
equivalent
the
cases
now
=
R(q,
(33)
=
R]. Consequently
given by
~
S(
~
arrangement,
e
is
distortion
no
~j
(~
(°~
=
@)
give
~
(R~)
2019
e
symmetric
in the
SURFACES
[25]
~ ~
(~
e
homogeneous sample is
non
expansion (22), similar
calculations
to be
~
+
=
If the
UNIFORM
axis is found
easy
~
°~
R)
NON
OF
~
~
+
e
axis
easy
contains
R(q,
a,
proportional
term
a
)~
(° ~
(34)
L).
to
~
R(q,
1° ~ ~)
vanishing
the
for
a
fl
=
interference
or
a
0
=
or
reduces
term
a
~.
=
L)
in the
that
Note
a,
(35)
anchoring
of strong
case
I-e- for L
-
0,
to
L
m
Rlq,
a,
0)
2
@)
~)~~~]
=
Rlq,
>
L).
a,
n=1
origin from a geometrical
c<), as expected.
An
proposed a few years ago by Yokoyama
interference
effective anchoring energy
term in the
was
variation
of the
surface
tilt angle experimentally
et al.
temperature
[23] to interprete the
observed in nematic samples. It is important to underline
that the analysis presented in section
holds only for a
fl « I, I-e- a
fl. In fact the parabolic expression
3 and in the present
one
deviations of the surface tilt angle
for the surface
used in the text is valid only for small
energy
ai « I and ii pi « I, and consequently a
fl. In
from the easy axis. This implies that ii
ai
pi
the general case of ii
I the analysis has to be performed by using for
I and ii
anchoring energy the Rapini-Papoular or more general expressions. In this frame
the surface
condition.
complicated, due to the non linear
character
of the boundary
the analysis is more
interference
surface
However
term is still expected in the
energy.
an
This
that
means
effect,
and
the
interference
only
vanishes
it
term
for
is always
present.
takes
It
anchoring (I.e.
weak
very
for
L
-
-~
-~
-~
-~
Conclusions.
5
We have
of
a
considered
periodic
the
effective
the
two
optical
easy
the effect
distribution
easy
axes
measurements
axis
on
with
is
plus
a
an
are
the
macroscopic surface
different
two
weigthed
In
term.
fact
More
axes.
(with respect
average
interference
involved.
easy
properties of
in
this
to
interference
This
case
a
nematic
a
liquid crystal
precisely we have shown that
the
geometrical extension) of
term
sample is
is
usually negligible
considered
uniform
if
if the
JOURNAL
2020
PHYSIQUE II
DE
N°11
Otherwise the sample is macroscopically
than the light wave-length.
goal of our analysis has to be modified. Our results confirm the model
characterized
by two
coworkers [23] relevant to surfaces
ago by Yokoyama and
also
be
applied
explain
recently
presented
in
this
directions.
The
analysis
to
paper
can
easy
induced by the
transitions
observed
surface
temperature in nematic samples limited by SiO
the
observed
evaporated surfaces
covered by lecithin [26]. It can be applied also to
connect
evaporated
liquid
crystal
limited
by
surfaces
containing
of
samples
surface
orientation
a
average
doped with an additive giving homeotropic alignment [27].
spatial regions
inhomogeneous
proposed long
thinner
are
the
and
Acknowledgements~
This
work
has
Strigazzi (Torino)
for
and
(Orsay)
to G.
Durand
Appendix
In
section
In
this
relevant
the
2
f(z)
TENIPUS
of the
Italian-Romanian
and of
programme
Many thanks are due to S. Faetti (Pisa) and A.
direction
discussions
the meaning of the equivalent easy
on
critical reading of the
manuscript.
useful
many
for
a
solution
the
Dirichlet
problem is given by
the
problem will be solved in a general
sample is solved by the function [28]
of
same
semi-infinite
a
ii
where
frame
I.
appendix
to
in the
realized
been
partially supported by MURST.
collaboration,
R(z,0)
=
Equation (Al
holds
in
our
z,
y)
?
f"
coincide
case
the
with
strong anchoring
in the
(j)~
=
case.
way.
Fourier
a
The
Dirichlet
expansion.
problem
jAi.1)
dj
~
direction
easy
Let
of
means
assume
us
the
(imposed by the surface).
f(() is given by,
function
f(z)=aforAn<z<On
E
n
f(z)
=
By putting the origin of the z-axis
this event equation (Al.I) gives
~~~ ~~
Simple
£Y
j fl
_
calculations
fl
for
On
in Ao
<
we
z
(Al.2)
Bn.
<
have An
~j arctg [~~ ~l
IO, c<)
n~, On
=
=
n~ + a, Bn
~~~ ~~
~)
~~~~~
~
~
=
(n
+
1)~.
In
~~~
~~
that
sho>v
(S)>
=
/~ S(z, y)dz
~
~~
=
~
~~
~~
IA lA)
o
S(~, y) along x is independent of y.
possible to determine the total energy, as done in section 2. We
means
complicated than the one given in section 2.
underline
however
that equation (Al.3) is more
direction imposed on a nematic
In section 3 the effect of the weak anchoring on the easy
liquid crystal has been analy2ed by means of Fourier expansion. In this appendix we show that
I-e-
the
By
average
of
value
(Al.3)
of
it is
DIRECTION
EASY
N°11
OF
NON
SURFACES
UNIFORM
2021
solve the
problem by means of general formula is very difficult. Let us consider the
same
has to satisfy the boundary
condition (10).
problem of the section 3. The harmonic
function
By indicating with f(z) the actual value of R(z, 0) and by using equation (Al.I) and equation
(10) we obtain for f(z) the integral equation
to
-) /~
/~~j~~di
)lf(z)
+
8(z)1
(A15)
°
=
because the kernel has a strong singularity. This equation is not
which is a singular equation,
integrable in the ordinary sense [29]. The appropriate solution can be expressed in term of
constructed
Cauchy's principal value. A complete theory ofsingular integral equations has been
for our aim,
Fourier
information
but it is difficult
However
expansion gives all the
to apply.
about
the physical mechanism, and it is simpler than to solve the singular integral equation.
used this technique in the text.
For this
reason
we
relation (14) let us
consider the periodic function
In order to show
Afor0<z<a,
g(Z)
"
z
<
~.
(Al.6)
[dncos(nqz)
+
ensin(nqz)],
(Al.7)
B for
By expanding g(z) in
Fourier
<
a
series
g(z)
£
+
=
obtain
we
~°
The
value
~
Aa +
B(~
a)
~
of
sin(nqa)
+
cos(nqa)
I
'
nq
over
~~
~~~
~~
~
g~(z) integrated
A~a
~~
~~~
~
'
~ ~'~~
nq
~ is
B~(~
a)
((
~
~ +
=
£
(d(
+
e() ~.
(Al.9)
~
By substituting
Appendix
(Al.8)
into
(Al.9)
we
obtain
easily (14).
2.
appendix the calculations reported in section 4, relevant to a sample of finite
characterized
by the
generalized. More precisely, we consider now a sample
given by equation (I) on a surface, and by a strong anchoring imposing a tilt angle
In this
will
other
be
one.
The
figure 4. The
the boundary
the
other
one.
thickness
of
the
sample will be
by
denoted
R(z, y) minimising the total
condition (10) on the lower surface,
function
energy
and
the
e.
reference
The
is still
harmonic
boundary
frame
and
condition
thickness
axis
easy
Rb on the
is shown in
it has
R(z,e)
satisfy
to
=
Rb
on
JOURNAL
2022
PHYSIQUE
DE
N°11
II
y
e
a
fl
0
Fig.
Finite
4.
liquid crystal
periodic
distribution
angle is fib-
Routine
I
a
of
thickness
are
limited
e
and
present
by
surfaces
solid
a
another
at
one
y
=
e
on
at
y
0
=
which
the
which
on
anchoring
give:
calculations
R(z, y)
sample of
directions
easy
x
~°
=
~
f
Cy
+
[Dncos(nqz)
+
+
Ensin(nqz)] sh[nq(e
(A2.I)
y)]
n=1
where
~
"
=
the
total
energy
nq
~~
~
that
If
we
the
last
whose
consider,
in F is
term
role is
as
done
in
independent
can
derive
~
~"~
section
~~~~
of Sb.
Hence
~j~)~
it
can
analysis.
z-uniform sample, the
~(j
be
~~~'~~
considered
as
4,
a
jk~~[
total
energy
is
found
~~~
expressions (A2.3) and (A2.4) we deduce that 8~ plays the role of the
conclusion
by applying the criterion proposed in section 4,
same
the
additive
an
in the
=
We
nq~flq~n
~
)
By comparing
j~~~"
~
fundamental
not
~~~)~~~
In qL
jA2_2)
~
constant
+ 2CL
is:
~~~~L~~~
Note
~~
~jj)~
~
that
fl(~
~
~~~
En
follows
+
2(a
Dn
It
aa
~
~
be:
to
(A2.4)
easy
axis.
connected
N°11
(R~). By
to
of
means
EASY
DIRECTION
(A2.I)
and
~~
(R~)
(A2.2)
~~/
=
+
NON
we
obtain:
~~~~
~
UNIFORM
OF
~~
~)
+
+
e
SURFACES
2023
~
e~ + Ri (q, a,
(A2.5)
L)
e
where
~~
The
~~
i
1~ ~l
~~
~
quantity
same
for
~~ll))1)t5Sii~
~
z-uniform
sample
~~
(S()
~)~
=
(A2.6)
series
The
interference
+
eRb(3L
converges
Ri
term
+
~~~
~
~)~
~ ~~~~
+
e)8e
~
~~'
~~~~~
is:
(~)
=
e
By comparing (A2.5) with (A2.7), the equivalent
e~8(
i
1~ ~l
~
easy
(A2.7)
e~
+
e
axis is
given by:
eR~Rb(3L + e) + IL + e)Ri
e~R(
rapidly. By approximating
it
by
means
of its
"
(A2.8)
0.
first
addendum,
the
becomes:
~~
~
e
p)2
(~
~~~'~~
2~r3
sample considered as uniform this term can be important or not according to the exa
the
perimental technique used to appreciate this fact. In the case of optical
measurements
sample will be considered as uniform if ~ is smaller than the wave-length of the visible light,
Consequently for samples studied by optical means, the
interference
I-e0.5 ~m.
term is
could be important for samples
negligible and Se
Be. On the contrary, the interference
term
studied e-g- by capacitance
in which the uniformity of the sample is not tested.
measurements
independently
interference
Of
if ~le is small the importance of the
term is also small,
course
of the technique employed.
conclusions
hold, of course, for the antisymmetric and symmetric cases
considered
The
same
For
-~
=
in
section
4.
References
[ii
Cryst. Liq. Cryst. 179 (1990) 217.
Physics of Liquid Crystalline Materials, I-C- Khoo and F. Simoni
1982) p.301.
H., &Iol. Cryst. Liq. Cryst. io8 (1988) 317.
Jerome B., Rep. Frog. Pliys. 54 (1991) 391.
Nehring J.,1<met2 A-R-, Shefer T-J-, J. Appl. Pliys. 47 (1976) 850.
Yang K-H-, Rosenblatt C., Appl. Pliys. Lent. 43 (1983) 62.
Yokoyama H., Van Sprang H-A-, J. Appl. Pliys. 57 (1985) 4520.
Bernasconi
J., Strassler S., Zeller H-R-, Pliys. Rev. A 22 (1980) 276.
Parson J-D-, J. Pllys.
France 37 (1976) 1187.
S.,
S.,
Breach,
[3] Yokoyama
Faetti
Mol.
[2] Faetti
[4]
[5]
[6]
[7]
[8]
[9]
[lo]
Sluckin
T-J-,
Texeira
P.,
to
be
published.
Eds.
(Gordon
and
JOURNAL
2024
PIIYSIQUE
DE
II
N°11
[1ii Rapini A., Papoular M.,
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
J. Pllys. Colloq.
France
30 (1969) C4-54.
P., Faetti S., Fronzoni L., J. Pllys. France
44 (1983) 1061.
Chiarelli P., Faetti S., Fronzoni L., Pliys. Lent.
ioiA (1984) 31.
Nobili M., Lazzeri C., Schirone A., Faetti S., Mol. Cryst. Liq. Cryst. 212 (1992) 97.
Rosenblatt
C., J. Pliys.
France
45 (1984) 1087.
Rosenblatt
C., Griffin A-C-, Uma Hari, Liq. Cryst. 7 (1990) 359.
Di Lisi G.,
D-W-, Pliys. Rev. Lent. 28 (1972) 1683.
Berreman
de Gennes P-G-, The Physics of Liquid Crystals (Oxford University Press, 1974).
Barbero
G., Durand G., J. Pllys. II France 1 (1991) 651.
Faetti S., Pbys. Rev. A 36 (1987) 408.
Monkade M., Boix hi.,
Durand
G., Europliys. Lent. 5 (1988) 697.
Cognard J., )[al. Cryst. Liq. Cryst. Suppl. 1 78 (1982)1.
Yokoyama H., I<obayashi S., I<amei H., J. Appl. Pllys. 56 (1984) 2645.
Jerome B., Pbys.
Rev. A 42 (1990) 6032.
The
total
unit length along the
z-axis, of a
z-uniform
sample
caracterized
by
energy
per
~~
~k~~~ +
@(z,0)
uniform
axis @e, VT E (0,1) and having.
0 is given by
easy
a
Chiarelli
=
~wu
2
(@su
z-uniform
w-r-t-
@su is
fe)~,
sample.
reached
where
Of
@su
"
course
for fsu
@u(-e)
in
this
Barbero
Illeman
Petrovski
case
~~
=
1+
(Lule)
is
wu
@u
depends
where
,
the
Lu
=
anchoring
linearly
on
energy
y.
The
of
the
minimum
2
e
considered
of
(22)
k/wu.
Cryst. Liq. Cryst. (to be published).
G., Beica T.,
Alexe-Ionescu
A-L-, Nloldovan R., Liq. Cryst. (to be published).
AI., Points, lignes, parois, (Les Edition de Physique, Les Ulis, 1977).
Moscow, 1975).
I-G-, Lectures on the Theory of Integral Equations (MIR Publishers
[26]1<omitov L., Sparavigna A., Striga22i A.,
[27]
[28]
[29]
and
=
1
Mol.

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