PHYSiCAL REVIE% A VOLUME
26,
NUMBER 3
SEPTEMBER
1982
Boundary-layer
phase
transition
in nematic
liquid
crystals
Ping Sheng
Cg
-'~".
-crate
Research Science Laboratories, Exxon Research
and
Engineering
Company,
P. O. Box
45,
Linden„Xeu
Jersey
07036
(Received 1 March
1982)
Phase-transition
properties
of
nematic
liquid crystals aligned
by
a short-range, arbitrary-
strength-substrate
potential
are examined in
the framework of
Landau
—
de
Gennes
theory.
It is shown
that
the substrate
potential,
which
can
arise from
surface
treatment
of
liquid-
crystal
display
cells,
not
only
induces a
boundary
layer
in
which
the order-parameter
values
can be
significantly
different
from that of the
bulk,
but also introduces
a
new
"boundary-
layer phase
transition"
which
occurs
at
temperatures
higher
than the
bulk-transition
tem-
perature.
This novel transition
is found to
take
place only
in
a
limited
range
of
substrate
potential strength.
For
4-pentyl-4-cyanobiphenyl
(PCS),
the
limiting
values of this
range
are
computed
to be
-0.
075
and
-0.
15
erg/cm
.
Calculations are
performed
for
both the
semi-infinite-sample
case
and
the
finite-thickness-sample
case. Various
phase
diagrams
are
presented
to
show the
effects
of
sample
thickness
and substrate
potential
on the
bulk
as
well
as the
boundary-layer phase-transition
temperatures.
The
paper
concludes with
a
dis-
cussion of
experimental possibilities.
I. INTRODUCTION
Substrate
alignment
of nematic
liquid
crystals
is
a
widely
used technique to
produce
uniform
or
twisted director
configurations
in
liquid-crystal
display
cells.
In
an
earlier
work,
'
it was
theoretical-
ly
predicted
that
if the
alignment
of
the nematic
liquid
crystals
were
anchored
at
the
nematic-
substrate interface,
then
there
is
a
"boundary layer",
extending
about
1000
A from the
substrate,
within
which
the
degree
of orientational
ordering
of
the
nematic
liquid
crystal
could be
significantly
dif-
ferent
from
that
of
the
bulk. It was further shown
that
if
the
liquid-crystal
cell were
comparable
to or
thinner
than
the
boundary-layer thickness,
then the
first-order
nematic-isotropic
phase
transition could
disappear
altogether.
In the
subsequent
work
by
Miyano,
the
existence of the
boundary layer
was
ex-
perimentally
verified,
although
the
difficulties in
preparing
nematic cells thinner
than
the
boundary
layer
have
prevented the
confirmation
of phase-
transition
modification
effects.
However,
optical
measurements
of the
boundary
layer
revealed
some
interesting
details. For
example,
the
experimental
data
are
consistent with
an order
parameter
value of
S
=
—
(P2(cos8)
)
=0.3
at the substrate-nematic
interface,
.
where
I'z
denotes
the
second
Legendre
polynomial,
9
is
the
angle
be-
tween
the
long
axis of
a
molecule and
the
director,
and
(
)
means
spatial
averaging.
Since the
inter-
face alignment
of
liquid
crystals
is due to
the
in-
teraction
between liquid-crystal
molecules and the
substrate,
the
relatively
small
value
of S
implies
that
the
strength
of the
substrate
potential
is much
weaker
than
what a
rigid
anchoring
condition
would require.
It follows
that the
previous
calcula-
tion based
on
the anchoring
condition is
no
longer
adequate
for
a
realistic description
of the experi-
mental
system.
It is
the
purpose
of
this note
to
ex-
amine the consequences
of a
short-range,
arbitrary-
strength-substrate potential.
The results of a
Landau
—
de Gennes
theory
calculation indicate that
while the
qualitative
behavior of the
boundary
layer
remains
the same
as
in
the
previous work,
there
is,
surprisingly,
a new
"boundary-layer
phase
transi-
tion"
that
occurs at a temperature
separate
from the
bulk
transition
temperature
Tz.
This
boundary-
layer
transition is
found
to
take
place only
in
a
lim-
ited
range
of the substrate
potential
strength
6,
and
can
be
observed either in
bulk
(semi-infinite)
sam-
ples
or
in
finite-thickness
samples
down to
a
thick-
0
ness of
—
1000
A
(at
which
point
the boundary-
layer
transition
merges
with the
bulk transition). In
what ensues,
the
nematic-substrate interaction and
the
nature of the
boundary-layer
phase
transition in
a
semi-infinite
sample
are examined
in Sec.
II.
The
phase-transition
properties
in
a
finite-thickness
sample
are
calculated
in Sec.
III. In Secs.
IV
and
V,
we
present
the
relevant
phase
diagrams
and
con-
1610
1982
The
American
Physical
Society
BOUNDARY-LAYER
PHASE TRANSITION
IN NEMATIC
LIQUID.
..
1611
elude with a brief discussion of
experimental
possi-
bilities.
II.
SEMI-INFINITE
SAMPLE
PROBLEM
Consider
a
sample
of
nematic
liquid
crystal
bounded
on
one side
by
a
substrate.
The
solid-
liquid-crystal
interface
is defined
as
z =0, and the
sample
is
assumed
to
be
uniform
in
the
x
and
y
directions. The
substrate is treated so
that the
nematic
liquid-crystal
molecules in
its immediate
vicinity experience
an uniaxial
aligning
potential
along
some fixed
spatial
direction
n. The
potential
felt
by
each
molecule can be
expressed
in
general
as
v(8,
z
)
=
G5—
(z)
[
P2
(cos8)
+
bP4
(cos8)
+cPb(cos8)
.
],
where 0
is
the
angle
between the
long
axis of the
molecule,
n,
z is
the normal distance
from
the
sub-
strate,
G is a constant
denoting
the
strength
of the
potential,
Pz„denotes
even-order
Legendre polyno-
mials,
and
b,
c,
etc.
,
are the series
expansion
coeffi-
cients for
the
angular
part
of
the
potential.
In
Eq.
(l),
we
have assumed
the
potential
to be
short
range
in
nature,
as evident
from the
delta
function
5(z).
If,
in
addition,
v(8,z)
is
truncated
to the
leading
term
of
the series
and
averaged
over
the
many
mol-
ecules
within
a
small
spatial
region,
then the
result-
ing
form
of the
macroscopic potential
is
given
by
V=
(
v(8,
z)
)
=
—
65(z)
(P
(cos8)
)
condition
of
minimum
free
energy.
The
minimiza-
tion
procedure
involves
two
steps:
First,
So
is held
fixed
and the
integral
in
Eq. (5)
is minimized
varia-
tionally
with
respect
to
S(z).
The
resulting
expres-
sion
for 4
is
then minimized
with
respect
to
Sp.
Implementation
of
the
first
step
results
in
the
Euler
equation
d
S
I''(S)
=2L
dz2
which can
be
integrated
once
to
yield
'2
dS
'
dz
=
f(S)+K
.
The
constant K is
determined
by
the
condition that
at
z
—
+no,
the
bulk
liquid crystal
is uniform
and,
therefore,
ds
dz
Z~
00
which
directly
implies
r
'2
2
dS
dz
=F(S)
F(Sb
)
~—
p
—
—
F(Sb)D
AaTg
Here
gp=(L/aT»)' is
the
correlation
length,
F(S)=f(S)/aT»,
and
Sb
is the bulk value
of
the
order
parameter. Substitution
of
Eq. (8)
into
Eq.
(5)
yields
=
—
65(z)S
.
(2)
So
+gp[2
I
QF(S)
F(Sb)dS
g—
Sp],
—.
To
study
the
thermodynamic
consequences
of
such
a
substrate
potential,
we start with the
Landau
—
de Gennes
free-energy
density
P=f(S)+L
—
—
5(z)S,
dS
G
(3)
dz
I
f(S)=a(T
T')S
+BS
+C—
S
(4)
where T is
the
temperature,
a,
T*,
8, C,
and L
are
material
parameters
which
can
be determined from
thermodynamic
and
fluctuation
measurements,
and
A
is
the
area
of the
planar
substrate. Given
P,
the
total
free
energy
4
is obtained
directly
by
integra-
tion over
z:
(9)
where D
is the
sample
thickness
(D/gp~
00
in
the
present
case),
and
g:
6/AgpaT»
is
—
the
dimension-
less
parameter
characterizing the
substrate
poten-
tial.
In
Eq. (9),
the
expression
for 4
clearly
consists
of
two
parts.
The first
term,
the bulk
free-energy
part,
is
proportional
to
D.
The
second
term,
the
boundary-layer
part
4qq,
can
be
picked
out
by
its
proportionality to
gp.
.
p
—
—
2
I
QF(S)
F(Sb)dS
—
gSp
.
—
AgpaT
b
S
+L
oo
dS
A dz
dz
—
—
Sp
(5)
G
(l0)
The
equilibrium value
of
So
is
determined
by
the
condition
d
@zan/dSO
—
—
0,
or
where
Sp
denotes
the value
of
S at z
=0.
To
deter-
mine
the equilibrium
form
of
S(z),
we
employ
the
F(Sp)=F(Sb)+
4
(Ila)
pINQ
SHFNG
]612
the
stypu]atjo
ion
that
js found
(11b)
S
)
'
F(S
jnj
mum
the
absolute m
re resents
t e
'
le
roots,
the
where min
p
.
(11)
has
multip
1
case
Eq
.
~
the
lowest
~
that one
wh'c
g'
S s
found,
correct
So
'
by
Eq
(10)
Qnc
o
of
BL
be obtalne
as
express
'
ed
through
the &n
the
S(~)
P
of
Eq.
(8),
«
s,
dS
)
s(,~
QF(S)
F(—
go
ed
result of
So
a
In
ig.
~
. the
calculate
e
for
nction
o e f e
p
a
-p
1-4'-cyanobiph
y
3
8=
—
0.
3
J/cm,
cm were use .ed.
It
is
se
h blkt
However,
a
perature
at
a
temperatu
hj
her
of
So
occurs
.
a ears
and
th
transit&on
o
o
t ansition
disapp
T At
g
~
g"
continuous
u
than
z
f
S
becomes
a
co
'tion
a
riatjon
o
f
th
e trans&
the vari
e
The nature
0
rves
closer
f
temperature
&~&g,
deser e
d
in
the
range
go&
behavior
of
the
g&g~
.
2
we show
o
't
is seen
tjn
In
~g
At T
=T~,
scru
&ny
f
0
004.
.
t uous
ayer
or
—
'
.
a djscon
oundary
d
S experience
ding
that b th
So
h
t o
curves
co
p
an
b
correspon
umP
asas
indjcated
by
t
the
transjtjon.
'ust
before and
3
S is
understan-
toS(~
j"s
of
So
and
b
by
~
l
eous transytjon
k
d
to the bulk
Sb
simu taneo
islin
e
o
2
in
able since
e
vain 0
0
g(dS/dz)
term
s ~represented
y
hmit
of
weak
the
elastic fore
nd
in
the
b
-energy density
.
S is
induced
the
«e e"
.
1
h
transitio»"
otential t
e
r,
however,
e
bulk
trans'tio .
er
behavior
is
p
llustrated
in
g
the bulk
transition
Alth
)
gh Sb
has
49 K
on the othe
stays
fixe
exhibits
a discon
d
d
the
~a]ue
of
So
"
'
d
(
0)
as
illus r
'le
Sb
stays
unc
g
first-order
tran-
sition whi
b
in this
case,
the
ll the
3(b).
Since in
layer,
we ca
in
F&g
'
nl
the
bounday
y
-layer
sltlon
'
Fj
involves on
y
~
3(b)
the bounda
y
han
e
shown in
g
this
occurrence
1S
phase
c
ang
. The
physics
o t rs
p
as
transjtfo
=0.
012
0.
0056=go
(g &g
=
0.
36
0.
34
0.
32
0.
30
0.
28
0.
26
0.
24
0.
22
0.
20
0.
18
0.
16
0.
14
0.
12
0.
10
g
0.
08
—
S
=
0.
20
0.
06
~
o
=
T'
+0.
143K
0.
04
T
g=0
0.
0056
0.
28
0.
32 0.
36
0.
20
0.
24
0.
12
0.
16
0.
02-
0.
04
0.
08
-0.
04 0
-T
'
(K)
of
temperature.
-0.
12
-0.
08
K
d s
a function
o
the
tic interfa
p
d
rameter value
a
e.
Values
o t e
c
PCB
order-parame
G
1.
Calculated
'
1
is
labeled
besi e
e
t ate
potentia
iMagnitude
offthesu
sr
figure.
BOUNDARY-LAYER
PHASE
TRANSITION
IN NEMATIC
LIQUID.
. .
1613
0.
40
0.
35—
0.
30
g
=
0.
004
T=
T'
K
PCB
0.
25—
0.
20—
0.
15-
0.
10-
0.
05-
20 40
60
I
80
I
100
I
120
140
Z/P
FIG. 2.
Variation
of
the
order
parameter
S
as
a
function
of
distance
z/go
away
from the substrate.
Upper
curve shows
S(z)
just
before the bulk
transition at
T
=
Tz,
and the
lower curve shows
S(z)just
after
the transition. Value of the
sub-
strate
potential
g
=0.
004
is noted in the
figure.
actually fairly simple.
The
layer
of nematic
mole-
cules
at
the liquid-crystal-substrate interface experi-
ences
two
forces: the elastic
force,
which connects
the surface
molecules
with
the
bulk,
and the
sub-
strate
aligning
force. When the substrate
potential
is
sufficiently
strong,
i.
e.
,
g
&gp
(but
g
&g,
),
the
in-
crease in the elastic
part
of
the
free
energy
caused
by
the
lowering
of
the bulk
order-parameter
value
at
Tx
cannot overcome
the surface
aligning
poten-
tial.
Therefore,
S0
stays
unchanged. However,
as
temperature
increases
beyond
T&,
there is
a
point
at
which
a trade
off
between the elastic free
energy
and the surface
potential
energy
becomes
advanta-
geous
and
a boundary-layer
transition occurs in
which the
gain
in
surface
potential
energy
(resulting
from
the decrease of
So)
is offset
by
the
lowering
of
the
elastic
free
energy
(and
vice versa when
T
is
lowered
through
the
transition
temperature).
The existence
of
an
upper
limiting
value of
g
for
the
boundary-layer
phase
transition can also
be
made
plausible.
From the
form
of
the substrate po-
tential
V=
—
G5(z)S,
it
is
clear
that
we
can
regard
the interface
alignment
as due to
the
application
of
a
localized
magnetic
field
H,
with
6
~H
. Since
it
is
well
known
that the
nematic-isotropic transition
.
has
a
critical
point
under a
strong
magnetic field,
it
follows
by
analogy
that
the
first-order
transition
in
So
should also
possess
a
critical
point
(Tz,
g,
)
at
which the
first-order
boundary-layer
phase
transi-
tion becomes
second
order. For
PCB,
the value of
g0
—
—
0.0056 and
g,
=0.
012 translate
into a substrate
potential
G/Ago
of
roughly
0.
15
and 0.3
J/cm,
respectively.
Since
pc=
"i/l.
/aTrr—
-5
A,
this
means
Gc/A
=0.
075
erg/cm
G,
/A
=0.15
erg/cm
III.
FINITE-
THICKNESS SAMPLE
PROBLEM
F(Sc)
Ii(Ss
)
=—
4
(13a)
Consider
a
sample
of
nematic
liquid crystal
of
uniform thickness 2D sandwiched
between two
identically
treated substrates situated
at
z=0
and
z=2D.
Instead of
the
condition
dS/dz(z=
cc
)=0
for
the
semi-infinite
sample,
in
this
case we
have
dS/dz(z
=D)
=0
due
to the
symmetry
of
the
prob-
lem.
Therefore,
if
we let
Sb
denote
the
order-
parameter
value at
middle
of the
sample,
i.
e.
,
z
=D,
then
we
are led
to
exactly
Eq.
(9),
where
4
now
stands
for
half of the total free
energy
of
the
sam-
ple.
However,
due to
the
fact
that D is
finite,
Sb
and
S0
can no
longer
be
decoupled
as
in the previ-
ous case.
Therefore, we
have
to
solve the
coupled
equations
i614
PCB
pgqo
SHENG
so
I"'(Sb)
I
—
D
Js,
~p(S)
F-(Sb)
=0.
26
0.35
9
=
0.008
(13b)
0.
30
0.
25-
0.
20-
Vl
0.
15-
0,
10-
0.
05-
0.
35-
0.
30-
9
=
0.008
T
= T'
+
0.
049K
K
0.25
0.20
0.
15
0.
10
0.
05
(b)
20 40
60
80 100
120 140
Z/(
Variation of
the order
parameter
S as
a
z)
ust before and
just
a
ter
e
f
he substrate
potential
=T
Magnitude
o
t
e su
X.
b tth
1 of
d
in the
figure.
n
r has a discontinuous
jump,
u
Variation
o as a
y
0
.
Substrate
potentia
is
e
T
=
T~+0.
049K.
u
rres
ond
to S
z
jus
t
before and
just
th'
th
'
n has occurred. Since in is
after
a transition
a
d
la
er,
it is labeledtransition invo ves
the
boundary-layer
phase
transition.
irs of solutioils
(
o~
b
~
S
),
en there
are
mu
p
is
always
that one
w
ic
g
ives the
the
correct
pair
is
a
(9).
It shouldas
expressed
by Eq
s
lowest
~'aiu
0 E
(l
3)
reduces
to
be
noted
that when
fo/D
l l for
the
semi-infinite
cas ~
Eq
(
f
th measured
parameter
Through
the use
o t
e m
1
ted
the variations ofalues
of PC~
we
have
calcu a
f
T,
and D.
ForS
as
a function
o,
g,
a
S~
and
Fi
.
4
that the transition
8
it
is
seen
from
Fig.
'th
decreasing
D.
As
temperature
b
of
S
increases wi
'
s
=300, it shows
two
r
t
transitions:
the
to
So,
at
D/go
,
—
—t
r
bo d
ryl
y
p
er hase transition
p
us
ui
the transition of
Sb.
e w
n which causes iscon
inu'
'
d
he
"bulk
transition,
in
a
D/g
. Si
f
d
characterization
as
-la
er transition
tempera
D
thwith
respect
to
p
tern
erature means
t a
p
=160,
below
w
ic
/p
——
disa
ears into
t
e
u
Q
11
fi
-o
d
Mi-
As D decreases
even
further,
t e
irs
-o
t
lly
turn into
S
and
S~
eventua
y
tions
in
both S t
y
—
0 012 lotted
r transitions
at
D/
0
——
ons with
g=
.similar calculatio
n contrast
to
the
semi-in
ini
in
Fig.
5. In con
has
no
abrupt
transition
a
at this value of
f
t I
thi
k
d
trate
otential,
for ini
e sa
'
nces a discon inu
h
first-order
transition in
b.
cident wit
t e
ir-
0.36
0.
32
0.28
0.
24-
0.
20-
X
0.16
0.~2
O
CC
0
0.
08-
0.
04-
I
I
I
0.40 0.44 0.48
0.52
0.
56
0.
60
0.
64
0.68
0.24
0.28
0.32
0.36
I
0.
08 0.12 0.16
0.20
-0.
12
-0.
08
-0.
04 0 0.04
TT
'
(K)
K
ure. The
(half) thickness
of
the
urve) as
a
function
of
temperature.
e
h
'
=o
'
'
h
f"
it
d
ofth
bt
t
ot ti 1
sample
is
la e e
'
bel
d
beside
each curve.
Magnitu
e
o
