PHYSICAL
REVIEW
A
VOLUME
28,
NUMBER
3
SEPTEMBER 1983
Anisotropic
steric
effects and
negative
(,
P4
)
in
nematic
liquid
crystals
Kean
Feng*
and
Chia-Wei
Woo~
Department
of
Physics, University
of
California, San
Diego,
I.a
Jolla,
California 92093
Ping Sheng
Exxon
Research
and
Engineering
Company,
Linden,
Xew
3ersey
07036
(Received 17
August
1982)
Starting
with
a
model intermolecular
potential
that includes
0, r terms to account for
anisotropic
steric
interactions,
we
carry
out
a statistical mechanical calculation
in
the
cell
approximation,
and
apply
it
specifically
to the isotropic-nematic
transition of
methoxybenzylidene
butylaniline
(MBBA).
The
potential
is determined to fit the
experimental
transition
temperature
T~~
and
the
discontinuity
in
the orientational
order
parameter
(P2
)
at transition.
(
P2 )
and
(
P4
)
are then calculated from
solving
a set
of
coupled
self-consistency
equations
for
the
orientational and
spatial
parts
of the
distri-
bution
function,
as
are
other phase-transition
properties.
There are
improvements
over model
calcu-
lations which
do
not
account for
anisotropic steric
effects,
but the improvements are
generally
less
than
significant.
The most
striking
result is that a
stable
nematic
phase
requires
(P4)
to
be
negative
at and near the transition. It
is
a
theoretical
result
qualitatively
consistent with
experimental
data
but has not been attained until now.
I.
INTRODUCTION
The
theory
of
liquid crystals
has
been
developing
in two
major
directions.
One,
the Landau
—
de Gennes
theory,
'
employs
a
phenomenological
approach.
The Helmholtz
free
energy
is
expanded
in
powers
of the order
parameter
and its
gradient, requiring
in
the
process
five
or more
ma-
terial
parameters
to
be
determined
by
experimental data.
The
theory
contains
a
simple
temperature
dependence
and
no
volume
dependence.
In
the
treatment
of
pretransitional
effects,
terms of
higher
order
than
quadratic
are
usually
omitted.
Such a
theory
is
physically
appealing,
mathematically convenient, and
qualitatively powerful.
Quantitatively,
however,
it encounters
difficulties
that
re-
sult
from
oversimplification.
The
other
approach
is the
"molecular"
theory,
defined
as orie that
begins
with
a
model,
be it
in the form
of rods
or
interparticle potentials.
Conventional statistical
methods
are
applied
to
obtain
distribution
functions,
free
energies,
thermodynamic
and phase-transition
properties,
and
dynamical correlations.
The
major
difficulty
in this
approach
is that
model construction and statistical
calcu-
lation
need
be carried out
simultaneously;
both the
mathematical
derivation
and the numerical
computation
can be
overwhelmingly demanding.
As
a
consequence,
too
many
simplifying
approximations
are made
—
in
the choice
of
models,
in
statistical mechanical
approximation
schemes, and in the evaluation of
thermodynamic quanti-
ties.
We shall
come back
to each
of these
issues
presently,
in
Sec. II.
In
our
group
we have devoted a
large
part
of our effects
on
liquid
crystals
to
the molecular theoretical
approach.
Using
relatively
simple
models
first
suggested
by
Kobayashi
and
McMillan
—
models without
anisotropic
steric
effects
—
we
have obtained
reasonably
good,
semi-
quantitative agreement
with
experiment
in
several
cases,
as have other authors.
But there remain
puzzling
discrepancies,
some
of
which have resulted in
a
certain
de-
gree
of unease toward the
usefulness of molecular
theories.
We
cite
here the
following
as
examples.
(i)
The calculated
temperature
dependence
of the
orien-
tational order
parameter
oz
=
—
(Pz )
(and
o4=
(P4))
is too
weak.
(ii)
oq
itself,
calculated
at
or
near the
isotropic-nematic
transition
temperature
Tl&,
is
too
large.
In
particular,
cal-
culations for
stable
nematic
phases always yield
positive
cr4,
while
experiment
on certain
nematic substances
[e.
g.
,
methoxybenzylidene butylaniline
(MBBA)]
gives
rise
to
negative
o4
near
TI&.
(iii)
The
derivative
d
1nTt~(p)/d
lnp,
or
d
lnT(p)/d
lnp
at constant
o.
2,
is
always
too
small,
by
a
factor
of 3
to
4.
(iv)
The calculated latent
heat
at
I-X
transition
is
usual-
ly
too
large,
by
a
factor
of
about 2
to
3.
II.
MODEL
AND
FORMALISM
A.
Model
We use a
potential
model.
The interaction
between
a
pair
of
cylindrically symmetrical nonchiral
molecules is
described as
u(i
j)=v(rJ,
Q;.
QJ,
Q;
rJ,
QJ
rj)
=
vo(rj)+u2(rz)P2(Q;
QJ)+v~(rj)P4(Q;
QJ)
+tv2(r
J
)[P2(Q;.
r
J
)+P2(QJ.
rtj)],
(1)
where
r;
and
Q;
denote the
position
of
the center of mass
of the ith molecule and
its
orientation,
respectively.
The
first
three terms
represent
the
popular
McMillan form
of
the
potential,
'
and the
last
term,
which
couples spatial
1983
The American
Physical Society
1588
KEAN
FENG,
CHIA-WEI
WOO,
AND
PING
SHENG 28
and orientational
variables,
stands
for
the
leading
anisotro-
pic
steric
correction.
In
an
earlier
paper
we
inspected
the
effect of a
weak
anisotropy
by
treating
wq(r)
as a
pertur-
bation. The
results obtained were not
particularly
infor-
mative. From
fitting potential profiles
obtained from
Eq.
(1)
to those obtained
with
rod
models,
we
learned that
rv2(r)
should
be
rather
large compared
to
v2(r)
and
u4(r)
even
for
rods which
are
short and
relatively
soft.
Thus,
there is
no
way
to avoid
treating
rv2(r)
on an
equal
basis
as
u2(r)
and
v4(r)
in a
full-fledged
statistical
calculation.
In
this work
we
shall
parametrize
the
potentials
as
fol-
lows:
ut(r)
=atexp(
Pr
—
),
l =0,
2,
4
rv2(r)=bzexp(
—
P
r
)
.
The
potential
parameters
are
ao, az, a4, b2,
and
/3.
B.
Statistical
formalism
Conventional statistical treatment of
a
classical
system
of N
particles
in
a volume V
(p—
:
N/V),
at temperature
T,
begins
with
the
Boltzmann distribution
P~(1,2,
. . .
,
N)
F=FO+
f
g
v(i
j)
d
1
dN
l
+J
P~(1,
2,
. .
.
,
N)
+kT
P~(1,2,
. . .
,
N)
Xln
d1
.
dS.
I'0 is
the ideal-gas
term. The
potential
energy,
i.e.
,
the
second
term,
can
be
reduced in
one
step
to an
integral
over
v(i,
j)pz(i,j).
The
last
term,
from
entropy,
however,
can-
not be reduced to a closed
form
involving
low order
P„(1,
. . .
,
n)
It
c. an
be
expanded
in a
cluster
series,
but
one
that
does not
converge
rapidly
when the
system
is
at
liquid
densities.
We
introduced in Ref.
8
a variational
approach
which
turns out
to
be most convenient for
treating
phase
transi-
tions associated with
nonspatial
degrees
of freedom. It
be-
gins
with
considering
the undetermined
X-particle
distri-
bution function
P~(1,2,
. . .
,
N)
and
a free-energy
func-
tional
NI
P~(1,2,
. . .
,
N)
=
'
exp
Z,
kT
where
Z
denotes the
partition
function
Z
=
f
exp
—
$
'
d
1
d2
. .
dN,
u(i
j)
kT
(3)
(4)
P~(1,
2,
.
. .
,
N)
w
IP~I=FO+
f
gu(i
j)
dl
.
dN
E
(J
P~(1,2,
. .
.
,
N)
+kT
NI
di—
:
dr;d
0;,
and
N.
is an
(arbitrary)
normalization
factor.
The
one-particle distribution function
Po&(1)=)of(rI,
Q&)=
—
f
exp
—
g
'
d2
.
dN,
l
(j
P~(1,2,
. . .
,
N)
gin dl
.
.
dN
.
Xf
Minimization of
the latter under
the constraint
f
P~(1,
2,
. .
.
,
N)d
l.
.
.
dN =N!,
(8)
is
also
ihe
density
function,
which contains information
on
macroscopic long-range
order
in the
system.
PI(1),
or
f
(r,Q),
is
the
weighting
function
used
for
evaluating the
order
parameter
(Pt
).
The two-particle distribution
func-
tion
yields
immediately the
Boltzmann
distribution,
Eq.
(3).
This
permits
us
to
"model"
P&(1,2,
.
. .
,
N)
in
various
stages
of
approximation.
For
example,
writing
P~(1,
2,
. . .
,
N)
as
N.
+,
&
Q(Q;),
and then
minimizing
~
with
respect
to
Q(Q),
leads
immediately
to
the
Maier-
Saupe
mean-field
approximation.
Writing
Ptv(1,
2,
. . .
,
N)
as
Pq(1,
2)
—
=
P
i
(1)p
i
(2)g
(1,
2)
gives
a measure of the short-range
pairwise
correlation
be-
tween
the molecules.
PI(1)
and
P2(1,
2)
can,
at
least
in
principle,
be calculated
by
means of cluster-expansion
pro-
cedures,
integral
equations,
molecular
dynamics,
or Monte
Carlo
methods.
The
Helmholtz free
energy
of
a
system
can
be
expressed
N!
+
Q(fl;)
C(rI,
r2,
.
. .
,
r~),
i=1
and then
minimizing ~
with
respect
to
Q
and
N,
gives
rise
to
a
formalism
in
which
the
orienting forces are
modified
by
pair
distribution
functions, which
are in turn
solved
for
orientationally
averaged
intermolecular
forces.
Such an
approximate
treatment
of
space-orientation
cou-
pling
was named
the
"orientationally
averaged
pair
corre-
lations"
theory (OAPC).
'
The
optimized
Ptv(1,
2,
.
. .
,
N),
when
substituted back
into
WIP&I,
gives
us
a
Helmholtz free
energy
which
is
exact
for
that
leuel
of
approximation,
without
the
need
of
a
seriously
questionable cluster-expansion
procedure.
28
ANISOTROPIC STERIC EFFECTS AND
NECrATIVE
(,P4)
IN. . .
1589
C. Cell
approximation
Spatial
correlations
play
an
important
role in the
theory
of high-density
fluids. For
liquid crystals,
one
way
to take
them
into
account
is
to
borrow
from
the
theory
of
classi-
cal
liquids,
as
in Refs.
5, 6,
and
8. The other is
to
use a
cell
approximation:
Use
a single-particle
approximation
and restrict
the motion of
molecules
to
individual cells.
The
fluidity
manifests itself
through
large
excursions
of
molecules from the centers
of their cells. In this
paper
we
I
N
P~(1,2,
. . .
,
N)=N!
ff
[g(Qr.
)P(r;
—
R;)],
(10)
where
IR;
I
represent
lattice
points
about
which
the
cells
are
centered.
Using
Eq.
(10)
in
Eq. (8),
we
find
adopt
the latter method
for
the
sake of
computational
economy.
The model for
Pz(1,
2,
. . .
,
N)
is then
given
by
~
Ig
4'I=P'0+
g
f
u(i,
j}g(Q;}g«)}gp(rk
—
Rk)dQ;dQ
dr,
.
dr
l
(J
k
+kT
g
f
g(Q;
}lng(Q;)dQ;+kT
f
+
p(r
—
R
)ln
+
p(rk
—
Rk)dr,
.
d
rz
=F0+
—,
g
f
u(i,
j
)p(r;
—
R;
)p(
ri
—
RJ
)d
r;d
rj
l
(J
+kTQ
f
g(Q;)lng(Q;)dQ;+kTQ
f
p(rk
—
Rk}in/(rk
—
Rk)drk,
with
u(i
j
)
=
t
u
(i
j)
g
(Q;
)g(QJ
)d
Q,
;d
Qi,
(12)
an
orientationally
averaged
pairwise potential.
Before
embarking
upon
a variational
treatment of
M
Ig,gj,
let us
require
that
g(Q)
be
dependent
only
on
Q.
n,
where
n denotes
the
director.
This
requirement
is
consistent
with the uniaxial
property
vf
nematics. On
the
other
hand,
it is not
obvious that the
solutions of the
vari-
ational
equations
are
necessarily uniaxial. We
shall return
to this
point
later and in
the
Appendix.
We
now
vary
~
[g,
PI
with
respect
to
g(Q)
and
P(
r
—
R)
under
the nor]nalization
constraints
f
g(Q)dQ=1
orientations
of molecules
k
and
1,
then
over all
possible
positions
of
molecule
k
in its
cell,
and
finally
summed
over all cells
except
the one
occupied
by
molecule
1.
The
Euler-Lagrange
equation
for
g(Q}
under the
as-
sumption
that
g
(Q)
=g (Q
n
)
yields
1 1
g
(Q)
=
exp
—
[
(G2cr2+K2)P2(Q.
n
)
Zg
kT
+G4o4P4(Q
n
)]
where
o]
——
f
g(Q)P](Q.
n)dQ,
l
=2,
4
1
Zg
—
—
exP
—
620.
2+K2
P2
01
n
kT
r
—
R
dr=1
.
(14)
—
V( r
—
R
]/kT
Zp
(15)
The
Euler-Lagrange
equation
for
]}]
(
r
—
R)
gives
rise
to
the
following solution:
+G4o4P4(Q]
n
)]
dQ],
G]=
g
yi(Rk,
R]),
l =0,
2,
4
k~1
(20)
(21)
with
y(r]
R])=
y
f
u(k,
1)y(rk
—
Rk)drk
k~1
+2
Q
+2(Rk Rl)
k~1
with
(22)
—
V(
r
—
R
)/kT
~~
Z~
—
—
e
dr .
V(r]
—
R])
is
clearly
seen
as a
mean
field
experienced
by
molecule
1,
obtained
by
first
averaging
u(k,
1)
over all
y](Rk,
R])=
f
ui(.
k])]}]](rk
—
Rk)P(r]
—
R])drkdr],
(23)
1590
KHAN
FENG,
CHIA-WEI
WOO,
AND
PING
SHEN' 28
l~z(Rk Rl)
I
~2(ykl)4(rk
Rk)
)&(t(r,
R—
,
)P2(r„,
.
n)drkdr,
.
The
Appendix
derives
these results for
a
cylindrically
symmetrical
P(
r
—
R).
D.
Reduction of
self-consistent
equations
The lattice that
we
choose to
work
with
is the
simplest
possible
which
can accommodate
a nematic
arrangement
that
accounts
for
anisotropic
steric
effects:
a
lattice
of
rectangular
parallelepiped
cells,
each of volume
dz
dy
dz
dx d,
=
1
/p.
In an
earlier
paper
we performed
a
similar
calcualtion
as
described
here
without the
presence
of
wz(r)
terms
in the potential.
In
that
case,
the
absence
I
of
anisotropic
steric effects
means, e.
g.
,
a
parallel
pair
of
molecules cannot
distinguish
a side-by-side
configuration
from
an
end-to-end
configuration. If
placed
in
cells
on
a
rectangular lattice,
the most stable
lattice would be cubic
(d„=d~=d,
)
by
symmetry
arguments.
Now that
(vz(r)
terms have
entered our
model,
this will
no
longer
be the
case. We
shall
have to consider
rectangular
lattices of
all
proportions,
d, /d„,
in order to
determine which structure
is
thermodynamically
the
most
stable.
Likewise,
instead of
employing
a
spherically
symmetri-
cal
P(
r
—
R),
as
in
Ref.
9,
we
must
now use a
cylindrically
symmetric form,
the
simplest
one
being
a
y
—
az[(x
—
X)2+(y
—
Y)2]
—
y
(z
—
Z)2
For
this
form of
P(
r
—
R)
y(ZI
~
(i3
+a
/2—
)(q„+q
)
(fl
+y
/2—
)qz
2
2
0
(i)2+az/2)(q
+q
)
—
(p
+y
/2)q
E2
——
dgx
dgydqz
C
2
3gz
2
—
—
1
M(q„,
q~,
q,
~
a,
y
),
Vx
+9'y
+9'z
(27)
where
a2d
2
/2
2
—
y
dz/2
(cosha
d„q„+cosha
d„q~)+e
'
coshy'd,
q,
—
ad
/2
—
ad /2
2
2
+
2e
"
[e
"
cosha
d„q„cosha
d„q~
M(q„,
qy,
q,
~
a,
y
)
=
e
v(i
j)
=
vo(rj
)+
vz(rj
)o2+
v4(r'z
)o4+2lvz(rz
)o2P2(yj.
n
),
'
for
use
in
Eq.
(16).
And
thus
2
2
2'
p
2 2
3'
V(r
—
R):
—
V(u
~
a,
y)=
dq dq
dq,
oo+azoz+a4o4+bzoz
—
—
1
—
00
qx+9y
+Qz
+e
'
(cosha
d„q„+cosha
d„q~)coshy
d,
q, ]
—
y'd,
'/2
2 2
—
azd„—
y
d
/2
2 2
+4e
" '
cosha
d„q„cosha
d„q~
coshy
d,
q,
+
Equations (18)
—
(20)
and
(26)
—
(28)
form
a
self-consistency
loop
for
g
(0),
given
P(r
—
R)
as
in
Eq.
(25).
Next,
from
Eqs.
(12)
and
(19),
using
the
addition
theorem,
the
Appendix yields
—
i)2(qzZ+q
+qZ)
—
az(q„+u„)2
—
aZ(q
+u )
—
y
(q,
+u
)2
Xe
~
'e
&M(q„+u„,
q~+u~,
q,
+u,
~
2a,
2y
)
.
—
V(
u
I
a,
y)/kTd~
Z
x
(31)
Substituting
Eq.
(30)
into
Eq.
(15)
and
taking
moments
with
Eq.
(25),
we find
I
Equations (30)
—
(32)
form
a self-consistency
loop
for
P(r
—
R),
in
the form
of
coupled equations
for a and
y,
given
o(
as
in
Eq.
(19).
The
two self-consistency
loops
are
intricately intertwined. We shall
show how
they
are
solved
numerically in Sec. III.
—
V(
u
j
a,
y)/kTd~
(32)
E. Helmholtz free
energy
Substitute
into
Eq.
(11)
the self-consistency solutions
for
the nematic and
isotropic
phases;
i.
e.
,
treat
Eq.
(11)
term
28 ANISOTROPIC STERIC EFFECTS
AND
NEGATIVE
(Pg)
IN.
.
.
1591
—
NkT
lnZg
—
NkT
lnZp,
t
(33)
FI
—
—
Fo
—
—
Go
—
NkT
lnZ~
—
NkT
lnZ~
.
N
I
I
2
(34)
These
expressions
will
be
used to
determine
which
phase
is
more
stable,
and,
in
turn,
the
phase-transition
temperature
at constant
density.
by
term
using
Eqs. (29),
(23),
(24), (21),
(22),
(18), (19),
(15),
and
(16),
in
that
order. The Helmholtz free
energies
reduce to
N
3N
N
2
N 2
F~
=
Fo
—
Go
—
(
G
2
cr~+
G
4
cr4)
—
2NK2
cry
2
2
a2
to negative values,
for
obvious
physical
reasons.
There
are
several
thermodynamic
variables.
p
and
T
will
be
chosen as independent
variables. The
ratio
d,
/d„
will
also
be varied
independently;
the
value that offers a
minimum Helmholtz free
energy
dictates a
thermodynam-
ically
stable
spatial
structure. Nematic
and
isotropic
phases
will,
of
course, prefer
different
spatial
structures
since
they
respond
differently
to steric effects.
In
particu-
lar,
the
optimum
value
of
d,
/d„
for the
isotropic
phase
is
necessarily unity.
For
every
given
p,
calculations need to
be carried out
for
a
range
of
T
and a
range
of
d,
/d„.
Parameters to
be
determined
by
the
self-consistency
equations,
for
every
set
of
the
potential
parameters
and
every
set of the
thermodynamic
variables,
include
62, 64,
and
K2
in
Q(Q),
and
a and
y
in
P(r
—
R).
III. CALCULATIONS FOR
MBBA
A. Outline
Much
information is available for
the nematic
liquid
crystal
MBBA. We
apply
our
present
molecular
theory
to
a
calculation
of
its
properties.
In several earlier
papers
' '
we
performed
molecular
theoretical calculations
for MBBA without
the inclusion
of
anisotropic
steric
terms
in
the
pairwise
potential.
Reference
9
gives
an
outline of the
procedure
when
a
cell
approximation
is
employed.
It
may
be
worthwhile
to
first
read Sec. IV
of
that
reference.
Let
us
sort out the
many
parameters
in
this
theory.
First of
all,
for
model
building we
have
in
Eq.
(2)
five po-
tential
parameters:
ao, a2,
a4, b2,
and
P.
We
need to
seek
a
combination
of
these
parameters
which
will
yield
the
correct phase-transition
temperature
Tz&
and order
pa-
rameter
o2
at
Tl&
at
a
given
density
p.
From
experiment,
at
p=0.
002315
A,
Trx
=317
—
318
K and
o2(Trav
)
=0.
33
—
0.
34.
Since
phase
transitions occur under
rather
stringent
conditions,
it is not
necessarily
an
easy
rnatter to
fit these data even with several
parameters
—
a
fact
that
will soon
become
appreciated.
In
our
case,
we shall
re-
strict
p
to values
close to
p'r,
ao
to positive
values,
and
B.
Actual
procedure
Ggo.
4
(35)
Kp
kT
Use them
in
Eqs.
(18),
(20),
and
(19)
to
calculate
crz
and
o
&.
Then
the
values
(
G2
/2k
T/o
2,
G—
—
4,
=
/4k
T/cr4,
K2
—
—
$2kT)
can be identified
immediately
as the set that
solves the
self-consistency
and
yields
that
particular
pair
of order
parameters.
By
tabulating
(g2,
g&,
g2)
with their
corresponding
(o2,
cr4),
one
thus
infers
self-consistent
sets
of
(G2,
G4,
K2).
Table I shows a
typical
segment
of the
table. If
one wishes
to
fix
o.
2
at,
e.
g.
,
its
experimental
value,
one chooses
the
line which
contains that
value for
cr2,
i.
e.
,
one of the lines marked
with asterisks.
Such
choices
are,
of
course,
far
from
unique.
They
permit
us
to
Actual
calculations
employ
a
procedure designed
to
minimize
computation.
We
begin
with the
recognition
that
any
set
of
the
paramters
(G2,
G4,
Kz)
would solve the
self-consistency
equation
for
Q(Q),
thus: Take at
given
T
Gp0.
2
kZ.
TABLE I.
Typical
segment
of tabulated results
for
the solution of
the
self-consistency
equation
for
g(fl).
G2o.
2
kT
—
0.90
—
0.90
—
0.
90
—
0.
90
—
0.
90
—
0.
90
64o4
kT
0.
99
0.
99
0.
99
0.
99
0.
99
0.
99
K2
'=
kr
—
1.26
—
1.
16
—
1.06
—
0.96
—
0.
86
—
0.76
0.367
0.351
0.335
0.319
0.303
0.286
O4
—
0.025
—
0.031
—
0.
042
—
0.050
—
0.057
—
0.064
G4/G2
15.954
11.
419
8.
787
7.
063
5.845
4.935
K2/G2
0.514
0.
453
0.
395
0.
340
0.289
0.242
—
1.
08
—
1.08
—
1.08
—
1.
08
—
1.
08
—
1.
08
1.78
1.
78
1.
78
1.
78
1.78
1.
78
—
1.32
—
1.
31
—
1.
30
—
1.29
—
1.28
—
1.27
0.339
0.338
0.
337
0.335
0.334
0.
333
—
0.112
—
0.112
—
0.113
—
0.114
—
0.114
—
0.115
5.017
4.968
4.
919
4.871
4.
824
4.778
0.415
0.410
0.
405
0.401
0.396
0.392
