Rep. Prog.
Phys.
55
(1992)
1241-1309.
Printed
in the UK
Phase
transitions
in
lyotropic colloidal
and
polymer
liquid
crystals
G J
Vroege
and H N W
Lekkerkerker
Van
't
Hoff
Laboratory,
University
of
Utrecht,
Padualaan
8,
3584
CH
Utrecht,
The
Netherlands
Abstract
An
overview
is
given
of
theory
and
experiments
on
liquid crystal phases which appear
in
solutions
of
elongated colloidal particles
or
stiff
polymers.
The
Onsager (1949)
virial
theory
for the
isotropic-nematic
transition
of
thin rodlike particles
is
treated
comprehensively along with extensions
to
polydisperse solutions
and
soft
interactions.
Computer simulations
of
liquid
crystal phases
in
hard particle
fluids are
summarized
and
used
to
assess
the
quality
of
statistical mechanical theories
for
stiff
particles
at
higher volume
fraction—like
the
inclusion
of
higher virial coefficients,
y-expansion,
scaled particle theory
and
density
functional
theory. Both computer simulations
and
density functional theory indicate formation
of
more
highly
ordered smectic phases.
The
range
of
experimental applicability
is
strongly widened
by the
extension
of the
virial
theory
to
wormlike chains
by
Khokhlov
and
Semenov
(1981,
1982).
Finally,
experimental results
for a
number
of
carefully
studied, charged
and
uncharged colloids
and
polymers
are
summarized
and
compared
to
theoretical results.
In
many cases
the
agreement
is
semi-quantitative.
This review
was
received
in
July
1991.
0034-4885/92/081241+69118.00
©
1992
IOP
Publishing
Ltd
1241
1242
G J
Vroege
and H N W
Lekkerkerker
Contents
Page
1.
Introduction 1243
2.
Virial
theory
of the
isotropic-nematic
transition 1244
2.1. Imperfect
gas
1244
2.2. Non-ideal solutions 1245
2.3. Virial coefficients
of
hard rods 1247
2.4.
The
isotropic-nematic transition
for
thin hard rods 1249
2.5.
Polydispersity
1255
2.6. Charged rods 1258
2.7. Attractive interaction 1262
3.
Computer simulations
and
hard-core models
for
liquid crystals
at
higher volume
fraction
1263
3.1. Introduction 1263
3.2. Computer simulations
of
hard-core models
for
liquid crystals 1264
3.3. Incorporation
of
higher
virial
coefficients into
the
Onsager
theory
of the
isotropic-nematic
transition 1266
4.
Density
functional
theory
of
liquid crystals 1272
4.1. Stability analysis 1274
4.2.
The
isotropic
direct correlation
function
1275
4.3. Density functional theory 1278
5.
Virial theory
of
partially
flexible
polymers 1285
5.1.
The
free
energy
for
semiflexible chains 1285
5.2.
The
isotropic-nematic
transition
for
semiflexible chains 1287
5.3. Alternative methods
and the
deflection length 1289
5.4. Intermediate chain lengths 1290
5.5.
Extensions
1291
6.
Comparison with experiments 1292
6.1. Rigid rodlike particles 1293
6.2. Semiflexible polymers 1295
7.
Concluding remarks 1299
Acknowledgments 1300
A A
Gaussian distribution
functions
1301
A B.
Configurational
entropy
of
wormlike chains 1302
References
1305
Phase
transitions
in fyotropic
colloidal
and
pofymer
liquid
crystals
1243
1.
Introduction
Although
liquid crystals have been known
for
more than
a
century (Reinitzer 1888)
a
real upsurge
in
interest
can be
traced
to the
last
25
years.
One
reason
for
this
is
the use of
liquid crystals
in
device applications such
as
displays. Another reason
for
an
increased study
of
liquid crystals
is the
wealth
of
different
possible phases
with
orientational order and/or partial positional order which appear beside
the
more
common
gaseous,
liquid
and
crystalline phases.
These
liquid crystal phases presented
the
opportunity
to
study
all
types
of
subtle problems
in the
theory
of
phase transitions,
in
particular critical phenomena (Pershan
1988).
In
this review
we
shall
not
treat
the low
molecular weight liquid crystals referred
to
above
but
their
high
molecular weight counterparts consisting
of
polymers
or
colloidal
particles
and of
those
only
the
liquid
crystalline phases which
are
formed
in
solutions
of
these
particles (the
so
called
fyotropic
liquid crystals).
In
parallel
with
the
history
of low
molecular weight
liquid
crystals,
tyotropic
liquid crystalline
phases were recognized
a
long
time
ago, namely
in
solutions
of
inorganic particles
(V
2
O
5
by
Zocher
(1925))
and
biological particles (tobacco mosaic virus
by
Bawden
et
al
(1936))
but
there
is an
increased interest over
the
last
decade.
Partially
this
originates
from
industrial applications like
the wet
spinning
of
ultrastrong
fibres
(e.g.
polyaramids,
for
reviews
see
Kwolek
et al
(1987)
and
Northolt
and
Sikkema
(1990)),
partially
from
biological problems like,
for
example,
the
dense packing
of DNA in
virus
heads (Earnshaw
and
Casjens
1980).
Apart
from
these applications,
the
phase
transitions
observed
for fyotropic
liquid
crystals
are of
interest
in
their
own
right
allowing
a
satisfactory
theoretical description
as a
result
of the low
concentrations
at
which these phases appear
and the
relatively simple
way in
which
the
interaction
between
the
particles
may be
described.
The
thermotropic
variety
of
liquid
crystalline
polymers (which show liquid crystal
phase transitions
by
variation
of the
temperature)
will
not be
covered
in
this review.
The
basic reason
for
this
is
that
the
vast amount
of
work which
has
been performed
since
the
discovery
of the
thermotropic main chain polymers (e.g. polyesters (Jackson
and
Kuhfuss
1976))
has
mainly
focused
on the
materials science, whereas
a
description
from
the
viewpoint
of
statistical
mechanics—as
is the
emphasis
in our
paper—has
proven
very
difficult
up to
now.
As
alluded
to
above
our
starting point
in
this review
is the
statistical mechanical
theory
of
phase transitions
in
liquid
crystals. This
field was
opened
by the
semi-
nal
work
of
Onsager
in the
1940s (Onsager 1942, 1949)
who
recognized that
the
isotropic-nematic
phase transition
for
stiff
slender particles
may be
treated within
a
virial
expansion
of the
free
energy—in
contrast
to the
gas-liquid
transition.
For
very
thin,
rigid,
hard particles
the
transition occurs
at
very
low
volume
fraction
and
the
virial expansion
may be
truncated after
the
second
virial
term,
leading
to an
exact
theory
for
infinitely
thin particles.
We
therefore describe
the
Onsager theory
in
depth
and
extension
in
section
2,
combineing
it
with
later work
within
the
second
virial
approach.
For
somewhat shorter particles
the
second virial theory
is no
longer
adequate,
so
section
3
deals
with
attempts
to go
beyond
it. In
recent
years, computer
simulations
have played
an
important role
in
this respect, which
we
also summarize
in
section
3.
Computer simulations
of
hard particles show that
it is
possible
to
obtain
more
highly
ordered
phases like
the
layered smectic phases without attractive inter-
actions between
the
particles.
Section
4
describes
the
density
functional
formalism
which
is, in
principle, able
to
include these more
highly
ordered
phases.
For the
1244
G J
Vroege
and H N W
Lekkerkerker
application
of
virial
theories
a
strong impediment
has
been
its
limitation
to rigid
par-
ticles.
This
has
changed
with
its
extension
to
semiflexible
(wormlike
chain) polymers
by
Khokhlov
and
Semenov
(1981,
1982),
which
is
treated
in
section
5.
Finally,
in
section
6 we
compare
the
theory
to
available experimental
results.
In
this review
we
have chosen
a
didactic presentation
for our
material. This forces
us to
refrain
from
presenting
all
lines
of
attack. Specifically,
we do not
discuss
the
lattice-based
theories
of
Flory
and
coworkers
(Flory
1956, Flory
and Abe
1978,
Abe
and
Flory 1978, Flory
and
Frost 1978,
Frost
and
Flory 1978, Flory 1978, Flory
and
Ronca
1979,
Airier
and
Flory 1980, Flory 1984)
or
alternative lattice
theories
like
DiMarzio
(1961).
Although
these
theories
present
an
ingenious model which
may
describe
many features
of
liquid crystalline polymers qualitatively (like e.g. behaviour
at
higher
densities,
with attractive interactions
and for
bidisperse systems), they
do
not
lead
to the
exact result
of
Onsager
for
infinitely
thin hard
rods.
There
are
also
some
other
models
based
on the
wormlike chain which
we can
only mention
here:
an
extension
of the
Flory theory
to
wormlike chains (Ronca
and
Yoon
1982, 1984, 1985)
and
theories
using
a
Maier-Saupe
potential
(Ten
Bosch
et al
1983, 1987,
\torner
et
al
1985).
As a
last
point
we
want
to
draw
attention
to
some monographs
and
review
papers
on
liquid crystals
(De
Gennes 1974, Stephen
and
Straley 1974,
Chandrasekhar
1977,
Luckhurst
and
Gray 1979, Vertogen
and De Jeu
1988, Frenkel 1991)
and
liquid
crystalline polymers (Straley
1973a,
Blumstein 1978, Grosberg
and
Khokhlov
1981,
Ciferri
et al
1982, Miller 1982,
Samulski
and
DuPré 1983, Gordon
and
Plate 1984,
Khokhlov
and
Semenov 1985,
Odijk
1986a,
Ciferri
and
Marsano 1987, Semenov
and
Khokhlov 1988, DuPré
and
Yang
1991),
which might
be
helpful
for the
reader.
2.
Virial
theory
of the
isotropic-nematic
transition
2.1. Imperfect
gas
At the
beginning
of
this century
the
virial expansion
was
introduced
by
Kamerlingh
Onnes
(1901)
as an
empirical
systematic
correction
to the
pressure
P of an
ideal
gas
at
higher (number) densities
p
2
3
... (1)
where
B
2
and
B
3
are the
second
and
third virial
coefficients
respectively. Later,
and
especially
in the
1930s,
much work
was
done
to
provide this density expansion
with
a
theoretical background, starting
from
statistical mechanics.
As
derivations
of
the
virial expansion
are
found
in
many
textbooks
of
statistical mechanics (Mayer
and
Mayer 1940, Hansen
and
McDonald 1986, McQuarrie 1976,
Feynman
1972)
we
briefly
summarize
some relevant formulae.
In the
following
we
assume
pairwise
interaction
u
between particles
so
that
the
potential energy
U can be
written
as a
summation
over pairs
(2)
Starting
either
from
the
canonical
or the
grand-canonical partition
function,
both
of
which
contain
the
Bolt/man
n
factor
exp(-[//fcT),
it was
shown that
a
virial
series
Phase
transitions
in fyotropic
colloidal
and
polymer
liquid
crystals
1245
for
the
pressure could
be
made.
In
view
of the
pairwise
potential energy
(2) we
expect functions
exp(—u(i,
j)/kT)
to
appear
in the
derived expressions
for the
virial
coefficients. They indeed show
up as the
so-called
Mayer
functions
(Mayer
and
Mayer 1940)
*(i,j)
=
exp(-u(i,j)/fcT)
- 1 (3)
which
vanish outside
the
range
of the
potential.
The
virial
coefficients
B
n
are
pro-
portional
to
irreducible cluster integrals
ß
n-l
of
these
Mayer functions.
The
second
virial coefficient contains interactions between
two
particles
(4)
as
it is the
coefficient
for the
quadratic term
in p. The
third virial
coefficient
involves
all
clusters
with
simultaneous interactions between three particles
Note that
for a
cluster,
in
which
one
pair does
not
interact,
the
corresponding Mayer
function
is 0 and
this cluster
does
not
contribute
to
B
3
(although
it
does
to
B
2
).
In
1937
and
later
it was
hoped that virial expansions could
be
used
to
describe
the
gas-liquid
phase transition (Mayer
and
Mayer 1940). This hope
was not
fulfilled,
but
we
shall
see
that
at the
same time Onsager (1942, 1949) recognized
it to be
useful
in
the
description
of the
isotropic-nematic
phase transition
in
solutions
of
thin rods.
2.2.
Non-ideal solutions
As Van
't
Hoff's
law
says
there
is a
striking similarity between
an
ideal
gas and an
ideal
solution when
we
replace
the
pressure
of the
gas,
P, by the
osmotic pressure
of the
solution,
H.
The
analogy also holds
for the
imperfect case when
we use the
potential
of
mean
force
u>(i,j)
between
two
solute particles instead
of the
direct potential
u(i,j)
(Onsager
1933,
McMillan
and
Mayer 1945, Hill
1956).
This procedure takes
an
average over
all
possible configurations
of the
solvent molecules accounting
for
interactions among themselves
and
with
the
solute particles. Using
w(i,j)
in the
Mayer
function
virial
expansion
(1) and
virial
coefficients
(4) and (5) now
hold
for the
osmotic
pressure.
For
phase transitions
it is
more appropriate
to use a
virial
expansion
of the
Helmholtz
free
energy
of the
solute
A F
AF
u°(T,»
n
,u,,
..)
NkT kT
where
^°
is the
reference chemical potential
of the
solute, which depends
on the
chemical potentials
of the
solvent components
/z
0
,/z
1
,...
and A is the de
Broglie
wavelength.
The
virial
series
for the
(osmotic) pressure
(1) is
obtained
by the
usual
thermodynamic
relation
,T,M,
..."
(8)