8Liquid crystals hydrodynamics
III Theoretical Physics of Soft Condensed Matter
8.2 Coarsening dynamics for nematics
We will discuss the coarsening dynamics for nematic liquid crystals, and indicate
how polar liquid crystals are different when appropriate. As before, we begin
with a completely disordered phase with
Q
= 0, and then quench the system
by dropping the temperature quickly. The liquid crystals then want to arrange
themselves. Afterwards, we local have
Q =
λ 0 0
0 −λ/2 0
0 0 −λ/2
with free energy
f
(
λ
). If the quench is deep enough, we have spinodal-like
instability, and we quickly get locally ordered. Coordinate-independently, we
can write
Q =
ˆ
λ
n
i
n
j
−
1
3
δ
ij
.
Since all this ordering is done locally, the principal axis
n
can vary over space.
There is then a slower process that sorts out global ordering, driven by the elastic
part of the free energy.
Compare this with Model B/H: at early times, we have
φ
:
±φ
B
, but we get
domain walls between the ±φ
B
phases.
x
φ
The late time dynamics is governed by the slow coarsening of these domain walls.
The key property of this is that it has reduced dimensionality, i.e. the domain
wall is 2 dimensional while space is 3 dimensional, and it connects two different
grounds states (i.e. minima of F ).
The domain wall can be moved around, but there is no local change that
allows us to remove it. They can only be removed by “collision” of domain walls.
Analogous structures are present for nematic liquid crystals. The discussion
will largely involve us drawing pictures. For now, we will not do this completely
rigorously, and simply rely on our intuitive understanding of when defects can
or cannot arise. We later formalize these notions in terms of homotopy groups.
We first do this in two dimensions, where defects have dimension
<
2. There
can be no line defects like a domain wall. The reason is that if we try to construct
a domain wall
then this can relax locally to become
On the other hand, we can have point defects, which are 0-dimensional. Two
basic ones are as follows:
q = −
1
2
q = +
1
2
The charge
q
can be described as follows — we start at a point near the
defect, and go around the defect once. When doing so, the direction of the order
parameter turns. After going around the defect once, in the
q
=
−
1
2
, the order
parameter made a half turn in the opposite sense to how we moved around the
defect. In the q = +
1
2
case, they turned in the same sense.
We see that
q ±
1
2
are the smallest possible topological charge, and is a
quantum of a charge. In general, we can have defects of other charges. For
example, here are two q = +1 charges:
hedgehog
vortex
Both of these are
q
= +1 defects, and they can be continuously deformed
into each other, simply by rotating each bar by 90
◦
. For polar liquid crystals,
the quantum of a charge is 1.
If we have defects of charge greater than
±
1
2
, then they tend to dissociate
into multiple
q
=
±
1
2
defects. This is due to energetic reasons. The elastic
energy is given by
F
ell
=
κ
2
|∇ · Q|
2
∼
κ
2
λ
2
|(∇ · n)n + n · ∇n|
2
.
If we double the charge, we double the
Q
tensor. Since this term is quadratic
in the gradient, putting two defects together doubles the energy. In general,
topological defects tend to dissociate to smaller q-values.
To recap, after quenching, at early stages, we locally have
Q → 2λ(nn −
1
2
1).
This
n
(
r
) is random, and tend to vary continuously. However, topological defects
are present, which cannot be ironed out locally. All topological defects with
|q| >
1
2
dissociate quickly, and we are left with q = ±
1
2
defects floating around.
We then have a late stage process where opposite charges attract and then
annihilate. So the system becomes more and more ordered as a nematic. We
can estimate the energy of an isolated defect as
˜κ
2
|(∇ · n)n + ∇n|
2
,
where ˜κ = κλ
2
. Dimensionally, we have
∇ ∼
1
r
.
So we have an energy
E ∼ ˜κ
Z
1
r
2
dr ' ˜κ log
L
r
0
,
where
L
is the mean spacing and
r
0
is some kind of core radius of the defect.
The core radius reflects the fact that as we zoom close enough to the core of the
signularity,
λ
is no longer constant and our above energy estimate fails. In fact,
λ → 0 at the core.
Recall that the electrostatic energy in two dimensions is given by a similar
equation. Thus, this energy is Coulombic, with force
∝
˜κ
R
. Under this force, the
defects move with overdamped motion, with the velocity being proportional to
the force. So
˙
R ∼
1
R
,
˙
L ∝
1
L
.
So
L(t) ∼ t
1/2
.
This is the scaling law for nematic defect coarsening in 2 dimensions.