8Liquid crystals hydrodynamics
III Theoretical Physics of Soft Condensed Matter
8.1 Liquid crystal models
We finally turn to the case of liquid crystals. In this case, our order parameter no
longer takes a scalar value, and interesting topological phenomena can happen.
We first write down the theory in detail, and then discuss coarsening behaviour
of liquid crystals. We will describe the coarsening purely via geometric means,
and the details of the model are not exactly that important.
Recall that there are two types of liquid crystals:
(i)
Polar liquid crystals, where the molecules have “direction”, and the order
parameter is p, a vector, which is orientational but not positional.
(ii)
Nematic liquid crystals, where the molecules are like a cylinder, and there
is an order parameter Θ.
We will do the polar case only, and just quote the answers for the nematic
case.
As before, we have to start with a free energy
F =
Z
a
2
p
2
+
b
4
p
4
+
κ
2
(∇
i
p
j
)(∇
i
p
j
)
dr ≡
Z
F dr.
The first two terms are needed for the isotropicpolar transition to occur. Note
that
(i) F [p] = F [−p], so we have no cubic term.
(ii)
A linear term would correspond to the presence of an external field, e.g.
magnetic field.
(iii)
The
κ
term penalizes splay, twist and bend, and this term penalizes them
roughly equally. This is called the one elastic constant approximation. If
we want to be more general, we need something like
κ
1
2
∇ · p
2
+
κ
2
2

ˆ
p · ∇ ∧ p
2
+
κ
3
2

ˆ
p ∧ p
2
.
Here
p
is not conserved, so
˙
p
is not of the form
−∇ · J
. Instead, (without flow)
we have
˙
p = −Γh, h =
δF
δp(r)
,
where
h
(
r
) is called the molecular field, and Γ is a constant, called the angular
mobility.
We now want to generalize this to the case when there is a field flow. We
can just write this as
Dp
Dt
= −Γh,
where D is some sort of comoving derivative. For the scalar field, we had
Dφ
Dt
=
˙
φ + v · ∇φ.
Note that the advective term is trilinear, being first order in
v
,
∇
and
φ
. For
p
,
there is certainly something like this going on. If we have a translation, then
p
gets incremented by
∆p = v · ∇p ∆t,
as for a scalar.
There is at least one more thing we should think about, namely if
v
is
rotational, then we would expect this to rotate
p
as well. We have the corotational
term
∆p = ω ∧ p ∆t,
where ω is the angular velocity of the fluid, given by
ω
i
=
1
2
ε
ijk
Ω
jk
, Ω
jk
=
1
2
(∇
i
v
j
− ∇
j
v
i
).
This part must be present. Indeed, if we rotate the whole system as a rigid body,
with v(r) = ω × r, then we must have
˙
p = ω ∧ p.
It turns out in general, there is one more contribution to the advection, given
by
∆p = −ξD · p∆t
with D
ij
=
1
2
(
∇
i
v
j
+
∇
j
v
i
) and
ξ
a parameter. This is the irrotational part of
the derivative. The reason is that in a general flow,
p
needn’t simply rotate with
ω. Instead, it typically aligns along streamlines. In total, we have
Dp
Dt
= (∂
t
+ v · ∇)p + Ω · p − ξD ·p.
The parameter
ξ
is a molecular parameter, which depends on the liquid crystal.
The ξ = 1 case is called no slip, and the ξ = 0 case is called full slip.
With this understanding, we can write the hydrodynamic equation as
Dp
Dt
= −Γh, h =
δF
δp
.
We next need an equation of motion for v. We can simply
ρ(∂
t
+ v · ∇)v = η∇
2
v − ∇P + ∇ · Σ
p
,
where Σ
p
is a stress tensor coming from the order parameter.
To figure out what Σ
p
should be, we can consider what happens when we
have an “advective” elastic distortion. In this case, we have
Dp
Dt
= 0, so we have
˙
p = −v · ∇p − Ω · p + ξD ·p,
The free energy change is then
δF =
Z
δF
δp
·
˙
p ∆t dr = ∆t
Z
h · p dr,
On the other hand, the free energy change must also be given by
δF =
Z
Σ
p
ij
∇
i
u
j
(r) dr,
the product of the stress and strain tensors. By matching these two expressions,
we see that we can write
Σ
p
ij
= Σ
(1)
ij
+ Σ
(2)
ij
+ Σ
(3)
ij
,
where
∇
i
Σ
(1)
ij
= −p
k
∇
j
h
k
, Σ
(2)
ij
=
1
2
(p
i
h
j
− p
j
h
i
), Σ
(2)
ij
=
ξ
2
(p
i
h
j
+ p
j
h
i
).
Analogous results for a nematic liquid crystal holds, whose derivation is a
significant pain. We have
DQ
Dt
= −ΓH, H
ij
=
δF
δQ
ij
−
Tr
δF
δQ
δ
ij
d
.
The second term in H is required to ensure Q remains traceless.
The total derivative is given by
DQ
Dt
= (∂
t
+v ·∇)Q + (Ω·Q −Q·ω) +ξ(D·Q +Q ·D) −2ξ
Q +
1
d
Tr(Q ·∇v).
The terms are the usual advection term, rotation, alignment/slip and tracelessness
terms respectively. The Navier–Stokes equation involves a stress term
Σ
Q
= Σ
Q,1
+ Σ
Q,2
,
where
∇
k
Σ
Q,1
k,`
= −Q
ij
∇
`
H
ij
Σ
Q,2
= Q · H −H · Q − ξ
2
3
H + 2
d
QH −2Q tr(QH)
,
with the hat denoting the traceless symmetric part. The rest of the Navier–Stokes
equations is the same.