7Model H

III Theoretical Physics of Soft Condensed Matter

7 Model H

Model B was purely diffusive, and the only way

φ

can change is by diffusion.

Often, in real life, fluid can flow as well. If the fluid has a velocity

v

, then our

equation is now

˙

φ + v · ∇φ = −∇·J.

The

v · ∇φ

is called the advection term. Our current

J

is the same as before,

with

J = −M

δF

δφ

+

p

2k

B

T MΛ.

We also need an evolution equation for

v

, which will be the Navier–Stokes

equation with some noise term. We assume flow is incompressible, so

∇ · v

= 0.

We then have the Cauchy equation with stress tensor Σ

TOT

, given by

ρ(

˙

v + v · ∇v) = ∇ · Σ

TOT

+ body forces.

We will assume there is no body force. This is essentially the momentum

conservation equation, where −Σ

TOT

is the momentum flux tensor.

Of course, this description is useless if we don’t know what Σ

TOT

looks like.

It is a sum of four contributions:

Σ

TOT

= Σ

p

+ Σ

η

+ Σ

φ

+ Σ

N

.

– The Σ

p

term is the pressure term, given by

Σ

p

ij

= −P δ

ij

.

We should think of this P as a Lagrange multiplier for incompressibility.

– The Σ

η

term is the viscous stress, which we can write as

Σ

η

ij

= η(∇

i

v

j

+ ∇

j

v

i

)

For simplicity, we assume we have a constant viscosity

η

. In general, it

could be a function of the composition.

– The Σ

φ

term is the φ-stress, given by

Σ

φ

= −Πδ

ij

− κ(∇

i

φ)(∇

j

φ), Π = φµ − F.

This is engineered so that

∇ · Σ

φ

= −φ∇µ.

This says a non-constant chemical potential causes things to move to even

that out.

– The final term is a noise stress with

hΣ

N

ij

(r, t)Σ

N

k`

(r

0

, t

0

)i = 2k

B

T η

δ

ik

δ

j`

+ δ

i`

δ

jk

−

2

3

δ

ij

δ

k`

δ(r−r

0

)δ(t−t

0

).

The last term

δ

ij

δ

k`

is there to ensure the noise does not cause any

compression. This is a white noise term whose variance is determined by

the fluctuation dissipation theorem.

We can then compute

∇ · Σ

TOT

= ∇ · Σ

P

+ ∇ · Σ

η

+ ∇ · Σ

φ

+ ∇ · Σ

N

= −∇P + η∇

2

v + −φ∇µ + ∇ · Σ

N

Hence Model H has equations

˙

φ + v · ∇φ = −∇ · J

J = −M∇µ +

p

2k

B

T MΛ

∇ · v = 0

ρ(

˙

v + v · ∇v) = η∇

2

v − ∇P − φ∇µ + ∇ · Σ

N

.

Compared to Model B, we have the following new features:

(i) −φ∇µ drives deterministic fluid flow.

(ii) Σ

N

drives a random flow.

(iii) Fluid flow advects φ.

How does this affect the coarsening dynamics? We will see that (i) and (iii) gives

us enhanced coarsening of bicontinuous states. However, this does not have any

effect on isolated/disconnected droplet states, since in a spherically symmetric

setting,

φ∇µ

and

∇P

will be radial, and so

∇ · v

= 0 implies

v

= 0. In other

words, for φ∇µ to drive a flow, we must have some symmetry breaking.

Of course, this symmetry breaking is provided by the noise term in (ii). The

result is that the droplets will undergo Brownian motion with

hr

2

i ∼ Dt

, where

D =

k

B

T

4πηR

is the diffusion constant.

If we think about the Ostwald process, even if we manage to stop the small

droplets from shrinking, they may collide and combine to form larger droplets.

This forms a new channel for instability, and separate measures are needed to

prevent this. For example, we can put charged surfactants that prevent collisions.

We can roughly estimate the time scale of this process. We again assume

there is one length scale

¯

R

(

t

), which determines the size and separation of

droplets. We can then calculate the collision time

∆t '

¯

R

2

D(

¯

R)

∼

¯

R

3

η

k

B

T

.

Each collision doubles the volume, and so

¯

R →

2

1/3

¯

R

. Taking the logarithm, we

have

∆ log

¯

R ∼

log 2

3

in time ∆t.

So we crudely have

∆ log

¯

R

∆t

∼

log 2

3

k

B

T

η

¯

R

3

.

If we read this as a differential equation, then we get

d log

¯

R

dt

=

1

¯

R

˙

¯

R ∼

k

B

T

η

¯

R

3

.

So we find that

¯

R

2

˙

¯

R ∼

k

B

T

η

.

So

¯

R(t) ∼

k

B

T

η

t

1/3

.

This is diffusion limited coalescence.

Recall that in the Ostwald process, droplets grew by diffusion of molecules,

and we had the same power of t. However, the coefficient was different, with

¯

R ∼

Mσ

φ

B

t

1/3

.

It makes sense that they have the same scaling law, because ultimately, we are

still doing diffusion on different scales.

Bicontinuous states

We now see what the fluid flow does to bicontinuous states.

L

Again assume we have a single length scale

L

(

t

), given by the domain size.

Again we assume we have a single length scale. As time goes on, we expect

L

(

t

)

to increase with time.

A significant factor in the evolution of the bicontinuous phase is the Laplace

pressure, which is ultimately due to the curvature

K

. Since there is only one

length scale, we must have

˙

K ∼ v.

The Laplace pressure then scales as ∼

σ

L

.

The noise terms Σ

N

matter at early times only. At late times, the domains

grow deterministically from random initial conditions. The key question is how

L(t) scales with time. The equation of motion of the flow is

ρ(

˙

v + v · ∇v) = η∇

2

v − ∇P − φ∇µ,

We make single length scale approximations as before, so that

v ∼

˙

L

and

∇ ∼ L

−1

. Then we have an equation of the form.

ρ

¨

L + ρ

˙

L

2

L

∼ η

˙

K

L

2

+ Lagrange multiplier +

σ

L

2

(∗)

where we recall that at curved interfaces,

µ ∼ ±

σ

R

. Here we have a single variable

L

(

t

), and three dimensionful parameters

ρ, η, σ

. We can do some dimensional

analysis. In d dimensions, we have

ρ = ML

−d

, η = ML

2−d

T

−1

, σ = ML

3−d

T

−2

.

We want to come up with combinations of these for length to depend on time,

and we find that in three dimensions, we have

L

0

=

η

2

ρσ

, t

0

=

η

3

ρσ

2

.

One can check that these are the only combinations with units

L, T

. So we must

have

L(t)

L

0

= f

t

t

0

.

We now substitute this into the equation (

∗

). We then get a non-dimensionalized

equation for (∗)

αf

00

+ βf

02

/f = γ

f

0

f

2

+

δ

f

2

,

with α, β, γ, δ = O(1) dimensionless numbers.

If we think about this, we see there are two regimes,

(i) The LHS (inertia) is negligible at small t/t

0

(or small f). Then we get

γf

0

f

2

+

δ

f

2

= 0,

so

f

0

is a constant, and so

L

grows linearly with

t

. Putting all the

appropriate constants in, we get

L ∝

σ

η

t.

This is called the viscous hydrodynamic regime, VH .

(ii)

For large

f

, we assume we have a power law

f

(

x

)

∼ x

y

, where

y >

0 (or

else f would not be large). Then

¯αx

y−2

+

¯

βx

y−2

= ¯γx

−y−1

+ δx

−2y

.

It turns out at large

f

, the

x

−y−1

term is negligible, scaling wise. So we

have y − 2 = −2y, or equivalently, y =

2

3

. So

L

L

0

∼

t

t

0

2/3

.

Putting back our factors, we have

L ∼

σ

ρ

1/3

t

2/3

.

This is called the inertial hydrodynamic regime, IH . In this regime, interfa-

cial energy is converted into kinetic energy, and then only “later” dissipated

by η.

Essentially, in the first regime, the system is overdamped. This happens until

the right-hand side becomes big and takes over, until the viscous term finally

takes over. In practice, it is difficult to reach the last regime in a lab, since the

time it takes is often ∼ 10

4

.

Droplet vs bicontinuous

In general, when do we expect a bicontinuous phase and when do we expect a

droplet phase?

–

In three dimensions, the rule of thumb is that if

ψ ∼

0

.

4

to

0

.

6, then

we always get a bicontinuous medium. If

ψ <

0

.

3 or

>

0

.

7, then we

get droplets always. In between these two regions, we initially have

bicontinuous medium, which essentially de-percolates into droplets.

–

In two dimensions, things are different. In the fully symmetric case, with a

constant

η

throughout, and

F

is strictly symmetric (

F

(

φ

) =

F

(

−φ

)), the

only case where we have bicontinuous phase is on the ψ =

1

2

line.