9Active Soft Matter

III Theoretical Physics of Soft Condensed Matter

9 Active Soft Matter

We shall finish the course by thinking a bit about motile particles. These are

particles that are self-propelled. For example, micro-organisms such as bacteria

and algae can move by themselves. There are also synthetic microswimmers. For

example, we can make a sphere and coat it with gold and platinum on two sides

PtAu

We put this in hydrogen peroxide H

2

O

2

. Platinum is a catalyst of the decompo-

sition

2H

2

O

2

→ 2H

2

O + O

2

,

and this reaction will cause the swimmer to propel forward in a certain direction.

This reaction implies that entropy is constantly being produced, and this cannot

be captured adequately by Newtonian or Lagrangian mechanics on a macroscopic

scale.

Two key consequences of this are:

(i) There is no Boltzmann distribution.

(ii)

The principle of detailed balance, which is a manifestation of time reversal

symmetry, no longer holds.

Example. Take bacteria in microfluidic enclosure with funnel gates:

In this case, we expect there to be a rotation of particles if they are self-

propelled, since it is easier to get through one direction than the other. Contrast

this with the fact that there is no current in the steady state for any thermal

equilibrium system. The difference is that Brownian motion has independent

increments, but self-propelled particles tend to keep moving in the same direction.

Note also that we have to have to break spatial symmetry for this to happen.

This is an example of the Ratchet theorem, namely if we have broken time

reversal symmetry pathwise, and broken spatial symmetry, then we can have

non-zero current.

If we want to address this type of system in the language we have been using,

we need to rebuild our model of statistical physics. In general, there are two

model building strategies:

(i)

Explicit coarse-graining of “micro” model, where we coarse-grain particles

and rules to PDEs for ρ, φ, P, Q.

(ii)

Start with models of passive soft matter (e.g. Model B and Model H), and

add minimal terms to explicitly break time reversal phenomenologically.

Of course, we are going to proceed phenomenologically.

Active Model B

Start with Model B, which has a diffusive, symmetric scalar field

φ

with phase

separation:

˙

φ = −∇ · J

J = −∇˜µ +

√

2DΛ.

We took

F =

Z

a

2

+

b

4

φ

4

+

κ

2

(∇φ)

2

dr.

To model our system without time reversal symmetry, we put

˜µ =

δF

δφ

+ λ(∇φ)

2

.

The new term breaks the time reversal structure. These equations are called

active Model B . Another way to destroy time reversal symmetry is by replacing

the white noise with something else, but that is complicated

Note that

(i)

(

∇φ

)

2

is not the functional derivative of any

F

. This breaks the free energy

structure, and

P

F

P

B

6= e

−β(F

2

−F

1

)

for any F [φ]. So time reversal symmetric is broken barring miracles.

(ii)

We cannot achieve the breaking by introducing a polynomial term, since if

g(φ) is a polynomial, then

g(φ) =

δ

δφ

Z

dr

Z

φ

g(u) du

!

.

So gradient terms are required to break time reversal symmetry. We will

later see this is not the case for liquid crystals.

(iii)

The active model B is agnostic about the cause of phase separation at

a < 0. There are two possibilities:

(a) We can have attractive interactions

(b)

We can have repulsive interactions plus motile particles: if two parti-

cles collide head-on, then we have pairwise jamming. They then move

together for a while, and this impersonates attraction. This is called

MIPS — mobility-induced phase separation. It is possible to study

this at a particle level, but we shall not.

(iv)

The dynamics of coarsening during phase separation turns out to be similar,

with L(t) ∼ t

1/3

. The Ostwald–like process remains intact.

(v)

The coexistence conditions are altered. We previously found the coexistence

conditions simply by global free energy minimization. In the active case,

we can’t do free energy minimization, but we can still solve the equations

of motion explicitly. In this case, instead of requiring a common tangent,

we have equal slope but different intercepts, where we set

(µφ − f)

1

= (µφ − f)

2

+ ∆.

This is found by solving J = 0, so

˜µ =

∂f

∂φ

− κ∇

2

φ + λ(∇φ)

2

= const.

(vi) There is a further extension, active model B+, where we put

J = −∇˜µ +

√

2DΛ + ζ(∇

2

φ)∇φ.

This extra term is similar to

∇

(

λ

(

∇φ

)

2

) in that it has two

φ

’s and three

∇

’s, and they are equivalent in 1 dimension, but in 1 dimension only. This

changes the coarsening process significantly. For example, Ostwald can

stop at finite R (see arXiv:1801.07687).

Active polar liquid crystals

Consider first a polar system. Then the order parameter is

p

. In the simplest

case, the field is relaxational with v = 0. The hydrodynamic level equation is

˙

p = −Γh, h =

δF

δp

.

We had a free energy

F =

Z

a

2

p

2

+

b

4

p

4

+

κ

2

(∇

α

p

β

)(∇

α

p

β

)

dr.

As for active model B,

h

can acquire gradient terms that are incompatible with

F

.

But also, we can have a lower order term in

∇

that is physically well-motivated —

if we think of our rod as having a direction

p

, then it is natural that

p

wants to

translate along its own direction at some speed

w

. Thus,

p

acquires self-advected

motion wp. Thus, our equation of motion becomes

˙

p + p · ∇p = −Γh.

This is a bit like the Navier–Stokes equation non-linearity. Now couple this to a

fluid flow v. Then

Dp

Dt

= −Γh,

where

Dp

Dt

=

∂

∂t

+ v · ∇

p + Ω · p − ξD · p + wp · ∇p.

The Navier–Stokes/Cauchy equation is now

(∂

t

+ v · ∇)v = η∇

2

v − ∇P + ∇ · Σ

(p)

+ ∇ · Σ

A

,

where as before,

∇ · Σ

(p)

= −p

i

∇

j

h

j

+ ∇

i

1

2

(p

i

h

j

− p

j

h

i

) +

ξ

2

(p

i

h

j

+ p

j

h

i

)

.

and we have a new term Σ

A

given by the active stress, and the lowest order

term is

ζp

i

p

j

. This is a new mechanical term that is incompatible with

F

. We

then have

∇ · Σ

A

= (∇ · p)p.

We can think of this as an effective body force in the Navier–Stokes equation.

The effect is that we have forces whenever we have situations looking like

or

In these cases, We have a force acting to the right for

ζ >

0, and to the left if

ζ < 0.

These new terms give spontaneous flow, symmetric breaking and macroscopic

fluxes. At high w, ζ, we get chaos and turbulence.

Active nematic liquid crystals

In the nematic case, there is no self-advection. So we can’t make a velocity from

Q. We again have

DQ

Dt

= −ΓH, H =

δF

δQ

traceless

.

where

DQ

Dt

is given by

DQ

Dt

= (∂

t

+ v · ∇)Q + S(Q, K, ξ).

Here K = ∇v and

S = (−Ω · Q − Q · Ω) − ξ(D · Q + Q · D) + 2ξ

Q +

1

d

Tr(Q · K)

Since there is no directionality as in the previous case, the material derivative

will remain unchanged with active matter. Thus, at lowest order, all the self-

propelled motion can do is to introduce an active stress term. The leading-order

stress is

Σ

A

= ζQ.

This breaks the free energy structure. Indeed, if we have a uniform nematic, then

the passive stress vanishes, because there is no elastic distortion at all. However,

the active stress does not since

ζQ 6

= 0. Physically, the non-zero stress is due to

the fact that the rods tend to generate local flow fields around themselves to

propel motion, and these remain even in a uniform phase.

After introducing this, the effective body force density is

f = ∇·Σ

A

= ζ∇ · Q ∼ ζλ(∇ · n)n.

This is essentially the same as the case of the polar case. Thus, if we see

something like

then we have a rightward force if ζ > 0 and leftward force if ζ < 0.

This has important physical consequences. If we start with a uniform phase,

then we expect random noise to exist, and then the active stress will destablize

the system. For example, if we start with

and a local deformation happens:

then in the

ζ >

0 case, this will just fall apart. Conversely, bends are destabilized

for

ζ <

0. Either case, there is a tendency for these things to be destabilized,

and a consequence is that active nematics are never stably uniform for large

systems. Typically, we get spontaneous flow.

To understand this more, we can explicitly describe how the activity parameter

ζ affects the local flow patterns. Typically, we have the following two cases:

ζ > 0

contractile

ζ < 0

extensile

Suppose we take an active liquid crystal and put it in a shear flow. A rod-like

object tends to align along the extension axis, at a 45

◦

angle.

If the liquid crystal is active, then we expect the local flows to interact with

the shear flow. Suppose the shear rate is v

x

= yg. Then the viscous stress is

Σ

η

= ηg

0 1

1 0

.

We have

Σ

A

∝ ζλ

nn −

1

d

= ζλ

0 1

1 0

if

n

is at 45

◦

exactly. Note that the sign of

ζ

affects whether it reinforces or

weakens the stress.

A crucial property is that Σ

A

does not depend on the shear rate. So in the

contractile case, the total stress looks like

g

Σ

TOT

In the extensile case, however, we have

g

Σ

TOT

g

∗

This is very weird, and leads to spontaneous flow at zero applied stress of

the form

Defect motion in active nematics

For simplicity, work in two dimensions. We have two simple defects as before

q = −

1

2

q = +

1

2

Note that the

q

=

−

1

2

charge is symmetric, and so by symmetry, there cannot be

a net body stress. However, in the

q

= +

1

2

defect, we have a non-zero effective

force density.

So the defects themselves are like quasi-particles that are themselves active.

We see that contractile rods move in the direction of the opening, and the

extensile goes in the other direction. The outcome of this is self-sustaining

“turbulent” motion, with defect pairs

±

1

2

are formed locally. The

−

1

2

stay put

and the +

1

2

ones self-propel, and depending on exactly how the defect pairs are

formed, the +

1

2

defect will fly away.

Experimental movies of these can be found in T. Sanchez Nature 491, 431

(2012). There are also simulations in T. Shendek, et al, Soft Matter 13, 3853

(2017).