Using a theory of nematic liquid crystals to model

swimming microorganisms Nigel Mottram

Department of Mathematics and Statistics University of Strathclyde

Background

  Swimming organisms, motivation:

  Behaviour of fish, sea mammals, interaction with man-made objects

  Smaller organisms, zooplankton, phytoplankton

  Interesting self-organisation

  Non-equilibrium fluid dynamics

Background

Behaviour of fish, sea mammals, interaction with man-made objects

Zooplankton

Copepod Krill

Zooplankton: small crustaceans and other animals that feed on other plankton

Phytoplankton

Phytoplankton: algae that live near the water surface where there is sufficient light to support photosynthesis.

Self-organisation

Flocking/shoaling:

A mathematical model considers "flocking" as the collective motion of a large number of self-propelled entities.

It is considered an emergent behaviour arising from simple rules that are followed by individuals and does not involve any central coordination.

Flocking

  The first model of flocking involved three relatively simple rules

  Separation - avoid crowding neighbours (short range repulsion)

  Alignment - steer towards average heading of neighbours

  Cohesion - steer towards average position of neighbours (long range attraction)

  A simpler model changes the direction of motion by averaging over neighbours

is the average orientation of neighbours, is a random fluctuation

Flocking

(a)  High noise, low density: particles move independently

(b)  Low noise, low density: particles form independent groups

(c)  High noise, high density: particles move with some correlation (d)  Low noise, high density: all particles move in same direction

Flocking and Ferromagnetism?

The part of this update rule looks like a model of a ferromagnet…

…but in a ferromagnet you can’t have a symmetry breaking event in 2d

The flocking model creates organisation because it is out of equilibrium.

Similarities to liquid crystal molecular dynamics

The Gay-Berne potential is used to model a group of elongated molecules…

Similarities to liquid crystal molecular dynamics

  Separation – repulsion as molecules approach

  Alignment – side-side alignment gives lower energy state

  Cohesion – presence of a minimum in the energy

Similarities to liquid crystal molecular dynamics

Coarsening – continuum limit

We move to a continuum model by thinking of the velocity at a point in space as being the average velocity of a (large) number of entities.

Possibly more plausible for microorganisms but has been used for larger organisms.

Governing equations are derived in a similar way to the Navier-Stokes but without the Galilean invariance.

We should probably also model the orientational order of the entities

A simpler model

A simpler model has been proposed, which does include orientational order.

In this model the “swimming” organisms are either “pushers” or “pullers”

An appropriate model is the Ericksen-Leslie with an extra term in the stress tensor

We will consider a simple 1d system to look at the basic properties of an active nematic

We look at three cases: (a) Spontaneous flow, (b) Flow induced through shear, (c) Backflow and kickback

extensile (pushers) contractile (pullers)

Spontaneous flow

The active nematic is initially aligned parallel to the bounding surfaces.

Flow is only considered in one direction and the director stays in the plane.

What happens?

The active nematic induces flow but if is constant there will be no contribution to the flow equation.

Will the system break the symmetry and create flow?

Spontaneous flow

Linearise around the initial state…

and consider the solution…

which leads to…

Spontaneous flow

Apply the boundary condition for ,

then using the equation for the velocity and the boundary condition…

we arrive at the following condition

Spontaneous flow

This condition determines when the initial state becomes unstable

This indicates that, for sufficiently small values of , the mode decays, leaving the initial state.

However, for a mode becomes unstable.

This plot also indicates other unstable modes and other critical values of the activity parameter.

Spontaneous flow

We can see this instability by solving the full nonlinear equations for different values of and for different initial conditions.

For the initial state decays.

Spontaneous flow

For the initial state does not decay.

Spontaneous flow

However, for we can also obtain an alternative state.

Spontaneous flow

We find at least three solutions, two of which seem to be (locally) stable.

For higher values of the activity parameter we would expect even more possible solutions. Further analysis of the bifurcations and solution stabilities is needed.

We would like to be able to find critical values of ζ for which different solutions exist.

If we now force a shear in the system there is no stable trivial state and the director prefers to align at the “flow-aligning” angle.

There are however, instabilities away from this state.

This system might be similar to a layer of active nematic on top of a moving immiscible fluid.

The induced flow from the active nematic may affect the mixing of the background fluid, nutrients, salinity etc.

Alignment in shear flow

Edwards and Yeomans numerically found different states but only considered single mode solutions.

Alignment in shear flow

The third case we consider is a classic example of director-flow coupling in liquid crystals.

There may be interesting parallels in active fluid systems.

Here we start with the same system as in the first case but with a different initial state.

The active nematic may have been aligned by a variety of external influences: magnetic field, light source, food source…

Backflow/Kickback

We first linearise about the state

The linearised governing equations are similar to the first case

and we seek solutions of the form

Backflow/Kickback

The modenumber and associated time constant are determined by,

Because is negative and is positive, the time constant is negative (i.e. all modes decay) when the activity parameter is not negative.

Backflow/Kickback

For we get a number of modes determined by

The high order modes decay, causing kickback and leaving a single mode.

Backflow/Kickback

For small positive or negative values of the activity parameter the decay is similar to the normal nematic.

Backflow/Kickback

For small positive or negative values of the activity parameter the decay is similar to the normal nematic.

Backflow/Kickback

The first mode disappears at critical values of the activity parameter

Backflow/Kickback

For more negative values of the activity parameter we get decay without kickback.

Backflow/Kickback

For more positive values of the activity parameter we get decay with more pronounced kickback.

Backflow/Kickback

For larger positive values of the activity parameter we get decay with more pronounced kickback.

Backflow/Kickback

Future questions

  What happens if density and order are included in models of active nematics?

  Most marine based microorganisms are polar; how does this break in symmetry affect the results?

  How realistic is it to use continuum models for large organisms?

  How do active species affect mixing?

  What happens in 2d or 3d?

Acknowledgements – Allan Sharkie, SAMS, MRC (for future funding)