Physics Project On Physical World, Units and Measurement
Using a theory of nematic liquid crystals to model swimming microorganisms
1. Using a theory of nematic
liquid crystals to model
swimming microorganisms
Nigel Mottram
Department of Mathematics and Statistics
University of Strathclyde
2. 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
6. 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.
7. 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
8. 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
9. 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.
10. Similarities to liquid crystal molecular dynamics
The Gay-Berne potential is used to model a group of elongated
molecules…
11. 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
13. 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
14. 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”
extensile (pushers) contractile (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
15. 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?
17. Spontaneous flow
Apply the boundary condition for ,
then using the equation for the velocity and the boundary condition…
we arrive at the following condition
18. 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.
19. 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.
22. 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.
23. Alignment in shear flow
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.
24. Alignment in shear flow
Edwards and Yeomans numerically found different states but only
considered single mode solutions.
25. Backflow/Kickback
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…
26. 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
27. 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.
28. Backflow/Kickback
For we get a number of modes determined by
The high order modes decay, causing kickback and leaving a single mode.
35. 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)
