Using a theory of nematic liquid crystals to model swimming microorganisms
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Transcript of Using a theory of nematic liquid crystals to model swimming microorganisms
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Using a theory of nematic liquid crystals to model
swimming microorganisms Nigel Mottram
Department of Mathematics and Statistics University of Strathclyde
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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
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Background
Behaviour of fish, sea mammals, interaction with man-made objects
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Zooplankton
Copepod Krill
Zooplankton: small crustaceans and other animals that feed on other plankton
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Phytoplankton
Phytoplankton: algae that live near the water surface where there is sufficient light to support photosynthesis.
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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.
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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
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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
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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.
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Similarities to liquid crystal molecular dynamics
The Gay-Berne potential is used to model a group of elongated molecules…
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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
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Similarities to liquid crystal molecular dynamics
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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
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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)
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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?
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Spontaneous flow
Linearise around the initial state…
and consider the solution…
which leads to…
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Spontaneous flow
Apply the boundary condition for ,
then using the equation for the velocity and the boundary condition…
we arrive at the following condition
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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.
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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.
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Spontaneous flow
For the initial state does not decay.
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Spontaneous flow
However, for we can also obtain an alternative state.
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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.
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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
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Edwards and Yeomans numerically found different states but only considered single mode solutions.
Alignment in shear flow
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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
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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
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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
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For we get a number of modes determined by
The high order modes decay, causing kickback and leaving a single mode.
Backflow/Kickback
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For small positive or negative values of the activity parameter the decay is similar to the normal nematic.
Backflow/Kickback
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For small positive or negative values of the activity parameter the decay is similar to the normal nematic.
Backflow/Kickback
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The first mode disappears at critical values of the activity parameter
Backflow/Kickback
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For more negative values of the activity parameter we get decay without kickback.
Backflow/Kickback
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For more positive values of the activity parameter we get decay with more pronounced kickback.
Backflow/Kickback
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For larger positive values of the activity parameter we get decay with more pronounced kickback.
Backflow/Kickback
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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)