Aparna Baskaran

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Self propelled particles From microdynamics to
hydrodynamics
Aparna Baskaran Syracuse University
In Collaboration with M. Cristina Marchetti
Supported by grants DMR-0705105 and DMR-0806511.
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Scope of todays talk
Self-propulsion Induced modifications
Interactions Short-ranged contact Long ranged
medium induced
Large scale collective behavior in
Self-Propelled Particles (SPPs) Swarming,
Pattern formation
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Part I Contact Interaction
Context

• SPPs typically have an assymmetric shape
  Excluded volume interactions induce ordering
• Move on/ through a medium Overdamped dynamics
• Paradigm Nematic liquid crystal
• What changes in the presence of self-propulsion?

Start with the canonical microscopic model for
liquid crystals Long thin rods
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Part I Contact Interaction
The Microscopic Model
Interaction ? Excluded volume - Energy and
momentum conserving collisions.
F
friction z
Everything else is white noise
Parameters of the model 1. v0 F/ ? self
propulsion velocity 2. Noise kBT a
nonequilibrium temperature 3. Length of the
rod l.
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Part I Contact Interaction
The Theory
Langevin Equations
Statistical mechanics
N-particle distribution function
Low density, neglect correlations
1 particle distribution function
Separation of time scales between velocity
relaxation and spatial relaxations set by the
friction constant
Smoluchowski Equation
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Part I Contact Interaction
The Theory- Smoluchowski equation
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Part I Contact Interaction
Emergent behavior at long wavelengths

• Enhanced ordering No polar state. Isotropic
  nematic transition occurs below the Onsager
  density
• Propagating modes in the isotropic state
  Density waves in certain regions of parameter
  space (Movie)
• Homogeneous nematic state destabilized as self
  propulsion velocity increases
• Large number fluctuations in the steady states
  compared to analogous equilibrium systems

? Anisotropic momentum transfer during
collisions associated with the self propulsion
velocity along the long axis of the particle.
? Convective fluxes induced by self
propulsion that couple to orientational
fluctuations.
? When self propulsion wins over diffusion, splay
fluctuations in the system drive this instability.
? Competition between diffusion and self
propulsion changes the dynamics to give this
effect.
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End of Part I Lead in to Part II

• Have outlined how contact interactions modified
  by self-propulsion
• Have listed outcomes and hinted at mechanisms
• A Baskaran and MC Marchetti PRE (2008), PRL
  (2008)
• Now change gears What is the effect of fluid
  mediated interactions?
• Rest of the talk Collective dynamics of SPPs
  in viscous fluid

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Part II Hydrodynamic Interaction
Context

• Typical numbers for bacteria v microns/s, size
  a microns
• Low Reynolds numbers Mass unimportant v
  Force (Aristotle not Newton)
• Interested in length-scales gtgt swimmer,
  timescales gtgt stroke period

L
Forces on fluid are a force dipole
stroke average
distances gtgt L
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Part II Hydrodynamic Interaction
Microscopic Model
f
?
u
f
?
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Part II Hydrodynamic Interaction
Nomenclature
f lt 0
f gt 0
Pulls fluid in at head and tail
contractile Hydrodynamic center toward direction
of motion - puller
Pushes fluid in at head and tail
tensile Hydrodynamic center toward direction of
motion - pusher
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Part II Hydrodynamic Interaction
The forces and torques

• Long ranged forces and torques
• Both nematic in nature
• Polar part decays faster
• Sphericaly symmetric and vanishes in mean field

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Part II Hydrodynamic Interaction
Emergent Behavior on long length scales
Scale free Growth of splay fluctuations due to
long ranged part of hydrodynamic torque Seen in
numerical work Scantillian et al (2008),
Underhill et al (2008)
unstable (bend)

• No ordered state with only far field
  hydrodynamic interactions
• Can get a nematic state by including excluded
  volume interactions
• Polar state appears to need external symmetry
  breaking chemotaxis

Scale free Long range torques destabilize
orientational fluctuations No threshold Generic
as it occurs from the nematic part movers and
shakers Identified from phenomenological
hydrodynamics in Simha and Ramaswamy 2002 All
instabilites suppressed when the stroke of the
swimmer is time reversal parity invariant.
tensile pusher
unstable
stable
stable
unstable
contractile puller
unstable (splay)
Diffusive Suppression of longitudinal diffusion
due to hydrodynamic force Seen in Cytoskeletal
models Kruse et al (2000), Liverpool et
al.(2003) Bundling
Isotropic
Nematic
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Concluding Remarks

• Unified derivation of hydrodynamics of active
  liquid crystals from physical microscopic
  models
• Characterized nature of fluctuations in bulk
  systems and underlying mechanisms
• All conclusions hold for all systems in this
  class because the only critical assumption is
  momentum conservation
• Observability depends on the Peclet number (f
  l/? D), what else is going on
• Next step same systematic approach to
  nonlinearities and boundaries

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On the horizon - Nonlinearities and Boundaries
Nonlinearities in Part I
May be seen in motility assays of short actin
filaments?
Nonlinear terms in polarization equation due to
interactions give rise to banding (with Shradha
Mishra)
See poster II. 6 by Volker Schaller
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On the horizon - Nonlinearities and Boundaries
Boundaries in Part II
Tensile parallel anchoring
Berke et al, 2008
v/v0
v/v0
h
h
Use method of images systematically to study
pattern formation