Swarming and swirling: from granular rods to bacterial colonies

1
Swarming and swirling from granular rods
tobacterial colonies

• Arshad Kudrolli
• Department of Physics, Clark University
• Collaborators K. Safford, G. Lumay (Clark),
• D. Volfson, L. Tsimring (UCSD)
• M. Kardar (MIT), Y. Kantor (Tel-Aviv)
• Supported by National Science Foundation
  DMR-0605664

2
Background
Bacillus subtilis colony in a peptone and agar
rich thin layer
9 mm long rod shaped, driven by flagella, change
direction by tumbling
Low agar (viscosity)
High agar (viscosity)
Delprato, Samadani Kudrolli (2001)
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Importance of shape in thermal systems
Beyond spherical particles
Entropy maximization leads to long range order
Onsager (1949)
Tobacco mosaic virus (X35,000) The American
Phytopathological Society
4
Elephant seal colonies
California, 2005
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Motivation

• Examine the structure and dynamics of athermal
  apolar or polar, rigid or flexible rods which
  interact only during contact
• Structure
• can the configurations be described by polymer
  models (Doi Edwards)
• are ordering transitions observed with density
• Dynamics
• how does the diffusion scale with rod length,
  density
• convection? vortices
• ratchet motion
• random walk, directed random walk
• Active particle hydrodynamics

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Structure and dynamics of particles driven by
vibration
Rigid or Flexible polar rods
Anisotropy
Rigid or Flexible apolar rods
Area/Volume fraction

• Flexible rods beaded chain with and without a
  head
• Rigid rods - Rods, dimers, Robo-bug

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Experimental apparatus
High Frame Rate Camera (1k x 1k pixels)
N 1024
a G sin(2p f t)

• Experimental parameters
• 3g, f 30 Hz
• Particle diameter d 3.12 mm, connected by links
  0 to 1.5 mm
• Bead number
• N 1 - 1024

-D. Blair
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Self-avoiding random walk polymer model
Characteristic size, Radius of gyration

• ¾ in two dimensions
• ½ for ideal chain

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Persistence length of the chain

• Assuming each step in the chain can turn with a
  maximum angle of p/4 and the angles are uniformly
  distributed between p/4 and -p/4
• Persistence length Nc 10
• using
• ltcos(qn q1)gt ltcos(q2 q1)gtn-1 exp(
  (n-1)/Nc)
• gives
• Nc 1/ln(ltcos(q2 q1)gt) with ltcos(q2 q1)gt
  2 sqrt(2)/ p

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Radius of Gyration
Experiment
RW
2
SAW
10
d
8
/
6
g
R
4
2
1
4
6
2
4
6
2
4
6
2
10
100
1000
N

• Random Walk and Self-Avoided Walks simulations
  performed by Yacov Kantor
• Walks have a persistence length and were
  confined to a circle

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Pair correlation
for q Rg lt 1
Unconfined SAW model
Doi, Edwards (1999)
for q Rg gt 1
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Comparison with a self avoided walk simulation
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Dynamics of the chain
center of mass of the chain
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Diffusion constant versus chain length
Rouse model - assumes that each monomer
experiences a viscous drag proportional to its
velocity
kT -gt ½ mltv2gt 0.5 J, the granular temperature
is constant ? 2.87 10-2 N-m-1s, the drag
coefficient
Thus, vibrated surface acts like a thermal fluid
which gives and takes energy
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Dynamic Structure

• In the limit of q Rg ltlt 1, rn rm gtgt Rg and t
  large, g(q,t) -gt N exp(-k2 t/ D)

Structure and dynamics of vibrated granular
chains Comparison to equilibrium polymers, K.
Safford, Y. Kantor, M. Kardar, and A. Kudrolli,
PRE (accepted)
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• Next examine diffusion as a function of number of
  chains

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Diffusion as a function of density
f 0.12
f 0.48
Chains of length N 8
Area fraction f n N (d/D)2
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Displacements parallel and perpendicular to
principal axis
Parallel Component
Perpendicular Component

• Diffusion decreases with area fraction
• Perpendicular component decrease faster

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Dynamics of non-spherical rigid particles on
vibrated plate
S. Dorbolo, D. Volfson, L. Tsimring, A.K., PRL
(2005)

• Frictional interaction with substrate and
  symmetry breaking leads to translation motion in
  rods and dimers
• Derived relation for translation speed as a
  function of aspect ratio, vibration amplitude and
  frequency

Also Stochastic dynamics of a rod bouncing upon
a vibrating surface, Wright et al, PRE (2006)
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Collective dynamics of rigid rods
Slip Reversal regime
Slip-Stick regime
where, Vz V0/2
- D. Blair, T. Neicu, A.K. (2003), D. Volson,
A.K., L. Tsimring, PRE (2004)
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Self-organized vortex patterns observed with rods

• Driving
• a/g
• f 50 Hz,
• N 8000
• nf 0.535

Rods l 6.2 mm d 0.5 mm N 0
10000 Thanks to Seth Fraden
G 4.2
N Number of rods Nmax Rods required to
obtain a vertically aligned monolayer
D. Blair, T. Neicu, A.K. (2003)
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Phase diagram
Vortex motion
Nematic

• Vortex motion observed when rods are tilted with
  respect to horizontal
• Convective motion is not observed when rods are
  horizontal in our case
• Isotropic-Nematic transition studied in detail
  by Galanis et al, PRL (2006)
• Particle currents observed in vibrated rods by
  Narayan et al, Science (2007). Aranson et al,
  (2007) observe swirls when horizontal vibration
  component added

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Asymmetric Dimer Ratchets Robo-Bug
Simplest examples of noise driven motors
Steel-Glass beads
Aluminum-Steel beads

• Symmetry breaking leads to directed motion
• Sano et al PRE (2003) observed directed motion
  with a bolt shaped particle

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Polar Rods
- Geoffroy Lumay
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Cooperative dynamics with polar rods
f 0.3, G 2

• Particles migrate to the boundary on a flat bed
  under low noise conditions

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Alignment at boundaries observed in bacterial
colonies?
Bacillus subtilis colony
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Cooperative dynamics with polar rods
f 0.3, G 4

• Particles are uniformly distributed at higher
  excitation

28
Event driven simulation model
Dmitri Volfson and Lev Tsimring
Simulation of polar rods moving on a substrate
inside a circular boundary
Event Driven Simulations
R60, dr 0.732, Lr/dr 3.5 (from tip to tip)
mrr mrw 0.3, err erw 0.9, Cvdamp 0.5 Also
Peruani et al, PRE (2006)
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Event driven simulation model
Dmitri Volfson and Lev Tsimring
R60, dr 0.732, Lr/dr 3.5 (from tip to tip)
mrr mrw 0.3, err erw 0.9, Cvdamp 0.5 Also
Peruani et al, PRE (2006)
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Cooperative Dynamics
G 2g
G 4g
G 3g
Boundary aggregation vanishes for small aspect
ratio
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Velocity field of the polar rods
G 3, t 5s
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Spatial velocity correlation
Velocityrod director correlation

• Velocity strongly correlated with director even
  in presence of rod-rod collisions

• Correlation length is small and therefore system
  is in disordered state

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Particle number fluctuations
N

• Tu Toner (1997), Toner (personal communication)
  predict fluctuations to scale as N7/12 for
  ordered polar self-propelled particles
• Fit gives an exponent close to 2/3 Open question

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Swarming and swirling in self-propelled granular
rods, A. K., G. Lumay, D. Volfson, and L.
Tsimring, PRL (2008)

• Rigid polar rods are trapped at the boundary
  under low noise conditions
• Incipient clustering observed due to interplay
  between directed motion and particle shape even
  in disordered regime

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Diffusion of flexible self-propelled polar
particleConstruction of a particle with a head
and a flexible tail
Tail

Head

• Two types of surfaces
• Sand-blasted surface with 50 mm roughness
• Layer of 1 mm steel beads glued on vibrated
  surface

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Trajectory on a smooth substrate

• Motion of the SPP over 2.5s (left), 2000s
  (right).
• Confinement becomes important over long times.

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Rough versus smooth substrate

• Motion on a rough substrate quickly becomes
  diffusive, but on smooth substrate motion appears
  super-diffusive because of persistent nature of
  motion

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Dynamics of self-propelled particles as a
function of area fraction in two dimensions
Area fraction 0.08
Area fraction 0.5
Area fraction 0.6
SPP appear trapped
SPP diffuse
(Black) Head 3 (White) beads Aspect ratio 7
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Mean square displacement

• Diffusion decreases with volume fraction, but
  consistent with normal diffusion below jamming
• Need to do experiments over longer time interval

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Conclusions

• Polymer models give good description of
  configurations and dynamics of vibrated granular
  chains
• Diffusion decreases in perpendicular direction as
  chain concentration is increased
• Self-propelled particles can be constructed by
  using asymmetric mass distributions, motion
  described by persistent random walk models
• Show novel aggregation patterns such as swarming
  ring without any potential attractant

http//physics.clarku.edu/akudrolli