Hydrodynamic cooperativity in
active suspensions:
Self organization and
cluster phase formation
I.
Pagonabarraga
University of Barcelona
1. Introduction
2. Micro-Swimmers
3. Swimmer suspensions
4. Chemical suspensions
5. Interacting bacterial colonies
6. Conclusions
2
1. Introduction
Active systems: collections of elements able to convert internal energy
into mechanical work (autonomous motion)
intrinsically out-of-equilibrium systems (even
in a steady state, if any, and without external forcing)
Examples
flocks of birds
bacterial colonies
artificial self-propelled objects
molecular motors
1. Introduction
Actuated colloids
Adding reactivity:
new propelling mechanisms
Different sets of micro/nano robots
Heterogeneous particles
Confinement + asymmetric mobility
no deformation
Desired structures, adaptive, capable of self repair
1. Introduction
Energy from small scales
Systems intrinsically out of equilibrium
Nonequilibrium distributions
no detailed balance
Galajda et al. J Bacter. (2007)
Ratchets
Kaiser et al. PRL (2012)
Emerging patterns and phases
2. Propellers
Flow measurements
Chlamydomonas
Time dependent flow
cycle averaged
Guasto et al. PRL (2010)
Thutupalli et al. NJP (2011)
emulsions
Flagellum
dipolar flow
6
Drescher et al. PNAS (2011)
2. Propellers
Dynamics of active particles
Low Reynolds numbers
Absence of external driving
closer to electrophoresis?
Relevance of swimming mechanism
Fluid flows with vorticity
Coupling translation/rotation
relevance of near field interactions
2. Propellers
Squirmers
Metachronal wave on Opalina, Paramecium.
Fixed tangential velocity profile on the
Opalina
surface (Lighthill, 1952; Blake, 1971)
Surface tangential velocity
β=B2/B1
Steady squirmer
(Pedley 1986)
Propulsion velocity
2. Propellers
Squirmers
Metachronal wave on Opalina, Paramecium.
Fixed tangential velocity profile on the
surface (Lighthill, 1952; Blake, 1971)
Opalina
Balantidium coli
Surface tangential velocity
oligotrich
β=B2/B1
Steady squirmer
(Pedley 1986)
Propulsion velocity
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chlamydomonas
v~1/r2
β>0
extensile/pullers
β<0
contractile/pushers
B1=0
B2≠0
β=0
Passive
squirmer
Apolar
v~1/r3
2. Propeller suspensions
Ishikawa et al. JFM (2006)
HI align propellers
translation/rotation coupling
effective interactions
Relevance near field couplings
Particle acceleration
Nematogenic order in transient clusters
Decay vacf
Transient aggregate formation
3. Squirmers with attractive interactions
Stokes Law, small Reynolds
η = 0.5, Rp = 2.3
Transition to an ordered phase:
LJ interaction strength is reduced and
B2 is not too big.
Squirmer attraction
enhances cohesion
destroys ordering
3. Cluster morphologies
Dynamic structures
Morphological
characterization?
Density fluctuations
Zhang et al. PNAS (2010)
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3. Squirmer suspensions: density fluctuations
∆N ∼ N α
0.8
2
0.75
1
β
0.7
0
α
Quantify degree of ordering
sensitive to active stresses
distinguish puller/pusher
0.85
3
0.65
-1
0.6
Effect on density fluctuations
favours large dynamic clusters
-2
0.55
-3
0.5
0
0.5
1
ξ −1
1.5
2
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3. Squirmer suspensions: density fluctuations
Need to reach large system sizes
Strong correlations
Significant finite size effects
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3. Squirmer suspensions: Cluster distributions
competition hydrodynamics/attraction
Power-law decay
Wide range
dynamic structures
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pullers β=3
pullers β=1/2
pushers β=-3
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3. Motility-induced phase separation
Active Brownian Particles
Particles trapped
regions low motility
Stenhammar et al. (2014)
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3. Motility-induced phase separation
Active Brownian Particles
Particles trapped
regions low motility
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Stenhammar et al. (2014)
3. Squirmer suspensions: Cluster distributions
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3. Squirmer suspensions
velocity distributions
impact of HI
wider distributions for pullers
Generic polar order inside clusters
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4. Colloidal phoresis: a multiscale “transport phenomenon”
Phoretic transport: motion of colloidal particle under the effect
of external field (electric field, concentration/temperature gradients)
(solute molecules size)/(colloidal particle size)
large scale aggregates
R
(Sen et al, Faraday Discuss. 143, 15 (2009))
need for coarse-graining
4. Modeling colloidal phoretic transport
The presence of the solute can be taken into account by means of an effective “slip
velocity” as boundary condition for the flow on the colloid surface
surface phoretic mobility
colloid/solute interaction potential (short-ranged)
velocity of a (spherical)
particle of radius R
(Anderson, ARFM 21, 61 (1985); Golestanian et al, New J. Phys. 9, 126 (2007))
4. Test case: chemotaxis
Directed motion of a colloidal particle
in a linear concentration profile
For constant mobility the propulsion velocity
can be calculated exactly
chemoattractant
chemorepellent
4. Autonomous motion
What if particles might generate concentration gradients?
self-propulsion!!!
(Paxton et al, JACS 126, 13424 (2004))
(Golestanian et al, PRL 94, 22081 (2005))
activity modelled by simple updating rule
non-conserved dynamics for φ!
particle surface activity
Inclusion of a global source term
(mimicks coupling with an external “bath” of concentration field)
Active suspensions: chemical swimmers
In particular we use an asymmetric surface activity of the form
Janus particle
The velocity of an isolated particle of
constant mobility µ can be computed exactly
4. Collective dynamics in “2D”: phoretic mobility?
Teurkhauff et al. PRL (2012)
Palacci et al. Science (2013)
4. Repulsive chemical swimmers
No hydrodynamics
Towards a crystal structure
Faster dynamics
larger number of “defects”
4. Attractive chemical swimmers
No hydrodynamics
Cluster formation
4. Density fluctuations
A proper indicator to distinguish dynamical regimes?
Use variance of Voronoi tesselation
4. Radial distribution functions
Clustering regime
With hydrodynamics
Without hydrodynamics
No Hydro: larger friction
4. Statistics and geometry of clusters
5. Bacterial colonies
Reproduction
Motility (D bacteria >> D colloids)
E. Coli
run and tumble
Bacterial interactions (collective behavior)
excluded volume
quorum sensing (signaling molecules)
Speed
v
Tumbling rate τ
External interactions (medium)
taxis (chemotaxis)
gravity/shearing
Deff ~ v2 τ/3
L~ 1 µm -> Deff ~ 102 µm2/s
Passive colloid -> D~ 1/2 µm2/s at 300 K
5. Bacterial colonies
Non interacting bacteria
Fisher equation
Vw = 2
Colony advances generating a front
Reaction to driving fields?
Gravity on a vessel
Non-equilibrium phase transition
Γ
Barret-Freeman et al. PRL (2008)
5. Interacting bacterial colonies
Interacting bacteria
simplest models?
physical ingredients?
Neighbours affect mean velocity
Lack of detailed balance
Run and tumble
Induced net motility
Density explored region
Passive colloids
“Effective free energy”
Effective bacteria diffusivity
5. Interacting bacterial colonies
Simplest functional form?
Negative diffusivity
Spinodal-like instability
Linear stability analysis
Identify relevant parameters
5. Interacting bacterial colonies
Banded phase
Bacteria migrate from low to high density phases
… but at high ρ high death rate
Arrested coarsening
Two different length scales
Relevant role of density environment
5. Interacting bacterial colonies
Circular domains in 2D
Phase diagram controlled by 2 parameters
Cates et al. PNAS (2010)
5. Interacting bacterial colonies
Starting from an inoculum
Reminiscent of observed
patterns
Connection to chemotaxis?
Chemotactic density slaved to bacteria evolution ?
Alternative (physical) mechanisms to colony morphologies
Helps identifying potential relevant parameters underlying colony structures
5. Sedimenting bacterial colonies
Sensitivity to external forcing?
λ
Sensitivity to external forcing?
Disappearance of
inhomogeneous patterns at large Γ
5. Sedimenting bacterial colonies
At low Γ steady patterns stable -> change their relative stability
New phases: unsteady
reminiscent of unsteady patterns
5. Sedimenting bacterial colonies
Order parameters
homogeneous/inhomogeneous
lamella / nematic
unsteady phases
Lamellar phases
defects
larger sensitivity to gravity
Raining phases
frequency controlled by external driving
5. Colonies in fluid flows
Microorganisms pushed by local advection
+ feedback of fluid:
active stresses
Need for consistent dynamic evolution
5. Colonies in fluid flows
Active flow
Fisher velocity
Non-equilibrium phase diagram
Perturbed morphologies
Unsteady patterns
Transition to chaos?
5. Colonies in fluid flows
Induced flows
symmetry of bacterial patterns
metabolical relevance
6. Conclusions
Active matter
Release energy at small scales (natural/synthetic)
intrinsically out of equilibrium
New mechanisms to develop patterns and structures
Competition between attraction/activity
Interplay hydrodynamics/attraction
Large density fluctuations
Macroscopic cluster
Induce polar ordering: dominant effect of translation/rotation
Dynamic clusters
Interplay hydrodynamics/attraction
Large density fluctuations
Interacting bacterial colonies
relevance of collective motility
chemotaxis without chemicals
identify essential parameters
Relevance of hydrodynamics in microorganisms
simple couplings set up aggregation schemes
collective transport and patterns at different scales
Acknowledgements
Francisco Alarcón
University of Barcelona
Ricard Matas
University of Barcelona
Davide Marenduzzo
Mike Cates
University of Edinburgh
Barcelona Supercomputing Center
PRACE
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