slides

Swimming at small scales:
mechanisms
and
emergent structures
I.
Pagonabarraga
University of Barcelona
1. Introduction
2. Swimming at small scales
3. Model swimmers
4. Collective swimming
5. Chemical swimmers
6. Molecular motors
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
Flagella
E.coli
Volvox
Amoeba
Cilia
Ciliated
Microorganisms
Paramecium
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
Ratchets
Emerging patterns and phases
2. Swimming
Re = uL/ν
2. Swimming
Re = uL/ν
L~µm
u~10 µm/s
ν~10-2 cm2/s
Inertia irrelevant
Reciprocal motion
(1 degree of freedom)
Purcell, Am.J.Phys. (1977)
Simple models: wave propagation
body deformation
Najafi et al
Purcells’ theorem
Reversibility: no net displacement
Bibette et al
2. Swimming
Re = uL/ν
L~µm
u~10 µm/s
ν~10-2 cm2/s
Inertia irrelevant
Reciprocal motion
(1 degree of freedom)
Purcell, Am.J.Phys. (1977)
Purcells’ theorem
Reversibility: no net displacement
Bibette et al
3. Model swimmers
Each move change in HI
Decompose cycle in 4 moves
Total displacement in a cycle
Mean velocity
Expansion in small displacement
3. Model swimmers
Each move change in HI
Decompose cycle in 4 moves
Low Reynolds
Hydrodynamic coupling
Oseen tensor
Force free motion
3. Actuated swimmers
Magnetic field: Local acceleration
Asymmetry + local bending: wave
Top view
Doublet vs isolated
Bibette et al
Confinement + asymmetric mobility
no deformation
Larger aggregate
3. Actuated swimmers
Difference in friction along cycle
Tierno et al. PRL 2010
Open trajectories: no net motion
Persistent ballistic motion
3. Actuated swimmers
Two regimes: optimal velocity
Different sensitivity to driving
General behavior for general assemblies
Paramagnetic colloids
3. Actuated swimmers
Pair velocities: Low Re
Constraint: COM + orientation
linear relation
Effective mobilities
Asymmetric particle/wall friction
For circular trajectory:
Net average velocity
3. Actuated swimmers
Angular velocity: Low Re
Kinematic constraint
Circularly polarized magnetic field
Characteristic frequency
3. Actuated swimmers
Good agreement
Fitting: height
rotational friction
Low Ω : increase rectification rate
change in orientation negligible
High Ω : doublet aligns parallel to wall
decrease in asymmetry
Controlled motion in microfluidic
devices
4. Cilliated swimmers
Squirmers
Metachronal wave on Opalina, Paramecium.
Fixed tangential velocity profile on the
surface (Lighthill, 1952; Blake, 1971)
Opalina
Surface tangential velocity
β=B2/B1
Propulsion velocity
4. Cilliated swimmers
Squirmers
Metachronal wave on Opalina, Paramecium.
Fixed tangential velocity profile on the
Opalina
surface (Lighthill, 1952; Blake, 1971)
Balantidium coli
Surface tangential velocity
oligotrich
β=B2/B1
Propulsion velocity
4. Cilliated swimmers
Collective distorsion of cillia
small compared with body object
effective waves
Time-dependent slip velocity
Solve Stokes equation
4. Cilliated swimmers
Force-free swimmer
sets the self-propelling velocity
Disregard normal oscillations + focus in average flow field
Active stress induced by the effective slip velocity?
4. Cilliated swimmers
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
chlamydomonas
v~1/r2
β>0
Contractile
β<0
Extensile
B1=0
B2≠0
β=0
Passive
squirmer
Apolar
v~1/r3
5. Squirmer suspensions
Time scales: analogous to passive colloids
τr ∼ R2/ν
τr << τm << τD
τm∼R/u∞
Diffusion induced by collisions
τD∼R2/D
Size~ 1-500 µm
Speeds < 100 µm/s
Neglect thermal fluctuations
fluid inertia (Re<0.01)
τD/τm = u∞ R/D= Pe > 1
Paramecium
Opalina
u∞ ~1000 µm/s u∞ ~100 µm/s
R ~200 µm
Re ~ 0.2
R ~200 µm
Re ~ 0.02
5. Squirmer suspensions
Intermediate times t ~ τm
HI align propellers
effective interactions
Relevance near field couplings
Particle acceleration
Decay vacf
Transient aggregate formation
Pedley 2006
5. Squirmer suspensions
Long times t ~ τD
φ=0.1
Relevance near field couplings
Particle acceleration
φ=0.3
Structure:
aggregates
orientational order
finite life time
5. Squirmer suspensions
Mean square displacement 2D
Super-diffusive regime
Asymetric driving
Low volume fractions
φ=0.1
φ=0.4
Flocking models
transitions at high φ
Percolating clusters
Additional interactions stabilize clusters
Detailed analysis
analogies with spps’ transitions
5. Squirmer suspensions
Contractile
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6.Colloidal phoresis
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))
6.Colloidal phoresis
Concentration interacts with a solid surface
delocalized membrane?
Concentration gradient parallel to the surface
asymmetric change in chemical potential
Generates pressure gradient
Due to asymmetry induced by wall
Surface-induced flow
30
6. Colloidal phoresis
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))
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
Autonomous motion
What if particles might generate concentration gradients?
self-propulsion!!!
(Paxton et al, JACS 126, 13424 (2004))
Janus
(Golestanian et al, PRL 94, 22081 (2005))
activity modelled by simple updating rule
non-conserved dynamics for φ!
particle surface activity
Janus particle
The velocity of an isolated particle of
constant mobility µ can be computed exactly
7.Collective dynamics in “2D”: phoretic mobility?
7. Density fluctuations
7. Repulsive chemical swimmers
No hydrodynamics
Towards a crystal structure
Faster dynamics
larger number of “defects”
7. Atractive chemical swimmers
No hydrodynamics
Cluster formation
37
7. Radial distribution functions
7. Statistics and geometry of clusters
7. Orientational radial distribution functions
where
being the angle between the orientation
vector of the reference particle and the
one of the generic particle in
Hydrodynamics:
induces particle rotation
Leads to disorder / induced flows
10. Conclusions
Active matter
Release energy at small scales (natural/synthetic)
intrinsically out of equilibrium
New mechanisms to develop patterns and structures
Variety of mechanisms to swim
Mesoscopic modeling simple swimmers / chemical swimmers
Identify chemical/hydrodynamic interactions
Induced flows
Cluster formation
Enhanced fluctuations / pumping
Phoretic mobility relevant role
(non-eq) transition from repulsive crystal to clustering
Chemical signaling mainly determines clustering
hydrodynamics limits cluster growth