Active dumbbells Leticia F. Cugliandolo Universit Pierre et Marie - - PowerPoint PPT Presentation
Active dumbbells Leticia F. Cugliandolo Universit Pierre et Marie Curie Sorbonne Universits Institut Universitaire de France leticia@lpthe.jussieu.fr www.lpthe.jussieu.fr/ leticia Work in collaboration with D. Loi & S. Mossa
Work in collaboration with
Kyoto, Japan, August 2015
Active dumbbell system
Diatomic molecule - toy model for bacteria Escherichia coli - Pictures borrowed from internet.
Active matter
Rabani, Ariel and Be’er 13
Diatomic molecule
Two spherical atoms with diameter σd and mass md Massless spring modelled by a finite extensible non-linear elastic force bet- ween the atoms Ffene = −
with an additional repulsive contribution (WCA) to avoid colloidal overlapping. Polar active force along the main molecular axis Fact = Fact ˆ
Purely repulsive interaction between colloids in different molecules. Langevin modelling of the interaction with the embedding fluid: isotropic viscous forces,−γvi, and independent noises, ηi, on the beads. Directional motion (active) and effective torque (noise)
Control parameters
Number of dumbbells N and box volume S in two dimensions: packing fraction
d N
Energy scales: Active force work Factσd thermal energy kBT Péclet number Pe = 2Factσd
Active force Factσd/γ viscous force γσ2
d/md
Reynolds number Re = mdFact
. We keep the parameters in the harmonic (fene) and Lennard-Jones (repulsive) potential fixed. Stiff molecule limit: vibrations frozen. We study the φ, Fact and kBT dependencies. Pe ∈ [0, 40], Re < 10−2
Phase segregation Fixed packing fraction ϕ and fixed activity Fact, vary kBT
Mixed Large density fluctuations Segregation
Pe = 2Factσd
increases →
Gonnella, Lamura & Suma 13
Phase diagram : from the distribution of local dumbbell density
0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200
φ Pe
Mechanism for aggregation: note the head-tail alignment in the cluster.
Phase diagram
0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 200
φ Pe
Focus on the dynamics in the homogeneous phase ; vary φ and Pe.
Active force switched-on, Fact ̸= 0
ballistic → diffusive → ballistic → diffusive
d/(2kBT)
Note that ta → ∞ at kBT → 0.
10−3 10−2 10−1 100 10−1 100 101 102 103 104
〈∆r2
cm〉/4t
t
tI ta
Pe=0 Pe=2 Pe=20 Pe=40
Active force switched on, Fact ̸= 0
d/(2kBT)
Note that ta → ∞ at kBT → 0.
10−3 10−2 10−1 100 10−1 100 101 102 103 104
〈∆r2
cm〉/4t
t
tI ta
Pe=0 Pe=2 Pe=20 Pe=40 10−3 10−2 10−1 10−2 10−1 100 101 102 103 104
〈∆θ2〉/2t t
tI ta
Pe=0 Pe=2 Pe=20 Pe=40
Centre-of-mass mean-square displacement
cm⟩ = ⟨[rcm(t + t0) − rcm(t0)]2⟩
10−5 10−4 10−3 10−2 10−1 10−3 10−2 10−1 100 101 102 103 104
〈∆r2
cm〉/4t
t
Pe=2
φ=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−5 10−4 10−3 10−2 10−1 10−3 10−2 10−1 100 101 102 103 104
〈∆r2
cm〉/4t
t
Pe=40
φ=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
tI ta t∗ tI t∗ ta
Pe and φ effect
Angular mean-square displacement
10−4 10−3 10−2 10−1 10−3 10−2 10−1 100 101 102 103 104
〈∆θ2〉/2t t
Pe=2
φ=0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−4 10−3 10−2 10−3 10−2 10−1 100 101 102 103 104
〈∆θ2〉/2t t
Pe=40
φ=0.1 0.2 0.3 0.4 0.5 0.6 0.7
tI ta t∗ tI t∗ ta
Pe and φ effect
cm⟩ ≃ 2dDAt
0.05 0.075 0.1 0.125 0.2 0.4 0.6
DA(Fact,T,φ)/kBT φ
Pe=4 Fact=0.1 T=0.05 Pe=4 Fact=0.5 T=0.25 Pe=4 Fact=1 T=0.5 a1=-0.76 a2=-0.41
2 4 6 8 0.2 0.4 0.6
DA(Fact,T,φ)/kBT φ
Pe=40 Fact=0.1 T=0.005 Pe=40 Fact=0.5 T=0.025 Pe=40 Fact=1 T=0.05 a1=-2.40 a2=1.66
Translational diffusion diminishes at increasing density at all Pe increases at increasing Pe at fixed φ Proposals for φ, Pe dependence Similar to what observed for e.g., Janus particles in H2O2 Zheng et al 13
0.2 0.4 0.6 0.2 0.4 0.6
DR(T,φ)/kBT φ
Pe=4 Fact=0.1 T=0.05 Pe=4 Fact=0.5 T=0.25 Pe=4 Fact=1 T=0.5
0.2 0.4 0.6 0.2 0.4 0.6
DR(T,φ)/kBT φ
Pe=40 Fact=0.1 T=0.005 Pe=40 Fact=0.5 T=0.025 Pe=40 Fact=1 T=0.05
Rotational diffusion enhanced at increasing density for large Pe Incipient clusters
Translational motion in the active-force driven regimes
10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
Pe=40 20 4 2
10-6 10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
Pe=40 20 4 2
〈∆r2
cm〉/4t
t
Pe=2
φ=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−5 10−4 10−3 10−2 10−1 10−3 10−2 10−1 100 101 102 103 104〈∆r2
cm〉/4t
t
Pe=40
φ=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Non-Gaussian at high Pe
Translational motion in the active-force driven regimes
10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
Pe=40 20 4 2
10-6 10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
Pe=40 20 4 2
Janus particles in H2O2
Same double peak at high Pe Zheng et al. 13
Translational motion in the active-force driven regimes
10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
III
φ=0.01 0.1 0.3 0.5 0.7
10-5 10-4 10-3 10-2 10-1 100 101
1 2 3 4
σxP(∆x) ∆x/σx
IV
φ=0.01 0.1 0.3 0.5 0.7
〈∆r2
cm〉/4t
t
Pe=40
φ=0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Pe = 40
Non-Gaussian & exponentail tails in III
Translational motion in super-cooled liquids and granular matter
i=1⟨δ(r − |⃗
van Hove correlation function delay-time shorter than the structural relaxation time t < tα
Exponential tails Chaudhuri, Berthier & Kob 07
Rotational motion in the active-force driven regimes
10-5 10-4 10-3 10-2 10-1 100
2 4
σθP(∆θ) ∆θ/σθ
Pe=40 20 4 2
10-5 10-4 10-3 10-2 10-1 100
2 4
σθP(∆θ) ∆θ/σθ
Pe=40 20 4 2
〈∆θ2〉/2t t
Pe=2
φ=0.1 0.2 0.3 0.4 0.5 0.6 0.7 10−4 10−3 10−2 10−3 10−2 10−1 100 101 102 103 104〈∆θ2〉/2t t
Pe=40
φ=0.1 0.2 0.3 0.4 0.5 0.6 0.7Low density
Gaussian
Rotational motion in the active-force driven regimes
10-5 10-4 10-3 10-2 10-1 100
2 4
σθP(∆θ) ∆θ/σθ
III
φ=0.01 0.1 0.3 0.5 0.7
10-5 10-4 10-3 10-2 10-1 100
2 4
σθP(∆θ) ∆θ/σθ
IV
φ=0.01 0.1 0.3 0.5 0.7
〈∆θ2〉/2t t
Pe=40
φ=0.1 0.2 0.3 0.4 0.5 0.6 0.7Pe = 40
Exponential tails for φ ≥ 0.7
Phase diagram
& Suma et al. work in progress
Diatomic molecule
Two spherical atoms with diameter σd and mass md Massless spring modelled by a finite extensible non-linear elastic force bet- ween the atoms Ffene = −
with an additional repulsive contribution (WCA) to avoid colloidal overlapping. Polar active force along the main molecular axis Fact = Fact ˆ
Purely repulsive interaction between colloids in different molecules. Langevin modelling of the interaction with the embedding fluid: isotropic viscous forces,−γvi, and independent noises, ηi, on the beads. Directional motion (active) and effective torque (noise)
Spherical particles
Spherical particle with diameter σtr and mass mtr Very low tracer density φtr ≪ φ No polar active force Ftr
act = 0
Purely repulsive interaction between colloids in different molecules & tracers. Langevin modelling of the interaction with the embedding fluid: viscous forces,−γtrvtr
α , and independent noises, ηtr α , on the tracers.
We will distinguish thermal γtr ̸= 0 from athermal γtr = 0 tracers
Spherical tracers to probe the dynamics of the “active bath"
Gonnella, Laghezza, Lamura, Mossa, Suma & LFC
Thermal vs. athermal
10−4 10−3 10−2 10−1 100 10−3 10−2 10−1 100 101 102 103
Pe=40 〈∆r2 〉/4t t
athermal tracer thermal tracer dumbbells Dtr(φ=0)=0.005
thermal
athermal
Study of the dependence on mtr, ϕ, and other parameters
Suma, Gonnella & LFC in preparation
Active density dependence of the tracer’s diffusion constant
Wu & Libchaber 00 bacteria Leptos et al. 09 algae
Is this captured by this model (with no hydrodynamics) for some parame- ters ?
Active dumbbell system
Four dynamic regimes even at finite φ
Complex (though simpler than in just passive colloids, cfr. Tokuyama & Oppenheim 94) dependence of translational diffusion constant on φ Enhanced rotational diffusion constant for increasing φ < 0.5 More complex than Pe2 corrections at finite φ
w/ Gonnella, Laghezza, Lamura, Mossa & Suma via FDT In progress : potential and kinetic tracers coupled to the active dumbbells, always in homogenous phase
Tracer’s velocities & effective temperature Spherical particles with mass mtr that interact with the active matter.
2 4 6 8
v mtr
1/2
10
10
10
p(v) / mtr
1/2
10 30 50 100 300 1000 3000
10
1
10
2
10
3
10
4
10
5
10
6
mtr
1 2 3 4 5 6 7 8
Teff (mtr ) mtr a) b)
Maxwell pdf of tracers’ velocities v at an effective temperature Teff(mtr)
Loi, Mossa & LFC 07-09
Passive Leannard-Jones system
T T b m tr m tr hv 2 z i 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 10 1 1 0.8 0.6 0.4 0.2The kinetic energy of a tracer particle (the thermometer) as a function of its mass (τ0 ∝ √mtr)
1 2mtr⟨ v2 z⟩ = 1 2kBTeff.
J-L Barrat & Berthier 00
Same measurement in active dumbbell sample to compare with measu- rements of Teff from FDT.