Active dumbbells Leticia F. Cugliandolo Universit Pierre et Marie - PowerPoint PPT Presentation

Active dumbbells Leticia F. Cugliandolo Université Pierre et Marie Curie Sorbonne Universités Institut Universitaire de France leticia@lpthe.jussieu.fr www.lpthe.jussieu.fr/ ˜ leticia Work in collaboration with D. Loi & S. Mossa (2007-2009) and G. Gonnella, G.-L. Laghezza, A. Lamura, A. Mossa & A. Suma (2013-2015) Kyoto, Japan, August 2015

Motivation & goals Active dumbbell system • Reason for working with this model • Main properties of the model - phase diagram • Translational and rotational collective motion • Dynamics of tracers in complex environments revisited. • Effective temperatures out of equilibrium

Active dumbbell Diatomic molecule - toy model for bacteria Escherichia coli - Pictures borrowed from internet.

Bacteria colony Active matter Rabani, Ariel and Be’er 13

Active dumbbells Diatomic molecule Two spherical atoms with diameter σ d and mass m d Massless spring modelled by a finite extensible non-linear elastic force bet- k r ween the atoms F fene = − with an additional repulsive contribution 1 − r 2 /r 2 0 (WCA) to avoid colloidal overlapping. Polar active force along the main molecular axis F act = F act ˆ n . Purely repulsive interaction between colloids in different molecules. Langevin modelling of the interaction with the embedding fluid: isotropic viscous forces, − γ v i , and independent noises, η i , on the beads. Directional motion (active) and effective torque (noise)

Active dumbbells Control parameters Number of dumbbells N and box volume S in two dimensions: φ = πσ 2 d N packing fraction 2 S Energy scales: Active force work F act σ d Pe = 2 F act σ d Péclet number k B T thermal energy k B T Active force F act σ d /γ Re = m d F act Reynolds number . viscous force γσ 2 σ d γ 2 d /m d We keep the parameters in the harmonic (fene) and Lennard-Jones (repulsive) potential fixed. Stiff molecule limit: vibrations frozen. We study the φ , F act and k B T dependencies. Pe ∈ [0 , 40] , Re < 10 − 2

Active dumbbells Phase segregation Fixed packing fraction ϕ and fixed activity F act , vary k B T k B T = 0 . 01 k B T = 0 . 003 k B T = 0 . 001 Mixed Large density fluctuations Segregation Pe = 2 F act σ d increases → k B T Gonnella, Lamura & Suma 13

Active dumbbells Phase diagram : from the distribution of local dumbbell density 1.0 0.8 0.6 φ 0.4 0.2 0.0 0 50 100 150 200 Pe Mechanism for aggregation: note the head-tail alignment in the cluster.

Active dumbbells Phase diagram 1.0 0.8 0.6 φ 0.4 0.2 0.0 0 50 100 150 200 Pe Focus on the dynamics in the homogeneous phase ; vary φ and Pe.

Single molecule limit Active force switched-on, F act ̸ = 0 ballistic → diffusive → ballistic → diffusive • The dynamics is accelerated by F act and a new ballistic regime in the centre- t ∗ = 16 t a / Pe 2 of-mass translational motion appears at t a = γσ 2 • Ballistic to diffusive crossover of the cm motion at d / (2 k B T ) Note that t a → ∞ at k B T → 0 . D A = k B T/ (2 γ ) (1 + Pe 2 ) • The diffusion constant is 10 0 Pe=0 Pe=2 Pe=20 Pe=40 10 −1 cm 〉 /4t 〈∆ r 2 10 −2 10 −3 10 −1 10 0 10 1 10 2 10 3 10 4 t t I t a ⟨ [ r cm ( t + t 0 ) − r cm ( t 0 )] 2 ⟩

Single molecule limit Active force switched on, F act ̸ = 0 • The dynamics is accelerated by F act and a new ballistic regime in the centre- t ∗ = 16 t a / Pe 2 of-mass translational motion appears at t a = γσ 2 • Ballistic to diffusive crossover of the cm motion at d / (2 k B T ) Note that t a → ∞ at k B T → 0 . • The rotational motion is not affected by the longitudinal active force. 10 0 10 −1 Pe=0 Pe=0 Pe=2 Pe=2 Pe=20 Pe=20 Pe=40 Pe=40 10 −1 cm 〉 /4t 〈∆θ 2 〉 /2t 10 −2 〈∆ r 2 10 −2 10 −3 10 −3 10 −1 10 0 10 1 10 2 10 3 10 4 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t t t I t a t I t a ⟨ [ r cm ( t + t 0 ) − r cm ( t 0 )] 2 ⟩ ⟨ [ θ ( t + t 0 ) − θ ( t 0 )] 2 ⟩

Finite density system Centre-of-mass mean-square displacement ⟨ ∆ r 2 cm ⟩ = ⟨ [ r cm ( t + t 0 ) − r cm ( t 0 )] 2 ⟩ 10 −1 10 −1 10 −2 10 −2 Pe=2 Pe=40 cm 〉 /4t cm 〉 /4t 10 −3 10 −3 〈∆ r 2 〈∆ r 2 φ =0 φ =0 0.1 0.1 0.2 0.2 10 −4 10 −4 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 10 −5 10 −5 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t t t I t a t ∗ t I t ∗ t a Pe and φ effect

Finite density system Angular mean-square displacement ⟨ ∆ θ 2 ⟩ = ⟨ [ θ ( t + t 0 ) − θ ( t 0 )] 2 ⟩ 10 −1 10 −2 Pe=2 Pe=40 10 −2 〈∆θ 2 〉 /2t 〈∆θ 2 〉 /2t 10 −3 10 −3 φ =0.1 φ =0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 10 −4 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t t t I t a t ∗ t I t ∗ t a Pe and φ effect

Diffusion constants ⟨ ∆ r 2 cm ⟩ ≃ 2 dD A t Translational diffusion Pe=4 F act =0.1 T=0.05 Pe=4 F act =0.5 T=0.25 Pe=4 F act =1 T=0.5 0.125 a 1 =-0.76 a 2 =-0.41 diminishes at D A (F act ,T, φ )/k B T 0.1 increasing density 0.075 at all Pe 0.05 increases at 0 0.2 0.4 0.6 φ increasing Pe Pe=40 F act =0.1 T=0.005 Pe=40 F act =0.5 T=0.025 Pe=40 F act =1 T=0.05 8 a 1 =-2.40 a 2 =1.66 at fixed φ D A (F act ,T, φ )/k B T 6 Proposals for φ , Pe dependence 4 Similar to what observed for 2 0 0.2 0.4 0.6 e.g., Janus particles in H 2 O 2 φ Zheng et al 13 D A k B T = f A ( Pe , φ )

Diffusion constants ⟨ ∆ θ 2 ⟩ ≃ 2 D R t 0.6 Pe=4 F act =0.1 T=0.05 Pe=4 F act =0.5 T=0.25 Pe=4 F act =1 T=0.5 0.4 D R (T, φ )/k B T 0.2 0 0 0.2 0.4 0.6 φ Rotational diffusion 0.6 enhanced at 0.4 D R (T, φ )/k B T increasing density 0.2 for large Pe Pe=40 F act =0.1 T=0.005 Pe=40 F act =0.5 T=0.025 Pe=40 F act =1 T=0.05 0 Incipient clusters 0 0.2 0.4 0.6 φ D R k B T = f R ( Pe , φ )

Fluctuations Translational motion in the active-force driven regimes p (∆ x ) = p ( x cm ( t + t 0 ) − x cm ( t 0 )) 10 1 10 1 Pe=40 Pe=40 20 20 10 0 10 0 4 4 2 2 10 -1 10 -1 σ x P( ∆ x) σ x P( ∆ x) 10 -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 10 -6 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 ∆ x/ σ x ∆ x/ σ x t ∗ < t < t a t a < t 10 −1 10 −1 ϕ = 0 . 1 10 −2 Pe=2 10 −2 Pe=40 cm 〉 /4t cm 〉 /4t σ x = ⟨ ∆ x 2 ⟩ 1 / 2 10 −3 10 −3 〈∆ r 2 〈∆ r 2 φ =0 φ =0 0.1 0.1 10 −4 0.2 10 −4 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 Non-Gaussian at high Pe 0.7 0.7 10 −5 10 −5 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t t

Fluctuations Translational motion in the active-force driven regimes p (∆ x ) = p ( x cm ( t + t 0 ) − x cm ( t 0 )) 10 1 10 1 Pe=40 Pe=40 20 20 10 0 10 0 4 4 2 2 10 -1 10 -1 σ x P( ∆ x) σ x P( ∆ x) 10 -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 10 -6 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 ∆ x/ σ x ∆ x/ σ x t ∗ < t < t a t a < t Janus particles in H 2 O 2 Same double peak at high Pe Zheng et al. 13

Fluctuations Translational motion in the active-force driven regimes p (∆ x ) = p ( x cm ( t + t 0 ) − x cm ( t 0 )) 10 1 10 1 φ =0.01 φ =0.01 0.1 0.1 III IV 10 0 10 0 0.3 0.3 0.5 0.5 0.7 0.7 10 -1 10 -1 σ x P( ∆ x) σ x P( ∆ x) 10 -2 10 -2 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 ∆ x/ σ x ∆ x/ σ x t ∗ < t < t a t a < t 10 −1 Pe = 40 10 −2 Pe=40 cm 〉 /4t σ x = ⟨ ∆ x 2 ⟩ 1 / 2 10 −3 〈∆ r 2 φ =0 0.1 10 −4 0.2 0.3 0.4 0.5 0.6 Non-Gaussian & exponentail tails in III 0.7 10 −5 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t

Fluctuations Translational motion in super-cooled liquids and granular matter G s ( r ) = N − 1 ∑ N i =1 ⟨ δ ( r − | ⃗ r i ( t + t 0 ) − ⃗ r i ( t 0 ) | ) ⟩ van Hove correlation function delay-time shorter than the structural relaxation time t < t α σ = ⟨ ∆ r 2 ⟩ 1 / 2 Exponential tails Chaudhuri, Berthier & Kob 07

Fluctuations Rotational motion in the active-force driven regimes p (∆ θ ) = p ( θ ( t + t 0 ) − θ ( t 0 )) 10 0 10 0 Pe=40 Pe=40 20 20 4 4 10 -1 10 -1 2 2 10 -2 10 -2 σ θ P( ∆θ ) σ θ P( ∆θ ) 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 -4 -2 0 2 4 -4 -2 0 2 4 ∆θ / σ θ ∆θ / σ θ t ∗ < t < t a t a < t 10 −1 10 −2 ϕ = 0 . 1 Low density Pe=2 Pe=40 10 −2 〈∆θ 2 〉 /2t 〈∆θ 2 〉 /2t 10 −3 σ θ = ⟨ ∆ θ 2 ⟩ 1 / 2 10 −3 φ =0.1 φ =0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 0.7 Gaussian 10 −4 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t t

Fluctuations Rotational motion in the active-force driven regimes p (∆ θ ) = p ( θ ( t + t 0 ) − θ ( t 0 )) 10 0 10 0 φ =0.01 φ =0.01 0.1 0.1 III IV 0.3 0.3 10 -1 10 -1 0.5 0.5 0.7 0.7 10 -2 10 -2 σ θ P( ∆θ ) σ θ P( ∆θ ) 10 -3 10 -3 10 -4 10 -4 10 -5 10 -5 -4 -2 0 2 4 -4 -2 0 2 4 ∆θ / σ θ ∆θ / σ θ t ∗ < t < t a t a < t 10 −2 Pe = 40 Pe=40 〈∆θ 2 〉 /2t σ θ = ⟨ ∆ θ 2 ⟩ 1 / 2 10 −3 φ =0.1 0.2 0.3 0.4 0.5 Exponential tails for φ ≥ 0 . 7 0.6 0.7 10 −4 10 −3 10 −2 10 −1 10 0 10 1 10 2 10 3 10 4 t