Long-range forces and phase separation in simple active fluids

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1 1/21 Long-range forces and phase separation in simple active fluids Alex Solon, PLS fellow, MIT Harvard Kid s seminar, September 19th 2017

2 Active matter 2/21 Stored energy Mechanical energy Self-propelled particles

3 Active matter 2/21 Stored energy Mechanical energy Self-propelled particles Found at all scales in living systems sub-cellular cellular macroscopic

4 Active matter 2/21 Stored energy Mechanical energy Self-propelled particles Found at all scales in living systems sub-cellular cellular macroscopic Active matter = Assemblies of active particles?

5 3/21 Active matter Nonequilibrium systems (reversed energy cascade) Rich phenomenology Many experiments

6 3/21 Active matter Nonequilibrium systems (reversed energy cascade) Rich phenomenology Many experiments Artificial self-propelled particles Vibrated disks Janus colloids

7 3/21 Active matter Nonequilibrium systems (reversed energy cascade) Rich phenomenology Many experiments Artificial self-propelled particles Vibrated disks Janus colloids Precisely controlled interactions Bacterial patterns Robot swarms Universality Look at minimal models

8 Simple active fluid 4/21 Constant speed with persistent orientation v v Run and Tumble Active Brownian

9 Simple active fluid 4/21 Constant speed with persistent orientation v v Run and Tumble Active Brownian Dry active matter: no momentum conservation

10 4/21 Simple active fluid Constant speed with persistent orientation v v Run and Tumble Active Brownian Dry active matter: no momentum conservation Isolated particle: diffusive on scales larger than l r D eff l r v Need interactions on the scale of l r

11 5/21 Outline Long-range forces between passive objects [Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez, PRL 2016] [Baek, Solon, Xu, Nikola, Kafri, arxiv 2017] Motility-induced phase separation [Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur, PRL 2015] [Solon, Stenhammar, Cates, Kafri, Tailleur, arxiv 2016]

12 Force on an asymmetric object 6/21 An active bath applies a net force on an asymmetric object (Aranson lab.) (DiLeonardo lab.)

13 Force on an asymmetric object 6/21 An active bath applies a net force on an asymmetric object (Aranson lab.) (DiLeonardo lab.) The force depends on the local curvature

14 Force on an asymmetric object 6/21 An active bath applies a net force on an asymmetric object (Aranson lab.) (DiLeonardo lab.) The force depends on the local curvature Instability of a flexible filament

15 Flow generated by an object 7/21 F

16 Flow generated by an object 7/21 F F Force exerted by the object on the bath Generates flows

17 Flow generated by an object 7/21 F F Force exerted by the object on the bath Generates flows Steady-state density given by D eff 2 ρ = µ (ρ V + G)

18 Flow generated by an object F F Force exerted by the object on the bath Generates flows Steady-state density given by D eff 2 ρ = µ (ρ V + G) Similar to an electric dipole ρ(r) ρ b + µ r p 2πD eff J(r) µ 2π r 2 [ p r 2 2(r.p)r r 4 p = F ] 6 3 y/l r x/l r 3 2 ρl 2 r 1 0 7/21

19 Two-body interactions 8/21 p 1 p 2 Long-range interaction µ r p 1 F 2 (r) = 2πD eff ρ b r 2 p 2 + R 2 J 1 (r) + O(r 3 )

20 Two-body interactions 8/21 p 1 p 2 = 0 Long-range interaction µ r p 1 F 2 (r) = 2πD eff ρ b r 2 p 2 + R 2 J 1 (r) + O(r 3 )

21 Two-body interactions 8/21 p 1 p 2 = 0 Long-range interaction µ r p 1 F 2 (r) = 2πD eff ρ b r 2 p 2 + R 2 J 1 (r) + O(r 3 ) µ r p 1 τ 2 (r) = 2πD eff ρ b r 2 T 2 + γ 2 J 1 (r) + O(r 3 )

22 Two-body interactions 8/21 p 1 p 2 = 0 Long-range interaction µ r p 1 F 2 (r) = 2πD eff ρ b r 2 p 2 + R 2 J 1 (r) + O(r 3 ) µ r p 1 τ 2 (r) = 2πD eff ρ b r 2 T 2 + γ 2 J 1 (r) + O(r 3 ) p i, T i : spontaneous force and torque R i, γ i : linear response coefficients Can be tuned (in principle) by designing the object Different dynamical phenomena

23 Synchronized rotations 9/21 Two semi-circles pinned at different points J A C J C A CA C τ AC = γ C J A (r) C AC A A Changing the rotational mobilities µ A R, µc R : transition between phase-locking and synchronized rotations

24 Alignment of moving semi-circles 10/21 Two semi-circles with different center of mass C I A I II II J A III II III IV III IV A C I Snake-like motion

25 11/21 Conclusion Far field: Generic long-range interaction Not limited to asymetric objects: quadrupolar order Tunable interactions

26 11/21 Conclusion Far field: Generic long-range interaction Not limited to asymetric objects: quadrupolar order Tunable interactions Interesting for self-assembly? Non-universal near-field interactions

27 12/21 Outline Long-range forces between passive objects [Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez, PRL 2016] [Baek, Solon, Xu, Nikola, Kafri, arxiv 2017] Motility-induced phase separation [Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur, PRL 2015] [Solon, Stenhammar, Cates, Kafri, Tailleur, arxiv 2016]

28 13/21 Motility-induced phase separation (MIPS) Feedback loop: Collisions/interactions = Particles slow down = Increase in density

29 13/21 Motility-induced phase separation (MIPS) Feedback loop: Collisions/interactions = Particles slow down = Increase in density Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011

30 13/21 Motility-induced phase separation (MIPS) Feedback loop: Collisions/interactions = Particles slow down = Increase in density Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011 Cohesive matter without attractive interactions

31 13/21 Motility-induced phase separation (MIPS) Feedback loop: Collisions/interactions = Particles slow down = Increase in density Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011 Cohesive matter without attractive interactions How can we understand the phase coexistence?

32 Microscopic models 14/21 Pairwise interactions Fily and Marchetti (PRL 2012), Redner et al (PRL 2013), Stenhammar et al (PRL 2013), Mallory et al (PRE 2014), Takatori et al (PRL 2014), Solon et al (PRL 2015)...

33 Microscopic models 14/21 Pairwise interactions Quorum sensing Fily and Marchetti (PRL 2012), Redner et al (PRL 2013), Stenhammar et al (PRL 2013), Mallory et al (PRE 2014), Takatori et al (PRL 2014), Solon et al (PRL 2015)... Tailleur and Cates (PRL 2008), Solon et al (EPJ 2015)

34 Microscopic models 14/21 Pairwise interactions Quorum sensing Fily and Marchetti (PRL 2012), Redner et al (PRL 2013), Stenhammar et al (PRL 2013), Mallory et al (PRE 2014), Takatori et al (PRL 2014), Solon et al (PRL 2015)... Tailleur and Cates (PRL 2008), Solon et al (EPJ 2015) Similar to liquid-gas phase separation

35 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ρ t = M(ρ) δf[ρ] δρ, F = [ dr f (ρ) + c(ρ) ] 2 ρ 2

36 Liquid-gas phase separation in equilibrium 15/21 Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ

37 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x

38 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ

39 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ f (ρ g ) = f (ρ l ) equality of chemical potential

40 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ f (ρ g ) = f (ρ l ) equality of chemical potential µ x l x g ρdx = x l x g f (ρ) ρdx + xl x g [ c 2 ρ 2 c ρ] ρdx } {{ } =0

41 15/21 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ f (ρ g ) = f (ρ l ) equality of chemical potential µ(ρ l ρ g ) = f (ρ l ) f (ρ g ) equality of pressure P = ρf f

42 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g x g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ f (ρ g ) = f (ρ l ) equality of chemical potential µ(ρ l ρ g ) = f (ρ l ) f (ρ g ) equality of pressure P = ρf f f ρ g ρ l ρ 15/21

43 Nonequilibrium phase separation 16/21 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ

44 Nonequilibrium phase separation 16/21 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ 0 (ρ) + λ ρ 2 κ ρ = Cst. = µ

45 Nonequilibrium phase separation 16/21 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ 0 (ρ) + λ ρ 2 κ ρ = Cst. = µ µ 0 (ρ g ) = µ 0 (ρ l ) equality of chemical potential

46 Nonequilibrium phase separation 16/21 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ 0 (ρ) + λ ρ 2 κ ρ = Cst. = µ µ 0 (ρ g ) = µ 0 (ρ l ) equality of chemical potential µ(ρ l ρ g ) = f (ρ l ) f (ρ g ) + df dρ = µ 0 xl x g [λ ρ 2 κ ρ] ρdx } {{ } 0

47 Nonequilibrium phase separation 16/21 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ 0 (ρ) + λ ρ 2 κ ρ = Cst. = µ µ 0 (ρ g ) = µ 0 (ρ l ) equality of chemical potential µ(ρ l ρ g ) = f (ρ l ) f (ρ g ) + df dρ = µ 0 xl Uncommon tangent construction [Wittkowski et al, Nat. Comm. 2014] x g [λ ρ 2 κ ρ] ρdx } {{ } 0 f ρ

48 Generalized thermodynamic constructions 17/21 µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x g x l ρ l x µ x l x g ρdx = x l x g µ 0 (ρ) ρdx + x l x g [λ(ρ) ρ 2 κ(ρ) ρ] ρdx

49 Generalized thermodynamic constructions 17/21 µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x g x l ρ l x µ x l x g Rdx = x l x g µ 0 (ρ) Rdx + x l x g [λ(ρ) ρ 2 κ(ρ) ρ] Rdx Effective density R(ρ) s.t. R = 2λ+κ κ R

50 17/21 Generalized thermodynamic constructions µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x g x l ρ l x µ x l x g Rdx = x l x g µ 0 (ρ) Rdx + x l x g [λ(ρ) ρ 2 κ(ρ) ρ] Rdx Effective density R(ρ) s.t. R = 2λ+κ κ R µ(r l R g ) = φ(r l ) φ(r g ), Thermodynamic pressure: P = µ 0 R φ dφ dr = µ 0

51 17/21 Generalized thermodynamic constructions µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x g x l ρ l x µ x l x g Rdx = x l x g µ 0 (ρ) Rdx + x l x g [λ(ρ) ρ 2 κ(ρ) ρ] Rdx Effective density R(ρ) s.t. R = 2λ+κ κ R µ(r l R g ) = φ(r l ) φ(r g ), Thermodynamic pressure: P = µ 0 R φ dφ dr = µ 0 Common tangent construction

52 Quorum sensing interactions 18/21 Particles moving at velocity v(ρ) v v g v l ρ

53 Quorum sensing interactions 18/21 Particles moving at velocity v(ρ) v v g ρ t = M(ρ) [µ 0 (ρ) κ(ρ) ρ], µ 0 = log(ρv), κ(ρ) = l 2 v v v l ρ

54 Quorum sensing interactions 18/21 Particles moving at velocity v(ρ) v v g ρ t = M(ρ) [µ 0 (ρ) κ(ρ) ρ], µ 0 = log(ρv), κ(ρ) = l 2 v v = effective density R(ρ) = ρ 1 κ(u) du v l ρ φ µ R g R l R

55 Quorum sensing interactions 18/21 Particles moving at velocity v(ρ) v v g ρ t = M(ρ) [µ 0 (ρ) κ(ρ) ρ], µ 0 = log(ρv), κ(ρ) = l 2 v v = effective density R(ρ) = ρ 1 κ(u) du v l ρ ρ φ µ R g R l R Simulations, 1d lattice Simulations, 2d RTP off-lattice Simulations, 2d ABP off-lattice Equ. construction Theory v g /v l

56 Pairwise interactions 19/21 More difficult. No expression for the interfacial terms. measured numerically (b) Pe Simulations Eq. (11) Equilibrium ρ

57 Conclusions 20/21 MIPS described by a generalized Cahn-Hilliard equation including non-equilibrium gradient terms Equilibrium relations are recovered after defining an effective density The phase diagram depends on the interfacial terms

58 Collaborators 21/21 Mehran Kardar (MIT) Julien Tailleur (Paris Diderot) Michael Cates (Cambridge) Joakim Stenhammar (Lund) Yariv Kafri s group (Technion) Yongjoo Baek Nikolai Nikola Xinpeng Xu Active particles on curved surfaces: Equation of state, ratchets, and instabilities Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez Phys. Rev. Lett. 117, (2016) Generic long-range interactions between passive bodies in an active fluid Baek, Solon, Xu, Nikola, Kafri arxiv 2017 Pressure and Phase Equilibria in Interacting Active Brownian Spheres Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur Phys. Rev. Lett. 114, (2015) Generalized Thermodynamics of Phase Equilibria in Scalar Active Matter Solon, Stenhammar, Cates, Kafri, Tailleur arxiv (2016)