Pressure and phase separation in active matter

Transcription

1 1/33 Pressure and phase separation in active matter Alex Solon Technion, November 13th 2016

2 Active matter 2/33 Stored energy Mechanical energy Self-propelled particles

3 Active matter 2/33 Stored energy Mechanical energy Self-propelled particles Found at all scales in living systems sub-cellular cellular macroscopic

4 Active matter Stored energy Mechanical energy Self-propelled particles Found at all scales in living systems sub-cellular cellular macroscopic Active matter = Assemblies of active particles? 2/33

5 Active matter 3/33 Nonequilibrium systems (reversed energy cascade) Rich phenomenology Simple models Universality Numerous experiments

6 Active matter 3/33 Nonequilibrium systems (reversed energy cascade) Rich phenomenology Simple models Universality Numerous experiments v Artificial self-propelled particles Vibrated disks Janus colloids

7 Active matter Bacterial patterns Robot swarms 3/33 Nonequilibrium systems (reversed energy cascade) Rich phenomenology Simple models Universality Numerous experiments v Artificial self-propelled particles Vibrated disks Janus colloids Precisely controlled interactions

8 Active matter Bacterial patterns Robot swarms 3/33 Nonequilibrium systems (reversed energy cascade) Rich phenomenology Simple models Universality Numerous experiments v Artificial self-propelled particles Vibrated disks Janus colloids Precisely controlled interactions Smart materials?

9 Outline Pressure in active fluids Solon, Fily, Baskaran, Cates, Kafri, Kardar, Tailleur, Nat. Physics (2015) Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez, Phys. Rev. Lett. (2016) Motility-induced phase separation Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur, Phys. Rev. Lett. (2015) Solon, Stenhammar, Cates, Kafri, Tailleur, arxiv (2016) 4/33

10 Active forces 5/33 Cell cortex Wound healing

11 Active forces 5/33 Cell cortex Wound healing Much simpler: active fluid in a box Mechanical pressure

12 From passive to active pressure 6/33 Equilibrium fluid: P = F wall S = Tr σ d = F V N

13 From passive to active pressure 6/33 Equilibrium fluid: P = F wall S = Tr σ d = F V = f (ρ 0, T ) N extensive F Equation of state

14 From passive to active pressure 6/33 Equilibrium fluid: P = F wall S = Tr σ d = F V = f (ρ 0, T ) N extensive F Equation of state Active fluid: no free energy P = F wall S =?

15 Mechanical pressure 7/33 Particles confined by a potential V w ρ(x) V w 0 x w x Mechanical pressure: P = 0 dxρ(x)v w(x)

16 Mechanical pressure 7/33 Particles confined by a potential V w ρ(x) V w 0 x w x Mechanical pressure: P = 0 dxρ(x)v w(x) Vw Ideal gas: ρ(x) = ρ 0 e (x)/kt P = ρ 0 kt

17 Mechanical pressure 7/33 Particles confined by a potential V w ρ(x) V w 0 x w x Mechanical pressure: P = 0 dxρ(x)v w(x) Vw Ideal gas: ρ(x) = ρ 0 e (x)/kt P = ρ 0 kt P independent of V w Equation of state P(ρ 0, T )

18 ABPs et RTPs 8/33 Constant velocity v (no momentum conservation) Two mechanisms for reorientation v v Run and Tumble Particles (RTP) α =tumble rate Active Brownian Particles (ABP) D r =rotational diffusion

19 ABPs et RTPs 8/33 Constant velocity v (no momentum conservation) Two mechanisms for reorientation v v Run and Tumble Particles (RTP) α =tumble rate Two effects from the wall Active Brownian Particles (ABP) D r =rotational diffusion Force V w + Torque Γ w V w Γ w

20 Ideal active gas 9/33 Master equation Compute P = 0 dxρ(x)v w(x)

21 Ideal active gas 9/33 Master equation Compute P = 0 dxρ(x)v w(x) P = ρ 0 kt eff v D r + α v 2 kt eff = 2(D r + α) Ideal gas: exact expression 0 dx 2π 0 dθ Γ w (x, θ) sin(θ)p(x, θ)

22 Ideal active gas 9/33 Master equation Compute P = 0 dxρ(x)v w(x) P = ρ 0 kt eff v D r + α v 2 kt eff = 2(D r + α) Ideal gas: exact expression 0 dx 2π 0 dθ Γ w (x, θ) sin(θ)p(x, θ) P depends on the interaction with the wall: No equation of state

23 Self-propelled ellipses 10/33 Ellipses, axes a and b, rotational diffusion (x xw )2 Harmonic wall V w (x) = λ 2 ŷ p ŷ θ ˆx p ˆx

24 Self-propelled ellipses 10/33 Ellipses, axes a and b, rotational diffusion (x xw )2 Harmonic wall V w (x) = λ 2 ŷ p ŷ θ ˆx p ˆx Torque: Γ w = λκ sin(2θ) κ = (a 2 b 2 )/8

25 Self-propelled ellipses 10/33 Ellipses, axes a and b, rotational diffusion (x xw )2 Harmonic wall V w (x) = λ 2 ŷ p ŷ θ ˆx p ˆx Torque: Γ w = λκ sin(2θ) κ = (a 2 b 2 )/8 P ρ 0v 2 2λκ ] [1 e λκ Dr

26 Ellipses: numerics 11/33 When Γ w increases, P decreases

27 Ellipses: numerics 11/33 When Γ w increases, P decreases Γ w

28 Active ideal gas - Summary 12/33 Spherical particles (Γ w = 0): equation of state Non-spherical particles Torques No equation of state

29 Active ideal gas - Summary 12/33 Spherical particles (Γ w = 0): equation of state Non-spherical particles Torques No equation of state Other ways to lose the equation of state?

30 Active ideal gas - Summary 12/33 Spherical particles (Γ w = 0): equation of state Non-spherical particles Torques No equation of state Other ways to lose the equation of state? Spherical particles with interactions

31 Hard-core interactions 13/33 20 P 10 0 λ = 2 λ = 4 λ = 6 λ = 0.1 λ = 1 λ = 10 non-interacting ρ Hard-core repulsion WCA potential

32 Hard-core interactions 13/33 20 P 10 0 λ = 2 λ = 4 λ = 6 λ = 0.1 λ = 1 λ = 10 non-interacting ρ Hard-core repulsion WCA potential Equation of state

33 Quorum sensing interactions 14/33 50 P λ = 0.1 λ = 1 λ = 10 non-interacting ρ Quorum sensing v(ρ), ρ local density

34 Quorum sensing interactions 14/33 50 P λ = 0.1 λ = 1 λ = 10 non-interacting ρ Quorum sensing v(ρ), ρ local density No Equation of state

35 Alignment 15/33 8 P 6 4 λ = 0.1 λ = 1 λ = 10 non-interacting 2 0 ρ Alignment

36 Alignment 15/33 8 P 6 4 λ = 0.1 λ = 1 λ = 10 non-interacting 2 0 ρ Alignment No equation of state

37 Alignment 15/33 8 P 6 4 λ = 0.1 λ = 1 λ = 10 non-interacting 2 0 ρ Alignment No equation of state Confusing Need a simple test

38 A simple test 16/33 Place an asymmetric mobile partition in the middle of a cavity Equilibrium the partition is static

39 A simple test - All cases 17/33 No equation of states Spontaneous compression

40 Self-organization at the edges 18/33 Force balance In equilibrium F 1 = F 2

41 Self-organization at the edges 18/33 Force balance Active particles F 1 F 2

42 18/33 Self-organization at the edges Force balance Active particles F 1 F ρ(x) m(x) x Force exerted by the substrate

43 Curved walls 19/33 Active particles accumulate in curved regions [Fily, Baskaran, Hagan, Soft Matter 2014]

44 Curved walls 19/33 Active particles accumulate in curved regions (Di Leonardo et al., PNAS 2010) [Fily, Baskaran, Hagan, Soft Matter 2014]

45 Curved walls 20/33 Inhomogenous pressure: Wall

46 Curved walls 20/33 Inhomogenous pressure: Wall P Equation of state in average

47 Curved walls 20/33 Inhomogenous pressure: Wall P Equation of state in average J y Wall Ratchet non-universal J y = µf y

48 Filaments 21/33 Semi-flexible filament in an active bath Instability for q < q i (T, κ s, κ b ) 200 y y y t = 10 2 t = 10 3 t = Coarsening x x x

49 Free filament 22/33 J. Harder, C. Valeriani, A, Cacciuto, Phys. Rev. E. (2014) A. Kaiser, H. Löwen, J. Chem. Phys. (2014) J. Shin, A. Cherstvy, W. K. Kim, R. Metzler, New. J. Phys. (2015)

50 Free filament 22/33 J. Harder, C. Valeriani, A, Cacciuto, Phys. Rev. E. (2014) A. Kaiser, H. Löwen, J. Chem. Phys. (2014) J. Shin, A. Cherstvy, W. K. Kim, R. Metzler, New. J. Phys. (2015) 200 D κ b = 250 κ b = L f

51 Free filament 22/33 J. Harder, C. Valeriani, A, Cacciuto, Phys. Rev. E. (2014) A. Kaiser, H. Löwen, J. Chem. Phys. (2014) J. Shin, A. Cherstvy, W. K. Kim, R. Metzler, New. J. Phys. (2015) 200 D κ b = 250 κ b = L f Macroscopic motion without external symmetry breaking Sorting mechanism

52 Pressure - Summary 23/33 The existence of a pressure is not trivial in active systems P depends in general on the probe

53 Pressure - Summary 23/33 The existence of a pressure is not trivial in active systems P depends in general on the probe Unusual properties Anisotropic pressure: P(θ) if v(θ) Inhomogeneous pressure: P(x) if v(x)

54 Pressure - Summary 23/33 The existence of a pressure is not trivial in active systems P depends in general on the probe Unusual properties Anisotropic pressure: P(θ) if v(θ) Inhomogeneous pressure: P(x) if v(x) Special cases with equation of state Thermodynamics of active systems Currents, ratchets, instabilities

55 Outline Pressure in active fluids Solon, Fily, Baskaran, Cates, Kafri, Kardar, Tailleur, Nat. Physics (2015) Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez, Phys. Rev. Lett. (2016) Motility-induced phase separation Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur, Phys. Rev. Lett. (2015) Solon, Stenhammar, Cates, Kafri, Tailleur, arxiv (2016) 24/33

56 Motility-induced phase separation (MIPS) 25/33 Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011

57 Motility-induced phase separation (MIPS) 25/33 Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011 Feedback loop: Collisions/interactions = Particles slow down = Increase in density

58 Motility-induced phase separation (MIPS) 25/33 Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011 Feedback loop: Collisions/interactions = Particles slow down = Increase in density Cohesive matter without attractive interactions

59 Motility-induced phase separation (MIPS) 25/33 Phase separation: dilute/fast vs dense/slow Buttinoni et al, PRL 2013 Liu et al, Science 2011 Feedback loop: Collisions/interactions = Particles slow down = Increase in density Cohesive matter without attractive interactions How can we understand the phase coexistence?

60 26/33 Microscopic models 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)...

61 26/33 Microscopic models 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)

62 26/33 Microscopic models 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

63 Liquid-gas phase separation in equilibrium 27/33 Cahn-Hilliard equation ρ t = M(ρ) δf[ρ] δρ, F = d r [ f (ρ) + c(ρ) ] 2 ρ 2

64 Liquid-gas phase separation in equilibrium 27/33 Cahn-Hilliard equation ] ρ [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ

65 27/33 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

66 27/33 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

67 27/33 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

68 27/33 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

69 27/33 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

70 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 27/33

71 Nonequilibrium phase separation 28/33 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ

72 Nonequilibrium phase separation 28/33 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ0 (ρ) + λ ρ 2 κ ρ = Cst. = µ

73 Nonequilibrium phase separation 28/33 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ0 (ρ) + λ ρ 2 κ ρ = Cst. = µ µ 0 (ρ g ) = µ 0 (ρ l ) equality of chemical potential

74 Nonequilibrium phase separation 28/33 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ0 (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

75 Nonequilibrium phase separation 28/33 Most general for a scalar conserved field ρ [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ0 (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

76 Generalized thermodynamic constructions 29/33 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

77 Generalized thermodynamic constructions 29/33 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

78 29/33 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 ), dφ dr = µ 0 Common tangent construction

79 29/33 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 ), dφ dr = µ 0 Common tangent construction Thermodynamic pressure: P = µ 0 R φ Equal-area construction

80 Quorum sensing interactions 30/33 Particles moving at velocity v(ρ) ρ t = M(ρ) [µ 0 (ρ) κ(ρ) ρ], µ 0 = log(ρv), κ(ρ) = l 2 v v

81 30/33 Quorum sensing interactions Particles moving at velocity v(ρ) 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 φ P 0 µ R g R l R P 1/R l 1/R g 1/R

82 Quorum sensing interactions Particles moving at velocity v(ρ) 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 φ P 0 µ R g R l R ρ P 1/R l 1/R g 1/R v v g v l ρ Simulations, 1d lattice Simulations, 2d RTP off-lattice Simulations, 2d ABP off-lattice Equ. construction Theory 0 v g /v l /33

83 Pairwise interactions 31/33 More difficult. No expression for the interfacial terms. measured numerically (b) Pe Simulations Eq. (11) Equilibrium ρ

84 Conclusions 32/33 MIPS described by a generalized Cahn-Hilliard equation including non-equilibrium interfacial terms Equilibrium relations are recovered after defining an effective density The phase diagram depends on the interfacial terms

85 33/33 References Julien Tailleur (Paris Diderot) Mehran Kardar (MIT) Michael Cates (Cambridge) Yariv Kafri (Technion) Nikolai Nikola (Technion) Raphaël Voituriez (UPMC) Joakim Stenhammar (Lund) Raphael Wittkowski (Dusseldorf) Aparna Baskaran (Brandeis) Yaouen Fily (Brandeis) Pressure is not a state function for generic active fluids Solon, Fily, Baskaran, Cates, Kafri, Kardar, Tailleur Nature Physics (2015) Pressure and Phase Equilibria in Interacting Active Brownian Spheres Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur Phys. Rev. Lett. 114, (2015) Active particles on curved surfaces: Equation of state, ratchets, and instabilities Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez Phys. Rev. Lett. 117, (2016) Generalized Thermodynamics of Phase Equilibria in Scalar Active Matter Solon, Stenhammar, Cates, Kafri, Tailleur arxiv: (2016)

86 33/33 References Julien Tailleur (Paris Diderot) Mehran Kardar (MIT) Michael Cates (Cambridge) Yariv Kafri (Technion) Nikolai Nikola (Technion) Raphaël Voituriez (UPMC) Joakim Stenhammar (Lund) Raphael Wittkowski (Dusseldorf) Aparna Baskaran (Brandeis) Yaouen Fily (Brandeis) Pressure is not a state function for generic active fluids Solon, Fily, Baskaran, Cates, Kafri, Kardar, Tailleur Nature Physics (2015) Pressure and Phase Equilibria in Interacting Active Brownian Spheres Solon, Stenhammar, Wittkowski, Kardar, Kafri, Cates, Tailleur Phys. Rev. Lett. 114, (2015) Active particles on curved surfaces: Equation of state, ratchets, and instabilities Nikola, Solon, Kafri, Kardar, Tailleur, Voituriez Phys. Rev. Lett. 117, (2016) Generalized Thermodynamics of Phase Equilibria in Scalar Active Matter Solon, Stenhammar, Cates, Kafri, Tailleur arxiv: (2016) Thank you for your attention