Thermodynamics of the motility-induced phase separation

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1 1/9 Thermodynamics of the motility-induced phase separation Alex Solon (MIT), Joakim Stenhammar (Lund U), Mike Cates (Cambridge U), Julien Tailleur (Paris Diderot U) July 21st 2016 STATPHYS Lyon

2 Active fluids 2/9 Fluid of self-propelled particles

3 Active fluids 2/9 Fluid of self-propelled particles Reorientation mechanisms v v Run and Tumble Particles Bacteria Active Brownian Particles Janus colloids

4 Motility-induced phase separation (MIPS) 3/9 Feedback loop Collisions/interactions = Particles slow down = Increase in density

5 Motility-induced phase separation (MIPS) 3/9 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

6 Motility-induced phase separation (MIPS) 3/9 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 How can we understand the phase coexistence?

7 Microscopic models 4/9 Hard core repulsion 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)...

8 Microscopic models 4/9 Hard core repulsion 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)

al (PRE 2014), Takatori et al (PRL 2014), Solon et al (PRL 2015).

9 Microscopic models 4/9 Hard core repulsion 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

10 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation t = M(ρ) δh[ρ] δρ, H = d r [ f (ρ) + c(ρ) ] 2 ρ 2

11 Liquid-gas phase separation in equilibrium 5/9 Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ

12 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g ρ g, ρ l? x l ρ l x

13 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ g ρ g, ρ l? x l ρ l x J = 0 = f (ρ) + c 2 ρ 2 c ρ = Cst. = µ

14 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ 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

15 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ 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 ρdx = x l f (ρ) ρdx + xl [ c 2 ρ 2 c ρ] ρdx } {{ } =0

16 5/9 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ 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

17 Liquid-gas phase separation in equilibrium Cahn-Hilliard equation ] [f t = M(ρ) (ρ) + c 2 ρ 2 c ρ ρ ρ 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 5/9

18 Nonequilibrium phase separation 6/9 Most general for a scalar conserved field [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ

19 Nonequilibrium phase separation 6/9 Most general for a scalar conserved field [ ] t = M(ρ) µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ J = 0 = µ0 (ρ) + λ ρ 2 κ ρ = Cst. = µ

20 Nonequilibrium phase separation 6/9 Most general for a scalar conserved field [ ] 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

21 Nonequilibrium phase separation 6/9 Most general for a scalar conserved field [ ] 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 [λ ρ 2 κ ρ] ρdx } {{ } 0

22 Nonequilibrium phase separation 6/9 Most general for a scalar conserved field [ ] 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] [λ ρ 2 κ ρ] ρdx } {{ } 0 f ρ

23 Generalized thermodynamic constructions 7/9 µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x l ρ l x µ x l ρdx = x l µ 0 (ρ) ρdx + x l [λ(ρ) ρ 2 κ(ρ) ρ] ρdx

24 Generalized thermodynamic constructions 7/9 µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x l ρ l x µ x l Rdx = x l µ 0 (ρ) Rdx + x l [λ(ρ) ρ 2 κ(ρ) ρ] Rdx Effective density R(ρ) s.t. R = 2λ+κ κ R

25 7/9 Generalized thermodynamic constructions µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x l ρ l x µ x l Rdx = x l µ 0 (ρ) Rdx + x l [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

26 7/9 Generalized thermodynamic constructions µ = µ 0 (ρ) + λ(ρ) ρ 2 κ(ρ) ρ ρ ρ g x l ρ l x µ x l Rdx = x l µ 0 (ρ) Rdx + x l [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

27 Quorum sensing interactions 8/9 Particles moving at velocity v(ρ) t = M(ρ) [µ 0 (ρ) κ(ρ) ρ], µ 0 = log(ρv), κ(ρ) = l 2 v v

28 8/9 Quorum sensing interactions Particles moving at velocity v(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

29 Quorum sensing interactions Particles moving at velocity v(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 /9

30 Conclusions 9/9 Hard core repulsion: more difficult but consistent results = Both models described by the same theory

31 Conclusions 9/9 Hard core repulsion: more difficult but consistent results = Both models described by the same theory 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

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