Universality in Soft Active Matter

Transcription

1 Universality in Soft Active Matter Chiu Fan Lee Department of Bioengineering, Imperial College London, UK M.C. Escher (1938) Day and Night

2 Universality in soft living matter Amyloid fibrils RNA granules Active matter 200nm Equilibrium: length distribution, breakage profile Non-equilibrium universality Brangwynne [Hyman Lab] Equilibrium: Lifshitz-Slyozov scaling & drop size distribution Non-equilibrium universality mbzusmak8 Fundamentally non-equilibrium Universality

3 Universality in soft living matter Amyloid fibrils RNA granules Active matter 200nm Equilibrium: length distribution, breakage profile Non-equilibrium universality Brangwynne [Hyman Lab] Equilibrium: Lifshitz-Slyozov scaling & drop size distribution Non-equilibrium universality mbzusmak8 Fundamentally non-equilibrium Universality

4 Plan Minimal system of active particles Motility-induced phase separation Universality in phase separation kinetics Modern concept of universality Incompressible polar active fluids Universality at criticality Universality in the ordered phase (2D) Summary & Outlook

5 Liquid-gas phase separation

6 Equilibrium phase separation Lennard-Jones liquid: Drag coeff. Gaussian noise Vapour U LJ (r) Liquid a r

7 Equilibrium vapour concentration x [Solute] Vapour conc. outside small drop is higher than vapour conc. outside large drop Gibbs-Thomson relation: ρ R = ρ 1 + ν R Size of drop x

8 Universality of surface-tension driven Oswald ripening droplet growth Lifshitz-Slyozov universal growth law: R(t) ~t 1/3 Universal droplet size distribution Sagui & Grant (1999) PRE

9 Wittkowski et al. (2013) Nat. Comm. Motility-induced phase separation

10 Motility-induced phase separation (MIPS) Active force Vapour U WCA (r) Gaussian noise Liquid Weeks-Chandler-Andersen potential Tailleur & Cates (2008) PRL; Fily & Marchetti (2012) PRL; Redner, Hagan & Baskaran (2013) PRL a r

11 Force balance at the interface in MIPS

12 Confined active point particles

13 Repulsive active point particles Active point particles

14 Forces on the wall Point particles Repulsive particles Force coming from particle repulsion [Swim pressure: Takatori, Yan & Brady (2014) PRL] High active force from the point particles due to orientation anisotropy

15 Sharp interface model Condensed phase: particles with isotropic orientation Vapour phase: point particles Min. force for MIPS: Lee, arxiv:

16 Gibbs-Thomson relation in MIPS

17 Circular drops: Caging effects Escape range 2σ R 2σ R Escape range Caging effect of neighbouring particles leads a curvaturedependent escape range: σ R = A + B R + O R 2

18 Circular drop: numerics ρ R (r) ρ R=20 ρ 60 ρ100 Gibbs-Thomson relation Lee, arxiv:

19 Universality of coarsening in MIPS 1. Active particles move diffusively across long distance 2. Gibbs-Thomson relation at the interface MIPS droplets lead to Lifshitz-Slyozov (LS) equilibrium coarsening: R(t) ~t 1/3 Universal (LS) droplet size distribution

20 Universality

21 Universality in physics Ferromagnetism with an easy axis at criticality Phase separation in multicomponent lipid membrane at criticality P. Colon s homepage Honerkamp-Smith et al. (2009) BBA Ising model: E = J As the measurement is at a larger and larger length scale (l ) <i.j> Proved by renormalisation group methods s i s j Kenneth Wilson, Nobel prize 1982

22 Renormalisation group transformation 1. Coarse grain out short wavelength (k > l 1 ) fluctuations 2. Rescale 3. Renormalise

23 Data from Hugo Duminil-Copin 1. Coarse grain Repeat 2. Rescale 3. Renormalise

24 Renormalisation group transformation 1. Coarse grain out short wavelength (k > l 1 ) fluctuations 2. Rescale 3. Renormalise l goes up further Level of coarse-graining Toy model H becomes the (asymptotically) exact model

25 Incompressible polar active fluids

26 Wound healing assay

27 Navier-Stokes description t v + v v = μ 2 v ρ 1 P Not enough!

28 Navier-Stokes description t v + v v = a + bv 2 v + μ 2 v ρ 1 P Missing elements: Not enough! Cells are motile -> Need active terms like a + bv 2 v

29 Navier-Stokes description t v + v v = a + bv 2 v + μ 2 v ρ 1 P + f Not enough! Missing elements: Cells are motile -> Need active terms like a + bv 2 v Fluctuations can be important -> Gaussian noise f

30 Navier-Stokes description t v + λ v v = a + bv 2 v + μ 2 v ρ 1 P + f Not enough! Missing elements: Cells are motile -> Need active terms like a + bv 2 v Fluctuations can be important -> Gaussian noise f No Galilean invariance -> prefactor of v v may not be 1

31 In fact, to be completely general There can be infinitely many more red terms t v + ρ 1 P f = λ v v a + bv 2 v μ 2 v +cv 4 v + ξ( 2 ) 2 v + With so many parameters, can we ever say something universal about the system

32 Energy The bottom has become very flat x y Adapted from QuantumDiaries.org Incompressible active fluids at criticality

33 Phase behaviour of incompressible active fluids v Critical region

34 Critical incompressible polar active fluids EOM: t v + P f = λ v v a + bv 2 v μ 2 v + cv 4 v + ξ( 2 ) 2 v + RG transformation t v + P + f l = λ l v v a l + b l v 2 v μ l 2 v Exact hydrodynamic EOM with TWO coefficients governing the model s scale-invariance properties: Level of coarse-graining g 1 (l) ~ D 2 lλ l, g μ 3 2 l ~ D lb l l μ 2 l

35 Advective term: g 1 v v Incompressible active matter (Today) g 1 l ~ D 2 lλ l μ ; g 3 2 l ~ D lb l l μ 2 l Chen, Toner, Lee (2015) New J. Phys. 17, Ferromagnets with dipolar interactions Aharony and Fisher (1973) Phys. Rev. Lett.

36 Randomly stirred fluids (Model B) Forster, Nelson & Stephen (1977) Phys. Rev. A Incompressible active matter (Today) g 1 l ~ D 2 lλ l μ ; g 3 2 l ~ D lb l l μ 2 l Chen, Toner, Lee (2015) New J. Phys. 17, Ferromagnets with dipolar interactions Aharony and Fisher (1973) Phys. Rev. Lett.

37 Randomly stirred fluids (Model B) Forster, Nelson & Stephen (1977) Phys. Rev. A g 1 l ~ D 2 lλ l μ ; g 3 2 l ~ D lb l l μ 2 l Chen, Toner, Lee (2015) New J. Phys. 17, Ferromagnets with dipolar interactions Aharony and Fisher (1973) Phys. Rev. Lett.

38 Randomly stirred fluids (Model B) Forster, Nelson & Stephen (1977) Phys. Rev. A Incompressible polar active fluids In 3D: v(r) v(r ) ~ r r (1.35±0.08) g 1 l ~ D 2 lλ l μ ; g 3 2 l ~ D lb l l μ 2 l Chen, Toner, Lee (2015) New J. Phys. 17, Ferromagnets with dipolar interactions Aharony and Fisher (1973) Phys. Rev. Lett.

39 Energy y x Adapted from QuantumDiaries.org Incompressible active fluids in the ordered phase

40 Ordered incompressible polar active fluids v Ordered phase Critical region Universality is more than criticality! Universal behaviour expected in the symmetry-broken phase of a continuous symmetry Belitz, Kirkpatrick, Vojta (2005) Rev. Mod. Phys.

41 EOM: Ordered phase of 2D incompressible active fluids 1. RG transformation Ordered phase: 2. Use streaming function h to enforce Incompressibility: 3. Equal-time (static) correlations in 2D active fluids are mapped onto the 2D smectic model: Chen, Lee, Toner, arxiv:

42 Universality of Kardar-Parisi-Zhang y Incompressible flock (living) Smectic layers (equilibrium) Kardar-Parisi-Zhang surface growth (nonequilibrium) Chen, Lee, Toner, arxiv: Time 2D Smectics = (1+1) KPZ [Golubović & Wang (1992) PRL] x

43 KPZ model:

44 2D incompressible active fluids in the ordered phase [Chen, Lee & Toner (2016) arxiv: ] Slide from J. Quastel (Toronto)

45 M.C. Escher (1938) Day and Night

46 Outlook

47 Periodic table of universality classes Equilibrium Hohenberg and Halperin (1977) Theory of dynamic critical phenomena Rev. Mod. Phys.

48 Periodic table of universality classes Equilibrium Non-equilibrium Randomly stirred fluids [Forster, Nelson & Stephen (1977) Phys. Rev. A] Directed percolation [Kinzel (1983)] Kardar-Parisi-Zhang [KPZ (1983) Phys. Rev. Lett.] Living matter Incompressible active polar fluids [Chen, Toner, Lee (2015) New J. Phys.] Hohenberg and Halperin (1977) Theory of dynamic critical phenomena Rev. Mod. Phys.

49 Periodic table of universality classes Equilibrium Non-equilibrium Directed percolation [Kinzel (1983)] Kardar-Parisi-Zhang [KPZ (1983) Phys. Rev. Lett.] Living matter Randomly stirred fluids [Forster, Nelson & Stephen (1977) Phys. Rev. A] Incompressible active polar fluids [Chen, Toner, Lee (2015) New J. Phys.] Hohenberg and Halperin (1977) Theory of dynamic critical phenomena Rev. Mod. Phys.

50 Periodic table of universality classes Equilibrium Non-equilibrium Transition New Universality class Ordered phase LSW KPZ Hohenberg and Halperin (1977) Theory of dynamic critical phenomena Rev. Mod. Phys.

51 Universalities in Biology Biological Motile bacteria Epithelial tissue Birds Biology Transition Ordered phase LSW New Universality class KPZ

52 Summary Universality is everywhere in biology Universal coarsening kinetics (LSW) in motility-induced phase separation [Lee, arxiv: ] Universal behaviour of incompressible polar active fluids At criticality (New universality class) [Chen, Toner, Lee (2015) New J. Phys.] Ordered phase in 2D (KPZ) [Chen, Lee, Toner, arxiv:1601:01924] Future direction: Categorisation of universality of living matter

53 Acknowledgement My group at Imperial College David Nesbitt (1 st yr PhD, EPSRC) Alice Spenlehauer (2 nd yr PhD, BBSRC) Jean David Wurtz (3 rd yr PhD, ICL) External collaborators Leiming Chen (China University of Mining and Technology) John Toner (Oregon) Thank you for your Attention!