1. Giant fluctuations at a granular phase separation threshold Baruch Meerson Racah Institute of Physics, The Hebrew University of Jerusalem, Israel in collaboration with Thorsten Pöschel and Thomas Schwager (Charité, Berlin) Pavel V. Sasorov (ITEP, Moscow) Southern Workshop on Granular Materials Pucon, Chile 2003

2. Outline 1.Phase separation in granular gas. 2.Hydrostatic theory of phase separation in a model system. 3.MD simulations: giant fluctuations at onset of phase separation. 4.Does granular hydrodynamics break down? 5.Summary.

3. Phase separation is common in granular flow solid phase liquid phase

4. Example 1. Granular Maxwell’s Demon Schlichting and Nordmeier (1996), Eggers (1999), Lohse et al. (2001), Brey et al. (2002),… original experiment more experiment + theory theory Phase separation is simpler in granular gas

5. Example 2. Phase separation in a granular monolayer driven by vertical vibrations Experiment: Olafsen and Urbach (1998) MD simulations: Nie et al. (2000), Cafiero et al. (2000) static (crystalline) cluster “gas” Movie: courtesy of Jeff Urbach

6. Example 3. Phase separation and coarsening in electrostatically driven metallic powders Aranson et al. (PRL 2000), Aranson, BM, Sasorov and Vinokur (PRL 2002), … 1.5mm up to 2 kV/cm

7. Why should we care about spontaneous phase separation in GMs? It provides a sensitive test to models of granular flows It contributes to our understanding of pattern formation far from equilibrium It is a cool but difficult problem

8. Phase separation in a prototypical model of inelastic hard disks vibrating/”hot” wall gravity = 0 elastic wall 00.20.40.60.81 0 0.1 0.2 0.3 0.4 X  elastic wall T Early work: Theory: Grossman et al. (1997). Experiment: Kudrolli et al. (1997) stripe state “hot”/vibrating wall

9. GH predicts instability/metastability of stripe state Livne, BM and Sasorov (2000,2002), Brey et al. (2002), Khain and BM (2002), Argentina et al. (2002), … Steady states describable by hydrostatic equations: heat conduction heat loss Instability/metastability Symmetry-broken hydrostatic solutions appear Stripe state: 1D solution of these equations p: pressure n: number density T: temperature

10. Symmetry-broken steady states A nonlinear Poisson solver, LMS (2000,2002). droplets bubble

11. Governing parameters of hydrostatic equations area fraction aspect ratio hydrodynamic heat loss parameter

12. f p Negative compressibility!  C Physics of stripe state p(f,  ): steady-state pressure of stripe state Khain and BM (2002), Argentina, Clerk and Soto (2002)

13. 00.060.120.18 0 0.01 0.02 0.03 0.04 0.05 f p Infinite layer Stripe state unstable within spinodal interval spinodal interval Spinodal interval everywhere critical point  C C:C:  C Argentina, Clerk and Soto (2002)  C Argentina, Clerk and Soto (2002): coexistence interval, powerful analogy with van der Waals gas

14. What if  is finite? Negative lateral compressibility: destabilizing effect f P Lateral heat conduction: stabilizing effect  c Critical lateral length:  c  C for instability Negative compressibility Livne, BM and Sasorov (2000,2002), Brey et al. (2002), Khain and BM (2002)

15. Supercritical bifurcation diagram: hydrostatic theory + hydrodynamic simulations Supercritical bifurcation diagram: hydrostatic theory + hydrodynamic simulations Y c : normalized y-coordinate of center of mass  |Y c | Δ = Δ c at fixed  and f, within spinodal

16. Δ<Δ c Δ>Δ c Δ>>Δ c Livne, BM and Sasorov (2002) Example of supercritical bifurcation observed in hydrodynamic simulations color density maps close-packed “droplet” hot wall

17. MD simulations MD simulations  c weakly fluctuating stripe state, hydrostatics is doing well Δ = 0.1 N=2·10 4 Δ c = 0.5 hot wall

18. MD simulations hydro simulations  c Nucleation, coarsening of droplets, one droplet survives. Hydrostatics is doing well. Livne, BM and Sasorov (2002) N=2·10 4 hot wall

19. Very large aspect ratio:  c MD simulations outside of spinodal interval, inside coexistence interval Argentina, Clerk and Soto (2002) fluctuations weak, hydrostatics OK

20. What happens around  c ? MD simulations Giant fluctuations. Transitions between the two states with broken symmetry? N=2·10 4 hot wall

21. Giant fluctuations. Hydrostatics fails in a wide region around  c Center-of-mass dynamics N=2·10 4

22. Symmetry-breaking transition occurs somewhere at 0.3 <  < 1, hydrostatics predicts  c = 0.51. Probability distribution of Y c at different 

23. Probability distribution maxima vs.  hydro simulation Systematic disagreement with hydrostatics within fluctuation- dominated region. Change of bifurcation character? hydrostatic bifurcation curve maxima of PDF

24. Y c vs. N at fixed , f and  c N=20000 N=15000 N=10000 N=5000 Low-frequency oscillations do not go down as N increases?

25. Anomalously wide fluctuation region N=2,300 MD simulations: Ramirez et al. (2000), Sunthar and Kumaran (2001) Hydrodynamic theory: He, BM and Doolen (2002), Khain and BM (2003) For comparison, another bifurcation: onset of thermal granular convection (same system + gravity). Doesn’t show large fluctuations

26. Where do giant fluctuations (GFs) come from? Scenario I. GFs are driven hydrodynamically, by either an instability, or a long-lived transient mode To check this scenario, more careful hydrodynamic simulations are needed (Livne, BM and Sasorov; work in progress).

27. Scenario II. GFs result from (a hydrodynamic) amplification of discrete-particle noise Here “Fluctuating Granular Hydrodynamics” is needed Now P, q and  include delta- correlated gaussian random noise terms, in the spirit of Landau and Lifshitz (1959). The noise terms drive the collective modes of the system Once it is computed one can start working… Surprise: noisy part of  is not known!

28. Summary 1.Phase separation phenomena are ubiquitous in granular flow, in granular gas. 2.Hydrostatic theory of phase separation in prototypical granular system works well far from finite-size threshold. 3.Giant fluctuations found at threshold: mechanism unknown, lot of work to do.