Giant fluctuations at a granular phase separation threshold

1. Southern Workshop on Granular Materials Pucon, Chile 2003 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)

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

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

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

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 Movie: courtesy of Jeff Urbach “gas”

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

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 elastic wall vibrating/”hot” wall stripe state gravity = 0 0.4 r 0.3 T elastic wall “hot”/vibrating wall 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 X Early work: Theory: Grossman et al. (1997).Experiment: Kudrolli et al. (1997)

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

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

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

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

13. Spinodal interval 0.05 p L<LC 0.04 L=LC spinodal interval 0.03 0.02 L>LC 0.01 f 0 0 0.06 0.12 0.18 L<LC everywhere LC: critical point Argentina, Clerk and Soto (2002) Stripe state unstable within spinodal interval Infinite layer Argentina, Clerk and Soto (2002): coexistence interval, powerful analogy with van der Waals gas

14. What if D is finite? L>LC P Negative compressibility Negative lateral compressibility:destabilizing effect f Lateral heat conduction: stabilizing effect Critical lateral length: D > Dc for instability Livne, BM and Sasorov (2000,2002), Brey et al. (2002), Khain and BM (2002)

15. Supercriticalbifurcation diagram: hydrostatic theory + hydrodynamic simulations Yc: normalized y-coordinate of center of mass • at fixed • and f, within spinodal |Yc | Δ= Δc D

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

17. MD simulations N=2·104 D << Dc Δ= 0.1 Δc = 0.5 hot wall weakly fluctuatingstripe state, hydrostaticsis doing well

18. D> Dc MD simulations hydro simulations Livne, BM and Sasorov (2002) N=2·104 hot wall Nucleation, coarsening of droplets, one droplet survives. Hydrostatics is doing well.

19. Very large aspect ratio:D>> Dc MD simulations outside of spinodal interval, inside coexistence interval Argentina, Clerk and Soto (2002) fluctuations weak, hydrostatics OK

20. What happens around D ~ Dc?MD simulations N=2·104 hot wall Giant fluctuations.Transitions between the two states with broken symmetry?

21. Center-of-mass dynamics N=2·104 Giant fluctuations.Hydrostatics fails in awideregion around c

22. Probability distribution of Yc at different D Symmetry-breaking transition occurs somewhere at 0.3 < D < 1,hydrostatics predicts Dc = 0.51.

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

24. Yc vs. N at fixed L,f and D<<Dc N=20000 N=15000 N=10000 N=5000 Low-frequency oscillations do not go down as N increases?

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

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 carefulhydrodynamic simulations are needed (Livne, BM and Sasorov; work in progress).

27. Scenario II. GFs result from (a hydrodynamic) amplificationof discrete-particle noise Here “Fluctuating Granular Hydrodynamics” is needed Now P, q andG 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 Surprise: noisy part of G is not known! Once it is computed one can start working…

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