Is jammed matter a new phase: Experiments in flowing foam

1. Is jammed matter a new phase: Experiments in flowing foam Michael Dennin U. C. Irvine Department of Physics and Astronomy

2. Jamming Phase Diagram • Plasticity in “molecular” systems • Glassy behavior in liquids • Flow of “multiphase” materials The “J-point” Liu and Nagel, Nature v 396, 1998

3. Definition of Terms Outer barrier moves with V Dr Strain:g =Dx/Dr Strain Rate: dg/dt = v/Dr Shear stress:sxy = F/L (two-dimensions)

4. stress flowing Ds elastic strain Stress – Strain Relations Stress drop:Ds

5. With a Yield Stress Stress fluctuations in Al-Mg: Yield stress Lebyodkin, et al. 1995

6. Without a Yield Stress Stress fluctuations in low density granular material: No yield stress? Hartley, 2002

7. Stress-rate of strain: Viscosity Viscosity:h = stress/(strain rate) yield stress: “power law” fluid:

8. Questions • How well does the jamming phase diagram describe these diverse systems? • What impact does the existence of the yield stress have on the nature of the plastic flow and stress fluctuations? • What is the fundamental “event” during flow in a disordered material? • How well is the average behavior under steady strain captured by continuum models? How important are fluctuations?

9. Fluctuations versus Average Behavior Central problem: fluctuations in driven systems • Fluctuations in stress • Velocity fluctuations: • Generalization of temperature Current state: characterization of fluctuations and correlation with nonlinear bubble rearrangements.

10. What is meant by a fundamental “event”? Flow in a crystal => dislocation motion Flow in an amorphous material => “T1” event ??

11. Dislocation Example Plastic deformation in solids => defect motion

12. Classic Foam Event: T1 event Ordered foams => T1 events known to be important Disordered foams => T1 events are observed Do they generalize?

13. “STZ” Zone: alternate view Many “T1 events” => very successful description of plasticity in amorphous metals (Proposed by Falk and Langer, picture from Schuh and Lund, Nature Materials, Vol 2, 2003)

14. Bubble Raft Two useful systems Bead Raft

15. Clear “elastic” regime Clear yield stress Stress fluctuations are non-Gaussian Well defined average stress drops “localized” flow behavior; possible discontinuity No “elastic” regime (?) No yield stress (?) Stress fluctuations are non-Gaussian Well defined average stress drops Flow behavior not measured Comparison of Two Systems Bead Rafts Bubble Rafts

16. Why do velocity curves probe the jamming transition? • Couette Geometry: average stress,s, proportional to 1/r2 • Yield stress,sy: => critical radius beyond which “rigid” body or elastic behavior, shear rate is a continuous function of r.

17. Sample stress curve Yield Stress

18. Ways this can break down • v(r) “localized” to a region much smaller than predicted => average stress doesn’t predict the jamming behavior. • Discontinuity in the strain rate => simple continuous models of stress as a function of strain rate don’t apply. • Critical radius not predicted by continuum mechanics => spatial inhomogeneities are important, “stress chains”? Impact of T1 events?

19. Shear Localization Debregeas, Tabuteau, Di Meglio, PRL 87 (2001)

20. Strain Rate Discontinuity Coussot, Raynaud, et al., PRL 88, 218301 (2002)

21. What do we find? • Discontinuous jamming transition, as in the three dimensional system. • Simple continuum picture does not predict the critical radius => spatial inhomogeneities are important • Role of T1 events unclear. • Convergence of fluctuating velocity profiles to average profiles.

22. Details of the Experiments • Use a Couette geometry • Use a bubble raft or bead raft • Make stress measurements or use video images to study bubble motions

23. Apparatus

24. Schematic of Apparatus Inner radius ri: 3.84 cm Outer radius ro: 7.43 cm Area fraction: 0.95 Boundary conditions: no slip at both walls, but inner cylinder is free to move.

25. Bubble Motions

26. Flow Response:

27. Effective Viscosity: stress/(strain rate)

28. Stress versus strain rc=6.7 cm rc=6.3 cm sy= 0.8 mN/m (1) Shear rate = 3 x 10-2 s-1 (2) Shear rate = 4 x 10-3 s-1

29. Continuum Mechanics Digression …

30. Continuum Facts: Part II Analytic Solutions for bubble motion: “Elastic” motion: W: outer cylinder rotation rate w: inner cylinder rotation rate G: elastic modulus of bubble raft k: torsion constant of wire ro: outer cylinder radius

31. “Elastic” behavior

32. “Flowing States”

33. Larger System Fit to power law.

34. T1 events and stress

35. Connection between stress and bubble motions.

36. Fluctuations of critical radius

37. T1 events and average velocity

38. Summary • Clear jamming transition with no exponential “shear localization” under slow steady strain. • Spatial inhomogeneities are central to understanding the transition from flow to jamming. • Critical radius does not correspond to yield stress within a continuum picture. • Possible stress discontinuity – may be the result of finite particle size.

39. Teaser Result

40. Stress Fluctuations:step strain versus continuous strain

41. Step strain stress response

42. Close up of stress relaxation

43. Stress drop distributions

44. Average Stress Drop

45. Summary of step strain • Qualitatively continuous strain and step strains are similar • Possible quantitative differences => important to understand when comparing to models

46. Thanks to … • Video images of bubble raft: John Lauridsen • Viscosity measurements: Ethan Pratt • Initial Bubble tracking software: Gregory Chanan • Stress measurements/plastic bead experiments: Michael Twardos • Funding: Department of Energy grant DE-FG02-03ED46071, Sloan Foundation, Petroleum Research Fund, and UCI UROP Discussions with D. Durian, C. Maloney, A. Kraynik, S. Cox, A. J. Liu, C. Yu, C. O’Hern