An introduction to the physics of. Granular Matter. Devaraj van der Meer

Transcription

1 An introduction to the physics of Granular Matter Devaraj van der Meer

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3 GRANULAR MATTER is everywhere: in nature: beach, soil, snow, desert, mountains, sea floor, Saturn s rings, asteroids... in industry: mining, pharmaceutical, food, construction, chemical...

4 Granular Matter can behave like a solid... a liquid... or a gas

5 When solid, Granular Matter is a special solid

6 Reynolds dilatancy Osborne Reynolds (1885): A strongly compacted granular medium dilates under pressure.

7 What causes the dilatancy? f = f = 0.785

8 When it behaves like a liquid, Granular Matter is a special liquid =?

9 hourglass - sand hourglass water klepsydra V out = k t V out = k t b t 2

10 Vibrated bidisperse mixture Segregation!

11 Brazil Nut Effect

12 Three explanations BNE 1. percolation: small grains percolate the empty spots between the large ones. 2. exclusion: while vibrating small grains fill space 3. convection: interaction with below the large ones, not vice versa. walls trigger convection rolls. large grains can follow the upward, but not the downward flow.

13 Reason clustering: Inelastic collisions Driving strength: high low And if Granular Matter behaves like a gas, it is a special gas

14 Flux model 1 2 Flux function

15 In five compartments:

16 Planet with rings

17 Phenomena Granular Solid: Packing density, dilatancy, force chains, compactification, pressure saturation (RJ law) Granular Fluid: Arching, blocking, convection, segregation Granular Gas: Clustering, non-equipartion

18 Why does Granular Matter behave so differently from other solids and fluids we know?

19 Definition: 1. GM is athermal Granular Matter = many body system in which the typical particle size > 100 µm (at room temperature) Thermal energy is negligible for such particles!

20 2. GM interacts through contact forces Chaotic network of contact points and forces!

21 3. GM interactions are dissipative v 2,n h 1 h 2 coefficient of normal restitution: v 1,n Grains have many internal degrees of freedom through which kinetic energy is dissipated. (sound, heat, deformation)

22 Implications 1. athermal Thermodynamic T irrelevant Define granular temperature 2. contact forces Ordered molecular-scale structures do not occur 3. dissipation Far-from-equilibrium system Constant energy supply is necessary to keep systems alive (i.e. T g >0)

23 Typical practical problems Production and handling: - cornflakes: filling - pill production: mixing - casting by sacrificial polystyrene Nature (geophysics): - dunes: movement - avalanches: ranges, volume, prediction - dikes: stability - seismology

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25 Casting by sacrificial polystyrene

26 solid fluid Granular packing (for spheres) 0.55, RLP = random loose packing 0.64, RCP = random close packing 0.74, crystal = perfectly hexagonal solid fraction

27 Compactification experiment regime 1: local reorganization regime 2: global reorganization

28 Analogy: car-parking in street Model (Ben-Naim): Initial state: randomly parked cars (no extra fit in) Start to move cars randomly. Whenever there is a large enough gap, a new car jumps in. regime 1: movement of a single car creates gap regime 2: more than one car has to move: required time for gap to open grows exponentially:

29 Force Chains (Bob Behringer, Duke)

30 In stalling flow, force chains manifest themselves as arches Segovia, Spain Pont du Gard, France Spain

31 Importance of sidewalls: Rayleigh-Janssen model Force parallelogram as unit cell of a 2D granular medium vertical forces horizontal forces balanced by sidewalls Lord Rayleigh: K = coefficient of redirection

32 Importance of sidewalls: Rayleigh-Janssen model (2) Slice experiences friction force with sidewalls: Vertical force balance on slice: Integration gives: Janssen s equation

33 Importance of sidewalls: Rayleigh-Janssen model (2) p v (hydrostatic regime) crossover: (saturated regime) h Janssen s equation

34 Effective weight of granulate in silo (decompaction parameter) Effective weight on bottom = F v (h) = p v (h) A (for large χ, i.e., large h) What happens to the remaining weight?

35 Collapsing silos Walls take this weight!

36 Decompactification through shaking Shaking: a sin(ω t); dim.less acceleration: vertical shaking Decompacted means: acceleration overcomes friction Force balance: acceleration gravity = (wall) friction: Sand moves freely if lhs > rhs!

37 Decompactification through shaking (threshold calculation) Total height of stack: h 0 Threshold condition lhs>rhs fullfilled from h t (<h 0 ) on. Γ = 1 means: h t = h 0 ; nothing can be fluidized Γ = 2 or larger: all can be fluidized

38 Decompaction: Experiment, Simulation and Theory Lan & Rosato, Phys. Fluids 7, 1818 (1995)

39 Decompaction: Experiment, Simulation and Theory

40 Vertically shaken granular matter: relevant dim.less parameters 1. Mildly fluidized: particles move with bottom, H = a Vigorously fluidized: take intrinsic l.s. H = R 2. Dissipation per particle: H typical (vertical) lengthscale dimensionless acceleration dimensionless shaking strength 3. Filling factor: (layers of particles) filling factor inelasticity

41 Vertically vibrated granular gas * Granular temperature: z T(z) * For dilute system: T roughly independent of z * Barometric height distribution: ρ(z)

42 Density and temperature in MD simulations z z ρ T

43 T follows from energy balance Energy input at bottom: For sawtooth driving: v out = -v in + 2v b v in v b

44 T follows from energy balance Energy input at bottom: Energy dissipation in the system Integral gives:

45 T follows from energy balance Energy input at bottom: Energy dissipation in the system Equating energy input and dissipation gives:

46 Flux function Flux through the hole is: F = density * velocity * area hole 1 2 Use: S h with:

47 Stability analysis 2 box system Around n 1 =1/2: B<B c : B>B c : stable unstable

48 Bifurcation diagram ~ B / N 2

49 Is a general hydrodynamic description of granular matter possible?

50 A) Hydrodynamic approach Coarse graining over small intervals Δx, Δt to define macroscopic quantities: density: velocity: temperature: Assuming local thermal equilibrium, one can derive mass, momentum, and energy conservation laws:

51 Conservation laws In the dilute limit, using the ideal gas law: expresses inelasticity In the stationary limit (u=0, d t =0) this becomes: These equations can be solved analytically:

52 Hydrodynamic solution: NOTE: ideal case (ε=0, no dissipation) has the solution: T(x) = T 0 ρ(x) = ρ 0 Using the boundary conditions: [constant T at left border] [elastic wall (no heat flux) at right border]

53 Particle dynamics solution: (using MD simulations)

54 B) Discrete description 2-particle collision with * momentum conservation: Before... * energy dissipation:...after collision. This implies: with:

55 Ideal case ε = 0: exchange of velocities. Finally all velocities will be given by the PDF of velocities on the left. Uniform distribution of particles, consistent with continuum description. Non-ideal case ε > 0: Numerical result very different from continuum result! 1 fast particle and (N-1) slow particles, clustering to the right and dissipating energy. Fast particle transports energy from left to right. No longer local thermal equilibrium! Breakdown of continuum approach!

56 Velocity center of mass (1) assume: v 0 = const = 1 (no random distribution) N N 1 N before first collision: after first collision (between N and N 1): after second collision (between N 1 and N 2):... after (N-1) th collision (between 2 and 1):

57 Velocity center of mass (2) Mean velocity of particles N,N-1,...,3,2: for large N * ε = 0, ideal case: * ε 0, real case: drift of cluster towards wall!

58 In an isolated 1D case granular hydrodynamics does not work. What about the general case?

59 Knudsen number l L λ λ = mean free path l = typical length at which macroscopic quantities vary L = typical system size Kn = λ/l Kn loc = λ/l (global Knudsen number) (local Knudsen number) Hydrodynamics work if Kn<<1! Molecular system: local Kn <<1 (not a Knudsen gas!) Granular system: local Kn large!

60 Granular Leidenfrost effect 1000 fps F = 16 layers

61 No separation of scales No separation of length scales: macroscopic quantities vary on the same scale as the mean free path! Flowing systems: mean velocity ~ thermal velocities = velocity fluctuations No separation of time scales: macroscopic quantities can change as fast as particle velocities

62 Conclusion There are many reasons why granular hydrodynamics should NOT work... The surprise is that nevertheless in many cases it DOES work!!!!

63 Smoluchowski-Feynman ratchet pawl vanes Marian Smoluchowski (1912) ratchet weight Richard P. Feynman (1963) Does not work in a molecular gas at thermal equilibrium!

64 But in a granular gas the ratchet works! Freshmen s physics project, University of Twente

65 Experimental setup Governing parameter: Rotational position sensor

66 Granular mill: typical time series θ [rad] ω [rad/s] t [s] t [s] ( N = 2000 particles, h = 51 mm, a = 1.5 mm, f = 110 Hz, S = 2.15 ) Look at probability distribution function of ω and θ

67 Mill: mild shaking P(ω) a = 3.5 mm f = 20 Hz S = 0.39 Vane motion is confined to a ω [rad/s] small region of phase space Kramers escape problem

68 Mill: moderate shaking P(ω) a = 1.5 mm f = 110 Hz S = 2.15 P(θ ) θ [rad] X-position is again favored ω [rad/s]

69 Mill: strong shaking P(ω) a = 3.5 mm f = 65 Hz S = 4.08 Vane motion sets ω up [rad/s] large scale gas circulation

70 Breaking the symmetry Possibility 1: Introduce ratchet and pawl on axis Disadvantage: granular gas properties may vary within the container Possibility 2: Break symmetry at the vanes: tape no tape geometrically by changing the collisional properties

71 Ratchet: moderate shaking a = 1.5 mm f = 110 Hz S = 2.15

72 Ratchet: moderate shaking a = 1.5 mm f = 110 Hz P(ω) S = 2.15 <ω > = 0.45 rad/s ω [rad/s]

73 Ratchet: strong shaking a = 3.5 mm f = 65 Hz S = 4.08

74 Ratchet: strong shaking P(ω) a = 3.5 mm f = 65 Hz S = 4.08 <ω > = 2.49 rad/s ω [rad/s]

75 Ratchet: ω vs. S

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