Jamming and Flow of Rod-Like Granular Materials

Transcription

1 Scott Franklin Undergraduate Researchers Kevin Stokely Ari Diacou Ken Desmond Jess Gallagher Scott Barker Jamming and Flow of Rod-Like Granular Materials PRE 67, (2003) PRE 73, (2006) Petroleum Research Fund ACS Research Corporation

2 Granular Basics 1 Granular materials: collections of macroscopic ( mm) particles interacting through contact forces (e.g. friction) Thermal energy irrelevant (k B T J P E 10 5 J) systems frozen in metastable state, unable to move to lower energy state without external help exhibit solid, liquid, gas-like behavior density of a granular material is not well defined (packing fraction: percentage of space occupied) 3-d spheres: 52%< ρ <74% 2-d disks: 78%< ρ <91% long, thin rods in 3-d: 5%< ρ <91% long, thin rods in 2-d:??< ρ <100%

3 Packing and Force Distribution 2 Ordered packing yields uniform force distribution Slight disorder produces localized force chains (Parke Rhoads, 1998) large force chains more common than Gaussian

4 Packing and Force Distribution 2 Ordered packing yields uniform force distribution Slight disorder produces localized force chains (Parke Rhoads, 1998) large force chains more common than Gaussian F y tube sand = W F y tube sand = µf sand tube x µ 0.2 = Fsand tube x 5W

5 Packing and Force Distribution 2 Ordered packing yields uniform force distribution Slight disorder produces localized force chains (Parke Rhoads, 1998) large force chains more common than Gaussian F y tube sand = W F y tube sand = µf sand tube x µ 0.2 = Fsand tube x 5W F F 2F sin (θ) = W = F = W 2 sin (θ)

6 Who cares? 3 40% loss in production from factories from handling granular materials great industrial significance petroleum: extract oil through (granular) soil chemical: maximize reaction rate between liquid and particulate reactants pharmaceutical: uniformly mix particulate constituents for medicine large percentage of GDP estimated to involve granular materials (33%???) Physicists: how to describe something both solid and liquid? Continuum theories lack stress-strain relation to close the equations Friction requires history dependence, MD simulations promising

7 3D Jamming 4 Particles (length L, width w) poured into acrylic tube of diameter D to height h 1/4 ball at bottom of tube pulled upwards by motor, connected to force sensor Scale lengths by particle length L: D = D/L Scale force by pile weight: F = F/[ρ(π(D/2) 2 )hφ]

8 Packing Largely Determined by Particle Geometry 5 φ depends on aspect ratio α, less so on D data agree with geometric model with one free parameter: average contact number c = 5.25 ± 0.03

9 Geometric Random Contact Model 6 Contacts between particles uncorrelated, pile essentially collection of independent pairs of rods in contact some particle orientations made inaccessible by existence of another (rods very close together must be aligned) Average number of contacts proportional to the product of density and excluded orientations: c = 1 f ex (ρ, r)ρ( r)d r 2 Approximate local density ρ( r) with average density ρ and ρ = 2 c v ex. (v ex = r 2 dr sin θ dθ dφf ex is long and ugly and derived by Onsager 50 years ago)

10 How Hard is it to Leave the Pile? 7 As D 1 maximum force needed increases. Particularly dramatic at lower aspect ratios (higher packing fractions).

11 Small α, Large D: Stick-slip motion 8 linearly increasing force (ball at rest) followed by very short bursts of motion maximum force comparable to pile weight Fluctuations due to granular rearrangements

12 Large α, Small D: Solid-body motion 9 Force can be many times greater than pile weight Fluctuations due to stick-slip motion on surface

13 Modest α, D: Transition Region 10 Transition region marked by small moments of solid-body like motion followed by particle rearrangments that bring the ball to a stop

14 Pile Phase Space: Tube Diameter vs. Aspect Ratio 11 In a single tube, increasing particle length moves system through transition region to solid-like motion. Whether this is due to decreased D or increased α is unknown (yet).

15 Pile Phase Space: Tube Diameter vs. Aspect Ratio 12 In a single tube, increasing particle length moves system through transition region to solid-like motion. Whether this is due to decreased D or increased α is unknown (yet).

16 Power-Law Scaling of Fluctuation Spectrum 13 fluctuations change from f 2 to f 1 ordinary granular materials scale as f 2 (Albert et al.), dry friction as f 1

17 Scaling Exponent: D = 5 cm 14 Transition region has larger fluctuations as system spends more time in solid-body (dry friction) state

18 Scaling Exponent: D = 2.5 cm 15

19 Scaling Exponent: D = 7.5 cm 16

20 Scaling Exponent Doesn t Simply Depend on D 17

21 Force Power Spectra Exponents 18 Power spectra of fluctuating force show power law tails with exponent characteristic of pile behavior

22 Flow of Particles Through Hoppers 19 Pour Release Measure mass Repeat

23 Prolate vs. Round Particles 20 Prolate granular materials (aspect ratio=α: length diameter = 16, 32) Random walk explains probability P (N) of arch w/n round particles Requires arch to be concave down

24 Random Walks in 2d Hopper Flow (To et al., 2001) 21 Left to right: π/2 > θ i > π/2 Concave down: θ 1 >... > θ i >... > θ n 1 No overlap: i k=1 r k j k=1 r k D Arch spans opening: X + D > R Pick number of disks in arch and simply add up all configurations that satisfy the constraints a n (x) = A n π/2 π/2 f 1 (θ 1 ) dθ 1... θn 1 β n 1 f n 1 dθ n 1 δ(x n 1 i=1 cos θ i )

25 Flow Mass (before jamming) Noisy quanity 22

26 Exponential Decay Distribution of Flow Mass 23 independent of aperture size

27 Uncorrellated Drop Model (Percolation, Random Walk) 24 Assume that the probability of a particle to exit the hopper is p, and doesn t depend on particle s neighbors (uncorrelated) Probability of exactly n particles exiting hopper before a jam (P (n))is P (n) = p p p... p (1 p) = p n (1 p) = log P (n) n log p If particles fall in groups of k and groups uncorrelated, P (k) p n k (1 p), scaling still holds Do we really believe rods fall in an uncorrelated arrangement?

28 Probability Scales with Mean Flow Size 25 apertures ranging in size by factor of six collapse onto single curve All details contained in mean flow

29 Mean Flow Diverges 26 Details of divergence unknown

30 Granular Flow in Rotating Drum 27 Another canonical system for study of granular systems

31 Critical Angles in Ordinary Granular Materials 28 Ordinary granular materials rotate until a critical angle, at which point an avalanche occurs, reducing the angle to a lower critical angle

32 Disappearance of Critical Angles w/increasing α 29 As aspect ratio increases, avalanches become less regular

33 Avalanches and Surface Flow 30 α = 4, L 1/4 α = 4, L = 1

34 Large Aspect Ratios and Pile Width W W/L 31 α = 16, W = 6 α = 16, W = 1/4

35 Conclusions and Future Work 32 Large aspect-ratio granular materials show a transition to solid-body like motion in both 2 and 3 dimensions For 3d experiments, both particle geometry and container size important Container size effects seen in increased angle of repose as pile thickness is decreased. Hopper flow intriguingly similar to ordinary granular materials MD simulations would complement increasing number of experimental systems we re investigating