1. Granular matter
김종현

2. Contents What is granular matter? A study about granular matter
Size segregation & Mixing
Conclusion & References

3. What is granular matter?
It consists of macroscopic particles of different size, shape, and surface properties
Granular flow
A flow with grains
A flow with powder in a vacuum(there is no fluid to support the particles)
A mixture of grains and fluid phase

4. Topics
Environment
Material behavior
Etc…
Segregation and clustering

5. Environment
Sand dunes
Astrophysical rings

6. Sand dunes
Their shape depends on the distribution of wind directions and ground behavior
Modeling of the morphology of sand dunes
Different time scales between wind field behavior and sand flux -> the calculation can be separated
The effect of perturbations of the wind field onto the stability of a dune

7. Global perturbation of the wind field onto a dune
The local interaction between the sand grains and the wind near the ground
Avalanches that maintains the sand transport due to gravity on the slip face

8. Astrophysical rings In simulation, viscosity increases with time
How the kinematic viscosity is depending on time and on the radial coordinate if there is some collisional dissipation

9. Material behavior Non-spherical particles Cohesion and sintering
Shear band formation
Hysteresis in the deformation of soils
Cohesion and sintering
Cohesive forces in granulates
Hydration kinetics of cement

10. Shear band formation
The deformations are not homogeneously distributed when apply pressure to confined granular materials
They are concentrated in thin layers of intensive shearing

11. Hysteresis in the deformation of soils

12. Cohesive forces in granulates
Particulate solids show sticky properties
Adhesive/cohesive forces acting between the particles
Reason : solid bridges
Solid bridges
Soluble particle material can form solid bridges itself through partial dissolution and re-solidification
Partial melting of particle with low melting point or at higher temperatures(sintering)

13. Yield loci depend on the loading history
Macroscopic cohesion is determined by the ability of the material to resist shear stress in static equilibrium without normal loading
When exceding a maximum stress(yield loci), the material begins to flow
Yield loci depend on the loading history

14. Hydration kinetics of cement
Ratio of hydrated and unhydrated numbers
x / (1 - x) ~ (t / t x) y
t < t x
Accelerated hydration ( y=2.5 )
Hydrates catalyse the hydration process
t > t x
Parabolic behavior ( y=0.5 )
They inhibit further hydration

15. Etc… Apollonian parking model Rheology of bi-disperse granular media
Studies on granular materials are confined to mono-disperse media
A real system : poly-dispersity in size and/or mass
Shock waves in dense gases

16. Apollonian parking model
Application
High performance concrete
Ceramics that have to endure extreme stress…

17. Segregation and clustering
Reverse size segregation and mixing
BNP and RBNP
Clustering and segregation
Pattern formation in vibrated granular media

18. Reverse size segregation & mixing
RBNP
Critical temperature Tc exist !!

19. BNP and RBNP BNP (Brazil Nut Problem) RBNP (Reverse BNP)
Hard spheres with large diameters segregate to the top when subjected to vibrations or shaking
RBNP (Reverse BNP)

20. Geometrical reorganization
Percolation effect
Smaller ones pass through the holes created by the larger ones
Geometrical reorganization
Small particles fill small openings below large particles
Global convection
Bring large particles up but not allow for reentry in the down stream

21. Define some parameters
Critical temperature Tc
Mass m and diameter d
Initial layer thickness t (in units of d)
Thickness of fluidized layer h

22. In D dimension, T > Tc T < Tc mv2/2 = DT/2 ~ mgdh At Tc , h=t
Tc ~ mgdt / t0 ( t0 is spatial dimension term )
T > Tc
The system is fully fluidized
T < Tc
A fraction of particles condenses at the bottom

23. In binary mixture of hard spheres, Tc(B) < T < Tc(A)
dA / dB =8 , mA / mB = 4 in 2D
dA / dB =2 , mA / mB = 2 in 3D

24. Crossover from the BNP to the RBNP

25. Crossover from the BNP to the RBNP
dA / dB =2 , mA / mB = 4 in 2D
dA / dB =2 , mA / mB = 6 in 3D

26. Crossover condition
y(D-1)= x 2D 3D

27. After… Does the RBNP exist? Reply
G.A. Canul-Chay et al. can not observe RBNP.
Temperature gradient exists along the vertical in their granular vibrating bed
Reply
The mixture is in contact with a thermal reservoir at a global temperature T (no temperature gradient)
Granular temperature (mean kinetic energy per particle) : balance between power input (vibration) and dissipation (inelastic collision)

28. A large, heavy grains rise, but equally large, light grains sink in a granular bed if the bed is deep and the amplitude of vibration is large
T. Shinbrot et al. “Reverse Buoyancy in Shaken Granular Beds”, PRL(1998)

29. Clustering and segregation
Reason : energy loss associated with particle-particle collisions
Necessary condition : net dissipation is strong
Example
Freely cooling bi-disperse mixture …

30. In freely cooling granular material, statistical fluctuations in density and temperature cause position dependent energy loss
Homogeneous state
Cluster growth
Clusters have reached system size

31. Pattern formation
A typical pattern that appears of vibrated materials (or external temperature gradient) is that of convection rolls
Pattern formation is cluster formation in a granular matter by vibration

32. f = 50 Hz
Γ = ω2A / g = 4.5

33. Magnetized sphere vibrates

34. Conclusion Study about granular matter is wide
RBNP is not yet completed

35. Reference Reverse Brazil Nut Problem, Daniel. C. et al. PRL, 86, 3423
Does the RBNP exist?, G.A.Canul-Chay et al. PRL, 89,
Comment on “RBNP : Competition between Percolation and Condensation”, H. Walliser PRL, 89,
Cluster-growth in freely cooling granular media, S. Luding, Chaos, 9, 673
Ordered Clusters and Dynamical states of Particles in a Vibrated fluid, Greg. A. et al., PRL, 88,