Granular matter 2003. 6. 11 김종현

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

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

Topics Environment Material behavior Etc… Segregation and clustering

Environment Sand dunes Astrophysical rings

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

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

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

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

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

Hysteresis in the deformation of soils

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)

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

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

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

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

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

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

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)

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

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

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

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

Crossover from the BNP to the RBNP

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

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

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)

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)

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

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

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

f = 50 Hz Γ = ω2A / g = 4.5

Magnetized sphere vibrates

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

Reference Reverse Brazil Nut Problem, Daniel. C. et al. PRL, 86, 3423 Does the RBNP exist?, G.A.Canul-Chay et al. PRL, 89, 189601 Comment on “RBNP : Competition between Percolation and Condensation”, H. Walliser PRL, 89, 189603 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, 234301