Critical scaling for the jamming transition of granular materials - PowerPoint PPT Presentation

Critical scaling for the jamming transition of granular materials M. Otsuki (Aoyama Gakuin Univ.) H. Hayakawa (Kyoto Univ.)

Granular materials Sand Saturn ring mustard seed Ginkaku-ji temple

Shear stress Shear stress Shear stress Shear stress Shear stress Shear stress Sheared granular materials No flow Homogeneous flow Inhomogeneous flow Gas Dense liquid Amorphous solid ( Φ = 0.12) ( Φ = 0.8) ( Φ = 0.85)

Shear stress Shear stress Shear stress Shear stress Jamming transition Transition point No flow Φ J Homogeneous flow Dense liquid Amorphous solid ( Φ = 0.8) ( Φ = 0.85)

Jamming transition for athermal materials Foam Colloidal suspensions Josephson junction array

δ Model of granular materials Φ < Φ J Φ > Φ J Ft Fn Ft Tangential force Fn • Friction coefficient : μ Normal force • F n = k δ Δ - η v n • F t < μ F n (Coulomb’s friction) Elastic part Dissipative part • Frictionless : μ = 0 • Δ = 1 (Disk) • Frictional : μ > 0 • Δ = 3 / 2 (Sphere)

Critical properties Frictionless case, Δ = 1 Shear modulus Φ < Φ J Φ > Φ J G ~ ( Φ - Φ J ) 1/2 Pressure P Φ J Viscosity η η ~ ( Φ - Φ J ) -3 P ~ ( Φ - Φ J ) Φ J Φ - Φ J

α α, β : Critical exponents Rheological property Frictionless case, Δ = 1 Shear stress σ σ ( γ , Φ ) / | Φ - Φ J | β scaling plot . γ | Φ - Φ J | - α . Shear rate γ Hatano, 2008 . non-linear transport property . σ ( γ , Φ ) = | Φ - Φ J | β S ± ( γ | Φ - Φ J | - α ) For Φ < Φ J , σ ∝ γ 2 (liquid) For Φ > Φ J , σ ≃ const (solid) . For Φ ≃ Φ J , σ ∝ γ y γ

Dynamics (constant shear rate) Φ = 0.80 < Φ J Φ = 0.85 > Φ J

Dynamics (velocity fluctuation) Φ = 0.80 < Φ J Φ = 0.85 > Φ J

Characteristic features Mean field theory Dimension D = 2, 3, 4 with the same exponents obtained from the theory. The critical exponents The critical exponents are depend on the type of the independent of the contact force. dimension. F n = k δ Δ

Effect of Friction σ σ Frictionless ( μ = 0.0) Frictional ( μ = 2.0) Hysteresis loop for frictional case

Effect of friction (pressure) Pressure P Pressure P Δ P Frictionless ( μ = 0.0) Frictional ( μ = 2.0) Continuous transition Discontinuous transition

Effect of friction (type of the transition) 0.005 Δ P 0.004 0.003 0.002 0.001 Δ P 0 0 0.5 1 1.5 2 Continuous transition Discontinuous transition Friction coefficient μ

Phase diagram P ~ ( Φ - Φ S ) Area of the hysteresis loop Φ C Φ S Δ P Φ S Φ C

Summary & Discussion • Jamming transition : Athermal transition from liquid-like states to solid-like states. • Critical exponents depend on the interaction. • Continuous transition for frictionless case, discontinuous transition for frictional case. • Hysteresis loop, many critical densities. • Our result may provide a better understanding of dynamics and non-linear transport properties of dense matters.