Driven Granular Fluids: Collective Effects W. Till Kranz MPI fr - PowerPoint PPT Presentation

Driven Granular Fluids: Collective Effects W. Till Kranz MPI für Dynamik und Selbstorganisation, Göttingen Institut für Theoretische Physik, Universität Göttingen Physics of Granular Flows, Ky¯ oto 2013

Acknowledgment Annette Zippelius ◮ Matthias Sperl ◮ Andrea Fiege ◮ Iraj Gholami ◮ Timo Aspelmeier ◮ W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Sandstorm for Experimental Physicists Abate & Durian, PRE 74 2006 ◮ Steel balls ( ∼ 7 mm ∅ ) on a sieve ◮ Driven by air flow ◮ Measurement of mean square displacement δ r 2 ( t ) = [ r ( t ) − r ( 0 )] 2 � � W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Inelastic, Smooth, Hard Spheres: A Model for Granular Particles Hard Spheres completely characterized by ◮ Mass m ◮ Radius a ◮ Coefficient of restitution ǫ ∈ [ 0 , 1 ] Collision law v t v 12 v ′ n = − ǫ v n , v ′ t = v t v n Energy Loss on average per collision ∆ E ∝ 1 − ǫ 2 W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Sandstorm for Theoretical Physicists Random Force ξ i ( t ) , gaussian distributed ◮ Average � ξ i � = 0 ξ 2 ◮ Driving power P D = � � i Stationary State as a balance between driving & dissipation Event Driven Simulations 10 000 particles Bidisperse to avoid crystallization Coefficient of Restitution ǫ = 0 . 9 I. Gholami et al. PRE 84 2011 Area Fraction η = 0 . 1–0 . 81 W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Sandstorm for Theoretical Physicists Random Force ξ i ( t ) , gaussian distributed ◮ Average � ξ i � = 0 ξ 2 ◮ Driving power P D = � � i Stationary State as a balance between driving & dissipation Event Driven Simulations 10 000 particles Bidisperse to avoid crystallization Coefficient of Restitution ǫ = 0 . 9 I. Gholami et al. PRE 84 2011 Area Fraction η = 0 . 1–0 . 81 W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Outline Static Structure & Momentum Conservation 1 Long-Time Tails 2 The Granular Glass Transition 1 3 1 See also the Lecture on Friday W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Static Structure: A Surprise 2 3.0 ε = 1.0 ε = 0.7, single 2.5 structure factor S q 2.0 ◮ Strong increase for k → 0 1.5 ◮ Highly Correlated on large 1.0 Length Scales 0.5 ◮ Implies so called 0.0 Giant Number Fluctuations 0.0 π 0.5 π 1.0 π 1.5 π 2.0 π 2.5 π wave number qd ◮ Volume Fraction ϕ = 0 . 2 ◮ N = 50 × 400 000 2 Kranz, Fiege, Zippelius, in preparation W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Toy Model ∂ t h ( r , t ) = η ∇ 2 h ( r , t ) + ξ ( r , t ) ≃ � ˆ ξ ( − k )ˆ ξ ( k ) � � � ˆ h ( − k )ˆ Correlation Function C ( k ) = h ( k ) Grinstein et al. , PRL 64 1990 η k 2 � � ξ ( − k )ˆ ˆ Random Force � ξ ( r ) ξ ( r ′ ) � ∝ δ ( r − r ′ ) ⇒ ξ ( k ) = 1. � � ∝ η k 2 due to FDT ξ ( − k )ˆ ˆ Equilibrium ξ ( k ) Local Pairs � ξ ( r ) ξ ( r ′ ) � ∝ Θ( r − r ′ − ℓ ) ⇒ � � ξ ( − k )ˆ ˆ ∝ ℓ 2 k 2 . ξ ( k ) W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Toy Model ∂ t h ( r , t ) = η ∇ 2 h ( r , t ) + ξ ( r , t ) ≃ � ˆ ξ ( − k )ˆ ξ ( k ) � � � ˆ h ( − k )ˆ Correlation Function C ( k ) = h ( k ) Grinstein et al. , PRL 64 1990 η k 2 � � ξ ( − k )ˆ ˆ Random Force � ξ ( r ) ξ ( r ′ ) � ∝ δ ( r − r ′ ) ⇒ ξ ( k ) = 1. � � ∝ η k 2 due to FDT ξ ( − k )ˆ ˆ Equilibrium ξ ( k ) Local Pairs � ξ ( r ) ξ ( r ′ ) � ∝ Θ( r − r ′ − ℓ ) ⇒ � � ξ ( − k )ˆ ˆ ∝ ℓ 2 k 2 . ξ ( k ) W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Toy Model ∂ t h ( r , t ) = η ∇ 2 h ( r , t ) + ξ ( r , t ) ≃ � ˆ ξ ( − k )ˆ ξ ( k ) � � � ˆ h ( − k )ˆ Correlation Function C ( k ) = h ( k ) Grinstein et al. , PRL 64 1990 η k 2 � � ξ ( − k )ˆ ˆ Random Force � ξ ( r ) ξ ( r ′ ) � ∝ δ ( r − r ′ ) ⇒ ξ ( k ) = 1. � � ∝ η k 2 due to FDT ξ ( − k )ˆ ˆ Equilibrium ξ ( k ) Local Pairs � ξ ( r ) ξ ( r ′ ) � ∝ Θ( r − r ′ − ℓ ) ⇒ � � ξ ( − k )ˆ ˆ ∝ ℓ 2 k 2 . ξ ( k ) W T Kranz Göttingen Driven Granular Fluids: Collective Effects

A Toy Model ∂ t h ( r , t ) = η ∇ 2 h ( r , t ) + ξ ( r , t ) ≃ � ˆ ξ ( − k )ˆ ξ ( k ) � � � ˆ h ( − k )ˆ Correlation Function C ( k ) = h ( k ) Grinstein et al. , PRL 64 1990 η k 2 � � ξ ( − k )ˆ ˆ Random Force � ξ ( r ) ξ ( r ′ ) � ∝ δ ( r − r ′ ) ⇒ ξ ( k ) = 1. � � ∝ η k 2 due to FDT ξ ( − k )ˆ ˆ Equilibrium ξ ( k ) Local Pairs � ξ ( r ) ξ ( r ′ ) � ∝ Θ( r − r ′ − ℓ ) ⇒ � � ξ ( − k )ˆ ˆ ∝ ℓ 2 k 2 . ξ ( k ) W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Divergence under Control 2.0 ε = 1.0 3.0 ε = 1.0 ε = 0.9 ε = 0.7, single ε = 0.8 2.5 ε = 0.7, paired ε = 0.7 structure factor S q 1.5 structure factor S q 2.0 1.0 1.5 1.0 0.5 0.5 0.0 0.0 0.0 π 0.5 π 1.0 π 1.5 π 2.0 π 2.5 π 3.0 π 0.0 π 0.5 π 1.0 π 1.5 π 2.0 π 2.5 π wave number qd wave number qd ◮ Volume Fraction ϕ = 0 . 4 ◮ Volume Fraction ϕ = 0 . 2 W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Insight can be used for Measurements 1.00 ε =0.9 ε =0.8 0.98 ε =0.7 shear viscosity η 0.96 ratio η int / η 10 0.94 0.92 ε =0.9 0.90 ε =0.8 ε =0.7 1 0.88 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 volume fraction ϕ volume fraction ϕ W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Long-Time Tails W T Kranz Göttingen Driven Granular Fluids: Collective Effects

The Velocity Autocorrelation Function 10 0 ◮ ψ ( t ) = � v s ( 0 ) | v s ( t ) � 10 -1 Long-Time Tails ψ ( t ) ∝ t − α | ψ (t)| 10 -2 (instead of exponential decay) ◮ In 3D elastic & inelastic hard spheres 10 -3 t -3/2 have α = 3 / 2 10 -2 10 -1 10 0 10 1 t W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Equation of Motion � t ∂ t ψ ( t ) + ω E ψ ( t ) + d τ M ( t − τ ) ψ ( τ ) = 0 0 Collision Frequency ω E Memory Kernel M ( t ) Incoherent Scattering Function φ s ( k , t ) contains more information about the tagged particle W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Mode-Coupling Approximation 3 ◮ Consider coupling of tagged particle to Collective Density Modes φ ( k , t ) Longitudinal Current Modes φ L ( k , t ) Transverse Current Modes φ T ( k , t ) ◮ Transverse Mode yields slowest decay (in 3D) � ∞ 2 ( k ) φ T ( k , t ) φ s ( k , t ) M ( t → ∞ ) = M T ( t → ∞ ) ∝ ( 1 + ε ) 2 dkj ′′ 0 0 and indeed ψ ( t → ∞ ) ∝ t − 3 / 2 ◮ Situation in 2D is very subtle 3 Kranz, Zippelius, in preparation W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Mode-Coupling Approximation 3 ◮ Consider coupling of tagged particle to Collective Density Modes φ ( k , t ) Longitudinal Current Modes φ L ( k , t ) Transverse Current Modes φ T ( k , t ) ◮ Transverse Mode yields slowest decay (in 3D) � ∞ 2 ( k ) φ T ( k , t ) φ s ( k , t ) M ( t → ∞ ) = M T ( t → ∞ ) ∝ ( 1 + ε ) 2 dkj ′′ 0 0 and indeed ψ ( t → ∞ ) ∝ t − 3 / 2 ◮ Situation in 2D is very subtle 3 Kranz, Zippelius, in preparation W T Kranz Göttingen Driven Granular Fluids: Collective Effects

The Glass Transition W T Kranz Göttingen Driven Granular Fluids: Collective Effects

The Glass Transition Amorphous Solid from either Supercooled Melt 1 Supersaturated Suspension 2 Dense Granular Fluid? 3 ◮ No Static Order Parameter Debenedetti & Stillinger, Nature 2001 W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Order Parameter: Plateau of the Scattering Function η ≪ η c φ q ( t ) � � Scattering Function φ ( q , t ) = ρ ∗ q ( τ ) ρ q ( τ + t ) log t η � η c independent of τ Density ρ ( r , t ) = � i δ ( r i ( t ) − r ) φ q ( t ) Order Parameter f q = φ ( q , t → ∞ ) ◮ Fluid: f q = 0 log t η ≥ η c ◮ Glass: f q > 0 φ q ( t ) log t W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Equation of Motion 4 � t ( ∂ 2 t + q 2 v 2 q ) φ ( q , t ) + d τ M ( q , t − τ ) ∂ τ φ ( q , τ ) = 0 0 Speed of Sound v q Memory Kernel M ( q , t ) 4 Kranz,Sperl,Zippelius, PRL 104 , 225701 (2010); PRE 87 , 022207 (2013) W T Kranz Göttingen Driven Granular Fluids: Collective Effects

Mode-Coupling Approximation � t ( ∂ 2 t + q 2 v 2 q ) φ ( q , t ) + d τ M ( q , t − τ ) ∂ τ φ ( q , τ ) = 0 0 ◮ Interpretation as interacting undamped sound waves � M ( q , t ) ≈ V qkp W qkp φ ( k , t ) φ ( p , t ) q = k + p Loss of Detailed Balance implies ◮ Rate of creation V qkp � = rate of annihilation W qkp ◮ Can still be calculated explicitly W T Kranz Göttingen Driven Granular Fluids: Collective Effects

The Granular MCT Glass Transition 0.59 Glas Volumenbruch η c 0.57 ◮ Percus-Yevick static structure factor 0.55 ◮ Iterative Numerical Solution 0.53 Fluid ◮ Standard Discretization Parameters 0.51 0 0.2 0.4 0.6 0.8 1 Restitution ε W T Kranz Göttingen Driven Granular Fluids: Collective Effects