Kuniyasu Saitoh Faculty of Engineering Technology, University of Twente, The Netherlands Physics of Granular Flows, the 25th of June, 2013, YITP, Kyoto University

I. Non-affine Response of Granular Packing

Contents 1.Introduction 2.Numerical model 3.Affine vs. Non-affine responses a.Response of overlaps b.Response of Probability distribution functions c.Scatter plots d.Parameters e.Non-affinity f.Langevin equation for the overlaps g.Anomalous relaxation of overlaps 4.Master equation a.Transition rate b.Closing & opening contact c.Markov-chain 5.Summary Kuniyasu Saitoh University of Twente Stefan Luding University of Twente Vanessa Magnanimo University of Twente

Introcution Static properties of granular packing Remarkable critical behavior near the jamming transition Mechanics of granular packing Response to macroscopic deformations is not affine (= non-affine). cf.) Affine displacement Particles’ positions Strain Question: How do the PDFs respond to the deformations? The PDF of forces [Metzger (2004)] Microscopic insights into granular packing Probability distribution functions (PDFs) of interparticle forces Response to a point loading [Ellenbroek et al. (2006)

Numerical model Molecular dynamics simulation Binary mixture of two-dimensional particles (50:50) filled in a square periodic box Force law Overlap Contact force Global damping System size The number of particles 10 samples for N=512, 2048, and 8192, and 2 samples for N=32768

Preparation of static packing We slowly grow the particles by controlling the mean overlap over all particles Static packing Critical scaling The mean overlapPressure The 1st peak of the radial distribution function jammed unjammed

Response of overlaps Rearrangement of particlesStatic state New static state Affine response of overlap Non-affine response Isotropic compression Rescaling of every radius Relaxation

Response of PDFs “Broadening” “Shift” Overlaps scaled by the mean value PDFs of overlaps Affine response Non-affine response

Scatter plots “Broadening”

Deviation from the affine response increases as increasing or decreasing Parameters Scattered data Red dots (●)Blue dots (●)

Non-affinity N= samples “Non-affinity” Mean overlaps after compression Mean variances Affine Non-affine and =constants

The distribution of is a Gaussian with the width (as shown in later). “Langevin equation” for the overlap Development of overlaps (>0) or forces mean valuefluctuation Development of overlaps (<0)

Anomalous relaxation of overlaps Cross-over decrease during relaxation increase during relaxation cf.) Bimodal character of stress transmission in granular packings F. Radjai et al. Phys. Rev. Lett. 90 (1998) 61.

Master equation Transition rate (Gaussian) Connect the PDFs cf.) Experiments of granular packing under compressions Majmudar & Behringer, Nature 435 (2005) Spatial correlations of forces were not found in their experiments under isotropic compressions.

Master equation (for negative overlaps) Transition rate (Cauchy dist.) Fitting parameter Closing contact Opening contact (could be fitted by…)

Markov-chain Increment of area fractions Initial condition is given by the MD simulation. Markov-chain

Summary We study microscopic responses of granular packing to compressions: Non-affine response Non-affine response of overlaps is described by the “Langevin equation”. The “broadening” of the PDFs is well quntified by Anomalous relaxation The overlaps increase during relaxation, while the overlaps decrease, where the cross-over scales Master equation We introduce the Master equation for the PDFs, where the transition rates are given by simple distribution functions. We confirm the Markov property with quite small compression rate.

Kuniyasu Saitoh University of Twente Ceyda Sanli OIST Graduate University Stefan Luding University of Twente Devaraj van der Meer University of Twente 1.Introduction 2.Experiments 3.Results 1.Large scale convection 2.Diffusion 3.Order Parameter 4.Dynamic Susceptibility 5.Width of Overlap Function 6.Correlation length 7.Dynamic Criticality 4.Summary Contents

Static structure Random, no difference between glass and liquid Dynamic structure or heterogeneity The displacements of particles are extremely heterogeneous in glass, while the displacements are homogeneous in liquid. We can make an analogy with critical phenomena by using the displacements as an order parameter. Introduction – Dynamic heterogeneity (DH) LiquidGlass

Introduction – DH in dense granular flows DH in driven granular systems Air-fluidized grains [Abate & Durian (2007)], Horizontally vibrated grains [Lechenault et al. (2008)], Sheared granular systems [Katsuragi et al. (2010), Hatano (2011)], etc. Control parameter Density & driving force Our aim: To investigate DH in granular convective flows Dense granular flows are (sometimes) more complex. eg.) granular convection

Setup 30 mm 40 mm Experiment Grains Polystyrene beads (polydisperse, mean diameter σ=0.62mm) Driving force Standing wave generated by the shaker Control parameter Area fraction of grains Interactions Capillary force (cohesive) Data Trajectories in 2 dimensions Units time = sec. length = σ

Convection Large scale convection Trajectories Velocity field Coarse graining function Fluctuation Transport by convection or 1 0 d r

1 2 Diffusion Mean square displacement : convection is dominant & almost ballistic : subdiffusion ( ) & normal diffusion ( ) Crossover time

1 0 a r Mobility Order parameter We normalize the displacementto [0,1]. or Overlap function Order parameter

Dynamic susceptibility & time scale has a single peak at Dynamic susceptibility = variance of the mobility Typical time scale cf.)

CG func.Overlap func.a (width of OF) None Step Gauss Step Gauss Step Gauss cf.) Horizontally vibrated grains (Lechenault & Dauchot et.al.) Width of overlap function Criteria for the width “a” The maximum amplitude of. CG func.Overlap func.Width “a”

4-point correlation function Correlation length Number density cf.) Horizontally vibrated grains (Lechenault & Dauchot et.al.) Correlation length satisfying

Dimension Horizontally Vibrated Grains Quasi-2D Air-fluidized Grains Quasi-2D Sheared Colloids44/33Quasi-2D Floating Grains Quasi-2D Exponents for Driven Granular Systems Dynamic Criticality of Floating Grains Dynamic criticality

Dynamic Heterogeneity in Floating Grains  Floating grains are driven by standing wave & flow with convection  CG method successfully subtracts the additional displacements  Crossover time diverges near the jamming point  Time scale given by susceptibility diverges near the jamming point  Dynamic correlation length diverges near the jamming point  Our results do not depend on the choose of CG & overlap functions Outlook  The maximum susceptibility  Self-intermediate scattering function  Self-van Hove function  Effect of convection Summary