Non- affine Response of G ranular P acking and Dynamic Heterogeneities in Dense Granular Flows

1. Physics of GranularFlows, the 25th of June, 2013,YITP, Kyoto University Non-affine Response of GranularPackingandDynamicHeterogeneities in DenseGranularFlows KuniyasuSaitoh Faculty of Engineering Technology, University of Twente, The Netherlands

2. I. Non-affine Response of Granular Packing

3. Contents • Introduction • Numerical model • Affine vs. Non-affineresponses • Response of overlaps • Response of Probability distribution functions • Scatter plots • Parameters • Non-affinity • Langevin equation for the overlaps • Anomalous relaxation of overlaps • Master equation • Transition rate • Closing & opening contact • Markov-chain • Summary Stefan Luding University of Twente Vanessa Magnanimo University of Twente KuniyasuSaitoh University of Twente

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

5. Numerical model Moleculardynamicssimulation 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

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

7. Response of overlaps Static state Rearrangement of particles New static state Isotropic compression Rescaling of every radius Affine response of overlap Relaxation Non-affine response

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

9. Scatter plots “Broadening”

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

11. Non-affinity N=8192 10 samples “Non-affinity” Mean overlaps after compression =constants Affine Non-affine Mean variances and

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

13. 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.

14. Master equation Connect the PDFs Transition rate (Gaussian) cf.) Experiments of granular packing under compressions Majmudar & Behringer, Nature 435 (2005) 1079. Spatial correlations of forces were not foundin their experiments under isotropic compressions.

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

16. Markov-chain Markov-chain Increment of area fractions Initialcondition is givenby the MD simulation.

17. Summary We studymicroscopic responses of granularpackingtocompressions: 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 overlapsincreaseduringrelaxation, while the overlaps decrease, where the cross-over scales Master equation We introduce the Master equationfor the PDFs, where the transition rates are given by simple distribution functions. We confirm the Markov property with quite small compression rate.

18. II. DynamicHeterogeneities in DenseGranularFlows

19. Contents • Introduction • Experiments • Results • Large scale convection • Diffusion • Order Parameter • Dynamic Susceptibility • Width of Overlap Function • Correlation length • DynamicCriticality • Summary Ceyda Sanli OIST Graduate University KuniyasuSaitoh University of Twente Stefan Luding University of Twente Devaraj van der Meer University of Twente

20. Introduction – Dynamicheterogeneity (DH) Liquid Glass Staticstructure Random, no difference between glass and liquid Dynamicstructureorheterogeneity The displacements of particles are extremelyheterogeneousin glass, while the displacements are homogeneous in liquid. We can make ananalogywithcriticalphenomena by using the displacements as an order parameter.

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

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

23. Large scaleconvection Convection Trajectories Velocity field Coarsegrainingfunction Fluctuation Transport by convection 1 or 0 r d

24. Diffusion 2 1 Mean square displacement : convection is dominant & almostballistic : subdiffusion ( ) & normaldiffusion ( ) Crossover time

25. Order parameter 1 1 0.5 0 0 r a We normalize the displacement to [0,1]. Mobility Overlap function or Order parameter

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

27. Width of overlap function Criteria for the width “a” 1. 2. The maximum amplitude of . CG func. Overlap func. Width “a” cf.) Horizontally vibrated grains (Lechenault & Dauchot et.al.)

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

29. Dynamiccriticality Dynamic Criticality of Floating Grains Exponents for Driven Granular Systems

30. Summary • Dynamic Heterogeneity in Floating Grains • Floating grainsare driven by standing wave & flow with convection • CG methodsuccessfully subtracts the additional displacements • Crossover timediverges near the jamming point • Time scalegiven by susceptibility diverges near the jamming point • Dynamic correlation lengthdiverges near the jamming point • Our resultsdo notdepend on the choose of CG & overlap functions • Outlook • The maximum susceptibility • Self-intermediate scattering function • Self-van Hove function • Effect of convection