Mode-coupling theory for sheared granular liquids Hisao Hayakawa (YITP, Kyoto Univ. , Japan), in collaboration with Koshiro Suzuki (Canon Inc.) 2013/5/22 Complex Dynamics in Granular Systems at Bejing (May 18-June 21, 2013) 1　　　 1

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 2　　　

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 3　　　

Complex Dynamics in Granular Systems Introduction Most of theories for granular physics are phenomenology. The kinetic theory based on Enskog equation only gives a microscopic basis, though the determination of pair correlation function is phenomenological. How can we go beyond the kinetic regime? 4 2013/5/22 Complex Dynamics in Granular Systems 4

Bagnold’s scaling and kinetic theory The results for granular flows are consistent with the prediction of the kinetic theory. N. Mitarai and H. Nakanishi, PRL94, 128001 (2005). 5 5 Complex Dynamics in Granular Systems

The limitation of kinetic theory   Complex Dynamics in Granular Systems 6

Background and objective Kinetic theory(Enskog) 　：φ<0.5 Liquid theory 　：0.5<φ<0.64 Constitutive equation = Bagnold law Const. Eq. = power law 　Bagnold⇒ power law? Application of MCT to sheared system 　：projection to momentum Mode-coupling(MCT) 　： noise excitation 2013/5/22 Complex Dynamics in Granular Systems 7　　　 7

Background and objective Jamming phase diagram Ikeda-Berthier-Sollich (2012) 1/ 2013/5/22 Complex Dynamics in Granular Systems 8　　　 8

Previous studies １ : vibrating beds Experiment（2D）： Plateau-like mean square displacement (MSD) appears. Application of MCT to white noise thermostat [Abate, Durian, PRE74 (2006) 031308] : Air-fluidized bed [Reis et al., PRL98 (2007) 188301] : Vibrating bed [Kranz, Sperl, Zippelius, PRL104 (2010) 225701]; [Sperl, Kranz, Zippelius, EPL98 (2012) 28001] broken：ε=1.0 red： ε=0.5 dotted： ε=0.0 φc 0.999φc 0.99φc 0.9φc Inelastic collision： Gaussian noise： Memory kernel is common. Plateau is independent of inelasticity. Actual noise is not Gaussian. 2013/5/22 Complex Dynamics in Granular Systems 9　　　

Previous studies ２ : sheared granules MD simulation for sheared granular liquids [Otsuki, Chong & Hayakawa., JPS2010] φ=0.63 [M.P.Ciamarra, A.Coniglio, PRL103 (2009) 235701] e=0.88 ・ No plateau for ordinary inelasticity（e<0.9） ・ Plateau appears in an elastic limit 2013/5/22 Complex Dynamics in Granular Systems 10　　　 10

Previous Studies 3 : MCT under shear MCT for sheared liquids There was only one trial for sheared granular liquids: Hayakawa & Otsuki, PTP 119, 381 (2008), but this might be incomplete. There are some studies of the application of MCT on sheared isothermal liquids: Miyazaki&Reichman (2002), Miyazaki et al. (2004) Fuchs & Cates (2002), (2009) Chong & Kim (2009) Suzuki & Hayakawa (2013) 2013/5/22 Complex Dynamics in Granular Systems

Complex Dynamics in Granular Systems Absence of the cage effect Schematic picture *The red particle loses its kinetic energy due to inelastic collisions. *The cage is destructed by the shear. ⇒　Can we understand this picture by MCT? 2013/5/22 Complex Dynamics in Granular Systems 12　　　 12

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 13　　　

Complex Dynamics in Granular Systems MCT for sheared granular liquids Preceding work this might be incomplete consider correlation to pair-density modes Our work extension of MCT for sheared molecular liquids consider correlation to current-density modes absence of the cage effect is realized calculation of flow curve is possible. [Hayakawa & Otsuki, PTP119,381(2008)] [Suzuki & Hayakawa, arXiv:1301.0866(2013) Powders &Grains 2013] [Suzuki & Hayakawa, PRE87,012304(2013)] [Hayakawa, Chong, Otsuki, IUTAM-ISIMM proceedings (2010)] 2013/5/22 Complex Dynamics in Granular Systems 14　　　 14

Complex Dynamics in Granular Systems Microscopic dynamics for grains SLLOD equation (Newton equation for uniform shear) Assumption for noise statistics is unnecessary Contact dissipation of frictionless grains ζ : viscous coefficient (mass/time) Boundary speed vb t=0 Equil. Relaxation to a steady state steady shear & dissipation y vb x 2013/5/22 Complex Dynamics in Granular Systems 15　　　 15

Complex Dynamics in Granular Systems Liouville equation (1) Time evolution of physical quantities A(q(t),p(t))   Formal solution 2013/5/22 Complex Dynamics in Granular Systems 16　　　 16

Complex Dynamics in Granular Systems Liouville equation (2) Time evolution of distribution function : Phase volume contraction Phase volume contraction Formal solution 2013/5/22 Complex Dynamics in Granular Systems 17　　　 17

Complex Dynamics in Granular Systems Projection operators Basis Projection op. density fluctuation current density fluctuation : we include projection onto the current 2013/5/22 Complex Dynamics in Granular Systems 18　　　 18

Complex Dynamics in Granular Systems Granular temperature We do not include the granular temperature in the basis of the projected space. The time scale of the dynamics is set to be much larger than the relaxation time of the granular temperature. Hence, the granular temperature is treated as a constant in the MCT equations. This is a strong assumption. It is determined to be consistent with the steady-state condition. 2013/5/22 Complex Dynamics in Granular Systems 19　　　 19

Time correlation functions including currents [Hayakawa, Chong, Otsuki, IUTAM-ISIMM proceedings (2010)] 2013/5/22 Complex Dynamics in Granular Systems 20　　　 20

Complex Dynamics in Granular Systems Mori equation Equations for time correlation functions Continuity equation Momentum Continuity equation noise Memory kernels So far, the argument is exact. We need a closure for explicit calculation 　⇒　MCT The kernel L can be neglected in weak shear cases. 2013/5/22 Complex Dynamics in Granular Systems 21　　　 21

MCT for sheared granular liquids Memory kernel 　Terms that originate from density-current projection Vertex functions D, V(vis) originate from dissipation（contain ζ） 2013/5/22 Complex Dynamics in Granular Systems 22　　　 22

Complex Dynamics in Granular Systems Vertex functions Vertex functions & associated functions ζ ： viscous coefficient 2013/5/22 Complex Dynamics in Granular Systems 23　　　 23

Isotropic & weak shear approx. Isotropic approximation Dissipation is almost isotropic For reducing the load of calculation （3D⇒1D） e.g. Density correlation Φ & cross correlation Ψ The validity can be checked from the results. Weak shear approximation We can ignore　　　　　　　which includes 4-body correlation. 2013/5/22 Complex Dynamics in Granular Systems 24　　　 24

MCT equation (isotropic approx.) Effective friction ζ ：　viscous coefficient 2013/5/22 Complex Dynamics in Granular Systems 25　　　 25

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 26　　　

Results: time correlation functions The density correlation Φ The cross correlation Ψ 1.0 3.0×10-2 0.8 2.0×10-2 0.6 Φq / Sq φ = 0.52 qd = 7.0 * = 10-2 T* = 1.0 Ψq / (vbd) 0.4 1.0×10-2 0.2 10-2 1 10 102 103 10-4 10-2 1 102 t / (d / vb) t / (d / vb) ζ < 1.0 (e > 0.9999)：plateau exists ζ > 2.0 (e < 0.9998)：plateau disappears　 ⇒　consistent with MD The disappearance of the plateau is connected to the cross correlation. For ζ > 2.0, t* << *-1 correlation disappears. Isotropic approximation is valid. ε = (φ - φc)/φc = +10-3 φc = 0.516 2013/5/22 Complex Dynamics in Granular Systems 27　　　 27

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state constitutive equation (flow curve) comparison with the kinetic theory Discussion Summary & Perspective 2013/5/22 Complex Dynamics in Granular Systems 28　　　

Complex Dynamics in Granular Systems Constitutive equation (1) Balance between shear heating and dissipation Granular temperature is determined Stress tensor Dissipation rate 2013/5/22 Complex Dynamics in Granular Systems 29　　　 29

Complex Dynamics in Granular Systems Steady-state condition (1) Generalized Green-Kubo formula Mixing property Work function 2013/5/22 Complex Dynamics in Granular Systems 30　　　 30

Complex Dynamics in Granular Systems Steady-state condition (2) Identity (equilibrium contributions) Exact condition Evaluation of the nonequilibrium contributions is necessary. 2013/5/22 Complex Dynamics in Granular Systems 31　　　 31

Complex Dynamics in Granular Systems Steady-state shear stress Stress formula under MCT approximation The stress formula in MCT is expected to be valid for an arbitrary sheared system. 2013/5/22 Complex Dynamics in Granular Systems 32　　　 32

Complex Dynamics in Granular Systems Steady-state dissipation rate Rayleigh’s dissipation function In contrast to the stress formula, the validity of the dissipation function in MCT is rather non-trivial. As a first trial, we adopt this expression and examine the results. 2013/5/22 Complex Dynamics in Granular Systems 33　　　 33

Complex Dynamics in Granular Systems Steady-state condition in MCT Steady-state condition in the Mode-coupling Approx. 2013/5/22 Complex Dynamics in Granular Systems 34　　　 34

Complex Dynamics in Granular Systems Problems If we formally use Green-Kubo formula, the resulting granular temperature is apparently unphysical, and the Bagnold law is not attained in unjammed conditions. If we regard the dissipation parameter ζ as an independent parameter, we still do not get Bagnold’s scaling. This problem might be related to our treatment in which temperature is not regarded as one of slow variables. To avoid this problem, we introduce a scaling property of the dissipation parameter ζ (of mass/time) with respect to the granular temperature. 2013/5/22 Complex Dynamics in Granular Systems 35　　　 35

Complex Dynamics in Granular Systems A possible choice of ζ Relation in the linear spring model Virial theorem (e : restitution coefficient) 2013/5/22 Complex Dynamics in Granular Systems 36　　　

Complex Dynamics in Granular Systems Scaling property of ζ Coefficients 2013/5/22 Complex Dynamics in Granular Systems 37　　　

Results: flow curve for the shear stress volume fraction 0.62 Qualitatively the same result as volume fraction 0.52. Shear stress Granular temperature Deviation from the Bagnold scaling is not obtained within this framework. 2013/5/22 Complex Dynamics in Granular Systems 38　　　

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 39　　　

Comparison with the kinetic theory Comparison of the kinetic theory with simulation has been previously performed. It has been shown that the kinetic theory (dashed lines) is invalid at volume fraction ～ 0.6. [Mitarai and Nakanishi, PRE75, 031305 (2007)] Granular temperature T* Dissipation rate Γ* Shear stress σ* 2013/5/22 Complex Dynamics in Granular Systems 40　　　

Results of ＭＣＴ(no fitting parameter) The results of MCT qualitatively agree with the simulation, but poorly agree in quantitative sense. The monotonicity of T is not obtained. Granular temperature T* Shear stress σ* The expression of the dissipation rate is presumably invalid, and a revision is necessary. 2013/5/22 Complex Dynamics in Granular Systems 41　　　

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 42　　　

Complex Dynamics in Granular Systems Discussion (1) So far, MCT does not give sufficiently useful results for sheared granular systems, though it gives some interesting and nontrivial results. The key point is how to determine the temperature, which is very different from the standard MCT under a constant temperature. See, K. Suzuki &HH, PRE87,012304 (2013). 2013/5/22 Complex Dynamics in Granular Systems

Complex Dynamics in Granular Systems Discussion (2) In other words, if we choose the correct relation for the temperature, we may discuss even a jammed state. How can we include the effect of the stress network or the rigidity of granular solids? Nevertheless, to extend the projection operator formalism for temperature is hopeless. So we need a physical input for this part. This is an apparent weak point of our MCT. 2013/5/22 Complex Dynamics in Granular Systems

Complex Dynamics in Granular Systems Contents Introduction MCT for sheared granular liquids basic formulation disappearance of the cage (plateau) steady-state properties (flow curve) comparison with the kinetic theory Discussion Summary 2013/5/22 Complex Dynamics in Granular Systems 45　　　

Complex Dynamics in Granular Systems Summary We have formulated MCT for sheared dense granular liquids. This MCT can capture the behavior of density correlation function: the disappearance of plateau because of current-density correlation. MCT can produce qualitative behavior of the flow curve below jamming transition, which gives a new theoretical tool. Nevertheless, agreement is not satisfactory. 2013/5/22 Complex Dynamics in Granular Systems