1. 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)
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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
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3. 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
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4. 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
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5. 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, (2005). 5 5
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6. The limitation of kinetic theory
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7. 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
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8. Background and objective
Jamming phase diagram
Ikeda-Berthier-Sollich (2012) 1/
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9. 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) ] : Air-fluidized bed
[Reis et al., PRL98 (2007) ] : Vibrating bed
[Kranz, Sperl, Zippelius, PRL104 (2010) ]; [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.
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10. 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) ]
e=0.88
・ No plateau for ordinary inelasticity（e<0.9）
・ Plateau appears in an elastic limit
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11. 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)
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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?
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13. 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
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14. 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: (2013)
Powders &Grains 2013]
[Suzuki & Hayakawa, PRE87,012304(2013)]
[Hayakawa, Chong, Otsuki, IUTAM-ISIMM proceedings (2010)]
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15. 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
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16. Complex Dynamics in Granular Systems
Liouville equation (1)
Time evolution of physical quantities A(q(t),p(t))
Formal solution
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17. Complex Dynamics in Granular Systems
Liouville equation (2)
Time evolution of distribution function
: Phase volume
contraction
Phase volume contraction
Formal solution
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18. Complex Dynamics in Granular Systems
Projection operators
Basis
Projection op.
density fluctuation
current density fluctuation
: we include projection onto the current
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19. 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.
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20. Time correlation functions
including currents
[Hayakawa, Chong, Otsuki,
IUTAM-ISIMM proceedings (2010)]
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21. 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.
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22. MCT for sheared granular liquids
Memory kernel
　Terms that originate from density-current projection
Vertex functions D, V(vis) originate from dissipation（contain ζ）
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23. Complex Dynamics in Granular Systems
Vertex functions
Vertex functions & associated functions
ζ ： viscous coefficient
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24. 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.
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25. MCT equation (isotropic approx.)
Effective friction
ζ ：　viscous coefficient
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26. 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
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27. 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 > )：plateau exists
ζ > 2.0 (e < )：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
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28. 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
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29. Complex Dynamics in Granular Systems
Constitutive equation (1)
Balance between shear heating and dissipation
Granular temperature is determined
Stress tensor
Dissipation rate
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30. Complex Dynamics in Granular Systems
Steady-state condition (1)
Generalized Green-Kubo formula
Mixing property
Work function
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31. Complex Dynamics in Granular Systems
Steady-state condition (2)
Identity (equilibrium contributions)
Exact condition
Evaluation of the nonequilibrium contributions is necessary.
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32. 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.
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33. 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.
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34. Complex Dynamics in Granular Systems
Steady-state condition in MCT
Steady-state condition in the Mode-coupling Approx.
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35. 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.
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36. Complex Dynamics in Granular Systems
A possible choice of ζ
Relation in the linear spring model
Virial theorem
(e : restitution coefficient)
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37. Complex Dynamics in Granular Systems
Scaling property of ζ
Coefficients
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38. 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.
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39. 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
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40. 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, (2007)]
Granular temperature T*
Dissipation rate Γ*
Shear stress σ*
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41. 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.
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42. 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
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43. 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, (2013).
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44. 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.
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45. 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
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46. 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.
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