Confidential 1 2013/5/22 Complex Dynamics in Granular Systems at Bejing (May 18-June 21, 2013) Hisao...

ConfidentialConfidential

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2013/5/222013/5/22

Complex Dynamics in Granular Systems at Bejing (May 18-June 21, Complex Dynamics in Granular Systems at Bejing (May 18-June 21, 2013)2013)

Hisao HayakawaHisao Hayakawa

(YITP, Kyoto Univ. , Japan),(YITP, Kyoto Univ. , Japan),in collaboration with Koshiro Suzuki in collaboration with Koshiro Suzuki (Canon Inc.)(Canon Inc.)

Mode-coupling theory Mode-coupling theory for sheared granular for sheared granular

liquidsliquids


2013/5/222013/5/22Complex Dynamics in Granular Complex Dynamics in Granular

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Contents IntroductionIntroduction MCT for sheared granular liquidsMCT for sheared granular liquids

basic formulationbasic formulation disappearance of the cage (plateau)disappearance of the cage (plateau) steady-state properties (flow curve)steady-state properties (flow curve) comparison with the kinetic theorycomparison with the kinetic theory

DiscussionDiscussion SummarySummary

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DiscussionDiscussion Summary Summary

33　　　　　　


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?


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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).

Complex Dynamics in Granular Systems Complex Dynamics in Granular Systems 55

ConfidentialConfidentialThe limitation of kinetic theory

Complex Dynamics in Granular Systems Complex Dynamics in Granular Systems 66


objectivebackground

Kinetic theory(Enskog) ： φ<0.5

Constitutive equation = Bagnold law

Mode-coupling(MCT)　： noise excitation

Liquid theory　： 0.5<φ<0.64

Const. Eq. = power law Bagnold ⇒ power law?

Application of MCT to sheared system　： projection to momentum

77　　　　　　

Background and objectiveBackground and objective


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88　　　　　　2013/5/222013/5/22Complex Dynamics in Granular Complex Dynamics in Granular

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Background and objectiveBackground and objective

Jamming phase diagram

Ikeda-Berthier-Sollich (2012)



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Previous studies Previous studies １１ : vibrating beds: 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]

Gaussian noise ：

broken ： ε=1.0red ： ε=0.5dotted ： ε=0.0

φc

0.999φc0.99φc0.9φc

Plateau is independent of inelasticity.

Inelastic collision ：

Memory kernel is common.

Actual noise is not Gaussian.



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Previous studies Previous studies ２２ : : sheared granulessheared granules MD simulation for sheared granular liquids

・ No plateau for ordinary inelasticity （ e<0.9 ）

[M.P.Ciamarra, A.Coniglio, PRL103 (2009) 235701]

e=0.88

[Otsuki, Chong & Hayakawa., JPS2010]

φ=0.63

・ Plateau appears in an elastic limit


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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Schematic picture

*The red particle loses its kinetic energy due to inelastic collisions.*The cage is destructed by the

shear.

1212　　　　　　

⇒ Can we understand this picture by MCT?

Absence of the cage effectAbsence of the cage effect


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1313　　　　　　



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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.

1414　　　　　　

MCT for sheared granular liquidsMCT for sheared granular liquids

[Hayakawa & Otsuki, PTP119,381(2008)]

[Suzuki & Hayakawa, arXiv:1301.0866(2013) Powders &Grains 2013]

[Hayakawa, Chong, Otsuki, IUTAM-ISIMM proceedings (2010)]

[Suzuki & Hayakawa, PRE87,012304(2013)]



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SLLOD equation (Newton equation for uniform shear)

1515　　　　　　

x

y

Microscopic dynamics for grainsMicroscopic dynamics for grains

ζ : viscous coefficient (mass/time)

t=0

Equil. Relaxationto a steady state

steady shear &dissipation

Assumption for noise statistics is unnecessary

Boundary speed vb

vb

Contact dissipationof frictionless grains



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Time evolution of physical quantities A(q(t),p(t))

1616　　　　　　

LiouvilleLiouville equationequation (1)(1)

Formal solution



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Time evolution of distribution function

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LiouvilleLiouville equationequation (2)(2)

Formal solution

: Phase volume contraction



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Basis

Projection op.

Projection operatorsProjection operators

current density fluctuation

1818　　　　　　

density fluctuation

: we include projection onto the current



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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.

Granular temperatureGranular temperature

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Time correlation functions

Time correlation functionsTime correlation functions

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Correlation functionsincluding currents[Hayakawa, Chong, Otsuki, IUTAM-ISIMM proceedings (2010)]



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Equations for time correlation functions

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Mori equationMori equation

So far, the argument is exact. We need a closure for explicit calculation 　⇒　 MCT The kernel L can be neglected in weak shear cases.

Momentum Continuity equation

Continuity equation

noiseMemory kernels



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Memory kernel

MCT for sheared granular liquidsMCT for sheared granular liquids

Terms that originate from density-current projection

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Vertex functions D, V(vis) originate from dissipation （ contain ζ ）


2323　　　　　　

Vertex functionsVertex functions

ζ ： viscous coefficient


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Isotropic approximation Dissipation is almost isotropic For reducing the load of calculation （ 3D⇒1D ）

e.g.


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Isotropic & weak shear approx.Isotropic & weak shear approx.

The validity can be checked from the results.

Density correlation Φ & cross correlation Ψ

Weak shear approximation We can ignore 　　　　　　　 which includes 4-

body correlation.



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MCTMCT equation (isotropic approx.)equation (isotropic approx.)

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Effective friction

ζ ：　 viscous coefficient



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Results: time correlation functionsResults: time correlation functions

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The density correlation Φ The cross correlation Ψ

ζ < 1.0 (e > 0.9999) ： plateau existsζ > 2.0 (e < 0.9998) ： plateau disappears　⇒ consistent with MD

For ζ > 2.0, t* << *-1 correlation disappears. Isotropic approximation is valid.

The disappearance of theplateau is connected to thecross correlation.

ε = (φ - φc)/φc = +10-3

φc = 0.516

φ = 0.52qd = 7.0 * = 10-2

T* = 1.0

t / (d / vb)10-2 1 10210 103 10-4

t / (d / vb)10-2 1 102

Ψq /

(v bd

)

3.0×10-2

2.0×10-2

1.0×10-2

0

1.0

0.8

0.6

0.4

0.2

0

Φq /

Sq



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basic formulationbasic formulation disappearance of the cage (plateau)disappearance of the cage (plateau) steady-state constitutive equation (flow steady-state constitutive equation (flow

curve)curve) comparison with the kinetic theorycomparison with the kinetic theory

DiscussionDiscussion Summary & PerspectiveSummary & Perspective

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Balance between shear heating and dissipation

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Constitutive equation (1)Constitutive equation (1)

Granular temperature is determined

Stress tensor

Dissipation rate



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Generalized Green-Kubo formula

Mixing property

Work function

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Steady-state condition (1)Steady-state condition (1)



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Identity (equilibrium contributions)

Exact condition

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Steady-state condition (2)Steady-state condition (2)

Evaluation of the nonequilibrium contributions is necessary.



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Stress formula under MCT approximation

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Steady-state shear stressSteady-state shear stress

The stress formula in MCT is expected to be valid for an arbitrary sheared system.



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Rayleigh’s dissipation function

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Steady-state dissipation rateSteady-state dissipation rate

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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Steady-state condition in the Mode-coupling Approx.

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Steady-state condition in MCTSteady-state condition in MCT



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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.

ProblemsProblems

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A possible choice of ζ Relation in the linear spring modelRelation in the linear spring model

Virial theoremVirial theorem

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(e : restitution coefficient)



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Scaling property of ζ CoefficientsCoefficients

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Results: flow curve for the shear stress volume fractionvolume fraction 0.620.62

Qualitatively the same result as volume fraction 0.52.Qualitatively the same result as volume fraction 0.52.

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Deviation from the Bagnold scaling is not obtained within this framework.

Shear stress Granular temperature



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Comparison with the kinetic theoryComparison with the kinetic theory Comparison of the kinetic theory with simulation Comparison of the kinetic theory with simulation

has been previously performed.has been previously performed. It has been shown that the kinetic theory (dashed It has been shown that the kinetic theory (dashed

lines) is invalid at volume fraction lines) is invalid at volume fraction ～～ 0.6.0.6.

4040　　　　　　

Granular temperature T* Dissipation rate Γ* Shear stress σ*

[Mitarai and Nakanishi, PRE75, 031305 (2007)]



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Results of ＭＣＴ (no fitting parameter)

The results of MCT qualitatively agree with the The results of MCT qualitatively agree with the simulation, but poorly agree in quantitative sense.simulation, but poorly agree in quantitative sense.

The monotonicity of T is not obtained.The monotonicity of T is not obtained.

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Granular temperature T* Shear stress σ*

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

0.5 0.52 0.54 0.56 0.58 0.6

gran

ular

tem

pera

ture

volume fraction

e=0.98

e=0.92

e=0.80

e=0.98 (MN)

e=0.92 (MN)

e=0.70 (MN)0

40

80

120

160

200

0.5 0.55 0.6

shea

r st

ress

volume fraction

e=0.98

e=0.92

e=0.80

e=0.98 (MN)

e=0.92 (MN)

e=0.70 (MN)

The expression of the dissipation rate is presumably invalid, and a revision is necessary.



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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).


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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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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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Confidential 1 2013/5/22 Complex Dynamics in Granular Systems at Bejing (May 18-June 21, 2013) Hisao...

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