Granular rheology

1
Granular rheology cooperative motion of grains

• Takahiro Hatano (Univ. Tokyo)

1. Granular rheology near jamming Bagnolds
scaling ? yield stress a critical scaling for
rheology
2. Velocity correlation growing time scales ??
rheology
3. Finite-size effect length matters! emergent
length scale near jamming point
2
granular rheology and a kinetic theory
angle?
(Silbert et al. PRE 2001)
depth
a kinetic theory (such as Garzo and Dufty, PRE
1999) cannot reproduce dense flow properties.
angle?
? due to velocity correlation ?
1. Jenkins on Tuesday (also Phys. Fluids 2006)
2. Kumaran on Wednesday
(Mitarai Nakanishi , PRE 2007)
3
granular rheology and jamming
If one deals with hard spheres, various
quantities diverge at a certain density. (jamming)
pressure
(Lois et al. 2005)
shear stress
(kinetic) temeprature
friction coefficient
? what if soft grains ?
4
Jamming acquisition of rigidity
No rigidity
Rigidity
Density (f)
solidlike
fluidlike
jamming density
Jamming density is unique for frictionless
particles
64 in 3D 84 in 2D
(OHern et al. PRE 2003)
NB. not unique for frictional particles.
60 64 in 3D 80 84 in 2D
i.e., depends on the history
(Aharonov Sparks, PRE 1999) (Zhang Makse, PRE
2003)
5
computational model
an assembly of soft frictionless inelastic spheres
(without attractive force)
diameter
i
Force only normal direction
j
elastic damping
mass
relative normal velocity
linear
normal vec.
overlap length
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computational system
Lees-Edwards Bound. Cond.
dynamics SLLOD eq.
diameter
i
j
mass
ensuring uniform shear flow
units
polydispersity
m1 (mass) d 1 (length)
shear rate is scaled with
for 3D
(time)
shear stress is scaled with
for 2D
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contents
1. granular rheology near jamming Bagnolds
scaling ? yield stress a critical scaling for
rheology
2. velocity correlation growing time scales ?
rheology
3. finite-size effect emergent length scale
8
Granular rheology near jamming
shear stress
(a 3D case)
1. At lower densities Bagnolds scaling
Yield stress
2. At higher densities Yield stress
Higher density
(Herschel-Bulkley)
3. At the jamming density
shear rate
(power-law fluid)
(TH, Otsuki, Sasa, J.Phys.Soc.Jpn. 2007)
is conjectured
9
jamming a critical phenomenon
(spontaneous) magnetization
yield stress
F
low temp.
high temp.
low density
high density
jamming
ferromagnetism
density ? temperature (T) shear rate
? magnetic field (H) shear stress ?
magnetization (M)
NB. indeed not correct
at a critical point,
??
(TH, Otsuki, Sasa, J.Phys.Soc.Jpn. 2007)
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a scaling law for (suspension) rheology
Olsson Teitel PRL 2007
just like conventional critical phenomena
order parameter (unjammed nonzero)
F
relevant variable S, F
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remarks on the model of Olsson and Teitel

• a 2D system
• Overdamp (massless) dynamics
• No inelastic collision

not a granular material
2, 3
(e.g. foams or suspensions at T0)
Newtonian viscosity instead of Bagnolds
scaling Different exponent for critical rheology
cf. granular matter
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a critical scaling of granular rheology
(TH, J.Phys.Soc.Jpn. 2008)
point J is a critical point
Technical remark We redo simulation at lower
shear rates. (as low as Otsuki Hayakawa, a
talk in yesterday)
3D system with N4000
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scaling ansatz in granular rheology
the identical expressions
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three regimes of granular rheology
critical rheology
yield stress
if
if
Bagnold Yielding
crossover to
Bagnold
no crossover
in arbitrary range of shear rate
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power-law fluid at the critical density
(TH, arXiv0804.0477)
shear stress
N80,000
monodisperse
NB Otsuki Hayakawa PRE 2009 (on Thursday )
shear rate
the origin of critical behavior?
? velocity correlation
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contents
1. granular rheology near jamming Bagnolds
scaling ? yield stress a critical scaling for
rheology
2. velocity correlation growing time scales ?
rheology
3. finite-size effect emergent length scale
17
velocity autocorrelation function
for tracer particle i
3D system with N4000
velocity relaxation time
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scaling of velocity relaxation time (1)
unjammed
scaling functions for x ltlt1
(unjammed)
jammed
(jammed)
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scaling of velocity relaxation time (2)
unjammed
At the jamming density,
jammed
N4000 is not enough (at all)
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relaxation time at fJ
N80000
NB.
velocity-field dynamics without structural change
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shear stress and relaxation time
From dimensional analysis, (assuming no
anomalous dimension mean-field-like)
can reproduce the rheological properties
22
velocity relaxation time rules diffusion
N80000
shear rate
V
1. superdiffusion for
(correlated motion)
2. diffusion coefficient
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partial summary (1)
1. Critical scaling for dense granular
rheology (jam unjam)
2. The rheological properties can be explained
from the vel. corr. time and the dimensional
analysis
3. Diffusion profiles are collapsed using vel.
corr. time.
24
contents
1. granular rheology near jamming Bagnolds
scaling ? yield stress a critical scaling for
rheology
2. velocity correlation growing time scales ?
rheology
3. finite-size effect emergent length scale
25
but, the length scale?
In analogy to (conventional) critical phenomena,
there must be the growing correlation length,
which is responsible for the growing correlation
time.
Ising model at Tc
Lechenault at al. EPL 2008
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spatial heterogeneity in velocity fluctuation
yellow grains large velocity fluctuations
monodisperse system at
gradient
flow
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Spatiotemporal dynamics of velocity field
(strain 1 )
(strain 50 during this movie)
large velocity
small velocity
intermediate
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collective motion
velocity vector field for a 2D system
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The growing correlation length
a correlation function for the spatial
heterogeneity
1 for slow particles 0 for fast particles
at
3D
2D
the exponent is dimensionality-independent!
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the finite-size effect in rheology
(TH, arXiv0804.0477)
smaller system higher stress
we have to recall shear rate ?
correlation length ?
If (correlation length) gt (system size), the
cooperative motion is suppressed.
? hardening
the length scale affects rheology!!
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finite-size scaling
data collapse
the correlation length
consistent with the velocity correlation!
(TH, arXiv0804.0477)
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stress fluctuation due to finite-size effect
0.01
(a 2D system)
Gaussian fluctuation
0
20
40
20
40
strain
strain
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distribution of instantaneous shear stress
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cage due to finite-size effect
diffusion profile
ltP(0)P(t)gt
3D
if (correlation length) gt (system
size) subdiffusive motion is observed. ?
force-chain avalanche?
35
Gutenberg-Richter law for force-chain avalanche
events
E
?E
energy drop distribution
t
elastic energy
sudden energy drop
b1.4 for 2D b0.7 for 3D (cf. b0.5-0.8 for
earthquakes)
(microfracture, acoustic emission)
distribution of
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power-law develops for larger N
A
B
A
N50
N100
B
C
C
N300
7 orders of magnitude!
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Partial summary (2)
1. smaller systems ? cooperative motion
impossible so that a granular system becomes
brittle.
2. Gutenberg-Richter law (and the likes) in
granular matter Hirata (JPSJ 1999), Bretz et al.
(EPL 2006), Behringer et al.(2009), etc
was indeed the finite-size effect.
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summary
1. Critical scaling for dense granular
rheology (jam unjam)
2. vel. corr. time dominates rheological
properties
3. correlation length matters. finite size effect
a. rheology changes. b. force chain avalanche ?
GR law
39
future works
derivation of
mean-field nature true?
relation between the dynamical heterogeneity in
thermal systems (glass formers)
and many more.
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(No Transcript)
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Back to J. Jenkins question
dissipation rate per unit volume
for unjammed hard spheres
Instead we obtain
For these expressions to be identical,
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The dynamic critical exponent
comparable with Abate et al. 2007 (2D
air-fluidized beads)
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remarks on the critical scaling
1. The exponent xF depends on the force
model. (Hookean / Hertzian)
2. The exponent ? depends on the dynamics
overdamp
(e.g. Olsson Teitel)
inertial
3. The exponents do NOT depend on the
dimensionality. (OHern et al. 2003 Wyart et
al. 2005 Otsuki et al. 2009)