Soft motions of amorphous solids

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Soft motions of amorphous solids

Matthieu Wyart
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Amorphous solids

• structural glasses, granular matter, colloids,
  dense emulsions
• TRANSPORT
• thermal conductivity
• ?????????few ?? ??molecular sizes
• ? phonons strongly scattered
• FORCE PROPAGATION
• L?

ln (T)
L?
Behringer group
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Glass Transition
Heuer et. al. 2001

• ????????e????????????????????
• ?????????

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Angle of Repose
h
???Rearrangements Non-local
Pouliquen, Forterre
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Rigidity
cage  effect
Rigidity toward collective motions more
demanding
Maxwell not rigid

Zd1 local
characteristic length ?
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Vibrational modes in amorphous solids?

• Continuous medium phonon plane wave
• ? Density of states D(?)? N(?) V-1 d?-1

D(?) ?2 Debye

• Amorphous solids
• - Glass excess of low-frequency modes.
• Neutron scattering ? Boson Peak (1 THz10
  K0)
• 
  Transport,
• Disorder cannot be a generic explanation
• Nature of these modes?

D(?)/?2
?
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Amorphous solid different from a continuous
bodyeven at L ??
Ohern, Silbert, Liu, Nagel
D(?) ?0

• Particles with repulsive, finite range
  interactions at T0
• Jamming transition at packing
• fraction ?c 0.63

Jammed, ? ? ?c Pgt0
Unjammed, ? ? ?c P0

Crystalplane waves Jamming??
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Jamming critical point scaling properties
Geometry coordination
zc2d
z-zc?z (???c)1/2

• Excess of Modes
• same plateau is reached for
• different ?
• Define D(?)1/2 plateau
• ? ?z B1/2

Relation between geometry and excess of modes ??
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Rigidity and soft modes
???????, Thorpe, Alexander
Not rigid ? soft mode
Rigid
Soft modes
for all contacts ltijgt
??Ri??Rj??nij0
Maxwell z rigid? constraints Nc
degrees of freedom Nd
?
z2Nc/N ? 2d gtd1 global
jamming marginally connected zc2d
isostatic
(Moukarzel,
Roux, Witten, Tkachenko,...)

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Isostatic D(?) ? 0
?????? lattice independent lines ? D(?) ? 0
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zgtzc
?
? 1/ ?z ? ? B1/2/L ?z B1/2

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Random Packing
Wyart, Nagel and Witten, EPL 2005

• main difference modes are not one dimensional
• ? 1/ ?z
• L lt L continuous elastic description bad
  approximation

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Ellenbroeck et.al 2006
Consistent with L ?z-1
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Extended Maxwell criterion
S. Alexander
f
X
?
dE k/L2 X2 - f X2 stability ? k/L2 gt
f
? ?z gt (f/k)1/2 e1/2 (???c)1/2
Wyart, Silbert, Nagel and Witten, PRE 2005
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Hard Spheres
V(r)
?
?cri?0.5
?c?0.64
???0.58
1

• contacts, contact forces fij

Ferguson et al. 2004, Donev et al. 2004
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Effective Potential
Brito and Wyart, EPL 2006

• discontinuous potential ? expand E?
• ? coarse-graining in time lt Rigt

fij(ltrijgt)?
h
1 d
Z?pi dhij e- phij/kT
pkT/lthgt
hijrij-1
Z?pi dhij e- fijhij/kT
Isostatic

• fijkT/lthijgt

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• fijkT/lthijgt

?
V( r) - ?kT ln(r-1) if contact V( r)0
else
G ??ij V( rij)
rijltRigt-ltRjgt

• weak ( ?z) relative correction throughout the
  glass phase

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Linear Response and Stability

• dynamical matrix dF M dltRgt
• ? Vibrational modes

?zgt C(p/B)1/2p-1/2

• Near ???and after a rapid
• quench just enough contacts
• to be rigid ? system stuck in
• the marginally stable region

???????
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vitrification
vitrification
Ln(?z)
Rigid
Equilibrium configuration
Unstable
Ln(p)
??????????????????
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Activation
?c
??

Point defects? Collective mode?
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Activation
?c
??
Brito and Wyart, J. phys stat, 2007
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Granular matter

• ????????
• Counting changes zc d1
• not critical z(p?0)? zc
  d1lt z lt2d
• z depends on ????and preparation
• 
  Somfai et al., PRE 2007
• 
  Agnolin et Roux, PRE 2008

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?start?h)
Hypothesis (i) z gt z_c (ii) Saturated
contacts ?zc.c. f(?/p) f(tan
(????(staron) (iii) Avalanche starts as ?z
?zc.c(?start) Consistent with numerics
(2d,?????? (somfai, staron) ?z0.2
?zc.c(?start) 0.16

?
h
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Rigidity criterion with a fixed and free boundary
wyart, arXiv 0807.5109
Fixed boundary ?z -gt ?z a/h
Free boundary ?z -gt ?z a'/h
a'lta
Finite h ?z -gt ?z (a-a')/h ???z (a-a')/h
f(tan ?? h? c0/ c1 tan ?????z

?????????? effect gt 2
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Acknowledgement
Tom Witten Sid Nagel Leo Silbert Carolina Brito
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Isostatic D(?) ? 0
Wyart, Nagel and Witten, EPL 2005

• just rigid remove m contactsgenerate m
• SOFT MODES
• ? High sensitivity to boundary conditions

L
Xi
L

• generate pLd-1 soft modes independent (instead
  of 1 for a normal solid)
• argument show that these modes gain a frequency
  ?L-1
• when boundary conditions are restored. Then

D(?) Ld-1/(LdL-1) L0
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• Soft modes extended,
• heterogeneous
• Not soft in the original system, cf
• stretch or compress contacts cut to
• create them
• Introduce Trial modes
• Frequency ? harmonic modulation of a
  translation,
• i.e plane waves?
  ? ?? ??????L-1
• ? D(?) ?0 (variational) ? Anomalous
  Modes

?????????????Ri?? sin(xi p/L) ??Ri?
x

L
??????????


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A geometrical property of random close packing
?z gt (???c)1/2
maximum density ? stable to the compression ?c ?
? relation density landscape // pair
distribution function g(r)
1(???c)/d
?z ? g(r) dr stable ? g(r) (r-1)-1/2
1
Silbert et al., 2005
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Glass Transition

• ??G ??????????????relaxation time

??????????????????????
Heuer et. al. 2001

• ????????e?????????????????????????????

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Vitrification as a buckling" phenomenum
? increases P increases
L