Soft motions of amorphous solids
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Transcript of Soft motions of amorphous solids
Soft motions of amorphous solids
Matthieu Wyart
Amorphous solids• structural glasses, granular matter, colloids, dense emulsions
TRANSPORT: thermal conductivity few molecular sizes phonons strongly scattered FORCE PROPAGATION:
L?
ln (T)
Behringer group
L?
Glass Transition
Heuer et. al. 2001
•e
Angle of Repose
h
RearrangementsNon-local
Pouliquen, Forterre
Rigidity``cage ’’ effect:
Rigidity toward collective motions more demanding
Z=d+1: local
characteristic length ?
Maxwell:not rigid
Vibrational modes in amorphous solids?
• Continuous medium: phonon = plane wave Density of states D(ω) N(ω) V-1 dω-1
• Amorphous solids: - Glass: excess of low-frequency modes. Neutron scattering ``Boson Peak” (1 THz~10 K0)
Transport, …
Disorder cannot be a generic explanationNature of these modes?
D(ω) ∼ ω2 Debye
D(ω)/ω2
ω
Amorphous solid different from a continuous bodyeven at L
Unjammed, c
P=0
Jammed, c
P>0
• Particles with repulsive, finite range interactions at T=0• Jamming transition at packing fraction c≈ 0.63 :
O’hern, Silbert, Liu, Nagel
D(ω) ∼ ω0
Crystal:plane waves :: Jamming:??
Jamming ∼ critical point: scaling properties
z-zc=z~ (c)1/2 Geometry: coordination
Excess of Modes:• same plateau is reached for different • Define D(ω*)=1/2 plateau
ω*~ z B1/2
Relation between geometry and excess of modes ??
zc=2d
Rigidity and soft modes
RigidNot rigid soft mode
Soft modes:
RiRjnij=0 for all contacts <ij>
Maxwell: z rigid? # constraints: Nc
# degrees of freedom: Nd
z=2Nc/N 2d >d+1 global
(Moukarzel, Roux, Witten, Tkachenko,...) jamming: marginally connected zc=2d “isostatic”
, Thorpe, Alexander
Isostatic: D(ω)~ ω 0
lattice: independent lines D(ω)~ ω 0
z>zc
*
* = 1/ z ω*~ B1/2/L*~ z B1/2
• main difference: modes are not one dimensional
* ~ 1/ z
L < L*: continuous elastic description bad approximation
Wyart, Nagel and Witten, EPL 2005Random Packing
Ellenbroeck et.al 2006
Consistent with L* ~ z-1
*
Extended Maxwell criterion
f
dE ~ k/L*2 X2 - f X2 stability k/L*2 > f z > (f/k)1/2~ e1/2 ~ (c)1/2
X
Wyart, Silbert, Nagel and Witten, PRE 2005
S. Alexander
Hard Spheres
c0.640.58 cri0.5
1
V(r)
• contacts, contact forces fij
Ferguson et al. 2004, Donev et al. 2004
• discontinuous potential expand E?• coarse-graining in time: < Ri>
Effective Potential
fij(<rij>)?
hij=rij-1
1 d:
Z=∫πi dhij e- fijhij/kT
fij=kT/<hij>
h
Isostatic:
Z=∫πi dhij e- phij/kT p=kT/<h>
Brito and Wyart, EPL 2006
V( r)= - kT ln(r-1) if contactV( r)=0 else
rij=||<Ri>-<Rj>||
G = ij V( rij)
fij=kT/<hij>
• weak (~ z) relative correction throughout the glass phase
•dynamical matrix dF= M d<R> Vibrational modes
z> C(p/B)1/2~p-1/2
Linear Response and Stability
•Near and after a rapid quench: just enough contactsto be rigid system stuck inthe marginally stable region
vitrification
Ln(z)
Ln(p)
Rigid
UnstableEquilibriumconfiguration
vitrification
Activationc
Point defects?Collective mode?
Activationc
Brito and Wyart, J. phys stat, 2007
Granular matter
:
- Counting changes zc = d+1
-not critical z(p0)≠ zc d+1< z <2d
- z depends on and preparation Somfai et al., PRE 2007 Agnolin et Roux, PRE 2008
starth)
h
Hypothesis:
(i) z > 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)
z≈0.2 zc.c(start) ≈ 0.16
Finite h: z -> z +(a-a')/hz +(a-a')/h = f(tan
h c0/ [ c1 tan z]
wyart, arXiv 0807.5109 Rigidity criterion with a fixed and free boundary
Free boundary : z -> z +a'/h
Fixed boundary : z -> z +a/h
a'<a
: effect > *2
Acknowledgement
Tom WittenSid NagelLeo SilbertCarolina Brito
XiL
L
• generate p~Ld-1 soft modes independent (instead of 1 for a normal solid)•argument: show that these modes gain a frequency ω~L-1
when boundary conditions are restored. Then:
D(ω) ~Ld-1/(LdL-1) ~L0
•``just” rigid: remove m contacts…generate m SOFT MODES: High sensitivity to boundary conditions
Isostatic: D(ω)~ ω 0
Wyart, Nagel and Witten, EPL 2005
• 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 ω L-1
D(ω)~ ω0 (variational) Anomalous Modes
R*isin(xi π/L)Ri
xL
z > (c)1/2
A geometrical property of random close packing
maximum density stable to the compression c
relation density landscape // pair distribution function g(r)
1
1+(c)/d
z ~ g(r) dr stable g(r) ~(r-1)-1/2
Silbert et al., 2005
Glass Transition=G relaxation time
Heuer et. al. 2001
•e
Vitrification as a ``buckling" phenomenum
increases
P increases
L