soft motions of amorphous solids
DESCRIPTION
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). L?. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/1.jpg)
Soft motions of amorphous solids
Matthieu Wyart
![Page 2: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/2.jpg)
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?
![Page 3: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/3.jpg)
Glass Transition
Heuer et. al. 2001
•e
![Page 4: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/4.jpg)
Angle of Repose
h
RearrangementsNon-local
Pouliquen, Forterre
![Page 5: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/5.jpg)
Rigidity``cage ’’ effect:
Rigidity toward collective motions more demanding
Z=d+1: local
characteristic length ?
Maxwell:not rigid
![Page 6: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/6.jpg)
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
![Page 7: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/7.jpg)
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:??
![Page 8: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/8.jpg)
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
![Page 9: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/9.jpg)
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
![Page 10: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/10.jpg)
Isostatic: D(ω)~ ω 0
lattice: independent lines D(ω)~ ω 0
![Page 11: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/11.jpg)
z>zc
*
* = 1/ z ω*~ B1/2/L*~ z B1/2
![Page 12: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/12.jpg)
• main difference: modes are not one dimensional
* ~ 1/ z
L < L*: continuous elastic description bad approximation
Wyart, Nagel and Witten, EPL 2005Random Packing
![Page 13: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/13.jpg)
Ellenbroeck et.al 2006
Consistent with L* ~ z-1
![Page 14: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/14.jpg)
*
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
![Page 15: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/15.jpg)
Hard Spheres
c0.640.58 cri0.5
1
V(r)
• contacts, contact forces fij
Ferguson et al. 2004, Donev et al. 2004
![Page 16: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/16.jpg)
• 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
![Page 17: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/17.jpg)
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
![Page 18: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/18.jpg)
•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
![Page 19: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/19.jpg)
vitrification
Ln(z)
Ln(p)
Rigid
UnstableEquilibriumconfiguration
vitrification
![Page 20: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/20.jpg)
Activationc
Point defects?Collective mode?
![Page 21: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/21.jpg)
Activationc
Brito and Wyart, J. phys stat, 2007
![Page 22: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/22.jpg)
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
![Page 23: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/23.jpg)
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
![Page 24: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/24.jpg)
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
![Page 25: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/25.jpg)
Acknowledgement
Tom WittenSid NagelLeo SilbertCarolina Brito
![Page 26: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/26.jpg)
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
![Page 27: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/27.jpg)
• 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
![Page 28: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/28.jpg)
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
![Page 29: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/29.jpg)
Glass Transition=G relaxation time
Heuer et. al. 2001
•e
![Page 30: Soft motions of amorphous solids](https://reader030.vdocument.in/reader030/viewer/2022013106/568167d4550346895ddd2822/html5/thumbnails/30.jpg)
Vitrification as a ``buckling" phenomenum
increases
P increases
L