!"#$%&'#()*+#,*-.! +/010#+)+!*)-0! 2-((#*$3 ! - - PowerPoint PPT Presentation
!"#$%&'#()*+#,*-.! +/010#+)+!*)-0! 2-((#*$3 ! 4,**)45*$!+1#*! $.-++)+!6#7%!0)-.#78 ! Patrick Charbonneau 9#()*+#,*-.!4,..-:,0-7,0+ ! Carolina Brito (Porto Alegre) Benoit Charbonneau (Waterloo) Eric Corwin (Oregon) Daan Frenkel
Carolina Brito (Porto Alegre) Benoit Charbonneau (Waterloo) Eric Corwin (Oregon) Daan Frenkel (Cambridge) Andrea Fortini (Bayreuth) Atsushi Ikeda (Montpellier) Jorge Kurchan (ENS Paris) Kunimasa Miyazaki (Nagoya) Koos van Meel (Vienna) Giorgio Parisi (La Sapienza) Gilles Tarjus (UPMC) Pierfrancesco Urbani (CEA) Francesco Zamponi (ENS Paris)
Pressure, βP ϕGCP ϕth ϕm ϕFCC ϕf ϕd ϕK Packing fraction, ϕ
Auer and Frenkel, Nature (2002)
>-'#/+!
Pfender and Ziegler, Notices AMS (2004)
4D surface free energy is 2-3 times larger than 3D at similar supersaturations! van Meel, Frenkel, Charbonneau, PRE (2009)
Freezing 0.288 Melting 0.337
D4 0.617
R
van Meel, Charbonneau, Fortini, Charbonneau, PRE (2009)
resemblance vanishes with dimension.
glasses are more easily accessible than crystal.
“[T]here were almost as many versions of the phlogiston theory as there were pneumatic chemists. That proliferation of versions of a theory is a very usual symptom of crisis.”
(1962)
Liu and Nagel Nature (1998), Liu and Kamien PRL (2007); PZ RMP (2010); CKUPZ Nat. Comm. (2014)
equilibrium liquid stable glass marginal glass J-line
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
d/p
4 5 6 7 8 9
2dϕ/d
4 5 6 7 8 9
φ Volume fraction, Pressure, P φ φ φ
th K GCP d
φ Σ
j
φ 0.1 0.2 0.3 0.4 0.5 0.6 150 100 50 0.55 0.6 (b) φ=0.635 free volume theory Carnahan-Starling φ=0.6465 free volume theory φ=0.640 free volume theory Rutgers, et al. Rintoul and Torquato Speedy e e al w st d a
d . a i-
Figure 1 A possible phase diagram for jamming. Temperature Load 1/Density Loose grains, bubbles, droplets etc. Jammed grains etc. G l a s s Liquid
CIPZ PRL (2011); CCPZ PRL (2012)
10 15 20 25 30
2
dϕ
0.05 0.1
1/p
fluid EOS (fit) threshold glass (extr.) ideal glass (RT)
0.1 0.2 0.3 0.4 20 21 22 23 24 25 26
2
dϕj
γ1/2
0.05 0.1 0.15
f
Decreasing γ 0.1, 0.03, ... , 0.00003 ϕK ϕGCP ϕd ϕth
Consistent with d=3 results Chaudhuri, Berthier, Sastry PRL (2010)
Between the liquid and jamming, something must happen, because How determined are the force contacts (and the rattlers)?
10−2
2 5
10−1
2 5
1 10−2
2 5
10−1
2 5
1
1− < fij >
102 103 104 105 106 107
p
102 103 104 105 106 107
d=4,6,8
CKUPZ Nature Comm. (2014)
ϕ < ϕg ϕ > ϕg
High ϕ > ϕg Low ϕ > ϕg Low ϕ > ϕg High ϕ > ϕg
!"# !$# !%# !&#
α
Vibrations
!'#
IIIb Ib IIb σ + O(p−µ) σ Z(r) r
∼ ∆φµ ∼ ∆φ0 ∼ ∆φ
¯ z (b)
CCPZ PRL (2012) (a)
d=3–10
5 10 15 20 5 10 15 20
Z(r)
10−12 10−9 10−6 10−3
r/σ − 1
As proposed by Wyart PRL (2012), two critical regimes can be identified. 1RSB solution does not: critical exponents are 0.
Silbert, Liu, Nagel, PRE (2005)
Eqs.
Color online The near-contact Z b l for a nearly
Donev, Stillinger, Torquato, PRE (2005) Skoge et al., PRE (2006)
!=0.5 !=0.4 HS SS
CCPZ PRL (2012); Lerner, Duering, Wyart Soft Matter (2013)
(a) rattlers
10−3
2 510−2
2 510−1 10−3
2 510−2
2 510−1
fraction
3 4 5 6 7 8 9 10
d
3 4 5 6 7 8 9 10
Suggest that upper and lower critical dimension d=2 (?) Agrees with finite-size scaling arguments
!=0.42(1)
(f)
d=3–10
∼ (r/σ − 1)1−α
10−5 10−3 10−1 101
r/σ − 1 (e)
d=3–10
∼ (r/σ − 1)1−α
10−3 10−1 101 103 10−3 10−1 101 103
Z(r) − Z(σ + σ10−7)
10−7 10−5 10−3 10−1 101
r/σ − 1
!=0.38(3) A/..><M!!=0.41269 HS SS
in- models. infinite-system- recent that diffi- ideal
will .
Donev, Stillinger, Torquato, PRE (2005) O’Hern et al. PRE (2003) CCPZ PRL (2012)
HS SS "=0.42(2)
(b) ∼ f 1+θ
10−2
2 5
10−1
2 5
100
G(f)
2 5 10−1 2 5
100
2 5
101
f/ ¯ f
2 5 10−1 2 5
100
2 5
101
(a) ∼ f 1+θ
10−2
2 5
10−1
2 5
100 10−2
2 5
10−1
2 5
100
G(f)
2 5 10−1 2 5
100
2 5
101
f/ ¯ f
2 5 10−1 2 5
100
2 5
101
0.0 0.5 1.0 0.0 0.5 1.0 Z−(x) 10−210−1 100 101 102 x
"=0.28(3) Embarrasingly different… HS SS
Lerner, Duering, Wyart Soft Matter (2013); DeGiuli et al. arXiv:1402.3834
HS
10
−6
10
−4
10
−2
10 10
2
10
−4
10
−3
10
−2
10
−1
f
1 0.17
N = 8000 N = 1000 N = 124
SS d=3 HS/SS d=2 d=4 HS/SS
10−3 10−2 10−1 100
P(Zbuck.)
2 4 6 8 10 12
d
2 4 6 8 10 12
CCPZ unpublished (2014)
10−6 10−4 10−2 100
g(f/f) ftot fA fN f 1.25 f 1.17462 f 1.42311
10−6 10−4 10−2 100
g(f/f)
10−5 10−3 10−1 101
f/f
10−5 10−3 10−1 101
ftot fA fN f 1.3 f 1.17462 f 1.42311
10−12 10−9 10−6 10−3 10−12 10−9 10−6 10−3
∆EA
102 104 106 108
p
102 104 106 108
d = 3 d = 4 d = 6 d = 8 ∼ p−κ ∼ p−3/2
![NEKN\]K
dt2
p = 102–108
10−14 10−10 10−6 10−2
∆(t)
10−8 10−4 1
t
10−8 10−4 1
10
1
10
2
10
3
10
4
< f >
10
10
10
10
10
N = 1024 N = 256 N = 256
< δR
2>
Brito and Wyart J. Chem. Phys. (2009)
![NE\ High pressure challenges:
reorganize
CKUPZ Nat. Comm. (2014)