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The New York Times, July 29, 2008
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Classical and Quantum Theory of Glasses
1. Ancient (1980’s)T.R. Kirkpatrick, D. Thirumalai, R. Hall, Y. Singh, J.P. Stoessel
2. Modern (2000’s) X.Y. Xia, V. Lubchenko, J. Stevenson, J. Schmalian,
R. Hall, R. Small
Peter G. Wolynes
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“…you had the impression they were trying to sell you an old car” --- Jean-Philippe Bouchard, as quoted in The New York Times, July 29, 2008
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The Architecture of Aperiodic Crystals
Model handbuilt by J.D. Bernal
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Crystallization vs. Glassy Dynamics
N1/3
crystallite
F
large surface cost
critical nucleus size
Free energy gap
F(N) = -Δƒ N + N2/3
(ΔEs - TSc),
N F(N) = - TscN + N1/2
Notice no energy gap.
Crystal nucleation barrier depends on TF - TGlassy Barrier depends on Tsc alone!
Δ F‡ = 2
4Tsc
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Lubchenko and Wolynes, Annu. Rev. Phys. Chem. 2007, 85:235-66.
€
τ =τ o expDTo
T −To
⎡
⎣ ⎢
⎤
⎦ ⎥
Tem
pera
ture
of
Van
ishi
ng E
ntro
py
Glassy Dynamics from a Mosaic of Energy Landscapes
Super Arrhenius temperature dependence of rates
SiO2
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RFOT theory predicts fragility parameter, m
m from RFOT
m from experiment
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RFOT predicts the non-exponentiality parameter from fragility and
thermodynamics
ξ
Mosaic picture
ξ=4.5a
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RFOT predictions of CRR size agree with experiment
22
4 TP
B
C
Tk χχΔ
≥
Berthier et al. Science (2005) 310, 1797
Data from:
Bohmer et al. J. Chem. Phys. (1993) 99, 4201
3/122
2
2)10ln(/ ⎟⎟
⎠
⎞⎜⎜⎝
⎛
Δ=
P
B
Ckm
ea
βπ
ξ
Berthier et al. inequality
34 )/( aξπχ =
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Levinthal Meets Kauzmann!
• Bare RFOT:
• In RFOT theory σCRR is a universal function of log(τα /τ0)
• Relaxation time = random search time of a correlated region
• Adam-Gibbs assumes – σCRR = constant
– (and small, typically)
€
τα =τ0eσ CRR (T ) /4kB
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Relaxation Time and the Complexity of Rearranging Regions
Capaccioli-Ruocco-Zamponi J. Phys. Chem. B (2008)
4,corrcCRR NS=σ
Mode coupling theory with RFOT instanton vertex
Bhattacharyya-Bagchi-Wolynes
2
2
2
4, ln
ln
)(⎟⎠
⎞⎜⎝
⎛Δ
=Td
deTC
kN
P
Bcorr
ατβ
Berthier et al Science (2005)
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Shapes of CRR’s
• Surface interaction energy favors compact shape• Shape entropy favors fractal shape
),(log),( 0int bNTkbvNTSbNF Bc Ω−+−=
Small surface area
Large surface area
Gebremichael et al. J. Chem. Phys 120, 4415
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Shape transition signals crossover temperature
Same as Hagedorn transition in string theory!
String Transition
Mode Coupling Transition
Sc(Tg)/Sc
Log(
Vis
cosi
ty ,
P)
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R.W. Hall and PGW
Self consistent phonon theory and liquid equation of state
∫ ′−′−⎟⎠⎞
⎜⎝⎛ ′−−′⎟
⎠⎞
⎜⎝⎛=′−− ))(exp()(
2exp)(exp( 2
2/3
RrrrVrdRrV eff αβ
π
αβ
*36.5* /Tρ
Intermolecular forces and the glass transition
CB SNk /14
Plots mV and mP on the one hour time scale using the MGC equation of state
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FIG. 1. (left) Plot of A,, K, and G versus nb. (right) Plot of 1n A, 1n K, and 1n G versus nb.
FIG. 2. Plot of log10(TATG) (scale shown on left axis) and TK/TG (scale shown on right axis) versus nb.
Microscopic Theory of Network Glasses
Randall W. Hall Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803-1804
Peter G. Wolynes
Department of Chemistry and Biochemistry, University of California-San Diego, La Jolla, California 92093-0332 (Received 27 July 2002; published 27 February 2003)
A theory of the glass transition of network liquids is developed using self-consistent phonon and liquid state approaches. The dynamical transition and entropy crisis characteristic of random first-order transitions are m apped as a function of the degree of bonding and density. Using a scaling relation for a soft-core model to crudely translate the densities into temperatures, theory predicts that the ratio of the dynamical transition temperature to the laboratory transition temperature rises as the degree of bonding increases, while the Kauzmann temperature falls explaining why highly coordinated liquids are " strong" while van der Waals liquids without coordination are "fragile."
DOI: 10.1103/PhysRevLett.90.085505 PACS numbers: 61.43.Fs, 64.70.Pf, 65.60.+a
nb nb nb
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Explicit magnetic analogies for structural glass
∑ −−⎟⎠⎞
⎜⎝⎛=i ii
ii Rrr ))(exp()( 2
2/3
απ
αρ
Small α liquid state
Large α frozen state
TSc
F(α)
α
α*
Self-Consistent Phonon Theory / Density functional Theory
Nucleation dynamics
F(m)
m<h>
Dynamics equivalent to random Ising system escaping from the metastable state
-Jacob Stevenson-Rachel Small-Aleksandra Walczak-PGW
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VCh Δ⇔Δ 2
∑∑ ↑↑
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
ijjijeff
i
iglass RVF );(log
2
3αβ
π
α
Compare to liquid state free energy
Glassy free energy from self consistent phonon theory
Recover the direct mapping:
cTSh ⇔
Coloring gives flipping cost. Blue is the most stable
Making the mapping explicit
P(Jij)P(hi)
∫ −+−=η
ηηρ
0
0 )1(lnd
ZNNNF EoSliq
( ) NFRVh liqj
jijeffi
i /;log2
3βαβ
π
α−+⎟
⎟⎠
⎞⎜⎜⎝
⎛= ∑ ↑
↑
( ) ( )↑↓↓↑ += jiijeff
jiijeff
ij RVRVJ αααα ,;,; 22
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Constructing explicit magnetic analogies for glass forming liquids
F(m)
m
<h>
;/Ths ic ⇔
TSc
F(q)
qq*
msN
qN
qi
ii
i =⇔= ∑∑ 11
Migliorini-Berker, 1998
Is there replica symmetry breaking?
Zero field phase diagram
2hCV Δ⇔Δ
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Relaxation time and free energy profile for
reconfiguration coordinate
N* = 130
6.01
12
≈Δ+
≈F
KWWβ With facilitation effects:
Xia-Wolynes, 2001
Escaped state
Transition state
Initial state
Sc = 1.1kB
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Increased mobility on free surfacesParticles on free surfaces feel reduced cage effect
dcbulk rTsrrF
3
44)( 2/3 ππ −= d
csurf rTsrrF3
4
2
14
2
1)( 2/3 ππ −=
Mismatch penalty
No mismatch penalty
F‡surf = F‡
bulk / 2
bulksurf τττ 0=
)(2
3 2
Ts
T
c
π=
Free surface
Stevenson-Wolynes (2008)
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Surface mobility leads to high stability vapor deposited glasses
F†surf = F†
bulk / 2 )(2
3 2
Ts
T
c
π=
On the same time scale, the surface layer can reach configurational entropy values half that of the bulk.
2
1
)(
)(1 ≤−=
−−
=gc
fc
kg
fgK Ts
TsTTTT
θ
Ediger et al J. Phys. Chem. B (2008)
⎟⎟⎠
⎞⎜⎜⎝
⎛≈−
fB
fsurf
Tk
TFk
)(exp0
1 τ
Vapor Deposited glasses can reach a maximum of twice the stability of bulk glasses
sc
IMC
Stevenson-Wolynes (2008)
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Vassily Lubchenko & PGW, JCP (2004) 121, 2852
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Non-equilibrium aging effect is predicted from fragility within RFOT theory
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After long-aging the mosaic is more heterogeneous
“Ultra-slow” relaxations
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Confrontation of Classical RFOT Theory with ObservationLindemann length Onset of Activated Behavior
1984 1987
√
√
neutron scattering plateau, excellent density from microscopic theory, OK
Entropy Crisis 1987 2003 2007
√ √ √
a. b. c.
density, temperature OK from microscopic theory, OK dependence on crosslinking follows from microscopics pressure dependence of Tg, Mv vs. Mp, well satisfied
VTF behavior in deeply supercooled regime
1989, 2000
√ √ √
a. b. c.
To vs. Tk, well satisfied. D = .32 kg/ Cp, well satisfied universality of Sc(Tg), well satisfied
Stretched Exponentiality
2001 √
a. b.
vs. D, OK vs. T, OK?
Correlation length 1987
1989 2000
√ √
a. b.
absolute (Tg) vs. particle size, well satisfied vs. T, OK?
Aging Behavior 2004 √
a.
b. c.
m vs. x, well satisfied vs. Teff
ultraslow relaxations
Crossover Temperature (deviations from VTF)
2006
√ √
a. b.
Tc vs. Tg "magical" relaxation time
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Some Relationships of RFOT Theory with Other Approaches
RFOT Theory
(Microscopic) Mode Coupling Theory
Phenomenological Mode Coupling Theory
Leutheusser, Götze
Facilitation
Andersen, etc.
Frustrated Phase Transitions- icosahedratics, etc
Nelson, Kivelson, etc.
Yes, but a higher order effect
Strings, Bhattacharya, Bagchi, PGW
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Local libraries lead to tunneling resonancesLubchenko & PGW
N*
ΔE=0
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Density of ResonancesgT
g
eT
n /
3
1)( ε
ξε =
31453
101
)( −−≈=≈ mJT
Pngξ
εε<<Tg
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Direct spectroscopic evidence of complex structure of 2LS
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Confrontation of Quantized RFOT Theory with Observation
Density of Two Level Systems
2001 √
€
P vs. Tg, excellent
Size of Two Level Systems
2001 √
, roughly OK?
Coupling Constants with Stress with EM Fields
2001
√ √
€
P g2= constant = mol , but in nontrivial manner
Boson Peak
2003 √
BP
Onset of Multilevel Behavior 2001 √
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Percolation clusters and strings
• The surface of percolation clusters and strings scales with volume: b=αN.
)28.1()( Bc kSTNNF −−=),(log),( 0
int bNTkbvNTSbNF Bc Ω−+−=
)13.1()( Bc kSTNNF −−=
Percolation:
Strings:
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RFOT theory predicts dynamic fragility from thermodynamics
0
0
0TT
DT
e −=ττ
LJm
m
PP
STH
moleCC
1)(
ΔΔ
=Δ
cTS
rF 0
203πσ
=+
PC
RD
Δ=32
20
20
2
20
0
25.1
/log
4
3
r
Tk
e
ra
r
Tk
B
B
=
=π
σ
Dm=590/(m-16)Bohmer, Ngai, & Angell, JCP, (1993)
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Classical and Quantum Glasses
• Energy Landscapes• Library Construction
– Nature of cooperatively rearranging regions
• Two Level Systems as Resonances• Boson Peak• Electrodynamics• Beyond Semi-Classical Theory – Quantum Melting
• X.Y. Xia, UIUC/McKenzie• Vas Lubchenko, UH• Jake Stevenson, UCSD• Joerg Schmalian, Iowa• R. Silbey, MIT
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πe
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RFOT predictions of CRR size agree with experiment
• Berthier et al. derived the relationship between the four point correlation function, χ4, and the dynamic susceptibility
• χ 4≥(kB/CP)T2 χT2
• Taking this as a rough equality the size of the cooperatively rearranging region is deduced
• ξ/a = ((3/4π)β2m2kB/ΔCP)1/3
Berthier et al. Science (2005) 310, 1797
Data from:
Bohmer et al. J. Chem. Phys. (1993) 99, 4201
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Shapes of CRR’s
• Include in the nucleation theory the possibility that the nucleating shapes be other than spherical.
• Surface interaction energy wants compact shape• Shape entropy wants fractal shape
),(log),( 0int bNTkbvNTSbNF Bc Ω−+−=
Small surface area
Large surface area
Gebremichael et al. J. Chem. Phys 120, 4415
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Percolation clusters and strings
• The surface of percolation clusters and strings scales with volume: b=αN.
)28.1()( Bc kSTNNF −−=),(log),( 0
int bNTkbvNTSbNF Bc Ω−+−=
)13.1()( Bc kSTNNF −−=
Percolation:
Strings:
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Crossover temperature
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Local libraries lead to tunneling resonancesLubchenko & PGW
N*
ΔE=0
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Density of Resonances
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Distribution of Barriers
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Electrodynamics of GlassesTwo level systems possess dipole moments, quadrupole moments, etc.
<μ2>= μ2mol(dℓ/a)2(ξ/a)3 At lab Tg, μT ≈ μmol
Vas Lubchenko, PGW, R. Silbey, Mol. Phys. (2005)
sinθ=dℓ/a
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Direct spectroscopic evidence of complex structure of 2LS
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Beyond the Semi-Classical Tunneling System: Quantum Melting
Level repulsion
Δ
Quantum melted resonances lead to deviations from standard tunneling model.
Quantum melted modes
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VCh Δ⇔Δ 2
( )⎥⎦
⎤⎢⎣
⎡−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛= ∑ ↓↓↑↑
jijij
Lii
i VVhπ
α
π
αlog
2
3log
2
3
2
1
[ ]↓↓↑↑↓↑↑↓ −−+= ijijijijij VVVVJ2
1
∑∑ +⎟⎟⎠
⎞⎜⎜⎝
⎛=
ijjiijeff
i
i RVF ),,(log2
3 ???
ααπ
α
∑∑ −=ij
jiiji
ii ssJshH2
1
Compare to
Free energy from self consistent phonon theory
Recover the direct mapping:
cTSh ⇔
Coloring gives flipping cost. Blue is the most stable
Making the mapping explicit
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5.3≈∑j
ijJ
7.3log2
3==∑ e
J lind
jij π
α
Specific microscopic calculations give
64.0
8.02
≈Δ⇒
≈
P
c
C
Sδ
Berker’s Random Ising Magnet Phase Diagram and structural glass analogy
Fluctuations in configurational entropy
Results:
Antiferromagnet
The RFOT theory microscopic calculations give
Zero temperature phase diagram
paramagnet
Ferromagnet 1 step RSB
Spin glass higher RSB
OTP