The New York Times, July 29, 2008

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

“…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

The Architecture of Aperiodic Crystals

Model handbuilt by J.D. Bernal

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

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

RFOT theory predicts fragility parameter, m

m from RFOT

m from experiment

RFOT predicts the non-exponentiality parameter from fragility and

thermodynamics

ξ

Mosaic picture

ξ=4.5a

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 =

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

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)

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

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)

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

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

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

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

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

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

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)

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)

Vassily Lubchenko & PGW, JCP (2004) 121, 2852

Non-equilibrium aging effect is predicted from fragility within RFOT theory

After long-aging the mosaic is more heterogeneous

“Ultra-slow” relaxations

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

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

Local libraries lead to tunneling resonancesLubchenko & PGW

N*

ΔE=0

Density of ResonancesgT

g

eT

n /

3

1)( ε

ξε =

31453

101

)( −−≈=≈ mJT

Pngξ

εε<<Tg

Direct spectroscopic evidence of complex structure of 2LS

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 √

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:

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)

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

πe

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

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

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:

Crossover temperature

Local libraries lead to tunneling resonancesLubchenko & PGW

N*

ΔE=0

Density of Resonances

Distribution of Barriers

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

Direct spectroscopic evidence of complex structure of 2LS

Beyond the Semi-Classical Tunneling System: Quantum Melting

Level repulsion

Δ

Quantum melted resonances lead to deviations from standard tunneling model.

Quantum melted modes

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

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