A theoretician working on glass

My principle aim in the original Hedrick lectures, as well as in this enlarged version was to show that (a) extremely simple observations are often the starting point of rich and fruitful theories and (b) many seemingly unrelated developments are in reality variations on the same simple theme.

A theoretician working on glass

Courtesy Roald Hoffmann

The Theory of Glasses

I. Experimental phenomenology of glass

II. Energy landscapes & random first order transitions

• T.R. Kirkpatrick, D. Thirumalai, R. Hall, Y. Singh, J.P. Stoessel, and P.G. Wolynes

III. The mosaic picture of random first order transitions

• X.Y. Xia, V. Lubchenko, J. Stevenson, J. Schmalian

IV. Quantum theory of glasses• V. Lubchenko

Dynamics and thermodynamics near the glass transition

Ediger, Angell & Nagel, JPC 1996

Super Arrhenius temperature dependence of rates

“strong”

“fragile”

14 o

rder

s of

m

agni

tude

0

0

0TT

DT

e Vogel-Fulcher Law

SiO2

OTPglycerol

The glass transition and the “Kauzmann Paradox”

ΔCP

Slower cooling leads to sharper change

ΔCP is larger for “fragile” liquids

Ediger, Angell & Nagel, JPC 1996

The 3rd law (?)

T

T

Pref

ref

dTT

CSS

T0Kauz = T0

VF (±10°K)!!!

Residual entropy diminishes with slower cooling

1/Tm 1/T0 1/T

Latent heat/Tm

1/Tg

Sliquid -Scrystal

Aging: dynamics continues, but slower, in the glassy state

Alegria et al. Macromolecules,

1998

V. Lubchenko & PGW, JCP (2004)

T

Ex

T

Ex

ge

)1(

0

“Non-linearity parameter” 0<x<1

Slow quench

Fast quench

Narayanaswamy, Tool, Moynihan

Glasses have more low energy excitations than crystals

Raychaudhuri and Pohl, PRB 1982

Stephens, PRB, 1973

Entropy of these excitations is still small

aTdTT

CS

TV 0"" Extrapolated to

300°K, this is ≈10-2kB

CV/T

Intercept

CV =aT+AdebyeT3

The “Standard Model” of Quantised amorphous solids

dxC

gH ii

i

i

22

2 1

2/0

02/

2/

2/

ddPddP )/log,( max

Two-level tunneling states

tunneling Strain int’n

Continuum phonons

~Assume a distribution of ε, Δ

Surprisingly small variation of (reflected in CV) P

W. Phillips, P.W. Anderson, Halprin, Varma

The Architecture of Aperiodic Crystals

Model handbuilt by J.D. Bernal

RFOT theory predicts dynamic fragility from thermodynamics

0

0

0TT

DT

e

LJm

m

PP

ST

HmoleC

C1

)(

cTS

rF 0

203

PC

RD

32

20

20

2

20

0

25.1

/log

4

3

r

Tk

e

ra

r

Tk

B

B

Dm=590/(m-16)Bohmer, Ngai, & Angell, JCP, (1993)

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

Berthier et al. Science (2005) 310, 1797

Data from:

Bohmer et al. J. Chem. Phys. (1993) 99, 4201

3/122

2

2)10ln(/

P

B

C

km

ea

Berthier et al. inequality

34 )/( a

Shapes of CRR’s

• Surface interaction energy wants compact shape

• Shape entropy wants fractal shape

),(log),( 0int bNTkbvNTSbNF Bc

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

)13.1()( Bc kSTNNF

Percolation:

Strings:

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)

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

After long-aging the mosaic is more heterogeneous

“Ultra-slow” relaxations

Local libraries lead to tunneling resonancesLubchenko & PGW

N*

ΔE=0

Density of ResonancesgT

g

eT

n /

3

1)(

31453

101

)( mJT

Png

ε<<Tg

Direct spectroscopic evidence of complex structure of 2LS

THE DEEPEST AND MOST INTERESTING UNSOLVED problem in solid state theory is probably the theory of the nature of glass and the glass transition. This could be the next breakthrough in the coming decade The solution of the problem of spin glass in the late 1970s had broad implications in unexpected fields like neural networks, computer algorithms, evolution, and computational complexity. The solution of the more important and puzzling glass problem may also have a substantial intellectual spin-off. Whether it will help make better glass is questionable.

P. W. AndersonJoseph Henry Laboratories of Physics

Princeton UniversityScience, 1995