1. Dynamics and Thermodynamics of the Glass Transition J. S
Dynamics and Thermodynamics of the Glass Transition J.S. Langer Workshop on Mechanical Behavior of Glassy Materials, UBC, July 21, 2007
What molecular mechanism is responsible for super-Arrhenius relaxation near the glass transition?
What is the connection between the nonequilibrium dynamics of slow relaxation and the equilibrium thermodynamics of glassy materials?

2. Dynamics: Angell’s classification of strong and weak glasses
(T) defines Tg: (Tg) = 1013 Poise (arbitrary definition)
1013 P corresponds to a relaxation time of about 100 seconds
Plot log η vs. Tg/T to see deviations from Arrhenius behavior.
Vogel-Fulcher-Tammann
approximation:
Fragility:

3. Super-Arrhenius Activation Energy for Metallic Glass Vitreloy I

4. Thermodynamics:The configurational entropy apparently
extrapolates to zero at low temperatures.
TK = Kauzmann temperature

5. Apparent connections between dynamics and thermodynamics
Adam – Gibbs
Fragility appears to be roughly proportional
to the jump in the specific heat at the glass
transition.

6. JSL Assumptions and Opinions
Small molecules with short-ranged, frustrated interactions
Basic problem: Compute transition rates between microstates (“inherent states”). Why do these rates become anomalously slow near the glass transition?
These transitions are thermally activated molecular rearrangements.

7. Initial inherent state

8. New inherent state

9. Assumptions and Opinions, cont’d.
The RFOT theories are inconsistent with this molecular picture. They use Gibbsian statistical mechanics in a mean-field approximation to compute properties of an entropically favored phase, and use a droplet of this phase as a transition state in computing rates. But Gibbsian ergodicity is valid only when the transitions between microstates are much faster than the rates being computed.
What, then, is the transition state? Why aren’t simple Arrhenius, activated processes effective?

10. Stability of an activated density fluctuation: Spontaneous formation of the glassy analog of a “vacancy-interstitial pair”
“Vacancy”
“Interstitial”
TA= the temperature at which the interstitial is as likely to move away
from the vacancy as it is to fall back in
= upper limit of the super-Arrhenius region.

11. Excitation Chains, Below TA
Thermally activated formation of a stable density fluctuation
-- e.g. a shear-transformation zone -- in a disordered material
Longer chains cost more energy, but there are more of them.
“Vacancy”
“Interstitial” R N
= chain of displacements
containing N links, extending
a distance R.

13. Probability of forming an excitation chain of length N, size R
Random walk
Localization
Self-exclusion
q = number of choices per step
e0 = energy per step
= disorder strength ~ density of frustration-induced defects
U = exclusion energy ~
(a la Flory)

14. Minimize -ln W with respect to R, then find maximum
as a function of N. That is, compute the “free energy”
barrier for activating an indefinitely long chain of
displacements.
-ln W(N,R*) N* N N* N
The result (for temperatures low enough that N* is
large) is the Vogel-Fulcher formula:
Critical length scale:

15. Thermodynamic Speculations:
Critical length scale:
An isolated region of size R < R* is frozen because it cannot
support a critically large excitation chain.
Regions larger than R* shrink to increase entropy.
Therefore: Correlations are extremely strong and long-lasting
on length scales of order R*.  Domains of size ~ R*
The fraction of the degrees of freedom that are unfrozen and
contribute to the configurational entropy is proportional
to the surface-to-volume ratio of regions of size R*, i.e.

16. Two-Component, Two-Dimensional, Slowly Quenched, Lennard
-Jones Glass with Quasicrystalline Components: Blue sites have low-
energy environments. (Y. Shi and M. Falk)

17. Theory and Experiment m = fragility, implies Kauzmann paradox with
Adam-Gibbs relation for viscous
relaxation time
m = fragility,
Relation between specific heat and fragility
independent of m
~ consistent with experiments of Berthier et al., Science 310, 1797 (2005).

18. A more general formula Arrhenius part Super-Arrhenius
for T near T0
Modify the self-exclusion term so that it is weaker for short chains.
Set parameters so that α(T) = 0 for T > TA where chains disappear.

19. Super-Arrhenius Activation Energy for Ortho-Terphenyl
Long-chain V-F limit
Chain length vanishes
at T=TA

20. Effective Disorder Temperature
Basic Idea:
During irreversible plastic deformation of an amorphous solid, molecular rearrangements drive the slow configurational degrees of freedom (inherent states) out of equilibrium with the heat bath.
Because those degrees of freedom maximize an entropy, their state of disorder should be characterized by something like a temperature.
The effective temperature has emerged as an essential ingredient in STZ theories of large-scale plastic deformation.

21. Durian, PRE 55, 1739 (1997) Numerical model of a sheared foam

22. Sheared Foam Teff Ono, O’Hern, Durian, (S.) Langer,
Liu, and Nagel, PRL (2002)
Effective temperature, measured in
several different ways (response-
fluctuation theorems, etc.), goes
to a nonzero constant in the limit
of vanishing shear rate.
Teff
More generally,
= intrinsic relaxation time

23. New Results from T. Haxton and A. Liu (cond-mat 0706.0235)
MD simulations of a glass in steady-state shear flow over a wide range of strain rates, and bath temperatures ranging from well below to well above T0
Direct measurements of Teff in all these steady states
Quantitative analysis by JSL and L. Manning using shear-transformation-zone (STZ) theory of amorphous plasticity and concepts from excitation-chain theory of the glass transition

24. Haxton and Liu

25. Super-Arrhenius behavior below the glass transition?
= molecular rearrangement rate
XC theory -> p = 2 in 2D

26. Haxton-Liu data at three temperatures below
the glass transition, replotted and fit by L. Manning
Super-Arrhenius
Arrhenius

27. Anomalous Diffusion and Stretched Exponentials in Heterogeneous Glass-forming Materials JSL and S. Mukhopadhyay, cond-mat/
Glassy domains are surrounded by fluctuating (diffusing) disordered boundaries (Shi-Falk picture).
A tagged molecule is frozen in a glassy region until it is encountered by a diffusing boundary. It then diffuses for a short time before becoming frozen again.
Therefore the molecule undergoes a continuous-time random walk with two kinds of steps.

28. Two-Component, Two-Dimensional, Slowly Quenched, Lennard
-Jones Glass with Quasicrystalline Components: Blue sites have low-
energy environments. (Y. Shi and M. Falk)

29. Summary of Results Waiting-time distribution in glassy region
Continuous range of stretched exponentials with indices in the range , depending on temperature or ISF wavenumber
Non-Gaussian spatial distributions