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Dynamics and Thermodynamics of the Glass Transition J.S. Langer Workshop on Mechanical Behavior of Glassy Materials, UB PowerPoint Presentation
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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?

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slide1
Dynamics and Thermodynamics of the Glass TransitionJ.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?
slide2

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:

slide3

Super-Arrhenius Activation Energy for Metallic Glass Vitreloy I

Super-Arrhenius

Arrhenius

slide4

Thermodynamics:The configurational entropy apparently

extrapolates to zero at low temperatures.

TK = Kauzmann temperature

apparent connections between dynamics and thermodynamics
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.

jsl assumptions and opinions
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.
assumptions and opinions cont d
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?
slide10

Stability of an activated density fluctuation: Spontaneous formation of the glassy analog of a “vacancy-interstitial pair”

“Interstitial”

“Vacancy”

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.

slide11

“Interstitial”

“Vacancy”

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.

R

N

= chain of displacements

containing N links, extending

a distance R.

slide13

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)

slide14

Critical length scale:

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.

-lnW(N,R*)

N

N*

N*

N

The result (for temperatures low enough that N* is

large) is the Vogel-Fulcher formula:

slide15

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.

slide16

Two-Component, Two-Dimensional, Slowly Quenched, Lennard

-Jones Glass with Quasicrystalline Components: Blue sites have low-

energy environments. (Y. Shi and M. Falk)

slide17

Theory and Experiment

implies Kauzmann paradox with

Adam-Gibbs relation for viscous

relaxation time

Relation between specific heat and fragility

m = fragility,

independent of m

~ consistent with experiments of Berthier et al., Science 310, 1797 (2005).

a more general formula
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.

slide19

Super-Arrhenius Activation Energy for Ortho-Terphenyl

Long-chain V-F limit

Chain length vanishes

at T=TA

effective disorder temperature

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.

slide22

Sheared Foam

Teff

Ono, O’Hern, Durian, (S.) Langer,

Liu, and Nagel, PRL 095703 (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

new results from t haxton and a liu cond mat 0706 0235
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
slide25

Super-Arrhenius behavior below the glass transition?

= molecular rearrangement rate

XC theory -> p = 2 in 2D

slide26

Haxton-Liu data at three temperatures below

the glass transition, replotted and fit by L. Manning

Super-Arrhenius

Arrhenius

slide27
Anomalous Diffusion and Stretched Exponentials in Heterogeneous Glass-forming MaterialsJSL and S. Mukhopadhyay, cond-mat/0704.1508
  • 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.
slide28

Two-Component, Two-Dimensional, Slowly Quenched, Lennard

-Jones Glass with Quasicrystalline Components: Blue sites have low-

energy environments. (Y. Shi and M. Falk)

summary of results
Summary of Results
  • Waiting-time distribution in glassy region
  • Continuous range of stretched exponentials with indices in the range 0.5 - 1, depending on temperature or ISF wavenumber
  • Non-Gaussian spatial distributions