Dynamics and Thermodynamics of the Glass Transition J.S. Langer Workshop on Mechanical Behavior of...
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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? • What is the connection between the nonequilibrium dynamics of slow relaxation and the equilibrium thermodynamics of glassy materials?
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Transcript of Dynamics and Thermodynamics of the Glass Transition J.S. Langer Workshop on Mechanical Behavior of...
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?
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.
0
00 TT
TDT exp)(
Vogel-Fulcher-Tammann approximation:
DdT
dTm
gTT
1 |ln
Fragility:
Super-Arrhenius Activation Energy for Metallic Glass Vitreloy I
Tk
TG
B
)(exp
Super-Arrhenius
Arrhenius
TK = Kauzmann temperature
Thermodynamics:The configurational entropy apparently extrapolates to zero at low temperatures.
Apparent connections between dynamics and thermodynamics
0TTK
00
1
TTTsT
const
Tk
TGT
cB
)(
.)()(ln
pcm
Adam – Gibbs
Fragility appears to be roughly proportional to the jump in the specific heat at the glass transition.
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.
Initial inherent state
New inherent state
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?
“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.
Stability of an activated density fluctuation: Spontaneous formation of the glassy analog of a “vacancy-interstitial pair”
Thermally activated formation of a stable density fluctuation -- e.g. a shear-transformation zone -- in a disordered materialLonger chains cost more energy, but there are more of them.
= chain of displacements containing N links, extending a distance R.
N
R
“Vacancy”
“Interstitial”
Excitation Chains, Below TA
TkURNRTkNeN BB eeeeqRNW ///),( 220
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 ~ )/(int32 RNTkB (a la Flory)
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.
0
2123
TT
TTW
/int
/ )(ln
)qlnk/(eT B00
The result (for temperatures low enough that N* is
large) is the Vogel-Fulcher formula:
NN*
-ln W(N,R*)
Critical length scale:0
21
TT
TTTR
/int )(
)(
NN*
Critical length scale:0
21
TT
TTTR
/int )(
)(
An isolated region of size R < R* is frozen because it cannotsupport 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 andcontribute to the configurational entropy is proportionalto the surface-to-volume ratio of regions of size R*, i.e.
21000/
intc )TT(
)TT(s
)T(R
s)T(s
Thermodynamic Speculations:
Two-Component, Two-Dimensional, Slowly Quenched, Lennard-Jones Glass with Quasicrystalline Components: Blue sites have low-energy environments. (Y. Shi and M. Falk)
21000/
intc )TT(
)TT(s
)T(R
s)T(s
implies Kauzmann paradox with 0TTK
)(
)(lnln
Ts
sTW
c
0
0
Adam-Gibbs relation for viscous
relaxation time
independent of m~ consistent with experiments of Berthier et al., Science 310, 1797 (2005).
3
)T(R g*
Theory and Experiment
0
)(ln
gT
20
ms
T
sTc c
p
m = fragility,
Relation between specific heat and fragility
A more general formula
)(exp0 TT
TZ
Arrhenius part Super-Arrhenius
0
2/1int
2/3 )()(
TT
TTT
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.
Super-Arrhenius Activation Energy for Ortho-Terphenyl
Chain length vanishes at T=TA
Long-chain V-F limit
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.
Durian, PRE 55, 1739 (1997) Numerical model of a sheared foam
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.
Sheared Foam
Teff
Teff
)( 0effeff TT
More generally,
0 = intrinsic relaxation time
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
Haxton and Liu
Super-Arrhenius behavior below the glass transition?
)( 0effeff TT )(.
)()(ln 0 effeffeff
eff TT
constTf
0 = molecular rearrangement rate
peffeff
effeff TT
constT
)(
.)(
0,
)(ln 0
effT
XC theory -> p = 2 in 2D
Haxton-Liu data at three temperatures below the glass transition, replotted and fit by L. Manning
Arrhenius
Super-Arrhenius
Anomalous Diffusion and Stretched Exponentials in Heterogeneous Glass-forming Materials
JSL 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.
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
• 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
te~
