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?

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:

Super-Arrhenius Activation Energy for Metallic Glass Vitreloy I

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

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 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?

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.

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.

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)

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:

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.

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

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).

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.

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

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

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) 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? = molecular rearrangement rate XC theory -> p = 2 in 2D

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