Colloquium: The Mysterious Glass Transition Seminars: Large-Scale Deformation of Glassy Solids

1. Adventures in the Theory of Glassy MaterialsJ.S. LangerLectures at the Lewiner Institute at the TechnionNovember, 2007 • Colloquium: The Mysterious Glass Transition • Seminars: Large-Scale Deformation of Glassy Solids (1) Why Structural Engineers Need New Ideas in Nonequilibrium Physics (2) What Do We Mean by the "Temperature" of a Glass?

2. What kind of theory do we need to build bridges between atomistic physics and practical applications? • Dynamic equations of motion – analog of Navier Stokes – for amorphous solids – that can be used to solve free-boundary problems and study failure mechanisms. • Internal state variables – i.e. space and time dependent order parameters – to carry information about the history of recent deformations. (necking example) The order parameters, and their equations of motion, should be fully consistent with the fundamental principles of physics and thermodynamics, and should emerge naturally from atomistic models.

3. Continuum Theory of Large-Scale Deformation vi= materialvelocity; p = pressure; sij = deviatoric stress; d/dt includes advection. Rate-of-deformation tensor Elastic part Plastic part includes advection and rotation; = effective disorder temperature . STZ constitutive relations

4. Shear-Transformation-Zone (STZ) TheoryM.Falk and JSL, 1998 • Solidlike starting point: Extension of flow-defect theories of Cohen, Turnbull, Spaepen, Argon, and others. • STZ’s are dilute, deformable “defects” whose populations and orientations determine the response to applied forces. • STZ’s are two-state systems that carry information about the history of recent deformations.

5. Falk’s “D-squared-min” method for visualizing STZ activity • Choose a group of atoms. • Compute their displacements in a short time interval. • Fit these displacements as accurately as possible to a strain tensor. (Only an “affine” deformation will fit perfectly.) • Compute the square of the minimum deviation D from the best fit. Large values of D2 imply large non-affine deformations, therefore possibly irreversible rearrangements.

6. 0-10 0-30 29-30 0-10 Reverse Stress

7. M. Falk, MD Simulation of Tensile Deformation STZ activity observed as non-affine deformation (D2min)

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

9. Cyclic Strain: Small Amplitude STZ transition can be reversed. But STZ disappears after larger strain.

10. Simple Two-State STZ Model For simplicity, assume that the deviatoric (shear) stress sij has a fixed orientation, and that the STZ’s are two-state systems aligned along the same axes. syy= - s sxx= + s R(s) (+) state (-) state R(s) is proportional to the rate at which (+) → (-) transitions occur in response to the stress s. Reverse (-) →(+) transitions occur at rate R(-s).

11. Plastic strain rate = rate of volume-conserving deformation: = number of ± STZ’s per molecular volume = characteristic time scale Low-temperature equation of motion for: noise strength = rate of energy dissipation per STZ = ?? = equilibrated STZ density = STZ creation rate; = STZ annihilation rate

12. What is the density of STZ’s in an amorphous solid? Use an effective temperature Teff to characterize the state of disorder. (The “hotter,” the more disordered.) In nonequilibrium systems at low temperatures, Teff may not be the same as T. In quenched glasses, Teff is sometimes the “fictive temperature.” Let ESTZ be the STZ formation energy. Then the STZ density is proportional to a Boltzmann factor: Teff is close to the free volume of Cohen, Turnbull, Spaepen, et al. Under some circumstances, Teffmaybe proportional to free volume; but Teff is the more general concept.

13. M. Falk, MD Simulation of Tensile Deformation STZ Activity Dilation/ Contraction Region of increased disorder, larger Teff

14. Equation of Motion for the 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 STZ’s are density fluctuations well out in the wings of the disorder distribution; therefore

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

16. Sheared Foam Ono, O’Hern, Durian, (S.) Langer, Liu, and Nagel, PRL 095703 (2002) Temperature, measured in several different ways (response-fluctuation theorems, etc.), goes to a nonzero constant in the limit of vanishing shear rate.

17. Equation of motion for χ: Energy balance Heat generated by deformation drives χ toward . c0is a dimensionless specific heat ~1. But we still must evaluate Γ(s).

18. Summary of Low-Temperature STZ Equations Rate of plastic deformation: Master equation for STZ number densities: Equation of motion for the effective temperature:

19. Evaluate  (s) At low temperatures, where thermal fluctuations do not spontaneously induce STZ transitions, and R(s<0)=0, the work of deformation is all dissipated as heat, some of which becomes configurational entropy, i.e. disorder. Pechenik’s conjecture: Γ(s) is proportionalto the dissipation rate per STZ. That is: Two important results of this conjecture: The proportionality factor s0 is the dynamical yield stress. kT/s0 ~ the fraction of the heat that is converted to configurational disorder. To derive these results, some (too much) algebra is required.

20. STZ Order Parameters(Internal State Variables) scaled, scalar density of STZ’s orientational bias of STZ’s traceless, symmetric tensor, consistent with volume conserving plasticity

21. Equations of motion for the state variables Λ and m Deformation rate: = dimensionless stress (in units of s0) and The stress-dependent functions are: and

22. Athermal STZ Theory: No thermal fluctuations No backward transitions: if Rate of energy dissipation per unit volume = Rate of work done during deformation therefore

23. Athermal STZ Equations

24. Athermal STZ Equations: Dynamic Yield Stress (always a stable fixed point) Stable, jammed, steady-state: Stable, flowing, steady-state: Exchange of stability at implies dynamic yield stress at s = s0

25. m 1 -1 1 -1

26. because STZ density must be small; because STZ density must be small; because STZ density must be small; therefore and equations are stiff, can be replaced by their stationary solutions. therefore and equations are stiff, can be replaced by their stationary solutions. therefore and equations are stiff, can be replaced by their stationary solutions. Athermal STZ Equations: Eliminate the fast variables

27. Final Low-Temperature STZ Equations for otherwise

28. Example from Bouchbinder, JSL, Procaccia, PRE 75, 036107 (2007) Uniform system, constant strain rate. Include elasticity. Express in units of the yield stress. (See more complete analyses of rheological data for a bulk metallic glass, JSL PRE 70,041502 or for simulated amorphous Si, Bouchbinder et al, PRE 75 036108.)

29. Another example from Bouchbinder, JSL, Procaccia, PRE 75, 036107 (2007) Stress-controlled experiment: Note memory and hysteresis.

30. Lu et al, Bulk Metallic Glass: Transients at Constant Strain Rate Experiment STZ Theory

31. Early results: Numerical simulations by Chris Rycroft using the athermal STZ theory and a level-set method for tracking the moving boundary. The two-dimensional bar is being stretched from the sides at constant pulling speed. The color scale is the effective temperature. The pulling speed is very slow.

33. Higher resolution: Differences probably are due to lack of surface tension.

34. Three different pulling speeds – all slow – the slowest on top

35. Concluding Remarks • The STZ analysis seems to be on the right track. Its success – not shown here in detail – in explaining experiments (metallic glass) and simulations (amorphous Si) – especially relaxation rates – is reassuring. • Rycroft is just beginning to develop algorithms for solving free boundary problems using the STZ theory. The goal of this project to is look at various failure modes including fracture. • The greatest theoretical challenges have to do with the roles played by the effective temperature and the interplay between it and the ordinary temperature. That’s the main theme of Lecture III.