1. Dislocation Structures: Grain Boundaries and Cell Walls Dislocations organize into patterns Copper crystal http://www.minsocam.org/msa/col l- ectors_corner/vft/mi4a.htm Polycrystal rotations expelled into sharp grain boundaries Plasticity Work Hardening Dislocation Tangles Cell Wall Structures

2. Crystals are weird No elegant, continuum explanation for wall formation Crystals have broken translational, orientational symmetries Translational wave: phonon, defect: dislocation Orientational wave, defect? Grain boundaries Continuous broken symmetries: magnets, superconductors, superfluids, dozens of liquid crystals, spin glasses, quantum Hall states, early universe vacuum states… Only crystals form walls* Why? *Smectic A focal conics, quasicrystals

3. Plasticity in Crystals 1 Plas-tic: adj [… fr. Gk. plastikos, fr. plassein to mold, form] … 2 a: capable of being molded or modeled (Webster’s) Bent Fork Metals are Polycrystals Crystals have Atoms in Rows How do Crystals Bend? Crystal Axis Orientation Varies between Grains

4. Crystals Broken Symmetry and Order Parameters Order Parameter Space is a Torus: U(x) maps physical space into order parameter space Crystals Break Translational Symmetry Order Parameter Labels Local Ground State: Displacement Field U(x) Residual lattice symmetry U(x)  U(x) + n v 1 + m v 2 Unit cell with periodic boundary micro

5. Dislocations Topology, Burger’s vector, tangling Burger’s vector: loop around defect, registry on lattice shifts (extra columns on top). Topological charge. Dislocation line: tangent t, Burger’s vector b Screw Edge Plastic Deformation: mediated by dislocation line motion, limited by dislocation entanglement climb glide

6. Crystals and Dislocations Missing Half-Plane of Atoms Dislocations in 3D are Lines (Screw, edge, junctions, tangles) Broken Symmetry, Order Parameters, Topological Defects At Dislocation, Order Parameter Winds Around Torus Winding Number =Topological Charge =Burgers Vector

7. Work hardening and dislocations 3D dislocations tangle up During plastic deformation under external stress, new dislocations form, tangle up. Harder to push through tangle – increases yield stress. Tangle ‘remembers’ previous maximum stress.

8. Grain boundaries and dislocations Dislocations form walls Low angle grain boundary wall of aligned dislocations, strength b, separated by d favored by dislocation interaction energy mediates rotation of crystal (  =b/d) strain field ~exp(-y/d) expelled from bulk energy~(b 2 /d)log(d/b) ~-b  log 

9. Cell Wall Structures Matt Bierbaum, Yong Chen, Woosong Choi, Stefanos Papanikolaou, Surachate Limkumnerd, JPS Dislocation tangles eventually organize also into ‘cell structures’ – fractal walls?

10. Cellular structures (Glide only) Plastic deformation, relaxing from random “dented” initial strain field DOE BES (Climb & Glide qualitatively sharper in 2D, but rather similar in 3D)

11. Avalanches when bending forks Small avalanches in Metal Micropillars Dislocation Tangle Structure Dislocation motion happens in bursts of all sizes Ice crackles when it is squeezed So, surprisingly, do other metals Avalanches at microscale Analogies to earthquakes Plasticity fractal in time and space? Kraft Stretch Avalanches in Ice Number Size 10 5 10 9 10 10 -10 1/1000 cm

12. Dislocation Structures: Grain Boundaries and Cell Walls Dislocations organize into patterns Copper crystal http://www.minsocam.org/msa/col l- ectors_corner/vft/mi4a.htm Polycrystal rotations expelled into sharp grain boundaries Plasticity Work Hardening Dislocation Tangles Cell Wall Structures

13. Power laws and scaling Renormalization-group predictions   R-R- Power law <  R   correlations cut off by initial random length scale  correlations ~ R   

14. Climb & Glide 2D Emergent scale invariance Self-similar in space; correlation functions Real-space rescaling Power law dependence of mean misorientations DOE BES Glide Only Climb & Glide 3D

15. Refinement Cell sizes decrease and misorientations increase RelaxedStrained Boundaries above  c Self-similar implies no characteristic scale! Size goes down as cutoff  c goes to zero. DOE BES

16. Compare with previous methods Fractal and non-fractal scaling analysis both realistic Fractal dimension d f ~1.5  0.1 (Hähner expt 1.64- 1.79) Refinement scaling collapses  av ~ 1/D av ~  0.26  0.14 (Hughes expt  0.5, 0.66 different function) DOE BES