Dislocation Structures Grain Boundaries and Cell Walls Dislocations

Dislocation Structures: Grain Boundaries and Cell Walls Dislocations organize into patterns Copper crystal http: //www. minsocam. org/msa/col lectors_corner/vft/mi 4 a. htm Polycrystal rotations expelled into sharp grain boundaries Cell Wall Structures Plasticity Work Hardening Dislocation Tangles

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

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) • Metals are Polycrystals • Crystals have Atoms in Rows • How do Crystals Bend? Bent Fork Crystal Axis Orientation Varies between Grains

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

Dislocations climb Topology, Burger’s vector, tangling glide Edge 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 Plastic Deformation: mediated by dislocation line motion, limited by dislocation entanglement

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

Work hardening and dislocations 3 D 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.

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 (q=b/d) • strain field ~exp(-y/d) expelled from bulk • energy~(b 2/d)log(d/b) ~-bq logq

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

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

Avalanches when bending forks Dislocation motion happens in bursts of all sizes 1/1000 cm 105 Number Kraft Avalanches in Ice Stretch 10 -10 10 Dislocation Size Tangle 109 Small avalanches in Metal Micropillars Structure Ice crackles when it is squeezed So, surprisingly, do other metals Avalanches at microscale Analogies to earthquakes Plasticity fractal in time and space?

Dislocation Structures: Grain Boundaries and Cell Walls Dislocations organize into patterns Copper crystal http: //www. minsocam. org/msa/col lectors_corner/vft/mi 4 a. htm Polycrystal rotations expelled into sharp grain boundaries Cell Wall Structures Plasticity Work Hardening Dislocation Tangles

Power laws and scaling qs <r(x) r(x+R)> R -s q 2 -s <(L(x)-L(x+R))2> Renormalization-group predictions Power law <r r>~R-h correlations cut off by initial random length scale <L L> correlations ~ R 2 -h

Emergent scale invariance Self-similar in space; correlation functions lide G & b m i Cl e id l G & ly b n Glide O im Only l e C id Gl Real-space rescaling 2 D 3 D Power law dependence of mean misorientations DOE BES

Refinement Cell sizes decrease and misorientations increase Boundaries above qc Self-similar implies no characteristic scale! Size goes down as cutoff qc goes to zero. Relaxed Strained DOE BES

Compare with previous methods Fractal and non-fractal scaling analysis both realistic Fractal dimension df~1. 5 0. 1 (Hähner expt 1. 641. 79) Refinement scaling collapses qav ~ 1/Dav ~ e 0. 26 0. 14 (Hughes expt e 0. 5, 0. 66 different function) DOE BES