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Dislocation Structures: Grain Boundaries and Cell Walls

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

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Dislocation Structures: Grain Boundaries and Cell Walls

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

  2. Crystals are weird • Crystals have broken translational, orientational symmetries • Translational wave: phonon, defect: dislocation • Orientational wave, defect? Grain boundaries No elegant, continuum explanation for wall formation 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 1Plas-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. micro 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)  U(x) + n v1 + m v2 Order Parameter Space is a Torus: U(x) maps physical space into order parameter space

  5. Edge Screw climb glide 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 Plastic Deformation: mediated by dislocation line motion, limited by dislocation entanglement

  6. Crystals and Dislocations Broken Symmetry, Order Parameters, Topological Defects Missing Half-Plane of Atoms Dislocations in 3D are Lines (Screw, edge, junctions, tangles) 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. 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~(b2/d)log(d/b) • ~-bqlogq Grain boundaries and dislocations Dislocations form walls

  9. Cell Wall Structures Matt Bierbaum, Yong Chen, Woosong Choi, Stefanos Papanikolaou, SurachateLimkumnerd, 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 (Climb & Glide qualitatively sharper in 2D, but rather similar in 3D) DOE BES

  11. 1/1000 cm 105 Avalanches in Ice Avalanches when bending forks Kraft Number Stretch 10-10 Size Dislocation motion happens in bursts of all sizes 10 109 Small avalanches in Metal Micropillars Ice crackles when it is squeezed So, surprisingly, do other metals Avalanches at microscale Analogies to earthquakes Plasticity fractal in time and space? Structure Tangle Dislocation

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

  13. Power laws and scaling R-s qs <r(x) r(x+R)> q2-s <(L(x)-L(x+R))2> Renormalization-group predictions Power law <rr>~R-hcorrelations cut off by initial random length scale <L L> correlations ~ R2-h

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

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

  16. Fractal dimension df~1.50.1 (Hähnerexpt 1.64-1.79) Refinement scaling collapses qav ~ 1/Dav ~ e0.260.14 (Hughes expte0.5, 0.66 different function) Compare with previous methods Fractal and non-fractal scaling analysis both realistic DOE BES

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