Dislocation Structures: Grain Boundaries and Cell WallsDislocations organize into patterns

Copper crystalhttp://www.minsocam.org/msa/

coll-ectors_corner/vft/mi4a.htm

Polycrystalrotations expelled into sharp grain boundaries

PlasticityWork Hardening

Dislocation Tangles

Cell Wall Structures

Crystals are weirdNo 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 Crystals1Plas-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

CrystalsBroken 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 v1 + m v2

Unit cell with periodic boundary

micro

DislocationsTopology, 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

climbglide

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 TorusWinding Number =Topological Charge =Burgers Vector

Work hardening and dislocations3D 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 dislocationsDislocations 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~(b2/d)log(d/b) ~-bq logq

Cell Wall StructuresMatt 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

DOE BES

(Climb & Glide qualitatively sharper in 2D, but rather similar in 3D)

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 squeezedSo, surprisingly, do other metalsAvalanches at microscaleAnalogies to earthquakesPlasticity fractal in time and space?

Kraft

Stretch

Avalanches in Ice

Num

ber

Size

105

1091010-10

1/1000 cm

Dislocation Structures: Grain Boundaries and Cell WallsDislocations organize into patterns

Copper crystalhttp://www.minsocam.org/msa/

coll-ectors_corner/vft/mi4a.htm

Polycrystalrotations expelled into sharp grain boundaries

PlasticityWork Hardening

Dislocation Tangles

Cell Wall Structures

Power laws and scalingRenormalization-group predictions

qs <r(

x) r

(x+R

)> R -s

Power law <r r>~R-h correlationscut off by initial random length scale

<L L> correlations ~ R2-h

q2-s <

(L(x

)-L(x

+R))2 >

Climb & Glide

Glide Only 2D

Emergent scale invarianceSelf-similar in space; correlation functions

Real-space rescaling

Power law dependence of mean misorientations

DOE BES

Glide OnlyClim

b & Glid

e

3D

RefinementCell sizes decrease and misorientations increase

Relaxed Strained

Boundaries above qc

Self-similar implies no characteristic scale! Size goes down as cutoff qc goes to zero.

DOE BES

Compare with previous methodsFractal and non-fractal scaling analysis both realistic

Fractal dimension

df~1.50.1 (Hähner

expt 1.64-1.79)

Refinement scaling collapses

qav ~ 1/Dav ~ e0.260.14

(Hughes expt e0.5, 0.66

different function)

DOE BES