Continuum Materials: FF and GG Functional Forms and Generalized Gauge Invariance

Interfaces •  GG: Where is the surface? •  GG: Where is the energy? •  FF: Surface angles: energy cusps and fracture strength* •  FF: Etching rates and faceting •  FF: Intergranular fracture

JPS, Valerie Coffman, Nick Bailey, Chris Myers, Markus Rauscher, Thierry Cretegny, …

Stress-dependent dislocation barriers •  GG: Where is the dislocation? •  FF: Saddle-node

*Valerie Coffman: All others unpublished

Edge dislocation: where is it? Generalized gauge invariance

Nick Bailey

σxy σxy

y

Position (x,y) ambiguous ~ ±a Long-range dislocation field unchanged, but environmental impact “elastic dipole” ΔV modified: ΔV = ΔV + b2

b

b

Glide path gauge dependent Fixed gauge Zero dipole gauge

Energy barrier EB is gauge invariant

EB determines •  Yield stress σc •  Creep rate ε=exp(-EB/kT)

Dislocation barrier: Saddle-node Functional Forms

Nick Bailey

Vertical position y

Ener

gy EB

Energy barrier EB to dislocation glide depends on shear σxy

Washboard Potential 2D Lennard Jones

All guaranteed by Washboard model

Tilted sinusoid: h = 1 Extra fitting parameters An

Fit σc(σxx,σyy), An(σxx,σyy)

Properties of Peierls barrier EB •  EB(-σxy) - EB(σxy) = σxy/b2 •  Saddle-Node Bifurcation at σc •  EB = α3/2 (σc-σxy)3/2 + α5/2 (…)5/2+…

Dislocation barrier: Saddle-node Environmental Dependence

Nick Bailey

Fixed σxx,σyy, fit σc, A1, A2

Full σ tensor dependence: 9 parameters

Yield stress Saddle-node

EB

σxy Peierls stress σc ~ 0.2% of shear modulus µ Between covalent (1% µ) and fcc metals (0.01% µ) 3D: dislocation kinks, pinning, tangles…

Why unpublished? Weirdness at low pressures…

Interfaces: Where is the surface? Generalized gauge invariance

Nielsen, JPS, Stoltze, Jacobsen, Nørskov

When undercooled by ΔT, it depends on where you put the interface!

Melting a Copper Cluster (1993) Crystal in a Supercooled Liquid:

What is the Surface Energy? Nielsen, JPS, Stoltze, Jacobsen, Nørskov

Agree on a Convention [Choose a Gauge]: •  Mathematical description more precise than nature •  Gauge-invariant quantities •  Gauge transformations when shifting descriptions

Interfaces: Where is the Energy? Generalized Gauge Invariance

Bulk physics invariant to adding total derivative terms to free energy Gauss’ law: divergence = altered surface energy

Blue Phases

Meiboom, Brinkman, JPS, Anderson

Microscopic origin: atomistic ambiguity of location of energy! Atoms or bonds? Extra term

Berry’s Phase, θ Vacuum, Aharanov-Bohm Effect

€

∇ • (n •∇n − n∇ • n)stabilizes disclinations

Energy on atoms or bonds?

Interfaces and Energy Cusps Functional Forms

Thierry Cretegny

Materials Properties anisotropic •  Surface energy depends on surface normal (two variables) •  Cusps at vicinal surfaces

Energy ~ step density ~ |θ-θ0| •  Cusps at many high-symmetry points

Broken Bond Model Unrelaxed interface: dangling bond b, surface normal n, b n bonds broken per unit cell

Cu: Anisotropic Surface Energy Fit with 2 parameters!

Why unpublished? Already known (but in angular coords)

Etching rates and faceting Functional Forms

Markus Rauscher, Thierry Cretegny, Melissa Hines, Rik Wind

111

100

Etching rate has cusps at low-index surfaces

Etching rate jumps are

associated with a faceting

transition

First-order: nucleation

Etching rates and faceting Functional forms

Markus Rauscher

•  Measure rate •  Fit it where smooth •  Simulate

Morphology looks promising, but asymptotically flat: wrong microphysics at facet edges?

Why unpublished? Simulation keeps flattening (expt doesn’t)

Grain boundary energy Functional form for symmetric tilt boundary

Valerie Coffman

Vicinal grain boundary = Extra dislocations

Extra θ log θ

Gra

in b

ound

ary

ener

gy

θ

Grain Boundary Fracture Functional form for peak stress

Valerie Coffman

Extra dislocations = Fracture nucleation

sites

Jump down in fracture strength with extra dislocation

Extra dislocation interactions explain nearby strengths

Gra

in b

ound

ary

ener

gy

θ

Grain Boundaries: General case 3D Energy and Fracture: Functional forms?

Valerie Coffman

θ2

θ1

High Symmetry Boundaries Sizes one can simulate using

Periodic Boundary Conditions

Grain Boundary Energies

Center line = perfect crystal

Peak Stress