Computational Materials Science Network

Grain Boundary Migration Mechanism:

Tilt Boundaries

Hao Zhang, David J. Srolovitz

Princeton Institute for the Science and Technology of Materials, Princeton University

Computational Materials Science Network

Reminder: elastically driven boundary migration

X

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

1122

33

1122

33

5 (001) tilt boundary

• Drive grain boundary migration with an elastic driving force• even cubic crystals are elastically anisotropic

equal strain different strain energy• measure boundary velocity deduce mobility

• Applied strain• constant biaxial strain in x and y• free surface normal to z iz = 0• note, typical strains (1-2%) not linearly elastic

• Measure driving force• apply strain εxx=εyy=ε0 and σiz= 0 to perfect crystals,

measure stress vs. strain and integrate to get the strain contribution to free energy

• includes non-linear contributions to elastic energy

0

0

1122 )(

dF Grainyy

Grainxx

Grainyy

Grainxx

Computational Materials Science Network

Symmetric boundary

Asymmetric boundary = 14.04º

Asymmetric boundary = 26.57º

Reminder: Simulation / Bicrystal Geometry

[010]

5 36.87º

Computational Materials Science Network

0 10 20 30 40 500

50

100

150

200

250

1400K 1200K 1000K

Mob

ility

(1

0-9 m

3 /Ns)

• Mobilities vary by a factor of 4 over the range of inclinations studied at lowest temperature

• Variation decreases when temperature ↑ (from ~4 to ~2)

• Minima in mobility occur where one of the boundary planes has low Miller indices

Reminder: Mobility vs. Inclination

Computational Materials Science Network

Approach• Look in detail at atomic motions as grain boundary moves a short distance

• Focus on one boundary (=22º), time = 0.3 ns, boundary moves 15 Å

• For every 0.2 ps, quench the sample (easier to view structure) – repeat 1500X

• X-Z (┴ to boundary) and X-Y (boundary plane) views – remember this

Trans-boundary Plane View

Boundary Plane ViewX

Y

Z

Grain Boundary

Free Surface

Free Surface

Grain

2G

rain 1

tilt a

xis Color - potential energy

Computational Materials Science Network

Interesting Observations 1

Atomic displacements: t=5ps Atomic displacements: t=0.4ps, t=30ps

Boundary Plane - XY

• Substantial correlated motions within boundary plane during migration

Computational Materials Science Network

Interesting Observations 2Trans-boundary plane XZ

Atom positions during a period in which boundary moves downward by 1.5 nm

Color – von Mises shear stress at atomic position – red=high stress

• Regular atomic displacements – periodic array of “hot” points

Computational Materials Science Network

Interesting Observations 3Trans-boundary plane XZ

Atom positions during a period in which boundary moves downward by 1.5 nm

Color time – red=late time, blue=early time

• Atomic displacements symmetry of the transformation

Computational Materials Science Network

Coincidence Site Lattice

Part of the simulation cell in trans-boundary plane view

• CSL unit cell• Atomic “jump” direction ▲,○ - indicate which lattice

Color – indicates plane A/BDisplacements projected onto CSL “Interesting” displacement patterns

Computational Materials Science Network

Atomic Path for 5 Tilt Boundary Migration

Translations in the CSL

Types of Atomic Motions

Type I

• “Immobile” – coincident sites -1 d1= 0 Å

Type II

• In-plane jumps – 2, 4, 5

d2=d4=1.1 Å, d5=1.6 Å

Type III

• Inter-plane jump - 3

d3=2.0 Å

12 3

45

Computational Materials Science Network

Simulation Confirmation

○ initial average position projected on trans-boundary plane

∆ final average position came from the same atoms in initial

Color – indicates plane A/B

Trans-boundary plane XZ

• The atoms that do not move (Type I) are on the coincident sites• Plane changing motions (Type III), are “usually” as predicted

Computational Materials Science Network

Simulation Confirmation - Type III Displacements

Atomic displacements: t=0.4ps, t=30ps Boundary Plane - XY Trans-boundary plane XZ

Color – von Mises shear stress at atomic position

• The red lines on the left ( XY-plane) indicate the Type III displacements • These are the points of maximum shear stress

Computational Materials Science Network

• How are these different types of motions correlated?

which is the chicken and which is the egg?

• What triggers the motions that lead to boundary translation?

• Can we use this information to explain how mobility varies

with boundary structure (inclination)?

The Big Questions

Computational Materials Science Network

1 2 3 45

Transition Sequence

1 2 3 4 5 11 2 3 4 5

Sequence is 1,3,4 then 2 + 5

Trans-boundary plane XZColors Time

Boundary Plane - XYColor- time blue- early time

Type III motion

Computational Materials Science Network

Type II DisplacementsTrans-boundary plane XZ

Atom positions during boundary moves downward by 1.5 nm

Color – Voronoi volume change – red= ↑over 10%, blue = ↓over 10%

• Excess volume triggers Type II displacement events

Computational Materials Science Network

Connection with Grain Boundary Structure

5 10 15 20 25 30 35 40

0.390

0.395

0.400

0.405

0.410

0.415

0.420

0.425

60

80

100

120

140

160

180

Exc

ess

Vol

ume

(A)

Mob

ility

(1

0-9 m

3 /Ns)

ANV /Volume Excess

• The higher the boundary volume, the faster the boundary moves• More volume easier Type II events faster boundary motion

Computational Materials Science Network

Type III DisplacementsBoundary Plane - XY

Atomic displacements: t=5ps

Computational Materials Science Network

Excess Volume Transfer During String Formation

• Colored by Voronoi volume

• In crystal, V=11.67Å3

Boundary Plane - XY

• Excess volume triggers string-like (Type III) displacement sequence

• Net effect – transfer volume from one end of the string to the other

• Displacive not diffusive volume transport

• Should lead to fast diffusion

Computational Materials Science Network

Correlation with Boundary Self-diffusivity

5 10 15 20 25 30 35 400.8

1.0

1.2

1.4

1.6

1.8

60

80

100

120

140

160

180

Mob

ility

(1

0-9 m

3 /Ns)

Dy (

10-1

3 cm

3 /s)

• Diffusivity along tilt axis direction is correlated with boundary mobility

• Diffusivity along tilt axis – indicative of Type III events

• Diffusivity much higher along tilt-axis direction than normal to it

Computational Materials Science Network

How Long are the Strings?

Boundary Plane - XY

• Display atoms in 0.4 ps time intervals with displacements larger than 1.0 Å

• Arrow indicates the direction of motion in the X-Y plane

• 3 or 4 atom strings are most common

• Some strings as long as the entire simulation cell-10 atoms

Computational Materials Science Network

1 2 3 45

Another Measure of Simulation Size Effect

10 15 20 25 30 35 401

2

3

4

5

6

Mig

rati

on R

ate

(m/s

)

Thickness (Angstrom)

• Strings (Type III events) cannot be longer than simulation cell size

• The boundary mobility drops rapidly for cell sizes smaller than 6 atom spacings (12 Å)

• What happens if we make the simulation cell thinner in the tilt axis direction?

Sequence is 1,3,4 then 2 + 5

Computational Materials Science Network

Migration Picture

12 3

45

Atomic Path

Transition Sequence

1. A volume fluctuation occurs at the boundary

2. A Type II displacement event occurs

3. Triggers a Type III (string) event

4. Transfers volume

Boundary translation1,3,4 then 2 + 5