Crystal Boundary

2D DEFECTS (Surface / Interface)

Crystal-Air surface Interphase boundary Crystal-Crystal Grain boundary Stacking Faults Low angle High angle Anti-phase Boundary Twin Boundary

Homophase

Based on angle of rotation Based on axis Based on Lattice Models Based on Geometry of the Boundary plane

Low angle High angle Twist Tilt Mixed Special

CSL/Other

Random Curved Faceted Mixed

Low angle grain boundaries (misorientation < 10º) Two extremes

TILT

An array of edge dislocations

 Rotation axis lies on the boundary plane

TWIST

An array of screw dislocations

 Rotation axis lies  boundary plane to the

Low angle tilt boundary.

It can be represented by a line of edge dislocations.

Low angle tilt boundary in YBaCuO.

The numbers indicate the number of lattice planes between dislocations.

Coincidence Site Lattice, CSL This description is only applicable to certain rotation angles; but these situations are useful as reference and nature tends to favor them as well. Rotation by 26.57

° (not 36.87°) Take a 2x1 rectangle diagonal: 5a, sides are 2x and x (5a) 2 = (2x) 2 + x 2 x = a √5 Area of CSL unit cell = = 5 * area of lattice unit cell

Rotation to Coincidence

• Red and Green lattices coincide Points to be brought into coincidence

S

5 relationship

Red and Green lattices coincide after rotation of

2 tan

-1

= 36.9

° (1/3)

Rotation to achieve coincidence

• Rotate lattice 1 [Bollmann, W. (1970). Crystal Defects and Crystalline Interfaces. New York, Springer Verlag.] until a lattice point in lattice 1 coincides with a lattice point in lattice 2.

• Clear that a higher density of points observed for low index axis.

(a) HAADF and (c) ABF images of a [001](210)Σ5 grain boundary in a CeO2 thin film.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

(a) HAADF-STEM image, (b) calculated most stable structure, (c) strains and (d) defect energetics of SrTiO3 [001](310)Σ5 grain boundary.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

(a) Most stable atomic structures of SrTiO3 [001](210) Σ5 grain boundary obtained by theoretical calculation and (b) the corresponding HAADF-STEM image.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

(a) HAADF STEM image of a pristine Σ31 [0001] tilt grain boundary in alumina.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

(a) HAADF-STEM image of the Y doped Σ31 [0001] tilt grain boundary in alumina.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

(a) The HAADF-STEM image of the Pr doped ZnO [0001] Σ7 tilt grain boundary.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

The HAADF-STEM image of a Pr doped ZnO [0001] Σ49 grain boundary.

Ikuhara Y J Electron Microsc (Tokyo) 2011;60:S173-S188

© The Author 2011. Published by Oxford University Press [on behalf of Japanese Society of Microscopy]. All rights reserved. For permissions, please e-mail: [email protected]

Coherent tilt boundary in Cu and in CuBi alloy.

Notice that the bright Bi atoms are all located in the grain boundary.

DSC (displacement shift complete) lattice.

Includes every lattice point of both lattices. The finest grid used to describe grain boundaries.

Shift in the GB can be described as a dislocation in the DSC lattice. Notice that the lattice sites do not change, only how far one or the other grain extends changes.

Homophase

Based on angle of rotation Based on axis Based on Lattice Models Based on Geometry of the Boundary plane

Low angle High angle Twist Tilt Mixed Special

CSL/Other

Random Curved Faceted Mixed

Low Index Plane Model Structural Unit Model

Low Index Model

• Create two surfaces in bulk, A & B – Energy to do this is g A • Glue them together + g B (+ve) – Energy to do this is g AB (-ve) • Total energy – E= – If g A g A + g B + g AB + g B small (low index facets), E is small

Bicrystal Geometry [010]

S

5 36.87º

Asymmetric boundary a a = 26.57º Asymmetric boundary = 14.04º Symmetric boundary

Structural Unit Model

• Create two surfaces in bulk, A & B – Energy to do this is g A • Glue them together + g B (+ve) – Energy to do this is g AB (-ve) • Total energy – E= g A + g B + g AB – If g AB small (atoms fit), E is small – Based upon atoms, not geometry (CSL) – Not always obvious that these are different

Homophase

Based on angle of rotation Based on axis Based on Lattice Models Based on Geometry of the Boundary plane

Low angle High angle Twist Tilt Mixed Special

CSL/Other

Random Curved Faceted Mixed

From g -plot to EQUILIBRIUM SHAPE OF CRYSTAL → the Wulff construction    Draw radius vectors from the origin to intersect the Wulff plot (OA in Figure) Draw lines  to OA at A (line XY) The figure formed by the inner envelope of all the perpendiculars is the equilibrium shape

Twins in Pt

Faceting

Plane

Silicon

\