Lecture 9: Two-Dimensional Defects

PHYS 430/603 materialLaszlo Takacs

UMBC Department of Physics

Surface defects that do not violate nearest neighbor coordination

• Stacking fault– fcc ABCABCACABCABCABCABC– hcp ABABABABCABABABABABA– ABABABCACACACACACACA

• Twinning (mirror image)– fcc ABCABCABCBACBACBACBA– hcp not for a (0 0 0 1) plane, more complex possible

• Energy: of the order of 0.1 J/m2 ~ 0.6 eV/nm2 - rather small

Al Cu (J/m2)Stacking fault 0.2 0.075Twin 0.12 0.045

Construction of a general grain boundary in 2-d

Step 1: Start with two identical copies of the same lattice

Step 2: Rotate one copy of the lattice relative to the other. The angle of rotation is a characteristic of the boundary.

Step 3: Overlay the two lattices. The relative positions (shift) has to be specified

Step 4: Define the position and direction of the grain boundary

Step 5: Remove one or the other copy of the lattice on both sides.

It is not always clear if an atom should be taken out or left in place. Also, this is a purely geometric procedure, energy relaxation results

in local distortions.

Definition of twist and tilt (asymmetric and symmetric) boundaries

Energy of symmetrical tilt boundaries in Al.The tilt axis is a <1 1 0> direction.

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.

A low angle twist boundary produced by a network of screw dislocations

Coincidence Site Lattice, CSLThis 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 rectanglediagonal: 5a,sides are 2x and x(5a)2 = (2x)2 + x2

x = a √5

Area of CSL unit cell = = 5 * area of lattice unit cell

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.

The Bernal polyhedra.The close-packed structures can be described as a combination of

only regular tetrahedra and octahedra.

A semi-coherent phase boundary

Surface forces acting at the edge of a liquid drop

Angles at the boundary of three phases