1. Lecture 9: Two-Dimensional Defects
PHYS 430/603 material
Laszlo Takacs
UMBC Department of Physics

2. 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 ( ) 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
Twin

3. Construction of a general grain boundary in 2-d Step 1: Start with two identical copies of the same lattice

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

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

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

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

8. Definition of twist and tilt (asymmetric and symmetric) boundaries

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

10. Low angle tilt boundary
Low angle tilt boundary. It can be represented by a line of edge dislocations.

11. Low angle tilt boundary in YBaCuO
Low angle tilt boundary in YBaCuO. The numbers indicate the number of lattice planes between dislocations.

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

13. 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 + x2
x = a √5
Area of CSL unit cell =
= 5 * area of lattice unit cell

14. Coherent tilt boundary in Cu and in CuBi alloy
Coherent tilt boundary in Cu and in CuBi alloy. Notice that the bright Bi atoms are all located in the grain boundary.

15. DSC (displacement shift complete) lattice
DSC (displacement shift complete) lattice. Includes every lattice point of both lattices. The finest grid used to describe grain boundaries.

16. Shift in the GB can be described as a dislocation in the DSC lattice
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.

17. The Bernal polyhedra. The close-packed structures can be described as a combination of only regular tetrahedra and octahedra.

18. A semi-coherent phase boundary

19. Surface forces acting at the edge of a liquid drop

20. Angles at the boundary of three phases