e
e /2
e /2
Ls = ZNa e / 2
Ls : sublimation heat Z : number of nearest neighbors Na : Avogadro’s number = 6.0221023 mol-1
High-indexed surface
higher or irrational { hkl } index ? stepped layers of close-packed planes
relative boundary energy 1.5 1.0 3 2
70.5 twin
0.5
1
0
misorientation, q deg.
20
40
60
80
0
misorientation, q deg.
60
120
180
rotation axis || <100>
(surf. tension = surf. energy) dimension: Jm-2
dG = gdA + Adg = FdA
if g = const., F = g
3.2 Solid / Vapour Interfaces what does the surface energy originate from? —— broken bonds
Tilt boundaries
q
grain boundary // rotation axis
q
<100>a
<001>a
<100>b <010>a
boundary plane
Low-angle tilt boundaries
q
b
D
D
q
b b D tan q q
edge dislocations
164 324 375 378 625 756 468 660 1080
T /° C 223 450 950 1000 925 1350 1450 1300 2000
gb/gsv
0.24 0.30 0.33 0.27 0.36 0.40 0.23 0.29 0.41
measured grain boundary energies in Al
10 ~ 15°
g gsv /3
for low-angle grain boundaries g q
misorientation, q
measured grain boundary free energies
Crystal Sn Al Ag Au Cu g-Fe d-Fe Pt W
gb/mJm-2
1/a a (cos q )/ a broken bonds
q
(sin|q |)/ a broken bonds
broken bonds per unit surface with a crystal plane at an angle q to the close-packed plane: (cos q + sin|q |) / a2
Equilibrium crystalline shapes
Sn particles on NaCl
NiCr crystal on ThO2
Ni crystal on ThO2
Ir crystal in He
g: a<b<c<d
3.3 Boundaries in Single-Phase Solids description of a grain boundary a) orientation difference between two grains 3 freedoms b) boundary orientation in the sample space 2 freedoms grain boundary energy origin : crystalline defects order : 0.1 eV/atom or 0.5 J/m2
grain boundary transition zone
a few atom planes thick
Large- & low-angle boundaries
Misorientation and boundary energy
random high-angle GBs boundary energy, g
g - plot g -plot of a section in
surface in real space normal to OA A
3D orientation space (001) (111) (110)
gA
O
Surface Equilibrium Surface Equilibrium of a Isolated Particle a) surfaces with lower energies specially oriented surfaces (close-packed orientations) b) reduction of the total surface tending to a sphere
Tin boundary rotation axis
screw dislocations
Twin boundaries (large-angle)
f
a coherent twin boundary
an incoherent twin boundary
Large-angle grain boundaries
(a / b interface)
Metallographic Surfaces
3.1 Interfacial Free Energy liquid film F A G = G0 + A g A + dA F
g : free energy per unit area
(word done by F )
Crystal Interfaces and Microstructure
Basic Interface Types
Free Surfaces
(solid / vapour interface) (a / a interface)
Grain Boundaries
Interphase Interfaces