Crystal interfaces and
microstructure
Brent Fultz James HoweTransmission Electron Microscopyand Diffractometry of Materials
Free surfaces of crystal (solid / vapour interface)
Grain boundaries (/ interfaces)
Interphase interfaces (/ interfaces)
Solid / Vapour interface
Surface energy arise due to broken bonds on the surface
One reason to support this idea is that melting point scales with surface energy
Broken-bond model for surface energy
Variation of surface energy as a function of
For whichever surface the surface energy is low will be stable
22)sin(cosa
E
Equilibrium shape of a crystal can be predicted by -plot
Construct a surface about an origin such that the free energy of any plane is equal to the distance between the surface and the origin in the direction normal to the plane
Variation of with surface orientation in 3-DPredict the equilibrium shape of an isolated single crystal
Equilibrium shape Aii = minimum
(1-10) section through -plot
Grain boundaries
Boundaries in single phase solids Low angle and high angle grain boundaries Energy of low angle grain boundary Coincidence Site Lattice (CSL) model for GBs
Grain growth and Recrystallization
Boundaries in single phase solids
Nature depend on misorientation between two adjoining grains and the orientation of the boundary plane
5-paramters required to define a grain boundary
Three to specify orientation of one grain with respect to other Two to specify orientation of grain boundary plane with respect to one of the grain
Orientation and misorientation
Orientation of grain can be expressed in terms of Euler angle () Three rotation to coincide local coordinate system with reference coordinate system
Orientation matrix (g) ( 9 elements)
1 b/n 100 & X1 b/n 100 & Y1 b/n 100 & Z
333231
232221
131211
333
222
111
coscoscoscoscoscoscoscoscos
ggggggggg
g
1. the first rotation is by an angle about the z-axis using D,2. the second rotation is by an angle about the former x-axis (now x) using C, and3. the third rotation is by an angle about the former z-axis (now z) using B .
Euler Angles - rotation
Misorientation can be expressed in terms of
Misorientation matrix M= g2g1-1, where g1 and g2 are orientation matrix of each grain
Angle/axis pair rotation about given axis by particular angle to coincide lattice of one grain with the adjoining grain
sin2/)(sin2/)(sin2/)(
2/1cos
21123
13312
32231
332211
ggrggrggr
ggg
a coherent twin boundary (in fcc) is a pure twist boundary, 60
Pure tilt
Axis of rotation is parallel to the plane of the boundary
Pure twist
Axis of rotation is perpendicular to the plane of the boundary
Simple grain boundaries
Low angle and high angle grain boundaries
Dislocation model of GBsas an array of dislocations
Low angle tilt
Array of parallel edge dislocations
Low angle twist
Cross grid of two sets of screw dislocations
Unsymmetrical tilt boundary
Dislocations of different Burgers vectors are required to accommodate the misfit
Energy of low angle grain boundary
Given by total energy of the dislocations within unit area of boundary
For simple array - Depends on spacing of the dislocations
As increases strain fields of the dislocations progressively cancel out -- increases at decreasing rate
bbbD
2/2/
)2/sin(2/
For small dislocation spacing is very largeGrain boundary energy is approximately proportional to the density of dislocation in the boundary (1/D)
When > 10-15o the dislocation spacing is so small that the dislocation cores overlap, grain boundary energy become almost independent of misorientation
Bubble raft model
Low angle grain boundary
High angle grain boundary
Physical model of GBs
Coherent twin Incoherent twin
Special High-angle grain boundaries
GB Energy
Crystal Coherent Twin Incoherent twin GB
Cu 21 498 623
Ag 8 126 377
Fe-Cr-Ni (SS304)
19 209 835
High angle GBs are high energy however, Special HAGBs have very low energies
Coincidence Site Lattice (CSL) model for GBs
Fraction of atoms in coincidence at a grain boundary Reciprocal of that is CSL boundary expressed by
Rotation axis // (100) Rotation axis // (110)
53.1o or 36.9o rotation on axis will give CSL of 5
{100} plane in fcc
53.1o + 36.9o = 90 !!!
22o or 38.2o rotation on axis will give CSL of 7
{111} plane in fcc
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