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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