Dislocation Model of Grain Boundaries Hirth and Lothe : chapter 19 ...

Dislocation Model of Grain Boundaries
Hirth and Lothe : chapter 19
Grain boundaries can be constructed by placing the two misoriented crystal
together along the desired grain boundary place rigidly in a standard reference
position and then relaxing the entire ensemble.
In this process, the atoms in the boundary region will relax their positions to
minimize the total energy by a rigid body translation without rotation ( )
In general, eight geometric parameters are required to give a complete
macroscopic specification of a grain boundary
3 parameters - crystal orientation
2 parameters - orientation of boundary
3 parameters - rigid translation of crystal 2 with respect to crystal 1.
: rotation axis
θ
The orientation of the boundary
3 parameters
a unit vector normal to the boundary plane

2 parameters
has 3 components 3 parameters
Tilt boundary : boundary plane is parallel to ( )
Twist boundary : boundary plane is perpendicular to
mixed boundary : intermediate case
( )
simple boundaries
a) (100) symmetric tilt boundary
: dislocation separation distance
In tilt boundary tilt angle θ is fixed then will be fixed
All (100) planes intersecting the surface region
within ABC must end as incomplete planes
at the boundary
number of incomplete planes
mean separation between dislocations
for small angle θ
θ
b) (100) non-symmetric tilt boundary
θ
θ

φ : angle between the tilt boundary and symmetry boundary
Two set of dislocations are required to construct even a simple boundary.
In the present case, both (100) and (010) planes must terminate in the boundary,
thus dislocations of both sets and are present
similarly
ψ
θ
θ ψ
ψ
θ θ
θ ψ
arbitrary large angle boundary
Frank's formula
recall definition of Burgers vector in perfect reference crystal
Net dislocation density in an arbitrary small angle boundary
consider a wall containing several sets of dislocations
in each set dislocations
parallel
every spaced
same burgers vector
ψ

: a vector in the GB in terms of crystallographic coordinates of grain
A
: the same vector in terms of crystallographic coordinates of grain B
if there should be no in the wall, and should be identical
In terms of sets of dislocations
-
: number of dislocations of burgers vector cut by
The above equation valid for wall of dislocations
If has positive component along ξ ( : a unit normal to the boundary
pointing from grain A to grain B )
a dislocation cut by is counted as a positive contribution to
If the dislocation wall is true grain boundary crystal B is merely rotated with
respect to crystal A and not deformed.
For small angle rotation
θ ( : a unit vector along axis of rotation)
θ should be small, θ θ
θ -
equation and : Frank's formula

Arbitrary large-angle boundary
The same as previous case
burgers circuit
Take circuit in right handed manner about
In fig 19-18 (a)
A crystal : rotation θ from reference crystal
B crystal : rotation θ from reference crystal
: in boundary plane
and = rotation by θ and θ with respect to
θ θ θ
Burgers circuit in fig. b closure failure
in fig. b the same as equation
For a given grain boundary, reference crystal can be oriented such that the grain
boundary is produced by rotations θ θ θ ( fig. c)

then
rotation matrix for A
rotation matrix for B
From equation Frank's formula
θ
θ θ
θ θ
θ θ
θ θ
θ θ θ θ
θ θ θ θ
θ θ θ

Dislocation spacing in the boundary
By introduction of a reciprocal vector notation, Frank's formula for general GB
can be reduced to convenient parametric expressions for dislocation density
: dislocation spacing between boundary dislocation
: number of dislocation/unit length in directioni normal to ξ
ξ : sense vector of dislocation type
: reciprocal vector lying in the boundary and perpendicular to ξ
ξ
ξ
Determination of for a given and
from and
θ
θ -
-
If this equation is satisfied for two independent direction of , it is satisfied for
all in the boundary.
With a given and , equation constitute 6 equations for determination
of values of
For the case of 3 sets of dislocations with independent Burgers vectors, 3
value represent 6 unknowns (as each has two components in the boundary)
Thus for 3 independent , can be uniquely defined.

This is equivalent to considering a mixed boundary as a mixture of a tilt
boundary
(1 set) and a twist boundary (2 sets)
Given and 3 independent , the solution for is as follows
Define reciprocal lattice vector as
δ
multiplying equation by gives
Since is any vector in the boundary, is component of in the
boundary
Note that the long range stress fields of the primary dislocations essentially fall
to zero at distances from the boundary larger than about the wavelength of the
boundary.

Grain boundary energy
Read and Shockley (1950)
assume
small angle symmetric tilt boundary
isotropic (elastically)
dislocation spacing
for small θ
θ θ
energy/unit length of edge disloaction
μ
π ν
: extent of dislocation stress field
μ
π ν θ
number of dislocation/unit length

: energy/unit area of boundary
θ
π ν θ
μ
π ν
π ν
μ
for twist boundary
μ
π
π
μ
θ
θ θ γ
When the boundary contains several arrays of dislocations the same form of
γ can be expected
consider the symmetric tilt boundary
: energy associated with a strip
( )
suppose boundary misorientation decreases by θ , leading to increases in and
given by
θ
θ

change of
If dislocation spacing is large enough , does not change
: volume of strip increases but elastic energy density decreases
elastic energy density (elastic strain) 2
elastic energy density
cylindrical element of area
: stress in this region depends on only the included dislocation
elastic energy density does not change
radius of the cylinder
The elastic stress field in the increased area
increase of radius of the cylinder :
increase of area of slip plane by
stress in any point : τ
work done τ
τ τ
integration of the above equation
τ θ
energy/unit area of boundary γ
γ τ θ θ
If boundary contains more than one array of dislocations
the arrays should be considered.
interaction between
From above, interaction energy contributes only to elastic strain energy for
each array
The interaction energy absorbed into the constants of integration
Then, γ θ θ
θ
θ

In this derivation - is uniformly spaced
(This is possible, dislocation spacing corresponds to some crystal repeat distance)
Uniformly spaced dislocations occur only at particular misorientation.
For intermediate angles, irregularities in dislocation spacings introduce additional
terms in energy (a partial type dislocation)
For small derivation δθ from the nearest rational angle θ
(for rotation about axis in cubic crystal
extra energy is of order
δθ
δθ ( Burgers vector : )
δθ , slope
γ
θ
becomes infinite

The above treatment is valid only for low angle boundary (10 15°)
Wolf's observation
Computer simulation for relaxed grain boundary energy at 0 K.
for several series of boundary, tilt and twist angle varied.
common features as follows
Read-Shockley formula fitted very well to the calculated results for the entire
misorientation range when θ was replaced to θ.
γ θ θ θ
according to Frank's formula, at large angle θ,

θ can be replaced as
θ but no θ