3.3 Boundaries in Single-Phase Solids (/)
Special High Angle Grain Boundaries
Some special high angle boundaries have significantly lower energy than ordinary high angle boundaries.
1. Twin Boundary
If the twin boundary is // to the twining, the atoms in the boundaries fit perfect!!
undistorted position
coherent
Extremely low energy
3.3 Boundaries in Single-Phase Solids (/)
In fcc twin, mistorientation of 70.5 degree about a
<110> axis.
Special Grain Boundary
3.3 Boundaries in Single-Phase Solids (/)
Equilibrium in Polycrystalline Materials
Intersection of Grain Boundary with Free Surface
3 12 2
13 1
23
sin sin
sin
cos 2
2
b sv
3.3 Boundaries in Single-Phase Solids (/)
Crystallographic View of Grain Boundary
12 21 13 31 23 32
33 22 11
a a : a a : a a W : V : U
2 / 1 a a a cos
Transformation matrix {h1k1l1}<p1q1r1>
{h2k2l2}<p2q2r2>
33 32
31
23 22
21
13 12
11
a a
a
a a
a
a a
a M
Orthogonality aijaik=jk
3 independent variables
3.3 Boundaries in Single-Phase Solids (/)
x1 x2 x3
x1 x2 x3
Grain boundary plane or inclination angles Lattice orientations
of two grains
Specimen coordinate Crystal coordinate
X’3
X’2
X’1
2
1
2 independent variables
3.3 Boundaries in Single-Phase Solids (/)
Thermally Activated Grain Boundary Migration
Net Force due to Surface Tension
Direction of Grain Boundary Migration during Grain Growth (2-D) Junction angle: 120
Grains with # of boundary > 6 → grain growth
Grains with # of boundary < 6 → grain shrink, disappear
➪ Reduce the # of grains, increase the mean grain size, reducing the total g.b. energy
➪ Called Grain growth, grain coarsening
3.3 Boundaries in Single-Phase Solids (/)
Thermodynamic Consideration
For grain growth, G is due to the grain boundary curvature.
For recrystallization, G is due to the difference of strain energy
R
G 2 Vm
: effect of pressure difference by curved boundary
x F V
x
G m
/
Released Energy
Free 2
Nm
V F G
m
3.3 Boundaries in Single-Phase Solids (/)
The effect of G on Kinetics of Boundary Migration
Effective Flux of Atoms
from grain 1 to 2 from grain 2 to 1
If and there is no net boundary movement If and there is no net boundary movement
1 2 1
1
2 n v exp ( G / RT ) m s
A a
1 2 2
2
1 n v exp (G G)/ RT m s
A a
0
G
1 1 2 2
2
1n v A n v
A
0
G
RT
G T
R v G
n A J
a
net 2 1 1 exp 1 exp
3.3 Boundaries in Single-Phase Solids (/)
If boundary velocity is v (Net flux = v/((Vm/Na))
The boundary migration is a thermally activated process.
High energy g. b.→ Open g. b. structure → high mobility
Low energy g. b. → closed (or dense) g. b. structure → low mobility
T R
H R
S T
R N
V v n M A
where
V G M
v or
V G T
R G T
R N
V v n v A
a a
a m m
m a
a m
exp exp
/ exp
2 1 1 2 2 1 1 2
3.3 Boundaries in Single-Phase Solids (/)
Grain Boundary Segregation
Impurities tend to stay at the grain boundary
Gb : free energy reduced when a solute is moved to GB from matrix.
T R X G
Xb b
0 exp
3.3 Boundaries in Single-Phase Solids (/)
The Kinetics of Grain Growth
Experimental Observation
Interaction with Particles
M K
where t
K D
D
t d
D d M D
v
4 2
0
2
t n
K D
f D r
r f D
r r f
r P f
3 4 2 3 2
2 3 2
3
max
2
3.4 Interphase Interfaces in Solids (/)
- Different X-tal structure and composition
Interface Coherency
A.Fully coherent Interface : When two crystal match perfectly at interface.
구조가 같기 때문에 chemical energy 만 존재한다 coherent ch
( ) (1 – 200 mJ/m2)
3.4 Interphase Interfaces in Solids (/)
Lattice 가 같지 않아도 coherent interface를 만들 수 있다.
→ Lattice Distortion → Coherency Strain
3.4 Interphase Interfaces in Solids (/)
B. Semi-coherent Interfaces
For sufficiently large atomic misfit
2 / ) (
d d
b
D b d or
d D d d
(as ↑, D )
Interfacial Energy
st
nt ch
semicohere
( )
st for small
(semi-coherent) = 200 – 500 mJ/m2
3.4 Interphase Interfaces in Solids (/)
C. Incoherent Interfaces
Very different atomic configuration ( > 0.25)
D. Complex Semicoherent Interfaces
Semicoh. Int. observed at boundaries formed by low-index planes (atom pattern and spacing are almost equal.)
(inherent) = 500 – 1000 mJ/m2
Nishiyama-Wasserman (N-W) Relationship
fcc bcc
fcc
bcc // (111) , [001] //[101] )
110 (
Kurdjumov-Sachs (K-S) Relationships
] 1 1 0 [ //
] 1 1 1 [ , ) 111 ( //
) 110 (
3.4 Interphase Interfaces in Solids (/)
Second-Phase Shape: Interface Energy Effects
A. Fully Coherent Precipitate
If , have the same structure
Happens during early stage of many ppt hardening
Good match → can have any shape → spherical
minimum
Aii GP(Guinier- Preston) Zone in Al – Ag Alloys
% 7 .
0
rA rB
3.4 Interphase Interfaces in Solids (/)
B. Partially Coherent Precipitates
, have different structure and one plane which provide close match
Coherent or Semi-coherent in one Plane ; Disc Shape (also plate, lath, needle-like shapes are possible)
Alloys Ag
Al in
ecipitates 4%
Pr
Alloys Cu
Al Phase
3.4 Interphase Interfaces in Solids (/)
C. Incoherent Precipitates
When , have completely diff. structures.→ Incoherent interfaces
Interface energy is high for all plane → spherical shape
Polyhedral shapes from coh. or semi-coh. interfaces
D. Precipitates on Grain Boundaries
alloys Cu
Al in
phase
3.4 Interphase Interfaces in Solids (/)
Second-Phase Shape in : Misfit Strain Effects
A. Fully Coherent Precipitates
Coherency strain
minimum
Ai i GS
a a Misfit a
ned
Unconstrai
a a Misfit a
d
Constraine
(α’β: strained para.)
1/3 0.5 3
2 E E
3.4 Interphase Interfaces in Solids (/)
Elastically Isotropic Materials V GS
4
2Elastically Anisotropic Materials
disc sphere
sphere Shape
Zone
Misfit Zone
Cu Zn
Ag Al
A radius Atom
o
0.7% 3.5 % 10.5 % )
(
28 . 1 : 38
. 1 : 44
. 1 : 43
. 1 : )
(
3.4 Interphase Interfaces in Solids (/)
B. Plate-Like Precipitates
V Misfit V
Volume
2 1
2 2
2 2
2
c z a
y a
x
) / 3 (
2 2
a c f V GS
3.4 Interphase Interfaces in Solids (/)
C. Plate-Like Precipitates
3.4 Interphase Interfaces in Solids (/)
Coherency Loss
) (
4 )
(
3 4 4 4
) (
2
2 3
2
st ch
ch
r coherent
non G
r r
coherent G
st crit
st crit
small for
r r
1 4
3
2
