Partial dislocations in FCC crystals
¾ Dislocations in FCC
¾ Perfect and partial dislocations
¾ Frank’s rule and splitting of dislocations into partials
¾ Stacking fault energy and separation of partial dislocations
¾ Shockley and Frank partial dislocations
¾ Intrinsic and extrinsic stacking faults
¾ Examples of dislocation reactions: Transformation of Frank loop
¾ Examples of dislocation reactions: Lomer-Cottrell lock
¾ Atmospheres of impurities/solutes
References:
Hull and Bacon, Ch. 5
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Dislocations in FCC
primitive vectors are the shortest
lattice vectors ⇒ Burgers vector of
the lowest energy dislocation is
r
r a
a
|
|
b
=
b =
110
2
2
r
the next best option is b = a 001
z
r
a
a1 = ( yˆ + zˆ )
2
r
a
a 2 = ( xˆ + yˆ )
2
r
a
a 3 = ( xˆ + zˆ )
2
r
| b |= a
x
but it has twice higher value of b2 ⇒ rarely observed
the slip planes for b = a/2<110> dislocations are {111}
close packed planes stacked in ABCABC… arrangement
y
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
C
Perfect and partial dislocations in FCC
r
b1
B
r
b2
A
displacement of atoms by b1 moves them to
identical sites ⇒ glide of a perfect dislocation
leaves perfect crystal structure
A
C
A
B
r
b3
displacement of atoms by b2 or b3 is not a
lattice vector ⇒ motion of partial dislocation
leaves an imperfect crystal (stacking fault is
created)
dissociation of perfect dislocation into 2
Shockley partial dislocations
a
a
a
[ 1 10 ] = [ 1 2 1 ] + [ 2 11 ]
2
6
6
Partial dislocation outlines a stacking fault area (planar
defect): ABCACABCA… instead of ABCABCA…
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Frank’s rule and splitting of dislocations into partials
Wdisl ~ b 2
2
2
2
Frank’s rule for dislocation reactions: the reaction is favorable if b2 + b3 < b1
r
b1
dissociation decreases (elastic) energy of the
dislocation per length, Wel
perfect dislocation:
Shockley partial:
r a
b = [110 ]
2
r a
b = [112 ]
6
a2 a2 a2
+
<
6
6
2
a2 2
b =
(1 + 12 + 0 2 ) =
4
a2 2
2
b =
(1 + 12 + 2 2 ) =
36
2
r
b3 r
b2
a2
2
a2
6
b22 + b32 < b12
However, we also have to account for the energy of the stacking fault:
r
b1
r
b3 r
b2
there is an extra energy per unit area of the
stacking fault - stacking fault energy γ
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Stacking faults
perfect a/2[110] dislocation
two Shockley partials
a
a
[ 211 ] + [12 1 ]
6
6
stacking fault
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Stacking fault energy and separation of partial dislocations
r
b1
r
b3 r
b2
d
the spacing between partials is defined by the balance
between repulsive forces acting between the partial
dislocations and attractive force due to the stacking
fault energy γ
forces between parallel dislocations in the same glide plane with b2 and b3 at 60º to each other
can be calculated by considering interactions between their screw and edge components:
(
)
r
Gb 2 e b3 e x x 2 − y 2
Gb 2 e b3 e
Fe =
=
2 π (1 − ν ) x 2 + y 2 2
2 π (1 − ν ) d
(
)
r r
r r
r
G ( b2 s ⋅ b3 s ) G ( b2 e ⋅ b3 e )
F =−
+
2 πd
2 π (1 − ν ) d
attraction
repulsion
r
Gb 2 s b3 s
Gb 2 s b3 s
x
Fs = −
=
−
2π x 2 + y 2
2 πd
neglecting (1-ν):
for x = d and y = 0
r r
r G ( b2 ⋅ b3 ) Gb 2 b3 cos α Gb p2
F ≈
=
=
2 πd
2 πd
4 πd
b p = a / 6 , α = π /3 < π /2 ⇒ F is positive (repulsive)
the stacking fault energy γ [J/m2 or N/m] acts against the expansion of the stacking fault region,
with γ being the force acting on a unit length of a dislocation. The approximate equilibrium
separation deq can be found as
Gb p2
Gb p2
γ≈
d eq ≈
4 π d eq
4 πγ
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Shockley partial dislocations in fcc crystals
(111) plane
12 slip systems in fcc: for any of the 4 {111} slip
planes there are 3 <110> directions
3 dissociation reactions can be written for each plane:
a
[ 1 10 ] =
2
a
[ 1 01 ] =
2
a
[ 0 1 1] =
2
a
a
[ 1 2 1 ] + [ 2 11 ]
6
6
a
a
[ 2 11 ] + [ 1 1 2 ]
6
6
a
a
[1 2 1] + [ 1 1 2 ]
6
6
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Movement of extended (dissociated into partial) dislocations
dislocation split into Shockley partials is still able to glide on
the same glide plane as the perfect dislocation ⇒ leading
partial creates stacking fault and trailing one removes it ⇒ two
partials are connected by a stacking fault ribbon of width d
Peierls potential is lower for partial dislocations
⇒ the motion of partials can occur at τ when
G
⎛ 2πa ⎞
exp
τ
=
⎜−
⎟
P
the motion of perfect dislocations stops
K
⎝ Kb ⎠
approximate values of stacking fault energies
[Hirth and Lothe, Theory of Dislocations, 1982]
γ , mJ/m
2
Ag
Au
Cu
Ni
Al
Pd
Pt
16
32
45
125 166 180 322
wider SF ribbons, difficult cross-slip
r
F
γ
r
F
TEM of extended dislocations in CuAl alloy (Hull & Bacon)
(stacking fault ribbons appear as fringe patterns)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Movement of extended (dissociated into partial) dislocations
atomistic simulations melting and generation of crystal defects in a
Ni target irradiated by a short laser pulse
Springer Series in Materials Science, Vol. 130, 43, 2010]
40 ps
80 ps
γ = 110 mJ m−2 predicted by the EAM Ni potential
γ = 125 mJ m−2 (experiment [Hirth, Lothe, Theory
of Dislocations, 1982])
100 ps
125 ps
dislocation with b = a/2<110> dissociated
into two a/6<112> Shockley partials
connected by a stacking fault ribbon
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Movement of extended (dissociated into partial) dislocations
while a perfect screw dislocation in fcc (b = a/2<110>) can glide in two {111} planes, a Shockley
partial dislocation with b = a/6<112> lies in only one {111} plane ⇒ an extended dislocation
cannot cross-slip without recombining into a perfect screw dislocation (formation of a
constriction)
r
b
II
I
II
I
II
II
I
I
formation of a constriction requires energy and is easier for materials with large γ (small d)
cross-slip is one of the mechanisms of dislocation multiplication and propagation ⇒ more
difficult cross-slip in materials with low γ leads to the build up of high internal stresses ⇒ may
lead to more brittle behavior
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Frank partial dislocation; Intrinsic vs. extrinsic stacking fault
¾ Frank partial dislocation outlines a stacking fault formed by inserting or removing a region of
close packed {111} plane.
¾ Frank partial has Burgers vector b = a/3<111> normal to the {111} plane.
¾ b = a/3<111> is not contained in one of the {111} glide planes ⇒ this is a sessile dislocation cannot glide, can only climb
¾ Depending on whether the material is removed or added, the stacking fault outlined is called
intrinsic or extrinsic
intrinsic: …ABCACABCA…
extrinsic: …ABCACBCABC…
(same as created by Shockley partial)
B
B
B
B
A
A
A
A
C
C
C
C
B
B
B
C
A
A
A
A
C
C
C
C
B
B
B
B
A
A
A
A
loop of positive Frank partial dislocation
loop of negative Frank partial dislocation
(can be produced by precipitation of interstitials)
(can be produced by collapse of platelet of vacancies)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Frank partial dislocation; Intrinsic vs. extrinsic stacking fault
Intrinsic SF: can be generated by both Shockley or Frank partial dislocations
two Shockley partials
r a
a
a
b = [110 ] = [ 211 ] + [12 1 ]
2
6
6
stacking fault
loop of negative Frank partial dislocation
(can be produced by collapse of platelet
of vacancies)
r a
b = [ 1 11 ]
3
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Examples of dislocation reactions: Transformation of Frank loop
negative Frank loop
loop of perfect dislocation
Peierls
potential
sides align along <110>
close packed directions
r a
b = [110 ]
2
r a
b = [11 2 ]
6
SF
r a
b = [111 ]
3
annealing
a
a
a
[11 2 ] + [111 ] = [110 ]
6
3
2
Shockley Frank perfect
condition: energy decrease due to removal of SF is larger
than energy increase due to 2 a 2
a2
2
b =
b =
3
2
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Examples of dislocation reactions: Lomer-Cottrell lock
a
a
a
[ 011 ] = [112 ] + [ 1 21 ]
2
6
6
( 1 1 1)
a
a
a
[ 2 1 1 ] + [11 2 ] = [10 1 ]
6
6
2
(111 )
a2 a2 a2
b :
+
>
6
6
18
r a
b = [110 ] is perpendicular to
the
r
6
dislocation line l = [1 1 0 ]
a
a
a
[ 1 21 ] + [ 2 1 1 ] = [110 ]
6
6
6
a
[112 ]
6
( 1 1 1)
2
a
[11 2 ]
6
(111 )
( 001 ) that contains both b and l is
not a slip plane
the dislocation is sessile
(stair-rod dislocation)
this sessile dislocation is called Lomer-Cottrell lock - it locks slip in the two slip planes
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Structure of the lattice mismatched interface in Cu-Ag layered system
rapid melting and
resolidification of
the interface
square 2D network of misfit dislocations
r 1
r
1
b1 = [110]
b2 = [1 10]
2
2
stacking fault pyramid structure stabilized by
Lomer-Cottrell and Hirth locks
Lattice-mismatched interface generated by rapid melting and resolidification of Cu (001)
substrate - Ag film has a three-dimensional structure consisting of a periodic array of stacking
fault pyramids outlined by stair-rod partial dislocations. This interfacial structure presents a
strong barrier for dislocation propagation ⇒ surface hardening.
MD simulations [Wu et al., Appl. Phys. A 104, 781, 2011]
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Structure of the lattice mismatched interface in Cu-Ag layered system
MD simulations [Wu et al., Appl. Phys. A 104, 781, 2011]
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Atmospheres of impurities/solutes
σ xx
small and large solute atoms tend to diffuse to the regions
of compressive and tensile stresses, respectively
⇒ formation of Cottrell atmosphere
adsorption of solute atoms on the stacking
fault can reduce energy of the stacking fault γ
(or energy of the solute + γ) ⇒ increase of d
⇒ decrease of energy of the split dislocation
⇒ formation of Suzuki atmosphere
implications for mechanical properties:
¾ extra stress is needed to separate the dislocation from its solute atmosphere
¾ heat treatment may result on reappearance of atmospheres ⇒ strain aging
¾ at high T and low dε/dt, solutes can repeatedly lock dislocations ⇒ dynamic strain aging
(Portevin - Le Chatelier effect)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei