partial dislocations in fcc crystals

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Partial dislocations in FCC crystals

Dislocations in FCCPerfect and partial dislocations Frank’s rule and splitting of dislocations into partialsStacking fault energy and separation of partial dislocationsShockley and Frank partial dislocationsIntrinsic and extrinsic stacking faultsExamples of dislocation reactions: Transformation of Frank loopExamples of dislocation reactions: Lomer-Cottrell lockAtmospheres of impurities/solutes

References:Hull and Bacon, Ch. 5


Dislocations in FCC

)ˆˆ(23 zxaa +=

r

)ˆˆ(21 zyaa +=

r

)ˆˆ(22 yxaa +=

r

x

y

zprimitive vectors are the shortest lattice vectors ⇒ Burgers vector of the lowest energy dislocation is

1102ab =

r

2|| ab =

r

the next best option is 001ab =r

ab =||r

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


A

BC A

BC

Perfect and partial dislocations in FCC

displacement of atoms by b1 moves them to identical sites ⇒ glide of a perfect dislocationleaves perfect crystal structure

1br

3br

2br

displacement of atoms by b2 or b3 is not a lattice vector ⇒ motion of partial dislocationleaves an imperfect crystal (stacking fault is created)

Adissociation of perfect dislocation into 2 Shockley partial dislocations

]112[6

]121[6

]101[2

aaa+=

Partial dislocation outlines a stacking fault area (planar defect): ABCACABCA… instead of ABCABCA…


Frank’s rule and splitting of dislocations into partials

2~ bWdisl Frank’s rule for dislocation reactions:

3br

2br1b

rthe reaction is favorable if b2

2 + b32 < b1

2

dissociation decreases (elastic) energy of the dislocation per length, Wel

]110[2ab =

r

2)011(

4

2222

22 aab =++=perfect dislocation:

Shockley partial: ]112[6ab =

r

6)211(

36

2222

22 aab =++=

However, we also have to account for the energy of the stacking fault:

3br

2br1b

r

there is an extra energy per unit area of the stacking fault - stacking fault energy γ

266

222 aaa<+ 2

123

22 bbb <+


Stacking faults

perfect a/2[110] dislocation

two Shockley partials

]112[6

]211[6

aa+

stacking fault


Stacking fault energy and separation of partial dislocations

3br

2br1b

r 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

d

dbGb

yxxbGbF ssss

s π−=

+π−=

2232

2232

r

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:

( )( ) d

bGbyx

yxxbGbF eeeee )1(2)1(2

32222

2232

ν−π=

+

−ν−π

=r

dbbG

dbbGF eess

)1(2)(

2)( 3232

ν−π⋅

+π⋅

−=rrrr

r

dGb

dbGb

dbbGF p

π=

πα

=π⋅

≈42

cos2

)( 23232

rrr

neglecting (1-ν):

⇒ F is positive (repulsive)

for x = d and y = 0

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

/2/3 ,6/ π<π=α= abp

eq

p

dGbπ

≈γ4

2

πγ≈

4

2p

eq

Gbd

attraction repulsion


Shockley partial dislocations in fcc crystals

(111) plane12 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:

]112[6

]121[6

]101[2

aaa+=

]211[6

]112[6

]011[2

aaa+=

]211[6

]121[6

]110[2

aaa+=


Movement of extended (dissociated into partial) dislocationsdislocation 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

TEM of extended dislocations in CuAl alloy (Hull & Bacon)(stacking fault ribbons appear as fringe patterns)

approximate values of stacking fault energies[Hirth and Lothe, Theory of Dislocations, 1982]

322180166125453216PtPdAlNiCuAuAg

2mJ/m ,γ

wider SF ribbons, difficult cross-slip

γ Fr

Fr

Peierls potential is lower for partial dislocations ⇒ the motion of partials can occur at τ when the motion of perfect dislocations stops ⎟

⎠⎞

⎜⎝⎛ π−=τ

Kba

KG

P2exp


Movement of extended (dissociated into partial) dislocations

125 ps100 ps80 ps40 ps

dislocation with b = a/2<110> dissociated into two a/6<112> Shockley partials connected by a stacking fault ribbon

γ = 110 mJ m−2 predicted by the EAM Ni potential

γ = 125 mJ m−2 (experiment [Hirth, Lothe, Theory of Dislocations, 1982])

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]


Movement of extended (dissociated into partial) dislocationswhile 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)

I

II

I

II

I

III

II

br

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


A

Frank partial dislocation; Intrinsic vs. extrinsic stacking fault

Frank partial dislocation outlines a stacking fault formed by inserting or removing a region of close packed {111} plane.

Frank partial has Burgers vector b = a/3<111> normal to the {111} plane.

b = a/3<111> is not contained in one of the {111} glide planes ⇒ this is a sessile dislocation -cannot glide, can only climb

Depending on whether the material is removed or added, the stacking fault outlined is called intrinsic or extrinsic

ABCABCAB

ABCABC

BA

ABC

CB

AB

C

ABCACAB

intrinsic: …ABCACABCA…extrinsic: …ABCACBCABC… (same as created by Shockley partial)

loop of negative Frank partial dislocation(can be produced by collapse of platelet of vacancies)

loop of positive Frank partial dislocation(can be produced by precipitation of interstitials)


two Shockley partials

]112[6

]211[6

]110[2

aaab +==r

stacking fault

Intrinsic SF: can be generated by both Shockley or Frank partial dislocations

Frank partial dislocation; Intrinsic vs. extrinsic stacking fault

loop of negative Frank partial dislocation

(can be produced by collapse of platelet of vacancies)

]111[3ab =

r


Peierlspotential

Examples of dislocation reactions: Transformation of Frank loopnegative Frank loop

]111[3ab =

r

loop of perfect dislocation

annealingsides align along <110> close packed directions

]110[2ab =

r

]211[6ab =

r

SF

]110[2

]111[3

]211[6

aaa=+

Shockley Frank perfect

2

22 ab =

3

22 ab =

condition: energy decrease due to removal of SF is larger than energy increase due to


Examples of dislocation reactions: Lomer-Cottrell lock

)111(

]110[2

]211[6

]112[6

aaa=+]211[

6]112[

6]011[

2aaa

+=

)111(

)111(

]211[6a]112[

6a

)111(

]110[6

]112[6

]211[6

aaa=+

]011[=lris perpendicular to the

dislocation line]110[

6ab =

r1866

:222

2 aaab >+

)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


square 2D network of misfit dislocations

Structure of the lattice mismatched interface in Cu-Ag layered system

[ ]11021

1 =br [ ]101

21

2 =br

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 stackingfault pyramids outlined by stair-rod partial dislocations. This interfacial structure presents a strong barrier for dislocation propagation ⇒ surface hardening.

rapid melting and resolidification of

the interface

MD simulations [Wu et al., Appl. Phys. A 104, 781, 2011]


Structure of the lattice mismatched interface in Cu-Ag layered system

MD simulations [Wu et al., Appl. Phys. A 104, 781, 2011]


Atmospheres of impurities/solutes xxσ

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:extra stress is needed to separate the dislocation from its solute atmosphereheat treatment may result on reappearance of atmospheres ⇒ strain agingat high T and low dε/dt, solutes can repeatedly lock dislocations ⇒ dynamic strain aging

(Portevin - Le Chatelier effect)

partial dislocations in fcc crystals

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