Movement of Dislocations - people.Virginia.EDUpeople.virginia.edu/~lz2n/mse6020/notes/D-mobility.pdf · Dislocation motion by kinks The barrier to move a kink along the line is much

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

Movement of Dislocations

Glissile and prismatic dislocation loopsPlastic strain through movement of dislocationsGlide and climbLattice resistance to glide: Peierls stressKinks and jogsMovement of dislocations with kinks and jogsGeneration of dislocationsFrank-Read mechanism, cross-slip

References:Hull and Bacon, Chs. 3.4-3.9, 10Allen and Thomas, Ch. 5, pp. 283-294


Review of some of the dislocation propertiesDislocation lines can not end within a crystal ⇒ they originate on surfaces, grain boundaries, dislocation nodes, etc., or form dislocation loops

Dislocation movement: Edge dislocation moves || to shear directionScrew dislocation moves ⊥ to shear direction

τ

x

y

br

τ2

22 GbGbL

WT disl ≈α≈=line tension:

balance of energy due to the increase of the dislocation line by ΔL and work done by the external stress to increase the slip area by ΔS:

Example: expansion of a glissile dislocation loop

dRRdRRL π=π−+π=Δ 22)(2

RdRRdRRS π≈π−+π=Δ 2)( 22

LTSb Δ=Δτ

dRTRdRb π≈πτ 22

RGb

bRT

c 2≈≈τ

R dR

for a dislocation loop of radius R to expand, the external stress should exceed the critical value τc

critical stress decreases as the radius of the loop increases


Glissile and prismatic dislocation loops

glissile dislocation loop(b is within the plane of the loop)

br

prismatic loop(b is not within the plane of the loop)

br

at high T and/or presence of vacancy sinks the prismatic loops of vacancy type will shrink

at vacancy supersaturation (c > c0) the loops can grows chemFr

climbFr

climb defined by balance of

and

prismatic loop can be formed by condensation of vacancies in a material with high supersaturation of vacancies (e.g., due to rapid quenching or irradiation by energetic particles) ⇒ b is normal to the plane of the loop of edge dislocation is formed.


Prismatic dislocation loops

prismatic loops can be formed from clusters of vacancies or interstitials

disappearance of prismatic dislocation loops from a thin sheet of Al heated to 102 ºC in TEM

these are partial dislocations and forces due to the stacking faults are contributing (together with the line tension forces) to the climb forces acting on the dislocations

t = 0 min t = 213 min

t = 793 min t = 1301 min

Tartour & Washburn, Phil. Mag. 18, 1257, 1968


Plastic strain through movement of dislocationsPlastic deformation is directly related to the motion of mobile dislocations

dislocations glide under stress as shown by arrows

when a dislocation moves distance dacross the slip plane, it contributes bto the total displacement D

if N dislocations cross the crystal, plastic deformation would be

a dislocation can also moves a distance xi < d and stop due to an obstacle, resulting in incomplete slip - a contribution to the total displacement is a fraction xi/d of b. Thus, in general,

hNbhD // ==ε

xbxhdb

hD

m

N

ii ρ===ε ∑

=1

∑=

=N

iix

dbD

1

∑=

=N

iix

Nx

1

1where - average distance covered by a dislocation

hldNlm /=ρ - density of mobile dislocations

strain rate: vbxbvbdtd mmm ρ≈ρ+ρ=ε≡ε && / where v is the average dislocation velocity


Plastic strain through movement of dislocations

… and to the plastic strain due to climb

strain rate: vb mρ=ε& - this equation is also applicable to screw and mixed dislocations

hΔ

h

σxx

σxx

if a dislocation climbs a distance of xi, it contributes bxi/d to the displacement Δh

the total plastic tensile strain is then:

xbxhdb

hh

m

N

ii ρ==

Δ=ε ∑

=1

climbvb mρ=ε&

d

ix

order of magnitude estimation of the maximum strain rate

at large stresses, most of the dislocations can be mobile: 21514 m1010 −−≤ρ≈ρm

dislocation velocity cannot exceed speed of sound: m/s 3000≤v17-103-214 s103 m103 m/s,10 ,m10 −×=ρ=ε⇒×===ρ vbbv mm &

the length doubles in ~3×10-8 s = 30 ns

real strain rates 14 s10 −≤ε& ⇒ relatively small ρm can explain even the fastest plastic deformation


climbA

A’

B

B’

Glide vs. ClimbGlide/slip: conservative movement within the slip plane

Climb: non-conservative movement away from the slip plane

let’s consider a dislocation AB that moves to A’B’ in the direction tr

- surface normal

lr

tlnrrr

×=

lr0=⋅ nb rr

0>⋅ nb rr

0<⋅ nb rr

- conservative motion, glide

- addition of material

- removal of material

trb

r

tr

lr

nr

tr

br

⊗tlnrrr

×= 0<⋅ nb rr

- positive climb crystal shrinks in direction parallel to slip planeresults from compressive strain

example:

in general, volume change due to climb is

tlbtlbVrrrrrr⋅×=×⋅=Δ


Glide vs. Climb(1) slip: conservative movement of dislocations perpendicular to b × l, i.e., within the slip plane

the motion is reversible - if the sign of τ is reversed, the dislocation can move in the opposite direction and eventually restore the original configurationmovement does not involve point defectsan edge or mixed dislocation has only one slip plane defined by b and lfor screw dislocation, the number of slip planes is defined by its orientation and structure of the crystal (typically 2-4 easy slip planes) motion of a screw dislocation is always conservative

(2) climb: non-conservative movement of dislocations away from the slip planethe motion cannot be easily reversed - simple change of sign of τ does not reverse the process since additional work has to be done to create point defectsmovement is only possible with the help of point defects (vacancies or interstitials)climb is slow (involves diffusion of point defects) and has a strong dependence on Tdirect contribution of climb to the deformation rate is typically small (except for mostly screw dislocation with small edge segments - jogs)climb plays an important role in plastic deformation since it enable dislocations circumvent otherwise insurmountable obstacles

climbvbρ=ε&


Lattice resistance to glide: Peierls stressperiodicity of lattice translates into the periodic variation of energy as a function of displacement of the dislocation core along a direction of high symmetry - Peierls-Nabarro potential

high energy state energy minimumenergy minimum

b

a

assuming a = b, ν = 1/3, K = 2/3 (edge): τP ≈ 1.2 × 10-4 G << τ0 ≈ G/6 … G/30

x

PW ⎟⎠⎞

⎜⎝⎛ π−

π=

Kba

KGbWP

2exp2

K = 1 for screw dislocationK = 1 - ν for edge dislocations

2max

max 1bW

dxdW

bbF P

Pπ

=−==τ

W approximate evaluation:

⎟⎠⎞

⎜⎝⎛ π

−=b

xWWW P 2cos20

slip tend to occur in most widely spaced planes (large a) and for small b

(W and WP are per unit length)

⎟⎠⎞

⎜⎝⎛ π−=τ

Kba

KG

P2exp

0 b b2 b3


Lattice resistance to glide: Peierls stress

WP and τP are larger for materials with angular dependence of interatomic interactions and smaller for spherically-symmetric long-range interactions:

for fcc and hcp τP ≤ 10-6 - 10-5 G (dissociation into partials additionally reduce τP)for bcc (mixed type of bonding) τP ~ 10-4 G for covalent materials τP ~ 10-2 G

the dependence of τP on b/a is very strong, e.g. changing b/a from 1 to 1.5 increases τP from 1.2×10-4 G to 2.8×10-3 G, i.e., 23 times ⇒ only a small number of slip systems with large a and small b are normally activated

edge dislocations tend to be more mobile than screw ones, e.g.,a = b, K = 2/3 (edge): τP ≈ 1.2 × 10-4 Ga = b, K = 1 (screw): τP ≈ 1.9 × 10-3 G

Peierls energy landscape defines special low core energy directions in which the dislocation prefers to lie

if dislocation is unable to lie in one minima of the Peierlspotential, it forms kinks that cross from one minimum to the next one - these are geometrically necessary kinks

shape/width of a kink is defined by a balance of Wdisl and WP




Peierls potential and shape of dislocations

Low-energy configuration of a dislocation is defined by 3 factors:(1) due to the line energy/tension (Wdisl ~ L), dislocations tend to be straight(2) due to the lower energy of screw components (Wdisl ~ 1/K), screw segments tend to be longer(3) due to the Peierls energy landscape (WP), dislocations tend to lie along closely-packed

directions for which the core energy is lowest (and τP is highest)

( )2

3GbbLWdisl ≈=per length of b: 343

1062exp GbKb

aK

GbbWP−×≈⎟

⎠⎞

⎜⎝⎛ π−

π=

(for a = b, K = 1)

LWW dislP /<<usually,

Frank-Read source in a Si crystal

from Hull and Bacon

the dislocation lines tend to lie along <110> directions, where the core has lower energy

what is the direction of the Burgers vector in this image?


Dislocation motion by kinks

The barrier to move a kink along the line is much smaller (WP2 << WP) and for metals with predominantly metallic bonding is negligible ⇒ lateral motion of kinks can take place at low τ << τP ⇒ small plastic strain (pre-yield microplasticity)

Kinks on a screw dislocation is a short segment with edge character ⇒ screw dislocation with a kink can slide in only one glide plane (the one that contain the kink). If it glides in a different plane, the kink serves as an anchor point for the screw dislocation.

Pτ<τKinks are steps of atomic dimension in the dislocation line that are contained in the glide plane of the dislocation

Motion of kinks can be studied in internal friction experiments (measurements of energy losses in a vibrating material): small oscillating stresses can generate reversible movement of kinks - since there is no irreversible deformation, the vibrations are still in the elastic regime. Frequency dependence of the elastic response contains information about the kinks.


Dislocation motion by double kink formation Pτ<τDouble kinks can form spontaneously due to thermal

fluctuations. Nucleation of a double kink corresponds to an energy increase:

lGblbWW kinkdk )1(2

24

2

υ−πα

−τ−=Δ

work done by τ work against attraction of the kinks

00

=∂Δ∂

=ll

dk

lW

bGbl >>τυ−π

α=

)1(20

kinkkinkdkdk WGbWllWW 2)1(2

22)( 30

max ≈υ−πτα

−==Δ=Δ

0.00080.00070.0010.0040.010.230.450.090.0850.10.20.311.52.2AlAgCuFeBiGeSi

eV ,bWP

eV ,kinkWrate of nucleation:

⎟⎟⎠

⎞⎜⎜⎝

⎛−ω≈

TkWR

B

kink2exp0

1120 s10 −≈ω

Double kink nucleation plays a role at low τ. At higher τ dislocations can move without help of double kink formation

The effect of lattice resistance to τ is only significant at low T

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei(the example in this page is for absorption, rather than emission of vacancies)

Climb of mostly screw dislocation with small edge segments

e

seff

llnav &2

climb =

Let’s consider a dislocation in a simple cubic lattice that consists of long screw segments of length ls >> a and short edge segments of length le ~ a

ϕ

x

y

zthe screw segments have 2 slip planes, (001) and (100)

the edge segments are simply one (le = a) or two (le = 2a)rows of atoms located above the edge segments along z-axis ⇒ these segments can only glide in [010] direction in (001) plane

motion in [001] direction would require climb of all edge segments by a. Such climb would allow the whole dislocation to move up by a ⇒ the effective climb velocity can be large

if the rate of vacancy absorption per unit length is , the effective climb velocity is n&

this climb velocity can be much larger than the climb velocity of an edge dislocation: nav edge &2climb =

)(climb2

climb ϕ== ctgvllnav edge

e

seff &

active climb can result in substantial increase in the concentration of point defects, typically vacancies (c > c0)

br

sl

brjogs

vacancies


Kinks and JogsReal dislocations are not straight - they always contain kinks and jogs

Kinks and jogs are steps of atomic dimension in the dislocation line• Kinks are contained in the glide plane of the dislocation• Jogs are not contained in the glide plane of the dislocation

x

y

z

kink

jog

kinks in edge and screw dislocations

jogs in edge and screw dislocations

Jogs always form during climb. Climb proceeds by movement of jogs through emission or absorption of point defects.

climb by emission of interstitials


Effect of kinks and jogs on dislocation motion Kinks:

Jogs of screw dislocations have edge character and can only glide along the line ⇒ movement in other directions involves climb ⇒ jogs impede glide and results in the generation of point defects (mostly vacancies since Ev

f < Eif)

Kinks do not impede glide of the dislocation in the plane of the kink, on the contrary, double kink formation can help dislocation to move at τ < τP

A screw dislocation with a kink can glide in a specific glide plane (the glide plane of the kink) ⇒in other planes the kink serves as an anchor point for the screw dislocation

Jogs:

Generation of kinks and jogs:

Geometrical kinks, thermally activated generation of double kinksGeneration of jogs by absorption or emission of point defects in response to Fchem (super-/under-saturation of point defects)Intersection of dislocations

when two dislocations intersect, each acquires a jog equal in direction and length to b of the other dislocation

1br

2br

2br

1br


Motion of dislocations with super-jogs

br

br

br

small (atomic) jog: τc > τ0 ⇒ screw drags the jog along, creating a trail of vacancies (or, less likely, interstitials)

longer jog: n point defects have to be generated for each step forward, τc < τ0 = nEf/b2ls ⇒ a dipole of edge dislocations of opposite sign is formed.

sc l

GbR

Gb=≈τ

2

When a point defect is created, the jogs moves forward one atomic spacing, ~b, resulting in work done by τ being τb2ls

If the point defect formation energy is Ef, the critical stress to move the dislocation can be obtained from the energy balance:

brsl

sf lbE 20 /=τ

for a segment of length ls the maximum stress is

even longer jog: the interaction between the dislocations in thedipole is weaker and the two edge segments can pass each other ⇒ two parts of the dislocation move independently from each other.


Jogs and prismatic loops

br

Prismatic loops can form(1) by condensation of point defects, e.g., the ones generated by the plastic deformation

(interstitials can diffuse and self-organize into loops at lower T as compared to vacancies)(2) pinch-off of dislocation dipoles formed during propagation of screw dislocations with jogs or

interaction between edge/mixed dislocations

br

br

(3) formation of loops due to multiple cross-slip (e.g., interaction of dislocations with obstacles)

br

I

II

I

II

I

II

III

I

II

III


Interaction of dislocations with obstacles

Orowan mechanismshear loop is formed

Hirsch mechanismcross-slip occurs 2-3 times, leading to formation of prismatic loops

from Hull and Bacon


0.000 0.005 0.010 0.015 0.020 0.025-50

0

50

100

150

200

250

300

350

She

ar s

tress

/ M

Pa

Shear Strain

strain rate 108

pinning of dislocation and release via cutting: combination of Orowan and cutting mechanisms

Molecular dynamics simulations of dislocation dislocation -- precipitate interactions in precipitate interactions in Al(2139) alloy

AlCu5Mg0.5Ag0.4 (Mn)

CuAl2 Ω phase has hexagonal plate-like shape with broad face aligned along the Al (111) planes

Thickness < 6 nm

D.W. Brenner, L. Sun, andM.A. Zikry, NCSU


Generation and multiplication of dislocationsAlthough there are no “equilibrium” dislocations at any T, dislocations are always present in crystals. They are introduced during crystal growth (e.g., due to internal stresses generated by impurity particles, ∇T, condensation of point defects into prismatic loops, at interfaces).

It is difficult to reduce dislocation density below ~1010 m-2

Plastic deformation results in a rapid increase in dislocation density up to ~1014 - 1015 m-2

Mechanisms of dislocation multiplication include: Frank-Read source, multiple cross-slip, emission of dislocation from grain boundaries, etc.

τ > αGb/Rmin

http://zig.onera.fr/DisGallery/index.html

Simulation of Frank-Read source using Dislocation Dynamics method

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Movement of Dislocations - people.Virginia.EDUpeople.virginia.edu/~lz2n/mse6020/notes/D-mobility.pdf · Dislocation motion by kinks The barrier to move a kink along the line is much