Transcript Visualization of Dislocations in a Nanoindentation Simulation

Visualization of Dislocations in a 3-D
Nanoindentation Simulation
Aree Witoelar
Computational Physics
Rijksuniversiteit Groningen
Outline
Dislocations
Simple dislocations
Stacking faults, partial dislocations
Nanoindentation Simulation
Results
Summary
2
Introduction
Imperfections in crystals are:
point defects,
line defects (dislocations),
planar defects and
volume defects.
Dislocations can be analyzed by atomic simulations, for example
a nanoindentation simulation.
3
What are Dislocations?
Dislocations are one-dimensional line defects in crystals.
Example: edge dislocation and screw dislocation on SC.
Slip
Slip
Edge dislocation: an extra half plane of
atom is inserted
H. Föll, Defects in Crystal, http://www.techfak.unikiel.de/matwis/amat/def_en/
Screw dislocation: atoms go
around in a ramp
4
Close-Packed Lattices
Close-packed lattices are comprised of layers of close-packed
planes of atoms.
FCC: ABCABCABC…
HCP: ABABAB…
H. Föll, Defects in Crystal, http://www.techfak.unikiel.de/matwis/amat/def_en/
5
Stacking Faults
A stacking fault occurs
when there is a
mismatch of closepacked planes.
ABCABC 
ABCBABC…
Perfect dislocations
may disassociate into
stacking faults as it has
lower energy.
Partial Dislocation
A
B
C
DISLOCATION
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Nanoindentation Simulation
A 3-D nanoindentation
experiment is simulated
using Molecular
Dynamics.
A spherical indenter is
pushed down into the
system gradually.
The position of each
atom in the system are
recorded at given times.
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Molecular Dynamics
The position and momentum of each atom are calculated at each
time step.
Equations of motion using Verlet algorithm:


τ 
τ 2 N 
rk(t  τ )  rk(t)  pk(t) 


m
2m l  k rk (t )


 N  
 
pk (t   )  pk (t )    
 

2 l  k  rk (t ) rk (t   ) 
Atom-atom potential and atom-indenter potential
1 I
1

      (rij )   ind (rij )
2 j i
i 1  2 j i

N
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Potential Models
Potential models:
Lennard-Jones (LJ) : two-body
potential.
 12  6 


 LJ (rij )  4       ; rij  rc
 rij 
 rij  

rij = distance between two atoms
Lennard-Jones Spline (LJS): adds a
term to soften discontinuity.
Embedded Atom Method (EAM):
embedded in an “electron sea”
1
2
 EA ( i )  d (d  1) (1   )ei ln i ,
rij  rc
2
 rc2  rij2 
1

 ,
i   ij 

2
2
ed (d  1) j i  rc  r0 
j i
0  rij  rc

 12   6 
 LJ  4         ; rij  rspl
r  
 LJS (rij )  
 rij 
 ij  


2
2 2
2
2 3
 spline  a2 (rc  rij )  a3 (rc  rij ) ; rspl  rij  rc
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Visualization of Dislocations
Detect dislocations using
coordinate number C (number
of nearest neighbors).
On a perfect crystal:
SC: 6 nearest-neighbors
FCC: 12 nearest-neighbors
Atoms with wrong C 
dislocated atoms!
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Simple Dislocations in SC
We make a perfect SC crystal and adjust the positions to make simple
dislocations analytically.
Edge dislocation
Screw dislocation


y
xy
arctan x  2(1   )(x 2  y 2 ) ,



b  1  2
x2  y2
2
2
uy  
ln(
x

y
)

,

2
2 
2  4(1   )
4(1   )(x  y ) 
ux 
b
2
u z  0.
Extra half-plane
u x  0,
u y  0,
uz 
b
y
arctan ,
2
x
Top view
Dislocation
Dislocation
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Dislocations in a Nanoindentation Simulation
We simulate FCC-type iron (Fe)
±120000 atoms in a box
Lennard-Jones potential between atoms
Side and bottom boundaries are fixed, top surface is free
The indenter is pushed into the system (loading) and then
pulled out (unloading).
Only atoms with C ≠ 12 (dislocated) and surface atoms will be
shown.
12
Loading
Yellow = surface atoms
Indenter
Black = dislocated bulk
atoms
17Å
Blue = dislocated surface
atoms
68Å
100Å
Atoms are dislocated
because of stress.
Dislocated atoms form
into loops.
100Å
13
Dislocation Loops
Loops indicate stacking faults.
Partial dislocations are the edges of the stacking fault.
Stacking
fault
Stacking
fault
Partial
dislocation
Dislocation loops by Kelchner1
1. C. L. Kelchner et al., Phys. Rev. B 58, 11085–11088 (1998)
Partial
dislocation
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Dislocation Loops Propagation
Loops may appear or
disappear during
loading.
Loops can move away
or towards the indenter
tip.
Loops always connect
to the surface or other
dislocations.
Loop
Loop
LJ, -3.1Å
LJ, -2.0Å
Loop grows
LJ, -8.4Å
New loop
LJ, -14Å
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Unloading
Yellow = surface
atoms
Black = dislocated
bulk atoms
Blue = dislocated
surface atoms
Some dislocated
atoms remain after
unloading.
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Dislocation Loops
Loops to the surface can travel as a unit.
“Hillock” (bump on surface)
Dislocations
Stacking fault
Simulation by
Rodriguez1
1. Rodriguez et al, Phys. Rev. Letters Vol. 88 Num.3, 36101 (2002)
Glides away as a unit
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Slip Direction
Top view
Hillock
Experiment1
Slip in  1 1 0 
direction
110 
Close-packed planes slip along <110> directions  shortest
dislocation on FCC
1. Rodriguez et al, Phys. Rev. Letters Vol. 88 Num.3, 36101 (2002)
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Comparing LJ and LJS-EAM
We compare simulations with the same parameters,
but two different potentials: LJ and LJS-EAM potentials.
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Comparing LJ and LJS-EAM
The two simulation create different patterns of dislocated atoms.
Lennard-Jones
Lennard-Jones
Spline with
Embedded Atom
Method
LJ, -0.9Å
LJ, -8.4Å
LJ, -20.0Å
LJS-EAM, -0.9Å
LJS-EAM, -8.4Å
LJS-EAM, -20.0Å
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Comparing LJ and LJS-EAM
Less atoms jump out of the surface in
LJS-EAM
 LJS-EAM has additional forces to
keep the atoms in the system.
Deeper dislocation loops in LJS-EAM.
 More atoms creates higher stress,
causing dislocations.
LJ, -20.0Å
LJS-EAM, -20.0Å
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Summary
We have shown simple dislocations on a SC crystal.
Dislocation loops are seen in the nanoindentation simulation,
indicating stacking faults.
The loops always reach the surface or other dislocations.
Small bumps on the surface (“hillocks”) are created along the
<110> directions.
Dislocation loops in LJS-EAM go deeper into the system than
in LJ.
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Credits
Dr. K. Michielsen
Prof. Dr. H. de Raedt
RUG Computational Physics Group
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The End