Atomistic Modelling of
Problems in Materials Science
Steven D. Kenny
Department of Mathematical Sciences
Overview
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•
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Multiscale Modelling
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•
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Atomistic Model
Multiscale model
Results
Long Timescale Dynamics
Conclusions
Nanoindentation
Atomistic Model
Fixed atom
Thermostat atom
Free atom
Spring
Support
Atomistic Model
•
A diamond cube corner indenter with the tip rounded is
used.
•
•
The C interactions are described by Tersoff potentials.
•
Spring constant taken to be the same as a typical
experimental value.
The tip-surface interactions are described by the ZBL
potential.
Pop-ins in the Force-Depth
Curve
(c)
(b)
Å
Phys. Rev. B 76 245405 (2003)
Surface pile-up on (100) Fe
(a)
(b)
(c)
<001>
Slip planes in fcc Ag
Phil. Trans. Roy. Soc. A 363
1949-1959 (2005)
Coupled Model
•
Uses atomistic region to describe the material near the
indentation site.
•
Uses finite elements to describe the long-range elastic
fields.
•
Both the atoms and the finite element nodes are
integrated forward in time using a velocity Verlet
algorithm.
•
The difficult part is the linking of the two models.
t the boundary. Atom 5 has a first nearest neighbour force from ato
nd a second nearest neighbour force from atom 3, and imaginary ato
Coupled Model
as a second nearest neighbour force from atom 4. It is these forces th
•
portioning
the
force
to
the
nodes
of
the
element
in
which
that
atom
lie
Forces from the atoms are assigned to the nodes so that
•
e node-atom, labeled 5 in figure 2, applies its whole force to the node wi
ve theModel
interaction
fromwith
the atomistic
region to the
continuum region, b
can deal
any short-ranged
potential.
Newton’s 3rd law is obeyed.
Atomistic Region
Continuum
Region
ich it corresponds.
The imaginary
atoms
use the linear shape functions
element, Ni to apportion their
force. Thus atom 6 applies two thirds
FE Nodes
force to node 5 and a third to node 8.
For a three-dimensional tetrahedral
element at the interface, containin
Imaginary
Atoms
atoms, the forces on its nodes are calculated by
Fnde =
na
!
i=1
Nnde |i Fi
(
n from the atomistic region to the continuum region, by
Communication
eled 5 in figure 2, applies its whole force to the node with
MD
to
FE
s. The imaginary atoms use the linear shape functions of
ce to the nodes of the element in which that atom lies.
3
3
3
pportion their force. Thus atom 6 applies
thirdsonof
are present
• Forcestwo
imaginary atoms due to real
1
2
1
2
nd a third to node1 8. 2
atoms.
1
1
1/2
1/3
2
0
1/2
1/3
nsional tetrahedral element at the interface,
containing
The forces are assigned to
•
on
are calculated
by1/3
3 its nodes
0
0
Fnde =
na
!
i=1
Nnde |i Fi
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nodes according to shape
function values.
The nodal force is then used
in the dynamical FE update.
(2)
ce on node nde, Nnde |i is the value of the shape function
ζ
Communication
Dimensional Multiscale Modelling
111
FE to MD
t they do not p
need to be recalculated •atLinear
each interpolation
step. The is used to
position imaginary atoms according
vp , wp ) of an imaginary atom isηcalculatedtousing
set of
nodalthe
displacements.
fractional coordinates and the nodal displacements of the
•
t.
ξ
up = u14 ξp + u24 ηp + u34 ζp + u4
An atom p with fractional
coordinates (ξp,ηp,ζp) has
displacements (up,vp,wp).
•
These equations keep the fractional
lement
is mapped
this
forcoordinates
simplicity constant.
vp =
v14 ξp +tov24
ηp standard
+ v34 ζp +element
v4
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ons in terms of the natural coordinates ξ, η and ζ.
This provides (6.9)
feedback from the FE
wp = w14 ξp + w24 ηp + w34 ζp + w4 .
model to the MD region.
ts are applied to the imaginary atoms after each FE time
Coupling
Coupled Model
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Ghost forces are added to the nodes near
the boundary.
•
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These give a zero initial force on these nodes.
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Linear finite element model is used - assumes
distortions are small at the boundary.
Imaginary atoms are moved according to the
distortion of the element that contains them.
E. McGee, R. Smith, S.D. Kenny Int J Mater Res 98 430-437 (2007)
Results for Ag
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Ackland potential used to describe the Ag
interactions.
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Tip was indented to a depth of 14 Å.
Coupled model results are compared to a
atomistic only system containing the same
number of atoms.
Force-Depth Curve
Au Slip Systems
Multiscale Model Results
• MD region contains 194,509 atoms, FE region contains
32,860 nodes - volume is equivalent to 13.1 x 106 atoms.
• Contact pressure from an atomistics only simulation is
19.6 GPa.
• Contact pressure from a coupled model simulation is
17.6 GPa.
• Contact pressure from a coupled model simulation to an
equivalent depth is 10.4 GPa.
Displacement Field
Displacement Field
Displacement Field
System Details
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ZnO
Bond order potential
1.05m atoms
226,000 nodes, 1.3m elements
150 nm x 73 nm x 150 nm region
Equivalent to over 140m atoms
Multiscale Model Results
• For Si at 300K and a depth of 1.5 nm
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MD only model - 24 GPa
Coupled model - 14.8 GPa
Experiment - 12 GPa
• For Al at 300K and a depth of 1.5 nm
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MD only model - 13.3 GPa
Coupled model - 4.7 GPa
Conclusions
• The atomistic-FE multiscale model allows length scales
to be bridged.
• The correct description of the long range elastic fields
gives a better agreement with experiment.
• The contact pressure is significantly reduced due to the
inclusion of the long range elastic field.
Further Work
• Non-linear elastic model in FE region
• Coulombic Interactions
• Layered systems
Self Cleaning coating
Antireflective coating
TCO
Low Emissivity material
TCO
Bond Coat
Glass
Modelling Long
Timescale Dynamics
• Modelling O diffusion in Er O
• Forms the bixbyite structure
• Radiation tolerant ceramic
• Interested in calculating the prefactor for
2
hTST using Vineyard method
3
Energy Landscape
1.8
1.6
1.4
Potential energy (eV)
1.2
1
0.8
0.6
0.4
0.2
0
-2
-1.5
-1
-0.5
0
-0.2
Separated distance (Å)
0.5
1
1.5
2
Energy Landscape
0.12
0.1
Potential energy (eV)
0.08
0.06
0.04
0.02
0
-0.8
-0.6
-0.4
-0.2
0
0.2
-0.02
Separated distance (Å)
0.4
0.6
0.8
Conclusions
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“Correct” path would not have double
negative eigenvalue.
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Do such features exist?
If so how would we deal with them?
Acknowledgements
Gheewala, Ed McGee, Lanchakorn
• Ismail
Kittiratanawasin, Marc Robinson
• Roger Smith
• EPSRC, LANL