3.22 Mechanical properties of materials
Modeling of Metals
xxx
Lecture 3/4
Markus J. Buehler
Outline: 4 Lectures on Molecular Dynamics (=MD)
„
Lecture 1: Basic Classical Molecular Dynamics
General concepts, difference to MC methods, challenges, potential and
implementation
„
Lecture 2: Introduction to Interatomic Potentials
Discuss empirical atomic interaction laws, often derived from quantum
mechanics or experiment
„
Lecture 3: Modeling of Metals
Application of MD to describe deformation of metals, concepts:
dislocations (fracture), analysis techniques, interatomic potentials
„
Lecture 4: Reactive Potentials
New frontier in research: Modeling chemistry with molecular dynamics
using reactive potentials
© 2006 Markus J. Buehler, CEE/MIT
Overview: MD properties
© 2006 Markus J. Buehler, CEE/MIT
5
5
4
4
Solid Argon
3
g(r)
g(r)
Radial distribution function: Solid versus liquid versus gas
2
0
0
0.2
Liquid Ar
(90 K)
3
Gaseous Ar
(300 K)
2
Liquid Argon
1
Gaseous Ar
(90 K)
1
0.4
distance/nm
0.6
0.8
0
0
0.2
0.4
distance/nm
0.6
0.8
Figure by MIT OCW.
Note: The first peak corresponds to the nearest
neighbor shell, the second peak to the second
nearest neighbor shell, etc.
In FCC: 12, 6, 24, and 12 in first four shells
N (r ± Δr2 )
g(r) =
Ω(r ± Δr2 ) ρ
© 2006 Markus J. Buehler, CEE/MIT
Velocity autocorrelation function
t '= ∞
1
D0 = ∫ < v(0)v(t ) >dt '
3 t '= 0
Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.
Describes correlation of velocities in time
http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm
© 2006 Markus J. Buehler, CEE/MIT
MD properties: Classification
„
Structural – crystal structure, g(r), defects such as vacancies and
interstitials, dislocations, grain boundaries, precipitates
„
Thermodynamic -- equation of state, heat capacities, thermal
expansion, free energies
„
Mechanical -- elastic constants, cohesive and shear strength, elastic
and plastic deformation, fracture toughness
„
Vibrational -- phonon dispersion curves, vibrational frequency
spectrum, molecular spectroscopy
„
Transport -- diffusion, viscous flow, thermal conduction
© 2006 Markus J. Buehler, CEE/MIT
MD updating scheme: Complete
(1) Updating method (integration scheme)
2
ri (t0 + Δt ) = −ri (t0 − Δt ) + 2ri (t0 )Δt + ai (t0 )(Δt ) + ...
Positions
at t0-Δt
Positions
at t0
(2) Obtain accelerations from forces
f i = mai
ai = Fi / m
Accelerations
at t0
“Verlet central difference method”
(5) Crystal (initial conditions)
Positions at t0
(3) Obtain forces from potential
dV (r )
F =−
d r
xi
Fi = F
r
(4) Potential
⎛ ⎡σ ⎤12 ⎡σ ⎤ 6 ⎞
φweak (r ) = 4ε ⎜ .⎢ ⎥ − ⎢ ⎥ ⎟
⎜ ⎣r⎦
⎟
r
⎣
⎦
⎝
⎠
Courtesy of Dr. Helmut Foell. Used with permission.
© 2006 Markus J. Buehler, CEE/MIT
Lennard-Jones potential
Attractive
⎛ ⎡σ ⎤12
⎡
σ
⎤
6 ⎞
φ
weak (r
) =
4ε
⎜
.
⎢ ⎥ − ⎢ ⎥ ⎟
⎜ ⎣
r
⎦
⎟
r
⎣
⎦
⎝
⎠
Repulsive
dV (r)
F =−
dr
xi
Fi = F
r
F
r
x2
x1
© 2006 Markus J. Buehler, CEE/MIT
The BIG problem …
Possible solution: Multi-scale modeling techniques based on hierarchies of overlapping scales
Macroscale
~1023 atoms
Bridge
Time
Mesoscale
MD
MEMS
QM
NEMS
Electronics
100..2 atoms
Length
Figure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Handshaking in multi-scale modeling
„
„
„
„
Hierarchical multi-scale modeling is based on the idea to
calculate one property with two different methods, typically a
very accurate, computationally expensive one, and a coarser
one – solve the same problem with two different methods
This requires some overlap in the accessible range of
predictions
When applicable, it enables the rigorous link from smaller to
larger scales
Typically, this procedure is done for a a selection of P
properties to solve for C unknown coefficients. We will
exemplify this method in determining the parameters of a
Lennard-Jones potential
© 2006 Markus J. Buehler, CEE/MIT
Cracked crystal
(1E4..1E9 atoms)
QM
Not accessible
to classical MD
Elastic
properties
(unit cell)
Dislocation
junctions
Not accessible
to DFT/QM
Overlap between QM and MD
Surface
energy
MD
Band structure
“overlap”
Electronic structure
© 2006 Markus J. Buehler, CEE/MIT
Determination of parameters for atomistic interactions
„
„
Often, parameters are determined so that the interatomic
potential reproduces quantum mechanical or experimental
observations
12
Example:
6
⎡⎛ σ ⎞
⎛σ ⎞ ⎤
φ (r) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥
⎝ r ⎠ ⎥⎦
⎢⎣⎝ r ⎠
∂ φ (r)
k=
2
∂r
2
Calculate k as a function of ε and σ (for LJ potential)
Then find two (or more) properties (experimental, for example),
that can be used to determine the LJ parameters
This concept is called potential or force field fitting (training)
Provides quantitative link from quantum mechanics to larger
length scales
© 2006 Markus J. Buehler, CEE/MIT
Determination of parameters for atomistic interactions
“Compression”
P
K=
ΔV / V0
Bulk modulus
Know how to calculate pressure P
Volume from geometry – find ε
⎡⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎤
φ (r ) = 4ε ⎢⎜ ⎟ − ⎜ ⎟ ⎥
⎝ r ⎠ ⎥⎦
⎢⎣⎝ r ⎠
σ from geometry:
Distance between
atoms
© 2006 Markus J. Buehler, CEE/MIT
Modeling vs. simulation
„
Modeling: Building a mathematical or theoretical
description of a physical situation; maybe result in a set of
partial differential equations
For MD: Choice of potential, choice of crystal structure,…
„
Simulation: Numerical solution of the problem at hand
(code, infrastructure..)
Solve the equations – e.g. Verlet method, parallelization (later)
„
Simulation usually requires analysis methods –
postprocessing (RDF, temperature…)
© 2006 Markus J. Buehler, CEE/MIT
Pair interaction approximation
U total =
12 ∑ U (rij )
1
i≠ j
5
2
All pair interactions of atom 1 with
neighboring atoms 2..5
4
3
1
5
2
3
All pair interactions of atom 2 with
neighboring atoms 1, 3..5
4
Double count bond 1-2
1
therefore factor 2
© 2006 Markus J. Buehler, CEE/MIT
Physical example: Surface structures
„
„
Example: Surface effects in some materials
Need a description that includes the environment of an atom
to model the bond strength between pairs of atoms
Pair potentials: All bonds
are equal!
Reality: Have environment
effects; it matter that there is
a free surface!
© 2006 Markus J. Buehler, CEE/MIT
Chemical bonding in metals
“metallic bonding”
„
„
„
„
Bonding between atoms with low electronegativity 1,2 or 3 valence
electrons, therefore there are many vacancies in valence shell.
When electron clouds overlap, electrons can move into electron cloud of
adjoining atoms.
Each atom becomes surrounded by a number of others in a threedimensional lattice, where valence electrons move freely from one valence
shell to another.
Delocalized valence electrons moving between nuclei generate a binding
force to hold the atoms together
positive ions in a sea of electrons
Thus:
„ Electron gas model
„ Mostly non-directional bonding, but the bond strength indeed depends on
the environment of an atom, precisely the electron density imposed by
other atoms
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Electron (q=-1)
+
Ion core (q=+N)
Properties of metals
Property
Physical/atomic reason
High density
Tightly packed FCC, BCC, HCP
High melting temperature
Strong forces between ion core and
delocalized electrons
Good conductors of heat
Vibration transport via delocalized
electrons (+phonons)
Good electrical conductors
Delocalized electrons
(flow in and out)
Many metals are ductile
Glide (and climb) of dislocations
Lustrous
Reflection of light by electron gas
© 2006 Markus J. Buehler, CEE/MIT
Why pair potentials fail…
„
In pair potentials, the strength of each bond is dependent only
on the distance between the two atoms involved:
The positions of all the other atoms are not relevant
(works well e.g. for Ar where no electrons are available for bonding and atoms are
attracted with each other only through the weak van der Waals forces)
„
„
„
However: QM tells that the strength of the bond between two
atoms is affected by the environment (other atoms in the
proximity)
As a site becomes more crowded, the bond strength will
generally decrease as a result of Pauli repulsion between
electrons.
The modeling of many important physical and chemical
properties depends crucially on the ability of the potential to
"adapt to the environment" Can not reproduce surface relaxation (change in electron
density)
http://www.fisica.uniud.it/~ercolessi/forcematching.html
© 2006 Markus J. Buehler, CEE/MIT
Modeling attempts: Multi-body potential
„
„
Multi-body potential depend on more than pairs of atoms, but instead
also on the environment of each atom
Important for metals due to existence of “electron gas”
new
1
φi = ∑
ϕ (rij ) + F ( ρ i )
j =1.. N neigh 2
6
5
ii
j=1
2
Pair potential
energy
4
ρi
3
ρi =
First proposed by Finnis, Sinclair, Daw, Baskes et al. (1980s)
Embedding
energy
as a function of
electron density
Electron density at atom I
based on a pair potential:
∑ π (r )
j =1.. N neigh
ij
© 2006 Markus J. Buehler, CEE/MIT
Numerical implementation of multi-body EAM potential
„
Requires two loops over atoms within each cell
r
O
O
Loop 1:
(i) Pair contributions (derivatives
and potential)
(ii) Calculate electron density
Loop 2:
(iii) Calculate embedding function
and derivatives
Due to additional (i) calculation of electron density and (ii)
embedding contribution EAM potentials are 2-3 times slower than
pure pair potentials
Crystal structure
„
„
„
Different crystal symmetries exist, depending on the
material considered.
For example, many metals have a cubical structure, such
as FCC=face centered cubic
http://home3.netcarrier.com/~chan/SOLIDSTATE/CRYSTAL/fcc.html
Figure by MIT OCW.
http://www.bss.phy.cam.ac.uk/~amd3/teaching/A_Donald/Crystalline_Solids_1.htm
„
How to deform crystals?
© 2006 Markus J. Buehler, CEE/MIT
Crystal structure and potential
„
„
The regular packing (ordering) of atoms into crystals is closely
related to the potential details
Several local minima for crystal structures exist, but materials
tend to go to the structure that minimizes the energy; often this
can be understood in terms of the energy per atomic bond and
the equilibrium distance (at which a bond features the most
potential energy)
N=4 bonds
N=6 bonds per atom
Square lattice
Hexagonal lattice
© 2006 Markus J. Buehler, CEE/MIT
Deformation of metals: Example
Image removed for copyright reasons.
See: Fig. 6 at http://www.kuleuven.ac.be/bwk/materials/Teaching/master/wg02/l0310.htm.
Image removed for copyright reasons.
See: Fig. 4 at http://www.kuleuven.ac.be/bwk/materials/
Teaching/master/wg02/l0310.htm
http://www.kuleuven.ac.be/bwk/materials/Teaching/master/wg02/l0310.htm
Deformation of materials:
Flaws or cracks matter
“Macro”
Stress σ
Failure of materials initiates at cracks
Griffith, Irwine and others: Failure initiates at defects, such as cracks, or
grain boundaries with reduced traction, nano-voids
© 2006 Markus J. Buehler, CEE/MIT
Deformation of crystals
„
Deformation of a crystal is similar to pushing a sticky tape
across a surface:
F~ τ ⋅ L
“homogeneous shear”
F ≈ Fripple
“localized slip (ripple)”
Lcrit ≈
Fripple
τ
Beyond critical length L it is easer to have a localized
© 2006 Markus ripple…
J. Buehler, CEE/MIT
Theoretical shear strength
„
Perfect crystal: Deformation needs to be cooperative
movement of all atoms; the critical shear stress for this
mechanism was calculated by Frenkel (1926):
τ th
=
b G
G
≈
a
2π
30
Figure by MIT OCW.
„
Although this is an approximation, the shear strength measured
in experiment is much lower:
τ exp
„
„
�
G
=
10
,000...100,000,000
Difference explained by existence of dislocations
by Orowan, Polanyi and Taylor in 1934
Confirmed by experiments with whiskers
(dislocation free crystals)
�
Figure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Ductile materials are governed by the motion of dislocations: Introduction
�
�
�
�
Figure by MIT OCW.
Dislocations are the discrete entities that carry plastic (permanent) deformation; measured by “Burgers vector”
http://www.people.virginia.edu/~lz2n/mse209/Chapter7.pdf
© 2006 Markus J. Buehler, CEE/MIT
Animation: Dislocation motion
Animation online:
http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/a5_1_1.html
© 2006 Markus J. Buehler, CEE/MIT
Geometry of a dislocation (3D view)
Image removed for copyright reasons.
See: Fig. 2 at http://www.kuleuven.ac.be/bwk/materials/Teaching/master/wg02/l0310.htm
Summary of important concepts
„ Reviewed
some analysis techniques and basic MD
concepts
„ Modeling vs. Simulation for Molecular Dynamics
„ Metallic bonding: Basics and motivation for multi-body
interactions
„ Models for metallic bonding – EAM (=embedded atom
method)
(electron gas etc.)
„ Plasticity and Concept of dislocation nucleation and
motion; at a crack tip: Dislocations are responsible to
carry plasticity
© 2006 Markus J. Buehler, CEE/MIT
Additional references
http://web.mit.edu/mbuehler/www/
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Buehler, M.J., Large-scale hierarchical molecular modeling of nano-structured biological materials. Journal of Computational and Theoretical
Nanoscience, 2006. 3(5).
Buehler, M.J. and H. Gao, Large-scale atomistic modeling of dynamic fracture. Dynamic Fracture, ed. A. Shukla. 2006: World Scientific.
Buehler, M.J. and H. Gao, Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature, 2006. 439: p. 307-310.
Buehler, M.J., et al., The Computational Materials Design Facility (CMDF): A powerful framework for multiparadigm multi-scale simulations. Mat.
Res. Soc. Proceedings, 2006. 894: p. LL3.8.
R.King and M.J. Buehler, Atomistic modeling of elasticity and fracture of a (10,10) single wall carbon nanotube. Mat. Res. Soc. Proceedings,
2006. 924E: p. Z5.2.
Buehler, M.J. and W.A. Goddard, Proceedings of the "1st workshop on multi-paradigm multi-scale modeling in the Computational Materials
Design Facility (CMDF)". http://www.wag.caltech.edu/home/mbuehler/cmdf/CMDF_Proceedings.pdf, 2005.
Buehler, M.J., et al., The dynamical complexity of work-hardening: a large-scale molecular dynamics simulation. Acta Mechanica Sinica, 2005.
21(2): p. 103-111.
Buehler, M.J., et al. Constrained Grain Boundary Diffusion in Thin Copper Films. in Handbook of Theoretical and Computational Nanotechnology.
2005: American Scientific Publishers (ASP).
Buehler, M.J., F.F. Abraham, and H. Gao, Stress and energy flow field near a rapidly propagating mode I crack. Springer Lecture Notes in
Computational Science and Engineering, 2004. ISBN 3-540-21180-2: p. 143-156.
Buehler, M.J. and H. Gao, A mother-daughter-granddaughter mechanism of supersonic crack growth of shear dominated intersonic crack motion
along interfaces of dissimilar materials. Journal of the Chinese Institute of Engineers, 2004. 27(6): p. 763-769.
Buehler, M.J., A. Hartmaier, and H. Gao, Hierarchical multi-scale modelling of plasticity of submicron thin metal films. Modelling And Simulation
In Materials Science And Engineering, 2004. 12(4): p. S391-S413.
Buehler, M.J., Y. Kong, and H.J. Gao, Deformation mechanisms of very long single-wall carbon nanotubes subject to compressive loading.
Journal of Engineering Materials and Technology, 2004. 126(3): p. 245-249.
Buehler, M.J., H. Gao, and Y. Huang, Continuum and Atomistic Studies of the Near-Crack Field of a rapidly propagating crack in a Harmonic
Lattice. Theoretical and Applied Fracture Mechanics, 2004. 41: p. 21-42.
Buehler, M. and H. Gao, Computersimulation in der Materialforschung – Wie Großrechner zum Verständnis komplexer Materialphänomene
beitragen. Naturwissenschaftliche Rundschau, 2004. 57.
Buehler, M. and H. Gao, Biegen und Brechen im Supercomputer. Physik in unserer Zeit, 2004. 35(1): p. 30-37.
Buehler, M.J., et al., Atomic plasticity: description and analysis of a one-billion atom simulation of ductile materials failure. Computer Methods In
Applied Mechanics And Engineering, 2004. 193(48-51): p. 5257-5282.
Buehler, M.J., F.F. Abraham, and H. Gao, Hyperelasticity governs dynamic fracture at a critical length scale. Nature, 2003. 426: p. 141-146.
Buehler, M.J., A. Hartmaier, and H. Gao, Atomistic and Continuum Studies of Crack-Like Diffusion Wedges and Dislocations in Submicron Thin
Films. J. Mech. Phys. Solids, 2003. 51: p. 2105-2125.
Buehler, M.J., A. Hartmaier, and H.J. Gao, Atomistic and continuum studies of crack-like diffusion wedges and associated dislocation
mechanisms in thin films on substrates. Journal Of The Mechanics And Physics Of Solids, 2003. 51(11-12): p. 2105-2125.
Buehler, M.J. and H. Gao. "Ultra large scale atomistic simulations of dynamic fracture"; In: Handbook of Theoretical and Computational
Nanotechnology. 2006: American Scientific Publishers (ASP), ISBN:1-58883-042-X.