3.22 Mechanical properties of materials
Introduction to Interatomic Potentials xxx
Lecture 2/4
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
Property calculation (part II)
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
„
Lecture 4: Reactive Potentials
New frontier in research: Modeling chemistry with molecular dynamics
using reactive potentials
© 2006 Markus J. Buehler, CEE/MIT
Motivation for atomistic modeling
Ductile versus brittle materials
BRITTLE
Glass Polymers
Ice...
DUCTILE
Copper, Gold
Shear load
Figure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Schematic of stress field (“forces”) around a single (static) crack
tensile stress
shear
¾ The stress field around a crack is complex, with regions of dominating
tensile stress (crack opening) and shear stress (dislocation nucleation)
© 2006 Markus J. Buehler, CEE/MIT
The problem to solve
„
„
In atomistic simulations, the goal is to understand and model the motion of
each atom in the material
The collective behavior of the atoms allows to understand how the material
undergoes deformation (metals: dislocations), phase changes or other
phenomena, providing links between the atomic scale to meso/macro
phenomena
Figures by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Molecular dynamics
„
MD generates the dynamical trajectories of a system of N
particles by integrating Newton’s equations of motion, with
suitable initial and boundary conditions, and proper
interatomic potentials
Particles with mass mi
N particles
ri(t)
z
vi(t), ai(t)
y
x
© 2006 Markus J. Buehler, CEE/MIT
Analysis methods
Last time, we discussed how to calculate:
„
Temperature
„
Potential energy
„
Pressure
2 K
T=
3 N ⋅ kB
U = U (
rj )
1
P = Nk BT −
3V
1 N 2
K = m∑ v j
2 j =1
Kinetic
energy
Æ This lecture
dV
< rij
>
∑∑
drij
i j <i
Kinetic
contribution
Volume
Force vector multiplied
by distance vector
Time average
Need other measures for physical and thermodynamic
properties
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function
The radial distribution function is defined as
Density of atoms (volume)
g (r ) = ρ (r ) / ρ
Local density
Provides information about the density of atoms at a given
radius r; ρ(r) is the local density of atoms
< N (r ± Δr2 ) >
g (r ) =
Ω(r ± Δr2 ) ρ
Average over all
atoms
Volume of this shell (dr)
g (r )2πr 2 dr = Number of particles lying in a spherical shell
of radius r and thickness dr
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function
considered volume
N (r ± ∆2r )
g (r ) =
Ω(r
± ∆2r ) ρ
Density
Ω(r ± ∆2r )
ρ = N /V
Note: RDF can be measured experimentally using neutron-scattering
techniques.
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function
Reference
atom
Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.
http://www.ccr.buffalo.edu/etomica/app/modules/sites/Ljmd/Background1.html
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function: Solid versus liquid
Image removed due to copyright reasons.
Screenshot of the radial distribution function Java applet.
Interpretation: A peak indicates a particularly
favored separation distance for the neighbors to a given particle
Thus: RDF reveals details about the atomic structure of the
system being simulated
Java applet:
http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function: JAVA applet
Image removed for copyright reasons.
Screenshot of the radial distribution function Java applet.��
Java applet:
http://physche
m.ox.ac.uk/~r
kt/lectures/liqs
olns/liquids.ht
ml
© 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
© 2006 Markus J. Buehler, CEE/MIT
Radial distribution function: Solid versus liquid
Image removed due to copyright reasons.
Screenshot of the radial distribution function Java applet.
Interpretation: A peak indicates a particularly
favored separation distance for the neighbors to a given particle
Thus: RDF reveals details about the atomic structure of the
system being simulated
Java applet:
http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html
© 2006 Markus J. Buehler, CEE/MIT
Mean square displacement (MSD) function
Liquid
Crystal
Relation to diffusion constant:
d
lim < Δr 2 >= 2dD
t→∞ dt
d=2 2D
d=3 3D
© 2006 Markus J. Buehler, CEE/MIT
Property calculation in MD
Time average of dynamical variable A(t)
t '=t
1
< A >= lim ∫ A(t ')dt '
t →∞ t
t '=0
Time average of a dynamical
variable A(t)
1
< A >=
Nt
Average over all time steps
Nt in the trajectory (discrete)
Nt
∑ A(t )
1
Correlation function of two dynamical variables A(t) and B(t)
1
< A(0) B(t ) >=
N
1 Ni
Ai (t k ) Bi (t k + t )
∑
∑
i =1 N i k =1
N
© 2006 Markus J. Buehler, CEE/MIT
Overview: MD properties
© 2006 Markus J. Buehler, CEE/MIT
Velocity autocorrelation function
1
< v(0)v(t ) >=
N
1
vi (t k )vi (
t k + t )
∑
∑
i =1 N i k =1
N
Ni
•The velocity autocorrelation function gives information about the atomic
motions of particles in the system
• Since it is a correlation of the particle velocity at one time with the
velocity of the same particle at another time, the information refers to how
a particle moves in the system, such as diffusion
Diffusion coeffecient (see e.g. Frenkel and Smit):
t '=
∞
1
D0 = ∫ < v(0)v(
t ) >dt '
3 t '=0
Note: Belongs to a the Green-Kubo relations can provide links between
correlation functions and material transport coefficients, such as thermal
conductivity
© 2006 Markus J. Buehler, CEE/MIT
Velocity autocorrelation function (VAF)
„
Liquid or gas (weak molecular interactions):
Magnitude reduces gradually under the influence of weak forces:
Velocity decorrelates with time, which is the same as saying the atom
'forgets' what its initial velocity was.
Then: VAF plot is a simple exponential decay, revealing the presence
of weak forces slowly destroying the velocity correlation. Such a result
is typical of the molecules in a gas.
„
Solid (strong molecular interactions):
Atomic motion is an oscillation, vibrating backwards and forwards, reversing their velocity at the end of each oscillation. Then: VAF corresponds to a function that oscillates strongly from positive to negative values and back again. The oscillations decay in time.
This leads to a function resembling a damped harmonic motion. © 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.
http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm
© 2006 Markus J. Buehler, CEE/MIT
The concept of stress
Force F
F
Undeformed vs. deformed
(due to force)
F
σ =
A
A = cross-sectional area
© 2006 Markus J. Buehler, CEE/MIT
Atomic stress tensor: Cauchy stress
„
„
„
Important:
How to relate the continuum stress with atomistic stress
Typically continuum variables represent time-/space averaged
microscopic quantities at equilibrium
Difference: Continuum properties are valid at a specific
material point; this is not true for atomistic quantities (discrete
nature of atomic microstructure)
Discrete fields u (x)
Displacement
only defined
at atomic
site
Continuous fields ui(x)
© 2006 Markus J. Buehler, CEE/MIT
Atomic stress tensor: Virial stress
Virial stress:
Contribution by atoms
moving through control
volume
1 ⎛
1
∂φ (r ) ri
σ ij = ⎜⎜ − ∑ mα uα ,i uα , j +
⋅ rj |r = rαβ
∑
2 α , β ,a ≠ β ∂r r
Ω⎝ α
⎞
⎟
⎟
⎠
Force Fi
x2
F
x1
rαβ
Atom β
Atom α
D.H. Tsai. Virial theorem and stress calculation in molecular-dynamics. J. of Chemical Physics, 70(3):1375–1382, 1979.
Min Zhou, A new look at the atomic level virial stress: on continuum-molecular system equivalence,
Royal Society of London Proceedings Series A, vol. 459, Issue 2037, pp.2347-2392 (2003)
Jonathan Zimmerman et al., Calculation of stress in atomistic simulation, MSMSE, Vol. 12, pp. S319-S332 (2004) and references in
those articles by Yip, Cheung et al.
© 2006 Markus J. Buehler, CEE/MIT
Virial stress versus Cauchy-stress
1⎛
1
∂φ (r ) ri
σ ij = ⎜⎜ − ∑ mα uα ,i uα , j +
⋅ rj |r = rαβ
∑
2 α , β ,a ≠ β ∂r r
Ω⎝ α
1
F1
F2
2
r21
⎞
⎟
⎟
⎠
3
r23
1 1
∂φ (r ) ri
σ ij =
⋅ rj |r = rαβ
∑
Ω 2 α , β ,a ≠ β ∂r r
Force between 2 particles:
∂φ (r ) ri
F =−
∂r r
1 1
(F1r21 + F2 r23 )
σ 11 =
Ω2
© 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
Limitations of MD: Electronic properties
„
There are properties which classical MD cannot calculate
because electrons are involved.
„
To treat electrons properly one needs quantum mechanics.
In addition to electronic properties, optical and magnetic
properties also require quantum mechanical (first principles
or ab initio) treatments.
Such methods will be discussed in the quantum simulation
part of the lectures.
© 2006 Markus J. Buehler, CEE/MIT
Atomic scale
„
„
Atoms are composed of electrons, protons, and neutrons.
Electron and protons are negative and positive charges of
the same magnitude, 1.6 × 10-19 Coulombs
Chemical bonds between atoms by interactions of the
electrons of different atoms
(see QM part
later in IM/S!)
“Point” representation
e-
ep+ o
n
+
p
+
o
p
no no n+
no + p
p
p+ no
e-
V(t)
e-
r(t)
y
x
ee-
a(t)
Figure by MIT OCW.
Figure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Atomic interactions
„
Primary bonds (“strong”)
Ionic,
Covalent,
Metallic (high melting point, 1000-5000K)
„
Secondary bonds (“weak”)
Van der Waals,
Hydrogen bonds
(melting point 100-500K)
„
Ionic: Non-directional
Covalent: Directional (angles, torsions)
Metallic: Non-directional
„
„
© 2006 Markus J. Buehler, CEE/MIT
Models for atomic interactions
„
Atom-atom interactions are necessary to compute the
forces and accelerations at each MD time integration
step: Update to new positions!
„
Usually define interatomic potentials, that describe
the energy of a set of atoms as a function of their
coordinates:
U total =
U total (
ri
)
„
Simple approximation: Total energy is sum over the
energy of all pairs of atoms in the system
U total =
12 ∑ U
(
rij
)
i ≠
j
© 2006 Markus J. Buehler, CEE/MIT
Pair interaction approximation
U total =
12
∑ U (
rij
)
i ≠
j
1
5
2
4
3
1
5
2
3
All pair interactions of atom 1 with
neighboring atoms 2..5
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
From electrons to atoms Electrons
Energy
Core
r
Distance
Radius
r
Governed by laws of quantum mechanics: Numerical solution by
Density Functional Theory (DFT), for example
© 2006 Markus J. Buehler, CEE/MIT
The interatomic potential
„
„
„
The fundamental input into molecular simulations, in addition to structural
information (position of atoms, type of atoms and their
velocities/accelerations) is provided by definition of the interaction potential
(equiv. terms often used by chemists is “force field”)
MD is very general due to its formulation, but hard to find a “good” potential
(extensive debate still ongoing, choice depends very strongly on the
application)
Popular: Semi-empirical or empirical (fit of carefully chosen mathematical
functions to reproduce the potential energy surface…)
Int
era
ctio
n
φ
“repulsion”
r
“attraction”
Atomic scale (QM) or
chemical property
Forces by dφ/dr
r
© 2006 Markus J. Buehler, CEE/MIT
Repulsion versus attraction
„
Repulsion: Overlap of electrons in same orbitals;
according to Pauli exclusion principle this leads to high
energy structures
Model: Exponential term
„
Attraction: When chemical bond is formed, structure
(bonded atoms) are in local energy minimum; breaking
the atomic bond costs energy – results in attractive force
„
Sum of repulsive and attractive term results in the typical
potential energy shape:
U = U rep + U attr
© 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
Lennard-Jones potential: Properties
⎛ ⎡σ ⎤12
⎡
σ
⎤
6 ⎞
φ
weak (r
) =
4ε
⎜ .
⎢ ⎥ − ⎢ ⎥ ⎟
⎜
⎣
r
⎦
⎟
r
⎣
⎦
⎝
⎠
ε: Well depth (energy per bond)
σ: Potential vanishes
Equilibrium distance between atoms D and maximum force
σ 2 =
D
6
Fmax,LJ =
2.394 ⋅ ε
σ
© 2006 Markus J. Buehler, CEE/MIT
Pair potentials
φ
i =
6
…
∑
ϕ
(
r )
j =
1.. N neigh
ij
5
Lennard-Jones 12:6
i
j=1
2
⎡
⎛ σ
⎞12 ⎛ σ ⎞ 6 ⎤
ϕ (rij )
=
4
ε ⎢⎜⎜ ⎟⎟ −
⎜⎜
⎟⎟
⎥
⎢⎝ rij
⎠
⎝
rij
⎠
⎥
⎦
⎣
4
3
rcut
Morse
ϕ (rij )
Reasonable model for noble
gas Ar (FCC in 3D)
© 2006 Markus J. Buehler, CEE/MIT
Numerical implementation of neighbor search:
Reduction of N2 problem to N problem
• Need nested loop to search for neighbors of atom i: Computational disaster
• Concept: Divide into computational cells (“bins”, “containers”, etc.)
• Cell radius R>Rcut (cutoff)
• Search for neighbors within cell atom
belongs to and neighboring cells
(8+1 in 2D)
• Most classical MD potentials/force fields
have finite range interactions
• Other approaches: Neighbor lists
• Bin re-distribution only necessary every
20..30 integration steps (parameter)
© 2006 Markus J. Buehler, CEE/MIT
Elastic deformation
Mechanical properties are often determined by performing
“tensile tests”, i.e. pulling on a specimen and measuring Force F
the displacement for each force level
From this data, one can calculate the stress versus displacement
Displacement is typically measured in strain:
L − L0 ΔL
=
ε =
L
L
F
σ =
A
Plot stress versus strain: Hooke’s law
σ = Eε
© 2006 Markus J. Buehler, CEE/MIT
Stress versus strain properties: 2D
Strain in [010] (y) -direction
5
5
4
4
Stress �
Stress �
Strain in [100] (x) -direction
3
2
1
0
0
3
2
1
0.1
Strain �xx
0.2
0
0
0.1
Strain �yy
0.2
LJ Solid
Harmonic Solid
Poisson ratio LJ solid
Tangent modulus Exx
Tangent modulus Eyy
80
80
60
60
40
40
20
20
0
0
0.1
Strain �xx
0.2
0
0
0.1
Strain �yy
0.2
Figure by MIT OCW.
© 2006 Markus J. Buehler, CEE/MIT
Stress versus strain properties: 3D
LJ
7
6
5
110
10
0
� 4
3
111
2
1
0
1
1.1
1.2
�
1.3
1.7
1.5
Figure by MIT OCW.
© 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
Summary
„
Discussed additional analysis techniques: “How to extract useful
information from MD results”
Velocity autocorrelation function
Atomic stress
Radial distribution function …
„
These are useful since they provide quantitative information about
molecular structure in the simulation; e.g. during phase
transformations, how atoms diffuse, elastic (mechanical) properties …
„
Discussed some “simple” interatomic potentials that describe the
atomic interactions; “condensing out” electronic degrees of freedom
„
Elastic properties: Calculate response to mechanical load based on
“virial stress”
„
Briefly introduced the “training” of potentials – homework assignment
© 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.