xxx

3.22 Mechanical properties of materials

Introduction to Interatomic Potentials 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 modelingDuctile versus brittle materials

BRITTLE DUCTILE

Glass Polymers Ice... Copper, Gold

Shear load

Figure by MIT OCW.

© 2006 Markus J. Buehler, CEE/MIT

Schematic of stress field (“forces”) around a single (static) crack

shear

tensile stress

shear

tensile stress

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

© 2006 Markus J. Buehler, CEE/MIT

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

The problem to solve

Figures by MIT OCW.

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

© 2006 Markus J. Buehler, CEE/MIT

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

vi(t), ai(t)

ri(t)

x y

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

xy

z

N particlesParticles with mass mi

Analysis methods

Last time, we discussed how to calculate:

N Temperature T =

2 KK =

1 m ∑v 2 j

Kinetic energy3 k N B ⋅ 2 j =1

Potential energy U = r U j ) Æ This lecture(

1 V 3

Pressure P = NkBT − ∑∑< rij dV

> i j <i drij

Kinetic contribution Force vector multiplied

Volume 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) r g ) = 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

Average over all

(rΩ

Volume of this shell (dr)

( (r g ) = < r N ± Δr ) >2 atoms ± Δr )ρ2

( 2r g )2πr dr = Number of particles lying in a spherical shell of radius r and thickness dr

© 2006 Markus J. Buehler, CEE/MIT

Radial distribution function

)( 2 rr ∆±Ω

considered volume

∆ r )( (r g ) = r N ±

∆ r 2

)ρ(r± Ω 2

Density

/ρ = V N

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

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

Image removed due to copyright reasons.

Screenshot of the radial distribution function Java applet.

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

Image removed for copyright reasons.

Screenshot of the radial distribution function Java applet.

Radial distribution function: Solid versus liquid versus gas

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

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

5

4

3

2

1

00 0.2 0.4 0.6 0.8

g(r)

distance/nm

Solid Argon

Liquid Argon

Liquid Ar(90 K)

Gaseous Ar(90 K)

Gaseous Ar(300 K)

Figure by MIT OCW.

Radial distribution function: Solid versus liquid

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

Image removed due to copyright reasons.

Screenshot of the radial distribution function Java applet.

Mean square displacement (MSD) function

Liquid Crystal

Relation to diffusion constant:

lim d Δ < r 2 >= 2dD d=2 2D

t→∞ dt d=3 3D

© 2006 Markus J. Buehler, CEE/MIT

Property calculation in MD

Time average of dynamical variable A(t)t t

< A >= lim1 '=

dt t A ' Time average of a dynamical t ∞ → t ∫ ( ') variable A(t)

t '=0

Nt1 ( Average over all time steps< A >= ∑ t A ) Nt in the trajectory (discrete)N t 1

Correlation function of two dynamical variables A(t) and B(t)N 1 Ni1( ( ) (< A (0) t B ) >= ∑ ∑ t B t A k + t )

N i =1 Ni k =1 i k i

© 2006 Markus J. Buehler, CEE/MIT

Overview: MD properties

© 2006 Markus J. Buehler, CEE/MIT

Velocity autocorrelation function

N 1 Ni1( ( ) (< v (0) t v ) >= ∑ ∑ t v t v k + t )N i = 1 Ni k = 1

i k i

•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):

1 t ' ∞ =

(D 0 = < v (0) t v ) > 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

')()0(3 1 '

0' 0 dtvD

t

t ∫ =

><= t v ∞ =

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 FForce F

Undeformed vs. deformed σ =(due to force)

A = cross-sectional area

FA

© 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 ⎛ iσ ij = ⎜⎜− ∑ uum α , j + 1 ∑

∂φ (r ) r ⋅ rj | rr αβ ⎟⎟

⎞

⎝Ω α α α ,i 2 βα ,a ≠β ∂ r r =

⎠,

Force Fi

x2

Atom βAtom βAtom βAtom β

x1 Atom αAtom αAtom αAtom α

Frαβrαβ

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

σ 1 ⎛ 1 ∑

∂φ ( r ) r ⋅ rj | rr αβ ⎟⎟

⎞ ij = ⎜⎜− ∑ um α , iu α , j 2 βαΩ ⎝ α

α + , , a ≠β ∂ r r

i =

⎠ F1F1 F2F2

1 32

r21r21 r23r23

σ ij = 11 ∑

∂φ ( r ) r i rr αβ

, , a ≠β ∂ r r ⋅ rj | =Ω 2 βα

Force between 2 particles:

F = − ∂φ

∂

( rr ) r

r Ω 2 1 21 2i σ 11 =

11 ( r F + r F 23 ) © 2006 Markus J. Buehler, CEE/MIT

MD properties: Classification

Structural – crystal structure, g(r), defects such as vacancies andinterstitials, 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

no p+ p+

p+

e -

e-

e-

e-

e-

e-

no

no no

no

p+ no

p+

p+

later in IM/S!) “Point” representation

V(t)

y x

r(t) 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 theforces and accelerations at each MD time integrationstep: Update to new positions!

Usually define interatomic potentials, that describethe energy of a set of atoms as a function of theircoordinates:

Utotal =Utotal (ri)

Simple approximation: Total energy is sum over theenergy of all pairs of atoms in the system

Utotal = ∑≠

12

i j U(rij)

© 2006 Markus J. Buehler, CEE/MIT

Pair interaction approximation

Utotal =12 U

i j ∑

≠

(rij)

2 5

43

1

2 5

43

1

All pair interactions of atom 1 with neighboring atoms 2..5

All pair interactions of atom 2 with neighboring atoms 1, 3..5

Double count bond 1-2therefore factor 1

2

© 2006 Markus J. Buehler, CEE/MIT

From electrons to atoms

Electrons

Core

Electrons

Core

Energy

Distance Radius

r

DistanceRadius

r

DistanceRadius

rr

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 theirvelocities/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…)

φ

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

Interaction

r

“repulsion”

“attraction”

Atomic scale (QM) orchemical property Forces by dφ/dr © 2006 Markus J. Buehler, CEE/MIT

Repulsion versus attraction

Repulsion: Overlap of electrons in same orbitals;according to Pauli exclusion principle this leads to highenergy structuresModel: Exponential term

Attraction: When chemical bond is formed, structure(bonded atoms) are in local energy minimum; breakingthe atomic bond costs energy – results in attractive force

Sum of repulsive and attractive term results in the typicalpotential energy shape:

= U U rep + Uattr

© 2006 Markus J. Buehler, CEE/MIT

Lennard-Jones potential

Attractive dV (r)12 6

4 ⎛⎜⎜ε

⎞⎟⎟. ⎤

⎥⎦

σ⎡⎢⎣

⎤σ− ⎡

r⎢⎣F=−weak ( )φ r ⎥⎦ d rr ⎠

Repulsive

r xFF i

i =

r

x1

x2

x1

x2

x1

x2

x1

x2F

= ⎝

© 2006 Markus J. Buehler, CEE/MIT

Lennard-Jones potential: Properties

12 6

4 ⎛⎜⎜ε

⎞⎟⎟. ⎤

⎥⎦

σ⎡⎢⎣

⎤σ− ⎡

r⎢⎣( )weakφ r = ⎥⎦r⎝ ⎠

ε : Well depth (energy per bond)

σ: Potential vanishes

Equilibrium distance between atoms D and maximum force

ε⋅394 .2 6σ 2 =D Fmax,LJ =σ

© 2006 Markus J. Buehler, CEE/MIT

Pair potentials

ϕ(∑i =φ rij ) j ..1= Nneigh

6 5

j=1

2 3

4

…

cutrcutr

i

Lennard-Jones 12:6

σrij

⎞⎟⎟⎠

12⎡ ⎤⎛⎜⎜⎝

⎞⎟⎟

⎛⎜⎜

σ⎢⎢⎣

ε4 ⎥⎥⎦

ϕ(rij )= −rij⎝ ⎠

Morse

Reasonable model for noble gas Ar (FCC in 3D)

)( ijrϕ

© 2006 Markus J. Buehler, CEE/MIT

6

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 FForce Fthe displacement for each force level

From this data, one can calculate the stress versus displacement

Displacement is typically measured in strain:

ε = = σ =FA

− 0 Δ

Plot stress versus strain: Hooke’s law σ = E ε

LL

© 2006 Markus J. Buehler, CEE/MIT

LL L

Stress versus strain properties: 2D

5

4

3

2

1

0 0

00

5

4

3

2

1

0 0

00

[ ] (x) S [ ] (y)

© 2006 Markus J. Buehler, CEE/MIT

0.1 0.2

0.1 0.2

0.1 0.2

20

40

60

80

Strain xx

Stre

ss

Stre

ss

Strain xx

0.1 0.2

20

40

60

80

Strain yy

Strain yy

Tangent modulus Exx

Strain in 100 -direction train in 010 -direction

Tangent modulus Eyy

LJ Solid Harmonic Solid Poisson ratio LJ solid

Figure by MIT OCW.

Stress versus strain properties: 3D

7

6

5

4

3

2

1

10

1.1 1.2 1.3 1.7 1.5

LJ

100

111

110

Figure by MIT OCW.

© 2006 Markus J. Buehler, CEE/MIT

© 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 6 ⎤

φ (r) = 4ε ⎢⎡⎜⎛σ

⎟⎞ − ⎜

⎛σ⎟⎞

⎥ ⎝ r ⎠ ⎥ Example: ⎢⎣⎝ r ⎠ ⎦

2∂φ (r)k = ∂ 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

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 referenceshttp://web.mit.edu/mbuehler/www/

1. Buehler, M.J., Large-scale hierarchical molecular modeling of nano-structured biological materials. Journal of Computational and Theoretical Nanoscience, 2006. 3(5).

2. Buehler, M.J. and H. Gao, Large-scale atomistic modeling of dynamic fracture. Dynamic Fracture, ed. A. Shukla. 2006: World Scientific. 3. Buehler, M.J. and H. Gao, Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature, 2006. 439: p. 307-310. 4. 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. 5. 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. 6. 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. 7. 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. 8. 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). 9. 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. 10. 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. 11. 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. 12. 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. 13. 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. 14. Buehler, M. and H. Gao, Computersimulation in der Materialforschung – Wie Großrechner zum Verständnis komplexer Materialphänomene

beitragen. Naturwissenschaftliche Rundschau, 2004. 57. 15. Buehler, M. and H. Gao, Biegen und Brechen im Supercomputer. Physik in unserer Zeit, 2004. 35(1): p. 30-37. 16. 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. 17. Buehler, M.J., F.F. Abraham, and H. Gao, Hyperelasticity governs dynamic fracture at a critical length scale. Nature, 2003. 426: p. 141-146. 18. 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. 19. 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. 20. 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.