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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008

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1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation

Part II - lecture 3

Atomistic and molecular methods

Content overviewLectures 2-10February/March

Lectures 11-19March/April

Lectures 20-27April/May

I. Continuum methods1. Discrete modeling of simple physical systems:

Equilibrium, Dynamic, Eigenvalue problems2. Continuum modeling approaches, Weighted residual (Galerkin) methods,

Variational formulations3. Linear elasticity: Review of basic equations,

Weak formulation: the principle of virtual work, Numerical discretization: the finite element method

II. Atomistic and molecular methods1. Introduction to molecular dynamics2. Basic statistical mechanics, molecular dynamics, Monte Carlo3. Interatomic potentials4. Visualization, examples5. Thermodynamics as bridge between the scales6. Mechanical properties – how things fail7. Multi-scale modeling8. Biological systems (simulation in biophysics) – how proteins work and

how to model them

III. Quantum mechanical methods1. It’s A Quantum World: The Theory of Quantum Mechanics2. Quantum Mechanics: Practice Makes Perfect3. The Many-Body Problem: From Many-Body to Single-Particle4. Quantum modeling of materials5. From Atoms to Solids6. Basic properties of materials7. Advanced properties of materials8. What else can we do?

Additional Reading

Books:

Allen and Tildesley: “Computer simulation of liquids”D. C. Rapaport (1996): “The Art of Molecular Dynamics Simulation”D. Frenkel, B. Smit (2001): “Understanding Molecular Simulation”J.M. Haile (Wiley, 1992), “Molecular dynamics simulation”

Additional lecture notes and publications (see MIT Server website)

Overview: Material covered

Lecture 1: Introduction to atomistic modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

Lecture 2: Basic statistical mechanics (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function, solution techniques: Monte Carlo and molecular dynamics)

Lecture 3: Basic molecular dynamics (advanced property calculation, chemical interactions)

II. Atomistic and molecular methods

Lecture 3: Basic molecular dynamicsOutline:1. Review lectures 1&2

1.1 Continuum-atomistic 1.2 Basic statistical mechanics

2. Advanced property calculation 2.1 Radial distribution function (RDF)2.2 Velocity autocorrelation function (VAF)2.3 Cauchy stress tensor – virial stress (VS)2.4 Closure – property calculation

3. Introduction: Chemical interactions4.

Goal of today’s lecture: Review of previous lectures to be up to speed for part IILearn how to characterize whether a system is a solid, liquid, or a gas, and how to determine what atomistic crystal structure it hasHow to calculate diffusivity using an alternative approach, how to calculate viscosity, how to calculate thermal conductivity, and how to calculate the stress tensor inside a materialLearn how to model chemical bonds in a material

1. Review lectures 1&2

1.1 Continuum-atomistic

Atomistic versus continuum viewpointContinuum:

Effective behavior (“constitutive laws”, coefficients, parameters), e.g. differential equationsCan be applied at “any” scale: Same theory for earth, microdevices, computer chip…Assume no underlying discreteness of matter

“Partial differential equations”

Atomistic viewpoint:Explicitly consider discrete atomistic structureSolve for atomic trajectories and infer from these about material properties & behaviorFeatures internal length scales (atomic distance)

“Many-particle system with statistical properties”

∂∂

Diffusion: Phenomenological description

Ink dotTissue

∂∂ 2nd Fick law

Image removed due to copyright restrictions. Please see http://www.uic.edu/classes/phys/phys450/MARKO/dif.gif

Continuum model: Empirical parameters

Continuum model requires parameter that describes microscopic processes inside the material

Need experimental measurements to calibrate

Microscopicmechanisms

Continuummodel

Length scale

Time scale“Empirical”or experimentalparameterfeeding

Atomistic model of diffusionDiffusion at macroscale (change of concentrations): result of microscopic processes, random motion of particles

Atomistic model provides an alternative way to describe diffusion

Enables us to directly calculate the diffusion constant from thetrajectory of atoms

Follow trajectory of atoms and calculate how fast atoms leave their initial position

txpDΔΔ

Follow this quantity overtime

Molecular dynamics: A “bold” idea

Follow trajectories of atoms

( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii

“Verlet central difference method”

Positions at t0

Accelerationsat t0

Positions at t0-Δt

mfa ii /=

vi(t), ai(t)

N particlesParticles with mass mi

Typical MD simulation procedure

Set particlepositions

Assign particlevelocities

Calculate forceon each particle

Move particles bytimestep Dt

Save current positions and

velocities

Reachedmax. number of

timesteps?

Stop simulation Analyze dataprint results

Example molecular dynamics

Average square of displacement of all particles

( )22 )0()(1)( ∑ =−=Δi

ii trtrN

Particles Trajectories

Mean Square Displacement function

Courtesy of the Center for Polymer Studies at Boston University. Used with permission.

Calculation of diffusion coefficient

1D=1, 2D=2, 3D=3

slope = D

( )22 )0()(1)( ∑ =−=Δi

ii trtrN

Position of atom i at time t

Position of atom i at time t=0

( ))(lim21 2 tr

∞→

Einstein equation

SummaryMolecular dynamics provides a powerful approach to relate the diffusion constant that appears in continuum models to atomistictrajectories

Outlines multi-scale approach: Feed parameters from atomistic simulations to continuum models

Continuummodel

Length scale

Time scale“Empirical”or experimentalparameterfeeding

Example: MD simulation results

liquid solid

Courtesy of Sid Yip. Used with permission.

Atomistic trajectory – information about material state

Courtesy of Sid Yip. Used with permission.

JAVA applet

http://polymer.bu.edu/java/java/LJ/index.html

Experimentally one measured the same value of D at a slightly different temperature.

Both simulation and experiment were carried out at the same density, but the experiment was conducted at 90K whereas the simulation temperature was 94.4 K.

This is therefore an example of accurate prediction of diffusion in a simple liquid.

Rahmann, 1960

Mean squared displacement function obtained for liquid argon bymolecular dynamics simulation. The curve is plotted as the meanof 64 simulations; data with the highest deviations are shown as points.

Figure by MIT OpenCourseWare. Adapted from Rahman, A. “Correlationsin the Motion of Atoms in Liquid Argon.” Physical Review 136 (October 19, 1964): A405-A411.

1.0 2.0 3.00

D = 2.43 x 10-5 cm2 sec-1Mea

Time in 10-12 sec

1.2 Basic statistical mechanics

How to calculate properties from atomistic simulation

Property calculation: IntroductionHave:

“microscopic information”

Want: Thermodynamical properties (temperature, pressure, stress tensor..)Material state (gas, liquid, solid) and structure, e.g. BCC, FCC, ..…

)(),(),( txtxtxatoms ∀

Macroscopic vs. microscopic states

NVPT ,,,

Macroscopic state is represented by many microscopic configurations

)(),(),( txtxtx

Microscopic states

Macroscopic state

Microscopic states

Microscopic states characterized by pr,

{ } { }iii xmpxr == , Ni ..1=

Hamiltonian (sum of potential and kinetic energy, total energy)

)()(),( pKrUprH +=

i rrU..1

)()( φ ∑=

Ensembles: Set of microscopic configurations that realize macroscopic states

Result of thermodynamical constraints, e.g. temperature, pressure…

MicrocanonicalCanonicalIsobaric-isothermalGrand canonical

NpTμTV

Heat bath (constant temperature)Large system

NVT = canonical ensembleSmall system

Physical realization of canonical ensemble

Why one can not use a single microscopic state to calculate macroscopic properties

)(131)(

2 tvmNk

Which to pick?

Specific (individual) microscopic states are insufficient to relate to macroscopic properties

How to calculate the macroscopic ensemble average of a quantity A from microscopic states

),( prA Property due to specific microstate

drdprprpAAp r∫ ∫>=< ),(),( ρ

• Ensemble average, obtained by integral over all microscopic states

• Proper weight - depends on ensemble ),( rpρ

2131)( Temperature

How to solve…

Probability density distribution

Virtually impossible to carry out analytically

Must know all possible configurations

Therefore: Require numerical simulation

Molecular dynamics OR Monte Carlo

Monte Carlo schemeConcept: Find simpler way to solve the integral

Use idea of “random walk” to step through relevant microscopic states and thereby create proper weighting (visit states with higher probability density more often)

Monte Carlo schemes: Many applications (beyond what is discussed here; general method to solve complex integrations)

Monte Carlo scheme to calculate ensemble average

Similar method can be used to apply to integrate the ensemble averageNeed more complex iteration scheme

E.g. Metropolis-Hastings algorithm

drdprprpAAp r∫ ∫>=< ),(),( ρ ∑><

A 1Want:

Metropolis-Hastings AlgorithmConcept: Generate set of random microscopic configurationsAccept or reject with certain scheme

Images removed due to copyright restrictions.Please see Fig. 2.7 in Buehler, Markus J. Atomistic Modeling of Materials Failure.New York, NY: Springer, 2008.

Metropolis-Hastings Algorithm: NVT

Images removed due to copyright restrictions.Please see Fig. 2.8 in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, 2008.

Arrhenius law - explanation

)()( AHBH −

iteration

⎥⎦

⎤⎢⎣

⎡ −−<

TkAHBHp

)()(exp

attempt frequency ⎥

⎤⎢⎣

⎡ −−

TkAHBH

)()(exp

)()( AHBH −

E.g. = 0.8 most choices for pwill be below, i.e. higherchance for acceptance

0.8 low barrier

high barrier0.1

)()( AHBH >E

)()( AHBH −

radius

Summary: MC scheme

drdprprpAAp r∫ ∫>=< ),(),( ρ ∑

1Have achieved:

Note:• Do not need forces, only energies• Only valid for equilibrium processes

Molecular dynamics

Ergodic hypothesis:

Ensemble (statistical) average = time average

All microstates are sampled with appropriate probability density over long time scales

∑∑==

=>=<>=<tA Nit

TimeEnsNiA

AAiAN ..1..1

)(1)(1

MD versus MC

Obtain ensemble average by random stepping schemeAcceptance/rejection algorithm leads to proper distribution of microscopic statesNo dynamical information about system behavior (only equilibrium processes) – no “real” time

Obtain ensemble average by dynamical solution of equation, that is, time historyTime average equals ensemble average (Ergodichypothesis)Dynamical information about system obtained

Monte Carlo Molecular Dynamics

∑∑==

=>=<>=<tA Nit

TimeEnsNiA

AAiAN ..1..1

)(1)(1

Property calculation: Temperature and pressure

Temperature

Pressure( )>⋅−<

Ω= ∑

iiiii rfvmP

Distance vector multipliedby force vector (scalar product)

Volume

Kinetic contribution

><= ∑=

iii vvv ⋅=2

>⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

=< ∑ ∑ ∑= = ≠=

=3..1 ..1 ,..1,

,, |)(21

i N aNrri

rrvvmP

α ββαααα αβ

2. Advanced property calculation

Structures – distinguish crystal structures, liquids, gas

Calculate properties such as stress, thermal conductivity, shear viscosity…

2.1 Radial distribution function (RDF)

MD modeling of crystals – solid, liquid, gas phase

Crystals: Regular, ordered structure

The corresponding particle motions are small-amplitude vibrations about the lattice site, diffusive movements over a local region, and long free flights interrupted by a collision every now and then.

Liquids: Particles follow Brownian motion (collisions)

Gas: Very long free paths

J. A. Barker and D. Henderson, Scientific American, 1981

Figure by MIT OpenCourseWare. After J. A. Barker and D. Henderson.

How to characterize material state (solid, liquid, gas)

Application: Simulate phase transformation (melting)

http://www.t2i2edu.com/WebMovie/1Chap1_files/image002.jpg

Solid State- Ordered, dense.- Has definite shape and volume.- Very slightly compressible.

Liquid State- Disordered; usually less dense than a solid.- Definite volume; takes shape of its container.- Slightly compressible.

Gas State- Disordered; much less dense than solids or liquids.- Indefinite shape and volume.- Highly compressible.

Figure by MIT OpenCourseWare.

How to characterize material state (solid, liquid, gas)

IdeaMeasure distance of particles to their neighborsAverage over large number of particlesAverage over time (MD) or iterations (MC)

Regular spacing

Neighboring particles found at characteristic distances

Irregular spacing

Neighboring particles found at approximate distances (smooth variation)

More irregular spacing

More random distances, less defined

Reference atom

Formal approach: Radial distribution function (RDF)

ρρ /)()( rrg =

Ratio of density of atoms at distance r (in control area dr) by overall density

= relative density of atoms as function of radius

Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.

http://rheneas.eng.buffalo.edu/wiki/LennardJones:Background:Lennard_Jones_Radial_Distribution_Function

Formal approach: Radial distribution function (RDF)

ρρ /)()( rrg =

The radial distribution function is defined as

Provides information about the density of atoms at a given radius r; ρ(r) is the local density of atoms

)()()(

±Ω>±<

=drrrg 22)( π Number of particles that lie in a spherical shellof radius r and thickness dr

Local density

Density of atoms (volume)

Volume of this shell (dr)

Number of atoms in the interval 2rr Δ±

Radial distribution function

ρ)()()(

±Ω±

VN /=ρDensity

Note: RDF can be measured experimentally using x-ray or neutron-scattering techniques

)( 2rr Δ±Ω considered volume

Radial distribution function: Solid versus liquid

Interpretation: A peak indicates a particularlyfavored separation distance for the neighbors to a given particleThus, RDF reveals details about the atomic structure of the system being simulatedJava applet:http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html

solid liquid

1.2 1.4Distance

1.6 1.8 2 2.2 2.4 10.80

1.2 1.4Distance

1.6 1.8 2 2.2 2.82.4 2.6

Radial distribution function: JAVA applet

Java applet:http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html

Images removed due to copyright restrictions.Screenshots of the radial distribution 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

00 0.2 0.4 0.6 0.8

distance/nm

00 0.2 0.4 0.6 0.8

distance/nm

Solid Argon

Liquid Argon

Liquid Ar(90 K)

Gaseous Ar(90 K)

Gaseous Ar(300 K)

Notes: Radial distribution function

Pair correlation function (consider only pairs of atoms)Provides structural informationCan provide information about dynamical change of structure, butnot about transport properties (how fast atoms move)

Additional comments:

Describes how - on average - atoms in a system are radially packed around each other Particularly effective way of describing the structure of disordered molecular systems (liquids) In liquids there is continual movement of the atoms and a singlesnapshot of the system shows only the instantaneous disorder it is extremely useful to be able to deal with the average structure

2.2 Velocity autocorrelation function (VAF)

Autocorrelation functions

Concept: Calculate how property at one point in time is related to property at a later point in time

Time-dependent quantity

Can represent transport information of the system, e.g. viscosity, thermal conductivity, diffusivity

Autocorrelation functions belong to a class of functions called the Green-Kubo relations

A: Dynamical properties

time t0 time t0+t

iii ttAtA

100 )()(1)()0(

• The velocity autocorrelation function gives information about the atomic motions of particles in the system: Relation of velocity at one point in time with velocity at another point in time

• 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 (by time integration)

+⋅>=<N

iii ttvtv

100 )()(1)()0(

Velocity autocorrelation function (VAF)

scalar product)()( 00 ttvtv ii +

How to calculate the VAF

iii tvtv

1000 )()(1)()0(

Δ+>=Δ+=<N

iii ttvtv

Ntttvv

1000 )()(1)()0(

Δ+>=Δ+=<N

iii ttvtv

Ntttvv

1000 )2()(1)2()0(

Δ+>=Δ+=<N

iii tntvtv

Ntnttvv

1000 )()(1)()0(

Harmonic oscillator (“crystal”)v

iii tvv

)()0(1)()0(

1t 2t 3t 4t0t

Velocity autocorrelation function

http://www.eng.buffalo.edu/~kofke/ce530/Lectures/Lecture12/sld010.htm

Δ+>=Δ+=<N

iii tntvtv

Ntnttvv

1000 )()(1)()0(

Damping due to dissipation - solid

Courtesy of the Department of Chemical and Biological Engineering of the University at Buffalo. Used with permission.

Velocity autocorrelation function: Solid, liquid, ideal gas

Liquid

Dense gas

Ideal gas

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.

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.

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.

VAF may be Fourier transformed to project out the underlying frequencies of the molecular processes: closely related to the infra-red spectrum of the system, which is also concerned with vibration on the molecular scale.

VAF used to calculate diffusion constant

')()0(31 '

dttvvDt

t∫∞=

Diffusion coefficient (see additional notes available on MIT Server “Additional notes re. velocity autocorrelation function”):

Diffusion coefficient can be obtained from both VAF and the MSD

Green-Kubo relations

')'()0(1

dttTUk t

xyxyB∫∞

><= σση

Diffusivity

Shear viscosity

Thermal conductivity

')()0(31 '

dttvvDt

t∫∞=

')'()0(1

0'2 dttqq

TUk tB∫∞

⎟⎠

⎞⎜⎝

⎛ += ∑ ∑= =

iijii rUvm

2 )(21

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+−Ω

= ∑ ∑= ≠=

yxxy rrr

..1 ,..1,,, |)(

DiffusivityTime integral over VAF

')()0(31 '

dttvvDt

t∫∞=

)()( 00 ttvtv ii +⋅=

Velocity vector of particle i at t0

Velocity vector of particle i at t0+t

Shear viscosity

')'()0(1

dttTUk t

xyxyB∫∞

><= σση

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+−Ω

= ∑ ∑= ≠=

yxxy rrr

..1 ,..1,,, |)(

Time integral over autocorrelation function of shear stresses

shear stresses

Definition of shear stresses:

Force –Fi, x-componentvolume

mass of particle α x/y velocity component of particle α

distance vector betweenα and β, y-component

Shear viscosity: Physical meaning

dydvAF xη=Resistance to shearing

shear viscosity

velocity gradient

velocity profile )( yvx

Thermal conductivity

')'()0(1

0'2 dttqq

TUk tB∫∞

><=λ ⎟⎠

⎞⎜⎝

⎛+= ∑ ∑

iijii rUvm

2 )(21

Time integral over autocorrelation function of quantity q (time derivative of Hamiltonian)

time derivative kinetic

energy of particle i

total potential energy (sum over all pairs of atoms)

potentialenergy

Thermal conductivity – physical meaning

T2dxdTA

dtdQH λ−== dt

dQFourier's law:

In 3D formulation (heat flux density)

TAq ∇−= λ Describes rate of heat flow across an interface with different temperatures (temperature gradient)

2.3 Virial stress (directional dependent pressure)

Stress tensor

“Forces inside the material”

surface normalforce direction

∆P∆P

∆P∆A x

∆A ∆Py

Figure by MIT OpenCourseWare. After Buehler.

σxz σxx

div σ + f = 0

Atomic stress tensor: Virial stress

Virial stress:

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.

Force -Fi

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+−Ω

= ∑ ∑= ≠=

jiij rrr

..1 ,..1,,, |)(

Contribution by atoms moving through control volume

Pressure (no directionality)

atom number direction

( )zzyyxxp σσσ ++−=31

>⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

=< ∑ ∑ ∑= = ≠=

=3..1 ..1 ,..1,

,, |)(21

i N aNrri

rrvvmP

Frαβ

Atom α

Atom β

2.4 Closure – property calculation

( )>⋅+<= ∑=

iiiii frvm

><= ∑=

iii vvv ⋅=2

∑∑==

+>=<iN

ttvtvNN

)()(11)()0(

>±Ω±

ρ)()()(

( )22 )0()(1)( ∑ =−>=Δ<i

ii trtrN

Temperature

Pressure

Stress

Direct

Diffusivity

Atomic structure (signature)

Diffusivity, phase state, transport properties

>⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

+−<Ω

= ∑ ∑≠

=α ββα

ααα αβ

jiij rrr

,,,, |)(

Property Definition Application

Summary – property calculation

Material properties: Classification

Structural – crystal structure, RDF, VAF

Thermodynamic -- equation of state, heat capacities, thermal expansion, free energies, use RDF, VAF, temperature, pressure

Mechanical -- elastic constants, cohesive and shear strength, elastic and plastic deformation, fracture toughness, use Virial Stress

Vibrational -- phonon dispersion curves, vibrational frequency spectrum, molecular spectroscopy, use VAF

Transport -- diffusion, viscous flow, thermal conduction, use MSD, temperature, VAF (viscosity, thermal conductivity)

3. Introduction: Chemical interactions

Molecular dynamics: A “bold” idea

( ) ...)()(2)()( 20000 +Δ+Δ+Δ−−=Δ+ ttattrttrttr iiii

Positions at t0

Accelerationsat t0

Positions at t0-Δt

mfa ii /=Forces between atoms… how to obtain?

vi(t), ai(t)

N particlesParticles with mass mi

Atomic scale – how atoms interact

Atoms are composed of electrons, protons, and neutrons. Electron and protons are negative and positive charges of the same magnitude, 1.6 ×10-19 CoulombsChemical bonds between atoms by interactions of the electrons ofdifferent atoms

“Point” representation

no p+ p+

r(t) a(t)

Figure by MIT OpenCourseWare. After Buehler.

Concept: Interatomic potential

“point particle” representation

Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)

Electrons

Energy Ur

1/r12 (or Exponential)

Repulsion

Radius r (Distancebetween atoms)

eAttraction

Interatomic bond - model

Attraction: Formation of chemical bond by sharing of electronsRepulsion: Pauli exclusion (too many electrons in small volume)

Harmonic oscillatorr0

r~ k(r - r0)2

Energy Ur

1/r12 (or Exponential)

Repulsion

Radius r (Distancebetween atoms)

eAttraction

Interatomic pair potentials

Morse potential

Lennard Jones potential

Harmonic approximation

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