Part I Lecture 12 Review Session—Preparation Quiz I

1.021, 3.021, 10.333, 22.00 Introduction to Modeling and Simulation

Part I – Continuum and particle methods

Review session—preparation quiz I Lecture 12

Markus J. Buehler Laboratory for Atomistic and Molecular Mechanics Department of Civil and Environmental Engineering Massachusetts Institute of Technology

1 Content overview

I. Particle and continuum methods Lectures 2-13 1. Atoms, molecules, chemistry 2. Continuum modeling approaches and solution approaches 3. Statistical mechanics 4. Molecular dynamics, Monte Carlo 5. Visualization and data analysis 6. Mechanical properties – application: how things fail (and how to prevent it) 7. Multi-scale modeling paradigm 8. Biological systems (simulation in biophysics) – how proteins work and how to model them

II. Quantum mechanical methods Lectures 14-26 1. It’s A Quantum World: The Theory of Quantum Mechanics 2. Quantum Mechanics: Practice Makes Perfect 3. The Many-Body Problem: From Many-Body to Single- Particle 4. Quantum modeling of materials 5. From Atoms to Solids 6. Basic properties of materials 7. Advanced properties of materials 8. What else can we do? 2 Overview: Material covered so far… Lecture 1: Broad introduction to IM/S

Lecture 2: Introduction to atomistic and continuum modeling (multi-scale modeling paradigm, difference between continuum and atomistic approach, case study: diffusion)

Lecture 3: Basic statistical mechanics – property calculation I (property calculation: microscopic states vs. macroscopic properties, ensembles, probability density and partition function)

Lecture 4: Property calculation II (Monte Carlo, advanced property calculation, introduction to chemical interactions)

Lecture 5: How to model chemical interactions I (example: movie of copper deformation/dislocations, etc.)

Lecture 6: How to model chemical interactions II (EAM, a bit of ReaxFF—chemical reactions)

Lecture 7: Application to modeling brittle materials I

Lecture 8: Application to modeling brittle materials II

Lecture 9: Application – Applications to materials failure

Lecture 10: Applications to biophysics and bionanomechanics

Lecture 11: Applications to biophysics and bionanomechanics (cont’d) 3 Lecture 12: Review session – part I Check: goals of part I You will be able to …

Carry out atomistic simulations of various processes (diffusion, deformation/stretching, materials failure) Carbon nanotubes, nanowires, bulk metals, proteins, silicon crystals, …

Analyze atomistic simulations

Visualize atomistic/molecular data

Understand how to link atomistic simulation results with continuum models within a multi-scale scheme

4 Topics covered – part I

Atomistic & molecular Property simulation calculation algorithms

Potential/ force field Applications models

5 Lecture 12: Review session – part I

1. Review of material covered in part I 1.1 Atomistic and molecular simulation algorithms 1.2 Property calculation 1.3 Potential/force field models 1.4 Applications 2. Important terminology and concepts

Goal of today’s lecture: Review main concepts of atomistic and molecular dynamics, continuum models Prepare you for the quiz on Thursday

6 1.1 Atomistic and molecular simulation algorithms 1.2 Property calculation 1.3 Potential/force field models 1.4 Applications

Goals: Basic MD algorithm (integration scheme), initial/boundary conditions, numerical issues (supercomputing)

7 Differential equations solved in MD

8 Basic concept: atomistic methods

2 G rd i rdU )( G PDE solved in MD: mi 2 −= G = rr j}{ = ..1 Ni = Fma )( dt rd i

Results of MD simulation: G G G iii tatvtr )(),(),( = ..1 Ni 9 Complete MD updating scheme

(1) Initial conditions: Positions & velocities at t0 (random velocities so that initial temperature is represented)

(2) Updating method (integration scheme - Verlet) 2 i 0 i 0 i 0 i 0 ( ttattrttrttr ) +Δ+Δ+Δ−−=Δ+ ...)()(2)()(

Positions Positions Accelerations at t0-Δt at t0 at t0 (3) Obtain accelerations from forces

, = amf ,ijjij = ,, / mfa jijij ∀ = ..1 Nj

(4) Obtain force vectors from potential (sum over contributions from all neighbor of atom j) 12 6 φ r)(d x ,ij ⎛ ⎡ ⎤ ⎡σσ⎤ ⎞ F −= ,ij = Ff Potential r = 4)( εφ⎜ − ⎟ d r r ⎜ ⎣⎢ ⎦⎥ ⎣⎢ rr ⎦⎥ ⎟ ⎝ ⎠ 10 Algorithm of force calculation

6 … 5 for i=1..N

for j=1..N (i ≠ j) j=1 i 4

[add force 2 3 contributions] rcut

11 Summary: Atomistic simulation – numerical approach “molecular dynamics – MD”

Atomistic model; requires atomistic microstructure and atomic positions at beginning (initial conditions), initial velocities from random distribution according to specified temperature Step through time by integration scheme (Verlet) Repeat force calculation of atomic forces based on their positions Explicit notion of chemical bonds – captured in interatomic potential

Loop

12 Scaling behavior of MD code

13 Force calculation for N particles: pseudocode

Requires two nested loops, first over all atoms, and then over all other atoms to determine the distance for i=1..N:

for j=1..N (i ≠ j): 0 t Δ 2

i j =N determine distance between and 0 t Δ Δt ~NΔt 1 0 calculate force and energy (if ~N·N Δt0

r < r , cutoff radius) total ij cut t Δ add to total force vector / energy

N2 computational disaster time ~ : 14 Strategies for more efficient computation

Two approaches

1. Neighbor lists: Store information about atoms in vicinity, calculated in an N2 effort, and keep information for 10..20 steps

Concept: Store information of neighbors of each atom within vicinity of cutoff radius (e.g. list in a vector); update list only every 10..20 steps

2. Domain decomposition into bins: Decompose system into small bins; force calculation only between atoms in local neighboring bins

Concept: Even overall system grows, calculation is done only in a local environment (have two nested loops but # of atoms does not increase locally) 15 Force calculation with neighbor lists

Requires two nested loops, first over all atoms, and then over all other atoms to determine the distance

for i=1..Nneighbors,i:

for j=1..N (i ≠ j): 0 t

determine distance between i and j Δ ~N

Δt1~NΔt0 total calculate force and energy (if t Δt0 Δ rij < rcut, cutoff radius)

add to total force vector / energy

time ~ N: computationally tractable even for large N 16 Periodic boundary conditions

Periodic boundary conditions allows studying bulk properties (no free surfaces) with small number of particles (here: N=3), all particles are “connected”

Original cell surrounded by 26 image cells; image particles move in exactly the same way as original particles (8 in 2D)

Particle leaving box enters on other side with same velocity vector.

Image by MIT OpenCourseWare. After Buehler. 17 Parallel computing – “supercomputers”

Images removed due to copyright restrictions. Please see http://www.sandia.gov/ASC/images/library/ASCI-White.jpg

Supercomputers consist of a very large number of individual computing units (e.g. Central Processing 18 Units, CPUs) Domain decomposition

Each piece worked on by one of the computers in the supercomputer Fig. 1 c from Buehler, M., et al. "The Dynamical Complexity of Work-Hardening: A Large-Scale Molecular Dynamics Simulation.” Acta Mech Sinica 21 (2005): 103-11. © Springer-Verlag. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. 19 Modeling and simulation

20 Modeling and simulation

What is a model? What is a simulation?

Modeling vs Simulation

• Modeling: developing a mathematical representation of a physical situation

•Simulation: solving the equations that arose from the development of the model.

© Google, Inc. All rights reserved. © Massachusetts Bay This content is excluded from our Transportation Authority. All rights Creative Commons license. For more reserved. This content is excluded information, see from our Creative Commons http://ocw.mit.edu/fairuse. license. For more information, see http://ocw.mit.edu/fairuse.

21 Modeling and simulation

22 Atomistic versus continuum viewpoint

23 Diffusion: Phenomenological description

Physical problem Ink dot

2 ∂c cd nd = D 2 Fick law (governing ∂t dx2 equation) Tissue BC: c (r = ∞) = 0 24 IC: c (r=0, t = 0) = c0 Continuum model: Empirical parameters

Continuum model requires parameter that describes microscopic processes inside the material

Need experimental measurements to calibrate

Time scale “Empirical” Continuum or experimental model parameter feeding Microscopic mechanisms ???

Length scale 25 How to solve continuum problem: Finite difference scheme ∂c 2cd i = = D Concentration at i space 2 at old time ∂t dx j = time

δt “explicit” numerical scheme cc += ()2 −− cccD + ,1, jiji δx 2 + − ,1,,1 jijiji (new concentration directly from concentration at earlier time)

Concentration at i Concentration at i-1 at old time at old time

Concentration at i Concentration at i+1 at new time at old time 26 Other methods: Finite element method Atomistic model of diffusion

Diffusion at macroscale (change of concentrations) is result of microscopic processes, random motion of particle

Atomistic model provides an alternative way to describe diffusion

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

Follow trajectory of atoms and calculate how fast atoms leave their initial position Concept: follow this quantity over Δx2 time −= pD Δt

27 Atomistic model of diffusion 2 G rd i rdU )( G mi 2 −= G = rr j}{ = ..1 Ni dt rd i

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

Δr2 Particles Trajectories

Average squares of Mean Squared displacements of all Displacement function particles 1 G G 2 2 tr )( =Δ ()trtr =− )0()( N ∑ ii i 28 Calculation of diffusion coefficient

2 1 2 tr )( =Δ ∑()ii trtr =− )0()( N i Position of Position of atom i at time t atom i at time t=0

(nm)

Einstein equation slope = D 2 1 d Δr D = (Δ 2 tr )(lim ) 2d t ∞→ dt

1D=1, 2D=2, 3D=3 (ps)

29 Summary Molecular dynamics provides a powerful approach to relate the diffusion constant that appears in continuum models to atomistic trajectories

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

Time scale “Empirical” Continuum or experimental model parameter feeding MD

???

Length scale 30 1.1 Atomistic and molecular simulation algorithms 1.2 Property calculation 1.3 Potential/force field models 1.4 Applications

Goals: How to calculate “useful” properties from MD runs (temperature, pressure, RDF, VAF,..); significance of averaging; Monte Carlo schemes

31 Property calculation: Introduction

Have: GG G iii tatvtr )(),(),( = ..1 Ni

“microscopic information”

Want: Thermodynamical properties (temperature, pressure, ..) Transport properties (diffusivities, shear viscosity, ..) Material’s state (gas, liquid, solid)

32 Micro-macro relation

tT )(

Which to pick?

N t 1 1 G2 tT )( = ∑ ii tvm )( Images courtesy of the Center for Polymer Studies 3 Nk at Boston University. Used with permission. B i=1 Specific (individual) microscopic states are insufficient to relate to macroscopic properties 33 Ergodic hypothesis: significance of averaging

>=< ∫∫ ρ ),(),( drdprprpAA property must pr be properly averaged Ergodic hypothesis:

Ensemble (statistical) average = time average

All microstates are sampled with appropriate probability density over long time scales 1 1 ∑ )( Ens AAiA Time =>=<>=< ∑ iA )( N A = ..1 Ni A Nt = ..1 Ni t

Monte Carlo MD Average over Monte Carlo steps Average over time steps34 NO DYNAMICAL INFORMATION DYNAMICAL INFORMATION Monte Carlo scheme

Concept: Find convenient way to solve the integral

>=< ∫∫ ρ ),(),( drdprprpAA pr

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)

35 Monte Carlo scheme: area calculation

Method to carry out integration (illustrate general concept)

Want: G ∫ )( dxfA Ω= Ω

E.g.: Area of circle (value of π)

2 πd π Ω AC = AC = d = 1 4 4 36 Monte Carlo scheme G Step 1: Pick random point x i in Ω Step 2: Accept/reject point based on criterion (e.g. if inside or outside of circle andG if in area not yet counted) xf = 1)( Step 3: If accepted, add i G to the total sum otherwise xf i = 0)( G )( dxfA Ω= C ∫ Ω d = 2/1 N A : Attempts made 1 G AC = ∑ xf i )( N A i

Courtesy of John H. Mathews. Used with permission. 37 See also: http://math.fullerton.edu/mathews/n2003/MonteCarloPiMod.html Area of Middlesex County (MSC)

100 km

2 ASQ = km000,10

100 km y

Image courtesy of Wikimedia Commons. Fraction of points that lie within MS County

1 G MSC = AA SQ ∑ xf i )( Expression provides area of MS County N A i 38 Detailed view - schematic

NA=55 points (attempts) 12 12 points within 1 G = AA xf 2 =⋅= km8.2181km22.010000)( 2 MSC SQ ∑ i 39 N A i Results U.S. Census Bureau

823.46 square miles = 2137 km2 (1 square mile=2.58998811 km2 )

Taken from: http://quickfacts.census.gov/qfd/states/25/25017.html

2 Monte Carlo result: AMSC = km8.2181

40 Analysis of satellite images

Spy Pond (Cambridge/Arlington)

Image removed due to copyright restrictions.

Google Satellite image of Spy Pond, in Cambridge, MA.

200 m 41 Metropolis Hastings algorithm

Images removed due to copyright restrictions. Images removed due to copyright restrictions. Please see: Fig. 2.7 in Buehler, Markus J. Atomistic Please see: Fig. 2.8 in Buehler, Markus J. Atomistic Modeling Modeling of Materials Failure. Springer, 2008. of Materials Failure. Springer, 2008.

>=< ∫∫ ρ ),(),( drdprprpAA pr 42 Molecular Dynamics vs. Monte Carlo

MD provides actual dynamical data for nonequilibrium processes (fracture, deformation, instabilities)

Can study onset of failure, instabilities

MC provides information about equilibrium properties (diffusivities, temperature, pressure)

Not suitable for processes like fracture

1 1 ∑ )( Ens AAiA Time =>=<>=< ∑ iA )( 43 N A = ..1 Ni A Nt = ..1 Ni t Property calculation: Temperature and pressure

Temperature N 1 1 G2 G2 G G T ∑ vm ii ><= ⋅= vvv iii 3 NkB i=1

~ vT 2

~ Tv

44 Temperature control in MD (NVT ensemble) Berendsen thermostat

2 i 0 i 0 i 0 i 0 ttattrtrttr +Δ+Δ−−=Δ+ ...)()()(2)(

N 1 1 G2 Calculate temperature T ∑ vm ii ><= 3 NkB i=1

~ Tv

Calculate ratio of current vs. desired temperature

45 Rescale velocities Interpretation of RDF

46 Formal approach: Radial distribution function (RDF) The radial distribution function is defined as Density of atoms (volume) = ρ rrg /)()( ρ

Local density

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

Δr Number of atoms in the interval r ± 2 < rN ± Δr )( > 1 rg )( = 2 Δr r ±Ω 2 )( ρ Volume of this shell (dr) 2 2)( π drrrg = Number of particles that lie in a spherical shell 47 of radius r and thickness dr Radial distribution function: Solid versus liquid versus gas

5 5 Gaseous Ar 4 4 (90 K) Solid Argon 3 3 Liquid Ar (90 K) Gaseous Ar g(r) g(r) (300 K) 2 2 Liquid Argon 1 1

0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 distance/nm distance/nm

Image by MIT OpenCourseWare.

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 48 RDF and crystal structure 1st nearest neighbor (NN) 5 3rd NN 4 nd Solid Argon 2 NN … 3 g(r) 2 Liquid Argon 1

0 0 0.2 0.4 0.6 0.8 distance/nm Image by MIT OpenCourseWare.

Peaks in RDF characterize NN distance, can infer from RDF about crystal structure 49 http://physchem.ox.ac.uk/~rkt/lectures/liqsolns/liquids.html Images courtesy of Mark Tuckerman. Used with permission. 50 Interpretation of RDF

1st nearest neighbor (NN) 2nd NN 3rd NN

copper: NN distance > 2 Å

X not a liquid (sharp peaks) 51 Graphene/carbon nanotubes

RDF

Images of graphene/carbon nanutubes: http://weblogs3.nrc.nl/techno/wp- content/uploads/080424_Grafeen/Graphene_xyz.jpg http://depts.washington.edu/polylab/images/cn1.jpg

Graphene/carbon nanotubes (rolled up graphene)

NN: 1.42 Å, second NN 2.46 Å … 52 Summary – property calculation

Property Definition Application

N 1 1 G2 G2 G G Temperature T ∑ vm ii ><= ⋅= vvv iii Direct 3 NkB i=1

2 1 2 MSD tr )( >=Δ< ∑()ii trtr =− )0()( Diffusivity N i rN ± Δr )( rg )( =< 2 > Atomic structure RDF Δr (signature) r ±Ω 2 )( ρ

53 Material properties: Classification

Structural – crystal structure, RDF

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

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

Transport -- diffusion, viscous flow, thermal conduction, use MSD, temperature

54 Bell model analysis (protein rupture)

55 Bell model analysis (protein rupture)

= ⋅ln),;( + bvaExvf a=86.86 pN bb b=800 pN

kBT a = (1) k =1.3806505E-23 J/K B a b xb Get and from fitting to the graphs ⎛ ⎛ ⎞⎞ kbT Eb b =− ln⎜ω 0 xb exp⎜ − ⎟⎟ (1) Solve (1) for a, use (2) to solve xb ⎝ kbT ⎠ ⎝ ⎠ for Eb

13 a=(800-400)/(0-(-4.6))) pN ω0 ×= sec/1101 56 =86.86 pN Æ xb=0.47 Å Determining the energy barrier

Tk ⎛ ⎛ E ⎞⎞ Tk ⎡ E ⎤ b −= b ⎜ω x expln ⎜− b ⎟⎟ −= b ln()ω x − b ⎜ 0 b ⎜ ⎟⎟ ⎢ 0 b ⎥ xb ⎝ ⎝ bTk ⎠⎠ xb ⎣ bTk ⎦

bxb Eb ln()ω0 xb −=− bTk bTk

Eb bxb ln()ω0 xb += bTk bTk

⎡ bxb ⎤ b = ⎢ln()ω0 xE b + ⎥ bTk ⎣ bTk ⎦

b=800 pN Æ Eb≈9 kcal/mol 1 kcal/mol=6.9477E-21 J

3 H-bonds Possible mechanism: Rupture of (H-bond energy 3 kcal/mol)57 Scaling with Eb : shifts curve

bb = ⋅ln),;( + bvaExvf

Eb ↑

⋅Tk ⋅Tk ⎛ E ⎞ B B ω xv exp⎜−⋅⋅= b ⎟ a = b −= ⋅ln v0 00 b ⎜ ⎟ xb xb 58 ⎝ b ⋅Tk ⎠ Scaling with xb: changes slope

bb = ⋅ln),;( + bvaExvf

xb ↓

⋅Tk ⋅Tk ⎛ E ⎞ a = B b −= B ⋅ln v ω xv exp⎜−⋅⋅= b ⎟ 0 00 b ⎜ ⋅Tk ⎟ xb xb 59 ⎝ b ⎠ 1.1 Atomistic and molecular simulation algorithms 1.2 Property calculation 1.3 Potential/force field models 1.4 Applications

Goals: How to model chemical interactions between particles (interatomic potential, force fields); applications to metals, proteins

60 Concept: Repulsion and attraction

Electrons

Energy U r

1/r12 (or Exponential)

Core Repulsion

Radius r (Distance between atoms) e r Attraction

1/r6

Image by MIT OpenCourseWare. “point particle” representation

Attraction: Formation of chemical bond by sharing of electrons Repulsion: Pauli exclusion (too many electrons in small volume) 61 Generic shape of interatomic potential

Energy U r Effective 12 1/r (or Exponential) interatomic Repulsion potential

Radius r (Distance Harmonic oscillator between atoms) r0 e )2 Attraction ~ k(r - r0 r

1/r6

Image by MIT OpenCourseWare. Image by MIT OpenCourseWare. Many chemical bonds show this generic behavior

Attraction: Formation of chemical bond by sharing of electrons Repulsion: Pauli exclusion (too many electrons in small volume) 62 Atomic interactions – different types of chemical bonds

Primary bonds (“strong”) Ionic (ceramics, quartz, feldspar - rocks) Covalent (silicon) Metallic (copper, nickel, gold, silver) (high melting point, 1000-5,000K)

Secondary bonds (“weak”) Van der Waals (wax, low melting point) Weaker bonding Hydrogen bonds (proteins, spider silk) (melting point 100-500K)

Ionic: Non-directional (point charges interacting) Covalent: Directional (bond angles, torsions matter) Metallic: Non-directional (electron gas concept)

Know models for all types of chemical bonds! 63 How to choose a potential

64 How to calculate forces from the “potential” or “force field” Define interatomic potentials, that describe the energy of a set of atoms as a function of their coordinates

“Potential” = “force field” G = {j } = ..1 Njrr

= rUU )( Depends on position of total total all other atoms

G −∇= K = ..1)( NirUF i totalri Position vector of atom i ⎛ ∂ ∂ ∂ ⎞ Change of potential energy G =∇ ⎜ ,, ⎟ due to change of position of ri ⎜ ⎟ 65 ⎝ ∂ ∂ ∂rrr ,3,2,1 iii ⎠ particle i (“gradient”) Force calculation

Pair potential Note cutoff!

66 Pair potentials: energy calculation

Simple approximation: Total energy is sum over the Pair wise energy of all pairs of atoms in the system interaction potential

φ rij )(

Pair wise summation of bond energies r12 avoid double counting N N r25 1 Utot = 2 ∑∑φ rij )( ==11, ≠ jii j

rij = distance between Energy of atom i i = φ rU ij )( ∑ 67 particles i and j j=1 Total energy of pair potential

Assumption: Total energy of system is expressed as sum of the energy due to pairs of atoms

N N 1 Utotal = 2 ∑∑φ rij )( ==11, ≠ jii j r23 r12 φij = φ rij )(

with

1 =0 Utotal ()+++++= φφφφφφ323123211312 2 =0 beyond cutoff 68 Force calculation – pair potential

Forces can be calculated by taking derivatives from the potential function

Force magnitude: Negative derivative of potential energy with respect to atomic distance φ r )(d F −= ij d rij

To obtain force vector Fi, take projections into the three axial directions i ComponentG of j vector rij xi r i = Ff ij rij i f x2 x2 G f2 = rr ijij f1 x1 69x1 Force calculation – pair potential

Forces on atom 3: only interaction with atom 2 (cutoff)

1. Determine distance between atoms 3 and 2, r23 2. Obtain magnitude of force vector based on derivative of potential 3. To obtain force vector Fi, take projections into the three axial directions ComponentG i of vector rij φ r )(d x G F −= 23 = Ff i = rr d r i r 2323 23 23 70 Interatomic pair potentials: examples

φ ij Dr (−= α ij − 0 )− (−α ij − rrDrr 0 )(exp2)(2exp)( ) Morse potential

12 6 ⎡⎛ ⎞ ⎛ σσ⎞ ⎤ Lennard-Jones 12:6 ⎢⎜ ⎟ ⎜ ⎟ ⎥ rij = 4)( εφ − potential (excellent model ⎢⎜ ⎟ ⎜ rr ⎟ ⎥ ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎦ for noble gases, Ar, Ne, Xe..)

6 ⎛ r ⎞ ⎛ σ ⎞ φ Ar exp)( ⎜−= ij ⎟ − C⎜ ⎟ ij ⎜ ⎟ ⎜ ⎟ Buckingham potential ⎝ σ ⎠ ⎝ rij ⎠

Harmonic approximation 71 Lennard-Jones potential – example for copper

0.2

0.1

0 6 σ 2 = r0 (eV) φ

-0.1 ε

-0.2 2 r0 3 4 5 o r (A)

Image by MIT OpenCourseWare.

LJ potential – parameters for copper (e.g.: Cleri et al., 1997) 72 Derivative of LJ potential ~ force

0.2

0.1 ο dφ(r) F = _ = -φ' dr 0

-dφ/dr (eV/A) -0.1 Fmax,LJ φ(r) -0.2 2 r0 3 4 5 o r (A)

Image by MIT OpenCourseWare.

EQ r0

73 Cutoff radius N i = ∑φ rU ij )( j=1

… 6 5 Ui = ∑φ rij )( = ..1 Nj neigh

j=1 i 4 φ LJ 12:6 ~σ potential 2 3

rcut ε0 rcut r

Cutoff radius = considering interactions only to a certain distance Basis: Force contribution negligible (slope) 74 Derivative of LJ potential ~ force

0.2

0.1 cut ο r force not zero dφ(r) F = ______dr 0

Potential shift -dφ/dr (eV/A) -0.1

-0.2 2 r0 3 4 5 o r (A)

Image by MIT OpenCourseWare.

Beyond cutoff: Changes in energy (and thus forces) small75 Crystal structure and potential

The regular packing (ordering) of atoms into crystals is closely related to the potential details Many 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)

LJ 12:6 2D example potential φ N=4 bonds N=6 bonds per atom ~σ

ε r

Square lattice Hexagonal lattice76 Example

Initial: cubic lattice Transforms into triangular lattice

Images courtesy of the Center for Polymer Studies at Boston University. Used with permission. time

http://polymer.bu.edu/java/java/LJ/index.html 77 All bonds are not the same – metals

stronger + different bond EQ distance Surface

Bulk

Reality: Have environment Pair potentials: All bonds are equal! effects; it matter that there is a free surface!

78 Embedded-atom method (EAM): multi-body potential

new 1 φi = ∑ ij + Fr ρφi )()( = ..1 Nj neigh 2

Pair potential Embedding energy energy as a function of electron density

ρi Electron density at atom i based on a pair potential:

= πρr )( i ∑ ij Models other than EAM (alternatives): = ..1 Nj neigh Glue model (Ercolessi, Tosatti, Parrinello) Finnis Sinclair 79 Equivalent crystal models (Smith and Banerjee) Effective pair interactions: EAM potential

EAM=Embedded 1 atom method Bulk 0.5 Can describe differences between Surface bulk and surface 0 Effective pair potential (eV ) pair potential (eV Effective

-0.5 2 3 4 o r (A)

Image by MIT OpenCourseWare.

Pair potential energy Embedding energy as a function of electron density 1 U = φ + Fr ρ )()( Embedding term: depends on ∑∑ ij ∑ i environment, “multi-body” 80 2 ij, ≠ ji i Effective pair interactions: EAM r

+ + + + + + Bulk 0.5 + + + + + + Change in + + + + + +

Surface NN distance 0 r Effective pair potential (eV ) pair potential (eV Effective

-0.5 2 3 4 o r (A) + + + + + + Image by MIT OpenCourseWare. + + + + + + + + + + + +

Can describe differences between bulk and surface 81 Daw, Foiles, Baskes, Mat. Science Reports, 1993 Model for chemical interactions

Similarly: Potentials for chemically complex materials – assume that total energy is the sum of the energy of different types of chemical bonds

= + + + +UUUUUU total Elec Covalent Metallic vdW −bondH 82 Concept: energy landscape for chemically complex materials

total Elec += Covalent + Metallic + vdW +UUUUUU −bondH

Different energy contributions from different kinds of chemical bonds are summed up individually, independently

Implies that bond properties of covalent bonds are not affected by other bonds, e.g. vdW interactions, H-bonds

Force fields for organic substances are constructed based on this concept:

water, polymers, biopolymers, proteins …

83 Summary: CHARMM-type potential =0 for proteins

total = Elec + Covalent + Metallic + vdW +UUUUUU −bondH

qq ji UElec : Coulomb potential φ rij )( = ε1rij 1 φ = − rrk )( 2 stretch 2 stretch 0 1 = + +UUUU φ k −= θθ)( 2 Covalent stretch bend rot bend 2 bend 0 1 φ k −= ϑ))cos(1( rot 2 rot 12 6 ⎡⎛ ⎞ ⎛ ⎞ ⎤ ⎜ ⎟ ⎜ σσ⎟ U : LJ potential rij = 4)( εφ⎢ − ⎥ vdW ⎢⎜ ⎟ ⎜ rr ⎟ ⎥ ⎣⎝ ij ⎠ ⎝ ij ⎠ ⎦ 12 10 ⎡ ⎛ R ⎞ ⎛ R ⎞ ⎤ ⎜ −bondH ⎟ ⎜ −bondH ⎟ 4 U −bondH : φ ij )( = Dr −bondH ⎢5 − 6 ⎥ θDHA )(cos ⎢ ⎜ r ⎟ ⎜ r ⎟ ⎥ 84 ⎣ ⎝ ij ⎠ ⎝ ij ⎠ ⎦ Summary of energy expressions: CHARMM, DREIDING, etc.

δ− Bond stretching δ+ Electrostatic δ+ interactions

δ− Angle Bending

vdW ONLY vdW Interactions between atoms of Bond Rotation different molecules

Image by MIT OpenCourseWare.

…. 85 Identify a strategy to model the change of an interatomic bond under a chemical reaction (based on harmonic potential) – from single to double bond φ 1 Energy φ φ = − rrk )( 2 stretch 2 stretch 0 Effective interatomic potential

Harmonic oscillator r0 )2 ~ k(r - r0 r r

Image by MIT OpenCourseWare. Energy φstretch focus on C-C bond Transition point sp3 BO=1

C-C distance sp2 BO=2 r sp2 sp3 Solution sketch/concept

1 2 stretch = kk stretch (BO) φstretch = stretch − rrk 0 )( 2 = rr 00 (BO)

Make potential parameters kstretch and r0 a function of the chemical state of the molecule (“modulate parameters”)

E.g. use bond order as parameter, which allows to smoothly interpolate between one type of bonding and another one dependent on geometry of molecule See lecture notes (lecture 7): Bond order potential for silicon

-5 Potential energy (eV) energy Potential

-10 0.5 1 1.5 2 2.5 3 3.5 o Distance (A)

Triple S.C. Double F.C.C. Fig. 2.21c in Buehler, Markus J. Atomistic Modeling Single of Materials Failure. Springer, 2008. Effective pair-interactions for various C-C (Carbon) bonds © Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. Image by MIT OpenCourseWare. Summary: CHARMM force field

Widely used and accepted model for protein structures

Programs such as NAMD have implemented the CHARMM force field

E.g., problem set 3, nanoHUB stretchmol/NAMD module, study of a protein domain part of human vimentin intermediate filaments

88 Overview: force fields discussed in IM/S

Pair potentials (LJ, Morse, harmonic potentials, biharmonic potentials), used for single crystal elasticity (pset #1) and fracture model (pset #2) Pair potential

EAM (=Embedded Atom Method), used for nanowire (pset #1), suitable for metals Multi-body potential

ReaxFF (reactive force field), used for CMDF/fracture model of silicon (pset #2) – very good to model breaking of bonds (chemistry) Multi-body potential (bond order potential)

Tersoff (nonreactive force field), used for CMDF/fracture model of silicon (note: Tersoff only suitable for elasticity of silicon, pset #2) Multi-body potential (bond order potential)

CHARMM force field (organic force field), used for protein simulations (pset #3) Multi-body potential (angles, for instance)

89 How to choose a potential

ReaxFF

EAM, LJ

CHARMM-type (organic) ReaxFF

CHARMM

90 SMD mimics AFM single molecule experiments

Atomic force microscope pset #3 G G v v k x x k

91 Steered molecular dynamics (SMD)

pset #3 G Steered molecular v dynamics used to apply f forces to protein Virtual atom G moves w/ velocity v k structures G x G G ⋅= − xtvkf )( ⋅ − xtv end point of SMD spring constant molecule G G G −⋅= xtvkf )(

Distance between end time point of molecule and SMD virtual atom deformation speed vector 92 Comparison – CHARMM vs. ReaxFF

Covalent bonds never break in CHARMM

93 M. Buehler, JoMMS, 2007 1.1 Atomistic and molecular simulation algorithms 1.2 Property calculation 1.3 Potential/force field models 1.4 Applications

Goals: Select applications of MD to address questions in materials science (ductile versus brittle materials behavior); observe what can be done with

MD 94 Application – mechanics of materials tensile test of a wire

Brittle Ductile

Necking Stress

Brittle Strain Ductile

Image by MIT OpenCourseWare. Image by MIT OpenCourseWare.

95 Brittle versus ductile materials behavior

96 Ductile versus brittle materials

BRITTLE DUCTILE

Glass Polymers Copper, Gold Ice...

Shear load

Difficult Image by MIT OpenCourseWare. to deform, Easy to deform breaks easily hard to break

97 Deformation of materials: Nothing is perfect, and flaws or cracks matter

“Micro (nano), local” “Macro, global” σ r)( r

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, other imperfections 98 Crack extension: brittle response

σ r)(

Large stresses lead to rupture of chemical bonds between atoms Thus, crack extends

99 Image by MIT OpenCourseWare. Basic fracture process: dissipation of elastic energy

δa

Undeformed Stretching=store elastic energy Release elastic energy dissipated into breaking chemical bonds 100 Limiting speeds of cracks: linear elastic continuum theory

Mother-daughter mechanism

Super- Sub-Rayleigh Rayleigh Intersonic Supersonic Mode II

Mode I Limiting speed v Cr Cs Cl

Subsonic Supersonic Mode III Limiting speed v Cs Cl

Linear Nonlinear

Image by MIT OpenCourseWare.

• Cracks can not exceed the limiting speed given by the corresponding wave speeds unless material behavior is nonlinear • Cracks that exceed limiting speed would produce energy (physically impossible - linear elastic continuum theory) 101 Summary: mixed Hamiltonian approach

Wi W2 W1 100%

0% x R-Rbuf R R+Rtrans R+Rtrans +Rbuf

Transition ReaxFF Tersoff layer

Transition region

Tersoff ghost atoms ReaxFF ghost atoms

Image by MIT OpenCourseWare.

FReaxFF−Tersoff = ( ReaxFF Fxw ReaxFF + − ReaxFF )1()( Fw Tersoff )

ReaxFF + Tersoff = 1)()( ∀xxwxw 102 Atomistic fracture mechanism

Image by MIT OpenCourseWare.

103 Lattice shearing: ductile response

Instead of crack extension, induce shearing of atomic lattice Due to large shear stresses at crack tip Lecture 9

τ τ Image by MIT OpenCourseWare.

τ τ

Image by MIT OpenCourseWare.

104 Concept of dislocations

additional half plane

τ τ

τ τ b

Image by MIT OpenCourseWare.

Localization of shear rather than homogeneous shear that requires cooperative motion

“size of single dislocation” = b, Burgers vector 105 Final sessile structure – “brittle”

Fig. 1 c from Buehler, M., et al. "The Dynamical Complexity of Work-Hardening: A Large-Scale Molecular Dynamics Simulation." Acta Mech Sinica 21 (2005): 103-11. 106 © Springer-Verlag. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse. Brittle versus ductile materials behavior

Copper, nickel (ductile) and glass, silicon (brittle)

For a material to be ductile, require dislocations, that is, shearing of lattices

Not possible here (disordered)

crack has stopped

slope=equal to crack speed

108 MD simulation: chemistry as bridge between disciplines

109 Application – protein folding

Combination of 3 DNA letters equals ACGT a amino acid Four letter E.g.: Proline – .. - Proline - Serine – Proline - Alanine - .. code “DNA” CCT, CCC, CCA, CCG Sequence of amino acids Transcription/ Folding “polypeptide” (1D (3D structure) translation structure)

Goal of protein folding simulations: Predict folded 3D structure based on polypeptide sequence

110 Movie: protein folding with CHARMM

de novo Folding of a Transmembrane fd Coat Protein http://www.charmm-gui.org/?doc=gallery&id=23

Polypeptide chain

Quality of predicted structures quite good Confirmed by comparison of the MSD deviations of a room temperature ensemble of conformations from the replica-exchange simulations and experimental structures from both solid-state NMR in lipid bilayers and solution-phase NMR on the protein in micelles) 111 2. Important terminology and concepts

112 Important terminology and concepts

Force field Potential / interatomic potential Reactive force field / nonreactive force field Chemical bonding (covalent, metallic, ionic, H-bonds, vdW..) Time step Continuum vs. atomistic viewpoints Monte Carlo vs. Molecular Dynamics Property calculation (temperature, pressure, diffusivity [transport properties]-MSD, structural analysis-RDF) Choice of potential for different materials (pair potentials, chemical complexity) MD algorithm: how to calculate forces, energies MC algorithm (Metropolis-Hastings) Applications: brittle vs. ductile, crack speeds, data analysis 113 3. Continuum vs. atomistic approaches

114 Relevant scales in materials λ(atom) << ξ(crystal) << d(grain) << dx,dy,dz << H,W, D y Ay σyy

n y σyx σ σyz xy

nx σxz σxx x Image from Wikimedia Commons, http://commons Image by MIT Ax Courtesy of Elsevier, Inc., Fig. 8.7 in: Buehler, M. z .wikimedia.org. OpenCourseWare. http://www.sciencedirect.com. Atomistic Modeling of Materials Used with permission. Failure. Springer, 2008. © Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Bottom-up http://ocw.mit.edu/fairuse. Top-down

Atomistic viewpoint: •Explicitly consider discrete atomistic structure •Solve for atomic trajectories and infer from these about material properties & behavior •Features internal length scales (atomic distance) “Many-particle system with statistical properties” 116 Relevant scales in materials Separation of scales RVE λ(atom) << ξ(crystal) << d(grain) << dx,dy,dz << H,W, D y Ay σyy

n y σyx σ σyz xy

nx σxz σxx x Image from Wikimedia Commons, http://commons Fig. 8.7 in: Buehler, M. Courtesy of Elsevier, Inc., Ax .wikimedia.org. http://www.sciencedirect.com. Atomistic Modeling of Materials z Image by MIT Used with permission. Failure. Springer, 2008. © OpenCourseWare. Springer. All rights reserved. This content is excluded from our Creative Commons license. For more information, see Top-down http://ocw.mit.edu/fairuse. Continuum viewpoint: •Treat material as matter with no internal structure •Develop mathematical model (governing equation) based on representative volume element (RVE, contains “enough” material such that internal structure can be neglected •Features no characteristic length scales, provided RVE is large enough 117 “PDE with parameters” Example – diffusion

118 Example solution – 2nd Fick’s law

∂c 2cd Concentration = D ∂t dx2 c0

c0 x time t

txc ),(

x BC: c (x = 0) = c0 IC: c (x > 0, t = 0) = 0

Need diffusion coefficient to solve for distribution! 119 2. Finite difference method

How to solve a PDE

∂c 2cd = D + BCs, ICs ∂t dx2

dc Note: Flux −= DJ dx 120 Finite difference method ∂c 2cd = D 2 ∂t dx discrete grid (space and time) Idea: Instead of solving for continuous field xtc ),( t solve for xtc ji ),(

j i = space j = time

),( = cxtc , jiji δt = ..1 Ni i x δx 121 Finite difference method ∂c 2cd = D ∂t dx2

Idea: Instead of solving for continuous field xtc ),( solve for xtc ji ),(

Approach: Express continuous derivatives as discrete differentials ∂c Δc ≈ ∂t Δt

122 Finite difference method ∂c 2cd = D ∂t dx2

Idea: Instead of solving for continuous field xtc ),( solve for xtc ji ),(

Approach: Express continuous derivatives as discrete differentials ∂c Δc ≈ ∂t Δt ⎛ Δc ⎞ Δ⎜ ⎟ ∂c Δc ∂2c Δt ≈ ≈ ⎝ ⎠ ∂x Δx ∂x2 Δx 123 Forward, backward and central difference ∂c c ∂x Forward difference ,tx ji forward

c + ,1 ji Δc c , ji

c − ,1 ji Backward difference backward x

xi−1 xi xi+1 c δx

Central difference c + ,1 ji

c , ji

c − ,1 ji central x

xi−1 xi xi+1 124 How to calculate derivatives with discrete differentials c ∂c 2cd = D 2 ∂c Δc c ≈ ∂t dx ji +1, c Δc ∂t Δt , ji δt

∂c + − cc ,1, jiji + − cc ,1, jiji ≈ = (1) t t t ∂t − tt δt j j+1 ,tx ji +1 jj

δt discrete time step ∂c ∂t ,tx ji

125 How to calculate derivatives with discrete differentials ∂c 2cd ∂c ∂c = D ⎛ Δc ⎞ 2 c ∂x ∂x 1 ,tx j ,tx 2 Δ⎜ ⎟ ∂t dx i+ i+ 1 j ∂c Δc ∂ c Δt 2 2 ≈ ≈ ⎝ ⎠ ∂x Δx ∂x2 Δx c + ,1 ji

c , ji

c − ,1 ji central ∂c + − cc ,,1 jiji + − cc ,,1 jiji ≈ = x ∂x ,tx +1 − xx ii δx x x x i+ 1 j i−1 i i+1 2 x 1 i+ 2

xi− 1 ∂c − cc − cc 2 ≈ − ,1, jiji = − ,1, jiji ∂x ,tx − xx ii −1 δx i− 1 j 2

126 How to calculate derivatives with discrete

differentials ∂c 2 ∂c 2cd ∂ c = D ∂x 2 ⎛ ∂c ⎞ ∂t dx2 ,tx ji ∂x Δ⎜ ⎟ ,tx ji ∂c Δc ∂2c ∂x ≈ ≈ ⎝ ⎠ ∂x Δx ∂x2 Δx c' 1 + 2 , ji

c' 1 − 2, ji central ∂c + − cc ,,1 jiji + − cc ,,1 jiji ≈ = x

∂x − xx δx xi− 1 x x 1 ,tx +1 ii 2 i i+ i+ 1 j 2 2 Apply central difference ∂c − cc − ,1, jiji − cc − ,1, jiji ≈ = method again… ∂x ,tx − xx ii −1 δx i− 1 j 2 ∂c ∂c − (2) 2 ∂x ,tx ∂x − 1 ,tx i+ 1 j 2 ji ()−−− cccc 2 −− ccc ∂ c 2 + − ,1,,,1 + − ,1,,1 jijijijijijiji 2 ≈ = 2 = 2 ∂x 1 − xx 1 δx δx 127 ,tx ji 2 ii −+ 2 Complete finite difference scheme

2 ∂c cd 2 ∂ c + 2 −− ccc − ,1,,1 jijiji = D 2 ≈ ∂t dx ∂x2 δx2 ,tx ji

∂c − cc ≈ + ,1, jiji ∂t δt ,tx ji

− cc ⎛ 2 −− ccc ⎞ + ,1, jiji = D + − ,1,,1 jijiji ⎜ 2 ⎟ δt ⎝ δx ⎠

128 Complete finite difference scheme

2 ∂c cd 2 ∂ c + 2 −− ccc − ,1,,1 jijiji = D 2 ≈ ∂t dx ∂x2 δx2 ,tx ji

∂c − cc ≈ + ,1, jiji ∂t δt Want ,tx ji

− cc ⎛ 2 −− ccc ⎞ + ,1, jiji = D + − ,1,,1 jijiji ⎜ 2 ⎟ δt ⎝ δx ⎠

δt cc += ()2 −− cccD + ,1, jiji δx2 + − ,1,,1 jijiji 129 Complete finite difference scheme ∂c 2cd i = = D Concentration at i space 2 at old time ∂t dx j = time

δt “explicit” numerical scheme cc += ()2 −− cccD + ,1, jiji δx 2 + − ,1,,1 jijiji (new concentration directly from concentration at earlier time)

Concentration at i Concentration at i-1 at old time at old time

Concentration at i Concentration at i+1 at new time at old time 130 Summary

Developed finite difference approach to solve for diffusion equation

Use parameter (D), e.g. calculated from molecular dynamics, and solve “complex” problem at continuum scale

Do not consider “atoms” or “particles”

131 Summary Multi-scale approach: Feed parameters from atomistic simulations to continuum models

Time scale “Empirical” Continuum or experimental model parameter feeding MD

Quantum mechanics

Length scale

132 MIT OpenCourseWare http://ocw.mit.edu

3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Spring 2012

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