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

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1 MIT OpenCourseWare J / 1.021J / J / J / 22.00J Introduction to Modeling and Simulation Markus Buehler, Spring 2008 For information about citing these materials or our Terms of Use, visit:

2 1.021/3.021/10.333/18.361/22.00 Introduction to Modeling and Simulation Part II - lecture 5 Atomistic and molecular methods 1

3 Content overview I. Continuum methods 1. Discrete modeling of simple physical systems: Equilibrium, Dynamic, Eigenvalue problems 2. Continuum modeling approaches, Weighted residual (Galerkin) methods, Variational formulations 3. Linear elasticity: Review of basic equations, Weak formulation: the principle of virtual work, Numerical discretization: the finite element method II. Atomistic and molecular methods 1. Introduction to molecular dynamics 2. Basic statistical mechanics, molecular dynamics, Monte Carlo 3. Interatomic potentials 4. Visualization, examples 5. Thermodynamics as bridge between the scales 6. Mechanical properties how things fail 7. Multi-scale modeling 8. Biological systems (simulation in biophysics) how proteins work and how to model them III. Quantum mechanical methods 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? Lectures 2-10 February/March Lectures March/April Lectures April/May 2

4 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) Lecture 4: Interatomic potential and force field (pair potentials, fitting procedure, force calculation, multi-body potentials-metals/eam & applications, neighbor lists, periodic BCs, how to apply BCs) Lecture 5: Interatomic potential and force field (cont d) (organic force fields, bond order force fields-chemical reactions, additional algorithms (NVT, NPT), application: mechanical properties basic introduction) Lecture 6: Application to mechanics of materials-ductile materials (significance of fractures/flaws, brittle versus ductile behavior [motivating example], basic deformation mechanisms (cracking, dislocations), modeling approaches: metals-eam, brittle-pair potential/reaxff (silicon)) Lecture 7: Application to mechanics of materials-brittle materials; case study: supersonic fracture (example for model building); case study: fracture of silicon (hybrid model) Lecture 8: Review session Lecture 9: QUIZ 3

5 II. Atomistic and molecular methods Lecture 5: Interatomic potential and force field (cont d) Outline: 1. Brief review: Pair potential, EAM potential for metals 2. Chemical complexity ( more chemical bonds, beyond metallic bonding) 2.1 Force fields for organic chemistry - how to model proteins 2.2 Bond order force fields - how to model chemical reactions 3. Additional MD algorithms (NVT, NPT), choice of time step 4. Introduction: mechanical properties brittle versus ductile materials Goal of today s lecture: Learn how to model chemical bonds in organic structures, exemplified in proteins (application: protein folding) Introduce a method to model chemical reactions Introduce additional MD algorithms (NVT, NPT), discuss how to choose time step in MD 4

6 1. Brief review: Pair potential, EAM potential for metals 5

7 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 { r } j = 1 N r = j.. U total = U total (r) Depends on position of all other atoms Position vector of atom i Fi = rutotal( r) i = 1.. N i r i = r 1, i, r 2, i, r 3, i Change of potential energy due to change of position of 6 particle i ( gradient )

8 Pair potentials: energy calculation Simple approximation: Total energy is sum over the energy of all pairs of atoms in the system Pair wise interaction potential φ( r ij ) 3 4 rij 3 = 1 r 12 2 r distance between particles i and j Pair wise summation of bond energies avoid double counting N N 1 2 ) i= 1, i j j= 1 U total = φ( r ij Energy of atom i N U i = φ( r ij ) j= 1 7

9 Interatomic pair potentials: examples ( 2α ( r r )) 2D exp( ( r )) φ( rij) = D exp ij 0 α ij r0 Morse potential φ( r ij ) = σ 4ε rij 12 σ rij 6 Lennard-Jones 12:6 potential (excellent model for noble Gases, Ar, Ne, Xe..) φ( r ij ) r exp ij σ A C σ r = ij 6 Buckingham potential Harmonic approximation 8

10 Derivative of LJ potential ~ force, cutoff r cut F = dφ( r) d r φ(r) potential shift Beyond cutoff: Changes in energy (and thus forces) small 9

11 Are all bonds the same? metallic systems Pair potentials: All bonds are equal! Reality: Have environment effects; it matter that there is a free surface! 2006 Markus J. Buehler, CEE/MIT Bonds in a molecular structure depend on the environment! 10

12 Effective pair interactions: EAM potential EAM=Embedded atom method Can describe differences between bulk and surface Effective pair potential (ev ) Bulk Surface r 0,bulk r r 0,surface o r (A) 3 4 Figure by MIT OpenCourseWare. 1 U = φ( r + ij ) F( ρi ) 2 i, i j Pair potential energy j i Embedding energy as a function of electron density Embedding term: depends on environment, multi-body 11

13 Limitations of EAM potentials Bonding is non-directional: Good for metals like copper, gold, nickel, but not ideal for transition metals (e.g. Fe) and covalent systems Alloys are difficult to model, due to the chemical complexity and uncertainties of defining electron density functions EAM potentials are not unique: many formulations exist for the same metal No interactions between metal atoms and organic substance (e.g. oxygen, water, carbon compounds CH 4..) Today: Models for other materials 12

14 2. Chemical complexity 13

15 Atomistic structure of enzyme (protein) O, N, C, H Image from Wikimedia Commons, 14

16 Atomistic structure of structural proteins O, N, C, H Image removed due to copyright restrictions. Please see Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, or Alpha-helical coiled-coil protein structure In cell s cytoskeleton, hair, hoof 15

17 Catalysis reactions, involves metals and organics Figure by MIT OpenCourseWare. Water formation on a Pt surface Buehler, M. J., A. Duin, T. Jacob, B. Merinov, and W.A. Goddard. Formation of Water at a Pt(111) Catalyst Surface: A Study Using the ReaxFF Reactive Force Field. MRS Symposium Proceedings 900E (2006): O

18 Oxidation of aluminum surface 17

19 Many realistic materials phenomena require models to deal with a variety of chemical interactions... Here we will focus on two aspects: Chemical complexity, that is, the presence of a variety of distinct chemical bonds in a system, or bonds that change their chemical nature over time Chemical reactivity, that is, the breaking and formation of chemical bonds, leading to molecule formation, rupture, conformation changes etc. Sen, D., and M. J. Buehler. "Chemical Complexity in Mechanical Deformation of Metals." International Journal for Multiscale Computational Engineering 5 (2007):

20 Atomic interactions different types of chemical bonds Weaker bonding Primary bonds ( strong ) Ionic (ceramics, quartz, feldspar - rocks) Covalent (silicon) Metallic (copper, nickel, gold, silver) (high melting point, ,000K) Secondary bonds ( weak ) Van der Waals (wax, low melting point) Hydrogen bonds (proteins, spider silk) (melting point K) Ionic: Non-directional (point charges interacting) Covalent: Directional (bond angles, torsions matter) Metallic: Non-directional (electron gas concept) 19

21 Recall the pair potential Assumption: Total energy of system is expressed as sum of the energy due to pairs of atoms U total r 12 r with 3 4 N N 1 2 ) i= 1, i j j= 1 U total = φ( r ij φ = φ ( r ij 1 = N N... φ 2 ( φ + φ + φ + φ... + φ + φ φ + + ) ij ) N 1, N 20

22 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 U total = U + U + U + U + U Elec Covalent Metallic vdw H bond 21

23 Concept: energy landscape for chemically complex materials U total = U + U + U + U + U Elec Covalent Metallic vdw H bond 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 22

24 2.1 Force fields for organic chemistry - how to model proteins 23

25 Significance of proteins Proteins are basic building blocks of life Define tissues, organs, cells Provide a variety of functions and properties, such as mechanical stability (strength), elasticity, catalytic activity (enzyme), electrochemical properties, optical properties, energy conversion Molecular simulation is an important tool in the analysis of protein structures and protein materials Goal here: To train you in the fundamentals of modeling techniques for proteins, to enable you to carry out protein simulations Explain the significance of proteins (application) 24

26 Human body: Composed of diverse array of protein materials Eye s cornea (collagen material) Muscle tissue (motor proteins) Skin (complex composite of collagen, elastin) Cells (complex material/system based on proteins) Image removed due to copyright restrictions. Human Body 3D View image of whole bodies. Nerve cells Blood vessels Tendon (links bone, muscles) Cartilage (reduce friction in joints) Bone (structural stability) Image courtesy of NIH. and 25

27 Cellular structure: Protein networks Cell nucleus Actin network Microtubulus (e.g. cargo) Vimentin (extensible, flexible, provide strength) = cytoskeleton Image courtesy of NIH. 26

28 Image courtesy of NIH. 27

29 Protein structures define the cellular architecture Image courtesy of NIH. Intermediate filaments Image removed due to copyright restrictions. Please see Fig in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, or Courtesy Elsevier, Inc., Used with permission. 28

30 How protein materials are made the genetic code Proteins: Encoded by DNA (three letters ), utilize 20 basic building blocks (amino acids) to form polypeptides Polypeptides arrange in complex folded 3D structures with specific properties 1D structure transforms into complex 3D folded configuration ACGT Four letter code DNA Combination of 3 DNA letters equals a amino acid E.g.: Proline CCT, CCC, CCA, CCG Transcription/ translation.. - Proline - Serine Proline - Alanine -.. Sequence of amino acids polypeptide (1D structure) Folding (3D structure) 29

31 Chemical structure of peptides/proteins Typically short sequence of amino acids Longer sequence of amino acids, often complex 3D structure Image removed due to copyright restrictions. Please see: R = side chain, one of the 20 natural amino acids 20 natural amino acids differ in their side chain chemistry 30

32 Forms peptide bond Nonpolar Amino Acids H 3 N + H C COO - H 3 N + CH 3 C COO - H H Glycine (Gly) G Alanine (Ala) A NE NE CH 3 S CH 3 CH 3 CH H Valine (Val) V 6.0 E CH 3 CH 3 CH CH 2 H C COO - 3 N + H C COO - 3 N + H Leucine (Leu) L 6.0 E H N CH 3 H 3 N + CH 3 CH 2 CH C COO - H Isoleucine (lle) l 6.0 E R CH 2 CH 2 There are 20 natural amino acids Difference in side chain, R H 3 N + CH 2 C COO - H Phenylalanine (Phe) F 5.5 E H 3 N + OH CH 2 C COO - H Serine (Ser) S 5.7 NE H 3 N + CH 3 C COO - H Methionine (Met) M 5.7 E Polar Amino Acids (Neutral) Acidic Amino Acids O H 3 N + C CH 2 C COO - H Aspartic acid (Asp) D 2.8 NE CH 3 HCOH H 3 N + C COO - H Threonine (Thr) T 5.6 E O O- O - C H 3 N + CH 2 CH 2 C COO - H Glutamic acid (Glu) E 3.2 NE H 3 N + δ- CH 2 CH 2 H C COO - 2 N + H Proline (Pro) P 6.3 NE OH CH 2 C COO - H Tyrosine (Tyr) Y 5.7 NE H 3 N + H 3 N + H 3 N + SH CH 2 C COO - H Cysteine (Cys) C 5.1 NE CH 2 C COO - H Histidine (His) H 7.6 E CH 2 C COO - H Tryptophan (Trp) W 5.9 E Basic Amino Acids charges HN δ+ + NH O H 3 N + C NH 2 CH 2 C COO - H Asparagine (Asn) N 5.4 NE H 3 N + + NH 3 CH 2 CH 2 CH 2 CH 2 C COO - H Lysine (Lys) K 9.7 E O H 3 N + C CH 2 NH 2 CH 2 C COO - H Glutamine (Gln) Q 5.7 NE H 3 N + Figure by MIT OpenCourseWare. NH 2 C NH CH 2 CH 2 + NH 2 CH 2 C COO - H Arginine (Arg) R 10.8 E 31

33 Chemistry, structure and mechanical properties are linked Chemical structure Cartoon Presence of various chemical bonds: Covalent bonds (C-C, C-O, C-H, C-N..) Electrostatic interactions (charged amino acid side chains) H-bonds (e.g. between H and O) vdw interactions (uncharged parts of molecules) 32

34 Concept: split energy contributions U total = U + U + U + U + U Elec Covalent Metallic =0 for proteins vdw H bond Covalent bond described as 1. Bond stretching part (energy penalty for bond stretching) 2. Bending part (energy penalty for bending three atoms) 3. Rotation part (energy penalty for bond rotation, N 4) Consider ethane molecule as elastic structure Ethane C 2 H 6 U = U + U + Covalent stretch bend U rotate 33

35 Force fields for organics: Basic approach U total = U + U + U + U + U Elec Covalent Metallic =0 for proteins vdw H bond U = U + U + U Covalent 1 φstretch = kstretch( r r0 ) 2 U = φ stretch pairs stretch 1 φbend = kbend( θ θ0) 2 U = φ bend triplets bend 1 φrot = krot(1 cos( ϑ)) 2 U = φ rot rot quadruplets 2 2 stretch bend rot Images removed due to copyright restrictions. Please see: Fig. 2.18a,b,c in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer,

36 Model for covalent bonds φ stretch = kstretch( r r0 ) φ bend = kbend( θ θ0) φ rot 1 = k 2 rot (1 cos( ϑ)) Courtesy of the EMBnet Education & Training Committee. Used with permission. Images created for the CHARMM tutorial by Dr. Dmitry Kuznetsov (Swiss Institute of Bioinformatics) for the EMBnet Education & Training committee ( 35

37 Force fields for organics: Basic approach U total = U + U + U + U + U Elec Covalent Metallic =0 for proteins vdw H bond partial charges U Elec UElec : Coulomb potential φ ( r ) = ij q q i ε r 1 ij j Images removed due to copyright restrictions. Please see Fig. 2.18d in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, ε = 1 4 πε 0 ε 0 electrostatic constant distance Coulomb forces = C F( r ij qiq ) = ε r 1 j 2 ij 36

38 Force fields for organics: Basic approach U total = U + U + U + U + U Elec Covalent =0 for proteins Metallic vdw H bond U vdw Images removed due to copyright restrictions. Please see Fig. 2.18e in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, UvdW : LJ potential φ( r ij σ ) = 4ε rij 12 σ rij LJ potential is particularly good model for vdw interactions (Argon) 6 37

39 H-bond model U total = U + U + U + U + U Elec Covalent =0 for proteins Metallic vdw H bond H 2 O D U H bond H 2 O θ DHA H A H-bond Evaluated between acceptor (A) /donor(d) pairs Between electronegative atom and a H- atom that is bonded to another electronegative atom Slightly modified LJ, different parameters H bond H bond 4 U H bond : φ( ) H bond 5 R 6 R r ij = D cos ( θdha) rij rij r 38 ij = distance between D-A

40 Summary =0 for proteins U total = U Elec +U Covalent +U Metallic +U vdw +U H bond U Elec : Coulomb potential q i q φ(r ) = ij ε 1 r ij j 1 φ = k ( ) 2 stretch 2 r r stretch 0 1 φ = k ( ) 2 bend θ θ 2 bend 0 1 φ rot = k (1 cos( ϑ)) 2 rot 12 σ σ 6 LJ potential φ(rij) = 4ε r r ij ij 12 R 10 φ(r ) = 5 H bond R D H bond 4 6 ij H bond cos (θ r r ij ij 1 φ rot = k ot (1 cos(ϑ)) U Covalent = U stretch +U bend +U rot U vdw : U H bond : 2 r DHA) 39

41 The need for atom typing Limited transferability of potential expressions: Must use different potential for different chemistry Different chemistry is captured in different tags for atoms: Element type is expanded by additional information on particular chemical state Tags specify if a C-atom is in sp 3, sp 2, sp or in aromatic state (that is, to capture resonance effects) Example atom tags: CA, C_1, C_2, C_3, C, HN, HO, HC, sp3 sp2 sp 40

42 Atom typing in CHARMM 41

43 VMD analysis of protein structure 42

44 Common empirical force fields for organics and proteins Class I (experiment derived, simple form) CHARMM CHARMm (Accelrys) AMBER OPLS/AMBER/Schrödinger ECEPP (free energy force field) GROMOS Harmonic terms; Derived from vibrational spectroscopy, gasphase molecular structures Very systemspecific Class II (more complex, derived from QM) CFF95 (Biosym/Accelrys) MM3 MMFF94 (CHARMM, Macromodel ) UFF, DREIDING Include anharmonic terms Derived from QM, more general

45 CHARMM force field Widely used and accepted model for protein structures Programs such as NAMD have implemented the CHARMM force field Problem set 2, GenePattern NAMD module, study of a protein domain part of human vimentin intermediate filaments 44

46 Application protein folding ACGT Four letter code DNA Combination of 3 DNA letters equals a amino acid E.g.: Proline CCT, CCC, CCA, CCG Transcription/ translation.. - Proline - Serine Proline - Alanine -.. Sequence of amino acids polypeptide (1D structure) Folding (3D structure) Goal of protein folding simulations: Predict folded 3D structure based on polypeptide sequence 45

47 Movie: protein folding with CHARMM de novo Folding of a Transmembrane fd Coat Protein Polypeptide chain Images removed due to copyright restrictions. Screenshots from protein folding video, which can be found here: 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 46 solution-phase NMR on the protein in micelles)

48 What about chemical reactions? CHARMM and other related force fields can not describe reactivity, that is, the formation and breaking of bonds 47

49 Many realistic materials phenomena require models to deal with a variety of chemical interactions... Here we will focus on two aspects: Chemical complexity, that is, the presence of a variety of distinct chemical bonds in a system?? Chemical reactivity, that is, the breaking and formation of chemical bonds, leading to molecule formation, rupture etc. 48

50 3.2 Bond order force fields - how to model chemical reactions 49

51 Challenge: chemical reactions sp3 sp2 Energy???? Transition point Distance sp 2 sp 3 CHARMM-type potential can not describe chemical reactions 50

52 Why can not model chemical reactions with CHARMM-like potentials? φ φ stretch = kstretch( r r0 ) bend = kbend( θ θ0) Set of parameters only valid for particular molecule type / type of chemical bond k stretch, sp 2 k stretch, sp 3 Reactive potentials or reactive force fields overcome these limitations 51

53 How can one accurately describe the transition energies during chemical reactions? Use computationally more efficient descriptions than relying on purely quantum mechanical (QM) methods (see part III, methods limited to 100 atoms) q Key features of reactive potentials H q C C q H + H 2 H A 2 C = C H A 2 involves processes with electrons?? A--B q q q A--B q q q B B 52

54 Key features of reactive potentials Molecular model that is capable of describing chemical reactions Continuous energy landscape during reactions (key to enable integration of equations) No typing necessary, that is, atoms can be sp, sp2, sp3 w/o further tags only element types Computationally efficient (that is, should involve finite range interactions), so that large systems can be treated (> 10,000 atoms) Parameters with physical meaning (such as for the LJ potential) 53

55 Theoretical basis: bond order potential Concept: Use pair potential that depends on atomic environment (similar to EAM, here applied to covalent bonds) Modulate strength of attractive part (e.g. by coordination, or bond order ) Image removed due to copyright restrictions. Please see: Fig. 2 in Brenner, D. "The Art and Science of an Analytical Potential." Physica Status Solidi (b) 217 (2000): Abell, Tersoff Changes in spring constant as function of bond order Continuous change possible = continuous energy landscape during chemical reactions 54

56 Theoretical basis: bond order potential Image removed due to copyright restrictions. Please see: Fig. 2 in Brenner, D. "The Art and Science of an Analytical Potential." Physica Status Solidi (b) 217 (2000): D. Brenner,

57 Concept of bond order (BO) r BO sp3 1 sp2 sp

58 Bond order based energy landscape Bond length Bond length Pauling Bond order Energy Energy Bond order potential Allows for a more general description of chemistry All energy terms dependent on bond order Conventional potential (e.g. LJ, Morse) 57

59 Historical perspective of reactive bond order potentials 1985: Abell: General expression for binding energy as a sum of near nieghbor pair interactions moderated by local atomic environment 1990s: Tersoff, Brenner: Use Abell formalism applied to silicon (successful for various solid state structures) 2000: Stuart et al.: Reactive potential for hydrocarbons 2001: Duin, Godddard et al.: Reactive potential for hydrocarbons ReaxFF 2002: Brenner et al.: Second generation REBO potential for hydrocarbons : Extension of ReaxFF to various materials including metals, ceramics, silicon, polymers and more in Goddard s group 58

60 Example: ReaxFF reactive force field William A. Goddard III California Institute of Technology Courtesy of Bill Goddard. Used with permission. Adri C.T. v. Duin California Institute of Technology 59

61 Paper posted on MIT Server 60

62 ReaxFF: A reactive force field E system = Ebond + EvdWaals + ECoulomb + Eval, angle + E tors + E over + 2-body E under 3-body 4-body multi-body Total energy is expressed as the sum of various terms describing individual chemical bonds All expressions in terms of bond order All interactions calculated between ALL atoms in system No more atom typing: Atom type = chemical element 61

63 Example: Calculation of bond energy E + E + E + system = Ebond + EvdWaals + ECoulomb + Eval, angle tors over E under be,1 ( ij ) Ebond = De BO exp be,1 1 BO p ij p Bond energy between atoms i and j does not depend on bond distance Instead, it depends on bond order 62

64 Bond order functions BO goes smoothly from sp3 sp2 sp Images removed due to copyright restrictions. Please see Fig. 2.21c in Buehler, Markus J. Atomistic Modeling of Materials Failure. New York, NY: Springer, (1) (2) (3) β β β σ π ππ r π ij r ππ ij r ij BOij = exp ασ exp α π exp α + + ππ r0 r0 r0 Characteristic bond distance All energy terms are expressed as a function of bond orders 63

65 Illustration: Bond energy Image removed due to copyright restrictions. Please see slide 10 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF." 64

66 vdw interactions E + E + E + system = Ebond + EvdWaals + ECoulomb + Eval, angle tors over E under Accounts for short distance repulsion (Pauli principle orthogonalization) and attraction energies at large distances (dispersion) Included for all atoms with shielding at small distances Image removed due to copyright restrictions. Please see slide 11 in van Duin, Adri. "Dishing Out the Dirt on ReaxFF." 65

67 Resulting energy landscape Image removed due to copyright restrictions. Please see Fig. 3 in van Duin, C. T. Adri, et al. "ReaxFF: A Reactive Force Field for Hydrocarbons." Journal of Physical Chemistry A 105 (2001): Contribution of E bond and vdw energy 66

68 Current development status of ReaxFF La Ac : not currently described by ReaxFF A--B Allows to interface metals, ceramics with organic chemistry: Key for complex materials, specifically biological materials Periodic table courtesy of Wikimedia Commons. A 67 B

69 References Bond Order/Bond Length relationship L. Pauling. "The Nature of the chemical Bond" (Cornell University Press, Ithaca, 1960), 3rd ed. Pauling, J. Am. Chem. Soc., 69, 542 (1947). Bond order potentials G. C. Abell, Phys. Rev. B 31, 6184 (1985). J. Tersoff, Phys. Rev. Lett. 56, 632 (1986); Phys. Rev. B 37, 6991 (1988). D. W. Brenner, Phys. Rev. B 42, 9458 (1990). Brenner, D.W., et al., A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. Journal Of Physics-Condensed Matter, (4): p Stuart, S.J., A.B. Tutein, and J.A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions. Journal Of Chemical Physics, (14): p ReaxFF potential Duin, A.C.T.v., et al., ReaxFF SiO: Reactive Force Field for Silicon and Silicon Oxide Systems. J. Phys. Chem. A, : p Duin, A.C.T.v., et al., ReaxFF: A Reactive Force Field for Hydrocarbons. J. Phys. Chem. A, : p Buehler, M.J., A.C.T.v. Duin, and W.A. Goddard, Multi-paradigm multi-scale modeling of dynamical crack propagation in silicon using the ReaxFF reactive force field. Phys. Rev. Lett., Strachan, A., et al., Shock waves in high-energy materials: The initial chemical events in nitramine RDX. Physical Review Letters, (9). Nielson, K.D., et al., Development of the ReaxFF reactive force field for describing transition metal catalyzed reactions, with application to the initial stages of the catalytic formation of carbon nanotubes. J. Phys. Chem. A., : p. 49. Han, S.S., et al., Optimization and application of lithium parameters for the reactive force field, ReaxFF. Journal Of Physical Chemistry A, (20): p Chenoweth, K., et al., Simulations on the thermal decomposition of a poly(dimethylsiloxane) polymer using the ReaxFF reactive force field. Journal Of The American Chemical Society, (19): p Strachan, A., et al., Thermal decomposition of RDX from reactive molecular dynamics. Journal Of Chemical Physics, (5). Cheung, S., et al., ReaxFF(MgH) reactive force field for magnesium hydride systems. Journal Of Physical Chemistry A, (5): p van Duin, A.C.T., et al., Application of ReaxFF reactive force fields to transition metal catalyzed nanotube formation. Abstracts Of Papers Of The American Chemical Society, : p. U1031-U1031. Tao, L., et al., Mixed hybrid Dreiding-ReaxFF calculations for modeling enzymatic reactions in proteins. Under submission, M.J. Buehler. H. Tang, A. C.T. van Duin, W.A. Goddard III, "Threshold Crack Speed Controls Dynamical Fracture of Silicon Single Crystals", Physical Review Letters, Vol. 99, p , 2007

70 Case studies / examples: Atomistic simulation of reactive systems 69

71 Coupling mechanics - chemistry Application: corrosion, degradation surface properties γ gas-phase properties chemistry (surface) Elasticity E, µ Example: Environmentally assisted cracking (coupling mechanical properties-chemistry) 70

72 Mg-water interaction: How to make fire with water Images removed due to copyright restrictions. Images from video showing explosive reaction of magnesium, silver nitrate, and water, which can be accessed here: Mg 71

73 Mg water interaction ReaxFF MD simulation 72

74 Movies of ReaxFF simulations Images removed due to copyright restrictions. Stills from videos found at Shock-induced RDX reaction RDX/AlxOy/Al NVE-simulation 73

75 Formation of water Motivation: Water formation is an important chemical reactions Water plays a critical role in biological systems Water formation important in fuel cell applications Goal: Develop an atomistic model that allows proper description of chemistry of water formation Objective: Use the reactive force field applied to this system Pt/no Pt 2H 2 + O > 2H 2 O 74

76 Simulation geometry Pt H 2, O 2 N i molecules (of each component i) Pt Figure by MIT OpenCourseWare. Figure by MIT OpenCourseWare. Buehler, M. J., A. Duin, T. Jacob, B. Merinov, and W.A. Goddard. Formation of Water at a Pt(111) Catalyst Surface: A Study Using the ReaxFF Reactive Force Field. MRS Symposium Proceedings 900E (2006): O control system with same volume, but no Pt catalyst 75

77 Chemical bonds weaker System includes all of these bonds plus chemical reactions 76

78 Effect of Pt catalyst 5 4 Number of H O molecules over time 600 K with Pt 600 K without Pt 2 Water molecules Time ( ns) Figure by MIT OpenCourseWare. Buehler, M. J., A. Duin, T. Jacob, B. Merinov, and W.A. Goddard. Formation of Water at a Pt(111) Catalyst Surface: A Study Using the ReaxFF Reactive Force Field. MRS Symposium Proceedings 900E (2006): O3.9. MD simulation clearly proves the effect of the catalyst in greatly enhancing the reaction rate It also leads to more controlled reaction conditions 77

79 Formation mechanism H 2 O forms at the Pt (111) surface O 2 close to Pt surface Chemisorption of O 2 (Pt-O-O) Dissociation Pt-O and formation of Pt-O-H (stable) Formation of Pt-O-H 2 as another H 2 approaches; thereby leads to water and H-O-O molecule Several water molecules interact via hydrogen bonds Figure by MIT OpenCourseWare. Buehler, M. J., A. Duin, T. Jacob, B. Merinov, and W.A. Goddard. 78 Formation of Water at a Pt(111) Catalyst Surface: A Study Using the ReaxFF Reactive Force Field. MRS Symposium Proceedings 900E (2006): O3.9.

80 Reaction rate versus temperature Water molecules K 1200K 1000K 1100K 900K Time (picoseconds) Observe formation of water molecules at a time scale of several picoseconds The higher the temperature, the higher the production rate of water molecules The rates depend on concentration: The higher the concentration, the higher the rates. Need to be in the right MD window (time scale) Figure by MIT OpenCourseWare. Buehler, M. J., A. Duin, T. Jacob, B. Merinov, and W.A. Goddard. Formation of Water at a Pt(111) Catalyst Surface: A Study Using the ReaxFF Reactive Force Field. MRS Symposium Proceedings 900E (2006): O

81 Steered molecular dynamics (SMD) Steered molecular dynamics used to apply forces to protein structures Virtual atom moves w/ velocity v k f x f = k( v t x) end point of molecule f = k( v t x) deformation speed vector time Distance between end point of molecule and virtual atom 80

82 f SMD mimics AFM single molecule experiments Atomic force microscope k x k f x f x 81

83 Protein unfolding F PnIB 1AKG F M. Buehler, JoMMS, 2007 ReaxFF modeling 82

84 Protein unfolding Covalent bonds don t break CHARMM modeling M. Buehler, JoMMS,

85 Comparison CHARMM vs. ReaxFF M. Buehler, JoMMS,

86 3. Additional MD algorithms (NVT, NPT), choice of time step 85

87 NVE, NVT, NPT and other ensembles NVE ensemble (microcanonical): Constant number of particles, constant volume and constant energy, obtained from Verlet integration NVT ensemble (canonical): Constant temperature but no energy conservation NPT ensemble (isobaric-isothermal): Constant pressure and temperature, no energy conservation, no volume conservation Various algorithms exist to obtain dynamics for different ensembles, as for example Berendsen, Nosé-Hoover, Langevin dynamics, Parinello- Rahman Basic concept: Change integration scheme (e.g. Verlet method) so that the integration leads to the particular thermodynamical constraint Energy minimization: Obtain ground state energy with no kinetic energy (zero temperature); various computational methods exist, such 86 as Conjugate Gradient (CG), GLOK etc.

88 NVT - Berendsen thermostat / velocity rescaling Concept: velocities of all atoms are rescaled to move towards the desired temperature The parameter τ is a time constant that determines how fast the desired temperature T set is reached τ = rise time, describes the strength of the coupling of the system to a hypothetical heat bath Recall temperature T N 1 1 = < miv 3 Nk Rescaling parameter Δt T η = τ T set Modification of velocities v = vη new B i= 1 2 i > v i = v i η,new Velocity rescaling does not strictly conform to the canonical ensemble, recommend use of Langevin dynamics, Nose-Hoover (more complex) 87

89 NPT algorithm Control pressure and temperature Parrinello-Rahman approach Size and shape of the simulation cell are allowed to vary (periodic system, otherwise pressure zero) Basic idea: Change cell size so that the pressure approaches the desired, prescribed pressure tensor (straining the cell size of a periodic system changes its pressure) Photo removed due to copyright restrictions. Prof. Parrinello ETH Zurich change cell size to move towards desired pressure 88

90 Time scale dilemma () t = u () t + u () t u coarse fine The atomic displacement field consists of a low-frequency ( coarse ) and high frequency part ( fine ) u(t) t 89

91 Time-discretization Time step Δt needs to be small enough to model the vibrations of atomic bonds correctly Vibration frequencies may be extremely high, in particular for light atoms Thus: Time step on the order of fs (10-15 seconds) Need 1,000,000 integration steps to calculate trajectory over 1 nanosecond: Significant computational burden Time step is (typically) not varied during simulation; it is fixed Total time scale O(ns) 90

92 4. Introduction: mechanical properties brittle versus ductile materials 91

93 Ductile versus brittle materials BRITTLE DUCTILE Glass Polymers Ice... Copper, Gold Shear load Figure by MIT OpenCourseWare. 92

94 Goals in upcoming lectures How to build atomistic models to describe differences between ductile and brittle materials Understand basic deformation mechanisms (at nanoscale) that control macroscale behavior Case studies: silicon, copper, hyperelastic model materials 93

95 Fracture simulation: domain decomposition X Y V V Figure by MIT OpenCourseWare. Define domains to assign virtual atom types different chemistry in different domains 94