Introduction to classical molecular dynamics (cont d) Mechanics of Ductile Materials

Transcription

1 Fom nano to maco: Intoduction to atomistic modeling techniques IAP 2007 Intoduction to classical molecula dynamics (cont d) xxx Mechanics of Ductile Mateials Lectue 3 Makus J. Buehle

2 Outline 1. Intoduction to Mechanics of Mateials Basic concepts of mechanics, stess and stain, defomation, stength and factue Monday Jan 8, 09-10:30am 2. Intoduction to Classical Molecula Dynamics Intoduction into the molecula dynamics simulation; numeical techniques Tuesday Jan 9, 09-10:30am 3. Mechanics of Ductile Mateials Dislocations; cystal stuctues; defomation of metals Tuesday Jan 16, 09-10:30am 4. Dynamic Factue of Bittle Mateials Nonlinea elasticity in dynamic factue, geometic confinement, intefaces Wednesday Jan 17, 09-10:30am 5. The Cauchy-Bon ule Calculation of elastic popeties of atomic lattices Fiday Jan 19, 09-10:30am 6. Mechanics of biological mateials Monday Jan. 22, 09-10:30am 7. Intoduction to The Poblem Set Atomistic modeling of factue of a nanocystal of coppe. Wednesday Jan 22, 09-10:30am 8. Size Effects in Defomation of Mateials Size effects in defomation of mateials: Is smalle stonge? Fiday Jan 26, 09-10:30am

3 Histoic MD efeences Alde, B. J. and Wainwight, T. E. J. Chem. Phys. 27, 1208 (1957) Alde, B. J. and Wainwight, T. E. J. Chem. Phys. 31, 459 (1959) Rahman, A. Phys. Rev. A136, 405 (1964) Stillinge, F. H. and Rahman, A. J. Chem. Phys. 60, 1545 (1974) McCammon, J. A., Gelin, B. R., and Kaplus, M. Natue (Lond.) 267, 585 (1977)

4 Outline and content (Lectue 3) Topic: Basic molecula dynamics (MD), inteatomic foces, popety calculation Examples: Movie of 1,000,000,000 atom simulation Simple Java applets Mateial coveed: Review, computing stategies, adial distibution function, diffusion, viial stess Impotant lesson: How to link micoscopic atomistic pocesses and popeties with macoscopic obsevables; fist simple model fo inteatomic potential Histoical pespective: Ealy MD simulations: Themodynamical popeties of wate o noble gases (1960s)

5 A simulation with 1,000,000,000 paticles Image emoved due to copyight estictions. (F. Abaham, et al.)

6 Molecula dynamics Paticles with mass m i z m y x d dt 2 j 2 = i (t) v i (t), a i (t) U ( j ) j = 1.. N j N paticles Total enegy of system E = K + 1 K = 2 U = U Coupled system N-body poblem, no exact solution fo N>2 System of coupled 2 nd ode nonlinea diffeential equations Solve by discetizing in time (spatial discetization given by atom size ) m U N 2 v j j= 1 ( j )

7 Solving the equations i ( t ) i ( t0 + Δt) i ( t0 + 2Δt) i ( t0 + 3Δt)... i ( t0 0 + nδt ) 2 )( Δ )... ( t0 + Δt) = ( t0 Δt) + 2 ( t0) + a ( t0 t + i i i i Positions at t 0 -Δt Velocities at t 0 Acceleations at t 0 How to obtain acceleations? Velet cental diffeence method f i = ma i a = f m Need foces on atoms! i i /

8 Time-discetization Time step Δt needs to be small enough to model the vibations of atomic bonds coectly Vibation fequencies may be extemely high, in paticula fo light atoms Thus: Time step on the ode of fs (10-15 seconds) Need 1,000,000 integation steps to calculate tajectoy ove 1 nanosecond: Significant computational buden u(t) t

9 Peiodic bounday conditions Sometimes, have peiodic bounday conditions; this allows studying bulk popeties (no fee sufaces) with small numbe of paticles (hee: N=3!) all paticles ae connected Oiginal cell suounded by 26 image cells; image paticles move in exactly the same way as oiginal paticles (8 in 2D) Paticle leaving box entes on othe side with same velocity vecto. Figue by MIT OCW. Afte Buehle.

10 Numeical implementation of MD

11 How ae foces calculated? Foces equied to obtain acceleations to integate EOM Foces ae calculated based on the distance between atoms; while consideing some inteatomic potential suface In pinciple, all atoms in the system inteact with all atoms: Need nested loop F = m d dt 2 j 2 = U ( j ) j = 1.. N j Foce: Patial deivative of potential enegy with espect to atomic coodinates

12 How ae foces calculated? Foce magnitude: Deivative of potential enegy with espect to atomic distance F = dφ( ) d To obtain foce vecto F i, take pojections into the thee axial diections F i = F x i F x 2 x 1 Assume pai-wise inteaction between atoms

13 Atomic scale Atoms ae composed of electons, potons, and neutons. Electon and potons ae negative and positive chages of the same magnitude, Coulombs Chemical bonds between atoms by inteactions of the electons of diffeent atoms (see QM pat late in IM/S!) e - e- Point epesentation e - p + n o n o p+ p + p + n o n o n o n o p + p + e - y (t) V(t) a(t) e - x e - Figue by MIT OCW. Figue by MIT OCW. Afte Buehle Makus J. Buehle, CEE/MIT

14 Atomic inteactions Pimay bonds ( stong ) Ionic, Covalent, Metallic (high melting point, K) Seconday bonds ( weak ) Van de Waals, Hydogen bonds (melting point K) Ionic: Non-diectional Covalent: Diectional (angles, tosions) Metallic: Non-diectional

15 Models fo atomic inteactions Atom-atom inteactions ae necessay to compute the foces and acceleations at each MD time integation step: Update to new positions! Usually define inteatomic potentials, that descibe the enegy of a set of atoms as a function of thei coodinates: U = U total total ( i ) Simple appoximation: Total enegy is sum ove the enegy of all pais of atoms in the system U = U ( ) 1 total 2 ij i j

16 Pai inteaction appoximation U = U ( ) 1 total 2 ij i j All pai inteactions of atom 1 with neighboing atoms All pai inteactions of atom 2 with neighboing atoms 1, Double count bond theefoe facto 2

17 Fom electons to atoms Electons Enegy Coe Distance Radius Govened by laws of quantum mechanics: Numeical solution by Density Functional Theoy (DFT), fo example

18 Repulsion vesus attaction Repulsion: Ovelap of electons in same obitals; accoding to Pauli exclusion pinciple this leads to high enegy stuctues Model: Exponential tem Attaction: When chemical bond is fomed, stuctue (bonded atoms) ae in local enegy minimum; beaking the atomic bond costs enegy esults in attactive foce Sum of epulsive and attactive tem esults in the typical potential enegy shape: U = U ep + U att

19 Lennad-Jones potential = ) ( σ σ ε φ Attactive Repulsive F d ) dφ( = x F F i i = x 1 x 2 F

20 The inteatomic potential The fundamental input into molecula simulations, in addition to stuctual infomation (position of atoms, type of atoms and thei velocities/acceleations) is povided by definition of the inteaction potential (equiv. tems often used by chemists is foce field ) MD is vey geneal due to its fomulation, but had to find a good potential (extensive debate still ongoing, choice depends vey stongly on the application) Popula: Semi-empiical o empiical (fit of caefully chosen mathematical functions to epoduce the potential enegy suface ) φ Inteaction epulsion Atomic scale (QM) o chemical popety Foces by dφ/d attaction

21 Lennad-Jones potential: Popeties φ ( ) = σ 4ε 12 σ 6 ε: Well depth (enegy pe bond) σ: Potential vanishes Equilibium distance between atoms 0 and maximum foce σ 6 2 = 0 F max,lj = ε σ

22 Pai potentials φ = ϕ( i ij j= 1.. N neigh ) 6 5 j=1 i Lennad-Jones 12:6 ϕ( ij σ ) = 4ε ij 12 σ ij 6 cut Reasonable model fo noble gas A (FCC in 3D) ϕ( ij ) Mose

23 Physical example: Suface stuctues Example: Suface effects in some mateials Need a desciption that includes the envionment of an atom to model the bond stength between pais of atoms Pai potentials: All bonds ae equal! Reality: Have envionment effects; it matte that thee is a fee suface!

24 MD updating scheme: Complete (1) Updating method (integation scheme) ( t0 + Δt) = ( t0 Δt) + 2 ( t0) Δt + a ( t0 t + f i = ma i i ai = Fi / i Positions at t 0 -Δt m i Positions at t 0 (2) Obtain acceleations fom foces i 2 )( Δ )... Acceleations at t 0 Velet cental diffeence method (5) Cystal (initial conditions) Positions at t 0 (3) Obtain foces fom potential F dv ( ) = d (4) Potential φ ( ) weak = F i = F σ 4ε. x 12 i σ 6 Coutesy of D. Helmut Foell. Used with pemission.

25 Neighbo lists Anothe bookkeeping device often used in MD simulation is a Neighbo List which keeps tack of who ae the neaest, second neaest,... neighbos of each paticle. This is to save time fom checking evey paticle in the system evey time a foce calculation is made. The List can be used fo seveal time steps befoe updating. Each update is expensive since it involves NxN opeations fo an N-paticle system. In low-tempeatue solids whee the paticles do not move vey much, it is possible to do an entie simulation without o with only a few updating, wheeas in simulation of liquids, updating evey 5 o 10 steps is quite common.

26 How ae foces calculated?

27 Analysis methods Last time, we discussed how to calculate: Tempeatue T = 3 2 K N k B K = N 1 2 Kinetic m v j 2 j= 1 enegy Potential enegy U = U ( j ) Pessue P = N / Vk Kinetic contibution B T Need othe measues fo physical and themodynamic popeties 1 3V i Volume j< i < ij dv d ij > Foce vecto multiplied by distance vecto Time aveage

28 MD modeling of cystals: Challenges of data analysis Cystals: Regula, odeed stuctue The coesponding paticle motions ae small-amplitude vibations about the lattice site, diffusive movements ove a local egion, and long fee flights inteupted by a collision evey now and then. MD has become so well espected fo what it can tell about the distibution of atoms and molecules in vaious states of matte, and the way they move about in esponse to themal excitations o extenal stess such as pessue. Figue by MIT OCW. Afte J. A. Bake and D. Hendeson. [J. A. Bake and D. Hendeson, Scientific Ameican, Nov. 1981].

29 Pessue, enegy and tempeatue histoy 6.25 Pessue Enegy Tempeatue Time Figue by MIT OCW. Time vaiation of system pessue, enegy, and tempeatue in an MD simulation of a solid. The initial behavio ae tansients which decay in time as the system eaches equilibium.

30 Pessue, enegy and tempeatue histoy 5.4 Pessue Enegy -5.0 Tempeatue Figue by MIT OCW. Time vaiation of system pessue, enegy, and tempeatue in an MD simulation of a liquid: Longe tansients

31 Analysis methods

32 Radial distibution function The adial distibution function is defined as g ( ) = ρ ( ) / ρ Local density Povides infomation about the density of atoms at a given adius ; ρ() is the local density of atoms g( ) = < N( ± Ω( ± Volume of this shell (d) Density of atoms (volume) Δ 2 Δ 2 ) > )ρ Aveage ove all atoms 2 g( )2π d = Numbe of paticles lying in a spheical shell of adius and thickness d

33 Radial distibution function consideed volume g( ) = N( ± Ω( ± Δ 2 Δ 2 ) )ρ ( Ω ± Δ ) 2 Paticle density ρ = N /V Note: RDF can be measued expeimentally using neutonscatteing techniques.

34 Radial distibution function Refeence atom Coutesy of the Depatment of Chemical and Biological Engineeing of the Univesity at Buffalo. Used with pemission.

35 Radial distibution function: Solid vesus liquid solid liquid A 2.5 B g(r) 3 g(r) Distance Distance Figue by MIT OCW. Intepetation: A peak indicates a paticulaly favoed sepaation distance fo the neighbos to a given paticle Thus: RDF eveals details about the atomic stuctue of the system being simulated Java applet:

36 Radial distibution function: JAVA applet Java applet: Image emoved fo copyight easons. Sceenshot of the adial distibution function Java applet.

37 Radial distibution function: Solid vesus liquid vesus gas Solid Agon 4 Gaseous A (90 K) g() Liquid Agon g() Liquid A (90 K) Gaseous A (300 K) distance/nm distance/nm Note: The fist peak coesponds to the neaest neighbo shell, the second peak to the second neaest neighbo shell, etc. Figue by MIT OCW. In FCC: 12, 6, 24, and 12 in fist fou shells

38 Mean squae displacement (MSD) function < Δ 2 >= 1 = i i N i ( ( t) ( t 0) ) 2 Position of atom i at time t Position of atom i at time t=0 If aveaged ove all paticles: Mean squae distance that paticles have moved duing time t (measue of the aveage distance a molecule tavels) MSD is zeo at t=0; gows like t 2 with a coefficient popotional to k B T/m Solid: Expect that MSD gows to a chaacteistic value (elated to fluctuations aound lattice site), then satuate Liquid: All atoms diffuse continuously though the mateial, as in Bownian motion Diffusion: Linea vaiation of MSD in time t

39 Mean squae displacement (MSD) function 25 8 x 102 B <Squae Displacement> A <Squae Displacement> Time Time Liquid Relation to diffusion constant: d lim t dt < Δ 2 >= 2dD Figue by MIT OCW. d=2 2D d=3 3D Cystal

40 Popety calculation in MD Time aveage of dynamical vaiable A(t) < A >= 1 lim t t t' = t t' = 0 A( t') dt' Time aveage of a dynamical vaiable A(t) < A >= 1 N t Aveage ove all time steps N t 1 A( t) N t in the tajectoy (discete) Coelation function of two dynamical vaiables A(t) and B(t) < A(0) B( t) 1 >= N N i= 1 1 N i N i k = 1 A ( t i k ) B i ( t k + t)

41 Oveview: MD popeties

42 Velocity autocoelation function < v(0) v( t) >= 1 N N i= 1 1 N i N i k = 1 v i ( t k ) v i ( t k + t) The velocity autocoelation function gives infomation about the atomic motions of paticles in the system Since it is a coelation of the paticle velocity at one time with the velocity of the same paticle at anothe time, the infomation efes to how a paticle moves in the system, such as diffusion Diffusion coeffecient (see e.g. Fenkel and Smit): t 1 ' = D0 = < v(0) v( t) > dt' 3 t' = 0 Note: Belongs to the Geen-Kubo elations can povide links between coelation functions and mateial tanspot coefficients, such as themal conductivity, diffusivity etc.

43 Velocity autocoelation function (VAF) Liquid o gas (weak molecula inteactions): Magnitude educes gadually unde the influence of weak foces: Velocity decoelates with time, which is the same as saying the atom 'fogets' what its initial velocity was. Then: VAF plot is a simple exponential decay, evealing the pesence of weak foces slowly destoying the velocity coelation. Such a esult is typical of the molecules in a gas. Solid (stong molecula inteactions): Atomic motion is an oscillation, vibating backwads and fowads, evesing thei velocity at the end of each oscillation. Then: VAF coesponds to a function that oscillates stongly fom positive to negative values and back again. The oscillations decay in time. This leads to a function esembling a damped hamonic motion.

44 Velocity autocoelation function solid t 1 ' = D0 = < v(0) v( t) > dt' 3 t' = 0 Coutesy of the Depatment of Chemical and Biological Engineeing of the Univesity at Buffalo. Used with pemission.

45 Velocity autocoelation function The velocity autoccoelation function fo an ideal gas, a dense gas, a liquid, and a solid

46 The concept of stess Foce F Undefomed vs. defomed (due to foce) σ = F A A = coss-sectional aea

47 Atomic stess tenso: Cauchy stess How to elate the continuum stess with atomistic stess Typically continuum vaiables epesent time-/space aveaged micoscopic quantities at equilibium Diffeence: Continuum popeties ae valid at a specific mateial point; this is not tue fo atomistic quantities (discete natue of atomic micostuctue) Discete fields u i (x) Displacement only defined at atomic site Continuous fields u i (x)

48 Atomic stess tenso: Viial stess Viial stess: σ ij Contibution by atoms moving though contol volume 1 1 = m + αuα iuα, j Ω α 2 α, β, a φ( ) i, j = β αβ Foce F i x 2 F αβ Atom β x 1 Atom α D.H. Tsai. Viial theoem and stess calculation in molecula-dynamics. J. of Chemical Physics, 70(3): , Min Zhou, A new look at the atomic level viial stess: on continuum-molecula system equivalence, Royal Society of London Poceedings Seies A, vol. 459, Issue 2037, pp (2003) Jonathan Zimmeman et al., Calculation of stess in atomistic simulation, MSMSE, Vol. 12, pp. S319-S332 (2004) and efeences in those aticles by Yip, Cheung et al.

49 + Ω = = α β β α α α α αβ φ σ a j i j i ij u u m,,,, ) ( Viial stess in 1D = Ω = β β α αβ φ σ a j i ij,, ) ( Foce between 2 paticles: F i = ) ( φ ( ) F F + Ω = σ F 1 F

50 Othe tanspot popeties

51 MD popeties: Classification Stuctual cystal stuctue, g(), defects such as vacancies and intestitials, dislocations, gain boundaies, pecipitates Themodynamic -- equation of state, heat capacities, themal expansion, fee enegies Mechanical -- elastic constants, cohesive and shea stength, elastic and plastic defomation, factue toughness Vibational -- phonon dispesion cuves, vibational fequency spectum, molecula spectoscopy Tanspot -- diffusion, viscous flow, themal conduction

52 Modeling vs. simulation Modeling: Building a mathematical o theoetical desciption of a physical situation; maybe esult in a set of patial diffeential equations Fo MD: Choice of potential, choice of cystal stuctue, Simulation: Numeical solution of the poblem at hand (code, infastuctue..) Solve the equations e.g. Velet method, paallelization (late) Simulation usually equies analysis methods postpocessing (RDF, tempeatue )

53 Limitations of MD: Electonic popeties Thee ae popeties which classical MD cannot calculate because electons ae involved. To teat electons popely one needs quantum mechanics. In addition to electonic popeties, optical and magnetic popeties also equie quantum mechanical (fist pinciples o ab initio) teatments.

54 What makes MD unique Unified study of all physical popeties. Using MD one can obtain themodynamic, stuctual, mechanical, dynamic and tanspot popeties of a system of paticles which can be a solid, liquid, o gas. One can even study chemical popeties and eactions which ae moe difficult and will equie using quantum MD. (adapted fom Sid. Yip, Nuclea Engg./MIT)

55 What makes MD unique Seveal hunded paticles ae sufficient to simulate bulk matte. While this is not always tue, it is athe supising that one can get quite accuate themodynamic popeties such as equation of state in this way. This is an example that the law of lage numbes takes ove quickly when one can aveage ove seveal hunded degees of feedom. (adapted fom Sid. Yip, Nuclea Engg./MIT)

56 What makes MD unique Unified study of all physical popeties. Using MD one can obtain themodynamic, stuctual, mechanical, dynamic and tanspot popeties of a system of paticles which can be a solid, liquid, o gas. One can even study chemical popeties and eactions which ae moe difficult and will equie using quantum MD. (adapted fom Sid. Yip, Nuclea Engg./MIT)

57 What makes MD unique Diect link between potential model and physical popeties. This is eally useful fom the standpoint of fundamental undestanding of physical matte. It is also vey elevant to the stuctue-popety coelation paadigm in mateials science. (adapted fom Sid. Yip, Nuclea Engg./MIT)

58 What makes MD unique Diect link between potential model and physical popeties. This is eally useful fom the standpoint of fundamental undestanding of physical matte. It is also vey elevant to the stuctue-popety coelation paadigm in mateials science. (adapted fom Sid. Yip, Nuclea Engg./MIT)

59 What makes MD unique Detailed atomic tajectoies. This is what one can get fom MD, o othe atomistic simulation techniques, that expeiment often cannot povide. This point alone makes it compelling fo the expeimentalist to have access to simulation. (adapted fom Sid. Yip, Nuclea Engg./MIT)

60 Summay Discussed additional analysis techniques: How to extact useful infomation fom MD esults Velocity autocoelation function Atomic stess Radial distibution function These ae useful since they povide quantitative infomation about molecula stuctue in the simulation; e.g. duing phase tansfomations, how atoms diffuse, elastic (mechanical) popeties Discussed some simple inteatomic potentials that descibe the atomic inteactions; condensing out electonic degees of feedom Elastic popeties: Calculate esponse to mechanical load based on viial stess Biefly intoduced the taining of potentials homewok assignment

61 Ductile vesus bittle mateials BRITTLE DUCTILE Glass Polymes Ice... Coppe, Gold Shea load Figue by MIT OCW.

62 Defomation of metals: Example Image emoved fo copyight easons. See: Fig. 4 at be/bwk/mateials/teaching/maste/wg02/l0310.htm. Image emoved fo copyight easons. See: Fig. 6 at be/bwk/mateials/teaching/maste/wg02/l0310.htm.

63 Defomation of mateials: Flaws o cacks matte Maco Stess σ Failue of mateials initiates at cacks Giffith, Iwine and othes: Failue initiates at defects, such as cacks, o gain boundaies with educed taction, nano-voids

64 Inglis solution: Elliptical hole and hole b σ σ 0 * y = * a σ b yy 2 a σ yy Model setup fo the Inglis/Kolosov poblem. Figue by MIT OCW. x Nomalized maximum stess Cicle (a = b) a >b b >a b a Cavity shape, b/a Plot of stess changes at the edge of elliptical cavities. Nomalized maximum stess is σ yy /σ 0 *; insets at top show ellipse oientations. The dashed hoizontal line shows the level of stess change in the plate without a cavity pesent. Aow shows stess concentation (3.0) fo the cicula hole (a =b). Figue by MIT OCW. A/4 F* A D=A/2 A/4 Geomety fo calculating stess in a plate with a cicula hole. Figue by MIT OCW. σ = * a σ 1 + ρ yy 2 0 Stess magnification