Introduction to classical molecular dynamics

Transcription

1 From nano o macro: Inroducon o aomsc modelng echnques IAP 2007 Inroducon o classcal molecular dynamcs xxx Lecure 2 Markus J. Buehler

2 Revew

3 Revew y P 1 P A z x P 2 Fgure by MIT OCW.

4 Revew Equlbrum Sran shear roaon shear Generalzed Hooke s law roaons

5 Revew Equlbrum for beam: q z =-ρga M y Soluon (negrang EQ eqs.):

6 See wre-up Topcs: Soluon beam problem

7 Oulne 1. Inroducon o Mechancs of Maerals Basc conceps of mechancs, sress and sran, deformaon, srengh and fracure Monday Jan 8, 09-10:30am 2. Inroducon o Classcal Molecular Dynamcs Inroducon no he molecular dynamcs smulaon; numercal echnques Tuesday Jan 9, 09-10:30am 3. Mechancs of Ducle Maerals Dslocaons; crysal srucures; deformaon of meals Tuesday Jan 16, 09-10:30am 4. Dynamc Fracure of Brle Maerals Nonlnear elascy n dynamc fracure, geomerc confnemen, nerfaces Wednesday Jan 17, 09-10:30am 5. The Cauchy-Born rule Calculaon of elasc properes of aomc laces Frday Jan 19, 09-10:30am 6. Mechancs of bologcal maerals Monday Jan. 22, 09-10:30am 7. Inroducon o The Problem Se Aomsc modelng of fracure of a nanocrysal of copper. Wednesday Jan 22, 09-10:30am 8. Sze Effecs n Deformaon of Maerals Sze effecs n deformaon of maerals: Is smaller sronger? Frday Jan 26, 09-10:30am

8 Oulne and conen (Lecure 2) Topc: Inroducon no basc molecular dynamcs (MD); underlyng heorecal conceps; numercal soluon Examples: Applcaon of molecular dynamcs o model fracure movang example o show how powerful he aomsc approach s for nsably problems Maeral covered: F=ma as bass for MD; Hamlonan; force calculaon from neraomc poenal; numercal negraon; hermodynamcal ensembles, defnons and ermnology; numercal ssues; me scale dlemma, preprocessng and npu parameers; compung sraeges; analyss and vsualzaon, daa exracon Imporan lesson: Applcably and challenges of classcal molecular dynamcs; beng able o mplemen your own MD code Hsorcal perspecve: Growh of compung power o enable MD modelng of larger sysems; developmen of poenals

9 Why molecular dynamcs?... f we have connuum mechancs avalable? Las / oday s lecure: Basc mechancs conceps (sress, sran, solvng some smple beam problems), elascy Soluons and conceps presened had parameers such as Young s modulus, a maeral propery relang sresses and srans; remaned unknown hroughou he lecure. These properes can be deermned expermenally alernave paradgm: All of hese properes can be solved by calculang sascal properes over a large number of parcles (aoms), whereby all aoms nerac accordng o specfc laws of neracon ha are conrolled by quanum mechancs (somemes also referred o as quanum chemsry) Laws of neracon beween parcles s ypcally referred o as poenals Here we presen an approach ha enables us o predc maeral properes based on fundamenal aomsc neracons, referred o as molecular dynamcs

10 Beam soluon q=-ρga

11 Connuum mechancs vs. aomsc vewpon Connuum vewpon no underlyng nhomogeneous mcrosrucure, ha s, maer can be dvded nfnely whou change of maeral properes (numercal mplemenaon: fne elemen mehod) Maeral properes bured n Young s modulus E Aomsc vewpon - consder he dscreeness of maer for example, he dscreeness of an aomc lace n a meal, where aoms are glued o her posons No spaal dscrezaon necessary gven by aomc dsances, e.g. lace Fgure by MIT OCW.

12 Objecve: Lnk nano o macro?? DFT or Emprcal or Sem-emprcal Fgure by MIT OCW. aomsc dscree hp:// /srucures/spheresprng_300_248.jpg connuum

13 Some remarks MD s no only suable for elascy problems; MD can also be used o solve plasc or fracure problems (dsspave), naurally, whou changng anyhng abou he procedure; s also capable of solvng he dynamcal evoluon of non-equlbrum processes Ths lecure wll help o apprecae MD as an alernave means of solvng mechancs problems, a he nersecons of: The meeng room s aomsc srucure common language! Bology Mechancs Chemsry Physcs Maerals scence

14 The problem o solve In aomsc smulaons, he goal s o model, analyze and undersand he moon of each aom n he maeral The collecve behavor of he aoms allows o undersand how he maeral undergoes deformaon, phase changes or oher phenomena, provdng lnks beween he aomc scale o meso or macro-scale phenomena Exracon of nformaon from aomsc dynamcs s ofen challengng Vbraon, change of locaon, connecvy and ohers hp:// Sprng connecs aoms Fgures by MIT OCW.

15 Molecular dynamcs MD generaes he dynamcal rajecores of a sysem of N parcles by negrang Newon s equaons of moon, wh suable nal and boundary condons, and proper neraomc poenals, whle sasfyng hermodynamcal (macroscopc) consrans Parcles wh mass m N parcles r () z x y v (), a ()

16 Molecular dynamcs Parcles wh mass m z m y x d d r 2 j 2 = r () v (), a () r U ( rj ) j = 1.. N j N parcles Toal energy of sysem E = K + 1 K = 2 U = U Coupled sysem N-body problem, no exac soluon for N>2 Sysem of coupled 2 nd order nonlnear dfferenal equaons Solve by dscrezng n me (spaal dscrezaon gven by aom sze ) m ( r j U N 2 v j j= 1 )

17 Solve hose equaons: Dscreze n me (n seps), Δ me sep: Taylor seres expanson ) (... ) 3 ( ) 2 ( ) ( ) ( n r r r r r Δ + Δ + Δ + + Δ Solvng he equaons )( )... ( 2 1 ) ( ) ( ) ( Δ + Δ + = Δ + a v r r Addng hs expanson ogeher wh one for : ( 0 ) r Δ )( )... ( 2 1 ) ( ) ( ) ( Δ + Δ = Δ a v r r

18 Solvng he equaons )( )... ( ) ( 2 ) ( ) ( Δ + Δ + Δ = Δ + a r r r ma f = Verle cenral dfference mehod )( )... ( 2 1 ) ( ) ( ) ( Δ + Δ + = Δ + a v r r )( )... ( 2 1 ) ( ) ( ) ( Δ + Δ = Δ a v r r + Posons a 0 Acceleraons a 0 Posons a 0 -Δ How o oban acceleraons? m f a / = Need forces on aoms!

19 Tme scale dlemma () = u () + u () u coarse fne The aomc dsplacemen feld consss of a low-frequency ( coarse ) and hgh frequency par ( fne ) Requres Δ fs or less u() Need o resolve!

20 Tme-dscrezaon Tme sep Δ needs o be small enough o model he vbraons of aomc bonds correcly Vbraon frequences may be exremely hgh, n parcular for lgh aoms Thus: Tme sep on he order of fs (10-15 seconds) Need 1,000,000 negraon seps o calculae rajecory over 1 nanosecond: Sgnfcan compuaonal burden Tme sep can (ypcally) no vared durng smulaon; s fxed Toal me scale O(ns)

21 Tme scale dlemma Calculae mely evoluon of large number of parcles (negrae usng Velocy Verle, for example) F = ma Polycrysal srucure Buld crysals, componens F Nano Need o resolve hgh frequency oscllaons, e.g. C-H bond (a nanoscale) Tme sep: fs Macro Fgure by MIT OCW. Tme scale range of MD: Pcoseconds o several nanoseconds Tmescale dlemma: No maer how many processors (how powerful he compuer), can only reach nanoseconds: can no parallelze me

22 Consequences of he me scale dlemma Very hgh sran raes n fracure or deformaon (dsplacemen km/sec) Lmed accessbly o dffusonal processes or any oher slow mechansms Unlke as for he scale problem (ably o rea more aoms n a sysem) here s no soluon n sgh for he me scale dlemma MD has o be appled very carefully whle consderng s range of valdy (wndow, nche: fracure deal, snce cracks move a km/sec) When vald, MD s very powerful and ncely complemens expermen and heory, bu has lmaons whch need o be undersood km/sec Yes Yes w/ lmaons No Fracure n model maerals Fracure n real maerals GB dffuson a hgh emperaures GB dffuson a low emperaures Plascy n model maerals Plascy n real maerals Fgure by MIT OCW. Afer Buehler, hp:// See also arcle by Ar Voer e al. on he me scale dlemma

23 Characerscs of MD Aomsc or molecular smulaons (molecular dynamcs, MD) s a fundamenal approach, snce consders he basc buldng blocks of maerals as s smalles eny: Aoms A he same, me, molecular dynamcs smulaons allow o model maerals wh dmensons of several hundred nanomeers and beyond: Allows o sudy deformaon and properes, mechansms ec. wh a very dealed compuaonal mcroscope, hus brdgng hrough varous scales from nano o macro possble by DNS Somemes, MD has been referred o as a frs prncples approach o undersand he mechancs of maerals (e.g. dslocaons are made ou of aoms ) Wh he defnon of he neraomc poenals (how aoms nerac) all maerals properes are defned (endless possbles & challenges )

24 Wha makes MD unque Unfed sudy of all physcal properes. Usng MD one can oban hermodynamc, srucural, mechancal, dynamc and ranspor properes of a sysem of parcles whch can be a sold, lqud, or gas. One can even sudy chemcal properes and reacons whch are more dffcul and wll requre usng quanum MD. Several hundred parcles are suffcen o smulae bulk maer. Whle hs s no always rue, s raher surprsng ha one can ge que accurae hermodynamc properes such as equaon of sae n hs way. Ths s an example ha he law of large numbers akes over quckly when one can average over several hundred degrees of freedom. Drec lnk beween poenal model and physcal properes. Ths s really useful from he sandpon of fundamenal undersandng of physcal maer. I s also very relevan o he srucure-propery correlaon paradgm n maerals scence. Complee conrol over npu, nal and boundary condons. Ths s wha gves physcal nsgh no complex sysem behavor. Ths s also wha makes smulaon so useful when combned wh expermen and heory. Dealed aomc rajecores. Ths s wha one can ge from MD, or oher aomsc smulaon echnques, ha expermen ofen canno provde. Ths pon alone makes compellng for he expermenals o have access o smulaon. (adaped from Sd. Yp, Nuclear Engrg./MIT)

25 Ergodc hypohess The converson of hs mcroscopc nformaon o macroscopc observables such as pressure, sress ensor, sran ensor, energy, hea capaces, ec., requres heores and sraeges developed n he realm of sascal mechancs Sascal mechancs s fundamenal o he sudy of many dfferen aomsc sysems, by provdng averagng procedure or lnks beween mcroscopc sysem saes of he many-parcle sysem and macroscopc hermodynamcal properes, such as emperaure, pressure, hea capacy ec. Temperaure Imporan: The Ergodc hypohess saes Ensemble average = Tme average (aomsc daa e.g. pressure usually no vald nsananeously n me and space)

26 Analyss of molecular dynamcs daa Temperaure: Pressure T = 3 2 K N k B Poenal conrbuon Knec conrbuon Why do we need nformaon abou emperaure and pressure? The nformaon on pressure, energy and emperaure s useful o make sure ha he sysem s well equlbraed and ha nohng srange s happenng durng he enre smulaon. Temperaure ec. are macroscopc properes, and hey do no ell us wha s happenng a he mcroscopc level (deals averaged ou)!

27 Mone Carlo (MC) echnques Mone Carlo (MC) echnques and alke have been developed o overcome some of he lmaons of dynamcal (MD) aomsc calculaons Insead of negrang he EOM, MC performs a random walk o measure properes: Randomly probng he geomery of he molecular sysem (confguraon space, accepance depends on cos funcon ) MC enables modelng of dffuson and oher slow processes (slow compared o he me scale of aomc vbraons) only hrough equlbrum There exs many dfferen flavors, ncludng Classcal MC (no nformaon abou dynamcs, only abou mechansms and seady sae properes, e.g. hermodynamcal varables) Knec MC (ge nformaon abou dynamcs) Advanced MD mehods (marrage beween MC and MD, e.g. Temp. Acc. Dyn.) Bas poenals (e.g. resrans) o faclae specfc evens by reducng he barrers Generally, MC echnques requre more knowledge abou he sysem of neres han MD hp:// D. Frenkel and B. Sm Undersandng Molecular Smulaons: from Algorhms o Applcaons, Academc Press, San Dego, 2nd edon (2002). hp://

28 Example: Measurng he average deph of he Charles Rver Classcal grd-based quadraure scheme: Dscreze problem and perform measuremens a grd pons Mone Carlo: Perform random walk hrough he rver; measuremens are performed only a acceped locaons Couresy of Google. Used wh permsson. Dfference o MD: Random walk s no real dynamcs; bu generaed arfcally hp:// hp://maps.google.com/

29 Remarks MD s an alernave approach o MC by samplng phase and sae space, bu obanng acual deermnsc rajecores; hus: Full dynamcal nformaon In long me lm, and for equlbrum properes, he resuls of MC correspond o resuls obaned by MD MD can model processes ha are characerzed by exreme drvng forces and ha are non-equlbrum processes, MC can no Example: Fracure

30 The numercal problem o solve Molecular dynamcs of mechancs applcaons can be compuaonally challengng, due o Complexes of force feld expressons (calculaon of aomc forces) Large number of aoms and hus large number of degrees of freedom n he sysem (3N) To model realsc (macro-engneerng) dmensons of maerals wh mcrosrucural feaures: Need sysem szes wh ~10 23 aoms (1 mole) Ths resuls n challenges for daa analyss and vsualzaon, or jus for daa handlng and sorage Much research has been done o advance daa analyss echnques and vsualzaon schemes (e.g., Vashshsa and coworkers a USC s cener for Advanced Compung and Smulaon, hp://cacs.usc.edu)

31 Dfferen hermodynamcal ensembles Inegrang he Verle equaons wll: Conserve oal energy (E=cons.) Keep number of parcles consan (N=cons.) Keep volume consan (V=cons.) Thus: Yelds an NVE ensemble ( mcrocanoncal ensemble ) Oher hermodynamcal ensembles can be realzed by changng he equaons of moon (e.g. NVT couplng o hea bah, canoncal ensemble ) K T = 1 2 = 3 2 m N 2 v j j= 1 K N k B Temperaure ~K 3 TNk B = K Thus: changng 2 veloces of aoms changes emperaure (effec of hea bah)

32 NVE, NVT and oher ensembles NVE ensemble: Consan number of parcles, consan volume and consan energy NVT ensemble (canoncal): Consan emperaure bu no energy conservaon NpT ensemble: Consan pressure and emperaure, no energy conservaon Varous algorhms exs o oban dynamcs for dfferen ensembles, as for example Nosé-Hoover, Langevn dynamcs, Parnello-Rahman and ohers Energy mnmzaon: Oban ground sae energy wh no knec energy (zero emperaure); varous compuaonal mehods exs, such as Conjugae Graden, GLOK ec.

33 NVT wh Berendsen hermosa Even smpler mehod s he Berendson hermosa, where he veloces of all aoms are rescaled o move owards he desred emperaure Rescalng sep Inegraon sep The parameer τ s a me consan ha deermnes how fas he desred emperaure s reached hp://

34 Nosé-Hoover NVT hermosa The negral hermosa mehod, also referred o as he exended sysem mehod nroduces addonal degrees of freedom no he sysem's Hamlonan Equaon of moon are derved for new Hamlonan. These equaons for he addonal degrees of freedom are negraed ogeher wh "usual" equaons for spaal coordnaes and momena. Nosé-Hoover: Reduce effec of bg hea bah aached o sysem o one degree of freedom number of degrees of freedom Couplng nera ransfer coeffcen hea beah hp://phycomp.echnon.ac.l/~phsorkn/hess/node42.hml

35 Numercal mplemenaon of MD

36 Typcal modelng procedure Se parcle posons Assgn parcle veloces Calculae force on each parcle Move parcles by mesep D Save curren posons and veloces Reached max. number of meseps? Sop smulaon Analyze daa prn resuls

37 Geomery of MD Typcally, have cubcal cell n whch parcles are placed n a regular or rregular manner gas (lqud) sold - crysal

38 Perodc boundary condons Somemes, have perodc boundary condons; hs allows sudyng bulk properes (no free surfaces) wh small number of parcles (here: N=3!) all parcles are conneced Orgnal cell surrounded by 26 mage cells; mage parcles move n exacly he same way as orgnal parcles (8 n 2D) Parcle leavng box eners on oher sde wh same velocy vecor. Fgure by MIT OCW. Afer Buehler.

39 How are forces calculaed? Recall: Forces requred o oban acceleraons o negrae EOM Forces are calculaed based on he dsance beween aoms; whle consderng some neraomc poenal surface (dscussed laer n hs lecure) In prncple, all aoms n he sysem nerac wh all aoms: Need nesed loop F = m d 2 d r 2 j = r U ( rj ) j = 1.. N j Force: Paral dervave of poenal energy wh respec o aomc coordnaes

40 How are forces calculaed? Force magnude: Dervave of poenal energy wh respec o aomc dsance F = dv ( r) d r To oban force vecor F, ake projecons no he hree axal drecons F = F x r F r x 2 x 1 Ofen: Assume par-wse neracon beween aoms

41 Mnmum mage convenon Snce n he par poenal approxmaon, he parcles nerac wo a a me, a procedure s needed o decde whch par o consder among he pars beween acual parcles and beween acual and mage parcles. The mnmum mage convenon s a procedure where one akes he neares neghbor o an acual parcle, regardless of wheher hs neghbor s an acual parcle or an mage parcle. Anoher approxmaon whch s useful o keep he compuaons o a manageable level s o nroduce a force cuoff dsance beyond whch parcle pars smply do no see each oher (see he force curve).

42 Mnmum mage convenon In order no o have a parcle nerac wh s own mage, s necessary o ensure ha he cuoff dsance s less han half of he smulaon cell dmenson. Couresy of Nck Wlson. Used wh permsson.

43 Neghbor lss Anoher bookkeepng devce ofen used n MD smulaon s a Neghbor Ls whch keeps rack of who are he neares, second neares,... neghbors of each parcle. Ths s o save me from checkng every parcle n he sysem every me a force calculaon s made. The Ls can be used for several me seps before updang. Each updae s expensve snce nvolves NxN operaons for an N-parcle sysem. In low-emperaure solds where he parcles do no move very much, s possble o do an enre smulaon whou or wh only a few updang, whereas n smulaon of lquds, updang every 5 or 10 seps s que common.

44 MD modelng of crysals: Challenges of daa analyss Crysals: Regular, ordered srucure The correspondng parcle moons are small-amplude vbraons abou he lace se, dffusve movemens over a local regon, and long free flghs nerruped by a collson every now and hen. MD has become so well respeced for wha can ell abou he dsrbuon of aoms and molecules n varous saes of maer, and he way hey move abou n response o hermal excaons or exernal sress such as pressure. [J. A. Barker and D. Henderson, Scenfc Amercan, Nov. 1981]. Fgure by MIT OCW. Afer J. A. Barker and D. Henderson.

45 Pressure, energy and emperaure hsory 6.25 Pressure Energy Temperaure Tme Fgure by MIT OCW. Tme varaon of sysem pressure, energy, and emperaure n an MD smulaon of a sold. The nal behavor are ransens whch decay n me as he sysem reaches equlbrum.

46 Pressure, energy and emperaure hsory 5.4 Pressure Energy -5.0 Temperaure Fgure by MIT OCW. Tme varaon of sysem pressure, energy, and emperaure n an MD smulaon of a lqud: Longer ransens

47 Ineraomc poenals

48 The neraomc poenal The fundamenal npu no molecular smulaons, n addon o srucural nformaon (poson of aoms, ype of aoms and her veloces/acceleraons) s provded by defnon of he neracon poenal (equv. erms ofen used by chemss s force feld ) MD s very general due o s formulaon, bu hard o fnd a good poenal (exensve debae sll ongong, choce depends very srongly on he applcaon) Popular: Sem-emprcal or emprcal (f of carefully chosen mahemacal funcons o reproduce he energy surface ) Parameers Lennard-Jones Ineracon r φ repulson Forces by dφ/dr aracon r Or more sophscaed poenals (mul-body poenals EMT, EAM, TB )

49 Aomsc mehods n mechancs Use MD mehods o perform vrual expermens Compuaonal mcroscope As long as vald, deal mehod o gan fundamenal undersandng abou behavor of maerals Have nrnsc lengh scale gven by he aomc scale (dsance) Handles sress sngulares nrnscally Ideal for deformaon under hgh sran rae ec., no accessble by oher mehods (FE, DDD..)

50 Movaon: Fracure Maerals under hgh load are known o fracure MD modelng provdes an excellen physcal descrpon of he fracure processes, as can naurally descrbe he aomc bond breakng processes Oher modelng approaches, such as he fne elemen mehod, are based on emprcal relaons beween load and crack formaon and/or propagaon; MD does no requre such npu Wha s fracure?

51 Schemac of sress feld around a sngle (sac) crack ensle sress shear The sress feld around a crack s complex, wh regons of domnang ensle sress (crack openng) and shear sress (dslocaon nucleaon)

52 The aomc vewpon If n some caaclysm all scenfc knowledge were o be desroyed and only one senence passed on o he nex generaon of creaures, wha saemen would conan he mos nformaon n he fewes words? I beleve s he aomc hypohess ha all hngs are made of aoms - lle parcles ha move around n perpeual moon, aracng each oher when hey are a lle dsance apar, bu repellng upon beng squeezed no one anoher. In ha one senence, you wll see here s an enormous amoun of nformaon abou he world, f jus a lle magnaon and hnkng are appled. --Rchard Feynman