Molecular Dynamics. r F. r dt. What is molecular dynamics?

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1 What s molecula dynamcs? Molecula Dynamcs Molecula dynamcs (MD) s a compute smulaton technque that allows one to pedct the tme evoluton of a system of nteactng patcles (atoms, molecules, ganules, etc.). The basc dea s smple. Fst, fo a system of nteest, one has to specfy: a set of ntal condtons (ntal postons & veloctes of all patcles n the system) nteacton potental fo devng the foces among all the patcles. Second, the evoluton of the system n tme can be followed by solvng a set of classcal equatons of moton fo all patcles n the system. Wthn the famewok of classcal mechancs, the equatons that goven the moton of classcal patcles ae the ones that coespond to the second law of classcal mechancs fomulated by S Isaac Newton ove 300 yeas ago: m a F dv dt F 2 d = o m = o m = 2 fo the th patcle dt F Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

2 Molecula Dynamcs Solvng a set of classcal equatons of moton fo all patcles n the system m a F dv dt F 2 d = o m = o m = 2 fo the th patcle If the patcles of nteest ae atoms, and f thee ae a total of N at of them n the system, the foce actng on the th atom at a gven tme can be obtaned fom the nteatomc potental U( 1, 2, 3,, Nat ) that, n geneal, s a functon of the postons of all the atoms: F = - U(, 1 2,,..., 3 N at Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle ) Once the ntal condtons and the nteacton potental ae defned, the equatons of moton can be solved numecally. The esult of the soluton ae the postons and veloctes of all the atoms as a functon of tme, () t, v ( t ) Advantages of MD: the only nput n the model descpton of nteatomc/ntemolecula nteacton no assumptons ae made about the pocesses/mechansm to be nvestgated povdes a detaled molecula/atomc-level nfomaton dt F Results of the computatonal expement may lead to the dscove new physcs/mechansms!

3 Schematc dagam of a basc MD code Defne ntal postons and veloctes ( t ) and v ( t ) 0 0 Calculate foces at cuent tme t n : F = - U(, 1 2,,..., 3 N at ) Solve equatons of moton fo all patcles n the system ove a shot tmestep Δt. ( t ) ( t ) n n+1 v ( t ) v ( t ) n tn +1 = tn + Δt n+1 Calculate desed physcal quanttes, wte data to tajectoy fle Is t n+1 > t max? Wte to the dsc fnal atomc confguaton & fnsh Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

4 Molecula Dynamcs Example collson of a doplet wth a substate (by Yasush Katsum, UVa) 500 m/s MD Intal condtons ae specfed, (t 0 ) and v (t 0 ) Snapshot fom MD smulaton at tme t n = 100 ps MD s a detemnstc technque: gven ntal postons and veloctes, the evoluton of the system n tme s, n pncple, completely detemned (n pactce, accumulaton of ntegaton and computatonal eos would ntoduce some uncetanty nto the MD output). MD can be also used as a statstcal mechancs method: t geneates a set of confguatons that ae dstbuted accodng to statstcal dstbuton functons. In many cases we ae not nteested n tajectoes of ndvdual atoms, we ae nteested n macoscopc popetes of the mateal. MD nfomaton can be aveaged ove all the atoms n the system and ove tme to obtan themodynamc paametes. The man stengths of the MD method s the ablty to study fast non-equlbum pocesses wth atomc-level esoluton (e.g. mcoscopc mechansms of damage/plastc defomaton due to a shock wave popagaton, dynamc factue and cack gowth, on bombadment, cluste mpact, Unvesty etc.). Fo many of Vgna, of these MSE poblems, 4270/6270: MD Intoducton method does to Atomstc not have an Smulatons, altenatve. Leond Zhgle

5 Lmtatons of the MD technque 1. Classcal descpton of nteatomc nteacton Electons ae not pesent explctly, they ae ntoduced though the potental enegy suface that s a functon of atomc postons only (Bon-Oppenheme appoxmaton). The potental enegy suface, n tun, s appoxmated by an analytc functon that gves the potental enegy U as a functon of coodnates. Foces ae obtaned as the gadent of a potental enegy functon, F = U(1, 2,..., N ) Potental enegy sufaces (solutons of electonc Schödnge equaton wthn the Bon- Oppenheme appoxmaton) ae not avalable fo pactcally nteestng systems. The choce of a potental functon that appoxmates the actual (unknown) soluton of the Schödnge equaton s a dffcult task. Desgn of the potental functon and choce of the paametes s often based on fttng to avalable expemental data (e.g. equlbum geomety of stable phases, cohesve enegy, elastc modul, vbatonal fequences, tempeatues of the phase tanstons, etc.). Avalablty of good potental functons s one of the man condtons fo expanson of the aea of applcablty of the MD smulatons to the ealstc quanttatve analyss of the behavo and popetes of eal mateals. The Bon-Oppenheme appoxmaton and dffeent types of the potental functons wll be dscussed late n the couse. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

6 Lmtatons of MD: Classcal descpton of atomc motons 2. Classcal descpton of atomc moton In classcal MD we eplace Schödnge equaton fo nucle wth classcal Newton equaton. One ndcato of the valdty of the eplacement s the de Bogle wavelength Λ. Quantum effects ae expected to become sgnfcant when Λ s much lage that ntepatcle dstance. Fo themal moton we can use the themal de Bogle wavelength: Fo T = 300 K we have Λ th = 1 Å fo a H atom (m H = 1 amu) Λ th = 0.19 Å fo a S atom (m S = 28 amu) Λ th = 0.07 Å fo a Au atom (m Au = 197 amu) Typcal nteatomc spacng n sold-state mateals s d ~ 1-3 Å. Theefoe: Λ th = h 2π mk All atoms, except fo the lghtest ones such as H, He, Ne, can be consdeed as pont patcles at suffcently hgh tempeatue (d >> Λ) and classcal mechancs can be used to descbe the moton. B T Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

7 Lmtatons of the MD technque The classcal appoxmaton s athe poo fo lght elements (e.g. H, He) and quantum coectons ae often supemposed on the classcal descpton of moton. Example: H 2 O Let s ty to use classcal equpatton pncple to calculate the heat capacty of wate vapo. Rotaton E th = 3 ½RT (tansl.) + 3 ½ RT (otat.) ½ RT (vb.) = 6 RT c v = 6R Vbaton Tanslaton But expemental c v s much smalle. At T = 298 K H 2 O gas has c v = 3.038R. What s the eason fo the lage dscepancy? ν (cm -1 ) The table shows the vbatonal fequences of wate along wth the populaton of the fst excted state at 600 K Exp(-hν/kT) 1.0 x x x 10-5 Fo the hgh fequency OH stetchng motons, thee should be essentally no molecules n the fst vbatonal state even at 600 K. Fo the lowe fequency bendng moton, thee wll be about 2% of the molecules excted. Contbutons to the heat capacty can be consdeed classcally only f E n ~ hν << k B T. Enegy levels wth E n kt contbute lttle, f at all, to the heat capacty. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

8 Lmtatons of the MD technque Example: H 2 O ν (cm -1 ) Exp(-hν/kT) Rotaton (contnued) x x x 10-5 Vbaton Tanslaton OH stetch and, n a bg pat, OH bend eman n the gound vbatonal state at any T easonable fo H 2 O. => It would be wong to study ths system classcally (enegy would leak nto the vbatonal modes and would gve a wong heat capacty and heat conducton). Quantum coecton n MD smulatons: H 2 O ae usually consdeed to be gd (bendng and stetchng motons ae fozen (e.g. SHAKE/RATTLE methods). Quantum effects can become sgnfcant n any system as soon as T s suffcently low (e.g. tempeatue dependence of the heat capacty below the Debye tempeatue can not be explaned n classcal appoxmaton). Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

9 Examples of MD smulatons of wate Substate-asssted lase-ntated ejecton of a poten molecule embedded n a wate flm A small bomolecule, enkephaln (574 Da) H 2 O flm Au substate Dou et al., J. Phys. Chem. B 107, 2362, Equatons of moton ae ntegated usng the velocty Velet algothm n conjuncton wth the RATTLE constant method to mantan fxed O-H bond lengths and a fxed H-O-H bond angle. Molecula mechansm of wate evapoaton Nagata et al., Phys. Rev. Lett. 115, , 2015 flexble POLY2VS and gd SPC/E foce felds fo wate ae used, ntegaton tme step s 0.4 fs, 500 molecules ae smulated fo 600 ps at 331 K. Smulatons ae epeated 32 tmes. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

10 3. Tme- and length-scale lmtatons Lmtatons of the MD technque The lmtatons on the sze of the MD computatonal cell (numbe of atoms) and tme of the smulaton constan the ange of poblems that can be addessed by the MD method. Tme-scale: The maxmum tmestep of ntegaton n MD smulaton s defned by the fastest moton n the system. Vbatonal fequences n a molecula system ae up to 3000 cm -1 whch coesponds to a peod of ~10 fs. Optcal phonon fequences ae ~10 THz - peod of ~100 fs. Theefoe, a typcal tmestep n MD smulaton s on the ode of a femtosecond. Usng moden computes t s possble to calculate tmesteps. Theefoe we can only smulate pocesses that occu wthn ns. Ths s a seous lmtaton fo many poblems that nvolve themally-actvated pocesses, cluste/vapo flm deposton, annealng of adaton damage, etc. Seveal methods fo acceleaton of nfequent themally actvated events have been developed by Vote (Los Alamos), Fchthon (Penn State), and othes. The methods have been appled to study pocesses of suface dffuson, flm deposton, evoluton of pont defects. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

11 Length-scale: Lmtatons of the MD technque The sze of the computatonal cell s lmted by the numbe of atoms that can be ncluded n the smulaton, typcally Ths coesponds to the sze of the computatonal cell on the ode of tens of nm. Any stuctual featues of nteest and spatal coelaton lengths n the smulaton should be smalle than the sze of the computatonal cell. To make sue that the fnte sze of the computatonal cell does not ntoduce any atfacts nto the smulaton esults, one can pefom smulatons fo systems of dffeent sze and compae the pedcted popetes. ecod MD smulatons Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

12 Lmtatons of the MD technque Due to the lmtatons on the sze of the MD computatonal cell, an mpotant aspect of any MD smulaton s an adequate descpton of the nteacton of atoms n the MD computatonal cell wth suoundng nfnte mateal. We have to defne bounday condtons and apply specal methods fo tempeatue and pessue contol n the MD cell (heat and wok exchange between the MD computatonal cell and the suoundngs). We wll dscuss these ssues late n the couse. Lage extenal system MD MD Lage extenal system Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

13 Lmtatons of MD: Small tme- and length-scales effectve/macoscopc mateal behavo and popetes (consttutve elatons) Numbe of atoms ~ (sze of the system) 3 Computatonal cost ~ (numbe of atoms) n n > 1 Ttan (Oak Rdge Leadeshp Computng Faclty) Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

14 Dect MD smulatons s nanostuctued mateals 1 2 (1) MD smulatons of ndvdual stuctual elements (nanofbes, nanopatcles, ntefacal egons, gan boundaes, etc.) dffcult to pedct macoscopc popetes of nanomateals (2) Dect lage-scale MD smulatons of nanomateals. Nanocystallne mateals - system wth tens of nanogans (~ atoms) can be smulated and the effectve popetes can be nvestgated, patculaly n the egme of ultafast mechancal loadng (e.g. shock wave) o heatng (e.g. by shot lase pulse). Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

15 Dect MD smulatons s nanostuctued mateals 2 Nucleaton, gowth, and coalescence of vods n dynamc falue n nanocystallne Cu subjected to shock pulse loadng. Dongae, Rajendan, LaMattna, Zky, Benne, J. Appl. Phys. 108, (2010) Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

16 Dect MD smulatons s nanostuctued mateals 2 T = 0.94 T m Lase meltng of thn Au flms: The knetcs of meltng ae elated to tme-esolved electon dffacton Unvesty of measuements. Vgna, MSE 4270/6270: Ln et al., J. Intoducton Phys. Chem. to CAtomstc 114, 5686 Smulatons, (2010) Leond Zhgle

17 Bdgng the gap: The need fo mesoscopc models mesoscopc models dslocaton stuctues CNT mateals and CNT-polyme matx nanocompostes Unvesty J. Eng. Mate. Technol. of Vgna, 131, MSE (2009) 4270/6270: Hennch Intoducton et al., PCCP to 4, 2273 Atomstc (2002) Smulatons, Leond Zhgle Phan et al., Acta Mate. 59, 2172 (2011) Phys. Rev. B 71, (2005); J. Phys. Chem. CD. 114, Qan 5513 et (2010); al., APL PRL76, 104, (2000) (2010)

18 Bdgng the gap: The need fo mesoscopc models mesoscopc models Examples of mesoscopc models: system/phenomenon specfc Dslocaton Dynamcs fo ealy stages of plastc defomaton Mesoscopc methods fo evoluton of gan stuctue n polycystallne mateals (e.g. phase feld models, cellula automata, knetc Monte Calo Potts models) Coase-ganed models fo molecula and bomolecula systems Mesoscopc models fo cabon nanotubes and nanofbous mateals Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

19 Fst MD smulatons The fst smulaton usng the MD method was epoted n 1957 by Alde and Wanwght [Phase tanston fo a had sphee system, J. Chem. Phys. 27, , 1957]. They nvestgated a sold-flud tanston n a system composed of had sphees nteactng by nstantaneous collsons. Fo a system of 500 patcles, smulaton of 500 nte-patcle collsons took ~ an hou on IBM 704 compute. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

20 Fst MD smulatons Contnuous epulsve Bon-Maye nteacton potental was used fo the fst tme n MD smulaton of adaton damage n a Cu taget pefomed at Bookhaven Natonal Lab. n 1960 [J.B. Gbson, A.N. Goland, M. Mlgam, and G.H. Vneyad, Dynamcs of adaton damage, Phys. Rev. 120, , 1960]. A constant nwad foce was appled to each atom on the bounday of the cystallte to account fo the attactve pat fo the nteatomc nteacton. Ths was pobably the fst applcaton of the MD method n mateals scence. Computatonal cell composed of 446 to 998 coppe atoms was smulated. One ntegaton step took about a mnute on an IBM 704 compute. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

21 Fst MD smulatons Aneesu Rahman n 1964 used Lennad-Jones potental to descbe both attactve and epulsve nteacton n a system of 864 agon atoms [Phys. Rev. 136, A405-A411, 1964 ]. The methods of the smulaton and analyss of the MD esults descbed n ths pape ae stll used n many pesent MD smulaton studes. Pa coelaton functon, velocty autocoelaton functon, and mean squae dsplacement calculated fo lqud A. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

22 Contol Data Copoaton (CDC) 3600 at NCAR ( ) beautful compute wth smoked glass panels and a sold and stunnng look fom seveal mllon dollas Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

23 Cuent applcatons of the MD smulaton technque Snce the tme the MD method was ntoduced, t has been used to nvestgate a wde ange of poblems n dffeent eseach felds, e.g. Chemsty and Bochemsty: molecula stuctues, eactons, dug desgn, vbatonal elaxaton and enegy tansfe, stuctue of membanes, dynamcs of lage bomolecules, poten foldng, Statstcal Mechancs and Physcs: theoy of lquds, coelated many-body moton, popetes of statstcal ensembles, stuctue and popetes of small clustes, phase tanstons.. Mateals Scence: pont, lnea, and plana defects n cystals and the nteactons, mcoscopc mechansms of factue, suface econstucton, meltng and facetng, flm gowth, fcton, Shockwave-nduced plastcty [B.L. Holan and Actn flaments, smulaton by Unvesty P.S. Lomdahl, of Vgna, Scence MSE 4270/6270: 280, 2085 Intoducton (1998)] to Atomstc W. Wgges, Smulatons, Unvesty Leond Zhgle of Illnos

24 MD smulaton of human cowds Is t possble to epoduce the collectve behavo of a cowd of pedestans wth a MD model? A socal potental can be ntoduced to descbe nteacton of humans wth the neghbos n a cowd. The nteacton should be velocty-dependent: acceleaton s vey dffeent fo head on and paallel tajectoes empcal potental based on statstcal analyss of cowd data sets: F j E( τ) = kτ 2 e τ/ τ 0 whee τ s tme to collson,.e., tme fo whch two pedestans could contnue walkng wth the cuent veloctes befoe colldng f pedestans = dscs wth ad R and R j, τ = (b - d 1/2 )/a, whee a = v j 2, b = -x j v j, c = x j 2 -(R + R j ) 2, and d = b 2 -ac, then: τ/ τ 2 0 ( ) 2 τ/ τ 2 1 v 0 = ( τ) = ( τ ) = ke j xj xjvj v + x E x k e v j j 2 2 j vj τ τ τ ( x ) ( ( + ) ) jvj vj xj R R j An addtonal dvng foce defnng the desed decton of moton s also assgned to each pedestan. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, PRL, Leond 113, Zhgle , 2014 j

25 MD: Recod smulatons (length-scale): Tllons (?) of atoms Tllon-atom molecula dynamcs Gemann and Kadau Int. J. Mod. Phys. C 19, 1315, μm 3 cube Smulaton of a cube composed of Lennad-Jones (LJ) atoms ( atoms, edge length of about 2.5μm) aanged nto smple cubc lattce was pefomed n 2008 by T. C. Gemann and K. Kadau [Int. J. Mod. Phys. C 19, 1315 (2008)] on 212,992 pocessos of LLNL s BlueGene/L cluste. The test un took ~30 mnutes fo 40 tmesteps of ntegaton. Ductle falue of a FCC sold unde tenson The system s a slab wth 1008 atoms along the thee othogonal sdes, the total numbe of atoms s 1,023,103,872, nteacton s descbed by Lennad-Jones potental. The total smulaton tme s 200,000 tmesteps. It takes 1.7 seconds pe tmestep fo a 4096-node smulaton on ASCI Whte compute (~fou clockdays of total smulaton tme). Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

26 MD: Recod smulatons (length-scale): Bllons of atoms MD smulaton of bubble nucleaton and gowth Watanabe, Suzuk, Ito, Comput. Phys. Commun. 184, 2775, 2013 LJ potental wth cutoff at 3σ, 1,449,776,020 atoms CDC-3600 (NCAR) agon atoms [Rahman, Phys. Rev. 136, A405, 1964 ] OLCF 1,000,000,000,000 atoms 2014 Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle

27 MD: Recod smulatons (tme-scale): quest fo mllseconds Poten foldng The poten-foldng poblem s one of the majo challenges of molecula bology. Functonalty of potens s dectly elated to the confomaton - the complex folds allow potens to latch onto othe molecules and cay out ts bologcal ole. To nvestgate the poten foldng poblem n MD smulaton one has to follow the evoluton of a lage molecule n a soluton fo at least mcoseconds. MD smulaton of the foldng of the vlln headpece, one of the fastest-foldng potens Natue 451, 240, 2008 Foldng@home: a dstbuted computng poject - people fom aound the wold download and un softwae to make a vtual supecompute, allowng fo smulatons of mllseconds of foldng tme. Unvesty of Vgna, MSE 4270/6270: Intoducton to Atomstc Smulatons, Leond Zhgle