Basic molecular dynamics

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1 1.1, 3.1, 1.333,. Inoducon o Modelng and Smulaon Spng 11 Pa I Connuum and pacle mehods Basc molecula dynamcs Lecue Makus J. Buehle Laboaoy fo Aomsc and Molecula Mechancs Depamen of Cvl and Envonmenal Engneeng Massachuses Insue of Technology 1

2 Conen ovevew I. Pacle and connuum mehods 1. Aoms, molecules, chemsy. Connuum modelng appoaches and soluon appoaches 3. Sascal mechancs 4. Molecula dynamcs, Mone Calo 5. Vsualzaon and daa analyss 6. Mechancal popees applcaon: how hngs fal and how o peven 7. Mul-scale modelng paadgm 8. Bologcal sysems smulaon n bophyscs how poens wok and how o model hem II. Quanum mechancal mehods 1. I s A Quanum Wold: The Theoy of Quanum Mechancs. Quanum Mechancs: Pacce Makes Pefec 3. The Many-Body Poblem: Fom Many-Body o Sngle- Pacle 4. Quanum modelng of maeals 5. Fom Aoms o Solds 6. Basc popees of maeals 7. Advanced popees of maeals 8. Wha else can we do? Lecues -13 Lecues 14-6

3 Goals of pa I pacle mehods You wll be able o Cay ou aomsc smulaons of vaous pocesses dffuson, defomaon/sechng, maeals falue Cabon nanoubes, nanowes, bulk meals, poens, slcon cysals, ec. Analyze aomsc smulaons make sense of all he numbes Vsualze aomsc/molecula daa bng daa o lfe Undesand how o lnk aomsc smulaon esuls wh connuum models whn a mul-scale scheme 3

4 Lecue : Basc molecula dynamcs Oulne: 1. Inoducon. Case sudy: Dffuson.1 Connuum model. Aomsc model 3. Addonal emaks hsocal pespecve Goals of oday s lecue: Though case sudy of dffuson, llusae he conceps of a connuum model and an aomsc model Develop appecaon fo dsncon of connuum and aomsc appoach Develop equaons/models fo dffuson poblem fom boh pespecves Develop aomsc smulaon appoach e.g. algohm, pseudocode, ec. and apply o descbe dffuson calculae dffusvy Hsocal pespecve on compue smulaon wh MD, examples fom leaue 4

5 1. Inoducon 5

6 Relevan scales n maeals λ aom ξcysal << << << dgan dx, dy, dz << H, W, D y A y σ yy σ yz n y σ yx σ xy σ xz n x σ xx x Image fom Wkmeda Commons, hp://commons.wkmeda.og. Boom-up Couesy of Elseve, Inc., hp:// Used wh pemsson. Fg. 8.7 n: Buehle, M. Aomsc Modelng of Maeals Falue. Spnge, 8. Spnge. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. z Image by MIT OpenCouseWae. A x Top-down Aomsc vewpon: Explcly consde dscee aomsc sucue Solve fo aomc ajecoes and nfe fom hese abou maeal popees & behavo Feaues nenal lengh scales aomc dsance Many-pacle sysem wh sascal popees 6

Relevan scales n maeals ξcysal RVE λ aom << << dgan << dx, dy, dz << H, W, D y σ yy A y σ yz n y σ yx σ xy Image fom

7 n: Buehle, M. Aomsc Modelng of Maeals Falue. Spnge, 8. Spnge. All ghs eseved.

Top-down Connuum vewpon: Tea maeal as mae wh no nenal sucue Develop mahemacal model govenng equaon based on epesenave

7 Relevan scales n maeals ξcysal RVE λ aom << << dgan << dx, dy, dz << H, W, D y σ yy A y σ yz n y σ yx σ xy Image fom Wkmeda Commons, hp://commons.wkmeda.og. Couesy of Elseve, Inc., hp:// Used wh pemsson. Fg. 8.7 n: Buehle, M. Aomsc Modelng of Maeals Falue. Spnge, 8. Spnge. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. Image by MIT OpenCouseWae. Top-down Connuum vewpon: Tea maeal as mae wh no nenal sucue Develop mahemacal model govenng equaon based on epesenave volume elemen RVE, conans enough maeal such ha nenal sucue can be negleced Feaues no chaacesc lengh scales, povded RVE s lage enough PDE wh paamees z σ xz n x A x σ xx x 7

8 . Case sudy: Dffuson Connuum and aomsc modelng 8

9 Dffuson as example Goal of hs secon Connuum descpon op-down appoach, paal dffeenal equaon Aomsc descpon boom-up appoach, based on dynamcs of molecules, obaned va numecal smulaon of he molecula dynamcs 9

10 Inoducon: Dffuson Pacles move fom a doman wh hgh concenaon o an aea of low concenaon Macoscopcally, dffuson measued by change n concenaon Mcoscopcally, dffuson s pocess of sponaneous ne movemen of pacles Resul of andom moon of pacles Bownan moon Hgh concenaon Low concenaon c m/v cx, 1

11 Ink dople n wae ho cold souce unknown. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. 11

12 Mcoscopc obsevaon of dffuson 1

Mcoscopc mechansm: Random walk o Bownan moon Bownan moon was fs obseved 187 by he Bsh boans Robe Bown 1773-1858 when sudyng pollen gans n wae Inally hough o be sgn of lfe, bu lae

13 Mcoscopc mechansm: Random walk o Bownan moon Bownan moon was fs obseved 187 by he Bsh boans Robe Bown when sudyng pollen gans n wae Inally hough o be sgn of lfe, bu lae confmed o be also pesen n noganc pacles The effec was fnally explaned n 195 by Albe Ensen, who ealzed was caused by wae molecules andomly smackng no he pacles. Publc doman mage. 13

14 Bownan moon leads o ne pacle movemen me Pacle slowly moves away fom s nal poson 14

15 Robe Bown s mcoscope Robe Bown's Mcoscope 187 Insumen wh whch Robe Bown suded Bownan moon and whch he used n hs wok on denfyng he nucleus of he lvng cell Insumen s peseved a he Lnnean Socey n London I s made of bass and s mouned ono he ld of he box n whch can be soed hp:// 15

16 Smulaon of Bownan moon hp:// 16

17 Macoscopc obsevaon of dffuson 17

18 Macoscopc obsevaon: concenaon change Ink do Tssue See also: hp:// souce unknown. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. 18

19 Bownan moon leads o ne pacle movemen Tme c 1 m 1 /V low c m /V hgh souce unknown. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. The McGaw Hll Companes. All ghs eseved. Ths conen s excluded fom ou Ceave Commons lcense. Fo moe nfomaon, see hp://ocw.m.edu/fause. 19

20 Dffuson n bology Image emoved due o copygh escons. See he mage now: hp:// hp://

21 .1 Connuum model How o buld a connuum model o descbe he physcal phenomena of dffuson? 1

22 Appoach 1: Connuum model Develop dffeenal equaon based on dffeenal elemen m L m R W1 H1 J x x x J: Mass flux mass pe un me pe un aea Concep: Balance mass [hee], foce ec. n a dffeenal volume elemen; much geae n dmenson han nhomogenees suffcenly lage RVE

23 Appoach 1: Connuum model Develop dffeenal equaon based on dffeenal elemen m L m R W1 H1 J x Pobably of mass m cossng he cenal bounday dung a peod s equal o p x x 3

24 Appoach 1: Connuum model Develop dffeenal equaon based on dffeenal elemen m L m R W1 H1 J x Pobably of mass m cossng he cenal bounday dung a peod s equal o p x x J J 1 p p 1 1 L m L R m R Mass flux fom lef o gh Mass flux fom gh o lef [J] mass pe un me pe un aea 4

25 Appoach 1: Connuum model Develop dffeenal equaon based on dffeenal elemen m L m R W1 H1 J x Pobably of mass m cossng he cenal bounday dung a peod s equal o p J J x 1 p p 1 1 L m L R m R x Mass flux fom lef o gh Mass flux fom gh o lef Effecve mass flux 1 p J m L m R 1 1 Moe mass, moe flux m L s ~ o numbe of pacles 5

26 Connuum model of dffuson Expess n ems of mass concenaons c m V m cv J p m L m R c L c R W1 H1 J x x x 6

27 Connuum model of dffuson Expess n ems of mass concenaons c m V m cv J p m L m R J 1 p 1 1 x L R c c c 1 1 V W1 H1 c L J c R x x x 7

28 Connuum model of dffuson Expess n ems of mass concenaons J m c V 1 p 1 1 x L R c c 1 1 c V c L c R Concenaon gaden W1 H1 J x J p cx p x c x x x expand x 8

29 Connuum model of dffuson Expess n ems of mass concenaons J m c V 1 p 1 1 x L R c c 1 1 c V c L c R Concenaon gaden W1 H1 J x J p cx p x c x x x J p x dc dx D dc dx D p x Paamee ha measues how fas mass 9 moves n squae of dsance pe un me

30 Dffuson consan & 1 s Fck law Reeae: Dffuson consan D descbes he how much mass moves pe un me Movemen of mass chaacezed by squae of dsplacemen fom nal poson Flux J p x dc dx D dc dx 1 s Fck law Adolph Fck, D p x D ~ p Dffuson consan elaes o he ably of mass o move a dsance x ove a me songly empeaue dependen, e.g. Ahenus 3

31 nd Fck law me dependence c dc J D 1 dx s Fck law J 1 J x c J J x 1 1 x x x 1 J: Mass flux mass pe un me pe un aea 31

32 nd Fck law me dependence c dc J D 1 dx s Fck law J 1 J x x x x 1 c J 1 J x 1 x D dc dx dc D dx x x x x1 J 1 J x x D dc dx x x 3

33 nd Fck law me dependence c J D dc dx 1 s Fck law J 1 J x c 1 x D dc dx J dc D dx x x x x1 x x x 1 c d dx d dx dc dx J D Change of concenaon n me equals change of flux wh x mass balance 33

34 nd Fck law me dependence c J D dc dx 1 s Fck law J 1 J x c 1 x D dc dx J dc D dx x x x x1 x x x 1 c d dx d dx dc dx J D Change of concenaon n me equals change of flux wh x mass balance c D d c dx nd Fck law PDE Solve by applyng ICs and BCs 34

35 Example soluon nd Fck s law c D d c dx Concenaon c c x me c x, BC: c x c IC: c x >, x Need dffuson coeffcen o solve fo dsbuon! 35

36 How o oban dffuson coeffcen? Laboaoy expemen Sudy dsbuon of concenaons pevous slde Then f he appopae dffuson coeffcen so ha he soluon maches Appoach can hen be used o solve fo moe complex geomees, suaons ec. fo whch no lab expemen exss Top down appoach 36

37 Machng wh expemen paamee denfcaon Concenaon c me expemen connuum model soluon c x, x f paamee D 37

38 Summay Connuum model eques paamee ha descbes mcoscopc pocesses nsde he maeal Typcally need expemenal measuemens o calbae Tme scale Mcoscopc mechansms??? Connuum model Empcal o expemenal paamee feedng Lengh scale 38

39 . Aomsc model How o buld an aomsc boom-up model o descbe he physcal phenomena of dffuson? 39

40 Appoach : Aomsc model Aomsc model povdes an alenave appoach o descbe dffuson Enables us o decly calculae he dffuson consan fom he ajecoy of aoms mcoscopc defnon Appoach: Consde se of aoms/molecules Follow he ajecoy and calculae how fas aoms leave he nal poson Follow hs quany ove me x D p Recall: Dffuson consan elaes o he ably of pacle o move a dsance x ove a me 4

41 Molecula dynamcs smulae ajecoy of aoms Goal: Need an algohm o pedc posons, veloces, acceleaons as funcon of me 41

42 4 To solve hose equaons: Dsceze n me n seps, me sep:... 3 n Solvng he equaons: Wha we wan

43 43 To solve hose equaons: Dsceze n me n seps, me sep:... 3 n Solvng he equaons Recall: Taylo expanson of funcon f aound pon a

44 44 To solve hose equaons: Dsceze n me n seps, me sep:... 3 n Solvng he equaons Recall: Taylo expanson of funcon f aound pon a Taylo sees expanson aound a x x a

45 45 To solve hose equaons: Dsceze n me n seps, me sep:... 3 n Solvng he equaons Recall: Taylo expanson of funcon f aound pon a Taylo sees expanson aound... 1 a v a x x a

46 46 Taylo expanson of... 1 a v... 1 a v a x x a a x x a

47 47 Taylo expanson of... 1 a v... 1 a v... a v v

48 48 Taylo expanson of... 1 a v... 1 a v... a v v... a

49 49 Taylo expanson of... 1 a v... 1 a v... a v v... a Posons a - Acceleaons a Posons a

50 Physcs of pacle neacons Laws of Moon of Isaac Newon : 1. Evey body connues n s sae of es, o of unfom moon n a gh lne, unless s compelled o change ha sae by foces mpessed upon.. The change of moon s popoonal o he move foce mpesses, and s made n he decon of he gh lne n whch ha foce s mpessed. 3. To evey acon hee s always opposed an equal eacon: o, he muual acon of wo bodes upon each ohe ae always equal, and deced o conay pas. d x f m d ma nd law 5

51 Vele cenal dffeence mehod a... Posons a Posons a - Acceleaons a How o oban acceleaons? f ma a f m / Need foces on aoms! 51

52 Vele cenal dffeence mehod f / m... Posons a Posons a - Foces a 5

53 Foces on aoms Consde enegy landscape due o chemcal bonds Enegy U 1/ 1 o Exponenal Repulson e 1/ 6 Aacon Radus Dsance beween aoms Image by MIT OpenCouseWae. Aacon: Fomaon of chemcal bond by shang of elecons Repulson: Paul excluson oo many elecons n small volume 53

54 How ae foces calculaed? Foce magnude: Devave of poenal enegy wh espec o aomc dsance f du d To oban foce veco f, ake pojecons no he hee axal decons f f x f x x 1 Ofen: Assume pa-wse neacon beween aoms 54

55 Noe on foce calculaon Foces can be obaned fom a vaey of models fo neaomc enegy, e.g. Pa poenals e.g. LJ, Mose, Buckngham Mul-body poenals e.g. EAM, CHARMM, UFF, DREIDING Reacve poenals e.g. ReaxFF Quanum mechancs e.g. DFT pa II Tgh-bndng wll be dscussed n nex lecues 55

56 Molecula dynamcs Follow ajecoes of aoms Vele cenal dffeence mehod a... a f / m Posons a Posons a - Acceleaons a 56

57 Summay: Aomsc smulaon numecal appoach molecula dynamcs MD Aomsc model; eques aomsc mcosucue and aomc poson a begnnng Sep hough me by negaon scheme Repeaed foce calculaon of aomc foces Explc noon of chemcal bonds capued n neaomc poenal 57

58 Pseudocode Se pacle posons e.g. cysal lace Assgn nal veloces Fo all me seps: Calculae foce on each pacle suboune Move pacle by me sep Save pacle poson, velocy, acceleaon Save esuls Sop smulaon 58

59 Aomc posons nal condons Typcally, have cubcal cell n whch pacles ae places n a egula o egula manne gas lqud sold - cysal 59

60 Aomsc descpon Back o he applcaon of dffuson poblem Aomsc descpon povdes alenave way o pedc D Smple solve equaon of moon Follow he ajecoy of an aom Relae he aveage dsance as funcon of me fom nal pon o dffusvy Goal: Calculae how pacles move andomly, away fom nal poson 6

61 JAVA apple Couesy of he Cene fo Polyme Sudes a Boson Unvesy. Used wh pemsson. URL hp://polyme.bu.edu/java/java/lj/ndex.hml 61

62 Lnk aomsc ajecoy wh dffuson consan 1D Dffuson consan elaes o he ably of a pacle o move a dsance x fom lef o gh ove a me D p x x Idea Use MD smulaon o measue squae of dsplacemen fom nal poson of pacles, : me 6

63 Lnk aomsc ajecoy wh dffuson consan 1D Dffuson consan elaes o he ably of a pacle o move a dsance x fom lef o gh ove a me D p x x MD smulaon: Measue squae of dsplacemen fom nal poson of pacles, : 63

64 Lnk aomsc ajecoy wh dffuson consan 1D Dffuson consan elaes o he ably of a pacle o move a dsance x fom lef o gh ove a me D p x x MD smulaon: Measue squae of dsplacemen fom nal poson of pacles, and no x. Replace D p x D 1 Faco 1/ no deconaly n equal pobably o move foh o back 64

65 Lnk aomsc ajecoy wh dffuson consan 1D MD smulaon: Measue squae of dsplacemen fom nal poson of pacles, : D D R ~ 65

66 Lnk aomsc ajecoy wh dffuson consan D/3D D p x Hghe dmensons D 1 1 Faco 1/ no deconaly n foh/back d Faco d 1,, o 3 due o 1D, D, 3D dmensonaly Snce: dd ~ dd C C consan does no affec D 66

67 Example: MD smulaon slope D D 1 d lm d d 1D1, D, 3D3 souce: S. Yp, lecue noes D 1 d lm d d.. aveage ove all pacles 67

68 Example molecula dynamcs Couesy of he Cene fo Polyme Sudes a Boson Unvesy. Used wh pemsson. Pacles Tajecoes Mean Squae Dsplacemen funcon 1 N Aveage squae of dsplacemen of all pacles 68

69 Example calculaon of dffuson coeffcen 1 N Poson of aom a me Poson of aom a me slope D D 1 d lm d d 1D1, D, 3D3 69

70 Summay Molecula dynamcs povdes a poweful appoach o elae he dffuson consan ha appeas n connuum models o aomsc ajecoes Oulnes mul-scale appoach: Feed paamees fom aomsc smulaons o connuum models Tme scale MD Connuum model Empcal o expemenal paamee feedng Quanum mechancs 7 Lengh scale

71 Mul-scale smulaon paadgm Couesy of Elseve, Inc., hp:// Used wh pemsson. 71

72 3. Addonal emaks 7

73 Hsocal developmen of compue smulaon Began as ool o explo compung machnes developed dung Wold Wa II MANIAC 195 a Los Alamos used fo compue smulaons Meopols, Rosenbluh, Telle 1953: Meopols Mone Calo mehod Alde and Wanwgh Lvemoe Naonal Lab, 1956/1957: dynamcs of had sphees Vneyad Bookhaven : dynamcs of adaon damage n coppe Rahman Agonne 1964: lqud agon Applcaon o moe complex fluds e.g. wae n 197s Ca and Panello 1985 and followng: ab-no MD Snce 198s: Many applcaons, ncludng: Kaplus, Goddad e al.: Applcaons o polymes/bopolymes, poens snce 198s Applcaons o facue snce md 199s o Ohe engneeng applcaons nanoechnology, e.g. CNTs, nanowes ec. snce md 199s- 73

74 MIT OpenCouseWae hp://ocw.m.edu 3.1J / 1.1J / 1.333J / J /.J Inoducon o Modelng and Smulaon Spng 1 Fo nfomaon abou cng hese maeals o ou Tems of use, vs: hp://ocw.m.edu/ems.

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma