Property calculation I

Transcription

1 .0, 3.0, 0.333,.00 Introducton to Modelng and Smulaton Sprng 0 Part I Contnuum and partcle methods Property calculaton I Lecture 3 Markus J. Buehler Laboratory for Atomstc and Molecular Mechancs Department of Cvl and Envronmental Engneerng Massachusetts Insttute of Technology

2 Content overvew I. Partcle and contnuum methods. Atoms, molecules, chemstry. Contnuum modelng approaches and soluton approaches 3. Statstcal mechancs 4. Molecular dynamcs, Monte Carlo 5. Vsualzaton and data analyss 6. Mechancal propertes applcaton: how thngs fal (and how to prevent t) 7. Mult-scale modelng paradgm 8. Bologcal systems (smulaton n bophyscs) how protens work and how to model them II. Quantum mechancal methods. It s A Quantum World: The Theory of Quantum Mechancs. Quantum Mechancs: Practce Makes Perfect 3. The Many-Body Problem: From Many-Body to Sngle- Partcle 4. Quantum modelng of materals 5. From Atoms to Solds 6. Basc propertes of materals 7. Advanced propertes of materals 8. What else can we do? Lectures -3 Lectures 4-6

3 3 Lecture 3: Property calculaton I Outlne:. Atomstc model of dffuson. Computng power: A perspectve 3. How to calculate propertes from atomstc smulaton 3. Thermodynamcal ensembles: Mcro and macro 3. How to calculate propertes from atomstc smulaton 3.3 How to solve the equatons 3.4 Ergodc hypothess Goal of today s lecture: Explot Mean Square Dsplacement functon to dentfy dffusvty, as well as materal state & structure Provde rgorous bass for property calculaton from molecular dynamcs smulaton results (statstcal mechancs)

4 4 Addtonal Readng Books: M.J. Buehler (008): Atomstc Modelng of Materals Falure Allen and Tldesley: Computer smulaton of lquds (classc) D. C. Rapaport (996): The Art of Molecular Dynamcs Smulaton D. Frenkel, B. Smt (00): Understandng Molecular Smulaton J.M. Hale (Wley, 99), Molecular dynamcs smulaton

5 5. Atomstc model of dffuson How to buld an atomstc bottom-up model to descrbe the physcal phenomena of dffuson? Introduce: Mean Square Dsplacement

6 Recall: Dffuson Partcles move from a doman wth hgh concentraton to an area of low concentraton Macroscopcally, dffuson measured by change n concentraton Mcroscopcally, dffuson s process of spontaneous net movement of partcles Result of random moton of partcles ( Brownan moton ) Hgh concentraton Low concentraton c = m/v = c(x, t) 6

7 7 Ink droplet n water hot cold source unknown. All rghts reserved. Ths content s excluded from our Creatve Commons lcense. For more nformaton, see

8 Atomstc descrpton Back to the applcaton of dffuson problem Atomstc descrpton provdes alternatve way to predct D Smple solve equaton of moton Follow the trajectory of an atom Relate the average dstance as functon of tme from ntal pont to dffusvty Goal: Calculate how partcles move randomly, away from ntal poston 8

9 Set partcle postons (e.g. crystal lattce) Assgn ntal veloctes Pseudocode For (all tme steps): Calculate force on each partcle (subroutne) Move partcle by tme step Δt Save partcle poston, velocty, acceleraton Save results Stop smulaton r ( t0 + Δt) = r ( t0) r ( t0 Δt) + a ( t0) Δt +... a = f / m Postons at t 0 Postons at t 0 -Δt Acceleratons at t 0 9

10 JAVA applet Courtesy of the Center for Polymer Studes at Boston Unversty. Used wth permsson. URL 0

11 tme Lnk atomstc trajectory wth dffuson constant (D) Dffuson constant relates to the ablty of a partcle to move a dstance Δx (from left to rght) over a tme Δt D = p Δx Δt Δx Idea Use MD smulaton to measure square of dsplacement from ntal poston of partcles, Δr ( t) : Δr ( t) = = N N ( r ( t) r ( t = 0) ) = [ ( r ( t) r ( t = 0) ) ( r ( t) r ( t 0) )] scalar product

12 Lnk atomstc trajectory wth dffuson constant (D) Dffuson constant relates to the ablty of a partcle to move a dstance Δx (from left to rght) over a tme Δt D = p Δx Δt Δx MD smulaton: Measure square of dsplacement from ntal poston of partcles, Δr ( t) : Δr ( t) = = N ( r ( t) r ( t 0) ) Δr t

13 3 Lnk atomstc trajectory wth dffuson constant (D) Dffuson constant relates to the ablty of a partcle to move a dstance Δx (from left to rght) over a tme Δt D = p Δx Δt Δx MD smulaton: Measure square of dsplacement from ntal poston of partcles, Δr ( t) and not Δx ( t). Replace D = p Δx Δt D = Δr Δt Δr Factor / = no drectonalty n (equal probablty to move forth or back)

14 4 Lnk atomstc trajectory wth dffuson constant (D) MD smulaton: Measure square of dsplacement from ntal poston of partcles, Δr ( t) : Δr D Δr = Δr = Dt R ~ t t t

15 5 Lnk atomstc trajectory wth dffuson constant (D/3D) D = p Δx Δt Hgher dmensons D = Δr Factor / = no drectonalty n (forth/back) d Δt Factor d =,, or 3 due to D, D, 3D (dmensonalty) Snce: ddδt ~ Δr dd r Δt + C = Δ C = constant (does not affect D)

16 6 Example: MD smulaton Mean Square Dsplacement functon slope = D D = d lm t d d t ( Δr ) Δr D=, D=, 3D=3 C Courtesy of Sd Yp. Used wth permsson. D = d lm t d d t Δr.. = average over all partcles

17 Example molecular dynamcs Partcles Δr Trajectores Mean Square Dsplacement functon Δr ( t) = = N ( r ( t) r ( t 0) ) Average square of dsplacement of all partcles Courtesy of the Center for Polymer Studes at Boston Unversty. Used wth permsson. 7

18 8 Example calculaton of dffuson coeffcent Δr ( t) = = N ( r ( t) r ( t 0) ) Poston of atom at tme t Poston of atom at tme t=0 Δr slope = D D = d lm t d d t Δr D=, D=, 3D=3

19 Length scale 9 Summary Molecular dynamcs provdes a powerful approach to relate the dffuson constant that appears n contnuum models to atomstc trajectores Outlnes mult-scale approach: Feed parameters from atomstc smulatons to contnuum models Tme scale MD Contnuum model Emprcal or expermental parameter feedng Quantum mechancs

20 Length scale 0 Summary Molecular dynamcs provdes a powerful approach to relate the dffuson constant that appears n contnuum models to atomstc trajectores Outlnes mult-scale approach: Feed parameters from atomstc smulatons to contnuum models Tme scale MD Contnuum model Emprcal or expermental parameter feedng Quantum mechancs

21 MD modelng of crystals sold, lqud, gas phase Crystals: Regular, ordered structure The correspondng partcle motons are small-ampltude vbratons about the lattce ste, dffusve movements over a local regon, and long free flghts nterrupted by a collson every now and then. Lquds: Partcles follow Brownan moton (collsons) Gas: Very long free paths Image by MIT OpenCourseWare. After J. A. Barker and D. Henderson.

22 Example: MD smulaton results lqud sold sold Courtesy of Sd Yp. Used wth permsson.

23 3 Atomstc trajectory Courtesy of Sd Yp. Used wth permsson.

24 4 Mult-scale smulaton paradgm Courtesy of Elsever, Inc., Used wth permsson.

25 . Computng power: A perspectve 5

26 Courtesy Elsever, Inc., Used wth permsson. 6

27 7 Hstorcal development of computer smulaton Began as tool to explot computng machnes developed durng World War II MANIAC (95) at Los Alamos used for computer smulatons Metropols, Rosenbluth, Teller (953): Metropols Monte Carlo method Alder and Wanwrght (Lvermore Natonal Lab, 956/957): dynamcs of hard spheres Vneyard (Brookhaven ): dynamcs of radaton damage n copper Rahman (Argonne 964): lqud argon Applcaton to more complex fluds (e.g. water) n 970s Car and Parrnello (985 and followng): ab-nto MD Snce 980s: Many applcatons, ncludng: Karplus, Goddard et al.: Applcatons to polymers/bopolymers, protens snce 980s Applcatons to fracture snce md 990s to 000 Other engneerng applcatons (nanotechnology, e.g. CNTs, nanowres etc.) snce md 990s-000

28 8 3. How to calculate propertes from atomstc smulaton A bref ntroducton to statstcal mechancs

29 9 Molecular dynamcs Follow trajectores of atoms (classcal mechancs, Newton s laws) )( Δ )... r ( t0 + Δt) = r ( t0 Δt) + r ( t0) Δt + a ( t0 t + Verlet central dfference method a = f / m Postons at t 0 -Δt Postons at t 0 Acceleratons at t 0

30 30 Property calculaton: Introducton Have: x( t), x ( t), x ( t) mcroscopc nformaton Want: Thermodynamcal propertes (temperature, pressure, stress, stran, thermal conductvty,..) State (gas, lqud, sold) (propertes that can be measured n experment!)

31 Goal: To develop a robust framework to calculate a range of macroscale propertes from MD smulaton studes ( mcroscale nformaton ) 3

32 3. Thermodynamcal ensembles: Mcro and macro 3

33 Macroscopc vs. mcroscopc states C C T, p, V, N C 3 C N Same macroscopc state s represented by many dfferent mcroscopc confguratons C 33

34 34 Defnton: Ensemble Large number of copes of a system wth specfc features Each copy represents a possble mcroscopc state a macroscopc system mght be n under thermodynamcal constrants (T, p, V, N..) Gbbs, 878

35 35 Mcroscopc states Mcroscopc states characterzed by r r, p = { x }, p = { m x } =.. N = p

36 Mcroscopc states Mcroscopc states characterzed by r r, p = { x }, p = { m x } =.. N = p Hamltonan (sum of potental and knetc energy = total energy) expressed n terms of these varables H ( r, p) = U( r) + K( p) U( r) = φ ( r) K( p) = =.. N =.. N p m K = m v 36

37 37 Ensembles Result of thermodynamcal constrants, e.g. temperature, pressure Mcrocanoncal Canoncal Isobarc-sothermal Grand canoncal NVE NVT NpT TVμ μ chemcal potental (e.g. concentraton)

38 3. How to calculate propertes from atomstc smulaton 38

39 Lnk between statstcal mechancs and thermodynamcs 39 Mcroscopc (atoms)????? Macroscopc (thermodynamcs)

40 Lnk between statstcal mechancs and thermodynamcs 40 Mcroscopc (atoms) Statstcal mechancs Macroscopc (thermodynamcs) Macroscopc condtons (e.g. constant volume, temperature, number of partcles ) translate to the mcroscopc system as boundary condtons (constrants) Macroscopc system: defned by extensve varables, whch are constant: E.g. (N,V,E)=NVE ensemble

41 Lnk between statstcal mechancs and thermodynamcs Mcroscopc (atoms) Statstcal mechancs Macroscopc (thermodynamcs) Macroscopc condtons (e.g. constant volume, temperature, number of partcles ) translate to the mcroscopc system as boundary condtons (constrants) Macroscopc system: defned by extensve varables, whch are constant: E.g. (N,V,E)=NVE ensemble The behavor of the mcroscopc system s related to the macroscopc condtons. In other words, the dstrbuton of mcroscopc states s related to the macroscopc condtons. To calculate macroscopc propertes (va statstcal mechancs) from mcroscopc nformaton we need to know the dstrbuton of mcroscopc states (e.g. through a smulaton) 4

42 Example: Physcal realzaton of canoncal ensemble (NVT) 4 Heat bath (constant temperature) Coupled to large system, allow energy exchange NVT small system embedded n large heat bath Constant number of partcles = N Constant volume = V Constant temperature = T

43 Macroscopc vs. mcroscopc states Canoncal ensemble C r, p C r, p N, V, T C 3 r 3, p 3 C N Same macroscopc state s represented by many dfferent mcroscopc confguratons r N, p N 43

44 44 Important ssue to remember A few sldes ago: To calculate macroscopc propertes (va statstcal mechancs) from mcroscopc nformaton we need to know the dstrbuton of mcroscopc states (e.g. through MD smulaton) Therefore: We can not ( never ) take a sngle measurement from a sngle mcroscopc state to relate to macroscopc propertes

45 Mcro-macro relaton Courtesy of the Center for Polymer Studes at Boston Unversty. Used wth permsson. T (t) Whch to pck? T ( t) = 3 Nk B N = m v ( t) Specfc (ndvdual) mcroscopc states are nsuffcent to relate to macroscopc propertes t 45

46 Averagng over the ensemble Rather than takng sngle measurement, need to average over all mcroscopc states that represent the correspondng macroscopc condton Ths averagng needs to be done n a sutable fashon, that s, we need to consder the specfc dstrbuton of mcroscopc states (e.g. some mcroscopc states may be more lkely than others) What about tryng ths. Property A Property A Property A 3 ( ) C C C3 r r r 3, p3, p, p A macro = A + A + 3 A 3 46

47 Averagng over the ensemble Rather than takng sngle measurement, need to average over all mcroscopc states that represent the correspondng macroscopc condton Ths averagng needs to be done n a sutable fashon, that s, we need to consder the specfc dstrbuton of mcroscopc states (e.g. some mcroscopc states may be more lkely than others) What about tryng ths. Property A Property A Property A 3 ( ) C C C3 A macro = A + A + 3 Generally, NO! A 3 47

48 48 Averagng over the ensemble C C 3 C Property A Property A Property A 3 ( ) 3 macro 3 A A A A + + = Instead, we must average wth proper weghts that represent the probablty of a system n a partcular mcroscopc state! (I.e., not all mcroscopc states are equal) ), ( ), ( ), ( ), ( ), ( ), ( macro p r A p r p r A p r p r A p r A A A A ρ ρ ρ ρ ρ ρ + + = + + = Probablty to fnd system n state C

49 How to relate mcroscopc states to macroscopc varables? 49 A( r, p) Property due to specfc mcrostate < A >= A( p, r) ρ( p, r) drdp p r Ensemble average, obtaned by ntegral over all mcroscopc states ρ( p, r) Proper weght - depends on ensemble

50 How to relate mcroscopc states to macroscopc varables? A( r, p) Property due to specfc mcrostate To measure an observable quantty from MD smulaton we must express ths observable as a functon of the postons and lnear momenta of the partcles n the system, that s, r, p Recall, mcroscopc states characterzed by r = { x }, p = { m x } =.. N r, p = p 50

51 How to relate mcroscopc states to macroscopc varables? < A >= A( p, r) ρ( p, r) drdp p r Probablty densty dstrbuton ρ( p, r) H ( p, r) = exp Q kbt Probablty to fnd system n state (p,r) Boltzmann constant k B = m kg s K Partton functon Q H ( p, r) = exp drdp k T p r B 5

52 5 Illustraton/example: phase space 6N-dmensonal phase space Image removed due to copyrght restrctons. See the second mage at r, p ρ( p, r)

53 Defnton of temperature Classcal (mechancs) many-body system: Average knetc energy per degree of freedom s related to temperature va Boltzmann constant: mv = N f =.. N mv = f k B T # DOF N f = 3N # partcles (each 3 DOF for veloctes) Based on equpartton theorem (energy dstrbuted equally over all DOFs) 53

54 Defnton of temperature Classcal (mechancs) many-body system: Average knetc energy per degree of freedom s related to temperature va Boltzmann constant: mv = N f =.. N mv = f k B T # DOF N f = 3N =p T ( p) N = 3 Nk B = m v m = A( p) Temperature 54

55 55 How to calculate temperature N m v < T >= ρ( p, r) drdp 3 Nk m p r B =???

56 56 How to solve < A >= A( p, r) ρ( p, r) drdp p r Vrtually mpossble to carry out analytcally Must know all possble mcroscopc confguratons correspondng to a macroscopc ensemble, then calculate ρ Therefore: Requre numercal smulaton (the only feasble approach )

57 Summary: How macro-mcro relaton works A < A > Mcroscopc (atomc confguratons) Statstcal mechancs Macroscopc (thermodynamcal ensemble) mcroscopc A (sngle pont measurement) < A >= A( p, r) ρ( p, r) drdp p r defnes 57

58 3.3 How to solve the equatons 58

59 59 Approaches n solvng ths problem Method of choce: Numercal smulaton Two major approaches:. Usng molecular dynamcs (MD): Generate mcroscopc nformaton through dynamcal evoluton of mcroscopc system (.e., smulate the real behavor as we would obtan n lab experment). Usng a numercal scheme/algorthm to randomly generate mcroscopc states, whch, through proper averagng, can be used to compute macroscopc propertes. Methods referred to as Monte Carlo

60 60 Monte Carlo (MC) scheme Concept: Fnd smpler way to solve the ntegral < A >= A( p, r) ρ( p, r) drdp p r Use dea of random walk to step through relevant mcroscopc states and thereby create proper weghtng (vst states wth hgher probablty densty more often) =ensemble (statstcal) average

61 6 MC algorthm result Fnal result of MC algorthm: Algorthm that leads to proper Dstrbuton of mcroscopc states Ensemble (statstcal) average < A >= A( p, r) ρ( p, r) drdp < A > N p r A A Carry out algorthm for N A steps Average results..done!

62 3.4 Ergodc hypothess 6

63 63 Ergodcty MC method s based on drectly computng the ensemble average Defne a seres of mcroscopc states that reflect the approprate ensemble average; weghts ntrnscally captured snce states more lkely are vsted more frequently and vce versa Egodcty: The ensemble average s equal to the tme-average durng the dynamcal evoluton of a system under proper thermodynamcal condtons. In other words, the set of mcroscopc states generated by solvng the equatons of moton n MD automatcally generates the proper dstrbuton/weghts of the mcroscopc states Ths s called the Ergodc hypothess: < A > =< A > Ens Tme

64 64 Ergodc hypothess Ergodc hypothess: Ensemble (statstcal) average = tme average All mcrostates are sampled wth approprate probablty densty over long tme scales N A =.. N A( ) =< A > Ens =< A > Tme = N =.. A N t t A( ) MC MD

65 65 Ergodc hypothess Ergodc hypothess: Ensemble (statstcal) average = tme average All mcrostates are sampled wth approprate probablty densty over long tme scales N N m =< > =< > = v A Ens A Tme mv N A =.. N A 3 NkB = Nt =.. N t 3 NkB = MC MD

66 Importance for MD algorthm Follow trajectores of atoms (classcal mechancs, Newton s laws) Verlet central dfference method )( Δ )... r ( t0 + Δt) = r ( t0 Δt) + r ( t0) Δt + a ( t0 t + a = f / m Postons at t 0 -Δt Postons at t 0 Acceleratons at t 0 It s suffcent to smply average over all MD steps < A > Tme = N t =.. N t A( ) 66

67 Molecular dynamcs Durng ntegraton of equatons of moton must mpose thermodynamcal constrants For example, Verlet central dfference method leads to a mcrocanoncal ensemble (NVE) Other ntegraton methods exst to generate NVT, NpT ensembles etc. )( Δ )... r ( t0 + Δt) = r ( t0 Δt) + r ( t0) Δt + a ( t0 t + a = f / m Postons at t 0 -Δt Postons at t 0 Acceleratons at t 0 67

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