ME346C Advanced Techniques for Molecular Simulations ...

ME346C Advanced Techniques for Molecular Simulations -- Syllabus Summer quarter, 2011-2012, 3 units, Stanford MW(F) 11:00-12:15pm, 200-305 Instructor: Wei Cai, [email protected], http://www.stanford.edu/~caiwei Office hour: see coursework homepage TA: William (Billy) Cash, [email protected] Office hour: see coursework homepage Course website: http://coursework.stanford.edu (official) (last modified 6/24/2012) Course objective The students will learn to perform free energy and free energy barrier calculations by molecular simulations, and understand the statistical mechanics basis of the computational methods. Methods for free energy calculation, such as thermodynamic integration (TI) and adiabatic switching (AS), will be applied to accurately predict the melting point of crystals. Methods for (free) energy barrier calculation, such as nudged-elastic band (NEB) and umbrella sampling (US), will be applied to predict the rate of rare transition events. Prerequisite: ME346A and B Relation with other courses The statistical mechanics basis has been introduced in ME346A Introduction to statistical mechanics. Basic molecular dynamics and Monte Carlo simulation techniques as well as how to use the MD++ program have been introduced in ME346B Introduction to molecular simulations. This is the first time ME346C is offered, even though the ME346ABC sequence have been contemplated for many years. Grading Policies The final grade is given based on a final score, which is computed from homework (20%), midterm project (40%) and final project (40%). Homework will be issued biweekly and due on Wednesdays in class unless otherwise announced. To be fair to all students, homeworks that are late by n days will have their grades multiplied by a factor f(n) = (0.8)n. Students are encouraged to discuss with and learn from others when solving the homework problems, but are required to write up the solution (including computer programs) individually. Each projects require a written report and a ppt presentation from each team. Students with a final score higher than 90 will receive an A.

http://www.stanford.edu/%7Ecaiwei

http://coursework.stanford.edu/

Homework problems involve programming and numerical calculations in Matlab, simulations using MD++, and theoretical problems. Homework reports are submitted in paper. Numerical codes are submitted electronically to coursework.stanford.edu. Textbooks and course materials Daan Frenkel and Berend Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd ed. Computational science series; v.1, Academic Press, (2002). QD461.F86 Course lecture notes will be distributed in class and posted on coursework. The MD++ program and tutorial materials are available through a link on coursework. Tutorials of MD++ will be given in TA sessions (location to be announced). Other suggested readings: Vasily V. Bulatov and Wei Cai, Computer Simulation of Dislocations, Oxford University Press. Companion website: http://micro.stanford.edu/ Course outline Part I. Free Energy Methods

1. Statistical Mechanics Basis (2 lectures) 2. Thermodynamic Integration (1 lectures) 3. Reversible Scaling (1 lecture) 4. Free Energy of Solids (1 lecture) 5. Free Energy of Fluids (1 lecture) 6. Predicting Melting Temperature (1 lecture)

Midterm project presentation Part II. Free Energy Barrier Methods

7. Rate of Rare Events, Nucleation Theory (2 lectures) 8. Energy Barrier Methods, Nudged Elastic Band (2 lectures) 9. Free Energy Barrier Methods, Unbrella Sampling (2 lectures)

Final project presentation

http://micro.stanford.edu/

Important dates

Mon Wed Fri Tentative topics 6/25 Overview Finding a computer to run

MD++ / Matlab 6/27 Stat Mech of Free Energy Start looking for team mates 6/29 Thermodynamic Integration Make-up lecture 7/2 Reversible Scaling Pick midterm project topics 7/4 ------------------------- Independence day, No class 7/6 ------------------------- Travel, No class 7/9 ------------------------- Travel, No class

Final day to add / drop 7/11 ------------------------- Travel, No class 7/13 Free Energy of Solids HW 1 due

Make-up lecture 7/16 Free Energy of Fluids 7/18 Melting Temperature 7/20 ------------------------- 7/23 Student presentation Midterm project presentation 7/25 Nucleation Theory Pick final project topics 7/27 ------------------------- Final project proposal due 7/30 Energy barrier methods 8/1 Energy barrier methods 8/3 ------------------------- Course withdraw deadline 8/6 Umbrella Sampling HW 2 due 8/8 Umbrella Sampling 8/10 ------------------------- 8/13 Applications 8/15 Applications 8/17 Student presentation Final project presentation 8/20 ------------------------- Final report due

*Exact dates of lectures may subject to change.

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ffiffi"{^* rn:-lpos;i;ve Whuf,

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frhr, pow{;hkw fl""rdv^ 'f +fu- b|t mstwrh,"tlt 9,,n (sl"*r41

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p, )-r*^*^', o.wy_*sl_ io Tsrh ge^nu."{ Ze [v 7a) iS nnt or^tr-n,r't^tol,e onaU";+t,,'1,!y

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a"nd wp- are rytler'-wfed irv th4- ra'e- "f4*f t- f^, t&L4 bl iT^-) l r , f tA | f f i " .

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if lJ C n) nt*r /t:s i..t ^lao ayyr"X'nr$<J bJ /lwzrrhrwtc Ar1tr,y.

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tr rnotKunof"n_

rSLJ :

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i

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1 1 8

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FINDING TRANS IT ION PATHWAYS

Nrt.o /10s(b)

Ft c' 7. 1 . (a) A two-dimensional potrential energy function U t (x, y). The twometa-stablestates, A for- r <0and B fbr ; r >0, areseparatedby ar iOg, atx:0.Srand Sz mark the locations of two saddles on the ridge. (b) Instaniurr"ous valuesof coordinate x in an MC simulartion at knT Q.tj. (c) The microstates (dors)visited during transitions between A and B plotted on top of the contour lines ofU (x,y), The solid (dashed,) line connecting two energy minima is the minimumenergy path obtained by sreepest descent from saddle sr (sz).

Saddle ridge

Tlw da'vida'ny sw,Jr,t* (rv faiJ {yy;rtgil , x:e, d,irir{,*;s *kt

r4 *,3 '{E"{ro PlS.ftnf' : A am I B.

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"i-*^W k^c(6 po,r.f f6

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* , {x< o)

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I ' a , \

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+ r9J ,ytlu'*., ('9. '

T - LSoo ; /*geuNt- zd

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"fTuec,irl p or'f S/ D-r Jz

S I arrt d S. a/fr- saddh p oiHfs,

r",f,;*h NTV fo-t'eni->J ?ruAyj rnhi,na.

uvithin thz- d"-vtdiy smlaxa ( X - r>).

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l'lar#iry frrn Sr fu J" ) p|t t ^ rimnL(ytUutwtia,{rrru -n d f #"*S tfu-:*eefwt ol*awnt tn1e*i

wil( lg"d tM fr ^,ft MII minimsary( h 6 { b )

ll m.tni%rrtuw e,rtlJt4q pafrn (ME'- )t:.-fnne<:finq O a,.^d fs Cwrn b-L

c.6rL4f"worcY fr,* tht A^rv sl*;;,fl"/tfoLuca-"* tri*r.*r;eo frtrru eax.LJ'a dd {.u po no* 6, s{ Jw s .

d,'rri#Y r*.r{"*

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ih f Ps , b'* wvl- en-ut4y

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ot*f- n"e ilJl- E- , - -{ r n

t- +Kr( y- Va)L- t - - -\_----)__/i3ruwd

t - T J -

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I k*T - KeT Z*l l L Z *L_*_

f ,&nf't^xt f,*#''tln * a^,v h**v*na,r' fuimtswt ucr/(at*h ) '''r

ME 3+6C Lu*"* E TP3/nYz= (tt, !t) ,

tq(y-Yi"

tf,LAl-

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- F \ \ +

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t^3 n"cre d

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C{cx,yt: u(5r ) t Kt,[x- n) ' +

oxpand L( t2(, y) 67;Yor*n d,

!:: e-FGr{{v-Ya:'+ 4\l-

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c: Vs.TJ-., . ,tl^_wz

I.,TST KsT- :'THn; fL

ffi--tr--- , lzlrs-\ ,fB'^1 -7.Y't-

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Adatom Diffusion on Au Surface Wei Cai 8/1/2012 Department of Mechanical Engineering, Stanford University, Stanford 94305-4040 1. Prepare Au (111) Surface The atomistic calculations below will be based on the EAM Foiles potential for Au. The tcl file can be download from ME346C Coursework web site.

bin/eam_mc2 scripts/work/au/au_surface_atom.tcl 0 A perfect FCC crystal of Au is created with dimensions: 4[1 -2 1] x 4[1 0 –1] x 3[1 1 1] (1152 atoms). A 1.4% bi-axial compressive strain is applied in the x and y directions. (Otherwise the surface is under a large tension stress and adatoms will tend to be incorporated into the top surface instead of staying on top of it.)

Figure 1. Au (111) surface. The top surface layer atoms are in orange. Click on the atoms to get the (scaled) coordinates of various sites on the (111) surface. These coordinates will be used in step 2. 2. Add One Adatom to Au (111) Surface

bin/eam_mc2 scripts/work/au/au_surface_atom.tcl 1 One more atom is added on top of the (111) surface. The total number of atoms is 1153. There are two different types of sites to put the adatom: type A and type B. We create four different structures: A, Ap, B, Bp. A and Ap are of the same type. B and Bp are of the same type. The structures are saved in folder runs/au-surf-1-0-0.

ME346C Lecture 10 8/1/2012 Cai 1

Bp Ap

AB

Figure 2. Orange atoms are the top layer of Au (111) surface. There are two types of sites (type A and type B) for adatom, both avoiding the orange atoms. A, Ap are equivalent sites. B, Bp are equivalent sites.

A B

Ap Bp

Figure 3. For different adatom structures created by MD++.

3. Find Minimum Energy Path The string method is used to search for the minimum energy path between state A (or Ap) and state B (or Bp).

bin/eam_mc2 scripts/work/au/au_surface_atom.tcl 2 A B or

bin/eam_mc2 scripts/work/au/au_surface_atom.tcl 2 A Bp Figure 4 shows the energy profile for minimum energy path (after 1000 step of stringrelax) between different states.

2

Plot energy profile: cd runs/au-surf-2-A-B

octave data=load('stringeng.out'); plot(data(end-10:end,3:5:end)', '*-'); xlabel(‘replica ID’); ylabel(‘Energy (eV)’);

Figure 4. The energy profile for adatom migration on Au (111) surface. From these calculations, we see that the energy difference between the two adatom sites is:

EB - EA = 0.065 eV The energy barriers for the jump are (approximately): E* - EA = 0.083 eV E* - EB = 0.018 eV The calculation can be repeated with more relaxation steps to make sure the results have converged. It is also of interest to find out whether there are multiple transition paths ways between meta-stable states (e.g. between A and B). Further Reading: R. C. Jaklevic and L. Elie, Scanning-Tunneling-Microscope Observation of Surface Diffusion on an Atomic Scale: Au on Au(111), Phys. Rev. Lett. 60, 120–123 (1988) http://prl.aps.org/pdf/PRL/v60/i2/p120_1

3

http://prl.aps.org/pdf/PRL/v60/i2/p120_1

au_surface_atom.tcl~/Codes/MD++.svn/scripts/work/au/

1/508/05/2012

# −*−shell−script−*−# compute energy barrier of surface atom diffusion in FCC Au## make eam build=R SYS=mc2

source "scripts/Examples/Tcl/startup.tcl"

#*******************************************# Definition of procedures#*******************************************proc initmd { status n m } {#MD++ setnologMD++ setoverwriteMD++ dirname = runs/au−surf−$status−$n−$mMD++ NNM = 200}

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−proc readeam { material } { #Read in MEAM potential (MD++ format)MD++ potfile = "~/Codes/MD++/potentials/EAMDATA/eamdata.${material}Foiles"MD++ eamgrid = 500 readeam NNM=300MD++ element0 = $material}

proc make_perfect_crystal { nx ny nz } { MD++ crystalstructure = face−centered−cubic latticeconst = 4.08 MD++ latticesize = \[ 1 −2 1 $nx 1 0 −1 $ny 1 1 1 $nz \] MD++ makecrystal #finalcnfile = perf.cn writecn #eval}

proc make_dislocation_loop { } { # create structure B’ set bz [expr −0.5/ [MD++_Get latticesize(11)] ] MD++ input = \[ 3 1 0 0 $bz 0.02 −0.25 0.25 0.3 −10 10 −10 10 1 1 −0.25 0.25 0 \] MD++ makedipole}

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−proc relax_fixbox { } { MD++ {# Conjugate−Gradient relaxationconj_ftol = 1e−4 conj_itmax = 1000 conj_fevalmax = 1000conj_fixbox = 1relax} }#end of proc relax_fixbox

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−proc relax_freebox { } { MD++ {# Conjugate−Gradient relaxationconj_ftol = 1e−4 conj_itmax = 1000 conj_fevalmax = 1000conj_fixbox = 0conj_fixboxvec = [ 0 0 1 1 0 1 0 0 0 ]relax} }#end of proc relax_freebox

proc setup_window { } { MD++ {#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−#colors for Central symmetry viewcolor00 = "red" color01 = "blue" color02 = "green"color03 = "magenta" color04 = "cyan" color05 = "purple"color06 = "gray80" color07 = "white" color08 = "orange"#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−# Plot Configuration#atomradius = [0.7 0.78] bondradius = 0.3 bondlength = 0 #2.8285 #for Siwin_width=600 win_height=600#atomradius = 0.9 bondradius = 0.3 bondlength = 0 #2.8285 #for Siatomcolor = cyan highlightcolor = purple bondcolor = red


2/508/05/2012

fixatomcolor = yellow backgroundcolor = gray70#atomcolor = lightgrey highlightcolor = purple bondcolor = darkgreyplot_color_axis = 2 NCS = 12plot_color_windows = [ 5 0.6 10 1 10 20 6 20 80 8 80 5000 5 0 0.6 4 ]#plot_limits = [ 1 −10 10 −10 0.35 −10 10 ]#plot_color_windows = 0 plot_atom_info = 1 # reduced coordinates of atoms#plot_atom_info = 2 # real coordinates of atoms#plot_atom_info = 3 # energy of atoms#plot_highlight = [ 0 0 1 2 3 4 5 6 7 8 9 ]plotfreq = 10##rotateangles = [ 0 0 0 1.7 ]#rotateangles = [ 0 −90 0 1.7 ]#openwin alloccolors rotate saverot plot#plot_color_axis = 0 input = [ −8 −3 10] GnuPlotHistogram#plot_color_axis = 2 input = [ 0.6 50 50 ] GnuPlotHistogram} }

proc openwindow { } { setup_windowMD++ openwin alloccolors rotate saverot eval plot}

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−proc exitmd { } { MD++ quit }#end of proc exitmd#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−proc setup_md { } { MD++ { equilsteps = 0 timestep = 0.001 # (ps)atommass = 196.96655 # (g/mol)DOUBLE_T = 1totalsteps = 1000saveprop = 1 savepropfreq = 100 openpropfile #runsavecn = 1 savecnfreq = 10000 openintercnfileplotfreq = 100 printfreq = 100ensemble_type = "NVT" integrator_type = "VVerlet" implementation_type = 0vt2 = 1e28 #1e28 2e28 5e28wallmass = 2e3 # atommass * NP = 14380boxdamp = 1e−3 # optimal damping for 216 atoms and wallmass 1e−3saveH # Use current H as reference (H0), needed for specifying stressfixboxvec = [ 0 0 1 1 0 1 0 0 0 ]output_fmt = "curstep EPOT KATOM Tinst HELM HELMP TSTRESS_xx TSTRESS_yy TSTRESS_zz H_11 H_22 H_33" } }#end of proc setup_md

#*******************************************# Main program starts here#*******************************************# status 0:# 1:# 2:## read in status from command line argumentif { $argc == 0 } { set status 0} elseif { $argc > 0 } { set status [lindex $argv 0]}puts "status = $status"


3/508/05/2012

if { $argc <= 1 } { set n 0} elseif { $argc > 1 } { set n [lindex $argv 1]}puts "n = $n"

if { $argc <= 2 } { set m 0} elseif { $argc > 2 } { set m [lindex $argv 2]}puts "m = $m"

if { $status == 0 } { # create initial structure with (111) surface MD++ setnolog initmd $status $n $m readeam Au

make_perfect_crystal 4 4 3

setup_window openwindow

relax_freebox MD++ eval MD++ finalcnfile = "au−perf.cn" writecn

# make surface MD++ input = \[ 3 3 0.5 \] changeH_keepR

# apply bi−axial compression MD++ input = \[ 1 1 −0.014 \] changeH_keepS MD++ input = \[ 2 2 −0.014 \] changeH_keepS

relax_fixbox MD++ eval plot

#setup_md #MD++ T_OBJ = 30 initvelocity totalsteps = 1000 run #relax_fixbox

MD++ finalcnfile = "au−surf.cn" writecn

# click on atoms to get (sx,sy) coordinates of sites A, B, C

MD++ sleep exitmd

} elseif { $status == 1 } { # put adatom on surface # prepare and relax dislocation structure MD++ setnolog initmd $status $n $m readeam Au

# allocate more memory than needed because we will add atom later MD++ allocmultiple = 2 MD++ incnfile = "../au−surf−0−0−0/au−surf.cn" readcn

set n [MD++_Get "NP"] set indx [expr $n*3 ] set indy [expr $n*3+1] set indz [expr $n*3+2] # add one more atom on surface (site A) MD++ NP = [expr $n + 1]


4/508/05/2012

setup_md

# put new atom on site A MD++ SR($indx) = 0.0416 MD++ SR($indy) = 0.0 MD++ SR($indz) = 0.35

setup_window openwindow

relax_fixbox MD++ T_OBJ = 30 initvelocity totalsteps = 1000 run relax_fixbox MD++ finalcnfile = "au−surf−A.cn" writecn MD++ finalcnfile = "au−surf−A.cfg" writeatomeyecfg

# put new atom on site B MD++ incnfile = "../au−surf−0−0−0/au−surf.cn" readcn MD++ NP = [expr $n + 1] MD++ SR($indx) = 0.0 MD++ SR($indy) = 0.0 MD++ SR($indz) = 0.35 relax_fixbox MD++ T_OBJ = 30 initvelocity totalsteps = 1000 run relax_fixbox MD++ finalcnfile = "au−surf−B.cn" writecn MD++ finalcnfile = "au−surf−B.cfg" writeatomeyecfg

# put new atom on site Ap MD++ incnfile = "../au−surf−0−0−0/au−surf.cn" readcn MD++ NP = [expr $n + 1] MD++ SR($indx) = −0.0208 MD++ SR($indy) = 0.0625 MD++ SR($indz) = 0.35 relax_fixbox MD++ T_OBJ = 30 initvelocity totalsteps = 1000 run relax_fixbox MD++ finalcnfile = "au−surf−Ap.cn" writecn MD++ finalcnfile = "au−surf−Ap.cfg" writeatomeyecfg

# put new atom on site Bp MD++ incnfile = "../au−surf−0−0−0/au−surf.cn" readcn MD++ NP = [expr $n + 1] MD++ SR($indx) = 0.0625 MD++ SR($indy) = 0.0625 MD++ SR($indz) = 0.35 relax_fixbox MD++ T_OBJ = 30 initvelocity totalsteps = 1000 run relax_fixbox MD++ finalcnfile = "au−surf−Bp.cn" writecn MD++ finalcnfile = "au−surf−Bp.cfg" writeatomeyecfg

MD++ sleep exitmd } elseif { $status == 2 } { # find minimum energy path for dislocation loop nucleation MD++ setnolog initmd $status $n $m readeam Au

MD++ incnfile = "../au−surf−1−0−0/au−surf−$n.cn" readcn saveH MD++ incnfile = "../au−surf−1−0−0/au−surf−$m.cn" readcn restoreH SHtoR setconfig2 MD++ incnfile = "../au−surf−1−0−0/au−surf−$n.cn" readcn readcn setconfig1

setup_window #openwindow

MD++ eval

MD++ fixallatoms constrain_fixedatoms freeallatoms MD++ chainlength = 25 totalsteps = 1000


5/508/05/2012

MD++ timestep = 0.01 printfreq = 2 MD++ allocchain initRchain

# nebspec = [ relax_surounding_atoms reparam_freq fix_left_end fix_right_end climb_image] MD++ { nebspec = [ 0 1 1 1 1 ] stringrelax } MD++ finalcnfile = stringrelax.chain.cn writeRchain

#MD++ sleep exitmd } elseif { $status == 3 } {

exitmd

} elseif { $status == 4 } {

exitmd

} elseif { $status == 10 } { # visualization MD++ setnolog initmd "view" $n $m readpot

MD++ incnfile = "../au−surf−1−0−0/au−surf−$n.cn" readcn saveH MD++ incnfile = "../au−surf−1−0−0/au−surf−$m.cn" readcn restoreH SHtoR setconfig2 MD++ incnfile = "../au−surf−1−0−0/au−surf−$n.cn" readcn readcn setconfig1

MD++ zipfiles = 0 for { set iter 0 } { $iter <= 25 } { incr iter 1 } { MD++ incnfile ="../w−disl−nuc−${strain_yz}/stringrelax.chain.cn" readRchain

# set chain number set chain_no $iter set total_no 25

MD++ input = $iter copyRchaintoCN eval

if { $iter == 0 } { openwindow } MD++ eval plot # write cn or cfg file MD++ finalcnfile = "chain_no_${chain_no}.cn" writecn MD++ finalcnfile = "chain_no_${chain_no}.cfg" writeatomeyecfg MD++ sleepseconds = 1 sleep

} MD++ sleepseconds = 100 sleep exitmd

} else { puts "unknown status = $status" exitmd

}

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ME346C Advanced Techniques for Molecular Simulations ...

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