DMC presentation 08_05_03

8/7/2019 DMC presentation 08_05_03

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Dynamic Monte CarloDynamic Monte CarloMethods:

Theory and Applications

ByByDaniela S. MainardiDaniela S. Mainardi


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OutlineOutline

1. Theoretical IntroductionDMC MethodProgram Carlos 4.0

2. Applications

Temperature Programmed SimulationsLateral InteractionsVoltammetric Scans


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THEORYTHEORY


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Introduction to DMC MethodsIntroduction to DMC Methods

Quantum Mechanical(Ab Initio, DFT)

Classical(Monte Carlo, Molecular Dynamics)

Laws of Quantum Mechanics E H =

Quantum effects determine all the chemistry

1. Do not explicitly include the presence of electrons2. Effective force fields represent the electronic effects3. Physical and thermodynamic properties obtained using

classical statistical mechanics and Newtons Law (MD)

Hardware and software not enough to fully trerelatively large systems!!!


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Introduction to DMC MethodsIntroduction to DMC Methods

MD simulations provide dynamic phenomena tthe order of nanoseconds.

For phenomena taking place in the range ofseconds,

Coarse-Grained methods

Dynamic Monte Carlo


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DMC MethodsDMC Methods

Advantages:1.treatment ofnon-equilibrium systems in which

conditions change over time

Importance reaction mechanisms forheterogeneous catalysis 2.It is able tosimultaneously simulate many differe

transitions or reaction types covering events tspan time scales up to theorders of seconds 3.Thereal-time behavior of a system is simulate


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DMC MethodsDMC Methods

Main Challenge:Reaction rates are needed as input

experiments or ab initio calculations


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How can we implement theHow can we implement theDMC method?DMC method?

Carlos 4.0Carlos 4.0Copyright Johan J. LukkienCopyright Johan J. LukkienOctober 1995October 1995

If we want to study chemicalreactions on crystal surfaces


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What is Carlos useful for?What is Carlos useful for?

Systems not at equilibrium like 1. Temperature Programmed systems 2. Voltammetricscans3. Oscillating systems, spatio-temporal patternformation4. Epitaxial growth

Non-homogeneous systems 1. Surface steps/defects

2. Lateral interactionsSystems with slow relaxation


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What do we need?What do we need?

1. Specify the chemical System we want tstudy

2. A stochastic model is built:* crystal surface,* the adsorbates,* a law to describe the microscopicreaction steps that change the surfaceconfiguration over time.

The behavior of this system over time isdetermined by the rates of these reactions(specified as probabilities).


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Thesurfacesurface is represented by a lattice: eachlattice point corresponds to a surface site

Elements in the model

The crystal surface

fcc (111) surfaceTopBridgeHollow

Atom

Unit cell


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Elements in the model

The evolution of the system over time is described by the chemicalmaster equationmaster equationThe law

( ) = cc cccc t cP k t cP k dt t cdP

''' ),(),'(

),(

P(c,t) probability of finding the system inconfigurationc at time t .k cc microscopic rate of the reaction that transfer cinto c

=T k

E r

B

act exp Arrhenius LawArrhenius Law

ratereactionr k

r k

dt

d

A

cc

AAAA

k A

~

*

'

==


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We could either give Carlos the macroscopic rates orEact and , and Carlos will compute the microscopicones to solve the Master equation

There are methods which are used internally to solvethis equation: First-Reaction Method (FRM) First-Reaction Method (FRM) Variable Step Size Method (VSSM) Variable Step Size Method (VSSM)

Random Selection Method (RSM) Random Selection Method (RSM)

How Carlos works?


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= t

t

t r dt )(exp ''

r r = the time dependent rate of a reaction= the time dependent rate of a reaction = a random number selected uniformly (0,1= a random number selected uniformly (0,1

Solving the Master equation

The way in which this equation issolved gives rise to the differentmethods

R.J. Gelten, R.A. VanSanten, A.P.J. Jansen, in: P.B. Balbuena,J.M. Seminario (Eds.), Molecular dynamics: From classical toquantum methods, Elsevier Science, Amsterdam, 1999, p. 737


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FRMFRM

It can be used forBOTHBOTH time-dependent or timeindependent rate constants.

According to this method, when the system igiven configuration , the set of all possiblereactions is determined, and a time of occurrt is generated forEACHEACH reaction

Carloslists all the reactions from the one withthe smallestt to the one with the largest tim


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- Then, the reaction with the- Then, the reaction with theSMALLEST SMALLEST t t is selected, the configuration is changeis selected, the configuration is changeaccordingly, and the timeaccordingly, and the timet t is incrementedis incremented

inint t - Finally, the set of possible reactions is- Finally, the set of possible reactions is

generated according to the newgenerated according to the newconfigurationconfiguration , and the procedure is, and the procedure isrepeatedrepeated

FRMFRM


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VSSMVSSM

It can be used for time-independent rateconstantsONLY

According to this method, when the system given configuration , the set of all possiblereactions is determined, andONLY ONEONLY ONE time ofoccurrence t is generated forALLALLreactions

Then,ONEONE reaction is selected with a probabiproportional to its rate, the configuration ischanged accordingly, the procedure is repea


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It can be used forBOTHBOTH time-dependent or time-independent rate constants.

ASITESITE is selected with a probability 1/N, beingtotal number of sites. Then, a given reaction isselected with probability proportional to its rate

When this reaction is possible on the site, it isexecuted. After each selection of a site, time isincremented by t which is selectedRAMDOMLYRAMDOMLYfroma FIXEDFIXED distribution

the configuration is changed accordingly, the

i

ir Nt exp1

RSM


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FRM and VSSM can be implementedvery efficientlyvery efficiently because information generated at some time step cabe re-used at subsequent steps.

The advantage ofVSSM and RSM over FRMis that thecost per generated transition (reaction) isindependentindependentof the sizeof the sizeof the grid used in the simulation, whereain FRM it scales as thelogarithmlogarithmof the number of gridpoints. Due to this, VSSM and RSM are more efficin memory use and execution time.

The advantage ofFRM over VSSM is that it is asuitable method to simulate reactions withtime-time-dependentdependenttransition probabilities.

The advantage ofRSM is that is easy to implement andfor some models (or reaction types) it is a veryfastfastmethod.

Comments


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Preparing Input Files...Preparing Input Files...

CARLOS needs two input files to run aCARLOS needs two input files to run asimulation:simulation:1.1. Initial Configuration:Initial Configuration: can be made by CARLOS,

.con file, or can be constructed directly using aeditorname.init name.init

2.2. System FileSystem File:: contains a complete description of simulation model

name.sim name.sim


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Geting outputs...Geting outputs...The logfileThe logfile:: namename(information recorded during the simulation)The parameter fileThe parameter file:: name.parname.par

(describing both the simulation model and relevant infoabout the simulation such as timing, memory and methoThe graph fileThe graph file:: name.xxxx.gifname.xxxx.gif(pictures of the lattice in gif format, if g option is not g The configuration fileThe configuration file:: name.conname.con(the final configuration if the -C option is given)

The system fileThe system file:: name.sysname.sys(a copy of the system file if the -S option is present )


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APPLICATIONSAPPLICATIONS


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OO22 desorption from Pt (111)desorption from Pt (111)

Temperature Programmed

simulations


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ONLY O2 desorption reaction isinvestigated:

Prefactor: 1.0 x 1013s-1Activation energy: 0.36 eV

1 O 2 ML

O2 desorption O 2(ads_b) O 2 + b b = empty bridge

O2 O

2O

2O

2O

2O2

O2

Initial configuration: full, i.e, 1 O2 ML is adsorbed on the surfa

T = 100.0 + 2.0 * tI increased T from 100K to 190 K:


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Final configuration

O2_des

0

0.2

0.4

0.6

100 120 140 160 180

T (K)

Reactio

ns/unit cell/sec

0

0.2

0.4

0.6

0.8

1

100 120 140 160 180

T (K)

O2 Concentration

Initial configuration

A. Winkler et al.,Surface Science 201(1988), 419 - 443

RESULTS


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OO22 adsorption, dissociation,adsorption, dissociation,desorption, and O diffusiondesorption, and O diffusionon Pt (111)on Pt (111)

Temperature Programmed

simulations


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TopBridgeHollowPt atom

O2

O2

O2 adsorption on Pt(111) occurs on bridge sites (b). Upon dissociation of O 2 , O atoms occupy hollow sites

(h).

Pt(111) surface

O2 adsorption O 2 + b O 2(ads_b)

O2 dissociation O 2(ads_b) + 2h 2O(ads_h) + b

O diffusion O (ads_h hcp ) + h fcc h hcp + O (ads_h fcc )

O2 desorption O 2(ads_b) O 2 + b

64 x 64 lattice

b = empty bridgeh = empty hollow

Initial configuration: empty Pt surface


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Reaction Eact(eV)

Eact/KB (K) (s -1 )

O2 ads 0.09 1044.4 F(T)

O2 diss 0.3 3481.33 1.20E+12O2 des 0.36 4177.6 1.00E+13O diff 0.13 1508.58 2.60E+12

Prefactor calculated using Hertz-Knudsen formula:

MkT P F

2=KTE

0site

a

eFSA += [P] = Pa[k] = J/K[M] = kg (= 0.032 kg/NA)[T] = K[F] = molecules/m2.sec


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64 x 64 Pt atoms

In a lattice with 64x64 Pt atoms we hav64x64 unit cells (due to the periodic

boundary conditions)There are 3 bridge, 2 hollow, and 1 topsites per unit cell, therefore,

assuming that all sites have the same ar

N sites per unit cell = 6Total N of sites = 6 x 64 x 64 = 24576

24576 sites have an area of 3.151 x 10-12 cm21 site 1.282 x 10-16 cm2

For my simulations I will consider:

Asite = Area of 1 adsorption site = 1.28 x 10-16 cm2 = 1.28 x 10-20 m2

Assuming a flat surface:

......

.

.

.

.

.

.

rPt = 1.387 = 1.387 x 10-8 cm2DPt = 2.774 x 10-8 cm2

W

W = 64 x DPt = 1.775 x 10-6 cm2 Area = W2 = 3.151 x 10-12 cm2


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P = 0.001 PaA = 1.28E-20m2

KTE

0site

a

eFSA +=

T S086.3801091 2.97E-01

92.3057527 2.95E-0196.4545828 2.85E-0198.3604058 2.71E-01

100 272093 2 55E 01

T range (K) O2ads ( s-1 )

100 - 108 4.80 E+03108 - 111 5.72 E+02111 - 120 2.46 E+02

120 - 135 7.88 E+01135 - 148 1.94 E+01148 - 154 7.42 E+00154 - 162 2.47 E+00162 - 170 1.18 E+00

values used in Carlos input

A. Winkler et al.,Surface Science 201(1988), 419 - 443

MkT P F

2=


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0.00E+00

1.00E+02

2.00E+02

3.00E+02

120 130 140 150 160 170

T

Prefactor (O2 ads)

0.00E+00

1.00E+03

2.00E+03

3.00E+03

4.00E+03

5.00E+03

6.00E+03

100 105 110 115 120

T

Prefac

tor (O2 ads)

KTE

0site

a

eFSA +

= Prefactor for O2 ads calculated usingPrefactor used in Carlos at each T range


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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

100 120 140 160T (K)

Reactions/unit cell/sec

O2_desO2_adsO2_dissO_diff

00.1

0.20.30.4

0.5

0.60.70.8

0.91

100 120 140 160

T (K)

Concentration

O2O

P=10-3 Pa

Results with the previousapproximation

Heating rate = 2K/s


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Using Discrete StepsUsing Discrete Steps

(DS)...(DS)...and the following fitting for S0(T) given by Winkler et al

0

0.02

0.040.06

0.08

0.1

0.12

0.14

0.16

100 150 200 250 300

T (K

Rate O2_ads (1/s)

W [O2ads]mathematical fitti

W [O2ads] (1/sec)using S0 Exp.

100 K < T < 111 K

T xS 20 1049.17526.1

=

= T

xS 100

exp1012.3 20

T > 111K 00.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

100 150 200 250 300

Rate O2_ads (1/s)

ReactRate given bCarlos

W [O2ads]mathematical fitting


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Using DS I gave Carlos the functional formUsing DS I gave Carlos the functional formof the rate for Oof the rate for O 22 ads as Sads as S 00FAFAsitesite

P = 10-3 Pa

0.3

0.4

0.5

/unit cell/sec O2_des

O2_adsO2_dissO_diff

Heating rate = 2K/s

00.1

0.20.30.40.5

0.6

0.70.8

0.91

100 120 140 160 180 200

T (K)

Concentration

OO2

00.1

0.20.30.40.5

0.60.70.8

0.91

100 120 140 160

T (K)

Concentration

O2O

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.160.18

0.2

100 120 140 160T (K)

Reactions/unit cell/sec O2_des

O2_adsO2_dissO_diff


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OO22 electroreduction on Ptelectroreduction on Pt(111)(111)

Voltammetric scans


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I O 2(gas) + e - + H + + b HO 2(ads_b)II HO 2(ads_b) + 2h + e

- + H + 2OH(ads_h) + b

III O 2(gas) + b + e- O2

-(ads_b)

IV O 2-(ads_b) + 2h O -(ads_h) + O(ads_h) + b

V O(ads_h hcp ) + h fcc hhcp + O(ads_h fcc )

VI O - (ads_h) + H + OH (ads_h)

VII O (ads_h) + e - + H + OH (ads_h)

VIII OH (ads_h) + t h + OH (ads_t)IX OH (ads_t) + H + + e - H2O(ads_t)

X OH(ads_h) + H + + e - H2O(liq) + h

XI H 2O(ads_t) H2O (liq) + t

Proposed Mechanism


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Lateral Interactions

HO2O2- H2O

Up to 3rd neighbors repulsions for O

1st neighbors repulsionsModel 1 Model 2

Model 3


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0.8

1

tion O2

OH

Results

Model 1Model 2

0.8

1

tion


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( )

+= T K

V eE r

B

a

ii

i 00

exp

e 0 = charge of an electron Eai = activation energy at V=0V = overpotential i0 = pre-exponential factor = transfer coefficient = 0.5 if the reaction consumelectrons = 0.5 if it produces e-

V is taken with respect tthe standard reversiblepotential for O

2

reduction, 1.23 V, on thehydrogen scale.

Voltammetric ScansIn the presence of an overpotentialV (defined as thedifference between the actual and the equilibriumpotential), the rate of a reactionr i is given by:

Current = (e- production rate) (e- consumption rate)The overpotential was decreased linearly from 0 until -1.12 V a rate of 50 mV/sec starting from a clean Pt(111) surface.


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0-1

1

1.1

HE)

Tafel plot

Model V0(V) Tafel Slope(mA/cm2)1 -0.36 53 182 -0.39 34 12

3 -0.39 41 15


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AcknowledmentsAcknowledments

Dr. Perla BalbuenaDr. Perla BalbuenaMolecular modeling group:Molecular modeling group: Dr. WangDr. Wang Dr. Martinez-limiaDr. Martinez-limia Yingchun ZhangYingchun Zhang Sergio CalvoSergio Calvo Zhihui GuZhihui Gu Francisco Tarazona VasquezFrancisco Tarazona Vasquez Eduardo LamasEduardo Lamas Tyler WattTyler Watt Kyle CorbinKyle Corbin

Charlotte CooperCharlotte Cooper Diego AltomareDiego Altomare

DMC presentation 08_05_03

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