FROM ATOMIC SCALE ORDERING TO MESOSCALE
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FROM ATOMIC SCALE ORDERING TO MESOSCALE SPATIAL PATTERNS IN SURFACE REACTIONS: HCLG. MULTISCALE MODELING WORKSHOP II (KRATZER, RATSCH, VVEDENSKY) IPAM - UCLA OCT 2005. Jim Evans 1,2 , Dajiang Liu 1 : Stat Mech & Multiscale Modeling 1 Chemical Physics Program, Ames Laboratory USDOE

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From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

FROM ATOMIC SCALE ORDERING TO MESOSCALE

SPATIAL PATTERNS IN SURFACE REACTIONS: HCLG

MULTISCALE MODELING WORKSHOP II (KRATZER, RATSCH, VVEDENSKY) IPAM - UCLA OCT 2005

Jim Evans1,2, Dajiang Liu1: Stat Mech & Multiscale Modeling

1Chemical Physics Program, Ames Laboratory USDOE

2Mathematics Dept., Iowa State University, Ames, Iowa

MULTISCALE MODELING

OF MESOSCALE REACTION

FRONT PROPAGATION IN

CO-OXIDATION ON Pd(100)

HETEROGENEOUS COUPLED

LATTICE-GAS (HCLG)

SIMULATION APPROACH

…parallel LG simulations coupled

via mesoscale CO surface diffusion

Phys. Rev. B 70 (2004) 193408; SIAM Multiscale Modeling Sim. 4 (2005) 424


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

OUTLINE

  • PART I: CO-OXIDATION - KINETICS AND FRONTS

  • Traditional Modeling: mean-field rate equations & reaction-diffusion equations (RDE)

  • Expts: kinetics and steady-states, electron microscopy  Limitations of mean-field !

  • PART II: CONNECTINGTHELENGTHSCALES FROM

  • LOCAL ORDERING TO MESOSCALE PATTERNS

  • HCLG Multiscale Modeling to describe spatial patterns & reaction fronts on a large

  • a characteristic length scale (microns) incorporating precise atomic scale information

  • Collective or chemical diffusion on surfaces: non-trivial Onsager transport problem

  • PART III: CANONICAL ATOMISTIC LATTICE-GAS MODEL

  • Adspecies ordering; kinetics & steady-states; percolative chemical diffusion; HCLG

  • PART IV: REALISTIC MODELING FOR CO+O/Pd(100)

  • Development of atomistic LG model; HCLG results


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

MEAN-FIELD RATE EQUATIONS & REACTION-DIFFUSION

EQUATIONS (RDE’s) FOR CO-OXIDATION ON SURFACES

CO(gas) +  CO(ads) CO-ADSORPTION

O2(gas) + “ 2 ”  2O(ads) O2-ADSORPTION

CO(ads)+O(ads)  CO2(gas) +2CO+O REACTION

CO(ads)  CO(gas) +  CO-DESORPTION

CO(ads) +  + CO(ads) RAPID CO-DIFFUSION

PCO

PO2

k

d

h

MEAN-FIELD RATE AND REACTION-DIFFUSION EQUATIONS

/t CO = PCOSCO - RCO+O - d CO + DCO2CO

/t O = 2PO2SO2 - RCO+Owhere  = surface coverages

SCO,O2 = sticking coeffts, RCO+O = reaction rate  k COO or… DCO h

REFINEMENTS: SURFACE RECONSTRUCTION PROVIDES ADDITIONAL DEGREE OF FREEDOM


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

Stable Inactive State

…near CO-poisoned

CO

CO

BISTABILITY

OF STEADY-STATES

PCO

Stable Reactive State

…low CO coverage

CO-partial pressure PCO

PREDICTIONS OF MF RATE & RD EQUN: CO-OXIDATION

CO(gas) +   CO(ads); O2(gas) + 2  2O(ads); CO(ads) + O(ads)  CO2(gas) + 2

Increase d, T

Non-Equilibrium Critical Point:

Bistability  Monostability

Reaction-Diffusion Phenomena:

Front Width & Velocity  (DCO)1/2

CO

Inactive

REACTION

FRONT

Spatial Non-Uniformity

@ fixed (small) PCO

Reactive

x


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

CO

LOW T

CO

HIGH T

PCO

PCO

EXPT STUDIES OF REACTION KINETICS: CO-OXIDATION ON Pt(111)

Berdau et al. J. Chem. Phys. 110 (1999) 11551


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

PHOTO-EMISSION ELECTRON MICROSCOPY (PEEM) STUDIES: CO-OXIDATION

  • CO-OXIDATION ON Pt(111)

  • a classic bistable system

  • Expansion of reactive state

  • into CO-poisoned state

  • facilitated by an “O-defect”

  • Temperature = 413 K

PEEM studies by Christmann &

Bloch groups, JCP 110 (99) 11551

380 m

  • CO-OXIDATION ON Pt(110)

  • system with oscillatory kinetics

  • due to surface reconstruction

  • Temperature = 400 K

400 m

Review: Imbihl & Ertl, Chem. Rev. 1995


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

O

25 m

CO

SHORTCOMINGS OF MEAN-FIELD RDE TREATMENT

LEEM

IMAGE

300 K

CO

O

KMC

300 K

“COMPLEX” REACTION FRONTS:

TITRATION OF PREADSORBED CO

ON Pt(100) BY EXPOSURE TO O

Tammaro, Evans, …Bradshaw, Imbihl,

Surf Sci 407 (1998); also 307 (1994)

ISLANDING & ORDERING IN

REACTIVE STEADY-STATES:

CO-OXIDATION ON Pd(100) @ 300K

Realistic atomistic lattice-gas modeling

Liu and Evans, PRB (04); JCP (05)

  • Adspecies are not well-stirred or

  • Randomly distributed (interactions)

  • Reaction rate  kCOO, etc.

    cf. Engel & Ertl. J. Cat. (1981)

Fronts do have smooth tanh–form

of MF RDE due to ordering & due to

COMPLEX NATURE OF CHEMICAL

DIFFUSION IN MIXED ADLAYERS


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

HETEROGENEOUS COUPLED LATTICE-GAS (HCLG) ANALYSIS

...for simple reaction model, J. Chem. Phys. (1995)

Exact Reaction-Diffusion Eqns

/t CO = RCO({CO,O}) - JCO

/t O = RO({CO,O})

where {CO,O} denotes the full

configuration of the adlayer

Simultaneous LG simulations distributed across reaction front.

Extract simultaneously reaction kinetics and CO chemical diffusivity.

CO(i) = “RCO t” + [JCO(i-1i) - JCO(ii+1)]t, O(i) = “ROt”

HCLG: Tammaro, Sabella, Evans JCP (95); Liu & Evans PRB (04); SIAM-MMS (05)

cf. Heterogeneous Multiscale Method E & Enquist (03); Gap-tooth Method Kevrekidis et al. (03)


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

“EXACT” TREATMENT OF CO SURFACE MASS TRANSPORT

EXTENSIVE STUDIES on CHEMICAL (COLLECTIVE) DIFFUSION in INTERACTING

SINGLE SPECIES ADLAYERS, e.g., Gomer, Rep. Prog. Phys. (1990), but here…

CHEMICAL DIFFUSION IN MIXED INTERACTING ADLAYERS

Low CO… percolative diffusion of CO(ads) through relatively immobile coads. O(ads)

JCO = -CO COfor Onsager coefft. CO= CO-conductivity/(kT)

…so in addition to reaction kinetics, parallel HCLG simulations must also

determine the (collective) CO mobility, CO, & CO chemical potential, CO

(e.g., via Widom insertion method). Numerical implementation via…

JCO(kk+1) = - CO(k+½)[CO(k+1)-CO(k)] with CO(k+½ )= ½ [CO(k)+CO(k+1)]

...fairly mobile O(ads)  local adlayer equilibration ?  CO= CO(CO, O)

…or no CO-CO or CO-O interactions  random CO  ditto

 JCO = - DCO,CO CO- DCO,O O

where DCO,CO & DCO,O = (thermodynamic factors)  CO

…second “cross-term” always ignored in traditional MF RDE modeling


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

CANONICAL ATOMISTIC LATTICE-GAS MODEL: CO-OXIDATION

PRL 82 (99) 1907; J Chem Phys 111 (99) 6579; PRL 84 (00) 955, JCP 113 (00); Chaos 12 (02); SIAM MSS 4 (05)

  • KEY MODEL FEATURES:

  • SQUARE-LATTICE OF

  • ADSORPTION SITES

  • FOR BOTH CO AND O

  • VERY STRONG NN

  • O-O REPULSION

  • NO O-O NN PAIRS

  • CHECKERBOARD

    C(2X2) ORDERING

  • EIGHT-SITE RULE

    FOR ADSORPTION

  • CONSIDER REGIME OF

  • RAPID DIFFUSION OF

  • CO: h >> other rates

  • CO IS RANDOMLY

    DISTIBUTED ON SITES

    NOT OCCUPIED BY O


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

STEADY-STATE BEHAVIOR

d=0

OXYGEN ADATOMS

REACTION KINETICS & STEADY-STATE BIFURCATIONS

d/dt CO = PCO(1-CO-O) - 4kOCOloc - dCO = RCO(CO,{O})

d/dt O = 2PO2SO2({O}, CO) - 4kOCOloc= RO(CO,{O}) where…

SO2= probability of 8-site ads ensemble; COloc=CO/(1-O)

SYMMETRY-BREAKING TRANSITION

FOR CHECKERBOARD ORDERING

…TO UNEQUAL POPULATIONS

OF THE TWO SUB-DOMAINS


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

SURFACE CHEMICAL DIFFUSION OF CO & EXACT RDE’S

/t CO = RCO(CO,{O}) - JCO, and /t O = RO(CO,{O})

where RCO= PCO(1-CO-O) - 4kOCOloc and RO= 2PO2SO({O}, CO) - 4kOCOloc and…

JCO = - DCO,COCO - DCO,O O(Onsager transport theory)

JCO = -CO CO for CO chem potential CO = kBT ln[CO/(1-CO-O)]

so… DCO,O = CO(1-O)-1 DCO,CO = COloc DCO,CO

Also DCO,CO = DCO(O) is independent of CO but decreases with O

i.e., many-particle CO chemical diffusion problem

reduces to a problem of single-particle percolative

diffusion for CO through a labyrinth of coadsorbed O


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

DCO

DIFFUSION

PATH for CO

*

O

ANALYSIS OF CO PERCOLATIVE DIFFUSION

LOW O: DIFFUSION AROUND ISOLATED OBSTACLES (ADSORBED O)

DCO = D0[1-a1 O - a2 (O)2 -…]  D0[1 - a1 O] Lifshitz-Sepanova-type density expansion

a1(monomer)=-1=2.14 (Ernst et al.) a1(dimer) = 2.96 (Liu & Evans)

HIGH O: PERCOLATIVE DIFFUSION (ALONG DOMAIN BOUNDARIES)

Cessation of diffusion  lack of percolation of domain boundary diffusion paths

 percolation of c(2x2) O-domains  symmetry-breaking in the O adlayer

DCO ~ D0 [*- O]where  = dynamic critical exponent for percolative transport

 = 1.3 (random percolation Alexander-Orbach)  = 1.4 (Ising HS: Liu & Evans)

O

O

0


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

DIFFUSION PATH AT THE

PERCOLATION THRESHOLD

WHEN PERCOLATION OCCURS

AFTER SYMMETRY BREAKING

Dynamical Critical Exponent  = 1.3

DIFFUSION PATH AT THE

PERCOLATION THRESHOLD

FOR SIMULTANEOUS

PERC & SYMM-BREAKING

Dynamical Critical Exponent =1.4


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

HETEROGENEOUS COUPLED LATTICE-GAS SIMULATION

Liu and Evans, SIAM Multiscale Modeling Sim. 4 (2005) 424

CO

k-1 k k+1

JCO(kk+1)= - DCO,CO(k+½)[CO(k+1)-CO(k)]/x - DCO,O(k+½)[O(k+1)-O(k)]/x

with D..(k+½ )= ½ [D..(k)+D..(k+1)]


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

SCALED VELOCITY

(changes sign

@ equistability)

DIRECT

SIMULATION

HCLG

DIFFUSION

PATH for CO

PROPAGATION VELOCITY OF REACTION FRONTS IN THE BISTABLE REGION

EQUISTABILITY POINT

HCLG

MF CONST. Dco

DIRECT

SIMULATION

with incr. hCO

SIMPLE

RDE

ANALYSIS OF PERCOLATIVE TRANSPORT OF CO(ads) THRU COADS. O(ads)

DCO,CO(O)

See also: Liu & Evans, PRL 84 (00) 955; JCP 113 (00) 10252


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

LATTICE-GAS MODEL DEVELOPMENT: CO+O/Pd(100)

CO

EQUILIBRIUM ORDERING: CO/Pd(100)

c(222)R45 CO @ bridge sites …CO<0.5

SEPN REPULSION

a/2 1CO =  (exclusion)

a 2CO = 0.17 eV * #GGA-PBE=0.22eV

2 a 3CO = 0.03 eV #GGA-PBE=0.02eV

10 a/2 4CO  0

#LEED, TPD (Behm et al 80) *QADS(King et al 97)

EQUILIBRIUM ORDERING: O/Pd(100)

p(22) and c(22) O @ 4f hollow sites …O<0.5

SEPN INTERACTION

a 1o = 0.36 eV (NN repulsion)GGA-PBE=0.37eV

2 a 2o = 0.08 eV (2NN repulsion)GGA-PBE=0.10eV

2 a 3o = -0.02 eV (3NN attraction)GGA-PBE= -0.04eV

LEED, TPD (Chang, Evans & Thiel, SS 89, Chang & Thiel JCP 88)

c(22)-O

p(22)-O

LG MODEL ANALYSES: KMC, Transfer Matrix – Finite Size Scaling


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

LATTICE-GAS MODEL DEVELOPMENT: CO+O/Pd(100)

KINETICS OF ADSORPTION:

Steering of CO to on-top sites (allow occupation of bridge, hollow and on-top sites)

Eight-site rule for dissociative adsorption of O (2NN ads. sites with 6 NN free of O)

KINETICS OF CO DESORPTION:

EbCO = 1.6 eV from bridge (low CO) with b = 1016/s (Behm et al. 80) GGA-PBE=1.9 eV

KINETICS OF DIFFUSION:

EdO = 0.65 eV - non-equil. ordering (LEED) GGA-PBE = 0.35 eV; EdCO ~ 0.2 eV (rapid CO diffusion)

ECO+O=1.0eV =0.19eV ECO+O=0.73eV =big

CO+O INTERACTION & REACTION:

Low coverages: CO(br)+O(4fh)CO2(gas)

High coverage reaction: CO forced to 4fh

site by p(2x2)- or c(2x2)-O …lower barrier

CO

CO

O O

“Typical” High-Coverage

Reaction Config. Reaction Config.

Zhang & Hu JACS 123 (2001) 1166 DFT

References:

CO/Pd(100): Liu, JCP 121 (04); Eichler & Hafner, PRB 57 (98) ; Behm et al. JCP (80)

O/Pd(100): Liu & Evans, SS 563 (04); Chang & Thiel, PRL (87) JCP (88); Evans, JCP (87)

CO+O/Pd(100): Liu & Evans, PRB 70 (04); JCP (05) submitted; Zhang & Hu, JACS (01)


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

“EXACT” STEADY-STATE BIFURCATION BEHAVIOR: BISTABILITY

STEADY-STATE BEHAVIOR (KMC)

for CO coverage vs. PCO for various T

BIFURCATION DIAGRAM (KMC)

for bistability region in (PCO,T)-plane

NON-EQUILIBRIUM CRITICAL POINT

(CUSP BIFURCATION)

STABLE INACTIVE STATES

Reactive

State only

UNSTABLE

STATES

Inactive

State only

PCO

STABLE REACTIVE STATES

CO =

O =

400K

PCO=0.07

Reactive state =p(2x2)-O + CO

Inactive state = c(222)R45 CO

+ small holes

  • PARAMETERS:

  • Total Pressure

  • ~ 10-3 Torr

  • Tot. Ads. Rate

    PCO + PO2  1 s-1

REACTIVE STATE INACTIVE STATE


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

300 K Reactive State (O = 0.39ML) 300 K Reactive State (O = 0.28ML)

300 K Reactive State (O = 0.16ML) 300 K Near-CO-Poisoned State ? (O = 0.02ML)


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

co,o

CO

O

~0.5 ML

~0.08 ML

~0.28 ML

~0 ML

JCO’s

-COCO

-DCO,COCO

-DCO,OO

CO

0.13  max for 0

0  max

x

RESULTS OF HCLG ANALYSIS: FRONTS AND TRANSPORT

INACTIVE STATE REACTIVE STATE

“Complex” profile

shape differs from

tanh - form of

standard MF RDE

Latter = analogue of

tanh-profile of Cahn

- Allenphase bndries

HCLG results validated

by comparison with

direct “brute force” KMC

(scaling up simulations

for lower CO hop rate)

SIMULATION CONDITIONS:

Temperature = 380 K

Adsorption rates:

PCO = 0.17 ML/s PO2 = 1 ML/s

(equistability between reactive &

inactive states  stationary front)

CO mobility


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

SUMMARY

♦MULTISCALE HCLG MODELING EFFECTIVELY

INCORPORATES ATOMIC SCALE INFORMATION INTO

DESCRIPTION OF MESOSCALE FRONT PROPAGATION

…compare with similar applied math multiscale methods:

Gap-tooth methods for hydrodynamic systems – Kevrekidis

Heterogeneous Multiscale Methods (HMM) – E & Enquist

♦ KEY FACTOR: CORRECT TREATMENT OF DIFFUSIVE

TRANSPORT – non-trivial, collective diffusion in interacting,

mixed species lattice-gas models for surface adlayers

♦ APPLICATION TO SPECIFIC SYSTEM: CO+O/Pd(100)

Challenge: to describe complex adlayer ordering mediated

by weak adspecies interactions; determined from expt & DFT


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

TPR STUDIES: COMPARISON OF MODEL WITH EXPERIMENT

TPR EXPERIMENTS: CO2 PRODUCTION

Below: Stuve et al., Surf. Sci. 146 (1984)

Also: Zheng & Altman, JPC B 106 (2002)

TPR SIMULATIONS: CO2PRODUCTION

ATOMISTIC LG REACTION MODEL

O = 0.25 ML

360K peak

O =

0.25

CO =

0.80

0.75

0.55

405

O =

0.25

CO=

0.24

0.11

0.05

0.03

0.01

0.005

405K peak

CO =

0.40

0.28

0.19

0.10

0.050

low-T peak

O

PROCEDURE:

300K deposit 0.25ML O  p(22)

100K deposit various CO amounts

Heat @ ~10K/s

Monitor CO2production versus T

CO

High CO>0.25: Eact=0.73 Low CO: Eact=1.0 CO>0.1: Eact=1.0+=1.2


From atomic scale ordering to mesoscale spatial patterns in surface reactions hclg

ATOMISTIC MODELING OF STM-BASED TITRATION STUDIES

Pre-deposit O at low T: create c(2x2) domains plus antiphase boundaries. Expt: Chang et al. PRL (87)

Then expose to CO @ 300K: titrates O(ads), initially preferentially reacting at domain boundaries.

KMC

CO+O/Pd(100) @ 300 K

O

Reaction rate ~ (O)m,

withm  0.6  1/2

CO

STM

Wintterlin et al.

Science 278 (1997)

JCP 114 (2001)

Chaos 12 (2002)

CO

CO+O/Pt(111) @ 300K

O


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