FROM ATOMIC SCALE ORDERING TO MESOSCALESPATIAL PATTERNS IN SURFACE REACTIONS: HCLG

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

1Chemical Physics Program, Ames Laboratory USDOE2Mathematics 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

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

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: CONNECTINGTHELENGTHSCALES 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

OUTLINE

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

MEAN-FIELD RATE EQUATIONS & REACTION-DIFFUSION EQUATIONS (RDE’s) FOR CO-OXIDATION ON SURFACES

MEAN-FIELD RATE AND REACTION-DIFFUSION EQUATIONS

/t CO = PCOSCO - RCO+O - d CO + DCO2CO

/t O = 2PO2SO2 - RCO+O where = surface coverages

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

PCO

PO2

k

d

h

REFINEMENTS: SURFACE RECONSTRUCTION PROVIDES ADDITIONAL DEGREE OF FREEDOM

PREDICTIONS OF MF RATE & RD EQUN: CO-OXIDATION

CO(gas) + CO(ads); O2(gas) + 2 2O(ads); CO(ads) + O(ads) CO2(gas) + 2

CO

PCO

Non-Equilibrium Critical Point: Bistability Monostability

Increase d, T

CO

xReactive

Inactive

Spatial Non-Uniformity@ fixed (small) PCO

REACTION FRONT

Reaction-Diffusion Phenomena:Front Width & Velocity (DCO)1/2

Stable Inactive State…near CO-poisoned

Stable Reactive State…low CO coverage

CO-partial pressure PCO

COBISTABILITY

OF STEADY-STATES

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

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

CO

PCO

LOW TCO

PCO

HIGH T

CO-OXIDATION ON Pt(111)- a classic bistable system

Expansion of reactive stateinto CO-poisoned statefacilitated by an “O-defect” Temperature = 413 K

380 m

PEEM studies by Christmann &Bloch groups, JCP 110 (99) 11551

CO-OXIDATION ON Pt(110)- system with oscillatory kinetics due to surface reconstruction

Temperature = 400 K

400

m

Review: Imbihl & Ertl, Chem. Rev. 1995

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

SHORTCOMINGS OF MEAN-FIELD RDE TREATMENT

25 mO

CO

“COMPLEX” REACTION FRONTS: TITRATION OF PREADSORBED CO ON Pt(100) BY EXPOSURE TO OTammaro, Evans, …Bradshaw, Imbihl, Surf Sci 407 (1998); also 307 (1994)

ISLANDING & ORDERING IN REACTIVE STEADY-STATES:CO-OXIDATION ON Pd(100) @ 300KRealistic atomistic lattice-gas modeling Liu and Evans, PRB (04); JCP (05)

LEEMIMAGE300 K

KMC300 K

Adspecies are not well-stirred or Randomly distributed (interactions)Reaction rate kCOO, etc. cf. Engel & Ertl. J. Cat. (1981)

Fronts do have smooth tanh–form of MF RDE due to ordering & due toCOMPLEX NATURE OF CHEMICALDIFFUSION IN MIXED ADLAYERS

COO

HETEROGENEOUS COUPLED LATTICE-GAS (HCLG) ANALYSIS

Simultaneous LG simulations distributed across reaction front.Extract simultaneously reaction kinetics and CO chemical diffusivity.

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

Exact Reaction-Diffusion Eqns /t CO = RCO({CO,O}) - JCO

/t O = RO({CO,O})

where {CO,O} denotes the fullconfiguration of the adlayer

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

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)

“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 CO for 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(kk+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

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

d/dt CO = PCO(1-CO-O) - 4kOCOloc - dCO = RCO(CO,{O})

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

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

d=0

STEADY-STATE BEHAVIOR

REACTION KINETICS & STEADY-STATE BIFURCATIONS

SYMMETRY-BREAKING TRANSITION FOR CHECKERBOARD ORDERING …TO UNEQUAL POPULATIONS OF THE TWO SUB-DOMAINS

OXYGEN ADATOMS

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

where RCO = PCO(1-CO-O) - 4kOCOloc and RO = 2PO2SO({O}, CO) - 4kOCO

loc and…

JCO = - DCO,COCO - DCO,O O (Onsager transport theory)

SURFACE CHEMICAL DIFFUSION OF CO & EXACT RDE’S

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 problemreduces to a problem of single-particle percolativediffusion for CO through a labyrinth of coadsorbed O

DIFFUSIONPATH for CO

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

DCO

*O

O0

DIFFUSION PATH AT THEPERCOLATION THRESHOLDWHEN PERCOLATION OCCURSAFTER SYMMETRY BREAKINGDynamical Critical Exponent = 1.3

DIFFUSION PATH AT THEPERCOLATION THRESHOLDFOR SIMULTANEOUSPERC & SYMM-BREAKINGDynamical Critical Exponent =1.4

HETEROGENEOUS COUPLED LATTICE-GAS SIMULATION

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

CO

JCO(kk+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)]

k-1 k k+1

DIFFUSIONPATH for CO

PROPAGATION VELOCITY OF REACTION FRONTS IN THE BISTABLE REGION

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

DCO,CO(O)

EQUISTABILITY POINT

HCLG

SIMPLE RDE

DIRECTSIMULATIONwith incr. hCO

SCALED VELOCITY (changes sign @ equistability)

HCLG

DIRECTSIMULATION

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

MF CONST. Dco

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

EQUILIBRIUM ORDERING: CO/Pd(100)c(222)R45 CO @ bridge sites …CO<0.5

SEPN REPULSION a/2 1

CO = (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(22) and c(22) O @ 4f hollow sites …O<0.5

SEPN INTERACTION a 1

o = 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)

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

c(22)-O

p(22)-O

CO

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

“Typical” High-Coverage Reaction Config. Reaction Config.

Zhang & Hu JACS 123 (2001) 1166 DFT

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 DIFFUSION:Ed

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

KINETICS OF CO DESORPTION:Eb

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

CO+O INTERACTION & REACTION:Low coverages: CO(br)+O(4fh)CO2(gas)High coverage reaction: CO forced to 4fhsite by p(2x2)- or c(2x2)-O …lower barrier

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)

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

CO CO

O O

“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

PARAMETERS:Total Pressure ~ 10-3 TorrTot. Ads. Rate PCO + PO2 1 s-1

REACTIVE STATE INACTIVE STATE

CO =O =

400KPCO=0.07

Reactive state = p(2x2)-O + COInactive state = c(222)R45 CO + small holes

NON-EQUILIBRIUM CRITICAL POINT(CUSP BIFURCATION)

ReactiveState only

Inactive State only

STABLE INACTIVE STATES

STABLE REACTIVE STATESPCO

UNSTABLESTATES

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)

RESULTS OF HCLG ANALYSIS: FRONTS AND TRANSPORT

“Complex” profileshape differs fromtanh - form ofstandard MF RDE

Latter = analogue of tanh-profile of Cahn- Allen phase bndries

co,o

JCO’s

CO

x

-DCO,COCO

-COCO

-DCO,OO

CO O~0.5 ML

~0.28 ML~0 ML

~0.08 ML

0 max

0.13 max for 0

HCLG results validated by comparison with direct “brute force” KMC (scaling up simulations for lower CO hop rate)

SIMULATION CONDITIONS:

Temperature = 380 KAdsorption rates:PCO = 0.17 ML/s PO2 = 1 ML/s(equistability between reactive &inactive states stationary front)

INACTIVE STATE REACTIVE STATE

CO mobility

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

TPR STUDIES: COMPARISON OF MODEL WITH EXPERIMENT

TPR EXPERIMENTS: CO2 PRODUCTIONBelow: Stuve et al., Surf. Sci. 146 (1984)Also: Zheng & Altman, JPC B 106 (2002)

TPR SIMULATIONS: CO2 PRODUCTION

ATOMISTIC LG REACTION MODEL

O =

0.25

CO=

0.24

0.11

0.05

0.030.01 0.005

405O =0.25

CO =

0.80

0.75

0.55

O = 0.25 ML

CO =

0.40 0.28 0.19 0.100.050

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

360K peak

405K peak

low-T peak

PROCEDURE:300K deposit 0.25ML O p(22)100K deposit various CO amountsHeat @ ~10K/sMonitor CO2 production versus T

CO

O

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.

Reaction rate ~ (O)m,

with m 0.6 1/2

CO+O/Pd(100) @ 300 K

CO+O/Pt(111) @ 300K

O

CO

KMC

STMWintterlin et al.Science 278 (1997)JCP 114 (2001)Chaos 12 (2002)

O

CO