from atomic scale ordering to mesoscale spatial patterns in surface reactions: hclg
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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
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