Molecular dynamics simulations of
mass transport in chromium oxide
scales
Jukka Vaari
VTT Technical Research Centre of Finland
06/02/2013
2
Introduction
 Thermal spray coatings provide corrosion resistance for low-alloy
materials in high-temperature applications
 Goal: component lifetime prediction
 Means: atomistic, finite-element and thermodynamic modelling
 Starting point: simple model systems (Fe-Cr-O)
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1D corrosion model
 Steel is divided into control volumes
 Chemical reactions obtained by assuming thermodynamic
equilibrium in each control volume
 Mass transfer between control volumes occurs via diffusion
steel
position
∂Ci ( x ) ∂ 
∂C ( x ) 
=  Di , p ( x ) ⋅ i 
∂t
∂x 
∂x 
gas
x
3
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Classical molecular dynamics
 Molecular dynamics is
 a computer simulation of physical movements of atoms and molecules
 however, movement of planets about sun can be done with MD
 a numerical solution of Newton’s equations for a system of interacting
particles
 the interaction is described in terms of a potential (a.k.a force field)
 practical for times up to ns-µs, and for 105-107 atoms
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Diffusion in solid crystals
 No mass diffusion in perfect lattice
 Diffusion requires defects
 0D: point defects
 1D: dislocations
 2D: surfaces, grain boundaries
 A random walk process driven by
thermal energy
”Like human defects, those of
crystals come in a seemingly
endless variety, many dreary and
depressing, and a few fascinating”
- Ashcroft & Mermin, Solid State
Physics, Ch. 30
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Defect structure of Cr2O3
 A perfect lattice possible only at T = 0 K
 Defects present at finite temperatures

 For intrinsic point defects in Cr2O3
 For certain extrinsic defects (such as
substitutional Mg2+) EF can be as low as 2 eV
 Impurities determine the point defect
concentration (ppm range)
 Real Cr2O3 is a doped semiconductor with
charge carrier concentration dictated by
impurity concentration
 Nature of charge carrier can be modeled by
writing out the defect reactions for mass and
charge balance
Schottky defect 5.6 eV
Cation Frenkel defect 7.8 eV
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Model of the Cr2O3 crystal
 Cr2O3 has an orthorhombic primitive cell
 Simulation model built using a triclinic lattice and a hexagonal unit
cell containing three primitive cells (12 Cr atoms, 18 O atoms)
 The model has 4000 hexagonal unit cells and 120000 atoms with
periodic boundary conditions
• Schottky defects formed by
randomly deleting two Cr atoms and
three O atoms to maintain charge
neutrality
• Measures vacancy diffusion in both
anion and cation lattices
• Defect concentrations 2e-4 … 8e-4
in each lattice
• No attempt to model defect
concentration
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Interaction potential
 A combined Buckingham-Coulomb potential has been widely used
to model ionic crystals

, rcut = 15 Å
 Potential parameters A, ρ and C available for many metal-oxygen
pairs
Parameter set 1
[Lewis and Catlow 1985, Catlow 1977]
Parameter set 2
[Minervini et al 1999]
A (eV)
r (Å)
C (eV⋅Å6)
A (eV)
r (Å)
C (eV⋅Å6)
Cr3+ – O2-
1734.1
0.301
0
1204.18
0.3165
0
O2- – O2-
22764
0.149
27.88
9547.96
0.2192
32
Cr3+ – Cr3+
Only Coulombic
Only Coulombic
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Other computational details
 Ionic diffusion constants determined from mean square
displacement vs time curve
ri 2 (t ) =
1
N
2
(
)
(
)
[
]
r
t
r
−
0
= 6 Dt
∑ i
i
N
 Simulation temperatures 1300 K – 2000 K
 NPT ensemble
 Simulation timestep 1 fs
 Typical simulation time 400 ps
 Software: LAMMPS
 Hardware: Linux cluster ’Smokey’ (Intel Xeon 8-32 core CPU’s,
3.1…3.5 GHz)
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Typical MSD curves
 Defect fraction 8.3⋅10-4, T=1500 K
10
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Parameter set 1: oxygen diffusion
 Defect fraction 8.3⋅10-4
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Parameter set 2: oxygen diffusion
 Defect fraction 8.3⋅10-4
Horita et al, Solid State Ionics 179 (2008) 2216-2221: Ea=1.4 eV
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Parameter set 2: chromium diffusion
 Defect fraction 8.3⋅10-4
Liu et al, Solid State Ionics 109 (1998) 247-257: Ea=0.3 eV
Betova et al, VTT-R-04098-07: Ea=0.45 eV
13
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Oxygen diffusion coefficient vs defect fraction
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Extrapolation to lower temperatures and defect fractions
Young et al, Journal of the Electrochemical Society:
Solid-State Science and Technology vol. 134 pp.
2257-2260
• Die pressed Cr2O3 powder, high-temperature
sintering
• Seebeck measurements
• p-type semiconductivity
• Electron hole concentration 2⋅10-4
• Chromium vacancy concentration 6.7⋅10-5
Experimental data from Tsai et al, Materials
Science and Engineering A212 (1996) 6-13.
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Conclusions
 Mass transport due to Schottky point defects in bulk α-Cr2O3
investigated using molecular dynamics
 Defect fraction a free parameter in the approach
 Charge carrier concentrations from literature used as guidance
 Results sensitive to the potential used
 Parameter set #2 more credible
 Diffusion constants approximately linearly dependent on defect
fraction
 Extrapolation to lower temperatures through Arrhenius plot
 Qualitative agreement with experiments