Molecular dynamics simulations of mass transport in chromium oxide scales

Jukka Vaari VTT Technical Research Centre of Finland

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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

position x

steel gas ( ) ( ) ( )

∂∂⋅

∂∂

=∂

∂x

xCxDxt

xC ipi

i,

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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 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)

( ) ( ) ( )[ ] DtrtrN

trN

iii 601 22 =−= ∑

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Typical MSD curves

Defect fraction 8.3⋅10-4, T=1500 K

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Parameter set 1: oxygen diffusion

Defect fraction 8.3⋅10-4

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Parameter set 2: oxygen diffusion

Horita et al, Solid State Ionics 179 (2008) 2216-2221: Ea=1.4 eV

Defect fraction 8.3⋅10-4

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Parameter set 2: chromium diffusion

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

Defect fraction 8.3⋅10-4

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Oxygen diffusion coefficient vs defect fraction

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Extrapolation to lower temperatures and defect fractions

Experimental data from Tsai et al, Materials Science and Engineering A212 (1996) 6-13.

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

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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