2005 Summer School on Computational Materials Science:
Kinetics and Kinetic MC via the CE
Anton van der Ven
Materials Science
University of Michigan
Phone: 1-734-615-3834
Email: avdv --> umich.edu
Lectures
Computer Lab
- Lab Notes (3Kb, .txt) -- helper notes for running the lab materials
References
- Van der Ven, A. and Ceder, G., "First principles calculation of the interdiffusion coefficient in binary alloys" Phys. Rev. Lett. 94 (4) 045901 (2005).
- Van der Ven, A. and Ceder, G., "Vacancies in ordered and disordered binary alloys treated with the cluster expansion" Phys. Rev. B 71 (5), 054102 (2005)
- F. M. Bulnes, V. D. Pereyra, and J. L. Riccardo, Phys. Rev. E, 58:1 (1998) (.pdf, 166Kb)
Other review type articles
- D. de Fontaine, "Cluster Approach to Order-Disorder Transformations in Alloys", Solid State Physics, Vol. 47, H. Ehrenreich and D.Turnbull, eds., Academic Press. pp. 33-176 (1994).
- Fähnle, M., “Thermodynamic properties from ab-initio calculations: New theoretical developments, and applications to various materials systems,” Phys. Stat. Sol. (b)242, 1159–73 (2005).
Details about Lectures and Computer Lab
Kinetics in alloys systems and the use of the Cluster Expansion for the energetics will be presented. In addition, the use of CE within a Kinetic Monte Carlo framework will be explored. We discuss diffusion, energy barriers for atomic diffusion, and effects of vacancy ordering in alloys.
Diffusion | PDF (8.8 MB) | PPT (3.6 MB)
- Diffusion in multicomponent solids
- Coarse graining time-Diffusion in a crystal
- Interstitial diffusion
- C diffusion in bcc Iron (steel)
- Li diffusion in transition metal oxide host
- O diffusion on Pt-(111) surface
- Irreversible thermodynamics: -interstitial diffusion of one component
- Notation
- M = number of lattice sites
- N = number of diffusing atoms
- vs = volume per lattice site
- x = N/M
- C=x/vs
- Interstitial diffusion: -one component
- Trajectories
- More familiar form
- Common approximation
- Interstitial diffusion -(two components)
- C & N diffusion in bcc Iron (steel)
- Li & Na diffusion in transition metal oxide host
- O & S diffusion on Pt-(111) surface
- Diffusion of two species on a lattice
- Alternative factorization
- Kinetic coefficients -(fcc lattice in dilute vacancy limit, ideal solution)
- Diffusion in an alloy:-substitutional diffusion
- Not interstitial diffusion
- Instead, diffusing atoms form the lattice
- Dilute concentration of vacancies
- Thermodynamic driving forces for substitutional diffusion
- Textbook treatment of substitional diffusion-Not Rigorous
- Lattice frame and laboratory frame of reference
- Diagonalize the D-matrix
- Physical meaning of modes l+ and l-
- Comparisons of different treatments
- Intercalation Oxide as Cathode in Rechargeable Lithium Battery
- Cluster Expansions
- First principles energies (LDA)-of different lithium-vacancy configurations
- Cluster expansion for LixCoO2
- Calculated LixCoO2 phase diagram
- Predicted phases confirmed experimentally
- Effect of metal insulator transition
- Diffusion
- Interstitial diffusion and configurational disorder-
- Individual hops:-Transition state theory
- Kinetically resolved activation barrier
- Migration mechanism in LixCoO2
- Migration barriers depend configuration and concentration
- Local Cluster expansion for divacancy migration barrier
- Calculated diffusion coefficient-(First Principles cluster expansion + kinetic Monte Carlo)
- Available migration mechanisms for -each lithium ion
- Diffusion occurs with a divacancy mechanism
- Diffusion and phase transformations in Al-Li alloys
- fcc Al-Li alloy
- Calculated thermodynamic and kinetic properties of Al-Li alloy
- Expand environment dependence of vacancy formation energy
- Equilibrium vacancy concentration-(Monte Carlo applied to cluster expansion)
- Vacancy surrounds itself by Al-Short range order around a vacancy
- Vacancies reside on lithium sublattice in L12
- Migration barriers for lithium and aluminum differ by ~150 meV
- Calculated interdiffusion coefficient
- Hop mechanisms
- Frequency of hop angles between successive hops
- Conclusion
- Green-Kubo formalism yields rigorous expressions for diffusion coefficients
- Discussed diffusion formalism for both interstitial and substitutional diffusion
- Intriguing hop mechanisms in multi-component solids that can depend on ordering
- Thermodynamics plays a crucial role!
Kinetic | PDF (3.9 Kb) | PPT (3.1 MB)
- Kinetic Monte Carlo
- Triangular lattice
- Diffusion
- Standard Monte Carlo to study diffusion
- Pick an atom at random
- Pick a hop direction
- Calculate
- Consider all hops simultaneously
- Time
- After hop k we need to update the time
- Two independent stochastic variables: the hop k and the waiting time
- Kinetic Monte Carlo
- Hop every time
- Consider all possible hops simultaneously
- Pick hop according its relative probability
- Update the time
- Triangular 2-d lattice, 2NN pair interactions
- Activation barrier
- Thermodynamics
- Kinetics
Computer Lab
- Diffusion in alloys based on CE will be explored in simplified simulation, such as 2D Kinetic Monte Carlo
- Lab Notes (3Kb, .txt) -- helper notes for running the lab materials
Materials Computation Center · UIUC · 104 South Goodwin Avenue · Urbana, IL 61801
Funded by DMR 03-25939