2005 Summer School on Computational Materials Science:
Thermodynamics, Phase Diagrams, and the Cluster Expansion
Dane Morgan
Materials Science
University of Wisconsin-Madison
Phone: (608) 265-5879
Email: ddmorgan --> wisc.edu
Lectures
Computer Lab
- Lab_MonteCarlo.pdf (77Kb)
- AMTB1.31_Tutorial.tar.gz (21Kb)
References
- Curtarolo, S., Morgan, D., et al. "Predicting Crystal Structures with Data Mining of Quantum Calculations," Phys. Rev. Lett. 91, 135503 (2003).
Details about Lectures and Computer Lab
General aspects of thermodynamic simulation based on the CE and Monte Carlo will be extended to include how to identify phase transition, use of thermodynamic integration to get free-energy, how to obtain the phase diagram.
Thermodynamics and Phase Diagrams from Cluster Expansions | PDF (408 Kb) | PPT (1.3 MB)
- Thermodynamics and Phase Diagrams from Cluster Expansions
- The Cluster Expansion and Phase Diagrams
- Outline
- Phase Diagram Basics
- What is A Phase Diagram?
- Thermodynamics of Phase Stability
- Stable Phases from Cluster Expansion
- the Ground State Problem
- Determining Possible Phases
- The Convex Hull
- Getting the Convex Hull of a Cluster Expansion Hamiltonian
- Phase Diagrams from Cluster Expansion:
- Semi-Analytical Approximations
- Semi-Analytic Expressions for F (F)
- High-Temperature Expansion
- (NN Cluster Expansion)
- Low-Temperature Expansion
- (NN Cluster Expansion)
- Transition Temperature from LT and HT Expansion
- Mean-Field Theory - The Idea
- Implementing Mean-Field Theory
- The Cluster Variation Method
- Writing f[{r(s)}].
- Factoring the Probability to Simplify r(s)
- Truncating the Probability Factorization
- = Mean Field
- The Mean-Field Potential
- The Modern Formalism
- The CVM Potential
- Simplest CVM Approximation - The Point
- (Bragg-Williams, Weiss Molecular Field)
- CVM Point Approximation - Bragg-Williams
- (NN Cluster Expansion)
- Bragg-Williams Approximation
- (NN Cluster Expansion)
- Comparison of Bragg-Williams and High-Temperature Expansion
- Critical Temperatures
- Limitations of the CVM (Mean-Field), High- and Low-Temperature Expansions
- Phase Diagrams from Cluster Expansion: Simulation with Monte Carlo
- What Is MC and What is it for?
- MC Sampling
- Problem with Simple MC Sampling
- r(s) is Very Sharply Peaked
- Better MC Sampling
- Detailed Balance and The Metropolis Algorithm
- The Metropolis Algorithm (General)
- Metropolis Algorithm for Cluster Expansion Model (Real Space)
- Obtaining Thermal Averages From MC
- Energy Vs. MC Step
- Measuring Accuracy of Averages
- Example of Autocorrelation Function
- Semiquantitative Understanding of Role of Correlation in Averaging Errors
- Methods to Test Convergence Efficiently
- Finding Phases With MC
- Thermodynamic Potentials in MC
- Thermodynamic Integration
- Example of Thermodynamic Integration
- Summary
Computer Lab
A 2D Monte Carlo based on simplified CE, and possibly (time permitting) ATAT toolkit, will be used to investigate phase transitions, hysteresis, thermodynamic integration, and phase diagram.
- Lab_MonteCarlo.pdf (77Kb)
- Computational Laboratory: Monte Carlo for Phase Stability Calculations
- Atomistic Modeling Toolbox (AMTB)
- Useful Matlab tricks
- Introduction
- Doing the labs
- Exercise 0: Exploring the Code
- Exercise 1: Convergence
- Exercise 2: Phase stability from direct MC: concentration, susceptibility
- Exercise 3: Phase stability from varying µ, thermodynamic integration and free energies
- Exercise 4: Varying µ for an ordering system
- AMTB1.31_Tutorial.tar.gz (21Kb)
Materials Computation Center · UIUC · 104 South Goodwin Avenue · Urbana, IL 61801
Funded by DMR 03-25939