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Prof J A Elliott

This course aims to give a basic introduction to some of the main techniques for physical and numerical modelling of materials properties, suitable for those with a physical sciences background but no prior programming experience. Using a combination of lecture material, case studies and precompiled applets, we will describe how to fit empirical models to numerical data and quantify their goodness-of-fit, before exploring use of Monte Carlo and Molecular Dynamics methods to predict behaviour of systems from their basic physical laws.

Using two Case Studies, we will demonstrate the application of physical modelling techniques to simple (but realistic) materials problems: grain growth in metallic systems solidifying from melt, and the diffusion of ions in fluorite (a fast ion conductor). These will illustrate both how to apply the methods in practice and also what inputs/outputs can be expected from modelling at the microstructural and atomistic levels, respectively.

This lecture course will cover:

  • Overall scope and objectives. The nine stages of modelling. Numerical hygiene. Units and dimensional analysis. SI units and prefixes. Reduced units. Visualising the results.
  • Fitting models by non-linear least-squares analysis. Chi-squared goodness of fit. Some pitfalls of regression analysis and overfitting.
  • Statistical sampling methods and Buffon's needle. Introduction to Monte Carlo method. Thermal importance sampling and the Metropolis algorithm. Example of the Ising model.
  • Monte Carlo case study: microstructural modelling of grain growth in metals. The application of statistical mechanical and network models to microstructural evolution with comparisons to experimental data and analytical theory.
  • Introduction to Molecular Dynamics method. Solving Newton's equations of motion for a thermally isolated system. Verlet method. Velocity rescaling and the use of constant temperature simulations.
  • Molecular Dynamics case study: fast ion conduction in fluorite.
  • Use of static defect energy calculations and the molecular dynamics method to study fast ion conduction in fluorite crystal structures.