Multilevel Modelling of Materials
FSI-9MMMAcad. year: 2020/2021
Multiscale modeling of materials is an essential approach to understanding the properties between microstructure and macroscopic physical properties of materials. Atomistic methods based on empirical and semi-empirical potentials and Monte Carlo methods represent effective and commonly used tools for computer simulations of nanostructures (thin fibers, nanotubes, epitaxial layers, graphene), studies of radiation damage, moving dislocations under tension, solid solutions, phase transformations in multiferroics, and the like. In this course, the students will gain knowledge about the methods of computer modeling of materials from atomic level to macroscopic studies based on the finite element method. The individual methods will be demonstrated on simple examples that either allow analytical solution or can be studied using a simple program. In a series of individual assignments, students will gain practical experience with the implementation of these algorithms using Python and with methods of data visualization. Personal experience with the implementation of these approaches is a prerequisite for the systematic use of commonly available simulation programs and for the independent solution of research projects.
Learning outcomes of the course unit
This course will provide students with a comprehensive overview of the most commonly used methods of modeling materials from atomic level, through mesoscopic description to simulation of macroscopic systems. It will teach the basics of programming in Python and enable its future use by the students to solve their individual problems.
The knowledge of mathematics at the 2nd year of FME (differentiation of functions of several variables, probability theory, numerical methods) and basic knowledge of programming is expected. Basics of programming in Matlab, Python, eventually. C or Fortran are an advantage, but not a requirement.
Recommended optional programme components
Recommended or required reading
M. P. Allen, D. J. Tildesley: Computer simulation of liquids. Clarendon Press (1987). (EN)
J. P. Sethna: Statistical mechanics: Entropy, order parameters, and complexity. Oxford University Press (EN)
K. G. Wilson: Problems in physics with many scales of length. Scientific American 241 (1979) 140–157. (EN)
D. Frenkel, B. Smith: Understanding molecular simulation. Academic Press (2002). (EN)
E. H. Stanley: Introduction to phase transitions and critical phenomena. Oxford Science Publications (1987) (EN)
Planned learning activities and teaching methods
The lectures will give a theoretical basis for each method and explain the range of spatial scales for which the model is applicable. Each method will be explained on a simple example that can be solved either analytically or using minimal computational resources. More complex simulations of real problems will be executed in the exercises using programs prepared in Python.
Assesment methods and criteria linked to learning outcomes
Each student will be assigned an individual work, which will be related to some of the discussed methods. The output of these assignments will be the creation or modification of an existing simulation program, its use for solving the given problem and a written report summarizing the problem formulation, solution and main results of these simulations. The exam will consist of an oral defense of this work.
Language of instruction
The aim of the exercises is to get acquainted with computer implementations of individual models, which will allow detailed understanding of inputs, methods and results obtained from commonly used commercial and open-source programs for simulating microstructure and physical properties of materials.
Specification of controlled education, way of implementation and compensation for absences
The attendance at exercises is mandatory and every absence must be excused. In case of no-show, the student is required to submit a written report from the exercises and proves to the lecturer that he/she understood the topic.
Type of course unit
20 hours, optionally
Teacher / Lecturer
1. Modeling of the relationships between microstructure and physical properties. History and presence of computer simulations of materials.
2. Equilibrium statistical mechanics, spin models. Analytical solution of the 1D Ising model and discussion on solutions in higher dimensions. Mean field method. Monte Carlo Method - non-Conservative (Metropolis-Hastings, Glauber) vs. conservative (Kawasaki).
3. Critical points of phase diagrams, critical phenomena, critical exponents, correlation length. Renormalization group for 2D Ising model.
4. Cellular automata for the study of microstructure evolution according to a finite set of rules.
5. Molecular statics. Pair potentials (especially Lennard-Jones potential), interaction radius, determination of atomic forces, energies and stresses.
6. More advanced interaction potentials for the study of materials with covalent, metallic, ionic and mixed bonds. More detailed insight into the theoretical description of the EAM method and Tersoff potential.
7. Molecular dynamics, stability of numerical integration of equations of motion, thermostats and barostats. Parrinello-Rahman method.
8. Symmetry, order parameter and mesoscopic description of materials. Phase field method, (non)existence of spatial scale and its relation to lattice vibrations. Landau-Ginzburg description of free energy of ferroelastic materials and its analytical solution.
9. Description of non-equilibrium problems using kinetic equations (Master equation). Activation energy, anticipated and rare processes.
10. Rare processes and their studies using transformation path calculations. Activation energy, determination of transformation (reaction) coordinate.
11. Synchronization and self-organization in interacting systems at microscopic to macroscopic level.
12. Ordering in systems of interacting magnetic nanoparticles. Magnetoelastic anisotropy, dipole interaction, shape anisotropy (stray fields), influence of magnetic field.
13. Models with separated scales, relevant vs. irrelevant degrees of freedom, configurational entropy. Scalability of models away from critical points.