Introduction to Materials

Lecturers: Inna Kurganskaya

Contact: Inna Kurganskaya

Content:

The aim of this course is to provide an introduction into chemical and physical properties of solids. The course material is mainly focused on the crystalline solids (minerals and metals), amorphous materials (glasses), soft matter (polymers, gels), cements, ceramics and nanocomposites are also discussed.
The main topics covered by the course:
- Electronic structure and chemical bonding
- Theoretical foundations of Quantum Chemistry, Molecular Dynamics and Statistical Physics.
- Modelling and computer simulations of materials structure atomic structure and properties by using Quantum Mechanical, Molecular Dynamics and Monte Carlo methods.
- Mechanical properties of solids: stress, strain, defects, deformation, elasticity
- Diffusion and random walk
- Multiscale structure, properties and simulations
- Basics of the material’s design


Lecture 1. 15/10. Prerequisites for the course. Hints and tips for the successful Master’s studies. Now to take lecture notes, work with the literature. Choice of textbooks. Reading research papers. Identifying gaps in your knowledge. Checking your knowledge. Social networking for the studies. The foundations of the scientific method. The scientific research workflow and generation of the scientific knowledge. Testing hypothesis and paradigms, making experiments, theory development. The subject of the Material’s Science. Types of materials. Basic properties of materials. Levels of structure (nuclear to macroscopic). Material’s Science as an interdisciplinary field. Relationships to the other disciplines. The course objectives. Topics covered. The mathematical prerequisites for the course: Calculus, Analytical Geometry, Linear Algebra, Differential equations, Probability Theory and Statistics. Useful knowledge: basics of Mathematical Analysis, Group Theory, Abstract Algebra.

Literature: Slides. “Introduction to Computational Materials Science” R. Lesar, Chapter 1.

Lecture 2. 21/10 Mathematical background for the course. Functions, variables, differentials, derivatives, integrals. Vectors and their properties. Vector basis. Cartesian coordinates, basis vectors in Cartesian 3D space. Linear dependence. Dot product, vector product. Matrices. Vectors as 1-dim matrices. Eigenproblem. Eigenvalues of a matrix. Characteristic polynomial of a matrix. Elements of the set and group theory. Group, associativity, identity element, inverse element. Abelian groups. Examples of groups. Fields, operation properties for fields. Vector field and vector space. Abstract vectors. Functions as abstract vectors. Basis functions. Fourier series functions.

Literature: “Mathematics of classical and quantum physics” by Frederick W. Byron; Robert W. Fuller. Chapters 3 and 4. (library hard copy)

Lecture 3. 28/10 Basics of thermodynamics and statistical mechanics. Macroscopic and microscopic systems. Macroscopic thermodynamic parameters. First, second and third thermodynamic laws. Energy, Entropy. Functional relationships between thermodynamic parameters. Carnot cycle. Macrostates and microstates. Phase space (q,p). Probability, probability density distributions, mean, average. Uniform and gaussian distributions. Weighted average. The partition function. Microcanonical, canonical and grand canonical ensembles. Boltzmann factor. Maxwell-Boltzmann distribution. 1-D random walker.

Literature: “Thermodynamics and Statistical Mechanics: An Integrated Approach” M. Scott Shell (library online copy). Chapters 2 and 4. “Introduction to Computational Materials Science” R. Lesar, Chapter 2 for random walkers, Appendix G (library hard copy). Advanced reading: “Thermal Physics. Thermodynamics and Statistical Mechanics for Scientists and Engineers” Robert F. Sekerka (library online). “Fundamentals of Statistical and Thermal Physics” by F. Reif, Chapter 1.

Lecture 4. 4/11 Simple models of solid materials. Atomic structure. Interaction potential between two particles (pairwise potential). The system’s potential. Ideal crystals as periodic systems. Coordinate vector representations of atoms in the unit cell. Energy of the cubic crystal. Lenard-Jones potential. Repulsive and attractive forces. Classical harmonic oscillator.

Literature: “Introduction to Computational Materials Science” R. Lesar, Chapter 3

Waves. Electromagnetic waves spectrum. From gamma-rays to radio-waves. Visible light spectrum. Black body radiation. Temperature-wavelength dependence relationships. Temperature-dependent radiation wavelength spectra. The spectrum of the Sun. Ultraviolet catastrophe.

Literature: “Introduction to quantum mechanics in chemistry, materials science, and biology” by Blinder (online library). Chapter 1.

Lecture 5. 11/11 History of quantum mechanics. Black body radiation: Rayleigh-Jeans law, it’s failure at high frequencies. The idea of Max Planck to solve the ultraviolet catastrophe problem. Planck’s constant. Einstein’s relationships between energy and frequency. Energy of a photon. Electromagnetic waves, radiation intensity as a function of electric and magnetic field amplitudes. Wave equations. Wave amplitude. Interference of waves, light and dark fringes as a result of wave superposition. The double slit experiment, diffraction and interference. Random positions of scintillations. Wave-particle duality. The De Broglie waves: combined particle and wave properties. Derivation of De Broglie wave equation from Planck’s and Einstein’s relations between energy and momentum. Schrodinger equation for De Broglie waves.

Literature: “Introduction to quantum mechanics in chemistry, materials science, and biology” by Blinder (online library). Chapter 2. “Molecular quantum mechanics” by Atkins, 4th edition. Available online. Chapter 0.

Lecture 5. 11/18 Theoretical foundations of quantum mechanics. Operators and their properties. Linear operators. Eigenfunction of an operator. Eigenvalues of operators. Complete set of functions. Representation of a function as linear superposition of linearly independent functions. Position and momentum operators. Commutation of operators, commutator. The kinetic and potential energy operators. Laplacian. Coulomb energy operator. Integrals over operators. Complex conjugate over complex function. Overlap integral. Normalization integral. Kronecker delta and orthonormal functions. Dirac bracket notation. Representation of integrals in terms of Dirac notations. Hermitian operators. Eigenvalues of Hermitian operators. Orthogonality of eigenfunctions for Hermitian operator.

Literature: “Introduction to quantum mechanics in chemistry, materials science, and biology” by Blinder (online library). Chapter 2. “Molecular quantum mechanics” by Atkins, 4th edition. Available online. Chapters 0, 1.

Postulates of quantum mechanics. 1. Wavefunction representation of a system state, properties of a wavefunction. 2. Physical meaning of a wavefunction, Born’s interpretation. 3. Operator representation of observables. Commutation relations between quantum operators. Diagram of commutation. 4. The mean value of the observable as an expectation value of the corresponding operator. 5. The Schrodinger equation, time-dependent form. From time-dependent to time-independent Schrodinger equation. Separation of variables. Solution for the time-dependent part. Construction of the full wave-function as a product of time-independent and time-dependent parts.

Literature: “Introduction to quantum mechanics in chemistry, materials science, and biology” by Blinder (online library). Chapters 2,4. “Molecular quantum mechanics” by Atkins, 4th edition. Available online. Chapter 1.



Skills:

Students are anticipated to acquire the following knowledge and skills:
- understanding of the chemical bonding and structure of solids at atomic-nano-micron scales
- easy navigation in modern computer modelling techniques used to calculate properties of solid materials
- ability to recognize potential material properties based on their composition and structure
- ability to formulate research projects dedicated to material properties and design


Exam:

written exam

Literature:

1. Richard Lesar, "Introduction to Computational Materials Science. Fundamentals to Applications". MRS, Materials Research Society and Cambridge University Press, 2013
www.cambridge.org/9780521845878

2. Allen and Tildesley, “Computer simulation of liquids”, 1987. Clarendon Press, Oxford.

3. Links below

Links
"Thermal physics: thermodynamics and statistical mechanics for scientists and engineers" by Sekerka
"Introduction to quantum mechanics in chemistry, materials science, and biology", Blinder
A.F. Voter "Introduction into Kinetic Monte Carlo methods"
D. Hull and D.J. Bacon, “Introduction into dislocations”
Molecular dynamics and Monte Carlo methods, theory:

Daan Frenkel and Berend Smit, Understanding Molecular Simulation: From Algorithms to Applications

Molecular dynamics and Monte Carlo methods, theory:

Daan Frenkel and Berend Smit, Understanding Molecular Simulation: From Algorithms to Applications

Molecular dynamics and Monte Carlo methods, theory:

Daan Frenkel and Berend Smit, Understanding Molecular Simulation: From Algorithms to Applications

Quantum chemistry for materials:

Prof. Dr. Rutger A. van Santen Dr. Philippe Sautet “Computational Methods in Catalysis and Materials Science: An Introduction for Scientists and Engineers” (2009)

A.M. Ovrutsky, A.S. Prokhoda and M.S. Rasshchupkyna, Computational Materials Science:
Surfaces, Interfaces, Crystallization

Mitchell, Brian S. “An Introduction to Materials Engineering and Science (2003)”

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