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. Atkins “Physical chemistry” Chapter 8 of 2005 edition/library, Focus 7 in 2011 edition and later

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. Atkins “Physical chemistry” Chapter 8 of 2005 edition/library, Focus 7 in 2011 edition and later.

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. Atkins “Physical chemistry” Chapter 8 of 2005 edition/library, Focus 7 in 2011 edition and later.

Lecture 25/11. Simple quantum mechanical models. Wave packets. Particle in a box. Solution for a 1-D case. Quantized energy states as a result of boundary conditions. Wavefunctions for particle in a box and probability densities for first 4 energy levels. Zero-point energy/ground state energy. Normalization integral for finding integration constant for wavefunctions. Expectation values for particle’s position and momentum. Heisenberg’s uncertainty for particle in the box model. Hamiltonian as an energy operator. Application of the model to pi-electrons in butadiene. Emission wavelength calculated vs experimental, use of De Broglie relation. Hamiltonians for 2D and 3D boxes. Quantum numbers. Degeneracy with respect to energy for nx and ny numbers. Classical harmonic oscillator. Parabolic potential. Approximation of potential well in chemical bonds. Attractive and repelling forces. Equilibrium bond length. Raman spectroscopy – inelastic electron scattering and Raman-Stokes shift in vibration wavelength/frequencies. Modes of vibration in molecules.

Literature: Atkins chapter 2, Blinder Chapters 3 and 5. Atkins “Physical chemistry” Chapter 9 of 2005 edition/library, Focus 7 in 2011 edition and later.
Web-site for 2D particle in the box
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_2-Dimensional_Box

Lecture 2.12.2019. Pop-Up Quiz, discussion of the solutions. Classical harmonic oscillator. Quantum Harmonic Oscillator in parabolic potential. Wavefunctions and energies.

Literature: Blinder, Chapter 5


Lecture 9.12.2019. Quantum rigid rotor. Rigid rotor in 2D. Separation of variables: center of mass and internal rotation. Hamiltonian in polar coordinates. Application of the rigid rotor condition. Periodicity as a boundary condition. Wavefunctions for 2D rigid rotors. Energy levels. 3D rigid rotor: motion on a sphere. Transition from Cartesian to spherical coordinates. Hamiltonian for 3D rigid rotor. Separation of variables for the angular part. Solutions of the second angular part in terms of associated Legendre polynomials. Spherical harmonics. The restrictions imposed into l and m quantum numbers. Shape of the spherical harmonics: zero order and higher.

Literature: Atkins “Physical chemistry” Chapter 9 of 2005 edition/library, Focus 7 in 2011 edition and later, Atkins “Molecular quantum mechanics”, Chapter 3, Blinder Chapter 6


Lecture 16.12.2019. Stochastic methods of statistical mechanics. Monte Carlo integration. The Kinetic Monte Carlo approach. Construction of the event’s list for surface reactions. Parameterization of the model: reaction rates as a function on activation energies of bond breaking. Basic bonds in silicates. Basic building blocks of silicates, types of bonding in carbonates, sulfates, sulfides, (hydr)oxides. Dislocations in crystals, perfect and imperfect crystals. Point defects, screw and edge dislocations. Dislocations as sources of etch pits. Morphologies of dissolved surfaces: etch pits on mica surfaces, anisotropy of dissolution rates in different crystallographic directions. Influence of bond topology (connectivity) onto bond hydrolysis activation energy. Surface speciation of carbonates: protonation, deprotonation, ion adsorption. Grand Canonical Monte Carlo simulations. pH control of dissolution rates: function on proton concentration.

Literature: A. Voter “Introduction into Kinetic Monte Carlo methods”, Hull and Bacon “Introduction into dislocations”, presentation slides


Lecture 06.01.2020. Part 1. Selected advanced topics on surface science. Transition State Theory, activation barriers and rates. Potential wells and saddle points. Transition state. Surface speciation: protonated sites, deprotonated sites, adsorbed ions at the solid-fluid interface. Surface speciation obtained from Grand Canonical Monte Carlo simulations. Protonation of sites in the presence/absence of electrolyte ions. Dissolution rate as a function on pH/surface speciation.
Part 2. Spherical harmonics as eigenfunctions on the Legendrian operator and Hamiltonian operator for 3D rigid rotor. Physical meaning of quantum numbers l and m. Magnitude and orientation of angular momentum in quantum system. Energy levels for rigid rotors. The Hamiltonian for Hydrogen atom: nucleus part, electronic part, Coulomb potential. Change of coordinates to the center of mass. New Hamiltonian for the internal rotational motion and Coulomb interaction. Radial and angular parts of the wavefunction.

Literature: A. Voter “Introduction into Kinetic Monte Carlo methods”, Blinder Chapter 6,7. Atkin’s Physical Chemistry “Atomic structure and spectra”, “Molecular Structure”, (Chapter 9 of 2005 edition/library, Focus 7 of 2011 edition and later). Advanced: Atkin’s “Molecular quantum mechanics”, Chapters 3,4


Lecture 13.01.2020 Electronic structure of atoms and the nature of chemical bonding. Schrodinger equation for the hydrogen atom. Expansion in spherical coordinates. Radial wavefunction and spherical harmonics. Solution for the radial part in terms of the associated Laguerre polynomials. Principal quantum numbers n and energy levels. Principal quantum numbers l and ml, and their corresponding electronic orbitals. S, p, d, f orbitals and their quantum numbers. Electron spin. Stern Gerlach experiment and discovery of the intrinsic angular momentum. Pauli principle regarding the existence of the spin as postulate of quantum mechanics. Symmetry/antisymmetry of wavefunctions for bosons and fermions. Spins for bosons and fermions. Pauli exclusion principle for orbital occupation. Orbital approximation for multi-electron systems. He atom: possible spin states of electrons, existence of only one wavefunction form due to the antisymmetry restriction onto the wavefunction. Electronic structure of the atoms in terms of the hydrogenic orbitals. Hund’s principle onto occupancy of orbitals. Valence bond theory and spin pairing. Positions of overlapping electronic clouds, sigma and pi bonds. N2 and H2O molecules.

Literature: Atkin’s Physical Chemistry “Atomic structure and spectra”, “Molecular Structure”, (Chapters 10-11 of 2005 edition/library, Foci 8 and 9 of 2011 and later editions), Blinder Chapters 8-11, Advanced Extra: Atkins “Molecular quantum mechanics” Chapters 4,7

Lecture 20.01.2020 Molecular orbital theory. Wavefunction for molecular orbital as a linear combination of electronic orbitals. Bonding and antibonding orbitals for H2 and He2 molecules. Molecular orbital diagram for O2 bond – bonding and antibonding pi and sigma orbitals. Variational principle to calculate bonding and antibonding orbitals for heteroatomic molecules. The value of the overlap integral and the information regarding orbital overlap. Multi-electron systems and their wavefunctions. Slater determinant. Hartree-Fock method: Hamiltonian for multi-electron Hartree-Fock system, separation of the electronic and nuclei part (Born-Oppenheimer approximation), system of Hartree equations, Self-Consistent variational approach to find wavefunctions. Basis set functions. Slater and Gaussian type wavefunctions (orbitals).

Atkins “Physical chemistry” Chapter 9 (2011 or later), Chapter 8 (library version).
Blinder Chapters 11-12, Chapter 12 on the Hartree-Fock method

Lecture 29.01.2020 Basic principles of Density Functional Theory. Molecular dynamics simulations. Newtonian equations of motions for many-body systems. Forces as potential energy gradients. Potential energy as a sum of pairwise interactions. Lenard-Jones potential, Born-Meyer potential. Attractive and repulsive forces. Finite time step approach to integrate Newtonian equations. Simple one-step time propagator, two-step Verlet propagator (algorithm). Examples of MD simulations: ice crystallization, water-oil phase separation, graphene used to desalinate water, NaCl dissolution.

Literature: DFT, basics: Blinder Chapter 12, MD simulations: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Advanced_Theoretical_Chemistry_(Simons)/7%3A_Statistical_Mechanics/7.3%3A_Molecular_Dynamics_Simulations




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. "Physical Chemistry" Atkins, 2005 or 2011 and later

2. Atkins “Molecular quantum mechanics”

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

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

5. 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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