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

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


Lecture 1. The scientific method. How do distinguish a scientific question from non-scientific question? Popper’s criterion. The workflow of scientific research: hypothesis, experiment, model, model verification, hypothesis test, model-driven experiment, change of paradigm, theory development. Subject of materials science. Types of materials. Basic properties of materials (mechanical, chemical etc.). Basic levels of structure (scale). What is the nanostructure? Electronic structure? Materials science as an interdisciplinary field. Which scientific fields are covered by materials science? Importance of mathematics for materials science research. What is the use of mathematical knowledge in materials science research? Give examples in simulations and theory.

Lecture 2. Models of an atom. Electronic structure of an atom (Hydrogen atom model). Bohr (planetary) model of a nucleus-electron system. Wave-particle duality nature of electrons. Quantum numbers. Energy levels and electronic orbits. Electronic shells and subshells. Physical meaning of l and m quantum numbers. Shapes of s,p,d orbits and relationships with l quantum number. Orientation of p and d orbits, relationships to angular momentum vector. Magnitude and orientation of angular momentum vector. Difference between linear and angular momentum. Stern-Gerlach experiment. Electron’s spin. Electronic configuration of atoms based on Hydrogen model. 1s2 2s2 2p5 … etc.

Lecture 3. Part1. Chemical bonding. Visualization of electronic orbitals. Probability density of finding an electron at different distances for n=1, n=2, n=3… Geometry of probability density distribution for 1s, 2s, 2px, 2py, 2pz electrons. Chemical bonding. Electronic nature of chemical bonding. Electronic structure of covalent bonds. Bond hybridization mechanisms. Sp3 hybridization in diamond and methane. Sp2 hybridization in graphene. Carbon nanotubes. Electronic structure of ionic bonds. Properties of ionic materials. Electronic structure of metallic bonds. Properties of metallic materials. Diatomic molecules. Degrees of freedom in diatomic molecules. Harmonic oscillator description of chemical bonds. Potential energy and forces in diatomic molecules: attractive, repulsive. Lenard-Jones potential. Equilibrium bond length (interatomic distance). Depth of the potential well and strength/energy of bonds.
Lecture 3. Part2. Basics of crystallography. A unit cell and unit cell parameters. Translation of a unit cell and periodicity of crystal lattice. 7 crystal classes (systems) defined by their symmetries: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. 14 3D Bravais lattices (7classes+ body centered, face centered, base-centered, primitive). Coordinates of points (atom centers) in crystallographic basis set (internal unit cell coordinates). How to calculate indexes for crystallographic directions. How to calculate crystallographic plane indexes. hkl notations. [] and () notations for directions and planes. Pole projections of crystal planes. Crystal structures of NaCl, diamond, graphite, CsCl, fluorite.

Lecture 4. Mineral dissolution process and Kinetic Monte Carlo modelling. Importance of mineral dissolution for geological sciences and industry. Interdisciplinarity of mineral dissolution kinetics studies. Mineral dissolution as a multi-scale problem. Dissolution of phyllosilicates (clay minerals). Structure of phyllosilicates (sheet silicates). Vertical Scanning Interferometry (VSI) – an optical technique for surface topography measurements. Basic physical principles of VSI. Change of surface topography for muscovite mica as a function on time. Variety of dissolution patterns. Measurements of dissolution rates using surface imaging technique: use of reference mask, volumetric approximation (no mask). Spatial distribution of rates and natural dissolution rate variance. Rate spectra concept. Basics of Atomic Force Microscopy. Kinetic Monte Carlo simulations of crystal dissolution: how rates and dissolution probabilities are related? Dissolution probability and number of bonds. Kinetic Monte Carlo simulations of mica dissolution. Which chemical bonds are considered? Pitting mechanism. Superposition and direction of pits in each mica layer. First and second coordination sphere. Geometry of etch pits. Atomic steps and stepwaves. Coordination spheres in quartz (SiO2). Surface structure of (001) face depending on choice of coordination sphere model. Etch pits and dissolution stepwaves on (101) face of quartz. Dissolution of (100) prism face: striations, V-shaped pits, etch tubes, rectangular pits. Mechanisms of dissolution hillock formation. Dissolution of calcium carbonate (calcite). Geometry of reactive sites. Etch pits and atomic steps of calcite surface. Straight and curved step – which ones are faster? Surface normal retreat – how is it measured? Material flux (dissolution rate in moles/cm2 sec) from carbonate surface and from individual reactive sites.

Lecture 5. Part1. Crystal packing. Model of crystal as closed pack structure of hard spheres. Coordination number (CN). CN for calcite, NaCl. How to calculate atomic packing factor (APF) for cubic structure if only length of a cubic cell (a0) is known? How to do that for SC, BCC, FCC structures? How many whole atoms make up a whole unit cell for SC, BCC, FCC structures? How to calculate this number? How to compute unit cell volume in terms of atomic radius r if crystal structure is known? How ionic radius (r) and length of a unit cell (a0) are related in SC, BCC and FCC structures? How density of a metal with cubic structure can be calculated, if ionic radius, atomic weight and crystal structure are known? (see example for Cu). CN for ZnS, CaF2 (fluorite). Silicate ceramics and basic building block of silicates (SiO4 tetrahedral unit). Classification of silicate minerals and materials based on connectivity of SiO4 tetrahedra. Isolated, 2-memberd clusters, 3-,5-, and 6-membered rings, polymer chains. 3D fully connected network as in quartz, cristobalite. Structure of layered silicates.
Lecture 5. Part2. Dislocations. Experimental study case: dissolution of calcium carbonate (calcite) in fluid-flow reactor with in-situ measurements under the objective of Vertical Scanning Interferometer. Etch pits on calcite surface growing from hollow cores of screw dislocations. Point defects: vacancies, interstitial defects, self-interstitial defects. Temperature-dependent rate of defect migration. Substitutional impurities. Schottky and Frenkel defects. Intermetallic compounds and alloys (solid solutions). Planar defects: stacking faults, grain boundaries. Line defects: screw and edge dislocations. Burger’s vector. Rule for vector sum of Burger’s vectors. Dislocation networks.

Lecture 6. Mechanical properties of materials, stress, strain, tensor algebra. Tension, compression, shear deformation. Definitions of stress and strain. Hook’s law. Stress, strain and Young’s modulus. Elastic deformation. Attractive and repulsive forces and energies. How bonding forces and elastic modulus are related? How to calculate elongation under the given value of stress for a specimen with a given original length and Young’s modulus? Bulk modulus. Parameters for isotropic materials: Poisson ratio, shear modulus. Value of Poisson ratio and material stiffness: stiff, rubbery and anti-rubbery materials. Relationships between strain and displacement vector. Elasticity theory: 9 components of stress tensor. Difference between scalars, vectors and tensors. Stress as force per unit area. Physical and mathematical meaning of tensors. Stress and strain tensors. Tensor (matrix) equations for stress-strain via elastic stiffness and elastic compliance tensors. How to convert a 4-rank stiffness/compliance tensor into a 6x6 matrix? Rank of stress, strain, stiffness and compliance tensors. Relationships between bulk modulus, Young’s modulus, shear modulus and Poisson ratio for isotropic materials.
Math breaks: Derivative of a function, tangent line, differential (dy), derivative as a limit, how to calculate extrema of functions, second order derivative conditions for minimum and maximum, function on several variables, partial derivatives, functions, tensors, rank of tensors.
Lecture 7. Part1. Diffusion. Fick’s law. Gas diffusion under pressure difference through a metal plate or membrane. Physical nature of diffusion (Brownian or random motion or particles), material flux in the presence of concentration or pressure gradient. Solid state diffusion, diffusion of gases. Fick’s first law. Diffusion coefficient. Driving force of diffusion (concentration/pressure gradient). Second Fick’s law. Derivation of the second Fick’s law from first Fick’s law by using mass balance and finite difference approximation. Solution of the second Fick’s law in error-function. Temperature dependence of diffusion coefficient. Activation energy (Arrhenius) of diffusion and its calculation from experimental data. Diffusion on one dimension from one point source. What happens with concentrations over time? Diffusion and drift. What is the difference between diffusion transport and drift transport? Random walk as a mathematical model of diffusion. Solid state diffusion in metal alloys at different temperatures.
Lecture 7. Part2. Ceramics. Silicate ceramics. Types of silicate structures. Isolated (monomer, dimer, 3,5,6 membered rings), chains and double chains, sheets and interlayer cations, 3D structures. Bioceramics. Osteoporosis and bioceramics for implants. Types of bioceramic-tissue attachments. Types of bioceramics and attachment. Hydroxyapatite as important bone implant material.
Math break: differential equations, ordinary differential equations (ODE), partial differential equations (PDE), solution of a differential equation, integration constant, types of numbers, initial and boundary conditions, numerical integration, Euler’s method for forward integration.
Lecture 8. Part1. Molecular Dynamics (MD) simulations for material’s research. Newton’s laws for a system of particles. Dot notations for time derivatives. Relationships between forces, masses, positions, accelerations and velocities. Forces as potential energy derivatives. Non-bonded interactions: Coulomb forces, external potential, Lenard-Jones potential. Bonded (intramolecular) interactions (potentials): harmonic (bond stretching), angular (bond bending), torsional. Calculation of forces for a particle. Example: calculation of bending interaction potential for a molecule containing 4 atoms. Integration of Newton’s equations. Numeral integration via finite difference approximation (Taylor expansion). Verlet algorithm. Velocity Verlet (leapfrog) version with half-time steps. Time scale of MD simulations. Periodic Boundary Conditions.
Lecture 8. Part2. Temperature dependent phase transitions. Velocity distributions of gas molecules and relationships to temperature. Basics of Statistical Mechanics. Microstates and macrostates. Phase space for classical (not quantum) molecular simulations. Microstate as a point in a phase space. Maxwell-Boltzmann statistics: probability to have a specific energy of a molecule. The partition function. Ergodicity hypothesis. Simple average and weighted average. Thermodynamic parameters of a system: intensive and extensive. Entropy as a function on microstate number. Physical meaning of thermodynamic parameters: entropy, energy, pressure, temperature, free energy, chemical potential. Physical meaning of gas pressure at the molecular scale. Thermodynamic ensembles: microcanonical, canonical, grand canonical. Grand Canonical Monte Carlo simulations.

Lecture 9. Part 1. Surface charge of nanoparticles. The source of surface charge in minerals: cation substitution (Si4+ -> Al3+), protonation and deprotonation of surface sites. Acidity constants and ratio between deprotonated and neutral sites. pH as a function on proton (H+) concentration or water acidity. Derivation of the pH-dependence for deprotonated sites fraction. Particle aggregation and disaggregation depending on their charge. Conditions for clay mineral nanoparticles aggregation. 3D charge maps of illite nanoparticles at pH=10, pH=7, pH=5. At which pH nanoparticles should aggregate? (acidic, alkaline, neutral). Number of surface protons as a function on pH. The reason for two inflection points on the surface titration (pH-dependent surface charge) curve.
Lecture 9. Part 2. Grand Canonical Monte Carlo simulations of carbonate surface protonation and Kinetic Monte Carlo simulations of carbonate dissolution pH-dependence. 2 protonation steps of surface carbonate groups. Deprotonation of water attached to surface Ca atoms. Numerical realizations (configurations) of surface with protons and bulk fluid with ions. Dependence of protonation of surface sites on pH and surface site coordination (kink sites-k, step sites-s, terrace sites-t). Influence of ions (Cl-) onto protonation probabilities. pH-dependence of dissolution rates as pH-dependence of surface site protonation.
Lecture 11. Definition of a phase and a phase boundary. Triple point and critical point. Coexistence curves. The phase rule (Gibbs rule). Degrees of freedom in phase diagrams. Cooling curves and supercooling effect. Binary systems. Compositional axis. Phase rule for a binary system. Liquidus and solidus. Phase coexistence. The lever rule for calculation of solid and liquid phase compositions.
Lecture 12. History of Quantum Mechanics. Black body radiation and ultraviolet catastrophe. Plank’s solution to ultraviolet catastrophe problem. Photoelectric (Einstein’s) effect and threshold light frequency. Bohr model of atom. De Broglie’s waves. Wave equation and its solution. Schrodinger’s time dependent equation and solution for the time-dependence part. Schrodinger’s time-independent equation in terms of kinetic and potential energy operators. Particle in a box model.
Lecture 13. Postulates of Quantum Mechanics. Operators and their eigenvalues. Particle in a box model. Normalization of a wavefunction. Expectation value of operators. Expectation value of particle’s position. Wavefunction and probability density of particle’s location. Classical and Quantum Rigid Rotor. Linear and angular momentum. Spherical Harmonics. Physical meaning of l and m quantum numbers. Magnitude and orientation of angular momentum vector in quantum systems. The Hydrogen model. Difference between rigid rotor and hydrogen atom model. Radial and angular parts of wavefunction for hydrogenic atoms. Energy levels. S, p, d and f electron orbitals. Shape of electron orbitals.

written exam

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