» MIT OpenCourseWare » Materials Science and Engineering » Fundamentals of Materials Science, Fall 2003

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Goal

To provide a direct, rational connection between microscopic understanding and macroscopic properties, reinforced 'just-in-time' with real-life examples in lectures and labs. To show how the energy of materials descends from the mechanics of the electrons, and how this energetics determines bonding and zero-temperature structures. Thermodynamics takes a statistical mechanics slant, where discrete microstates become different realizations of the energetics, and equilibrium properties different averages on the appropriate ensembles of microstates. Macroscopic thermodynamic functions then summarize this microscopic complexity. Energy and entropy vie for macroscopic stability from the freezing of water to the folding of proteins. Several ideas are reinforced across disciplines, or presented in an unexpected environment (e.g. reducing a many-body problem to a mean field, or discussing an ideal gas of excitations). We support this program with a math core that provides a consistent description of the mathematical concepts needed in the lectures, and covers in depth the formal tools that are part of the professional competence of a materials scientist.

Note: S/B = Structure and Bonding Components; Thermo = Thermodynamics Components

SES #	ACTIVITY	READINGS	KEY DATES
1-3	Orientation
4	Introduction to the Course Overview of this Block of the Core and Review of Administrative Details What are We Going to Teach You, and How Does it Connect to What You Think of as Materials? Objectives for the Term Lectures First Lecture of S/B: An Introduction to Structure First Lecture of Thermo: An Introduction to the Overall Objectives of Thermodynamics, a First Look at Entropy
5	Safety Orientation Part 2 Recitation		Homework 1 Out
6	Lectures S/B: The Ionic Bond. Cohesive energy is defined by Coulomb's Law and bonding is non-directional. Structure of lowest energy is that with maximum packing of species of negative change about those with positive charge and vice versa Concept of "Ionic Radii" - Assumptions and Starting Points for Various Schemes (Pauling and Shannon-Prewitt) Connection Between Coordination Numbers and Chemical Composition for Simple Binary Compounds Stability Fields for A_n X_m Compounds as a Function of Radius Ratio Thermo: Fundamental Concepts (continued) Thermodynamic Variables, Systems, and Functions Definitions for Describing Materials Identification of Processes	S/B: Bransden, and Jochain. "Fundamental Constants, Atomic Units, and Conversion Factors." In Physics of Atoms and Molecules. Thermo: Refresher on Differentials and Partial Derivatives of Multivariate Functions.
7	Recitation
8	Lectures S/B: Linkage of Coordination Polyhedra and Pauling’s Rules for Stable Ionic Structures. More Quantitative Modern Extension to Bond Valence Sums Ionic Structures in Terms of Close-packing of the Large Species with Smaller Species in Available Interstices. FCC, HCP and BCC Packing. Examples: NaCl, ZnS, Spinel, BaTiO₃, AgI, LaCu₂O₄. Stacking Polytypes Thermo: The First Law Work and Heat Change the Internal Energy of a System Heat and Work in Reversible Processes State Functions vs. Path-dependent Functions Example of the Ideal Gas	Thermo: Further Discussion of the First Law, Work, Heat, and Reversible vs. Irreversible Processes
9	Lectures S/B: Linkage of Polyhedra in Chain, Sheet and Network Structures. Amorphous Structures. (Examples: SiO₂, B₂O₃, Polymers) Voronoi Polyhedra. Random-walk Model for Chain Length Structural Nature of Phase Transformations. Change in Primary Versus Secondary Coordination. Rapid/Displacive versus Reconstructive Transformations Thermo: Temperature, Heat, and Entropy Defining Temperature Consequences of the Relation between Temperature, Heat, and Entropy: Heat Capacity Calculations with Heat Capacities
10	Recitation		Homework 1 In Homework 2 Out
11	Lectures S/B: Description of Structures in Terms of Symmetry. Basic Mapping Operations in 3D-translation, Reflection, Rotation and Inversion (Changing Sense of Zero, One, Two, or All Three Coordinates Exhausts Possibilities) Translation Operations and Lattice Points. Notation for Coordinates, Rational Directions and Rational Planes. Forms Rotation Operation: Limited to α = 2π/n for Discrete Sets of Motifs. Concept of Group and Group Multiplication Table Combination of Rotation with Translation Restricts n to 1, 2, 3, 4 or 6 and, conversely, Results in Oblique, Square and Hexagonal Nets as Unique 2-D Lattices Thermo: Heat Storage and Release at Phase Transitions Heat Stored and Released at Phase Transitions Accounting for Thermal Energy in a Material: Introduction of Enthalpy Application Example: Phase Change Materials Technology Extra: Discovery of Latent Heat
12	Recitation
13	Lectures S/B: Combination of Translation and Reflection Restricts Lattice Nets to Primitive and Centered Rectangular and Leads to Discovery of the Glide Plane. (Thus only five kinds of 2D Llattices are possible) Combination of Rotation and Reflection about a Fixed Point in 2D: the Ten 2D Point Groups. International and Schonflies Notation Thermo: Measuring Thermodynamic Quantities in the Laboratory Direct and Indirect Measurement Thermal Expansion/Contraction, and Mechanical Expansion/Compression Maxwell Relations	Thermo: Bent, H. A. Survey of Molar Entropies.
14	Recitation		Homework 2 In Homework 3 Out
15	Lectures S/B: Interaction of Waves with Periodic Structures The Laue Equations and Demonstration of their Equivalence to Bragg's Law and "Reflection" from Lattice Planes Scattered Intensity for Structure with One Atom/Cell and Extension to Structures with Arbitrary Numbers of Atom/Cell Amplitude of Scattered Beam, after Clearing Out Physical and Geometric Factors, is the Structure Factor Thermo: Examples of Work Important in Materials Science and Engineering Polarization of Materials Magnetic Work Chemical Work
16	Recitation Quiz 1
17	Lectures S/B: Ewald’s Construction as a Nomogram for Interpretation of Bragg’s Law. Usefulness of the Concept of the Reciprocal Lattice in the Interpretation of Diffraction (Examples: Various Possibilities for Moving the Crystal into Diffracting Orientation, Range of hkl that May be Recorded, Total Number of Diffraction Peaks that can be Obtained, Reason for "missing" Reflections) Powder Diffraction and Single Crystal Methods Thermo: Thermodynamic Driving Forces Fundamental Equations for the Internal Energy and Entropy Components of Internal Energy and Entropy Determining the Fundamental Equation for a Materials System	Thermo: Dill, and Bromberg. "Thermodynamic Driving Forces." Ch. 7 in Molecular Driving Forces. Pp. 105-109. ———. "How to Write a Fundamental Equation." In Molecular Driving Forces. Pp. 153-155.
18-22	Lab Session 1
23	Lectures S/B: Principles of Combination of 2-D Point Groups and 2D Lattices to Obtain Plane Groups. Space Group Properties: General and Special Positions and their Use to Describe Structure Combination of Rotation, Reflection and Inversion in 3D: The Three Dimensional Point Groups Thermo: Equilibrium and the Second Law Introduction to the Second Law Applying the Second Law Thermal Equilibrium Mechanical Equilibrium	Thermo: Excerpt from Callen Showing the Proof that the Second Law Requires Internal Energy is Minimized at Equilibrium if Entropy is Fixed.
24	Recitation		Homework 3 In Homework 4 Out
25	Lectures S/B: Stacking Plane Groups to Obtain the 3D Space Lattices (Gives All Except Cubic) Addition of 3D Point Groups to Space Lattices. Leads to Screw Axes Thermo: Free Energy: Applying the Second Law in Laboratory Conditions Two New Thermodynamic Functions for Lab Conditions Helmholtz Free Energy Gibbs Free Energy Determining Equilibrium for Experiments in the Lab	Thermo: Dill, and Bromberg. "Laboratory Conditions and Free Energies." Ch. 8 in Molecular Driving Forces.
26	Recitation
27	Lectures S/B: Physical Assumptions Made for Properties that Must be Described as Tensors. Examples of First, Second, Third and Fourth-rank Tensor Properties Specification of Change in Cartesian Reference Axes in Terms of Direction Cosine Matrix Transformation of Tensor Elements to a New System of Reference Axes Symmetry Restrictions on Tensors: Tensor must Remain Invariant, Element by Element, to any Change of Axes that Corresponds to a Symmetry Transformation that Leaves the Crystal Unchanged Thermo: Chemical Potentials and the Gibbs Free Energy Describing Multi-phase/Multi-component Systems Molar and Partial Molar Quantities Chemical Potentials in Multi-phase Materials at Equilibrium
28	Lectures S/B: Anisotropy: Variation of Second-rank Tensor Property (a Scalar) with Direction The Representation Quadric Use of Properties of the Quadric (The "Radius-normal" Property) to Find the Extreme Values of a Physical Property and the Directions in which They Occur. (A Nice Example of an Eigen-value Problem that can be Formulated Entirely within a Geometrical and Physical Context!) Thermo: Models of the Chemical Potential; Chemical Reactions Chemical Potential of the Ideal Gas General Solution Model of the Chemical Potential Equating Chemical Potentials During Chemical Reactions		Homework 4 In Homework 5 Out
29	Recitation Quiz 2
30	Lectures S/B: Stress and Strain Tensors Higher-rank Properties and Anisotropy (Example: Variation of the Longitudinal Piezoelectric Modulus with Direction for Quartz) Thermo: Chemical and Electrochemical Reaction Equilibria Reactions in Condensed Phase Solutions Application Example: Designing a Battery Using The Electrochemical Potential	Thermo: Dill, and Bromberg. "Electrostatic Potential." Ch. 21 in Molecular Driving Forces.
31-35	Lab Session 2
36	Lectures S/B: Energy of Electrons and Atoms: What is an Atom: Electrons/Nuclei, and Why Electrons Behave Mostly as Waves, and Nuclei Mostly as Particles From Classical to Quantum: Description of a State (Position/Momentum → Wavefunction) Action on a State (Operators) Ground States and Evolution (Hamiltonian → Schroedinger Equation) Application Examples: Wave-like Properties of Photons, Electrons, and Even Fullerenes; Reminder of Classical Dynamical Trajectories Thermo: Gibbs Free Energy: Shapes of Things, and Stability Stable, Metastable, and Unstable Equilibria Le Chatelier’s Principle Requirements for the Shape of Free Energy Curves
37	Recitation		Homework 5 In Homework 6 Out
38	Lectures S/B: Mechanics of Electrons The Mechanics of Waves – Interference and Wavepackets Schroedinger Equation: Time-independent Problem, Separation of Time and Space Simplest Case: The Free Particle Evolving in Space Introduction to the Postulates Application Example: Electrons through Slits: Diffraction and Interference Thermo: Phase Changes and Phase Diagrams of Single-component Materials Behavior of the Chemical Potential/Molar Free Energy in Single-component Materials Phases and Phase Diagrams of Single-component Materials Constraints on the Shape of Coexistence Curves: The Clausius Clapeyron Equation	Thermo: Lupis. "Chemical Thermodynamics of Materials." (Excerpt)
39	Recitation
40	Lectures S/B: The Conceptual Framework Information Contained in a Wavefunction The Result of a Measure Classical Quantities as Quantum-mechanical Operators (e.g. The Kinetic Energy) Application Example: Indetermination Principle, and the Natural Size of an Atom Thermo: Phase Changes and Phase Diagrams of Single-component Materials (continued) An Example: Walking along Lines of Constant Temperature or Pressure in a Single-component Phase Diagram Introduction to Second-order Transitions: Order-disorder Transitions
41	Lectures S/B: Quantization of Energies, and the Birth of Interactions Boundary Conditions → Quantization of Energies: The Infinite Well Finite Well – Electrons Spread Out Two Wells – Bonding and Antibonding States Application Example: Stationary Waves in Organ Pipes and Drums. Tunneling Behavior of Electrons (STM) Thermo: Thermodynamics of Solutions Solutions in Materials Science and Engineering Graphical Constructions of the Free Energy in Mixtures and Solutions Melting Point Depression	Thermo: Zallen. "The Glass Transition." Ch. 1.4 in Physics of Amorphous Solids. Pp. 16-23. Gaskell. "The Free Energy of Solution." Ch. 11.5 in Introduction to Metallurgical Thermodynamics.
42	Recitation		Homework 6 In
43	Lectures S/B: The Hydrogen Atom Separation of Variables in a Central Potential, Leads to Radial and Angular Quantum Numbers Application Example: Visualizations of Atomic Orbitals, and of the Scattering of a Wavepacket from an Atom Thermo: Free Energy of Multi-phase Solutions at Equilibrium Free Energy Diagrams of Multi-phase Solutions Common Tangent Construction and the Lever Rule Introduction to Binary Phase Diagrams	Thermo: Lupis. "Binary Phase Diagrams." Ch. 8 in Chemical Thermodynamics of Materials.
44	Recitation Quiz 3		Homework 7 Out
45	Lectures S/B: Mean Field and the Periodic Table A Central Potential is All What is Needed for the n, l, m Structure of Any Many-electron Atom: Angular Momenta and Spherical Harmonics The Idea of Mean Field – Many-electron Atoms can Still be Looked as Single Electrons in a Central Potential: The Periodic Table, and the Empirical Rules that Summarize it Application Example: The Periodic Table Thermo: Binary Phase Diagrams Binary Solutions with Limited Miscibility in the Solid State: Eutectic Systems The Phase Rule Applied to Binary Phase Diagrams	Thermo: Excerpt from Lupis on Binary Phase Diagrams.
46	Recitation
47-53	Lab Session 3
54	Lectures S/B: The Variational Principle Minimization of the Energy as a Practical and Conceptual Tool to Characterize Ground States. E=<phi\|H\|phi/<phi\|phi> Adiabatic Separation Between Ions and Electrons, where Ions Follow Hellmann-Feynman Forces Why do Materials Take the Lowest Energy State? Forces and Stresses on Aggregates of Atoms Application Example: Tunnelling in the Ammonia Molecule. Suppression of Ferroelectricity by Quantum Fluctuations Thermo: Binary Phase Diagrams (continued) Invariant Points in Binary Systems Intermediate Compounds, and Examples in Ceramic Systems Other Types of Phase Diagrams Example Binary Phase Diagrams	Thermo: Excerpt from McCallister on Binary Phase Diagrams and the Connection to Microstructure.
55	Recitation		Homework 7 In Homework 8 Out
56	Lectures S/B: The Hydrogen Molecule We Understand and Determine the Ground State by Varying the Combinations of Orbitals, or the Shape of the Orbitals: 1. LCAO 2. Trial Wavefunctions Thermo: Connecting Events at the Atomic/Molecular Level to Macroscopic Thermodynamic Behavior Statistical Mechanics and Models of Materials The Microscopic Definition of Entropy Testing the Microscopic Definition of Entropy	Thermo: Bent, H. A. "Boltzmann's Relation." Ch. 20 in The Second Law. (Excerpt)
57	Lectures S/B: Molecules: Simple Bonding, Complex Structures The Lowest Energy Configuration Bond Formation, Bonding and Anti-bonding Orbitals The Structure of Dimers Sigma and Pi Bonds Linear Combination of Atomic Orbitals, and the Structure of Small Molecules and Functional Groups Application Example: Functional Groups have Similar Physical/Chemical Properties in Different Environments (e.g. C-H Vibrational Frequencies, or Strength of Single/Double/Triple Bonds) Thermo: Connecting Events at the Atomic/Molecular Level to Macroscopic Thermodynamic Behavior (continued) Temperature and the Occupation of States The Boltzmann Factor and Partition Function Application Example: The Einstein Solid	Thermo: Dill, and Bromberg. "The Boltzmann Distribution Law." Ch. 10 in Molecular Driving Forces. Pp. 171-191.
58	Recitation
59	Lectures S/B: Solids: Simple Structures, Complex Bonding Driving Forces for the Structure of Metals, Semiconductors, Oxides Relation between the Electronic Structure and the Strength and Shape of Equilibrium Geometries Application Example: Carbon, Starting with sp3, leading to diamond, and to sp2 (graphite, fullerene, nanotubes) Thermo: Molecular Degrees of Freedom that Make up Entropy The Einstein Solid, continued Degrees of Freedom in Molecular Models Application Example: The Curie Law of Paramagnetism
60-62	Lab Session 4
63	Recitation		Homework 8 In
64	Lab Session 4 (continued)
65	Lectures S/B: A Modern View Independent Electrons: Electrostatics (Hartree Potential) and Spin Spin-statistics Connection: The Pauli Principle and the Fermi-Dirac Distribution Application Example: Singlet vs. Triplet Excitations Thermo: Lattice Models of Materials Lattice Models for Translational Degrees of Freedom Application Example: Derivation of the Ideal Gas Law from a Molecular Model Lattice Models of Solutions: The Regular Solution Model	S/B: Binning, Gerd, and Heinrich Rohrer. "Scanning Tunneling Microscopy - From Birth To Adolescence." In Nobel Lecture. December 8, 1986. Thermo: Dill, and Bromberg. "Solutions and Mixtures." Ch. 15 in Molecular Driving Forces.
66	3.014 Final Presentations
67	Lectures S/B: A Modern View Pauli Principle for Independent Electrons Provides all the Fundamental Interactions: Classical Electrostatic Forces, Quantum Exchange, Orthonormality Application Example: High-pressure Alkali become Insulators to Avoid Electron Overlap Thermo: Thermodynamics of Adsorption: Macromolecules and Biomacromolecules Applying a Lattice Model to the Thermodynamics of Adsorption The Langmuir Adsorption Isotherm Adsorption Behavior of Polymers and Proteins	Thermo: Dill, and Bromberg. "Adsorption, Binding, and Catalysis." Ch. 27 in Molecular Driving Forces. Pp. 515-521.
68-72	Finals Week Quiz 4

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