Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Affordable and clean energy

Academic Year 2021/2022

Learning outcomes

At the end of the course the student will learn the microscopic physics (at the atomic and molecular level) behind thermodynamic equilibrium in different scenarios, from large systems to nanostructures and biomaterials. The student will be able to analyze non-equilibrium kinetic processes such as diffusion, catalyzed reactions, energy conversion, nucleation and spinodal decomposition.

Course contents

Equilibrium Thermodynamics
Equilibrium physical properties and their symmetry. Conditions for equilibrium in an isolated system. Heterogeneous systems with one component (pure substances): Clapeyron equation, vapor pressure, and construction of the phase diagram.

Multi-component homogeneous systems: partial thermodynamic quantities. Gibbs-Duhem equation. Solid solutions: ideal, diluted and regular solutions.

Multi-component heterogeneous systems: equilibrium conditions. Microscopic model of a binary system: long-range order and miscibility gap versus temperature. The chemical potential. Construction of binary phase diagrams: the common tangent method and the lever rule. The Gibbs phase rule. Basic features of phase diagrams: invariant points, liquidus, solidus, and solvus lines. Examples of binary phase diagrams. Introduction to ternary phase diagrams.

Multi-component heterogeneous reacting systems: affinity and van 't Hoff equation. Richardson-Ellingham diagrams for oxidation.

Kinetic Mechanisms
Entropy and entropy production. Basic postulate of irreversible thermodynamics. The Onsager reciprocity relations: symmetry of transport properties. Thermoelectric effects. Driving forces and fluxes in atomic diffusion: the diffusion potential. The diffusion equation. Link between the macro-and microscopic viewpoints: thermally activated jumps, random walk and diffusion coefficient. Atomistic diffusion mechanisms in solids: vacancies, interstitials, diffusion in a concentration gradient (Kirkendall effect). Defects and diffusion in ionic solids.

Structure of interfaces and surfaces in materials. Driving forces for interface motion. Curvature and diffusion potential. Morphological evolution due to capillary forces. Surface smoothing via surface diffusion or vapour transport. Anisotropic surface free energy and faceting. Wulff constructoin for the equilibrium crystal shape. Coarsening of microstructures (Ostwald ripening): mean-field theory in source-limited and diffusion-limited cases. Morphological evolution due to applied stress.

Introduction to phase transformations: conserved vs non-conserved order parameter. Continuous transformations: spinodal decomposition and order-disorder transition. Cahn-Hilliard and Allen-Cahn equations.

Discontinuous transformations: the classical nucleation theory. Heterogeneous nucleation. Nucleation and growth: Johnson-Mehl-Avrami kinetics and time-temperature- transformation (TTT) diagrams. Precipitation in the Cu-Co and Al-Cu systems: effects of elastic strains on nucleation. Martensitic transformations and shape-memory alloys.

2-dimensional materials

Introduction to graphene, discovery and fundamental properties. Crystal structure. Band structure in tight-binding approximation. Production methods: exfoliation, epitaxial growth, chemical vapour deposition. Production of graphene-like materials and 2-dimensional materials beyond graphene: boron nitride, transition metal dicalchogenides, semiconductors. Characterization of 2-dimensional materials: scanning and transmission electron microscopy, Raman spectroscopy.


The slides and the lecture notes are available on IOL.

For further readings the following books are recommended:

· R.W. Balluffi, S.M. Allen, W.C. Carter, Kinetics of Materials, Wiley

· R. DeHoff, Thermodynamics in Materials Science, Taylor and Francis

Teaching methods

Frontal lectures, both at the blackboard and with the aid of a videoprojector.

Assessment methods

Oral examination. The student is allowed to start the exam illustrating a topic of his/her choice among the ones treated during the course.

Teaching tools

The course teaching material is available in the on-line repository.

Office hours

See the website of Luca Pasquini

See the website of Vittorio Morandi