87926 - Microscopic Kinetics and Thermodynamics

Learning outcomes

At the end of the course, the student will learn the foundational laws that govern thermodynamic equilibrium in materials and systems of increasing complexity. He/she will be able to analyse non-equilibrium kinetic processes such as mass and charge transport by atomic diffusion, phase transformations (e.g. nucleation and spinodal decomposition), growth of nanostructures, and hydrogen absorption in metals. The student will learn the basics of energy conversion and storage processes that are vital for sustainable energy technologies.

Course contents

MODULE 1 (4 Credits)

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.

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. Precipitation in the Cu-Co and Al-Cu systems: effects of elastic strains on nucleation.

MODULE 2 (2 credits)

Electrochemical energy conversion and storage

Introduction to electrodic processes. Potentials of phases and electrochemical thermodynamics. Electrode kinetics (Butler-Volmer model, One-step and multi-step processes, Marcus theory of charge transfer). Mass transfer mechanisms. Electrochemical double layer. Potentiostatic and galvanostatic electrochemical methods. Electrochemical impedance spectroscopy. Electrodeposition of nanostructures. Electrochemical devices for energy conversion and storage. Introduction to semiconductor photoelectrochemistry.

Readings/Bibliography

The slides and the lecture notes are available on Virtuale.

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. Interactive anonymous questionnaires to boost engagement.

Assessment methods

The exam consists of an oral interview divided into two parts, one for each module. The duration of the part related to Module 1 is approximately 25 minutes, while the part for Module 2 lasts about 15 minutes.

In each part, the student begins by presenting a topic of their choice among those covered during the course. In-depth discussions developed independently by the student, based on the recommended bibliography or other scientific literature, are welcome but not mandatory. The presentation must be given at the blackboard or on paper, without slides or notes, and must not exceed 15 minutes for Module 1 and 10 minutes for Module 2. During the presentation, the instructor may ask questions to clarify or explore certain aspects of the topic. The presentation is expected to be thorough: the student should demonstrate knowledge of the mathematical derivations as well as a solid understanding of the physical principles and any approximations used.

Following the chosen-topic presentation, the instructor will ask the student to briefly discuss another topic from the course (see "Course Contents"). In this second part, the student is expected to demonstrate a good understanding of the main physical concepts, but derivations are not required. In particular, the student should be able to outline the initial assumptions, highlight any approximations, and explain how the results are applied.

The part related to Module 1 will conclude with the analysis of a binary phase diagram selected at random by the instructor. The student must explain all the information that can be extracted from the diagram, such as melting points, invariant points, miscibility gaps, and phase fields of pure substances.

Grading Criteria
The final grade is approximately the weighted average of the grades from Module 1 (weight 2/3) and Module 2 (weight 1/3). The exam is considered passed if the grade in each module is ≥18/30. In both modules, the grade is awarded according to the following criteria:

• 29–30L: Thorough preparation on all topics, strong analytical skills, full command of subject-specific terminology

• 26–28: In-depth knowledge of the chosen topic and fair to good preparation on the rest; good analytical skills and use of appropriate terminology and language

• 22–25: Good preparation on the chosen topic and fair knowledge of the rest; analytical skills demonstrated mainly with guidance from the instructor; correct use of language

• 18–21: Fair preparation on the chosen topic and sufficient understanding of the rest; analytical skills emerging only with guidance; overall acceptable language.

The grade obtained following the oral exam may be rejected a maximum of two times.

Teaching tools

The recorded lectures are available on Virtuale until the end of the course.

Office hours

See the website of Luca Pasquini

See the website of Raffaello Mazzaro