Theory of phase transition

Context

Theory of phase transition is a master-level course I taught at ETH during spring semesters 2020 and 2021.

Introduction

Phase transitions are observed whenever the potential energy density of a system is a non-convex function of one of its state variables. Illustrative examples are:

• Solidification, where the solid phase of a material grows at the expense of its liquid phase through the propagation of the solid/liquid interface. We aim at understanding how the solidification front propagates, how it its subject to instabilities (e.g., formation snowflakes).

• Solid-solid phase transitions, a prominent example of which is the martensitic transformation in shape memory alloys. Shape memory alloys are materials whose crystallographic-structure changes under mechanical or thermal loading. At the macroscale, this transformation is responsible for two exotic behaviors, namely, pseudoelasticity and shape memory effect.

• Ferroelectric switching, whereby the spontaneous electric polarization of ferroelectric materials is reversed by the application of an electric field. The switching observed at the macroscale results from the nucelation and growth of domains of polarization.

Course content

We address existing models for the evolution of the microstructure in phase transitions problems.

• Mechanical modeling of the solid-solid phase transformation in 1D: Mechanics of bars and stability of equilibria; The Ericksen's bar; Phase boundary motion and dissipation; Driving force and kinetic relation; Resulting macroscopic stress-strain response.

• Review of classical thermodynamics: First and second laws, thermodynamic potentials; Minimum principles; First order phase transition.

• Thermomechanical modeling of the solid-solid phase transformation in 3D: Mechanical, energy and entropy (im)balances in the presence of a surface of discontinuity; Entropy production at the interphase; kinetic relation.

• Sharp-interface modeling of solidification: Interfacial structure, Energy and entropy (im)balances; Stefan equation; Generalized Gibbs-Thomson equation, stability of a solidification front.

• Phase-field modeling of solidification: Formulation; Equilibrium profile and parameters; Allen-Cahn evolution equation.

Some material

• Lecture notes

• Slides of the first class (general overview)

• Problem Set on variational calculus and the stability of equilibria

• Problem set on the solid/solid phase transformation in 1D

• Problem set in classical thermodynamics on the measurability of the energy

• Problem set in classical thermodynamics on the internal energy as a fundamental relation

• Problem set on the formulation of a sharp-interface model for solidification using classical thermodynamics

• Problem set on the stability of a solidification front

• Problem set on a phase-field model for solidification

References

• Evolution of Phase Transitions: A Continuum Theory, R. Abeyaratne & J.K. Knwoles, Cambridge University Press (2006) [Chapters 1 to 3 and 6]

• Thermomechanics of Evolving Phase Boundaries in the Plane, M. Gurtin, Oxford University Press (1993) [Sections I.1, I.2 and III.14 to III.17 ]

• Thermodynamics and an Introduction to Thermostatistics, H. Callen, Wiley (1985) [Chapters 1 to 6 and 9]

• Phase-Field Models, M. Plapp, In: Multiphase Microfluidics: The Diffuse Interface Model, Springer (2012) [Sections 1 to 3]

Course on ETH catalogue

Link