Incompatible sets of gradients and metastability

TL;DR

This paper provides a mathematical analysis of how incompatibility of energy wells induces metastability in phase transformations, showing that certain phases remain stable due to energetic costs of transition layers.

Contribution

It introduces a rigorous framework for understanding metastability caused by incompatibility, including characterizations of incompatible matrix sets and transition layer estimates.

Findings

Metastability arises from energetic costs of transition layers due to incompatibility.

Parent phase remains a local energy minimizer in an $L^1$ neighborhood.

Connections are made to experimental observations and broader metastability research.

Abstract

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.