A hierarchy of catastrophes as a succession of stability limits for the crystalline state

Abstract

MANY theories of melting have focused on some form of instability or catastrophe delimiting the range of stability of the crystalline state. These are exemplified by the vibrational instability proposed by Lindemann¹ and the elastic shear instability of Born². Fecht and Johnson³ suggested that an entropy catastrophe delimits the stability range of both a superheated crystal and a supercooled liquid. Drawing on Kauzmann's idea⁴ that the glass transition pre-empts a catastrophe point at which the entropy of supercooled liquid equals that of the crystal, they suggest that a second catastrophe point occurs when the entropy of the superheated crystal equals that of the liquid phase, and that this is pre-empted by melting. Here I argue that these entropy catastrophes are an outer bound on the stability limits of the crystalline and liquid phases, and that a hierarchy of inner catastrophes occurs at reduced superheatings or supercoolings.