Polyamorphism is the ability of a substance to exist in several different amorphous modifications. It is analogous to the polymorphism of crystalline materials. Many amorphous substances can exist with different amorphous characteristics (e.g. polymers). However, polyamorphism requires two distinct amorphous states with a clear, discontinuous (first-order) phase transition between them. When such a transition occurs between two stable liquid states, a polyamorphic transition may also be referred to as a liquid–liquid phase transition. Overview Even though amorphous materials exhibit no long-range periodic atomic ordering, there is still significant and varied local structure at inter-atomic length scales (see structure of liquids and glasses). Different local structures can produce amorphous phases of the same chemical composition with different physical properties such as density. In several cases sharp transitions have been observed between two different density amorphous states of the same material. Amorphous ice is one important example (see also examples below). Several of these transitions (including water) are expected to end in a second critical point. Liquid–liquid transitions Polyamorphism may apply to all amorphous states, i.e. glasses, other amorphous solids, supercooled liquids, ordinary liquids or fluids. A liquid–liquid transition however, is one that occurs only in the liquid state (red line in the phase diagram, top right). In this article liquid–liquid transitions are defined as transitions between two liquids of the same chemical substance. Elsewhere the term liquid–liquid transition may also refer to the more common transitions between liquid mixtures of different chemical composition. The stable liquid state unlike most glasses and amorphous solids, is a thermodynamically stable equilibrium state. Thus new liquid–liquid or fluid-fluid transitions in the stable liquid (or fluid) states are more easily analysed than transitions in amorphous solids where arguments are complicated by the non-equilibrium, non-ergodic nature of the amorphous state. Rapoport's theory Liquid–liquid transitions were originally considered by Rapoport in 1967 in order to explain high pressure melting curve maxima of some liquid metals. Rapoport's theory requires the existence of a melting curve maximum in polyamorphic systems. Double well potentials One physical explanation for polyamorphism is the existence of a double well inter-atomic pair potential (see lower right diagram). It is well known that the ordinary liquid–gas critical point appears when the inter-atomic pair potential contains a minimum. At lower energies (temperatures) particles trapped in this minimum condense into the liquid state. At higher temperatures however, these particles can escape the well and the sharp definition between