  Faceted interfaces are a key feature in self-resembling morphologies of many
microstructures generated from solid state phase transformations.
Interpretations, predictions and simulations of the faceted morphologies remain
a challenge, especially for systems in which irrational orientation
relationships (ORs) between two phases and irrational interface orientations
(IOs) are preferred. In terms of structural singularities, this work suggests
an integrated framework, which possibly encompasses all candidates of faceted
interfaces. The structural singularities are identified in a matching pattern,
a dislocation structure and/or a ledge structure. The resultant singular
interfaces are discrete IOs, described with low index g's (rational
orientations) and/or Delta-g's (either rational or irrational orientations).
Elimination of defects exerts restrictions on the OR and the lattice
parameters. Various models are grouped according to their determined results
regarding the OR and IO, and the links between the models are clarified in the
integrated framework. An irrational IO is usually due to elimination of
dislocations in one direction, i.e., an O-line interface. Analytical methods
using both 3D and 2D models for quantitative determinations of O-line
interfaces are reviewed, and a detailed example showing the calculation for an
irrational interface is given. The association between structural singularities
and local energy minima is verified by atomistic calculations of interfacial
energies in fcc/bcc alloys where it is found that the calculated equilibrium
cross-sections are in a good agreement with observations from selected alloys.
