  The dynamics of phase field crystal (PFC) modeling is derived from dynamical
density functional theory (DDFT), for both single-component and binary systems.
The derivation is based on a truncation up to the three-point direct
correlation functions in DDFT, and the lowest order approximation using scale
analysis. The complete amplitude equation formalism for binary PFC is developed
to describe the coupled dynamics of slowly varying complex amplitudes of
structural profile, zeroth-mode average atomic density, and system
concentration field. Effects of noise (corresponding to stochastic amplitude
equations) and species-dependent atomic mobilities are also incorporated in
this formalism. Results of a sample application to the study of surface
segregation and interface intermixing in alloy heterostructures and strained
layer growth are presented, showing the effects of different atomic sizes and
mobilities of alloy components. A phenomenon of composition overshooting at the
interface is found, which can be connected to the surface segregation and
enrichment of one of the atomic components observed in recent experiments of
alloying heterostructures.
