  Due to it's simplicity the diffuse mismatch model (DMM) remains a popular
description of phonon transmission across solid-solid boundaries. However, it
remains unclear in which situations the DMM should be expected to be a valid
model of the underlying physics. Here, its validity is investigated
mode-by-mode using a 3-dimensional extension of the frequency domain, perfectly
matched layer (FD-PML) method, to study the interface between face-centered
cubic solids with interdiffused atoms. While submonolayer levels of
interdiffusion are found to increase the number of available modes for
transmission, consistent qualitatively with the DMM, we do not find
quantitative or qualitative convergence toward the DMM at higher levels of
interdiffusion. In particular, contrary to the fundamental assumption of the
DMM, modes are not found to lose memory of their initial polarization and
wavevector. The transmission coefficients of randomly interdiffused and
smoothly-graded interfaces are also compared. While smoothly graded interfaces
show strong anti-reflection properties, selection rules still prohibit
transmission of many modes, whereas interdiffused interfaces are not subject to
such rules and achieve similar thermal interface conductance by transmitting
with lower probability but using a wider range of modes.
