  Phonon boundary scattering is typically treated using the Fuchs-Sondheimer
theory, which assumes that phonons are thermalized to the local temperature at
the boundary. However, whether such a thermalization process actually occurs
and its effect on thermal transport remains unclear. Here we examine thermal
transport along thin films with both thermalizing and non-thermalizing walls by
solving the spectral Boltzmann transport equation (BTE) for steady state and
transient transport. We find that in steady state, the thermal transport is
governed by the Fuchs-Sondheimer theory and is insensitive to whether the
boundaries are thermalizing or not. In contrast, under transient conditions,
the thermal decay rates are significantly different for thermalizing and
non-thermalizing walls. We also show that, for transient transport, the
thermalizing boundary condition is unphysical due to violation of heat flux
conservation at the boundaries. Our results provide insights into the boundary
scattering process of thermal phonons over a range of heating length scales
that are useful for interpreting thermal measurements on nanostructures.
