  We report spatially resolved (FWHM$\sim3.8-4.6"$) mid-IR imaging observations
of the planetary nebula (PN) NGC 7027 taken with the 2.5-m telescope aboard the
Stratospheric Observatory for Infrared Astronomy (SOFIA). Images of NGC 7027
were acquired at 6.3, 6.6, 11.1, 19.7, 24.2, 33.6, and 37.1 $\mu\mathrm{m}$
using the Faint Object Infrared Camera for the SOFIA Telescope (FORCAST).The
observations reveal emission from Polycyclic Aromatic Hydrocarbon (PAH) and
warm dust ($T_D\sim90$ K) from the illuminated inner edge of the molecular
envelope surrounding the ionized gas and central star. The DustEM code was used
to fit the spectral energy distribution of fluxes obtained by FORCAST and the
archival infrared spectrum of NGC 7027 acquired by the Short Wavelength
Spectrometer (SWS) on the Infrared Space Observatory (ISO). Best-fit dust
models provide a total dust mass of $5.8^{+2.3}_{-2.6}\times10^{-3}$
$\mathrm{M}_\odot$, where carbonaceous large ($a=1.5$ $\mu$m) and very small
($a \sim12\AA$) grains, and PAHs ($3.1\AA<a<12\AA$) compose 96.5, 2.2, and 1.3
$\%$ of the dust by mass, respectively. The 37 $\mu$m optical depth map shows
minima in the dust column density at regions in the envelope that are
coincident with a previously identified collimated outflow from the central
star. The optical depth minima are also spatially coincident with enhancements
in the 6.2 $\mu$m PAH feature, which is derived from the 6.3 and 6.6 $\mu$m
maps. We interpret the spatial anti-correlation of the dust optical depth and
PAH 6.2 $\mu$m feature strength and their alignment with the outflow from the
central star as evidence of dust processing and rapid PAH formation via
grain-grain collisions in the post-shock environment of the dense
($n_H\sim10^5\,\mathrm{cm}^{-3}$) photo-dissociation region (PDR) and molecular
envelope.
