  Shockwaves driven by supernovae both destroy dust and reprocess the surviving
grains, greatly affecting the resulting dust properties of the interstellar
medium (ISM). While these processes have been extensively studied
theoretically, observational constraints are limited. We use
physically-motivated models of dust emission to fit the infrared (IR) spectral
energy distributions of seven Galactic supernova remnants, allowing us to
determine the distribution of dust mass between diffuse and dense gas phases,
and between large and small grain sizes. We find that the dense ($\sim 10^3
\,{\rm cm}^{-3}$), relatively cool ($\sim 10^3 \, {\rm K}$) gas phase contains
$>90\%$ of the dust mass, making the warm dust located in the X-ray emitting
plasma ($\sim 1 \,{\rm cm}^{-3}$/$10^6 \, {\rm K}$) a negligible fraction of
the total, despite dominating the mid-IR emission. The ratio of small
($\lesssim 10 \, {\rm nm}$) to large ($\gtrsim 0.1 \, {\rm \mu m}$) grains in
the cold component is consistent with that in the ISM, and possibly even
higher, whereas the hot phase is almost entirely devoid of small grains. This
suggests that grain shattering, which processes large grains into smaller ones,
is ineffective in the low-density gas, contrary to model predictions.
Single-phase models of dust destruction in the ISM, which do not account for
the existence of the cold swept-up material containing most of the dust mass,
are likely to greatly overestimate the rate of dust destruction by supernovae.
