  We present an infrared (IR) characterization of the born-again planetary
nebulae (PNe) A30 and A78 using IR images and spectra. We demonstrate that the
carbon-rich dust in A30 and A78 is spatially coincident with the H-poor ejecta
and coexists with hot X-ray-emitting gas up to distances of 50$''$ from the
central stars (CSPNs). Dust forms immediately after the born-again event and
survives for 1000 yr in the harsh environment around the CSPN as it is
destroyed and pushed away by radiation pressure and dragged by hydrodynamical
effects. Spitzer IRS spectral maps showed that the broad spectral features at
6.4 and 8.0 $\mu$m, attributed to amorphous carbon formed in H-deficient
environments, are associated with the disrupted disk around their CSPN,
providing an optimal environment for charge exchange reactions with the stellar
wind that produces the soft X-ray emission of these sources. Nebular and dust
properties are modeled for A30 with Cloudy taking into account different
carbonaceous dust species. Our models predict dust temperatures in the 40-230 K
range, five times lower than predicted by previous works. Gas and dust masses
for the born-again ejecta in A30 are estimated to be
$M_\mathrm{gas}=(4.41^{+0.55}_{-0.14})\times10^{-3}$ M$_\odot$ and
$M_\mathrm{dust}=(3.20^{+3.21}_{-2.06})\times10^{-3}$ M$_\odot$, which can be
used to estimate a total ejected mass and mass-loss rate for the born-again
event of $(7.61^{+3.76}_{-2.20})\times10^{-3}$ M$_{\odot}$ and
$\dot{M}=[5-60]\times10^{-5}$ M$_{\odot}$ yr$^{-1}$, respectively. Taking into
account the carbon trapped into dust grains, we estimate that the C/O mass
ratio of the H-poor ejecta of A30 is larger than 1, which favors the very late
thermal pulse model over the alternate hypothesis of a nova-like event.
