We study the dust evolution in the supernova remnant Cassiopeia A. We follow the processing of dust grains that formed in the Type II-b supernova by modelling the sputtering of grains. The dust is located in dense ejecta clumps crossed by the reverse shock. Further sputtering in the inter-clump medium once the clumps are disrupted by the reverse shock is investigated. The dust evolution in the dense ejecta clumps of Type II-P supernovae and their remnants is also studied. We study oxygen-rich clumps that describe the ejecta oxygen core, and carbon-rich clumps that correspond to the outermost carbon-rich ejecta zone. We consider the dust components formed in the supernova, several reverse shock velocities and inter-clump gas temperatures, and derive dust grain size distributions and masses as a function of time. We find that non-thermal sputtering in clumps is important and accounts for reducing the grain population by ~ 40% to 80% in mass, depending on the clump gas over-density and the grain type and size. A Type II-b supernova forms small grains that are sputtered within clumps and in the inter-clump medium. For Cas A, silicate grains do not survive thermal sputtering in the inter-clump medium. Our derived masses of currently processed silicate, alumina and carbon grains in Cas A agree well with values derived from observations. Grains produced by Type II-P supernovae better survive the remnant phase. For dense ejecta clumps, dust survival efficiencies range between 42% and 98% in mass. For the SN1987A model, the derived surviving dust mass is in the range ~ 0.06-0.13 Msolar. This type of dense supernovae may efficiently provide galaxies with dust. Specifically, silicate grains over 0.1 micron and other grains over 0,05 micron survive thermal sputtering in the remnant. Therefore, pre-solar grains of supernova origin possibly form in the dense ejecta clumps of Type II-P supernovae.
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