We investigate the formation of silicon carbide (SiC) grains in the framework of dust-driven wind around pulsating carbon-rich Asymptotic Giant Branch (C-rich AGB) stars in order to reveal not only the amount but also the size distribution. Two cases are considered for the nucleation process; one is the LTE case where the vibration temperature of SiC clusters $T_{rm v}$ is equal to the gas temperature as usual, and another is the non-LTE case in which $T_{rm v}$ is assumed to be the same as the temperature of small SiC grains. The results of hydrodynamical calculations for a model with stellar parameters of mass $M_{ast}$=1.0 $M_{odot}$, luminosity $L_{ast}$=10$^{4}$ $L_{odot}$, effective temperature $T_{rm eff}$=2600 K, C/O ratio=1.4, and pulsation period $P$=650 days show the followings: In the LTE case, SiC grains condense in accelerated outflowing gas after the formation of carbon grains and the resulting averaged mass ratio of SiC to carbon grains of $sim$ 10$^{-8}$ is too small to reproduce the value of 0.01-0.3 inferred from the radiative transfer models. On the other hand, in the non-LTE case, the formation region of SiC grains is inner than and/or almost identical to that of carbon grains due to the so-called inverse greenhouse effect. The mass ratio of SiC to carbon grains averaged at the outer boundary ranges from 0.098 to 0.23 for the sticking probability $alpha_{rm s}$=0.1-1.0. The size distributions with the peak at $sim$ 0.2-0.3 $rm{mu}$m in radius cover the range of size derived from the analysis of presolar SiC grains. Thus the difference between temperatures of small cluster and gas plays a crucial role in the formation process of SiC grains around C-rich AGB stars, and this aspect should be explored for the formation process of dust grains in astrophysical environments.
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