  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.
