  We present a model of the growth of water ice on silicate grains in the
circumstellar envelopes of Asymptotic Giant Branch (AGB) stars and Red Super
Giants. We consider the growth of ice by gas grain collisions, the thermal
evaporation of ice from a grain, and sputtering. Our model contains several
improvements compared to earlier models, including a detailed treatment of the
effects of sputtering, a detailed calculation of the radiation pressure on the
grain, and the treatment of subsonic drift velocities. In terms of drift
velocity between the grains and gas in the envelope, we find that the ice
formation process can be divided into three regimes: (i) a sputtering dominated
regime where ice growth is heavily suppressed, (ii) an intermediate regime with
moderately efficient condensation and (iii) a thermally dominated regime where
ice formation is highly efficient. Sputtering is the critical factor which
determines if ice formation can occur at all. We find that in Red Supergiants,
ice formation is suppressed, while the winds of OH/IR stars allow for efficient
condensation and can convert significant fractions of the available water vapor
(tens of percent) into ice mantles on grains. Population II AGB stars hardly
form ice due to their low dust to gas ratios. We also modify an analytical
equation describing condensation and depletion (Jura and Morris, 1985) in order
to give reasonable results for high and low drift velocities. Initially, ice
will condense in crystalline form, but continuing condensation at low
temperatures, and damage caused by interstellar UV photons favor the production
of amorphous ice as well. We predict that a significant fraction of the ice
formed will be amorphous.
