  We present a new gas-grain chemical model to constrain the effect of grain
size distribution on molecular abundances in starless and pre-stellar cores. We
introduce grain-size dependence simultaneously for cosmic-ray (CR)-induced
desorption efficiency and for grain equilibrium temperatures. We keep explicit
track of ice abundances on a set of grain populations. We find that the
size-dependent CR desorption efficiency affects ice abundances in a highly
non-trivial way that depends on the molecule. Species that originate in the gas
phase follow a simple pattern where the ice abundance is highest on the
smallest grains (the most abundant in the distribution). Some molecules, such
as HCN, are instead concentrated on large grains throughout the time evolution,
while others (like $\rm N_2$) are initially concentrated on large grains, but
at late times on small grains, due to grain-size-dependent competition between
desorption and hydrogenation. Most of the water ice is on small grains at high
medium density ($n({\rm H_2}) \gtrsim 10^6 \, \rm cm^{-3}$), where the water
ice fraction, with respect to total water ice reservoir, can be as low as $\sim
10^{-3}$ on large (> 0.1 $\mu$m) grains. Allowing the grain equilibrium
temperature to vary with grain size induces strong variations in relative ice
abundances in low-density conditions where the interstellar radiation field and
in particular its ultraviolet component are not attenuated. Our study implies
consequences not only for the initial formation of ices preceding the starless
core stage, but also for the relative ice abundances on the grain populations
going into the protostellar stage. In particular, if the smallest grains can
lose their mantles due to grain-grain collisions as the core is collapsing, the
ice composition in the beginning of the protostellar stage could be very
different to that in the pre-collapse phase.
