We compute the desorption rate of icy mantles on dust grains as a function of the size and composition of both the grain and the mantle. We combine existing models of cosmic ray (CR) related desorption phenomena with a model of CR transport to accurately calculate the desorption rates in dark regions of molecular clouds. We show that different desorption mechanisms dominate for grains of different sizes, and in different regions of the cloud. We then use these calculations to investigate a simple model of the growth of mantles, given a distribution of grain sizes. We find that modest variations of the desorption rate with grain size lead to a strong dependence of mantle thickness on grain size. Furthermore, we show that freeze-out is almost complete in the absence of an external UV field, even when photodesorption from CR produced UV is taken into consideration. Even at gas densities of $10^4$ ${rm cm^{-3}}$, less than 30% of the CO remains in the gas phase after $3times 10^5$ years for standard values of the CR ionization rate.
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