Extremely large deuteration of several molecules has been observed towards prestellar cores and low-mass protostars for a decade. New observations performed towards low-mass protostars suggest that water presents a lower deuteration in the warm inner gas than in the cold external envelope. We coupled a gas-grain astrochemical model with a one-dimension model of collapsing core to properly follow the formation and the deuteration of interstellar ices as well as their subsequent evaporation in the low-mass protostellar envelopes with the aim of interpreting the spatial and temporal evolutions of their deuteration. The astrochemical model follows the formation and the evaporation of ices with a multilayer approach and also includes a state-of-the-art deuterated chemical network by taking the spin states of H$_2$ and light ions into account. Because of their slow formation, interstellar ices are chemically heterogeneous and show an increase of their deuterium fractionation towards the surface. The differentiation of the deuteration in ices induces an evolution of the deuteration within protostellar envelopes. The warm inner region is poorly deuterated because it includes the whole molecular content of ices while the deuteration predicted in the cold external envelope scales with the highly deuterated surface of ices. We are able to reproduce the observed evolution of water deuteration within protostellar envelopes but we are still unable to predict the super-high deuteration observed for formaldehyde and methanol. Finally, the extension of this study to the deuteration of complex organics (COMs), important for the prebiotic chemistry, shows a good agreement with the observations, suggesting that we can use the deuteration to retrace their mechanisms and their moments of formation.
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