Radial profiles of the dust grain size at the peak of the size distribution, a_peak (see the main text for further details) and the grain-grain differential velocity of grains with size a_peak from the turbulence models (see Appendix A.1), noted δv_apeak, for the two snapshots shown in Fig. 1: models ℳ₀ and ℳ₁ at t_0.4M⊙. Left panels: Radial profiles built by computing the mean a_peak in concentric shells from the a_peak map (for further details, see Fig. G.1 and the text). The mean dust grain size clearly increases with decreasing envelope radius, up to ~10 μm. The power law in model ℳ₁ is higher than for model ℳ₀. This shows that the grains grow faster in model ℳ₁ since the gas turbulence tends to slow the collapse, leaving more time for grains to grow. Right panels: Dust grain populations (color-coded by a_peak) at all envelope radii, plotted against the δv_apeak they experience at this envelope location (in individual cells in the model). A vertical gray line represents the 100 au radius, approximating the ice line expected for such solar-type protostars. On each side of this line, orange and blue shaded areas represent the locations where the dust grains are expected to fragment (see text for further details).