E. I. Vorobyov, V. G. Elbakyan, A. Skliarevskii, V. Akimkin, I. Kulikov

Aims. Dust enrichment and growth during the initial stages of protoplanetary disk formation were numerically investigated. A particular objective was to determine the effects of various growth barriers, which were mimicked by setting a series of upper permissible limits on maximum dust sizes. Methods. We used the ngFEOSAD code to simulate the three-dimensional dynamics of gas and dust in the polytropic approximation starting from the gravitational collapse of a slowly rotating Bonnor-Ebert sphere to $\approx 12$ kyr after the first hydrostatic core and disk formation. Results. We found that dust growth starts in the contracting cloud in the evolution stage that precedes disk formation and the disk begins to form in an environment that is already enriched in grown dust. The efficiency of dust growth in the disk is limited by dust growth barriers. For dust grains with maximum size < 100 $\mu$m these are likely electrostatic or bouncing barriers, and for larger grains the fragmentation and drift barriers play the major role. The disk midplane becomes quickly enriched with dust, while the vertically integrated distribution of dust shows notable local variations around the canonical 1:100 dust-to-gas mass ratio. These positive and negative deviations are likely caused by local hydrodynamic flows, since the globally integrated dust-to-gas ratio deviates negligibly from the initial 1:100 value. We note that care should be taken when considering models with a fixed dust size, as it may attain a profound negative radial gradient already in the very early stages of disk formation. Models with a constant Stokes number may be preferable in this context. Conclusions. The early dust enrichment and growth may facilitate planet formation as suggested by observations of protoplanetary disk substructures.