Growing and trapping pebbles with fragile collisions of particles in protoplanetary disks

[abridged] Recent laboratory experiments indicate that destructive collisions of icy dust particles occur with much lower velocities than previously thought. When these new velocities are considered from laboratory experiments in dust evolution models, a growth to pebble sizes in protoplanetary disks (PPDs) is difficult. This may contradict (sub-)mm observations and challenge the formation of planetesimals and planets. We investigate the conditions that are required in dust evolution models for growing and trapping pebbles in PPDs when the fragmentation speed is 1ms$^{-1}$ in the entire disk. We distinguish the parameters controlling the effects of turbulent velocities, vertical stirring, radial diffusion, and gas viscous evolution, always assuming that particles cannot diffuse faster (radially or vertically) than the gas. To form pebbles and produce effective particle trapping, the parameter that controls the particle turbulent velocities must be small ($\delta_t\lesssim10^{-4}$). In these cases, the vertical settling can limit the formation of pebbles, which also prevents particle trapping. Therefore the parameter that sets the vertical settling of the grains must be $\delta_z<10^{-3}$. Our results suggest that different combinations of the particle and gas diffusion parameters can lead to a large diversity of millimeter fluxes and dust-disk radii. When pebble formation occurs and trapping is efficient, gaps and rings have higher contrast at mm-emission than in the NIR. In the case of inefficient trapping, structures are also formed at the two wavelengths, producing deeper and wider gaps in the NIR. Our results highlight the importance of obtaining observational constraints of gas and particle diffusion parameters and the properties of gaps at short and long wavelengths to better understand basic features of PPDs and the origin of the structures that are observed in these objects.