  Multi-wavelength observations of protoplanetary disks in the sub-millimeter
continuum have measured spectral indices values which are significantly lower
than what is found in the diffuse interstellar medium. Under the assumption
that mm-wave emission of disks is mostly optically thin, these data have been
generally interpreted as evidence for the presence of mm/cm-sized pebbles in
the disk outer regions. In this work we investigate the effect of possible
local optically thick regions on the mm-wave emission of protoplanetary disks
without mm/cm-sized grains. A significant local increase of the optical depth
in the disk can be caused by the concentration of solid particles, as predicted
to result from a variety of proposed physical mechanisms. We calculate the
filling factors and implied overdensities these optically thick regions would
need to significantly affect the millimeter fluxes of disks, and we discuss
their plausibility. We find that optically thick regions characterized by
relatively small filling factors can reproduce the mm-data of young disks
without requesting emission from mm/cm-sized pebbles. However, these optically
thick regions require dust overdensities much larger than what predicted by any
of the physical processes proposed in the literature to drive the concentration
of solids. We find that only for the most massive disks it is possible and
plausible to imagine that the presence of optically thick regions in the disk
is responsible for the low measured values of the mm spectral index. For the
majority of the disk population, optically thin emission from a population of
large mm-sized grains remains the most plausible explanation. The results of
this analysis further strengthen the scenario for which the measured low
spectral indices of protoplanetary disks at mm wavelengths are due to the
presence of large mm/cm-sized pebbles in the disk outer regions.
