Size-dependent charging of dust particles in protoplanetary disks. Can turbulence cause charge separation and lightning?

Protoplanetary disk are the foundation of planet formation. Lightning can have a profound impact on the chemistry of planetary atmospheres. The emergence of lightning in a similar manner in protoplanetary disks, would substantially alter the chemistry of protoplanetary disks. We aim to study under which conditions lightning could emerge within protoplanetary disks. We employ the ProDiMo code to make 2D thermo-chemical models of protoplanetary disks. We included a new way of how the code handles dust grains, which allows the consideration of dust grains of different sizes. We investigate the chemical composition, dust charging behaviour and charge balance of these models, to determine which regions could be most sufficient for lightning. We identify 6 regions within the disks where the charge balance is dominated by different radiation processes and find that the emergence of lightning is most probable in the lower and warmer regions of the midplane. This is due to the low electron abundance ($n_{\rm e}/n_{\rm\langle H \rangle}<10^{-15}$) in these regions and dust grains being the most abundant negative charge carriers ($ n_{\rm Z}/n_{\rm\langle H \rangle}>10^{-13}$). We find that $\rm NH4^+$ is the most abundant positive charge carrier in those regions at the same abundances as the dust grains. We then develop a method of inducing electric fields via turbulence within this mix of dust grains and $\rm NH_4^+$. The electric fields generated with this mechanism are however several orders of magnitude weaker than required to overcome the critical electric field.