Mean charge  ⟨ Z ⟩  of spherical dust grains and the standard deviation as a function of the grain radius a _∗  (left panel). The upper abscissa relates a_∗ to a BCCA agglomerate of N monomers assuming m_d = const. The electron fraction x_e, the ion concentration x_i (with m ⁺  as the dominant ion) and the mean charge of the dust  ⟨ Z ⟩  per dust grain are shown in the right panel using solid, dashed, and dotted lines. The profiles shown are obtained at R = 10   AU and z/h_g = 0 applying the modified Oppenheimer-Dalgarno network for spherical grains. The lines correspond to the values obtained by numerical integration while the values marked by (blue) filled circles are obtained using the semi-analytical solution. To be compared with Fig. 5, which is associated with the corresponding modified Oppenheimer-Dalgarno model for BCCA agglomerates.