Polydispersity Exponent in Homogeneous Droplet Growth

The homogeneous growth of $D$-dimensional hyperspherical droplets on a $d$-dimensional surface produces a bimodal distribution in the sizes of the droplets, with a roughly monodispersed distribution of larger droplets superimposed on a highly polydisperse distribution of smaller ones. The polydisperse regime is characterized by an exponent, which also determines the total droplet density. The exponent is evaluated, and it is deduced that $d=3\mathrm{}$ represents an upper critical dimensionality.