The phase diagrams and equilibrium 2D-morphologies of Co(Ni)MoS nano-crystallites with various promoter contents are determined by periodic DFT calculations coupled to a thermodynamic model. The edge energies of the crystallite depend on two key parameters: the chemical potential of sulfur (Delta mu s), representing the sulfo-reductive conditions of the gas phase, and the number and distribution of promoter atoms at the edges. High values of Delta mu s, corresponding to strongly sulfiding conditions, stabilize edges with a full promoter content. At very low values of Delta mu s, i.e. strongly reductive environment, the mixed Co(Ni)MoS active phase is destabilized with respect to MoS2 due to the total segregation of promoter atoms to the edges. For intermediate Delta mu s, including industrial HDS conditions, it is found that the M-edge is partially decorated by Co and Ni, and that the S-edge is partially decorated by Ni. The existence of mixed Co-Mo or Ni-Mo edge sites is thus thermodynamically possible. The Gibbs-Curie-Wulff equilibrium morphologies depend on the type of promoter. These quantitative DFT results are consistent with Le Chatelier's principle and are crucial to understand the thermodynamic balance between the active phase and the reaction conditions impacting the nature of active edge sites. (C) 2007 Elsevier B.V. All rights reserved.