First Principles Phase Diagram Calculation for the 2D TMD system $WS_{2}-WTe_{2}$

First principles phase diagram calculations, that included van der Waals interactions, were performed for the bulk transition metal dichalcogenide system $(1-X) \cdot WS_{2} - (X) \cdot WTe_{2}$. To obtain a converged phase diagram, a series of cluster expansion calculations were performed with increasing numbers of structure-energies, ($N_{Str}$) up to $N_{Str}=435$, used to fit the cluster expansion Hamiltonian. All calculated formation energies are positive and all ground-state analyses predict that formation energies for supercells with 16 or fewer anion sites are positive; but when $\approx 150 N_{Str} \leq 376$, false ordered ground-states are predicted. With $N_{Str} \geq 399$, only a miscibility gap is predicted, but one with dramatic asymmetry opposite to what one expects from size-effect considerations; i.e. the calculations predict more solubility on the small-ion S-rich side of the diagram and less on the large-ion Te-rich side. This occurs because S-rich low-energy metastable ordered configurations have lower energies than their Te-rich counterparts.