Supporting information for: On the energetics of cation ordering in tungsten-bronze-type oxides Gerhard Henning Olsen, Sverre Magnus Selbach, and Tor Grande ∗ Department of Materials Science and Engineering, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway E-mail: tor.grande@ntnu.no Derivation of thermodynamic equations Here we give a more detailed derivation of equation (3) in the main text. We start by explicitly evaluating the free energy expression: ΔG int = α x + β x 2 + RT ∑ s b s ∑ i x i,s ln x i,s , (S.1) where x is the degree of cation interchange, b s is the multiplicity of site (or sublattice) s, and x i,s is the fractional occupancy of species i (cations or vacancies) on site s. For the chemical equilibrium in equation (1) in the main text, we write out the entropy part as follows: ΔS int = −R ∑ s b s ∑ i x i,s ln x i,s = −R  2  1 + x 2 ln 1 + x 2 + 1 − x 2 ln 1 − x 2  + 4  4 − x 4 ln 4 − x 4 + x 4 ln x 4  (S.2) S1 Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics. This journal is © the Owner Societies 2015