Lattice parameter dependence on long-range ordered degree during order–disorder transformation

The order–disorder transformations of stoichiometrical Ni 3Al, NiAl, Cu 3Au, and TiAl are studied by using the Bragg–Williams model based on embedded atom method potentials. The calculated relationship between the lattice parameter and the long-range order (LRO) parameter should be linear for the completely first order transition such as Ni 3Al, non-linear for the second order transition such as NiAl, and a mixture of non-linear (in the range of 1> σ≥ σ 0) and linear (in σ 0> σ≥0) for common first order transition such as Cu 3Au. The calculated linear dependence of lattice parameter on LRO for Ni 3Al is consistent with the existing experimental results. The driving force of the lattice parameter change is the free energy change due to changing of LRO parameter. The relationship between the lattice parameter of a stable ordered structure with a given LRO parameter and its free energy is linear. The alternate distribution of pure Ti and pure Al atomic planes in TiAl leads to strong directional d-bonding between nearest neighbour Ti atoms in the pure Ti plane and the polarization of p-electrons at Al sites aligning with the [001] direction, resulting in an enhancement of the Ti–Al bonding. When LRO=0 the structure becomes cubic fcc, and the directional bonding is eliminated. As LRO increases, the lattice parameter a and the volume of the unit cell are reduced while the lattice parameter c is increased. The calculated change of lattice parameter proportionally corresponds to the change of the free energy of the ordered phase for all four alloys.