# Ab Initio Study of Martensitic Transformation in NiTiPt High Temperature Shape Memory Alloys
## 2 Model and Methodology  
NiTi B2, orthorhombic B19, and monoclinic B19\({}^{\prime}\) structures taken from our previous work [47] are adopted in the present study. Note that we employ the parent NiTi body-centered tetragonal (bct) lattice. The structural parameters of bct are \(a=a_{0}\), \(b=c=\sqrt{2}a_{0}\), here \(a_{0}\) is the lattice constants of cubic B2 NiTi structure. In addition, we focus on the B19 and B19\({}^{\prime}\) structures for the martensite phase reported in the literature [3; 5; 48]. The structures and parameters of B19 and B19\({}^{\prime}\) are very close to each other, except the monoclinic distortion from 90\({}^{\circ}\) to \(\beta=98.26^{\circ}\). Considering the symmetry of the cell, 3 \(\times\) 2 \(\times\) 2 supercells of the B2, B19, and B19\({}^{\prime}\) are constructed containing 48 atoms for the present simulation. The different site occupations of Ni atoms in the austenite B2 3 \(\times\) 2 \(\times\) 2 supercell Ni\({}_{24}\)Ti\({}_{24}\) structure, shown in Figure 1, are indicated by smaller blue spheres and labeled with numbers. A survey of the literature [9] reveals that the addition of Pt prefers to occupy the Ni rather than Ti sites in TiNiPt alloys. In this study, replacing nickel with platinum, we adopt the Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys (\(x\) = 0, 6.25, 8.33, 10.42, 12.5, 18.75, 25) alloys as our models. Geometry optimizations were implemented for all possible doping structures. The most stable structures were calculated with the principle of the lowest energy. Table 1 presents the site occupation of Pt in the parent phase of Ti-rich Ni-Ti-Pt alloys. Figure 1: Ni occupancy in the austenite B2 3 \(\times\) 2 \(\times\) 2 supercell Ni\({}_{24}\)Ti\({}_{24}\) structure. Smaller blue spheres denote Ni atoms; larger gray spheres denote Ti atoms. The 3 \(\times\) 2 \(\times\) 2 supercells of B19 and B19\({}^{\prime}\) are also used, and structures are similar to that of B2. The present calculations are performed within the framework of DFT as implemented in the Vienna ab initio simulation package [49; 50; 51] (VASP) code, applying the generalized gradient approximation [52] (GGA) and a projected augmented wave [53] (PAW) basis. We chose a 500 eV plane-wave energy cutoff and the 4 \(\times\) 4 \(\times\) 4 \(k\)-points [54] for the 3 \(\times\) 2 \(\times\) 2 supercells of the B2, B19, and B19\({}^{\prime}\) phases in TiNiPt alloys. In order to analyze the phase stabilities of B2, B19, and B19\({}^{\prime}\) of Ni-Ti-Pt alloys with the increasing of the Pt content, we evaluate the energies of formation, \(E_{\rm f}\)[55; 56; 57]. Formation energies per atom of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys are calculated as:  
\[E_{\rm f}=\frac{E_{\rm tot}(\rm Ti_{50}Ni_{50-x}Pt_{x})-50E_{\rm Ti}-(50-x)E_{ \rm Ni}-xE_{\rm Pt}}{100} \tag{1}\]  
where \(E_{\rm tot}(\rm Ti_{50}Ni_{50-x}Pt_{x})\) is the total energy of TiNiPt alloys. \(E_{\rm Ni}\), \(E_{\rm Ti}\), and \(E_{\rm Pt}\) denote the energies per atom of the pure Ni, Ti, and Pt in their bulk states, respectively. The transformation behaviors as a function of the doping concentration can be described by the total energy differences \(\Delta E^{\rm A-M}\) between the austenite and martensite phases. According to the laws of thermodynamics, the Gibbs free energy \(G\) are defined as:  
\[G=U-TS+PV \tag{2}\]  
where \(U\), \(T\), \(S\), \(P\), and \(V\) are the internal energy, temperature, entropy, pressure, and volume, respectively.