# Ab Initio Study of Martensitic Transformation in NiTiPt High Temperature Shape Memory Alloys
## 3 Results and Discussion
### Energy Differences between the Austenite and Martensite  
After studying the phase transition path of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys (\(x\) = 0, 6.25, 8.33, 10.42, 12.5, 18.75, 25), we set out to research the concentration dependence of \(M_{\text{s}}\) of these alloys using the density functional theory. At first, we invoke the total energy differences, \(\Delta E\), between B2 and martensite (B19 or B19\({}^{\prime}\) ) phases. According to the previous studies [47; 58; 59; 60], there is a strong relationship between \(M_{\text{s}}\) and \(\Delta E\) from first-principles calculations, i.e., a larger \(\Delta E\) corresponds to a higher \(M_{\text{s}}\). All these energies are normalized per atom. They are calculated (per atom) as:  
\[\Delta E_{\text{B19}}=E_{\text{B2}}-E_{\text{B19}} \tag{9}\]  
\[\Delta E_{\text{B19}^{\prime}}=E_{\text{B2}}-E_{\text{B19}^{\prime}} \tag{10}\]  
with \(E_{\text{B2}}\), \(E_{\text{B19}}\), and \(E_{\text{B19}^{\prime}}\) being the total energies per atom of B2, B19, and B19\({}^{\prime}\) structures of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys, respectively. At first, the energy differences \(\Delta E_{\text{B19}}=E_{\text{B2}}-E_{\text{B19}}=34\) meV/atom and \(\Delta E_{\text{B19}^{\prime}}=E_{\text{B2}}-E_{\text{B19}^{\prime}}=44\) meV/atom for an equiatomic binary NiTi alloy are obtained, which is consistent with other theoretical values [48; 61; 64; 68]. The \(\Delta E_{\text{B19}}\) and \(\Delta E_{\text{B19}^{\prime}}\) are both positive, and the \(\Delta E_{\text{B19}^{\prime}}\) is a little higher than \(\Delta E_{\text{B19}}\) which revealed that the monoclinic B19\({}^{\prime}\) is more energetically stable than orthorhombic B19 structures. Then, the calculated concentration dependence of \(\Delta E_{\text{B19}}\) and \(\Delta E_{\text{B19}^{\prime}}\) for Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys (\(x\) = 0, 6.25, 8.33, 10.42, 12.5, 18.75, 25) are presented in Figure 6. Finally, in order to confirm the experimental observations of alloy composition dependence of \(M_{\text{s}}\) in Figure 7,we plot the total energy differences, \(\Delta E\), as well as \(M_{\rm s}\) as a function of the Pt concentration in the Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys. \(M_{\rm s}\) of Ni-Ti-Pt is cited from Ref. [35]. As presented in Figure 6, the \(\Delta E_{\rm B19}\) and \(\Delta E_{\rm B19^{\prime}}\) are both positive. There is a crossover at the \(\Delta E\) curve for Pt near 15 at.% concentration, which corresponds to that of \(E_{\rm f}\) in B19 and B19\({}^{\prime}\) (see Figure 4), most likely due to the marginal difference in the Ni-Ti bond length previously mentioned. For Pt \(<\) 15 at.%, the \(\Delta E_{\rm B19^{\prime}}\) is higher than \(\Delta E_{\rm B19}\), which indicates that the monoclinic B19\({}^{\prime}\) phase is more stable than orthorhombic B19 structures; whereas for Pt \(>\) 15 at.%, the \(\Delta E_{\rm B19}\) is higher than \(\Delta E_{\rm B19^{\prime}}\), which suggests orthorhombic B19 structures are more stable than the monoclinic B19\({}^{\prime}\) phase. These are in line with the present computed results of \(E_{\rm f}\) of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys (\(x\) = 0, 6.25, 8.33, 10.42, 12.5, 18.75, 25) (see Figure 4). Figure 6: The energy differences between austenite and martensite (\(\Delta E\)) as a function of the doping concentration in Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys. Figure 7: Compositional dependence of \(\Delta E\) and phase transformation start temperature (\(M_{\rm s}\)) of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys. The data of \(M_{\rm s}\) refer to the right axis and those of \(\Delta E\) to the left axis. \(M_{\rm s}\) is cited from Ref. [35]. According to the data of \(\Delta E\) of the most stable martensite phase, \(\Delta E\) and \(M_{\rm s}\) of Ti\({}_{50}\)Ni\({}_{50-x}\)Pt\({}_{x}\) alloys are presented in Figure 7 as a function of Pt content.