Local lattice strain around alloying element and martensitic transformation in titanium alloys

Local strain is introduced into the lattice around solute atom due to the size mismatch between solute and solvent atoms in alloy. In this study, local lattice strains are calculated for the first time in titanium alloys, using the plane-wave pseudopotential method. As an extreme case, the local lattice strain around a vacancy is also calculated in various bcc, fcc and hcp metals. It is found that the local strain energy is very high in both bcc Ti and bcc Fe, where the martensitic transformation takes place. From a series of calculations, it is shown that the magnitude of the strain energy stored in the local lattice is comparable to the thermal energy, kBT, where kB is the Boltzmann constant and T is the absolute temperature. Therefore, the presence of local lattice strains in alloy could influence the phase stability that varies largely depending on temperatures. For example, the local lattice strain correlates with the martensitic transformation start temperature, Ms, in binary titanium alloys.


Introduction
As illustrated in Fig.1 (a) and (b), it is well known that local strains are introduced into the lattice around solute atom due to the size difference between solute and solvent atoms in alloy.
However, there are very few investigations to treat of this local strain problem in a quantitative way. In an extreme case, as shown in Fig.1 (c), a vacancy also introduces the local strain into the crystal lattice, but the information is very limited, too. Recently, local lattice strains have been calculated around a variety of alloying elements in hcp Mg [1,2].
In the present study, local lattice strains are calculated for various alloying elements, M, in both hcp Ti and bcc Ti, using the plane-wave pseudopotential method. For comparison, such local lattice strains are also evaluated around a vacancy in typical bcc, fcc and hcp metals.

Calculation Procedure
Using the plane-wave pseudopotential method (CASTEP code), the electronic structures are calculated with the supercells shown in Fig.2 (a) for hcp, (b) for bcc and (c) for fcc metals. Supercells are made by stacking each unit cell by (a) (3x3x2), (b) (3x3x3) and (c) (2x2x2) along the crystal axes. An alloying element, M, is substituted for a mother metal as shown in Fig.2. Also, a vacancy is located at the M position in any metal. The plane-wave cutoff energy is chosen to be 380 eV for all the calculations of bcc Ti and of the vacancy, and 350 eV for the calculation of hcp Ti, where 400~480 eV is used for rare earth elements, M (e.g., 480 eV for M=Eu).
In every calculation, the lattice parameter is first optimized for pure metal, and then the positions of the first-or the second-nearest-neighbour mother metals from M (or vacancy) are relaxed under the periodic boundary condition, while keeping the optimized lattice parameter constant. A detailed explanation of the calculation method is given elsewhere [1]. The calculated local strains around M agree reasonably with the experiments [1].

Local lattice strain around a vacancy in pure metals
The calculated results of the local strain energy around a vacancy are shown in Fig.3 (a) bcc, (b) fcc and (c) hcp metals. Here, the local strain energy is obtained by taking the total energy difference before and after the lattice relaxation around a vacancy. It is evident from Fig.3 that the strain energy is one order larger in bcc metals than in the close-packed fcc and hcp metals. Among the bcc metals, in particular, it is large in bcc Fe and bcc Ti, where the martensitic transformation takes place.
In case of bcc Fe, the spin-polarized calculation is also performed, and its strain energy denoted by symbol (○) is lower than that of the non-spin-polarized calculation denoted by symbol (•). This implies that the formation of ferromagnetic iron martensite (α') phase is assisted by the onset of the ferromagnetsism around 1043 K in the course of quenching of the high-temperature austenitic γ phase.
The actual local lattice strain around a vacancy is illustrated in Fig.4 (a) bcc Ti, (b) fcc Ni and (c) hcp Ti. In bcc Ti, the first-nearest-neighbour Ti (1) atoms from a central vacancy are displaced toward a vacancy along the <111> direction. The local strain along the <111> direction is defined as√3Δa 3 , where Δa 3 is the strain along the crystal axis. Then, √3Δa 3 is divided by the lattice parameter, a , and √3Δa 3 /a (=-6.30%) is obtained. The second-nearest-neighbour Ti (2) atoms are displaced apart from a vacancy along the <100> direction. The local strain along the crystal axis is This is a simple arithmetic average by weighting the coordination numbers around a vacancy. In case of bcc Ti, the average strain is about -3.16 %, so that the local lattice is shrunk around a vacancy in bcc Ti.
For fcc Ni shown in (b), the 12 first-nearest-neighbour Ni atoms are displaced toward a vacancy along the <110> direction. The local strain along the <110> direction is defined as √2Δa 5 and the value of √2Δa 5 /a is about -0.98%, indicating that the local lattice is shrunk slightly around a vacancy in fcc Ni. (Δa/a + Δc/c) =-0.82 %. Thus, the local lattice is shrunk around a vacancy in hcp Ti. This negative strain value in hcp Ti is, however, much smaller than -3.16 % in bcc Ti, as might be expected from the lower strain energy in hcp Ti as shown in Fig.3 (c). The existence of these local strains around a vacancy will influence the self-diffusion in metals.

Local strain around alloying element in hcp Ti
The local lattice strains around M=V are shown in Fig.5 (a). Here, the notation used is the same as in Fig.4(c). The total strain is (Δa/a +Δc/c)=-1.82 % around V and it is more negative than the value of -0.82% around a vacancy. This is simply interpreted as due to the strong interaction operating between V and Ti atoms so as to shorten the V-Ti bond distance. A similar trend is also seen in other 3d transition metals (e.g., M=Fe) as shown in Fig.5 (b). But, the local lattice is expanded to some extent around rare earth elements, (e.g., M=Ce).

Local strain around alloying element in bcc Ti
The local strain around M=Fe is illustrated in Fig.7 (a). The first-nearest-neighbor Ti (1) atoms are displaced toward a central Fe atom along the <111> direction. But the second-nearest-neighbor Ti (2) atoms are displaced apart from a central Fe atom along the <100> direction so as to keep the Ti (1) -Ti (2) distance close to the first-nearest-neighbor distance of pure Ti. As shown in Fig.7 (b), as to the strain directions of Ti (1) and Ti (2) atoms, the other M is the same as Fe except for Zr and Hf, which appear to behave like larger elements than Ti.