Relative strength of phase stabilizers in titanium alloys

Titanium alloys exhibit three distinct crystal structures: $\alpha$, $\beta$, and $\omega$. For various applications alloying elements can be used to stabilize the desired phase. Extensive data exist to determine the thermodynamic equilibrium phase, typically phase coexistence. However, the normal state of commercial alloys is a quenched solid solution. While alloy designers have well-established rules of thumb, rigorous theory for nonequilibrium single-phase crystal stability is less well established. We develop a theory to predict which phase a particular alloy will adopt, as a function of minor element concentration. We use two different methods based on density functional theory with pseudopotentials and plane waves, with either explicit atoms or the virtual crystal approximation (VCA). The former is highly reliable, while the latter makes a number of drastic assumptions that typically lead to poor results. Surprisingly, the agreement between the methods is good, showing that the approximations in the VCA are not important in determining the phase stability and elastic properties. This allows us to generalize, showing that the single-phase stability can be related linearly to the number of $d$ electrons, independent of the actual alloying elements or details of their atomistic-level arrangement. This leads to a quantitative measure of $\beta$ stabilization for each alloying transition metal.

Figure 1

Phase stability of Ti-based alloys against number of valence electrons, calculated with the supercell method (dark symbols) and with VCA (light symbols). Open symbols denote the hcp phase and filled symbols denote the $\omega$ phase. Bcc is the reference. Alloying elements are coded on a color scale, with blue circles, Al; red squares, Sc; violet diamonds, V; orange up triangles, Cr; green left triangles, $\mathrm{Cr}+2\mathrm{V}$; magenta down triangles, Nb; brown right triangles, Mo. Pluses are charged hcp and crosses are charged $\omega$.

Figure 2

Phase diagram showing temperature against composition plotted as the number of valence electrons. Points show transition temperatures calculated with the supercell method (dark symbols) and with VCA (light symbols). Alloying elements are coded on a color scale as in Fig. 1. Approximate phase boundary lines are guides for the eye. The vertical dashed line is pure Ti, the reference. Binary diagrams can be deduced from this figure by considering only the data concerning a single alloying element and rescaling the $x$ axis according to its valence.

Figure 3

(Left) Density of electronic states for alloys in the different phases, calculated with pure Ti (solid black line), $\mathrm{Ti}+25\%$ Sc (dashed red line), $\mathrm{Ti}+33\%$ V (long-dashed violet line), $\mathrm{Ti}+33\%$ Nb (dot-dot-dashed magenta line), $\mathrm{Ti}+16\%$ Cr (dot-dashed orange line), $\mathrm{Ti}+16\%$ Mo (dash-dash-dotted brown line) and $\mathrm{Ti}+8\%$ $\mathrm{Cr}+16\%$ V (dot-long-dashed green line). The inset shows the DOS for $\mathrm{Ti}+16\%$ Al (dotted blue line), compared with pure Ti. (Right) The corresponding plots for VCA, calculated with $-$0.2$e$ charged Ti (dotted blue line), $\mathrm{Ti}+20\%$ Sc (dashed red line), pure Ti (solid black line), $+$0.4$e$ charged Ti (dot-dashed orange line), and $\mathrm{Ti}+40\%$ V, (long-dashed violet line).

Figure 4

Elastic stability of bcc Ti-based alloys against number of valence electrons, calculated using VCA. The straight line, derived from simply adding electrons to a Ti supercell, gives $C^{\prime }=-11.6+42\Delta n_d$.