Composition- and temperature-dependence of β to ω phase transformation in Ti-Nb alloys

Phase diagram and Gibbs energy

The system considered is a binary Ti-Nb single crystal. The calculated Ti-Nb phase diagram has shown a ω + β two-phase region, as shown in Figure 1. The bulk chemical energies of homogeneous β and ω phases at different temperatures are obtained from the literature^[27] and can be formulated as follows:

Figure 1. Thermodynamic results of Ti-Nb binary system. (A) Calculated equilibrium phase diagram (blue solid lines) and metastable phase diagram considering the β and ω phases (red solid line), where the spinodal decomposition occurs within the regions below the chemical spinodal. The T₀ curve for metastable β and ω equilibrium is also shown by the black shot dash line; the Ms line for β → α" is based on^[4]; (B) Gibbs energy curves of the metastable β and ω phases at different temperatures; (C) The second derivatives of the Gibbs energy of the β phase at 573 K and 673 K, respectively, which define the spinodal regions (24.0 at.% < c < 76.0 at.%) at 573 K and (31.0 at.% < $$ \bar{c} $$ < 69.0 at.%) at 673 K; (D) The critical concentration value, c₀(T), obtained from the intersection point of the β and ω Gibbs energy curves at different temperatures (c₀ = 26.7 at.% and 17.7 at.% at 573 K and 673 K, respectively); (E) The chemical free energy surface f_ch(c,η) of the Ti-Nb system in both the composition and structural order parameter space at 673 K, showing G and G at η = 0 and η = 1, respectively; (F) Enlarged chemical free energy surface at η =0, showing spinodal decomposition of the β matrix.

where ϕ represents the β or ω phase, c is the mole fractions of Nb, R is the gas constant, and T is the absolute temperature. $$ { }^{0} G_{N b}^{\phi} $$ and $$ { }^{0} G_{T i}^{\phi} $$ are the Gibbs energies of pure elements Nb and Ti in the ϕ state based on the SGTE database. I^ϕ is the regular solution parameter having I^β = 13,075 J/mol and I = 2,510 J/mol. For numerical calculation in the following phase field simulations, the Gibbs energies of the metastable β and ω phases can be fitted as a function of Nb concentration and temperature:

where a_i and b_i are polynomial coefficients [Table 1].

Phase field model of β → ω transformation in Ti-Nb

The orientation relationship (OR) between the β and ω phases derived from the selected area electron diffraction patterns (SADP) is^[1], {111}_β||{0001}_ω, <110>_β||<1120>_ω, and <211>_β|| <0110>_ω, as shown in Figure 2. The OR is accomplished by a combination of a {211}<111>_β huffle and a {211}<111>_β shear and is the same in the athermal and isothermal ω^[21]. There are four possible <111>_β planes parallel to the {0001}_ω, so there exist four crystallographic variants of the ω. The β ↔ ω transformation is thought to be a shuffle-dominated process^[28] since it does not change the specific volume and has small temperature hysteresis; thus, the non-conserved structural order parameter is the shuffle degree η. A set of four structural order parameters η_p (p = 1,2,3,4) is used to describe the four variants of the ω phase, η_p = 0 represents the parent phase β, 0 < η_p < 1 represents the partially collapsed ω, and η_p = 1 represents the fully collapsed ω.

Figure 2. Schematic representations of the crystallographic relationship between the β and ω phases in their relevant crystallographic directions.

Therefore, the ω + β two-phase microstructure is characterized by two sets of order parameters, i.e., Nb concentration (mole fraction) c and shuffle degree η, and the total free energy of the system is formulated as:

where f_ch, f_gr, and E_el represents the chemical free energy density, the gradient energy density, and the elastic energy, respectively, and d³r means the integral of the chemical free energy density and gradient energy density of the computational cells. The chemical free energy surface f_ch(c, η) in total free energy F has been adopted by using an interpolation function h(η)^[29]:

where h(ƞ_p) = ƞ_p³(6ƞ_p² - 15ƞ_p + 10) satisfies that h(0) = 0, h(1) = 1, and dh/dη = 0 at η = 0 and 1. As shown in Figure 1E and F, η = 0 corresponds to the Gibbs energy curve of the metastable β phase (Nb-lean β₁ + Nb-rich β₂),G_β, and η = 1 corresponds to the Gibbs energy curve of the ω phase, G_ω. The ϕ₁ represents the energy barrier between two different ω variants along the minimum energy path on the local chemical-free energy landscape, and ϕ₂ characterizes the energy barrier between the β and ω phases. The gradient energy density is^[30]:

where k_c and k_η are the gradient coefficients of concentration and structure fields, respectively. The elastic energy is calculated via the microelasticity theory of Khachatuyan under the homogeneous modulus assumption (i.e., the β and ω phases are assumed to have the same elastic modulus)^[31]:

where C_ijkl is the elastic constants of the phase C₁₁ = 128.5 GPa, C₁₂ = 115.5 GPa, and C₄₄ = 14.9 GPa^[32]; $$ \Omega_{i j}^{-1} $$(n) = C_ijkln_kn_l, and n is a unit vector in the reciprocal space. $$ \sigma_{i j}^{\omega} $$(p) = C_ijklε_kl(p) and ε_ij(p) denotes the stress-free transformation strain (SFTS) of the pth variant for the β → ω transformation; then, the strain caused by β → ω transformation can be described in terms of the order parameter field, i.e., $$ \varepsilon_{i j}=\sum_{p=1}^{4} \varepsilon_{i j}^{\omega}(p) \eta_{p}^{2} $$. The four SFTS tensors are calculated using the lattice constants α_β = 3.273 Å for the β phase and α_ω = 4.659 Å, c_ω = 2.819 Å for the ω phase in Ti-22 at.% Nb^[32], and the Burgers correspondence between the β and ω phases [Figure 2], as shown in Table 2. In the reference coordinate N1, i.e., $$ x \|[2 \overline{11}0]_{\omega} $$, $$ y \|[\overline{1} 2 \overline{1}0]_{\omega} $$, and z||[0001]_ω in the basis of the ω phase, the transformation matrix that describes a homogeneous lattice deformation from β to ω is:

In the reference coordinate N2, i.e., x||[100]_β, y||[010]_β, and z||[001]_β in the β phase, the SFTS of the four variants can be calculated by $$ \varepsilon^{\omega}(i)=T(i) \varepsilon_{N 1}^{\omega} T(i)^{-1} $$, where i = 1, 2, 3, and 4, and T(i) is the rotation matrix mapping between N1 and N2 coordinates for the four variants:

Considering the 2D simulation is on the {011}_β plane (i.e., a typical zone axis SADP to observe ω structure)^[33-35], the final results of SFTSs corresponding to the (011)_β plane are:

Note that the SFTS for the β → ω transformation used in our simulation is assumed to be independent of concentration.

The stochastic Cahn-Hillardand time-dependent Ginzburg-Landau equations^[30,36] are used to govern the equations of the concentration and structural order parameters, respectively.

where $$ M=\frac{1}{R T} \exp \left(\frac{\phi}{R T}\right) $$ is the chemical mobility and derived from the interdiffusion coefficients, ϕ = (1-c)ϕ^Ti + cϕ^Nb + c(1-c)ϕ^Ti,Nb is activation enthalpy, ϕ^Ti and ϕ^Nb are the mobility parameters for pure Ti and Nb, respectively, ϕ^Ti,Nb is binary interaction parameter in the Ti-Nb binary system^[37], and L = 8.82 × 10^-11 m³·J^-1·s^-1[38] is the structural mobility of the structural order parameter. ζ_c and ζ_η are the Langevin stochastic noise terms characterizing thermal fluctuations and are used to describe the nucleation process through a Gaussian-distributed random number generator ρ_i^[39]:

where l₀ is the grid size, d is the dimensionality of the space, Δt is the time step, and ρ is a d-dimensional vector.

For nondimensionalization, the energy normalization factor E_normal is 40 KJ·mol^-1, the dimensionless energy barriers are Ф₁^* = 0.06 and Ф₂^* = 0.01^[40], and the dimensionless gradient energy coefficients are chosen to be k_c^* = 6.0 and k_η^* = 0.025^[41], guaranteeing a diffused interface profile and yielding dimensionless interfacial energy between the β and ω phases, γ^* = 0.095; thus, the numerical grid size is determined to be l₀ ~ 1 nm based on the relation $$ l_{0}=\frac{\gamma}{\gamma^{*} \cdot\left(E_{\text {normal }} / V_{m}\right)} $$^[41], where the interfacial energy γ of (111)/(0001) is ~380 mJ·m^-2[20], and the molar volume V_m = 1 × 10^-5 m³·mol^-1 is assumed for the system. The dimensionless structural mobility parameter is L^* = 1.0 to ensure a diffusion-controlled process^[41]. Except for ζ_c = 0 upon aging at 573 K in Ti-26 at.% Nb, the amplitude of dimensionless noise ζ_c^* is [-0.00025, 0.00025] to produce a minor concentration variation relative to the initial Nb content of the β matrix. The amplitude of dimensionless noise ζ_η^* is [-0.0022, 0.0022], which is determined via the numbers of the athermal nuclei at transformation temperature (ω_s) in Ti-26 at.% Nb upon water quenching, i.e., the athermal ω nuclei uniformly distribute in the computational cells, and the ζ_η^* is the same value upon aging. The system size used in the simulations is 512l₀ × 512l₀. Periodic boundary conditions are applied along all three dimensions. All the dimensionless parameters used in the simulations are listed in Table 3.

Gibbs energy curves of ω and β phases

Figure 1B shows the Gibbs energy curves of the ω and β phases as functions of Nb concentration and temperature. A metastable β miscibility gap can be found in the Ti-Nb binary system with a critical temperature of ~786 K [Figure 1A], and the spinodal regions are 24.0 at.% < $$ \bar{c} $$ < 76.0 at.% at 573 K and 31.0 at.% < c < 69.0 at.% at 673 K [Figure 1C]. The critical concentration c₀(T) at the intersection point of the ω and β Gibbs energy curves shows a parabolic increase with decreasing temperature, as shown in Figure 1D. When the alloy composition is less than c₀(T), congruent β → ω transformation will take place, followed by the rejection of Nb from the phase. When the alloy composition is slightly larger than c₀(T) with a Δc = c - c₀ ≤ 2 at.%^[40], peudospinodal decomposition could occur, which is defined as a phase transformation mechanism with continuous composition changes of non-isostructural phases. When the alloy composition is larger than c₀(T) with a Δc > 2 at.% and outside the spinodal regions at the two temperatures considered, only the conventional nucleation-and-growth mechanism can be activated for the β → ω transformation.

Microstructural evolution in Ti-26 at.% Nb

Formation of athermal ω

2D simulations of athermal ω transformation in Ti-26 at.% Nb are first carried out upon water quenching, and the results are shown in Figure 3. The water quenching process is from 1,173 K to 300 K. To show the detailed evolution of the ω, we zoom into the initial microstructures according to Figure 3A and show the enlarged images in Figure 3B. Figure 3B1-B4 shows the 2D simulation results of concentration fields, while Figure 3B5-B8 shows the corresponding structural order parameter fields. Athermal ω nanodomains with ellipsoidal shapes are formed via diffusionless structural transformation upon cooling since the concentration fields keep the initial Nb content. The number and average size of ω precipitates increase with decreasing temperature, and the final average size of ω nanodomains is less than 5 nm at 300 K. Thermodynamically, a spinodal region of T < 604 K exists at Ti-26 at.% Nb [Figure 1A], but the absence of the spinodal decomposition indicates the lack of driving force to defeat the gradient energy or dynamic difficulty for interdiffusion upon water quenching in Ti-26 at.% Nb. Moreover, athermal ω nanodomains have been found in Ti-19 at.% V without solute partitioning or composition fluctuations after water quenching from 1,173 K to room temperature via transmission electron microscope (TEM) and APT observations^[42], although a larger spinodal region of T < 969 K theoretically exists in the Ti-19 at.% V^[27]. Based on the above analysis, we believe that the smaller driving force is the major cause for the absence of phase separation in the alloy system with a lower Nb content.

Figure 3. 2D simulations of athermal ω transformation in Ti-26 at.% Nb upon water quenching. (A) Schematics of zooming into the microstructure; (B) Enlarged images showing microstructures of athermal ω nanodomains in the β matrix, as indicated in (A), where (B1-B4) show the concentration fields, and (B5-B8) show the structural order parameter fields. The four ω variants are presented in different colors, and the β matrix is represented in blue; (C) Change in volume fraction of athermal ω upon cooling; (D) Size distribution of athermal ω nanodomains; (E) Variation of the average shuffle degree of athermal ω nanodomains.

The volume fraction of the athermal nanodomains is shown in Figure 3C and presents that the athermal ω phase in Ti-26 at.% Nb increases gradually upon cooling. In the diffusionless β → athermal ω transformation, the actual transformation temperature ω_s will be below the T₀ [Figure 1A] because supercooling is needed to provide the additional driving force for overcoming interfacial and elastic energies caused by the structural transformation of a new phase. In our current model, this condition has been considered by introducing gradient and elastic energies. The simulated start transition temperature ω_s of Ti-26 at.% Nb is ~375 K, which has a good agreement with previous experimental studies estimated by the electrical resistivity curve^[17]. Figure 3D shows the related statistical distribution of the ω size and indicates that the size distribution is relatively uniform for the homogeneous ω nanodomains. The evolution of the average shuffle degree of ω upon cooling is shown in Figure 3E and displays that the athermal ω has a partially collapsed structure, and the shuffle degree increases with decreasing temperature and could help us to explain that the intensities of ω diffraction spots become higher at a lower temperature for Ti-Nb based alloys from [011]_β SADP^[8].

Formation of isothermal ω

According to the Gibbs energy curves in Figure 1B, c₀ (= 26.7 at.%) is higher than the alloy concentration; thus, the Gibbs energy of the ω phase is lower than the β phase at 573 K. As a result, a spontaneous nucleation process of isothermal ω can be found at 573 K via congruent transformation followed by Nb depletion from ω particles to β matrix, and four ω variants nucleate randomly with an equal probability at the beginning. Figure 4 shows the microstructural evolution of particles during isothermal aging at 573 K in Ti-26 at.% Nb alloy, where Figure 4A shows the concentration evolution during aging at 573 K, Figure 4B shows the corresponding one-dimensional (1D) concentration waves along the white short dash lines of the computational cell, and Figure 4C shows the morphological evolution during aging at 573 K. Ellipsoidal ω precipitates with a finer size have formed after a short time aging of 0.2 h, but the corresponding Nb content of the isothermal ω is same as the β matrix. Such rapid congruent transformation from the β matrix has also been found in α precipitation for alloy systems with lower solute content during aging^[40,43]. With increasing aging time, the number and size of isothermal particles increase; meanwhile, the Nb content of ω gradually decreases. The size distribution of ω particles is found to be time-dependent, as shown in Figure 4D, and gradually becomes wider with increasing aging time. The larger ω particles with a small volume fraction can grow up to > 40 nm, while the smaller ω nuclei with a size < 10 nm can also coexist at a longer aging time. Moreover, both the number and average size of isothermal ω formed at 573 K for 1 h are larger than that of athermal ω nanodomains [Figure 3], which is consistent with a previous experimental study in Ti-26 at.% Nb^[5].

Figure 4. Calculated microstructural evolution of isothermal ω aged at 573 K for different aging times in Ti-26 at.% Nb. (A) Concentration fields; (B) One-dimensional Nb concentration profiles along the white short dash line of the computational cells shown in (A); (C) Structural order parameter fields; (D) Size distribution of isothermal particles.

But at 673 K, the formation of isothermal ω in Ti-26 at.% Nb is activated by conventional nucleation-and-growth mechanism, which microstructural evolution is shown in Figure 5. The alloy composition of Ti-26 at.% Nb is much larger than c₀ (= 17.6 at.%) and outside the spinodal region at 673 K [Figure 1]; thus, the congruent transformation and other mechanisms assisted by spinodal decomposition cannot be accomplished. Redistribution of Nb via long-range diffusion is needed for the nucleation of isothermal particles, and the Nb content of initial ω nuclei has been found near the equilibrium composition of ω (c_ω = 10.2 at.%), which is a typical characteristic for a conventional nucleation and growth process^[40]. The nucleation density of the isothermal ω particles formed at 673 K is visibly low than that formed at 573 K [Figure 4], and the particle density also decreases with increasing aging time with a corresponding larger size at 673 K. The phenomena are universal for decreased density and coarsening at higher aging temperatures and longer aging time and have also been found in other precipitation such as Ni₄Ti₃^[44] and α^[45]. Moreover, previous literature has observed high-density finer ω precipitates after aging at 673 K for 1 h in Ti-26 at.% Nb^[5]. But in our current simulations, the isothermal ω is not found during aging at 673 K within 2.5 h, and we deduce that the reported ω particles may be athermal ω nanodomains formed upon cooling from 673 K to room temperature after aging at 673 K for 1 h. The simulations and reasoning can help us to understand that^[5] the average size of particles formed after aging at 673 K for 1 h is smaller than that formed after aging at 573 K for 1 h.

Figure 5. Calculated microstructural evolution of isothermal ω aged at 673 K for different aging times in Ti-26 at.% Nb. (A) Concentration fields; (B) One-dimensional Nb concentration profiles along the white short dash line of the computational cells shown in (A); (C) Structural order parameter fields; (D) Size distribution of isothermal ω particles.

Microstructural evolution in Ti-50 at.% Nb

Formation of athermal ω

For alloy composition within the spinodal region, thermal processing, including solution treatment and isothermal aging, can create a special transformation pathway, i.e., concentration modulations produced by the spinodal decomposition of the β phase followed by ω formation. Phase field simulations of athermal formation in Ti-50 at.% Nb upon water quenching from 1,173 K to 300 K are shown in Figure 6. A wide temperature range (300 K < T < 786 K) for spinodal regions exists in Ti-50 at.% Nb [Figure 1A], leading to precursory concentration modulations with a wavelength of ~ 10 nm before the formation of athermal ω, which is different from that in Ti-26 at.% Nb upon water quenching [Figure 3]. The Nb-rich β₂ regions (red color) form a continuous matrix while the Nb-lean β₁ regions (blue color) form uniformly distributed islands, and the amplitude of spinodal decomposition increases upon cooling at T ≥ 500 K. But at T < 500 K, no significant changes can be found in the spatial distribution and amplitude of spinodal decomposition by virtue of the low mobility at low temperatures^[37]. Figure 6C exhibit that finer dispersed ω nanodomains form at the solute-lean β₁ islands, which number density and size are smaller than that formed in Ti-26 at.% Nb upon water quenching [Figure 3]. The unchanged concentration fields at T < 500 K indicates that the β → ω phase transformation in Ti-50 at.% Nb upon water quenching also is diffusionless transformation, and the formed ω structure is athermal ω nanodomains assisted by the precursory spinodal decomposition. The ω formed in Nb-lean β₁ regions has also been found in compositionally modulated Ti-23Nb-2O (at.%) alloy via APT and TEM observations after solution treatment^[14].

Figure 6. Calculated microstructural evolution of athermal ω formed upon water quenching in Ti-50 at.% Nb. (A) Concentration fields; (B) One-dimensional Nb concentration profiles along the white short dash line of the computational cells shown in (A); (C) Structural order parameter fields.

Formation of isothermal ω

c₀ (= 26.7 at.%) is much higher than the equilibrium concentration of a β₁ phase (~ 10.0 at.% at 573 K) at 573 K[Figure 1]; thus, the phase has a lower Gibbs energy in the β₁ regions for Ti-50 at.% Nb. Figure 7 shows the microstructural evolution of ω precipitates during isothermal aging at 573 K in Ti-50 at.% Nb alloy. The spinodal decomposition preferentially occurs with a wavelength of ~ 8 nm smaller than that obtained upon water quenching, which amplitude gradually increases with aging time. Spinodal decomposition is a general state and has been experimentally found in many aged Ti-Nb-based alloys^[3,24,46,47] and other aged alloy systems such as CuNi-based alloys^[48], Fe-Cr alloys^[49], U-Zr alloy^[50], and high-entropy alloys^[51,52] via X-ray diffraction (XRD), TEM, scanning transmission electron microscopy (STEM), APT, and small-angle neutron scattering (SANS) analysis, showing temperature dependence and time dependence.

Figure 7. Calculated microstructural evolution of isothermal ω aged at 573 K for different aging times in Ti-50 at.% Nb. (A) Concentration fields; (B) One-dimensional Nb concentration profiles along the white short dash line of the computational cells shown in (A); (C) Structural order parameter fields.

When the concentration of β₁ regions is lower than c₀, these regions transform into a mixture of ω + β phases with fine-scale via displacive transformation. The concentration of the β₂ regions is much higher than c₀ and gradually evolves towards the equilibrium composition of the β phase ( c_β = 91.3 at.% at 573 K); thus, the β₂ regions become -free zones. ω embryos have been detected in solute-lean regions caused by spinodal decomposition in aged alloy^[25,26]. Moreover, the β₁/β₂ interfaces present high structural stability during aging at 573 K [Figure 7A], during which spatial distribution remains virtually unchanged upon the aging process and has a significant geometric constraint on the growth of ω domains; thus, a confined size of ω particles is found in the smaller β₂ islands even after long-time aging. Therefore, the morphology patterns of spinodal decomposition regulate the spatial distribution of ω precipitates in Ti-50 at.% Nb.

At 673 K, the equilibrium concentration of the Nb-lean β₁ regions (~18.3 at.%) is slightly larger but very close to c₀ (= 17.6 at.%). These regions will transform into a mixture of ω + β phases via the pseudospinodal mechanism[Figure 8]. The Nb-lean β₁ regions are uniformly distributed channels formed at 673 K. The amplitude of the spinodal decomposition gradually increases until the equilibrium concentrations are reached, then the wavelength increases from the initial ~13 nm to ~20 nm with further aging. Previous experimental studies have reported that the wavelength increases with increasing aging temperature^[24], and our simulations supply that the aging temperature can also regulate the spatial distribution of β₁ regions, which can help us to explain that^[2,23,24] the morphology of α" changes from spherical nanodomains to a plate-like structure with increasing temperature.

Figure 8. Calculated microstructural evolution of isothermal ω aged at 673 K for different aging times in Ti-50 at.% Nb. (A) Concentration fields; (B) One-dimensional Nb concentration profiles along the white short dash line of the computational cells shown in (A); (C) Structural order parameter fields.

The transformation sequence, i.e., wavelength increases followed by the isothermal ω formation, is attributed to the fact that no barrier exists in the spinodal decomposition, but thermal fluctuations of concentration and structure are needed for the ω formation via the pseudospinodal mechanism. A thermal fluctuation of concentration is required to bring local concentration in β₁ to the left of c₀, and a thermal fluctuation of structure is also required for structural transformation from bcc to hexagonal. Then, isothermal ω precipitates can be nuclear and grow, which concentration gradually decreases towards the equilibrium composition of the ω phase (c_ω = 10.2 at.% at 673 K) with increasing aging time. The distribution of isothermal ω particles is also confined in the β₁ channels at 673 K. This phase transformation pathway via pseudospinodal has also been found in precipitation with precursory spinodal decomposition^[40].

Furthermore, it is noteworthy that the size and shape of particles are decided by two main reasons. First is the morphological patterns of the concentration modulations produced by the spinodal decomposition. The interconnected β₁ channels obtained at 673 K provide more space for the growth of ω than the dispersed β₁ islands formed at 573 K. Thus, an elongated shape of ω precipitates can be found overall during 673 K aging. The β₁ channels formed at 673 K have a certain degree of orientation. The ω precipitates with their habit planes parallel to the channels have extensive growth and elongated shape, while those with their habit planes perpendicular to the channels (marked by yellow arrows in Figure 8A) experience limited growth and show spherical-like shape due to the geometric constraint of the solute-rich β₂ matrix. Moreover, experimental observations have found that^[19] the oxygen could regulate the morphology of isothermal ω in Ti-20Nb-xO (at.%) alloys aged at 723 K for 144 h, i.e., initial ellipsoid shape changes to elongated rod shape as oxygen content increasing from 0.1 at.% to 3.4 at.%. Theoretical study^[53] has shown that the spatial distribution of solute-lean β₁ can change from spherical islands to interconnected channels in Ti-23Nb-xO (at.%) aged at 1,073 K with increasing oxygen content from 1 at.% to 5 at.%. From our simulations and the above analysis, we summarize that the morphological patterns of spinodal decomposition can regulate the size and shape of the ω precipitates. Second is the nucleation sites of ω variants. Each ω variant has the same probability of nucleating by thermal fluctuations. When several ω variants (generally ≥ 3) nucleate in adjacent positions of β₁ channels (marked by yellow circles in Figure 8C), these ω particles tend to experience limited growth, whereas the dispersed variants have a faster growth rate and larger size (marked by red circles in Figure 8C).

Mechanisms of ω formation during isothermal aging

The above simulations indicate that the formation mechanism for isothermal ω depends sensitively on alloy composition and aging temperature. To examine more details of the microstructure during aging at two different aging temperatures, volume fraction, number, average area, and average shuffle degree of the isothermal ω particles at two alloy compositions are plotted as a function of aging time, and the results are exhibited in Figure 9. The statistical system size is 512 nm 512 nm. Figure 9A shows the volume fraction of the isothermal ω precipitates for four aging conditions. This volume fraction curve in Ti-26 at.% Nb aged at 573 K enables us to split the ω dynamics into two characteristic stages of nucleation regime (≤ 10 h) and growth regime (> 10 h), whereas the other curves can be divided into three stages, including incubation period without ω phase (stage I), nucleation regime (stage II), and growth regime (stage III). The incubation time increases with increasing alloy composition (~40 h for Ti-50 at.% Nb at 573 K) and aging temperature (i.e., ~5 h for Ti-26 at.% Nb at 673 K and ~120 h for Ti-50 at.% Nb at 673 K). During the incubation period, Ti-26 at.% Nb aged at 673 K experiences composition redistribution [Figure 5], while Ti-50 at.% Nb produces spinodal decomposition at two aging temperatures [Figures 7 and 8]. During the nucleation stage, multiple nuclei can form within a smaller time window that increases with increasing alloy composition.

Figure 9. Statistical analysis of isothermal ω precipitates at different aging temperatures for different average compositions. (A) Volume fraction change of ω; (B) The number of ω particles; (C) The average area of ω particles, where the inset describes the area evolution of Ti-50 at.% Nb during aging at 573 K; (D) Variation of the average shuffle degree of the isothermal ω precipitates.

Ti-26 at.% Nb alloy has a larger volume fraction of isothermal ω particles at two aging temperatures, and the volume fraction decreases with increasing aging temperature, which indicates the congruent transformation process promotes the formation of ω particles with a larger volume fraction as compared with the conventional nucleation-and-growth reaction. The evolution rates of ω volume fractions are similar in Ti-26 at.% Nb at two aging temperatures but have a significant difference in Ti-50 at.% Nb. For Ti-50 at.% Nb, the ω volume fraction gently evolves at 573 K and has limited growth at stage III (> 70 h), but it develops more sharply at 673 K. This difference is attributed to the fact that the β₁ regions formed at 673 K have lower Nb content with larger centralized spaces [Figure 8] than that formed at 573 K [Figure 7].

The number and average area of isothermal precipitates are shown in Figure 9B and C. From alloy composition, the number of isothermal ω particles formed in Ti-50 at.% Nb is overall larger than that formed in Ti-26 at.% Nb, but the average ω area in Ti-50 at.% Nb is noticeably smaller than that in Ti-26 at.% Nb. From aging temperature, the number of ω particles formed at 673 K gradually decreases with increasing aging times in two alloy compositions, and the corresponding average areas gradually increase as the aging process. But at 573 K, Ti-26 at.% Nb has a linear increase in the number of ω particles at the nucleation regime, then gradually decreases during the growth stage and shows a continuous increase of the average particle area. Whereas, Ti-50 at.% Nb exhibits a monotonic but non-linear increase in the number of ω particles and a sluggish increase of particle size during the nucleation and growth regimes.

The evolution of the average shuffle degree of the isothermal ω is shown in Figure 9D. Overall, the average shuffle degree of the isothermal does not reach completion (= 1) within 256 h aging but is noticeably larger than that of the athermal ω [Figure 3E]. Moreover, the shuffle degree of ω particles increases sharply at a certain time window (> nucleation stage) and then evolves gently at a longer aging time. It can be understood from the evolution results of concentration fields. According to the Gibbs energy curves [Figure 1B], a lower Nb content decreases the Gibbs energy of the whole system. Experimental observations have also shown that the collapse degree of {222}_β planes in athermal ω increases with decreasing solute content^[17,42]. At the initial formation stage of the isothermal ω, including the nucleation regime, a conspicuous depletion of Nb from ω particles can be found, which can cause a relatively sharp increase of shuffle degree in ω particles, but the Nb content of the ω particles will be close to the equilibrium composition of ω phase at a longer aging time, the evolution of concentration fields will become gentle thus slow the increase rate of shuffle degree. This description can also explain that a higher aging temperature accelerates the increase of shuffle degree and brings about a larger shuffle value [Figure 9D] since a higher temperature has a larger diffusion coefficient and accelerates the Nb depletion from ω precipitates^[37].

Comparison with O' and α" nanodomains

O' and α" nanodomains have also been found in Ti-Nb-based ferroelastic systems upon cooling and change the sharp, discontinuous martensitic transformation into a continuous transition over a wide temperature range^[2,4]. Theoretical calculations have shown no energy barrier in the transformation of β → O'/α" nanodomains^[4], but the β → athermal ω transformation in Ti-Nb binary alloys has an energy barrier and thus is a first-order transition^[32]. Experimental results have also demonstrated that a thermal hysteresis of ~3 K exists in the β ↔ athermal ω, but no temperature hysteresis can be detected in the transformation of β ↔ O'/α" nanodomains^[17]. Therefore, it is different in the transformation path for the formation of athermal ω and O'/α" nanodomains upon cooling.

Moreover, the required initial Nb concentration of the β matrix is different for the transformations to athermal ω and O'/α" nanodomains. A wider concentration range of ≥ 21 at.% Nb has been found in the β → athermal ω^[16], whereas the β → O'/α" nanodomains need a higher Nb content of ≥ 25 at.% in the Ti-Nb binary alloys having concentration modulation generated by spinodal decomposition^[4]. The shape of athermal ω in the Ti-(21~27) at.% Nb alloys seems concentration-insensitive and shows ellipsoidal nanodomains^[5,16], but that of the α" changes from plates in Ti-(15~24) at.% Nb alloys into spherical nanodomains in Ti-(≥ 25) at.% Nb alloys^[4]. The APT analysis in solution-treated Ti-23Nb-2O (at.%) alloy has shown that the athermal ω has lower Nb content than the O' nanodomains^[14]. The above discussion indicates a relatively weak effect of concentration fields on the formation of athermal ω nanodomains in Ti-Nb alloys with lower Nb content, but the concentration modulation is necessary for the formation of O'/α" nanodomains in Ti-Nb alloys with higher Nb content. As for the metastable phases formed during isothermal aging, we found that the isothermal ω has better phase stability than the O' and α" phases. Previous studies presented the only isothermal ω in Ti-(20-26) at.% Nb alloys after aging at 573 K and 673 K^[5,19], and oxygen doping or alloying could increase the phase stability of O' and α" during aging^[14,23]. In our current simulations, the competition among the ω, O', and α" is ignored and needs to be further studied.