Coupled nucleation of dual-phase lamellar structure

dual-phase


INTRODUCTION
Nucleation marks the beginning of a chain of transformations, such as crystallization, condensation, and solid-state phase transition.The classical nucleation theory (CNT) has originated from ideal models.][10][11] Tammann was the first to study lamellar formation in a eutectic system, proposing that the two lamellar phases solidified alternately. 12Hull et al. then built up a picture that envisaged the generation of a lamellar colony as successive nucleation and growth of plates. 13Zener and Tiller developed a formal theory for forming dual-phase lamellae, which described the sequential alternate nucleation sequence and edgewise growth of both lamellar phases. 14,15Hillert modified this theory, and Chadwick conducted a further review. 16,17According to the recent book by Aaronson et  al., the nucleation process for dual-phase systems is an alternate one. 18ince the initial proposal of Tammann, the alternative crystallization of dualphase lamellae, all subsequent theories on lamellar nucleation have been based on his deduction.As Chadwick noted, the formation of the lamellar structure occurs through a process of sidewise nucleation, in which phase A nucleates first, followed by the subsequent nucleation of phase B adjacent to the first. 17Despite the plausibility of the idea of Tammann about lamellar nucleation, it lacked a solid experimental foundation.Subsequent studies have relied on indirect experimental evidence, such as crystallographic orientation relationship, nucleation rates, etc., to propose phenomenological theories and mathematical models based on the initial hypothesis of Tammann.Nevertheless, the indirect evidence has led to conflicting viewpoints among researchers.For instance, while Hull et al. believed cementite is the preliminary plate for pearlite, Hillert et al. maintain that both cementite and ferrite can serve as the preferred nucleation phase.
We report here a new coupled nucleation phenomenon in a dual-phase lamellar structure, where two distinct lamellar phases nucleate simultaneously from the matrix in pairs, with each pair having an independent and heterogeneous nucleation process.Our proposition of the coupled nucleation concept relies on the alternative segregation of Nb characterized by atom probe tomography (APT) in a lamellar Ti-Al system.Subsequently, we intro-duce a modified theoretical model that corrects the classical model for plausibility, explaining this coupled nucleation process.Theoretical findings are further endorsed by molecular dynamics (MD) simulations and in situ highenergy synchrotron X-ray diffraction (HEXRD), confirming our experimental observations.Our proposed theory and model provide an add-on to the classical nucleation theory and give insights into the nucleation phenomenon of dual-phase lamellar structures.

Coupled formation of TiAl lamellae characterized by APT
Here, we used lamellar TiAl alloy as the model material to study the formation mechanism of lamellar structures.0][21][22] The TiAl lamellar structure is generated from the solid-state phase transition of α → α 2 + γ (α is disordered hexagonal close packed (h.c.p.), α 2 and γ are ordered D0 19 -type h.c.p. and ordered L1 0-type facecentered cubic (f.c.c.), respectively), through which the α 2 and γ plates are formed in eutectoid transformation according to the Ti-Al phase diagram. 23,24 third element, Nb, is used as a tracer marker in tracing the formation process of TiAl lamellae.This is because Nb dissolves and partitions to the product γ phases as the eutectoid transformation proceeds.[24][25][26] Moreover, Nb atoms are substitutional in TiAl with a much slower diffusivity than those of Ti and Al atoms.27 So, it can label the nucleated γ and α 2 phases and trace the ensuing phase change.20  10 Figure 1A and B shows the transmission electron microscopy (TEM) images of TiAl lamellae operating at high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) mode.In this figure, the dark phase is γ-TiAl lamellae, and the bright phase is α 2 -Ti 3 Al lamellae.The crystal orientation of α 2 and γ followed the Blackburn relationship as (0001) α2 ‖ (111) γ , <11 > α2 ‖ <1 > γ , 28 as shown by the diffraction spot in the fast Fourier transformation (Figure 1B).A detailed discussion of the interfaces among α, α 2 , and γ is given in Text S1 and Figure S1.
The APT reconstructed lamellar structure of a ternary Ti-Al-Nb alloy is shown in Figure 1C-E.The α 2 + γ dual-phase structure can be recognized from the concentration difference between Ti and Al in Figure 1C.It shows that the Ti-rich α 2 lamellae and Al-rich γ lamellae are arranged alternatively with a distinct lamellar thickness.Figure 1D shows the atom map and the isosurface of Nb, from which three Nb segregation areas can be seen as indicated by arrows.The Nb segregation is found to be parallel to the phase interfaces shown in Figure 1C.After decomposing the complex molecular ions, the one-dimensional (1D) concentration profiles were obtained perpendicular to the interfaces, as shown in Figure 1E.Significantly, the Nb concentration peaks are located only at the α 2 /γ interfaces but not at the γ/α 2 interfaces, as indicated by the arrows in Figure 1E.Combined with Figure 1D, Nb is proven to be segregated at the α 2 /γ interfaces.
We note that this is quite different from previous studies, [29][30][31] in which Nb is believed to be a stabilizing element of γ and favors partition in the γ phase.The reason for the difference may be related to the alloy processing.In our work, the Ti-Al-Nb alloy was tested in the as-cast condition.This is expected to be at a kinetic non-equilibrium state compared with the heat-treated state in previous studies, 30,32 where all Nb may have dissolved in the γ phase.More importantly, the concentration profiles in the upper curves of Figure 1E also show that all three γ lamellae can be regarded to have the same Ti/Al atomic ratio, however, the four α 2 lamellae did not.In the latter, the Ti concentration changes greatly.It means the α 2 lamellae are non-equilibrium because of the heavy precipitation of the O-phase in the α 2 phase. 33he segregation of Nb at the α 2 /γ interfaces can be eliminated after a twostep annealing (1523 K for 24 h and 1173 k for 30 min), as shown in Figure 2, from which Nb is found to partition to the γ phase.This was seen both in the APT and the scanning electron microscopy-energy-dispersive X-ray spec-troscopy (SEM-EDS) results.The annealed Ti-Al-Nb sample proves that the alternative segregation of Nb at the α 2 /γ interfaces was not caused by APT running condition.We also tested several different lamellar Ti-Al-Nb samples with different lamellar orientations and lamellar thicknesses (Text S2 and Figure S2).These results show that Nb is alternatively segregated at the α 2 /γ interfaces, and only the enrichment degree differs.We then believe that the formation of TiAl lamellae follows a coupled nucleation and growth manner.A theoretical model is proposed below.

Coupled nucleation model
Because the nucleation barrier is to create new interfaces, we start by discussing the interfacial structure and energy.Figure 3 depicts a pair of lamellae α 2 and γ forming simultaneously in the matrix of α.There are five types of interfaces among these three phases, labeled clockwise from 1 to 5, as depicted in Figure 3A and B. Types 2, 3, and 4 are the broad interfaces, and Types 1 and 5 are the edge or circumference interfaces.A detailed discussion of these interfaces is given in Text S1.Below, the interfacial energy per unit area of Type 1 to 5 will be represented by σ 1 to σ 5 , respectively.
We consider homogeneous nucleation of a disk-like coupled nuclei of γ and α 2 in the matrix of the α phase, as shown in Figure 3C.The nucleus has a radius of "r" and thicknesses of "t 1 " and "t 2 ", respectively, for the γ and α 2 .Based on CNT, 34 the standard free energy of formation of the nucleus is where ΔG V-γ and ΔG V-α2 are the energy gain for creating γ and α 2 lamellae, respectively, in the transformation of α → γ + α 2 .For simplicity, let t 1 = t 2 = t, ΔG V = (ΔG V-γ + ΔG V-α2 )/2, and σ 2 = σ 3 = σ 4 (considering type 2, 3, and 4 interfaces are all low energy), and we have Assuming that r and t are independent variables, we obtain the dimensions of the critical nucleus,

REPORT
The Innovation Materials 1(3): 100043, December 13, 2023 3 Thus, the free energy for forming the critical nucleus is However, homogeneous nucleation is known to be rare in real phase changes.Therefore, we convert it to heterogeneous nucleation.To do so, we introduce a flat surface in Figure 3C, which cuts across the coupled nucleus with an angle "θ".Then, we have where f(θ) = (θ -sinθ•cosθ) / π < 1 is a modified factor.The detailed derivation can be seen from Text S3.Based on this model, the total interface area for coupled nucleation is smaller than that of separate nucleation and sequential alternate nucleation, as discussed in Text S3 and shown in Figure S1E.Therefore, the free energy barrier for coupled nucleation is the smallest, making the coupled nucleation the easiest nucleation manner for lamellae.The growth of α 2 and γ should also follow the coupled formation manner because the α 2 phase is interfacial isotropy when precipitating from α matrix; if not accompanied by the γ plate, it should nucleate and grow as spherical, which cannot grow into lamellae (Text S4).

MD simulations
We performed MD simulations of α → γ + α 2 solid-state phase transition for TiAl lamellae to further corroborate our coupled nucleation prediction.All simulations were conducted with the large-scale atomic/molecular massively parallel simulator (LAMMPS) software.Two typical embedded-atom method (EAM) potentials were used to simulate the α phase below the melting point (see materials and methods for details) to enable cross-validation of the results.The temperature was then lowered to investigate the early-stage nucleation of α 2 and γ phases from the α matrix.Figure 4A shows that the new phases nucleate at 5.48 ns with the nuclei located on the surface of the parent phase.As time passes, lamellar phases of f.c.c. and h.c.p. structures begin to nucleate and grow simultaneously from the surface into the parent phase.(The f.c.c. and h.c.p. nuclei correspond to the γ-TiAl and α 2 -Ti 3 Al phases, respectively).Furthermore, with Ti atoms colored in white and Al atoms in green, it is shown that ordered α 2 -Ti 3 Al and γ-TiAl lattices appear simultaneously at the initial nucleation moment 5.48 ns (Figure 4B).Such ordered α 2 and γ phases during initial nucleation are consistent with our HEXRD results (Figure 5), further confirming the reliability of our coupled nucleation.
On the other hand, the radial distribution functions (RDFs) are normalized by the number density of particles within a given cutoff radius, which is often used to identify ordered structures and distinguish different phases. 35Since γ and α 2 phases have different nearest-neighbor atom distances, we used RDFs to distinguish γ and α 2 phases during nucleation.The calculated RDFs of the simulation system exhibit low long-range order at 5.47 ns, but a second-peak splitting appears at 5.82 ns (Figure 4C).This indicates the formation of more stable phases with long-range order.We compared the RDFs of the nucleation zone at 5.60 ns with perfect γ + α 2 (Figure 4D) to ascertain the configuration of the long-range order phases.It is shown that the nucleation zone has two small splitting peaks at the first main peak, corresponding to the perfect γ and α 2 phases, respectively.Furthermore, we calculated the RDFs of the perfect γ and α 2 phases (Figure 4E and F), which show only one main peak in both RDFs of two phases.This indicates that different axis lengths of the crystal lattice do not affect the RDFs.This proves that the two small peaks of the first main peak represent γ and α 2 phases from the side.The same nucleation process is further verified by another potential (Figure S3).All the MD simulation results are consistent with our prediction and experimental observations.

In situ HEXRD measurements
Our nucleation model is set up with the prediction that the α 2 and γ nuclei precipitate from the α matrix directly in anisotropic and ordered structures.This prediction was confirmed by in situ HEXRD measurements.The measured reflection peaks versus temperature are shown in Figure 5, show-

Materials
Figure 3.The coupled nucleation model for TiAl lamellae (A and B) The three-dimensional (3D) schematic sketch (A) and two-dimensional (2D) sketch (B) for the coupled formation of α 2 + γ lamellae from α matrix in the α → α 2 + γ phase transition.The interfaces formed in this process are labeled from 1 to 5, within which the habit planes of 2, 3, and 4 are indicated in red.(C) Coupled nucleation of one pair of α 2 + γ lamellae starts from a grain boundary (G.B.), which cuts across the coupled nucleus with an angle of "θ".The nucleus has a radius of "r" and thickness of "t 1 " and "t 2 ", respectively, for the γ and α 2 .The right side in (C) is a 2D sketch of a single disk marked with radius "r" and inclination angle "θ".(D) Coupled growth of the α 2 + γ lamellae leads to Nb periodical enrichment at α 2 /γ phase interfaces.

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The Innovation Materials 1(3): 100043, December 13, 2023 5 ing the continuous change in the phase intensity versus temperature upon cooling.The two super-lattice reflection peaks, marked by the red font, correspond to the α 2 (10 ) plane and γ (002) as well as γ (200) planes, respectively, representing the appearance of ordered α 2 and γ phases.From the continuous change in the phase intensity versus temperature, the ordered α 2 and γ phases are formed directly from the α matrix with their super-lattice reflection peaks of α 2 10  and γ 002 as well as γ 200 appearing at the very beginning of precipitation during continuous cooling.Further confirmation of ordered crystalline structure can be seen from Text S6 and Figure S4.This corresponds to the results of Appel and Zhang et al. that the ordered L1 0 ⇒ structure is formed directly at the onset of the lamellar decomposition (α α 2 + γ) in TiAl alloys rather than via the intermediate disordered f.c.c.structure. 24,36This result is consistent with the atomic electron tomography (AET) result of Zhou et al., who found that the early-stage nuclei are anisotropic and ordered. 9

DISCUSSION
Tammann and all other researchers believed that the formation of lamellae follows the sequential alternate nucleation manner, where there should be a primary phase during lamellar formation, either phase A or B. 12,17 However, this deduction cannot explain the periodic segregation of Nb at phase interfaces in our work.The only reasonable explanation is that the lamellae are formed pair by pair, and each pair is independent of the others as they nucleate from the matrix.Therefore, the α 2 and γ lamellae are cooperative in nucleation and growth, making the coupled growth of an α 2 /γ pair different from the uncoupled ones, i.e., the pair of γ/α 2 .The coupled formation process of lamellae can be described as follows.
Before the α → α 2 + γ transformation, Nb distributes uniformly in the α phase.During the transformation, Nb can partition among the γ phase, the α 2 phase, and the interface between them.According to the phase equilibria of the Ti-Al-Nb ternary system, the partitioning of Nb prefers γ. 37 Therefore, Nb will be enriched in γ more than in α 2 .Because the board interface between them has misfit strain, it dissolves Nb to reduce the strain energy.Thus, the segregation of Nb can identify the coupled growth, see Figure 3D.The important finding in Figure 1E revealed by the Nb marker is that it has identified the coupled growth of a pair of α 2 + γ by the interfacial segregation of Nb.There is another set of γ/α 2 interfaces between γ and α 2 without Nb segregation, which means their growths are uncoupled.On the other hand, both Ti and Al atoms are substitutional in all phases of the TiAl alloy.Hence, lattice diffusivity is sluggish, and only grain boundary diffusion and interfacial diffusion can take place during phase transformations. 23,25We consider the edgewise growth of a single pair of α 2 /γ lamellae in α, as depicted in Figure S1D.The horizontal arrows in Figure S1D depict the interdiffusion of Ti and Al along the growth front or the α/α 2 and α/γ interfaces, with Ti going from the latter to the former and Al going in the opposite direction.We note that both are uphill diffusion; therefore, they are driven by chemical potential gradient, not concentration gradient.The Al atomic flux is bigger than that of Ti, which is based on the fact that the concentration gradient of Al is bigger than that of Ti at α 2 /γ interfaces, as shown in Figure 1E.Thus, the Kirkendall effect occurs with a vacancy flux accompanying the Ti flux.These excess vacancies reach the α/α 2 interface and are absorbed by the growing α 2 phase.They assist the nucleation and growth of the O-phase in the α 2 lamellae.

CONCLUSION
In summary, we developed a coupled nucleation model of a dual-phase lamellar system based on the classical nucleation theory.In this model, two different lamellar phases nucleate simultaneously from the matrix in pairs, and each pair of lamellae nucleates separately and independently.The model was formulated based on observing alternative Nb segregation using APT in a lamellar Ti-Al system and further verified by HEXRD and MD simulations.Our model is applicable to other systems beyond TiAl.The coupled nucleation model predicts that the lamellar nuclei precipitate directly from the disordered matrix in anisotropic and ordered structures.By considering the interfacial anisotropy of the ordered phase and modifying the shape of the nucleus in the classical nucleation theory, it is theoretically and experimentally feasible to nucleate the ordered L1 0 phase or L1 2 phase from a disordered f.c.c.structure in FePt. 9Additionally, by introducing an anisotropic energy component in the nucleation equation of colloidal crystals, the model also enables the prediction and explanation of nonspherical nuclei. 10,11Therefore, our model can be applicable to other systems or phase transitions that involve anisotropic, ordered, or lamellar structures, offering a valuable addition to the existing nucleation theory.

Structural characterization
Structural characterization of the lamellar structure was carried out utilizing an FEI TITAN G2 60-300 instrument operating at 300 kV for TEM and high-resolution TEM (HRTEM) observations.Thin foils for TEM observations were prepared by cutting, mechanically grinding, and final electrochemical polishing in a solution of methanol, butan-1-ol, and perchloric acid.SEM and EDS characterization were conducted using a Zeiss Auriga crossbeam (FIB-SEM) workstation.

Atom-probe data acquisition and analysis
APT sample preparation was conducted in the FIB-SEM workstation, which was equipped with a Cobra gallium column and a Gemini electron column using the FIB lift-out method.The APT analyses were performed in a local electrode atom probe (CAMECA LEAP 4000X-Si) equipped with a ~ 10 picosecond, 355 nm ultraviolet (UV) laser.The needle-shaped specimen was cooled to a temperature set point of 50 K and then run with a laser pulsing energy of 60 pJ, a frequency of 250 kHz, and a detection rate of 0.015 per pulse.The Integrated Visualization and Analysis Software (IVAS 3.6.8)was used to reconstruct the data.

MD simulations
Two EAM potentials are employed for the interatomic potential between Ti-Al atoms, which can accurately predict the high-temperature properties.The simulations in the main text adopt the Zope-potential, and the simulations in the supplemental information adopt the Farkas-potential. 38,39The initial atomic configuration is prepared by an h.c.p. structure with randomly distributed Ti and Al atoms and an atomic ratio of 6:4.The simulation model is a Ti-40Al ball with a radius of 3.46 nm in a 11.54 nm × 9.99 nm × 9.28 nm box, containing 10409 atoms.The system is equilibrated via the Nose-Hoover thermostat with a time step of 1 fs. 40,41The simulation is started at 1600 K (melt point of 1760 K for γ and 1890 K for α 2 ).Then, the temperature of the system decreases linearly at every time step, from 1600 to 300 K, with a cooling rate of 0.1 K/ps.The atomistic structures are visualized by the postprocessing software Open Visualization Tool (OVITO). 42The polyhedral template matching (PTM) is used to clarify the evolutions of different structures. 43

In situ high-energy synchrotron X-ray diffraction measurement
A binary Ti-39.6Alalloy with its composition close to the eutectoid point was prepared in this part of the work to verify the coupled nucleation model.The in situ HEXRD experiment was conducted utilizing the P07 high-energy materials science beamline operated by Helmholtz-Zentrum Geesthacht at PETRA III at the Deutsches Elektronen-Synchrotron (DESY) in Hamburg, Germany.A beam size of 0.5 mm by 0.5 mm and a photon energy of 100 keV

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The Innovation Materials 1(3): 100043, December 13, 2023 7 (λ =0.124 Å) was used for measurement.The beam center and detector to sample distance (1839 mm) were calibrated using a LaB6 standard.The TiAl sample was heated to 1200℃ (the uni-phase area of α) at 20℃/s, held for 10 min, and then slowly cooled at a rate of 1℃/min to 750℃ (the two-phase region of α 2 + γ) while the HEXRD measurements were taken.

Figure 1 .
Figure 1.TEM and APT analysis of the TiAl lamellar structure (A and B) HAADF-STEM images of the α 2 and γ lamellae, while (B) is the magnified area marked in (A).Inset (B) is the diffraction spot acquired by fast Fourier transformation.(C) the combined atom map of Ti and Al with Ti in blue and Al in yellow; a region of interest (ROI) that represents the analysis box is shown inside, with its analysis direction indicated by an arrow.(D) The atom map combined with the iso-surface of Nb in red.(E) One-dimensional (1D) concentration profiles of Ti, Al, and Nb along the arrow indicated in (C).The errors are s.d., 2σ.

Figure 2 .
Figure 2. SEM and APT reconstructed TiAl lamellar structures after annealing (A and B) atom maps of Ti, Al, and Nb.(A) The combined atom maps of Ti and Al with Ti in blue and Al in yellow; an ROI that represents the analysis box is shown inside (A) with its analysis direction indicated by an arrow.(B) The atom map of Nb. (C) 1D concentration profiles of Ti, Al, and Nb along the arrow indicated in (A).(D) The SEM image of TiAl lamellae, in which gray phases represent γ and dark phases represent α 2 .(E) EDS line scanning along the arrow indicted in (D).Both characterization methods show that Nb partitions to the γ phase.

Figure 4 .
Figure 4. MD simulations of dual-phase TiAl lamellar nucleation (A) Evolution processes of lamellar nucleation and growth at different simulation times since the start of the simulations.All structures are distinguished by the polyhedral template matching (PTM) algorithm, the f.c.c.structure is yellow, the h.c.p. structure is blue, the intermediate structure is dark blue, and the unidentified structure is red.(B) With Ti atoms shown in white and Al atoms shown in green, ordered γ and α 2 lattice were observed to be coupled at the initial nucleation moment (5.48 ns).The right inset represents the perfect atomic configuration of α 2 -Ti 3 Al and γ-TiAl phases.(C) Radial distribution functions (RDFs) of the MD-simulated sample at 5.47 and 5.82 ns, where g(r) is the RDF and r is the pair separation distance.(D) RDFs of nucleation zone in the dashed box at 5.60 ns and the perfect γ + α 2 .(E and F) represent RDFs of perfect γ (E) and α 2 (F) phases, respectively.

Materials 11 Figure 5 .
Figure 5.In situ HEXRD measurement of binary Ti-Al alloy during continuous cooling The two-theta angle is plotted versus temperature T-ramp (T-ramp at the top).The intensity is plotted in a logarithmic grey scale.The reflection peaks are identified on the right.The super-lattice reflection peaks of the α 2 10 peak and γ 002 as well as γ 200 peaks are marked in red.