The ordered L1₂-γʹ phase precipitation-strengthened Co-based superalloys have attracted many interests due to their excellent properties at high temperature, such as good resistance to corrosion, oxidation, and fracture (Sato et al., 2006; Yan et al., 2014). Thus, instead of the Ni-based superalloys, the Co-based superalloy containing γʹ-Co₃ (Al _x W_1-x ) precipitates will be served as the next-generation high-temperature materials (Suzuki et al., 2008). The superior performance of Co-based superalloys at elevated temperatures is originated from the γʹ phase, which has a strong obstacle to the slip of dislocation (Nathal et al., 1987). The stability of the γ+γʹ microstructure is dependent on their retarding effect of coarsening, so the kinetic evolution of the γʹ phase in Co-based superalloys is significant on the mechanical properties. Extensive studies found that cobalt-based superalloys with a high γʹ volume fraction and slow coarsening rate are typically expected (Boothmorrison et al., 2009; Gupta et al., 2020; Liu et al., 2020).

As is known, γʹ-Co₃ (Al, W) precipitates with a L1₂ ordered f.c.c (face-centered cubic) structure have a special microstructure called the antiphase domains (APDs). The microstructure is introduced to the superalloys in the representation of the region of four equivalent ordered states, obtained by the movement of the crystal lattice from its initial position to [0, 0, 0], [1/2, 1/2, 0], [1/2, 0, 1/2], and [0, 1/2, 1/2], respectively (Li et al., 1998). If these four types of APDs impinge on each other during growth and coarsening, a structural L1₂ planar defect called antiphase domain boundary (APB) will be generated. However, the interaction between those “out-of-phase” particles (OPPs) is crucial to produce the high-energy APBs. In contrast, the contact among the particles which happen to be “in phase” (IPPs) with respect to each other is quite different. By definition, the IPPs generate a single perfect ordered domain rather than the high-energy APBs (Wang et al., 1995). Therefore, the coalescence events almost occurred in IPPs rather than in OPPs.

There are several studies relating to the APBs and their effects on the evolution kinetics of the γ′ phase in Ni-based alloys. Yang et al. (2017) studied the effects of the APDs and the elastic energy on the morphological evolution and coarsening behavior of γ′ precipitates in Ni-based alloys using the phase-field method. It was found that the presence of the APDs reduces the coarsening of γ′ precipitates. The studies of Wang et al. (1998); Chen et al., 2004) show that when the APB energy is greater than twice the γ/γʹ coherent interfacial energy, no APBs can exist between γʹ precipitates, except when the alloys are subjected to high-temperature annealing or severe plastic deformation (Yang et al., 2021).

Although some studies have been carried out on the coarsening kinetics of the γ′ phase in cobalt-based superalloys (Sauza et al., 2016; Azzam et al., 2018; Shi et al., 2020; Wang et al., 2020), the effect of APBs on the evolution kinetics of the γ′ phase from nucleation to growth is not disclosed, especially for an elastically strain system like Co-Al-W alloy. This work is focused on the formation of APBs, and the relationship between the APBs and coarsening behavior of γ′ phases in Co-10Al-10W at% alloy aged at 1173 K. The Kim–Kim–Suzuki (KKS) model (Kim et al., 1999) is introduced to the phase-field model in this work. Coupling with the KKS model, the extra potential at interfaces will be removed to guarantee the smooth interfacial diffusion between γ/γʹ phases. The advantage of the KKS model is that it distinguishes the different crystallographic variants of the γʹ phase and therefore can account for effects associated with antiphase boundaries. In sum, it is conventional to reveal the effect of the APBs on the free energy as well as the coarsening rate from the perspectives of energy and kinetics.

The formation of APBs and their impacts on the evolution kinetics of γʹ phases are studied in Co–10Al–10W at% superalloys at 1173 K aging for 1 h. According to the three order parameter fields, the emergence of the APBs can be described by the four types of APDs. The four APDs of the γʹ phase are distinguished from the perspective of order parameters, which are η _max (1, 1, 1), η _max (1, −1,−1), η _max (−1, −1,1), and η _max (−1,1,−1) (η _max = η _1max = η _2max = η _3max) (Wang et al., 2007). In these four APDs, the IPPs and OPPs are considered as two types of particles. In addition, we also take the distinction of γʹ phases with a single domain and four kinds of APDs on the process of growth and coarsening.

The order parameter fields are shown pictorially in Figure 1A–C, where the dark areas represent the γ matrix, and other color areas represent the γʹ phases with different orientations. Thus, particles with the same color are defined as IPPs, while OPPs represent particles with different colors.

Figure 1D shows the mesoscopic microstructure of Co–10Al–10W at% alloy aged for 1 h at 1173 K, in which the white areas represent the γʹ phases with equilibrium composition field of W.

In Figure 1A–D, each γʹ phase has reached the equal value 1 or –1 of order parameters at 1 h. As indicated in the red block, four APDs with different order parameters η _max collide with the neighbor one to form the APBs. Previous studies have shown that there are mainly two types of APBs in superalloys, namely, [010] APB and [111] APB (Saal et al., 2016). Of course, the energies of these two types of APBs are different due to their degree of ordering, which has been demonstrated by the density-functional theory (DFT) calculations (Karnthaler et al., 1996).

The presence of APBs plays an essential role in the morphology evolution of the γʹ phase. Therefore, the formation of APBs and their effects on the morphology evolution of the γʹ phase are studied. As is shown in Figure 2A, the variations of free energy are illustrated from the γ matrix to the γʹ phase, the data of each curve are obtained from the position signed with three points of corresponding colors in Figure 1D. As the aging goes on, the free energy in the γʹ phase decreases to the minimum to ensure it is in a steady state. On the contrary, the energy in the γ matrix keeps at a relatively high level in order to satisfy the law of conservation of energy (Lifshitz et al., 1969). In particular, it is indicated in the red circle that free energy in the γ matrix, the γʹ phase, and the APBs all have a decline at the beginning stage. However, a sharp increase in regions of APBs occurs to overcome the energy barrier for the formation of APBs. During the stage of APB formation, the γ′ phases with different APDs keep growing and eventually impinging to form an APB.

Further investigation has been carried out to clarify the relationship of free energy change and the particle’s edge-to-edge distance Δd, as shown in Figure 2B, the free energy between two neighboring γ′ phases rises with the increasing of the Δd. When Δd ₁ = 5nm, the energy reaches the maximum, which is in good agreement with the width of APBs in experimental results of superalloys (Hemker et al., 1993; Mebed et al., 2003). This supports the fact that the free energy in the APBs reaches its peak shortly after the formation of the APBs. In addition, when the Δd >7 nm, almost no energy fluctuations are produced due to the weak interaction of γʹ phases. In this state, the position is not the interface regions of γʹ phases, while it becomes the matrix phase.

The previous study found that the emergence of the high-energy APBs inevitably has a crucial impact on the coarsening of the γ′ precipitates. Therefore, the problem is studied from different perspectives. Figure 3 shows the variations of the γ′ area fraction with aging time t for the single domain and four APDs, which indicates the effect of APBs on slowing down the nucleation and growth of the γʹ phases. In particular, when t < 0.15 h, the slope of the γ′ area fraction is k ^APB = 5.77 and k ^single = 9.28, so the volume fraction of the single domain increases faster than that of the four APDs in the early stage of aging, and reaches a steady state when its volume fraction is around 3% higher than that of the four APDs. Then, the difference of volume fractions becomes small at 1.5%, which is shown in the inserted picture. Above all, during the whole aging time, APBs make the certain influence on the γ′ area fraction because of the decline of the coarsening rate and the variation of the coarsening mode.

As discussed before, the effect of APBs on the kinetics evolution of the γ′ phase from nucleation to growth is not investigated. Usually, the Ostwald ripening and coalescence coarsening are two types of coarsening mechanisms in experimental TEM or SEM morphology (Mebed et al., 2003), while the evolution of APBs is not detected clearly in the experimental morphology. Therefore, the effects of the APBs on the growth and coarsening of the ordered γʹ phase need a further study.

In this work, the particles’ number density in unit volume is defined as N V = N A 2 r , where N _A is the particles’ number of the unit area and <r> is the average size of the γ′ phase (Shi et al., 2020). As shown in Figure 4A, B, the variations of N _V of the γʹ phase for four APDs and single domain are exhibited with aging time at 1173 K. At the stage of coarsening, the N _V declines with the decrease in particles’ number, the time exponent of N _V for APB is n ₃ ^APB = −0.80 ± 0.03 and for single domain is n ₃ ^single = −0.99 ± 0.02. The time exponents approach to –1 of Kuehmann–Voorhees (KV) theory (Kuehmann et al., 1996) in ternary alloy. At the nucleation and growth stage, the time exponents N _V are n ₁ ^APB = 1.23 ± 0.03 and n ₁ ^single = 1.52 ± 0.07, n ₂ ^APB = 0.49 ± 0.05, and n ₂ ^single = 0.28 ± 0.01.

However, the time for nucleation and early growth stage is 0.37 h in possession of APBs, which is much longer than that of 0.19 h with a single domain. Therefore, the existence of APBs in four APDs significantly elongated the period of nucleation and growth. Furthermore, compared with the single-domain γ′ phase, the time exponent of N _V becomes small at the coarsening stage for the γ′ phase with four APDs, which reveals that the coalescence is inhibited due to the presence of APBs. In addition, the coarsening mode of the γ′ phase with APBs is Ostwald ripening, while for single domains, there is also the coalescence coarsening. So the decline of the coarsening rate in the presence of APBs is caused by the interface structure change of OPPs, whose duration of forming a single γʹ phase is noticeably longer than that of coalescence (Chen et al., 2021). It is indicated that more participation of Ostwald ripening occurs in the late stage of coarsening when APBs exist.

As mentioned before, the Ostwald ripening is considered as the dominant form of coarsening with the existence of APBs. Figure 5 shows the movement of APBs during the process of Ostwald ripening between two neighboring OPPs. The purple and pink ones represent the two γʹ precipitates with different orientations, delegated by η _eq = –1, η _eq = 1, respectively, and the black one delegates the γ matrix with η _eq = 0. The distribution of order parameter fields with aging time shows that the APB keeps moving toward the small particle until absorption by the larger one; this process is similar to the Ostwald ripening where small particles dissolve and large particles grow up. However, this is not a simple process of coalescence or Ostwald ripening, but a coupling behavior of Ostwald ripening and APBs mobility due to the extremely narrow distance between the two γʹ phases with different order parameters. It is hard to capture this interesting phenomenon in the experimental morphology, so the phase-field simulation with crystal structure information will play a predictive role in the further study.

In this work, the antiphase boundaries and their effects on the evolution kinetics of the γ′ phase were studied with the KKS model to reveal the feature of free energy variation during the formation of APBs, as well as the reason for the sharp increase in free energy. From the new perspective of free energy, the relationship between the particle edge-to-edge distance and the APB formation, and the width of APBs are revealed in this simulation. In addition, the presence of antiphase domains can slow down the rate of growth and coarsening of the γ′ phase. Finally, a novelty phenomenon is explained in the late stage of coarsening of the OPP-typed γʹ phase with APBs, where the Ostwald ripening is the nature of APB migration with the interface structure reconstruction between two γʹ phases.