2.1 Phase-field variables

In phase-field methods, the evolution of the microstructure is interpreted through a set of phase-field variables^[29]. Phase-field variables are divided into ‘conserved variable’ and ‘non-conserved variable’. A representative example of a conserved variable is the molar concentration $c_k(\vec{r}, t)$ (unit: mol m^-3), where $k$ denotes the chemical element. A typical example of a non-conserved variable is the order parameter $\eta_i(\vec{r}, t)$, which characterizes the state of matter and depends on space and time. For simplicity, the order parameter and the molar concentration are hereafter denoted by $\eta_i$ and $c_k$, respectively. This study aims to simulate the precipitation of three types of β-Nb rich precipitates by considering interfacial coherency with the α-Zr matrix. To distinguish different interfacial coherency conditions, the order parameter $\eta_i (i = 1, 2, 3)$ was employed, where $i$ denotes the interfacial coherency type. Here, $\eta_i$ denotes the local fraction of the β-Nb phase at a given position. Where $\eta_i = 1$ indicates the presence of the Type $i$ body-centered cubic (bcc, β) phase, and $\eta_i = 0$ signifies the presence of the hexagonal close-packed (hcp, α) phase. At the interface between two phases, $\eta_i$ takes continuous values between 0 and 1.

In the Zr–Nb binary system, the sum of the molar concentrations of Zr and Nb must equal 1, as constrained by Eq. (1). Therefore, only the molar concentration of Nb, $c_{Nb}$, is an independent variable. Where $c_{Zr}$ and $c_{Nb}$ denote the molar concentrations of Zr and Nb, respectively.

2.2 Free energy functional

In the phase-field method, the evolution of the microstructure proceeds in the direction that reduces the total free energy $F(c_{Nb}, \eta_i, \kappa_i; T)$. Eq. (2) represents the total free energy of the Zr–Nb system at a given temperature $T$.

The first term, $f(c_{Nb}, \eta_i; T)$ is the molar chemical free energy describing the bulk thermodynamic contribution of each phase. When two phases, α phase and β phase, coexist with distinct free energies, the molar chemical free energy of the system can be expressed using an interpolation function $h(\eta_i)$ as defined in Eq. (3).

As shown in Eq. (3), $h(\eta_i)$ denotes an interpolation function that ensures a smooth variation of material properties or thermodynamic quantities across the diffuse interface, typically varying continuously from 0 to 1 between two phases, and is defined as $h(\eta_i) = \eta_i^3(6\eta_i^2 - 15\eta_i + 10)$. $g(\eta_i)$ is the double-well potential used to stabilize phase separation, defined as $g(\eta_i) = \eta_i^2(1 - \eta_i)^2$, where $W$ specifies the height of the double-well potential and is fixed to 1 in this study to enhance numerical stability. $f_\alpha(c_{Nb}; T)$ and $f_\beta(c_{Nb}; T)$ denote the molar chemical free energy of the α phase and β phase respectively, and are taken from the SGTE (Scientific Group Thermodata Europe) database by A. T. Dinsdale^[30] and the thermodynamic assessment of the Zr–Nb system by A. Fernández Guillermet^[31]. The thermodynamic description of the Zr–Nb system is summarized in Table 1.

The second term, $\frac{1}{2} \kappa_i |\nabla \eta_i|^2$, corresponds to the interfacial energy contribution, where $\kappa_i$ denotes the gradient energy coefficient that penalizes sharp spatial variations of the order parameter $\eta_i$, thus governing the interface thickness and interfacial energy of the interface. $\kappa_i$ and $\eta_i$ introduced to account for different interfacial coherency between two phases. The detailed explanation is provided in Section 2.5. The last term, $f_n$, represents the discrete nucleation term, which is introduced to simulate the artificial nucleation of the β-Nb rich precipitates.

2.3 Governing equation

Molar chemical free energy (Eq. (3)) is solved using the time evolution equations for the molar concentration of Nb ($c_{Nb}$) and the order parameter ($\eta_i$), namely Cahn–Hilliard equation (Eq. (4)) and Allen–Cahn equation (Eq. (5)), respectively.

The Cahn–Hilliard equation^[32] is applied to conserved variables, the molar concentration $c_{Nb}$. Where $M_{Nb}$ is the mobility of Nb. The evolution of the non-conserved order parameter ($\eta_i$) is governed by the Allen–Cahn equation^[33], in which $L$ denotes the kinetic coefficient.

In this study, both the α phase and β phase are explicitly considered in the simulations, as they coexist in the Zr–Nb alloy. Both the Cahn–Hilliard (Eq. (4)) and Allen–Cahn (Eq. (5)) equations were solved, and the Kim–Kim–Suzuki (KKS) model^[34] was additionally employed to accurately describe the equilibrium compositions of the two phases. The KKS model extends the phase-field method to multiphase and multicomponent systems with diffuse interfaces while maintaining equal diffusion potentials $\mu_{Nb}^\alpha(c_{Nb}^\alpha)$ between coexisting phases. Diffusion potentials introduce separate molar concentrations $c_{Nb}^\alpha$ and $c_{Nb}^\beta$ for each component Nb in α phase and β phase.

By satisfying the diffusion potential equality, the KKS model avoids unphysical free energy penalties inside interfaces, making it highly suitable for simulations where large composition differences exist between phases.

2.4 Classical nucleation theory

Previous study on the formation of β-Nb rich precipitates in Zr–Nb alloys has shown that β-Nb rich precipitates form when the Nb concentration exceeds approximately 1.0 wt%^[35]. To simulate β-Nb rich precipitates at low Nb concentration, we employed classical nucleation theory (CNT) to model the nucleation phenomenon of β-Nb rich precipitates^{[36, 37]}.

The driving force for nucleation ($\Delta G_V$), given in Eq. (8), represents the volumetric chemical free energy change associated with the transformation from the α phase and β phase^[36]. $\Delta G_V$ is evaluated based on the equilibrium molar solvus concentration of Nb in the α phase, $c_{Nb}^{\alpha, e}$, and the molar concentration of Nb in the β phase, $c_{Nb}^\beta$

The nucleation rate ($I$), given in Eq. (9), is based on classical nucleation theory^[36]. In nucleation rate, $N_v$ is defined as the number of available nucleation sites normalized by volume, $Z$ is the Zeldovich factor accounting for the stability of critical nuclei, $\beta^*$ is the rate at which atoms attach to the nucleus, $\gamma_i$ represents the interfacial energies between α-Zr matrix and β-Nb rich precipitates, and $r^*$ is the critical nucleus radius determined by the balance between the surface and volume energy contributions. Furthermore, $r^*$, $Z$, and $\beta^*$ can be expressed based on the classical nucleation theory as follows^[36].

Where $V_\alpha$ denotes the atomic volume in the α-Zr matrix, $a$ is the lattice constant of the β-Nb rich phase, and $D_{Nb}^\alpha$ represents the diffusivity of Nb in the α-Zr matrix. Since the volume of an atom is extremely small, $V_\alpha$ was normalized in this study. Therefore, the Zeldovich factor is interpreted in a relative sense rather than as an absolute quantity.

2.5 Influence of interfacial coherency on the interfacial energy between the α-Zr matrix and β-Nb rich precipitate

According to a study that calculated the interfacial energy between the α-Zr matrix and the β-Nb rich precipitate using the Embedded Atom Method (EAM), the interfacial energy ($\gamma_i$) was reported to vary depending on the interfacial coherency (semi-coherent, coherent, and incoherent) between the two phases^[38].

In this study, based on the interfacial energy values obtained from previous study, the gradient energy coefficients ($\kappa_i$) for each case were calculated using Eq. (13) ^[34].

Based on the magnitude order of the calculated $\kappa_i$ values, the case with the smallest value was named Type 1 (Semi-coherent), the intermediate value was named Type 2 (Coherent), and the largest value was named Type 3 (Incoherent). Here, $\gamma_i$ and $\kappa_i$ correspond to the interfacial energy and gradient energy coefficient for Type 1 (Semi-coherent), Type 2 (Coherent), and Type 3 (Incoherent), respectively. Interface structures for Type 1–3 are shown in Fig. 1.

Fig.1. Atomistic configurations of the α-Zr/β-Nb interface for (a) Type 1 (Semi-coherent), (b) Type 2 (Coherent), and (c) Type 3 (Incoherent) misorientation relationships. Blue circles represent Zr atoms and green circles represent Nb atoms.

In the phase-field method, the gradient energy coefficient is a parameter that governs the interfacial energy and interface width, and is therefore employed to represent the interfacial energy in the present simulations. Also, the gradient energy coefficient value is often adjusted from the physical magnitude to ensure numerical stability and computational efficiency^[39]. The gradient energy coefficients calculated from interfacial energies obtained using the EAM were excessively large for direct implementation in the phase-field method. Accordingly, a scaling factor was applied to preserve their relative magnitudes while ensuring numerical stability in the simulations. The interfacial energies, scaling factor, and gradient energy coefficients before and after scaling used in this study are summarized in Table 2.

2.6 Simulation setup

When producing Zr–Nb alloys, an artificial annealing process is performed at a specific temperature to precipitate β-Nb rich precipitates^{[40, 41]}. The simulations were performed at 873 K, which corresponds to the annealing temperature for the β-Nb rich precipitates in the Zr–Nb alloy^[41]. The parameters, material properties, and equations used in this study are as follows.

3.1 Nucleation of β-Nb rich precipitates with different interface types

Simulations were performed under Type 1 (Semi-coherent), Type 2 (Coherent), and Type 3 (Incoherent) interfacial conditions for Zr–Nb alloys with Nb concentrations of 1.0, 1.25, and 1.5 mol%. The simulations were conducted in two stages. Nucleation was performed up to t = 25 s, after which the post-nucleation evolution of precipitates was simulated up to t = 100000 s.

Fig. 2 illustrates the temporal evolution of β-Nb rich precipitates in the Zr–1.25 mol% Nb system under the Type 1 (Semi-coherent) interfacial condition. At the early stage (t = 4 s), numerous small nuclei are formed, whereas at later times (t = 100000 s), the precipitates grow and coalesce, leading to reduced number of precipitates. β-Nb rich precipitates nucleate and subsequently grow through coalescence with surrounding precipitates. In this study, precipitates whose $c_{Nb}$ exceeded 30 mol% were defined as β-Nb rich precipitates once they entered the growth stage^[6].

Fig. 2. Nb concentration fields at (a) the initial stage (t = 4 s) and (b) a later stage (t = 100000 s for Type 1 (Zr–1.25 mol% Nb), where c denotes the Nb concentration).

According to CNT, the nucleation rate (Eq. (9), $I$) is significantly influenced by the driving force for nucleation (Eq. (8), $\Delta G_V$), and the interfacial energy ($\gamma_i$). Fig.3–Fig.5 plot the total number of precipitates, the average area per precipitate, and the total area of precipitates with respect to time. Fig. 3, which shows the total number of precipitates, reveals that Type 3 (Incoherent) precipitates do not form at Nb concentrations of 1.0 and 1.25 mol%, but begin to precipitate and grow at 1.5 mol% Nb. The increased Nb concentration provides sufficient driving force for nucleation, enabling the stabilization of nuclei at the Type 3 (Incoherent) interface characterized by a high interfacial energy.

Fig. 3. Evolution of the number of β-Nb rich precipitate with respect to time in Zr–Nb system. (a) 1.0 mol% Nb, (b) 1.25 mol% Nb, and (c) 1.5 mol% Nb

Fig. 4. Evolution of the average area per β-Nb rich precipitate with respect to time in Zr–Nb system. (a) Zr–1.0 mol% Nb, (b) Zr–1.25 mol% Nb, and (c) Zr–1.5 mol% Nb

Fig. 5. Evolution of the total area of β-Nb rich precipitates with respect to time in Zr–Nb system. (a) Zr–1.0 mol% Nb, (b) Zr–1.25 mol% Nb, and (c) Zr–1.5 mol% Nb

As shown in Fig. 5, the total area of precipitates increases with increasing Nb concentration. With increasing Nb concentration in the domain, sufficient Nb becomes available for precipitate growth, resulting in an increase in the total precipitate area.

The result of simulations performed using the chemical free energies and parameters described in Section 2.6 were compared with thermodynamic calculations carried out using Thermo-Calc^[43]. Table 4 compares the Nb concentration in the β-Nb rich phase in Zr–Nb alloys calculated using Thermo-Calc and the maximum Nb concentration within the β-Nb rich precipitate achieved in this study. In this study, it was confirmed that when β-Nb rich precipitates formed, the two values showed similar results.

3.2 Nucleation of β-Nb rich precipitates with coexisting interface types

In the previous Section 3.1, simulations were conducted assuming precipitates possess a single interface type. However, in actual Zr–Nb alloys, β-Nb rich precipitates form with diverse interface types^{[38, 44]}. Therefore, in this study, the nucleation and subsequent evolution of precipitates were simulated under conditions where interface types Type 1 (Semi-coherent), Type 2 (Coherent), and Type 3 (Incoherent)—as defined in Section 2.5—coexist within a single domain. Because the nucleation rate of the β-Nb rich precipitates is evaluated under identical conditions and driving forces for all interface types, multiple precipitation events may occur at the same spatial location. To suppress the occurrence of multiple precipitation events at the same spatial location, a modeling constraint was introduced such that only one precipitate corresponding to a single interface type is allowed to form per time step.

Fig. 6 plots the order parameter ($\eta_i$) of precipitates Type 1 (Semi-coherent), Type 2 (Coherent), and Type 3 (Incoherent) with different interface types. Precipitates with different interface types (Type 1, Type 2, and Type 3) were generated through nucleation up to t = 25 s, after which the post-nucleation evolution was simulated up to t = 100000 s. As defined in Section 3.1, precipitates with $c_{Nb}$ exceeding 30 mol% were defined as β–Nb rich precipitates.

Fig. 6. Order parameters ($\eta_i$) for β-Nb rich precipitates plotted for each interface type. Blue circles denote Type 1 (Semi-coherent), red circles denote Type 2 (Coherent), and green circles denote Type 3 (Incoherent).

Fig. 7 plots the change in the total number of precipitates after nucleating β-Nb rich precipitates with three interface types. As the Nb concentration increases, the total number of precipitates also shows an increasing trend. Furthermore, precipitates with the Type 3 (Incoherent) interface type, which did not form at an Nb concentration of 1.0 mol%, began to precipitate as the Nb concentration increased to 1.25 mol% and 1.5 mol%, due to the increased driving force provided by the higher Nb concentration.

Fig .7. Evolution of the number of β-Nb rich precipitates with coexisting Type 1–3 interfaces with respect to time in the Zr–Nb system. (a) Zr–1.0 mol% Nb, (b) Zr–1.25 mol% Nb, and (c) Zr–1.5 mol% Nb

Fig.8 plots the change in the total area of precipitates after nucleating β-Nb rich precipitates with three interface types. An increase in the total area of precipitates was observed as the Nb concentration increased. The nucleation and growth of Type 3 (Incoherent) precipitates, which had the highest interfacial energy, were the most restricted, while the nucleation and growth of Type 1 (Semi-coherent) precipitates, which had the lowest interfacial energy, were the most dominant.

Fig. 8. Evolution of the total area of β-Nb rich precipitates with coexisting Type 1–3 interfaces with respect to time in the Zr–Nb system. (a) Zr–1.0 mol% Nb, (b) Zr–1.25 mol% Nb, and (c) Zr–1.5 mol% Nb

As shown in Fig.8, the total precipitates area increases as precipitation proceeds. However, as the simulation time approaches 100000 s, we observe that Type 2 (Coherent) and Type 3 (Incoherent) precipitates, which have relatively large interfacial energies, are eliminated, while Type 1 (Semi-coherent), which has the smallest interfacial energy, grows. Because the system minimizes its total free energy by eliminating Type 2 (Coherent) and Type 3 (Incoherent) with large interfacial energies and promoting the growth of Type 1 (Semi-coherent) with a small interfacial energy ^[45].