Phase-Field Simulation of Grain Growth in Uranium Silicide Nuclear Fuel

Abstract

Uranium silicide (U₃Si₂) is regarded as a viable fuel option for improving the safety of nuclear power plants. In the present work, phase-field simulations were employed to investigate grain growth phenomena, encompassing both isotropic and anisotropic grain growth. In simulations of isotropic grain growth, it is commonly assumed that the energy and mobility of the grain boundaries (GBs) remain constant, represented by average values. The calculated grain growth kinetic rate constant, K, exhibits a close correspondence with the experimental measurements, indicating a strong agreement between the two. In simulations of anisotropic grain growth, the values of GB energy and mobility are correlated with the angular disparity between GBs. The simulation results demonstrated that the growth rate of U₃Si₂ can be influenced by both the energy anisotropy and mobility anisotropy of GBs. Furthermore, the anisotropy in mobility results in a greater prevalence of low-angle GB distribution in comparison to high-angle GBs. However, the energy anisotropy of GBs does not impact the frequency distribution of the angle difference between GBs.

1. Introduction

As the predominant fuel utilized in light water reactors (LWRs), UO₂ exhibits a significant drawback in low thermal conductivity, which diminishes as the temperature rises [1,2]. During the operation of the reactor, the low thermal conductivity of UO₂ can potentially lead to excessive core temperatures. Following the Fukushima nuclear accident in Japan in 2011, there has been an increased demand at the international level for enhanced safety and reliability standards for both new generation and existing reactors. In addition to the design of fault tolerance performance in nuclear power plants, the development of accident tolerant fuel (ATF) is considered a crucial step towards enhancing the inherent safety and accident tolerance of the fuel. This measure plays a significant role in improving the overall safety of nuclear power plants [3,4]. Currently, the UO₂ fuel design has reached its technological performance limit. Therefore, it is imperative to identify appropriate alternative fuel pellet materials that can either maintain or enhance fuel performance under normal operating conditions, operational transitions, and events that exceed the design basis and during the design basis [5]. The thermal conductivity of U₃Si₂ (98%TD) exhibits a remarkably high value, measuring at 14.7 W/(m·K) [6] at a temperature of 573 K. In comparison, UO₂ (95%TD) demonstrates a lower thermal conductivity of 6.5 W/(m·K) [7] at the same temperature. Due to its high thermal conductivity, the heat generated within the core will be efficiently dissipated, thereby maintaining a low operating temperature inside the core. This characteristic significantly mitigates the risk of core melting caused by excessive temperatures. Meanwhile, U₃Si₂ exhibits a notable uranium density [8], thereby enabling an extension of the core refueling period and enhancing the overall economic efficiency of the reactor. Therefore, U₃Si₂ is regarded as one of the most appropriate advanced test fuels (ATFs) for replacing UO₂.

U₃Si₂ fuel pellets can be fabricated using spark plasma sintering technology [9,10,11,12], during the sintering process, the grain size of U₃Si₂ increases, typically resulting in fuel pellets with micrometer-sized grains [9,13,14]. Polycrystalline U₃Si₂ materials consist of numerous grains exhibiting various orientations. The grain boundary (GB) refers to the regions where the grains possessing distinct orientations intersect. Grain growth can be understood as a fundamental mechanism involving the displacement of GBs. Under typical conditions, the development of these grains exhibits a gradual and slow progression. The driving force behind grain growth is the reduction in GB energy, which is the disparity in interfacial energy before and following the process of grain growth. The driving force in the grain growth phase-field model established in this article is the surface tension of the GB. Under the action of this force, the GB will move towards the direction of the resultant force of the surface tension. Grain growth leads to the continuous disappearance of small grains and the continuous growth of large grains, resulting in a decreasing area of grain boundaries in the system and a continuous decrease in energy. The conventional approach for quantifying grain growth involves measuring the alteration in the average grain size and subsequently determining the correlation coefficient using grain growth kinetics. As a highly effective approach in simulating the evolution of mesoscale structures, the phase-field method offers comprehensive models for simulating grain growth [15,16,17,18,19,20,21], Furthermore, notable advancements have been achieved in simulating U₃Si₂ grain growth using this method [22,23,24]. Cheniour et al. [22,23] developed a grain growth model for U₃Si₂ utilizing experimental data and phase-field simulation. In the proposed model, the grain growth constant and activation energy were derived from isothermal annealing experiments conducted on U₃Si₂ lamellae. Subsequently, the grain growth geometric constant was calculated using multiple 3D phase-field grain growth simulations. Finally, the GB mobility was determined by combining experimental data, GB energy, and phase-field simulations. Ma et al. [24] conducted a study on the nucleation and evolution of grains in U₃Si₂ using phase-field simulations. They analyzed the distribution of grains and found that the grain size evolution of U₃Si₂ followed a Rayleigh distribution. However, most phase-field simulations conducted on U₃Si₂ grain growth have primarily focused on isotropic grain growth. It is important to note that the grains of U₃Si₂ are anisotropic, and the current phase-field simulations for U₃Si₂ grain growth have not yet considered this aspect. To accurately simulate the grain growth phenomenon in U₃Si₂, it is imperative to consider the impact of anisotropy on the grain growth process and determine the kinetic parameters associated with grain growth in anisotropic conditions.

Phase-field simulations are employed to investigate the phenomenon of grain growth in U₃Si₂, encompassing both isotropic and anisotropic grain growth. In Section 2, we present a phase-field model for simulating the grain growth of U₃Si₂. This model involves the determination of the free energy density and the calculation of the correlation between the phase-field parameters and material property parameters. In Section 2.2, a dimensionless method is proposed to streamline the computer programming. Section 3 presents the simulation results, with a particular emphasis on the process of isotropic grain growth in Section 3.1. The calculation of the correlation coefficient for grain growth kinetics was performed by analyzing the average changes in grain size simulated through the phase-field method. Additionally, the obtained simulation results were examined by varying the phase-field parameters after applying dimensionless transformation. Section 3.2 delves into the process of anisotropic grain growth. An anisotropic model was employed to simulate the impact of GB energy anisotropy and mobility anisotropy on the process of grain growth. The analysis focused on the variations in average grain size and the distribution of angle differences in the presence of anisotropic conditions. The Section 4 serves as the conclusion of the simulation.

2. Phase-Field Model of U₃Si₂ Grain Growth

2.1. Establishment of Phase-Field Model

In this article, a mesoscale simulation method known as the phase-field method is employed to simulate the grain growth process of U₃Si₂ fuel. This method offers significant advantages in accurately simulating the microstructure evolution of materials. A significant benefit of employing the phase-field method is its ability to track interface changes without the need for additional parameters. The phase-field variables undergo continuous changes at the GB, resulting in different values of the same phase-field variable at the GB compared to inside the grain. The determination of whether the position is situated at the GB or within the grain can be achieved by evaluating the value of a phase-field variable or a combination of multiple phase-field variables.

To date, a multitude of phase-field models have been put forth by researchers to simulate the internal grain growth process of materials. Each phase-field model is specifically designed to tackle distinct issues or enhance the limitations of preceding models. In the present study, the phase-field model chosen is the well-established, classic phase-field model [25,26]. One justification for choosing this model lies in its simplistic mathematical structure, which allows for a comprehensive simulation of grain growth through straightforward interactions between grains and realistic parameters. This model has been widely cited by researchers to study and design grain growth problems. Its accuracy and reliability have been established and can be guaranteed. Another significant factor is the robust physical basis of the model, whereby all parameters of the phase-field model can establish meaningful correlations with the physical parameters of real materials. This characteristic proves advantageous in the analysis of simulation outcomes and facilitates their comparison with experimental results.

The order parameter η_i (r, t) (i = 1, 2, …, p) is utilized as the phase-field variable in the U₃Si₂ grain growth phase-field model to represent various grain orientations. There is a multitude of possible orientations for grains in real-world materials. In the context of the simulation study, the accurate simulation of the grain growth process can be achieved by selecting a limited yet sufficient number of orientations. In the present study, a comprehensive analysis was conducted involving a total of 32 distinct orientations, denoted as p = 32. The model employed in this study can be accurately depicted as follows, which refers to the model in reference [27]:

$$F=\int \left\{m\left[\sum _{i=1}^p\left({\displaystyle \frac{{\eta }_i^4}{4}}-{\displaystyle \frac{{\eta }_i^2}{2}}\right)+a_{gb}\sum _{i=1}^p\sum _{j\ne i}^p{\eta }_i^2{\eta }_j^2+{\displaystyle \frac{1}{4}}\right]+{\displaystyle \frac{\kappa }{2}}\sum _{i=1}^p{\left|\nabla {\eta }_i\right|}^2\right\}dV$$

(1)

The symbol F represents the overall free energy of the system, which can be divided into two components. The initial term within the integral sign on the right side of the equation denotes the density of bulk free energy, while the subsequent term represents the density of gradient energy. In the context of bulk free energy density, the symbol “m” denotes a physical quantity that is associated with energy density and is measured in units of J/m³. The coefficient of grain interaction, denoted as a_gb, is a dimensionless physical quantity. The inclusion of the term 1/4 in the bulk free energy density ensures that the minimum value of the bulk free energy density is equal to 0. Additionally, in the gradient energy density, the coefficient κ represents the gradient energy coefficient.

The variation in GB order parameters over time can be controlled by the Allen–Cahn equation:

$${\displaystyle \frac{\partial {\eta }_i}{\partial t}}=-L{\displaystyle \frac{\delta F}{\delta {\eta }_i}}=-L\left(m\left({\eta }_i^3-{\eta }_i+2a_{gb}{\eta }_i\sum _{j\ne i}^p{\eta }_j^2\right)-\kappa {\nabla }^2{\eta }_i\right)$$

(2)

where L represents the GB mobility coefficient.

As previously stated, all the phase-field parameters utilized in the phase-field model employed in this study can be correlated with tangible physical quantities, such as GB energy. According to the theory proposed by Cahn and Hilliard et al. [28], the GB energy of a system can be defined as the difference in energy between a system containing GBs and a system without GBs. Therefore, the GB energy between a grain with orientation i and a grain with orientation j can be mathematically expressed as follows:

$$\begin{gathered} {\gamma }_{gb}=\underset{-\infty }{\overset{+\infty }{\int }}\left[f^{bulk}\left({\eta }_i,{\eta }_j\right)-f_{min}^{bulk}+{\displaystyle \frac{\kappa }{2}}{\left({\displaystyle \frac{d{\eta }_i}{dx}}\right)}^2+{\displaystyle \frac{\kappa }{2}}{\left({\displaystyle \frac{d{\eta }_j}{dx}}\right)}^2\right]dx \\ =\underset{-\infty }{\overset{+\infty }{\int }}\left[m\left[{\displaystyle \frac{{\eta }_i^4}{4}}-{\displaystyle \frac{{\eta }_i^2}{2}}+{\displaystyle \frac{{\eta }_j^4}{4}}-{\displaystyle \frac{{\eta }_j^2}{2}}+a_{gb}{\eta }_i^2{\eta }_j^2+{\displaystyle \frac{1}{4}}\right]+{\displaystyle \frac{\kappa }{2}}{\left({\displaystyle \frac{d{\eta }_i}{dx}}\right)}^2+{\displaystyle \frac{\kappa }{2}}{\left({\displaystyle \frac{d{\eta }_j}{dx}}\right)}^2\right]dx \end{gathered}$$

(3)

where the minimum value of the bulk free energy density $f_{min}^{bulk}$ equals to 0. The equation is simplified by defining the following expression:

$$f={\displaystyle \frac{{\eta }_i^4}{4}}-{\displaystyle \frac{{\eta }_i^2}{2}}+{\displaystyle \frac{{\eta }_j^4}{4}}-{\displaystyle \frac{{\eta }_j^2}{2}}+a_{gb}{\eta }_i^2{\eta }_j^2+{\displaystyle \frac{1}{4}}$$

(4)

When the system is in equilibrium, the following applies:

$$\begin{gathered} {\displaystyle \frac{\partial {\eta }_i}{\partial t}}=-L{\displaystyle \frac{\delta F}{\delta {\eta }_i}}=-L\left(m{\displaystyle \frac{\delta f}{\delta {\eta }_i}}-\kappa {\nabla }^2{\eta }_i\right)=0 \\ {\displaystyle \frac{\partial {\eta }_j}{\partial t}}=-L{\displaystyle \frac{\delta F}{\delta {\eta }_j}}=-L\left(m{\displaystyle \frac{\delta f}{\delta {\eta }_j}}-\kappa {\nabla }^2{\eta }_j\right)=0 \end{gathered}$$

(5)

According to the principle of variational calculus,

$$mf={\displaystyle \frac{\kappa }{2}}\left({\left({\displaystyle \frac{d{\eta }_i}{dx}}\right)}^2+{\left({\displaystyle \frac{d{\eta }_j}{dx}}\right)}^2\right)$$

(6)

Therefore, GB energy can also be expressed as follows:

$${\gamma }_{gb}=2{\int }_{-\infty }^{+\infty }mfdx=2{\int }_0^{1}mf{\displaystyle \frac{dx}{d{\eta }_i}}d{\eta }_i$$

(7)

According to Equation (6), the following can be concluded:

$${\displaystyle \frac{dx}{d{\eta }_i}}=\sqrt{{\displaystyle \frac{\kappa }{2mf}}\left[1+{\left({\displaystyle \frac{d{\eta }_j}{d{\eta }_i}}\right)}^2\right]}$$

(8)

Substitute Equation (8) into Equation (7), as follows:

$${\gamma }_{gb}=2{\int }_0^{1}mf{\displaystyle \frac{dx}{d{\eta }_i}}d{\eta }_i={\int }_0^1\sqrt{2\kappa mf\left[1+{\left({\displaystyle \frac{d{\eta }_j}{d{\eta }_i}}\right)}^2\right]}d{\eta }_i$$

(9)

In Equation (9), solving the integral on the right side of the equation poses a significant challenge due to the lack of knowledge regarding the specific relationship between η_j and η_j. However, assuming that the relationship between η_i and η_j is given by η_j(x) = 1 − η_j(x) [29], where x represents the position coordinates, we can derive the following relationship:

$${\displaystyle \frac{d{\eta }_i}{dx}}=-{\displaystyle \frac{d{\eta }_j}{dx}}, {\displaystyle \frac{d^2{\eta }_i}{{dx}^2}}=-{\displaystyle \frac{d^2{\eta }_j}{{dx}^2}}$$

(10)

Substitute Equation (10) into Equation (5),

$${\displaystyle \frac{\delta f}{\delta {\eta }_i}}=-{\displaystyle \frac{\delta f}{\delta {\eta }_j}}$$

(11)

Substitute Equation (4) into Equation (11), and remember that η_j(x) = 1 − η_i(x),

$$-\left(1+2a_{gb}\right){\eta }_i^3+\left(3+2a_{gb}\right){\eta }_i^2-2{\eta }_i=-\left(1+2a_{gb}\right){\eta }_i^3+4a_{gb}{\eta }_i^2-\left(2a_{gb}-1\right){\eta }_i$$

(12)

This requirement can only be fulfilled at every point along the interface when a_gb = 1.5. According to the research conducted by Moelans et al. [30], the complexity of solving Equation (9) increases significantly when the value of a_gb is not equal to 1.5. The primary research objective of this article is to investigate the anisotropic grain growth of U₃Si₂. Notably, this study aims to avoid the utilization of complex parameters that could potentially complicate the resolution of the program. Therefore, by selecting a_gb = 1.5, the accuracy requirements can be met and the calculation can be simplified.

With a_gb = 1.5 and η_j(x) = 1 − η_i(x), Equation (9) can be rewritten as follows:

$${\gamma }_{gb}={\displaystyle \frac{\sqrt{2}}{3}}\sqrt{\kappa m}$$

(13)

The interface in the phase-field model is characterized as a diffusion interface, with a width that is typically wider than the actual interface but still several orders of magnitude smaller than the grain size. The objective behind this action is to amplify the temporal and spatial dimensions of the simulation. There are typically multiple approaches available to define and calculate the width of a GB. In this article, the method of derivative absolute value is employed to calculate the width of the diffusion interface. Based on the assumption presented in the article by Ahmed et al. [31], it is postulated that the values of η_i(x) and η_j(x) exhibit continuous and uniform variation at a flat interface. Consequently, the width of the flat interface, denoted as δ, can be defined as follows:

$$\delta ={\displaystyle \frac{1}{\left|{\left(d{\eta }_i/dx\right)}_{x=0}\right|}}$$

(14)

where η_j(x) is a diagonal line with a fixed slope at the interface. Define x = 0 as the center position of the GB, so η_i(x = 0) = η_j(x = 0) = 0.5. Substitute Equation (8) into Equation (14), as follows:

$$\delta =\sqrt{{\displaystyle \frac{\kappa }{mf\left({\eta }_i={\eta }_j=0.5\right)}}}=\sqrt{{\displaystyle \frac{8\kappa }{m}}}$$

(15)

By integrating Equation (13) and Equation (15) and eliminating m and κ, respectively, the following can be obtained:

$$\kappa ={\displaystyle \frac{3}{4}}{\gamma }_{gb}\delta , m={\displaystyle \frac{6{\gamma }_{gb}}{\delta }}$$

(16)

From the analysis and derivation process, it becomes evident that GB energy plays a crucial role in establishing a connection between phase-field parameters and actual material property parameters. To ensure the precision of the simulation outcomes, it is imperative to utilize precise energy parameters for GBs. Currently, the public literature is scarce regarding the investigation of U₃Si₂ GB energy. However, a greater number of published studies have focused on determining the average GB energy of U₃Si₂. There is a limited body of studies in the literature available for studying GB energy from various perspectives. When investigating the phenomenon of isotropic grain growth, greater emphasis is placed on the average GB energy. When researching on anisotropic grain growth, it is imperative to consider the GB energy associated with varying angle differences. Beeler et al. [32] employed molecular dynamics (MDs) to simulate the cellular arrangement of U₃Si₂ and determine the GB energy of U₃Si₂. Additionally, they computed the average GB energy at various temperatures, as presented in

Table 1. Average GB energy of U₃Si₂ at different temperatures [32].

Temperature (K)	${\overline{\mathit{\gamma }}}_{\mathit{g}\mathit{b}}$ (J/m2)
0	0.84
400	0.89
800	0.96
1200	1.04
1600	1.22

. The average GB energy (referred to as free energy in Ref. [32]) results from the sum of the enthalpy of formation per unit area H(T) (referred to as GB energy in Ref. [32]) and the entropy of formation per unit area S(T) contributions according to [33],

$${\overline{\gamma }}_{gb}=H\left(T\right)-TS\left(T\right)$$

(17)

The average GB data are most readily fitted to a second order polynomial function [32], by computer fitting, we can rewrite Equation (17) into the following:

$${\overline{\gamma }}_{gb}\left(T\right)=1.21\times {10}^{-7}{T}^2+3.46\times {10}^{-5}T+0.847$$

(18)

2.2. Dimensionalization of Parameters

The temporal development of grains is governed by the Allen–Cahn equation (Equation (2)), and the coefficient of GB mobility, denoted as L, can be computed using the method proposed by [34], as follows:

$$L={\gamma }_{gb}{M}_{gb}/\kappa$$

(19)

The term “M_gb” represents the mobility of GB with the unit of m⁴/(J·s), and a detailed explanation of its calculation will be provided later. The unit of L can be expressed as m³/(J·s) according to Equation (19), while the unit of κ_η is J/m. The parameters in phase-field models typically have a small order of magnitude, and their units must be considered during calculations. This undoubtedly adds complexity to the calculations and can impact their accuracy. In order to enhance operational convenience and ensure operational accuracy, a dimensionless approach is employed to facilitate the handling of phase-field parameters involved in the operation. In our model, we chose a characteristic length l* and a characteristic time t*, so that the dimensionless length λ = l/l^* and dimensionless time τ = t/t* can be easily obtained, l and t are the actual length and time. The dimensionless gradient operator is $\tilde{\nabla }=l^{*}\nabla$, therefore, Equation (2) can be rewritten as follows:

$${\displaystyle \frac{\partial {\eta }_i}{\partial \tau }}=-t^{*}L\left(m\left({\eta }_i^3-{\eta }_i+2a_{gb}{\eta }_i\sum _{j\ne i}^p{\eta }_j^2\right)-{\displaystyle \frac{\kappa }{{\left(l^{*}\right)}^2}}{\tilde{\nabla }}^2{\eta }_i\right)$$

(20)

Note that the left side of the equation is dimensionless, and the right side of the equation can be dimensionless by introducing an energy density eigenvalue e*,

$$\begin{gathered} {\displaystyle \frac{\partial {\eta }_i}{\partial \tau }}=-t^{*}{e}^{*}L\left({\displaystyle \frac{m}{e^{*}}}\left({\eta }_i^3-{\eta }_i+2a_{gb}{\eta }_i\sum _{j\ne i}^p{\eta }_j^2\right)-{\displaystyle \frac{\kappa }{{e^{*}\left(l^{*}\right)}^2}}{\tilde{\nabla }}^2{\eta }_i\right) \\ =\tilde{L}\left(\tilde{m}\left({\eta }_i^3-{\eta }_i+2a_{gb}{\eta }_i\sum _{j\ne i}^p{\eta }_j^2\right)-\tilde{\kappa }{\tilde{\nabla }}^2{\eta }_i\right) \end{gathered}$$

(21)

where $\tilde{L}=t^{*}{e}^{*}L$, $\tilde{m}=m/e^{*}$, and $\tilde{\kappa }=\kappa /\left({e^{*}\left(l^{*}\right)}^2\right)$ are dimensionless phase-field variables, typically 10⁻¹ − 10² in size in actual calculations, and the selection of dimensionless feature values follows the following principles:

$$e^{*}={\displaystyle \frac{m}{\tilde{m}}}, { t}^{*}={\displaystyle \frac{\tilde{L}}{e^{*}L}}, { l}^{*}=\sqrt{\kappa /\left(\tilde{\kappa }{e}^{*}\right)}$$

(22)

3. Results

Existing literature predominantly focuses on the isotropic grain growth of U₃Si₂ and has yielded significant findings [22,23]. The initial phase of the research primarily centers on investigating isotropic grain growth to validate the model and parameters. The subsequent phase delves into the anisotropic grain growth of U₃Si₂, aiming to analyze and elucidate the growth patterns of U₃Si₂ from an anisotropic standpoint, thereby providing additional insights to the existing body of research. In the present article, the grain growth process of U₃Si₂ is simulated under two-dimensional conditions using finite difference programming. The Allen–Cahn equation (Equation (2)) is solved utilizing the explicit Euler algorithm. The dimensions of the grid are dx = dy = 0.5, and time interval dt = 0.05. In this study, the width of GB δ is chosen to be 20 nm.

3.1. Isotropic Grain Growth of U₃Si₂

When examining the isotropic grain growth model, this article considers the averaging of pertinent parameters, including GB energy, GB energy mobility, and gradient coefficient of order parameters. These parameters are fixed constants that remain unchanged regardless of the type of GB. According to the equation describing the kinetics of grain growth,

$$D^n-D_0^n=2\alpha {\overline{M}}_{gb}{\overline{\gamma }}_{gb}t=Kt$$

(23)

where n is the growth exponent, which usually takes a value between 2 and 4; α is a geometric constant that varies depending on the grain topology, and according to the research of Cheniour et al. [23], the value α of in this article is 0.96; K is a rate constant and its value is related to temperature, according to Arrhenius expression,

$$K\left(T\right)=K_0{e}^{-\left(Q/k_{B}T\right)}$$

(24)

where K₀ is pre-coefficient factor and Q is activation energy; k_B is the Boltzmann constant; Q and K₀ were determined from measurements of the changing average grain size in U₃Si₂ at various temperatures [23]; and Q is 0.33 eV and K₀ is 8.77 × 10⁻¹⁸ m²/s.

The GB mobility coefficient L is related to the GB energy mobility M_gb, which can be calculated using the Arrhenius expression, as follows:

$$M_{gb}\left(T\right)=M_0{e}^{-\left(Q/k_{B}T\right)}$$

(25)

where M₀ is the pre-factor. It can be obtained that using Equation (23) to Equation (25), the GB energy mobility M_gb at 673K is as follows:

$$M_0{\displaystyle \frac{K\left(T\right)}{2\alpha {\overline{\gamma }}_{gb\left(T\right)}}}{e}^{\left(Q/k_{B}T\right)}=4.81\times {10}^{-18}{m}^4/\left(\mathrm{J}\mathrm{s}\right)$$

(26)

In the context of isotropic simulation, the calculation of GB energy involves taking the average value. Similarly, the determination of GB mobility is performed utilizing Equation (25), which is also based on the average value. Other phase-field parameters can be determined utilizing the energy and mobility of GBs. Firstly, the selected dimensionless parameters are $\tilde{L}=1$, $\tilde{m}=1$, and $\tilde{\kappa }=1$. The dimensions of the simulation system are 1280 nm × 1280 nm, the grid of the simulation area is set to 512 × 512, so the number of grid points corresponding to the grain boundaries is 4. At the initial moment, there are 5000 small grains, which are evenly divided into 32 different grain orientations and the temperature being simulated is 673 K.

The temporal evolution of grain microstructure is depicted in Figure 1. In the case of isotropic grain growth simulation conditions, it has been observed that the growth of grains demonstrates isotropy. Furthermore, the rate of grain growth is not influenced by grain orientation, but rather by the curvature of GBs. The simulation results exhibit similarities to those observed in numerous simulations of isotropic grain growth.

The growth of grains adheres to the kinetics described in Equation (23). To validate the accuracy of the model presented in this article, a series of simulations were performed under identical conditions. The average grain size within the simulation interval was then calculated at various time points. Given that the simulation conducted in this article does not account for the potential impact of external stress or particle pinning on grain growth, the exponent n in the growth kinetics equation is determined to be 2. By utilizing the simulated data and employing Equation (23) for data fitting, the growth rate K was determined. Figure 2 illustrates the fitting result of the K value with four groups of simulation data, which has been determined to be K(673K) = 2.80 ± 0.102 × 10⁻²⁰ m²/s. Additionally, the K value obtained from the experiment is K(673K) = 2.97 ± 0.11 × 10⁻²⁰ m²/s. In this article, it is posited that the simulation results exhibit a close resemblance to the experimental results, as the margin of error falls within an acceptable range. Therefore, the model and parameters employed in the article exhibit a reasonable approach. Building upon this foundation, the subsequent text establishes an anisotropic grain growth model, ensuring its rationality.

The selection of phase-field parameters is not unique, as it must satisfy convergence during the operation process. In this study, various sets of simulations were employed by manipulating different phase-field simulation parameters to examine the potential impact of parameter selection on the simulation outcomes. Irrespective of the chosen set of phase-field parameters, the computed results must adhere to the dynamic principles governing grain growth. Therefore, the anticipated outcome of this article is that the simulation results will remain unaffected by the selection of phase-field parameters.

Table 2. The size of K values calculated under different dimensionless conditions.

Value of Phase-Field Parameters			K (nm2/s)	Error (nm2/s)
$\tilde{\mathit{L}}$	$\tilde{\mathit{m}}$	$\tilde{\mathit{\kappa }}$
1.0	1.0	0.5	0.0253	0.00157
1.0	1.0	1.0	0.0312	0.00159
1.0	1.0	2.0	0.027	0.0021
0.5	1.0	0.5	0.0261	0.00141
2.0	1.0	0.5	0.032	0.00124
1.0	0.5	0.5	0.0332	0.00288
1.0	2.0	0.5	0.02445	0.00175

presents the calculated magnitudes of the K values for various phase-field parameters and provides clear evidence that the calculated K values for various phase-field parameters exhibit minor variations, indicating a lack of significant error. Additionally, during the investigation of the impact of a specific parameter on the K value using the control variable method, it was observed that the K value does not exhibit a consistent pattern of change in response to variations in the said parameter. The variation in the calculated K values for different parameters can be primarily attributed to simulation errors.

3.2. Anisotropic Grain Growth of U₃Si₂

Section 3.1 presents the phase-field simulation method for modeling isotropic grain growth. In this approach, the phase-field parameters were chosen to have the same gradient coefficient and migration coefficient for grains of different orientations. Additionally, the average values of GB energy and GB migration rate were employed. The process of grain growth in materials exhibits anisotropy. Various types and orientations of GBs are associated with distinct GB energies, and the rates of GB migration vary depending on the angle [35,36,37,38,39]. The anisotropy of grains is attributed to the variation in the angle between GBs. The determination of the critical angle difference, denoted as θ_m, between high-angle gr GBs and low-angle GBs is a significant parameter within the context of the anisotropic model. Experimentally, the angle θ_m has been observed to range between 10° and 30°, with variations depending on the material. From the information provided in publicly available literature, the critical angle difference θ_m was chosen as 15° in this article [40,41,42].

U₃Si₂ has a tetragonal structure, with one side having a lattice constant different from the other side (a = b ≠ c) [43], so the grain orientation θ in two-dimensional simulation can be considered to be randomly within the range 0° ≤ θ ≤ 180°. The angle difference of grains is defined as the orientation difference between GBs with different orientations, ${\theta }_{ij}=\mathrm{m}\mathrm{i}\mathrm{n}\left(\left|{\theta }_i-{\theta }_i\right|,180-\left|{\theta }_i-{\theta }_i\right|\right)$ where θ_i is the orientation of the ⁱth oriented grain.

The relationship between GB energy and angle difference can be expressed as follows [40,44]:

$${\gamma }_{ij}\left({\theta }_{ij}\right)=\left\{\begin{array}{c}{{\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\gamma }_m\left[1-\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\right)\right] for {\theta }_{ij}<{\theta }_m \\ {\gamma }_m for {\theta }_{ij}\ge {\theta }_m\end{array}\right.$$

(27)

where γ_m is the GB energy corresponding to high-angle GBs; through Equation (27), it can be found that the GB energy of high-angle GBs takes the same value, while the GB energy of low-angle GBs is related to the value of angle difference.

The relationship between GB mobility and angle difference is [41] as follows:

$$M_{ij}\left({\theta }_{ij}\right)=\left\{\begin{array}{c}{M}_m\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left\{-5{\left({\theta }_{ij}/{\theta }_m\right)}^4\right\}\right] for {\theta }_{ij}<{\theta }_m \\ M_m for {\theta }_{ij}\ge {\theta }_m\end{array}\right.$$

(28)

The variable M_m represents the mobility of GBs at high angles. Similarly, it is widely accepted in the field that the mobility of high-angle GBs remains constant, whereas the mobility of the low-angle GBs is influenced by the difference in angles between the boundaries. Figure 3 illustrates the correlation ratio between the energy of low-angle GBs and their mobility, considering various angle differences and high-angle GBs. The energy and mobility of GBs with low angles are comparatively lower than those with high angles. Furthermore, as the angle difference decreases, the GB energy and mobility also decrease.

The calculation of GB energy and migration rate for high-angle GBs in this article is conducted using an integral form that establishes a relationship with the average GB energy and migration rate.

$${\displaystyle \frac{{\int }_0^{90}{\gamma }_{ij}\left({\theta }_{ij}\right)d{\theta }_{ij}}{{\int }_0^{90}{\theta }_{ij}d{\theta }_{ij}}}={\overline{\gamma }}_{gb}, {\displaystyle \frac{{\int }_0^{90}{M}_{ij}\left({\theta }_{ij}\right)d{\theta }_{ij}}{{\int }_0^{90}{\theta }_{ij}d{\theta }_{ij}}}={\overline{M}}_{gb}$$

(29)

When examining the anisotropic grain growth model, the phase-field parameters are computed using the same methodology as the isotropic model. The distinction lies in the fact that the parameters governing anisotropic grain growth are associated with variations in local orientation, and it is crucial to consider the modifications in simulation parameters when performing simulation calculations. In the simulation of anisotropic grain growth, the width of the GB (δ) is set to 20 nm, and the grain interaction coefficient (a_gb) is set to 1.5. This specific configuration is chosen to maintain consistency between the phase-field parameters and the actual parameters, as well as to ensure that the dimensionless derivation aligns with the description provided for isotropy in Section 3.1.

In the simulation conducted in this study, it is possible to consider the anisotropy of GB energy and migration rate as independent factors, as they can be analyzed separately. We performed three sets of simulations, which were designed to investigate the effects of different combinations of GB energy anisotropy and mobility isotropy, GB energy isotropy and mobility anisotropy, and GB energy anisotropy and mobility anisotropy. The phase-field parameters associated with anisotropy are as follows:

$${\tilde{\kappa }}_{ij}\left({\theta }_{ij}\right)=\left\{\begin{array}{c}{{\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\tilde{\kappa }}_m\left[1-\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\right)\right] for {\theta }_{ij}<{\theta }_m \\ {\tilde{\kappa }}_m for {\theta }_{ij}\ge {\theta }_m\end{array}\right.$$

(30)

$${\tilde{m}}_{ij}\left({\theta }_{ij}\right)=\left\{\begin{array}{c}{{\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\tilde{m}}_m\left[1-\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\theta }_{ij}}{{\theta }_m}}\right)\right] for {\theta }_{ij}<{\theta }_m \\ {\tilde{m}}_m for {\theta }_{ij}\ge {\theta }_m\end{array}\right.$$

(31)

$${\tilde{L}}_{ij}\left({\theta }_{ij}\right)=\left\{\begin{array}{c}{\tilde{L}}_m\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left\{-5{\left({\theta }_{ij}/{\theta }_m\right)}^4\right\}\right] for {\theta }_{ij}<{\theta }_m \\ {\tilde{L}}_m for {\theta }_{ij}\ge {\theta }_m\end{array}\right.$$

(32)

where ${\tilde{L}}_m$, ${\tilde{m}}_m$, and ${\tilde{\kappa }}_m$ are the gradient energy coefficient, barrier height, and diffusion coefficient corresponding to the high-angle GB. When the GB energy or mobility is isotropic, the corresponding phase-field variables correspond to the average GB energy or average mobility.

To investigate the influence of anisotropic conditions on grain growth, a simulation was conducted to simulate the grain growth process. The simulation area size was set to 1280 nm × 1280 nm, the grid of the simulation area is set to 512 × 512, so the number of grid points corresponding to the grain boundaries is 4. The simulation temperature was maintained at 673 K, and a total of 64 grain orientations were considered. The initial number of grains in the simulation was set to 5000. The dimensionless parameters are ${\tilde{L}}_m=1$, ${\tilde{m}}_m=1$, and ${\tilde{\kappa }}_m=0.5$. Figure 4 shows the morphology distribution of grains at 35,000 steps under different anisotropic conditions.

From the analysis of Figure 4, it is evident that in cases where both the GB energy and migration rate exhibit isotropy, the grains exhibit a regular morphology, with no noticeable presence of irregular grains. When the energy or migration rate of GB exhibits anisotropy, it results in the formation of irregular grains with an irregular appearance. The presence of these grains can be attributed solely to variations in the migration rate and GB energy among grains with different orientations. This discrepancy in the movement speed of different boundaries within the same grain during the simulation leads to the formation of irregular grains.

Figure 5 illustrates the distribution of grain size under various conditions. In the simulation, the initial number of grains is 180. The size of the simulation area is 2560 nm × 2560 nm, the corresponding grid size is 1024 × 1024, and the simulation temperature is maintained at 673 K. Figure 6 illustrates the frequency distribution of grain sizes after 10,000 simulation steps. From Figure 6, it can be observed that the grain size frequency distribution remains similar across the four different conditions of grain growth. The majority of grain sizes are concentrated around the average grain size, indicating a tendency towards homogeneity. Furthermore, the specific distribution of grain sizes does not necessarily exhibit a correlation with anisotropic conditions. Therefore, the phenomenon of anisotropic grain growth does not exert a substantial influence on the size distribution of grains.

To investigate the impact of anisotropic conditions on the average grain size and angle difference distribution, this section primarily focuses on three distinct conditions, anisotropy in GB energy, anisotropy in mobility, and anisotropy in both GB energy and mobility. This study involves simulating and analyzing the variations and distribution patterns in the average grain size and angle difference. Figure 6 illustrates the temporal evolution of the average grain size in the simulation system under three distinct simulation conditions. It is evident from Figure 6 that the temporal evolution of grain size exhibits significant variations under different anisotropic conditions. The average movement speed of GBs increases as the grain size of the system increases. When the simulation condition specifies that only GBs can exhibit anisotropy, the average growth rate of grains is observed to be the highest. Conversely, when the simulation condition includes anisotropy in both GB energy and migration rate, the average growth rate of grains is found to be the lowest. This observation suggests that the energy and migration rate of low-angle GBs impose significant constraints on the process of grain growth. When taking into account both the energy of GBs and the migration rate of low-angle GBs, it can be observed that grain growth is occurring at a relatively slow pace.

Figure 7 illustrates the frequency distribution of angle differences under various anisotropic conditions while simulating an equal number of steps. When solely considering the anisotropy of GB energy, the variation in angle difference follows a uniform distribution. When examining the anisotropy of mobility, it is observed that the distribution of angle differences experiences substantial alterations, resulting in a significantly higher proportion of the low-angle GBs compared to high-angle GBs. Based on the analysis of the model and simulation results, it is posited in this article that the consideration of GB energy anisotropy alone does not exert a substantial influence on the migration rate of GBs in the model. Therefore, the migration rate between low-angle and high-angle GBs does not exhibit a significant difference, and the distribution of angle differences tends to be uniform. However, when considering the anisotropy of the migration rate, it is observed that the migration rate of high-angle GBs is significantly higher than that of low-angle GBs. Based on the principle of minimizing free energy, the area of the GBs in the simulation system consistently diminishes. Due to the higher mobility of high-angle GBs, these GBs will persistently migrate until they transform into low-angle GBs. As a result, the migration velocity decreases, facilitating their retention within the system. Consequently, there is an observed increase in the proportion of low-angle GBs. The rationality of the explanation can also be demonstrated through the analysis of the temporal variation in the average angle difference, as depicted in Figure 8. Therefore, in this article, it is posited that the distribution of angle difference and the magnitude of average angle difference can be influenced only when considering the anisotropy of mobility.

4. Conclusions

A mesoscale phase-field method was employed to simulate the grain growth process of U₃Si₂ nuclear fuel. Respective isotropic and anisotropic grain growth simulations were conducted. The calculation of grain growth kinetic parameters under isotropic conditions was conducted. Anisotropic grain growth simulation was performed, incorporating angle differences to introduce GB energy anisotropy and mobility anisotropy into the phase-field model. In the present study, the morphology, grain size, and variations in grain angle, size, and distribution were investigated under different conditions of anisotropy, including anisotropy in grain boundary (GB) energy, mobility, and GB energy/mobility. The obtained conclusions are as follows:

(1) The growth of isotropic grains was investigated using the phase-field model, and the kinetic constant of grain growth was determined to be K(673K) = (2.80 ± 0.102) × 10⁻²⁰ m²/s. This finding aligns with the experimental results reported in the available literature. The analysis also considered the impact of parameter selection in phase-field simulation on the calculation results.

(2) The simulation results for anisotropy demonstrate that the anisotropy of GB energy has an impact on the rate of grain growth, but it does not influence the distribution of grain size. The anisotropy of GB energy does not have an impact on the alteration of the distribution of average angle differences.

(3) The anisotropy of mobility can also impact the grain growth rate. However, it does not influence the distribution of grain size. The anisotropy of mobility can simultaneously result in a higher proportion of low-angle differences compared to high-angle differences, thereby causing a reduction in the average angle difference.