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# Phase-Field Simulation of δ Hydride Precipitation with Interfacial Anisotropy

1 School of Materials Science and Engineering, Collaborative Innovation Center of Ministry of Education and Shanxi Province for High-Performance Al/Mg Alloy Materials, North University of China, Taiyuan, 030051, China

2 Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing, Beijing, 100083, China

3 Institute of Materials Intelligent Technology, Liaoning Academy of Materials, Shenyang, 110004, China

* Corresponding Author: Yuhong Zhao. Email:

*Computers, Materials & Continua* **2023**, *77*(2), 1425-1443. https://doi.org/10.32604/cmc.2023.044510

**Received** 01 August 2023; **Accepted** 31 October 2023; **Issue published** 29 November 2023

## Abstract

Previous studies of hydride in zirconium alloys have mainly assumed an isotropic interface. In practice, the difference in crystal structure at the interface between the matrix phase and the precipitate phase results in an anisotropic interface. With the purpose of probing the real evolution of hydrides, this paper couples an anisotropy function in the interfacial energy and interfacial mobility. The influence of anisotropic interfacial energy and interfacial mobility on the morphology of hydride precipitation was investigated using the phase-field method. The results show that the isotropy hydride precipitates a slate-like morphology, and the anisotropic hydride precipitates at the semi-coherent and non-coherent interfaces exhibited parallelogram-like and needle-like, which is consistent with the actual experimental morphology. Compared with the coherent interface, the semi-coherent or non-coherent interface adjusts the lattice mismatch, resulting in lower gradient energy that is more consistent with the true interfacial state. Simultaneously, an important chain of relationships is proposed, in the range of ( or ), with the increase of the anisotropic mobility in the y-axis, the gradient energy increases (decreases), the tendency of the non-coherent (semi-coherent) relationship at the interface, and the precipitation rate of hydride decreases (increases). Furthermore, the inhomogeneous stress distribution around the hydride leads to a localized enrichment of the hydrogen concentration, producing a hydride tip. The study of interfacial anisotropy is informative for future studies of hydride precipitation orientation and properties.## Keywords

Zirconium alloy is widely used in nuclear fuel rod cladding for nuclear reactors, motivated by excellent mechanical properties, corrosion resistance, and low neutron adsorption cross-section [1]. In nuclear reactors, at constant temperatures, the concentration of hydrogen in a solid solution is lower than the final solid solubility of the hydride precipitate [2]. When the hydrogen concentration reaches its solid solution limit, the zirconium alloy absorbs hydrogen to produce brittle

The nature of the interface between the

The phase-field method is a good remedy for the inability of experiments to observe the phase transformation process dynamically and continuously [12–14]. This method has been used to simulate and predict the microstructure and tissue morphology evolution during the phase transformation of materials by solving the controlling equations [15–17]. It helps to understand the precipitation mechanism of solid phase change processes [18–24]. The phase-field method has been widely used to simulate the microstructure of hydride in zirconium alloy [25–29]. Usually, the majority of research on stress-induced (internal stress and external load) nucleation, growth, stacking, and reorientation behavior of

In fact, due to the difference in crystal structure between the hexagonal close-packed (HCP) matrix phase and the face-centered cubic (FCC)

The phase-field simulation for

Cahn-Hilliard phase-field equation:

Allen-Cahn phase-field equation:

where

Total free energy of the system [44]:

where

The chemical free energy density

where

The strain energy can be calculated according to the Khanchaturyan theory of micro-elastic strain energy [46–48]:

where

The lattice difference between the FCC

where

In the KKS (Kim, Kim, and Suzuki) phase field model, interfacial mobility is related to interfacial energy [52]. The interfacial mobility

where

The relevant parameters are shown in the following Table 1:

3 Simulation Results and Analysis

The crystal structure and lattice constant of the hydride precipitate phase and matrix phase are different, resulting in a semi-coherent or non-coherent interface. Differences in interfacial states will produce differences in interfacial energy, resulting in interfacial anisotropy of the hydride, which affects the precipitate morphology and orientation of the hydride. Fig. 3 is a schematic diagram of the hydride variants with different precipitation orientations. The formation of

To obtain accurate hydride morphology characteristics, the different variants were simulated and tested individually. Fig. 5 shows the precipitated morphology, equivalent forces diagram, and the circumferential length and radial height curves with time for different time steps of the individual

where

In general, the interfacial mobility coefficient is closely related to the interfacial energy. Therefore, anisotropic interfacial energy leads to the anisotropy of interfacial mobility. The change of hydride precipitated morphology is observed by changing the ratio of x and y-axis coefficients in interfacial mobility. Fig. 6 shows the hydride morphology and equivalent force diagrams for different anisotropic mobility ratios. The variation curves of circumferential length and radial height and aspect ratio of hydride with anisotropic mobility ratio are shown in Fig. 7. As the y-axis mobility ratio increases, the circumferential length peaks significantly at

The magnitude of energy will affect the nucleation and growth of hydride. The gradient energy coefficient is related to interfacial energy, and the magnitude of anisotropic interfacial energy will affect the variation of gradient energy. Under the action of anisotropic interfacial energy, hydride forms parallelogram-like and needle-like morphology. Figs. 10 and 11 are isotropic and anisotropic hydride gradient energy diagrams and curves. Interfacial anisotropy results in a decrease in gradient energy. Due to the influence of interfacial anisotropy between the hydride and matrix phase, the structure change at the interface is larger than the concentration change. Therefore, the gradient energy related to the structure tends to decrease significantly compared with the gradient energy related to the concentration.

Interfacial anisotropy leads to different hydride morphology, and there are differences in the stress states around different hydride morphology. Fig. 12 shows the stress distribution of anisotropic hydride. The evolution of the equivalent stress with time step is shown in Fig. 12a. The stress distribution shows that the equivalent stresses are concentrated at the hydride tips, and the stresses are less at the sides and inside the hydride. Figs. 12b–12d show the positive stress in the x direction, the positive stress in the y direction, and the shear stress in the xy direction of the hydride, respectively. The stress value around the hydride is negative, which is manifested as compressive stress, and the positive stress value shows tensile stress. From Figs. 12b and 12c, it can be seen that

Under the effect of anisotropic interfacial energy and interfacial mobility, hydrides precipitate needle-like and parallelogram-like morphologies (Fig. 4b). Yuan et al. [6] obtained

Interfacial isotropy assumes that the interface is coherent. However, in practice, the semi-coherent or non-coherent interfacial state between HCP and FCC crystal structures will generate anisotropy. The effect on hydride morphology was investigated by varying the y-axis anisotropic interfacial mobility, as shown in Fig. 6. It was found that the y-axis mobility coefficient

Hydride precipitated morphology and orientation depend on the interfacial energy and interfacial mobility. The gradient energy coefficient is related to the interfacial energy [59,62]. Under the action of anisotropic interfacial energy and interfacial mobility, variant 1 hydride precipitates needle-like morphology, and variant 2 and variant 3 hydride precipitates parallelogram-like morphology. Figs. 10 and 11 show the gradient energy diagram and energy curve of the interfacial isotropy and anisotropy hydride in Fig. 3. The interfacial anisotropy gradient can be reduced compared to the interfacial isotropy. As the interfacial anisotropy adjusts the lattice mismatch, it reduces the energy difference due to the lattice mismatch, resulting in a lower gradient energy. The structural difference changes significantly compared to the concentration difference. Therefore, the structure gradient energy decreases significantly more than the concentration gradient energy. It shows that the non-coherent or semi-coherent interface conforms to the interfacial state between the precipitate hydride phase and the matrix phase. It is more advantageous to replace the coherent interface with a semi-coherent or non-coherent interface close to the real situation.

Differences in the precipitated morphology and growth characteristics of anisotropic hydride affect the change in the stress state around the hydride. Fig. 12 represents the stress state of the anisotropy hydride, where the equivalent force is concentrated at the tip. Due to the mismatch between the two phases lattice, the matrix is strained to accommodate the growing hydride. The matrix part around the hydride is compressed, the rest is stretched, and the hydrogen atoms tend to diffuse toward the tension zone [25]. The internal stress of the hydride precipitation in the x-axis direction is small, and the tensile stress and compressive stress at the tip are symmetrical. Since the regions above and below the hydride are in a compressed state, the hydrogen depletion region is formed, while the region near the hydride edge is in a stretched state, thus attracting hydrogen atoms. The tension region also tends to promote hydride precipitation, while compression impedes it, thus causing the hydride shape to be elongated.

In this study, phase-field simulation was performed on

1. The interfacial isotropic hydride precipitated morphology is slat-like. The addition of interfacial anisotropy, under the combined effect of interfacial energy and anisotropic interfacial mobility, transforms the hydride morphology from slat-like to parallelogram-like and needle-like morphology. This study optimizes the model to be closer to the real state, which is consistent with the experimental results.

2. As the y-axis mobility coefficient

3. The gradient energy coefficient is related to the interfacial energy. The anisotropy of the non-coherent or semi-coherent interface reduces the lattice mismatch between the precipitate phase and the matrix phase interface. Reduce the energy difference caused by lattice distortion, resulting in a decrease in gradient energy related to interfacial anisotropy. The semi-coherent or non-coherent interface instead of the coherent interface conforms to the real interfacial state between the precipitate phase and the matrix phase.

4. The hydride morphology affects the stress state around the hydride. Due to the lattice mismatch between the two phases, the hydride in the matrix is under compression on both sides, hindering the absorption of hydrogen atoms. The tip and inside are stretched, promoting the absorption of hydrogen atoms, resulting in stress concentration at the tip and elongation of the hydride. At the same time, the uneven stress distribution around the hydride causes local enrichment of hydrogen concentration and produces tip morphology.

Acknowledgement: The authors especially acknowledge Prof. Sanqiang Shi of Hong Kong Polytechnic University and Dr. Zhihua Xiao of Peking University.

Funding Statement: The authors gratefully acknowledge financial support from the National Natural Science Foundation of China (Nos. 52375394, 52074246, 52275390, 52205429, 52201146), National Defense Basic Scientific Research Program of China (JCKY2020408B002, WDZC2022-12), Key Research and Development Program of Shanxi Province (202102050201011, 202202050201014), Science and Technology Major Project of Shanxi Province (20191102008, 20191102007) and Guiding Local Science and Technology Development Projects by the Central Government (YDZJSX2022A025, YDZJSX2021A027).

Author Contributions: Investigation: H.L. Nie, X.C. Shi, W.K. Yang, K.L. Wang, Y.H. Zhao; data collection: H.L. Nie, X.C. Shi, W.K. Yang; analysis and interpretation of results: H.L. Nie, X.C. Shi, W.K. Yang, K.L. Wang, Y.H. Zhao; draft manuscript preparation: H.L. Nie, X.C. Shi, W.K. Yang, K.L. Wang, Y.H. Zhao; software and funding: Y.H. Zhao. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: All data generated or analysed during this study are included in this article and are available from the corresponding author upon reasonable request.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

H. Nie, X. Shi, W. Yang, K. Wang and Y. Zhao, "Phase-field simulation of δ hydride precipitation with interfacial anisotropy,"*Computers, Materials & Continua*, vol. 77, no.2, pp. 1425–1443, 2023.