Phase-field simulation of sintering process of ceramic composite fuel

Liao Yu-Xuan; Shen Wen-Long; Wu Xue-Zhi; La Yong-Xiao; Liu Wen-Bo

doi:10.7498/aps.73.20241112

Abstract

Authors and contacts

FullText HTML

1. Introduction
2. Model equation
3. 研究结果
3.1 扰动振幅对FPUT现象的影响
3.2 扰动信号与泵浦光之间相位差对FPUT现象稳定性的影响
3.3 噪声对FPUT现象稳定性的影响
4. 总结与讨论
数据可用性声明

References

Abstract

Due to the limitation of existing experimental techniques, it is difficult to observe the evolution of microstructure in the sintering process in real time, resulting in a lack of in-depth understanding of the sintering mechanism of two-phase composite fuels. Therefore, it is greatly important to carry out theoretical simulation studies in the sintering process of composite fuels. In this work, a phase-field model of the two-phase sintering process of ceramic composite fuel is established, and the sintering process of UN-U₃Si₂ composite fuel is simulated by using this method. The simulation results show that during the formation of sintering neck, the surface deformation of the grains with higher surface energy is significant. The size of the final equilibrium dihedral angle formed by the two-phase double grains depends on the ratio of the grain boundary energy to the surface energy of the two phases. The phenomenon of large grains swallowing small grains does not occur between the two unequal double grains. Subsequently, the pore shrinkage and the properties of the trident grain boundary among the two-phase three grains are investigated in the sintering process. It is found that the angle of the trident grain boundary formed by the two-phase three grains deviates from 120°. The high-energy barrier at the grain boundary hinders the diffusion of the pore vacancies along the grain boundary, resulting in a slow shrinkage rate of the pore vacancies at the trident grain boundary. In addition, the simulation results of the microstructure evolution of two-phase polycrystalline sintered tissue with different volume fraction ratios show that the grain boundary diffusion plays a major role in the two-phase sintering process. The grain growth of the phase with a higher volume fraction is dominant, and there exists a hindrance to the migration of grain boundaries between two-phase grains. The phenomenon of grain migration exists between grains of the same phase.

Keywords:

PACS:

78.55.-m	(Photoluminescence, properties and materials)
33.50.-j	(Fluorescence and phosphorescence; radiationless transitions, quenching (intersystem crossing, internal conversion))
42.70.-a	(Optical materials)

Authors and contacts

1.
Institute of Nuclear Science and Technology, Xian Jiaotong University, Xi’an 710049, China
2.
China institute of atomic energy, Beijing 102413, China
3.
Shaanxi Key Laboratory of Advanced Nuclear Energy and Technology, Xi’an Jiaotong University, Xi’an 710049, China

Corresponding author: Liu Wen-Bo, liuwenbo@xjtu.edu.cn

Funds: Project supported by the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (NSAF Joint Fund) (Grant No. U2130105) and the Innovative Scientific Program of China National Nuclear Corporation (CNNC).

FullText HTML

1. Introduction

In the early 1950s, Italian physicists Fermi, Pasta, Ulam and their student Tsingou simulated the evolution of a nonlinear system composed of 64 harmonic oscillators on the first generation of large computers, hoping to prove the principle of energy equipartition in statistical physics. At the initial moment, all the energy of these harmonic oscillators is concentrated on one oscillator, and the initial energy of other oscillators is zero. According to the principle of energy equipartition, the system should finally reach an equilibrium state, that is, the energy should be equally divided. ^{[ 1 ]} The discovery of this phenomenon has greatly promoted the development of nonlinear physics. The early research on FPUT phenomenon was mainly concentrated in the field of fluid mechanics. Because of the high cost of experimental observation requiring large ships and other experimental equipment, the research on FPUT phenomenon was gradually shelved. With the development of optical precision instruments and the progress of experimental technology, the observation of FPUT phenomenon in optical fibers was first realized in the laboratory in 2001. ^{[ 2 ]} Since then, the nonlinear physics community has set off a wave of research on the FPUT phenomenon. ^{[ 3 – 15 ]} .

The formation of FPUT phenomenon is related to the double periodic wave state and modulation instability of nonlinear system. ^{[ 3 – 15 ]} Modulation instability describes the exponential growth of a small disturbance applied to the plane wave of a system with the increase of propagation distance, which can effectively explain the initial growth process of the amplitude of the first-order disturbance signal with the increase of propagation distance in FPUT phenomenon. ^{[ 16 – 18 ]} , the growth of the amplitude of higher order perturbation components can be explained by the cascading modulation instability ^{[ 6 , 19 , 20 ]} When the amplitude of multiple perturbations increases to its maximum, the amplitude decays rapidly, but the reason has not yet been effectively explained. In theory, the relationship between different types of FPUT phenomena and modulation instability has also been studied. ^{[ 21 ]} The influence of high-order effect on FPUT and the frequency spectrum characteristics of FPUT under high-order effect are discussed. ^{[ 22 , 23 ]} The analysis of FPUT phenomenon by three-wave truncation method is explored. ^{[ 13 ]} In recent years, it has also been found that the FPUT phenomenon can be excited not only in the modulation instability region (called conventional modulation instability), but also outside the modulation instability region (called unconventional modulation instability). ^{[ 24 – 26 ]} This may be related to the fact that the nonlinear effect of perturbation is neglected in the calculation of modulation instability. In addition, the study of FPUT phenomenon has been extended to two coupled nonlinear systems. Based on the Manakov model equation, it is found that the two coupled systems have the characteristics of asymmetric spectrum. The main results include the analytical calculation of the related asymmetric spectrum. ^{[ 27 ]} Non-degenerate Akhmediev breather solution ^{[ 28 ]} And complex higher-order modulation instability evolution ^{[ 29 ]} In recent years, different techniques have been used to observe the FPUT phenomenon in optical fibers, such as the destructive "cutback" measurement. ^{[ 30 ]} All-fiber measurement method based on OTDR system ^{[ 31 – 33 ]} Cyclic fiber loop method based on periodic amplification ^{[ 16 , 34 ]} And other different technical means.

Stable nonlinear wave is the key to experimental observation. However, as far as we know, there are very few studies on the stability of FPUT phenomenon. The relevant studies mainly discuss the stability of FPUT phenomenon by analyzing the stability of a specific double-periodic wave. ^{[ 26 , 35 ]} The analysis results are limited to a specific analytical two-periodic wave solution, and the stability of FPUT phenomenon can not be directly analyzed from the actual experimental conditions. In this paper, the actual experimental conditions are considered, and the stability of FPUT phenomenon in optical fiber is analyzed by numerical simulation method with the sinusoidal disturbance plane wave with a certain noise as the incident condition. In the study, we are not limited to a specific value of the "disturbance" amplitude and "disturbance" frequency (many related literatures in the study.

2. Model equation

The governing equation describing the propagation of a slowly varying electric field envelope in a nonlinear single-mode fiber is the classical nonlinear Schrodinger equation, namely。

${\mathrm{i}}\frac{{\partial a}}{{\partial z}} \pm \frac{1}{2}{\beta _2}\frac{{{\partial ^2}a}}{{\partial {t^2}}} + \gamma {\left| a \right|^2}a = 0 ,$

(1)

Among a Represents the slowly varying envelope amplitude of the electric field; z Represents the transmission distance; t Represents the time coordinate in the group velocity reference frame; β ₂ Is the group velocity dispersion parameter, which can be positive or negative ( β ₂ 0 stands for normal dispersion, β ₂ < 0 stands for anomalous dispersion); γ Nonlinear coefficient representing the Kerr effect of an optical fiber.

Without loss of generality, for the convenience of analysis, we can put the model equation ( 1 ) is normalized to。

${\mathrm{i}}\frac{{\partial A}}{{\partial Z}} \pm \frac{1}{2}{{\mathrm{sgn}}} \left( {{\beta _2}} \right)\frac{{{\partial ^2}A}}{{\partial {T^2}}} + {\left| A \right|^2}A = 0 ,$

(2)

Among。

$\begin{split}& A = \sqrt {\gamma {L_{\text{d}}}} a , \;Z = \frac{z}{{{L_{\text{d}}}}} ,\; T = \frac{z}{{{T_0}}} , \\ &{L_{\text{d}}} = \frac{{T_0^2}}{{\left| {{\beta _2}} \right|}} ,\;{\mathrm{sgn}}(\beta_{2}) = \pm 1 . \end{split}$

(3)

( 3 ) In the formula L _d Is the dispersion length, T ₀ Is the initial pulse width.

The analysis later in this paper will be based on the normalization equation ( 2 ) to expand the discussion. Equation ( 2 The plane wave solution for) is。

$A = {A_0}\exp \left( {{\mathrm{i}}A_0^2Z} \right) ,$

(4)

Among A ₀ Represents the amplitude of the incident plane wave. According to the linear stability analysis method, in the case of normal dispersion, that is, when β ₂ 0 (or sgn ( β ₂ ) = 1), the fiber has no modulational instability, and in the case of anomalous dispersion, that is, when β ₂ 0 (or sgn ( β ₂ ) = – 1), the fiber has modulation instability, based on the normalized model equation ( 2 The fiber gain spectrum of。

$g = \left| {{{\mathrm{Im}}} \left( K \right)} \right| = \frac{1}{2}\left| \varOmega \right|\sqrt {4A_0^2 - {\varOmega ^2}} ,\;\; \varOmega < {\varOmega _{\mathrm{c}}} = 2{A_0} ,$

(5)

Among K Is a sinusoidal disturbance beam; Ω Is the sinusoidal perturbation frequency; Im ( K ) representative K The imaginary part of the optimal gain is。

${g_{\max }}\left( {\varOmega = {\varOmega _{\text{m}}} = \sqrt 2 {A_0}} \right) = A_0^2 .$

(6)

We adopt in this paper initial conditions that are relatively easy to implement in experiments, namely。

$\begin{split} A\left( {Z = 0,T} \right) =\;& A_0^{}\big[ 1 + \sqrt \delta \cdot {{\mathrm{e}}^{{\mathrm{i}}\varOmega t + {\mathrm{i}}{\varphi _0}}} \\ \;& + \sqrt \delta \cdot {{\mathrm{e}}^{ - {\mathrm{i}}\varOmega t + {\mathrm{i}}{\varphi _0}}} + {\mathrm{Noise}} \big] , \end{split}$

(7)

Where the equation ( ) The second and third terms on the right side represent two sinusoidal perturbation signals applied on either side of the pump light frequency ( $\pm \varOmega$ Frequency components) whose amplitudes are all $\sqrt \delta$ , φ ₀ Represents the phase difference between the sinusoidal signal and the pump light, and the third term represents the inevitable white noise under the experimental conditions. In the simulation, let。

${\mathrm{Noise}} = 0.01 \times \left( {1 - 2 \cdot {\text{rand}}} \right) ,$

(8)

Where rand represents a series of random numbers uniformly distributed in the range [0,1], and the amplitude of the noise is 1% of the amplitude of the pump light.

3. 研究结果

本文采用的模拟方法为经典的谱方法与自适应步长的龙格库塔方法相结合的模拟方法. 在本文的模拟结果中, 不失一般性, 令平面波振幅A₀ = 1(对应的最佳扰动频率为 ${\varOmega _{\text{m}}} = \sqrt 2$ , 临界扰动频率为 ${\varOmega _{\text{c}}} = 2$ ), 为使结果具有普适性, 同时不激发高阶谐波, 在我们的研究中, 正弦扰动频率Ω > Ω_c/2 = 1, 其值不仅可以位于调制不稳定性区域内(Ω < 2), 也可以位于调制不稳定性区域外(Ω > 2); 正弦扰动振幅δ不局限于无穷小量, 其值任意.

3.1 扰动振幅对FPUT现象的影响

通过大量的数值模拟, 可以发现对一给定扰动频率, 扰动振幅可以极大地影响FPUT现象的稳定性. 图1(a)和图1(b)分别反应当扰动频率Ω = 1.5, φ₀ = 0.5π, 扰动振幅δ = 0.01时扰动平面波随传输距离的时空和频率谱演化, 可看出在该参数条件下, 扰动平面波振幅可周期性的回归到初始态(z = 0处), 此即FPUT现象, 稳定的FPUT现象可传输至约z = 15处, 当z > 15, FPUT现象逐渐消失, 波形很快变得杂乱无章; 随着扰动振幅δ的增加, 发现FPUT现象的稳定性逐渐增强, 当增加至δ = 0.1, FPUT现象的稳定性增至最强, 如图1(c)和图1(d)所示(δ = 0.1), 稳定的FPUT现象可传输至约z = 22.5处; 随着扰动振幅的进一步增加, FPUT现象稳定性逐步减弱, 如图1(e)和图1(f)所示, 其中 δ = 0.25, 这时稳定的FPUT现象只可传输至约z = 15.

$图 1 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, φ0 = 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25\r\nFig. 1. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.$

图 1 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, φ₀ = 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25

Fig. 1. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 1.5, φ₀ = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.

下载: 全尺寸图片幻灯片

由图1可看出, 随着传输距离的增加, 规则的双周波演化(也即FPUT现象)最终都变得杂乱无章, 这应该与调制不稳定性及级联调制不稳定性相关. 由级联调制不稳定性效应所出现的高阶频率谱成分也具有调制不稳定性增益, 当这些高频成分的振幅长到一定值, 彼此之间及与低频率成分之间就产生相互作用, 从而改变波形演化的动力学特征; 另一方面, 噪声在实验中不可避免, 而噪声中也存在众多具有非零增益的频率成分, 这些频率成分的振幅随传输距离增加而快速增长, 当其增加到一定程度, 彼此之间都会产生相互作用. 故图1中规则的双周期波随着传输距离的增加都会变得杂乱无章. 这里的论述可用于解释下文中所有相关图中规则双周波演化最终都变得杂乱无章的原因.

通过比较图1可看出: 第一, 在给定的扰动频率下, 随着扰动振幅的增加, FPUT现象的稳定性先增强再减弱; 第二, 在给定的扰动频率下, 存在一个临界的扰动振幅δ_cr, 当δ < δ_cr时, 随着δ的增加, FPUT现象的稳定性逐渐增强; 当δ = δ_cr时, FPUT现象稳定性增至最强; 当δ > δ_cr时, 随着δ的增加, FPUT现象稳定性逐渐减弱.

上述结论也适用于扰动频率位于调制不稳定性带外的情况, 如图2所示(所用参数为Ω = 2.2 > Ω_c = 2, φ₀ = 0.5π, 扰动振幅从上到下依次是δ = 0.01, 0.25, 0.5). 当δ = 0.01时, 稳定的FPUT现象只可传输至距离约z = 9处, 当δ = 0.25时, 稳定的FPUT现象可传输距离增至约z = 22处, 当δ = 0.5时, 稳定的FPUT现象可传输距离减至约z = 13处.

$图 2 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, φ0 = 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5\r\nFig. 2. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 2.2, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5.$

图 2 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, φ₀= 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5

Fig. 2. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (1% noise) with Ω = 2.2, φ₀ = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.25; (e), (f) δ = 0.5.

下载: 全尺寸图片幻灯片

临界扰动振幅δ_cr值与扰动频率密切相关, 其值随着扰动频率的增加而增加, 如图3所示. 此结论不仅适用于扰动信号与泵浦光之间相位差φ = 0.5π的情况, 还适用于其他相位差的情况, 主要区别是随着φ的减小, 较小扰动频率所需的临界扰动振幅δ_cr相对较小(比较图3中Ω < 1.7时φ = 0.5π和φ =0.3π对应的δ_cr值曲线), 反之则比较大(比较图3中Ω > 1.7时φ = 0.5π和φ =0.3π对应的δ_cr值曲线).

$图 3 临界扰动振幅δcr与扰动频率的依赖关系(1%噪声)\r\nFig. 3. The dependence of the critical perturbation amplitude (δcr) on the perturbation frequency (1% noise).$

图 3 临界扰动振幅δ_cr与扰动频率的依赖关系(1%噪声)

Fig. 3. The dependence of the critical perturbation amplitude (δ_cr) on the perturbation frequency (1% noise).

下载: 全尺寸图片幻灯片

3.2 扰动信号与泵浦光之间相位差对FPUT现象稳定性的影响

扰动信号与泵浦光之间相位差(φ₀)是影响FPUT现象演化规律的另一个非常重要的物理参量, 在图1和图2中, φ₀ = 0.5π, 下面将探讨扰动振幅一定时, 相位差φ₀从0逐渐增加至0.5π时所引起的对FPUT现象稳定性的影响, 为突出相位差φ₀的影响, 我们在该部分的研究中令扰动振幅保持不变.

图4展示了当扰动频率位于调制不稳定性带内时相位差φ₀对FPUT现象稳定性的影响. 如图4(a)所示, 当φ₀ = 0.1π时, FPUT现象可比较稳定的传输到z = 25处; 当相位差φ₀增至0.3π时, FPUT现象只可以较稳定的传输到z = 15处(图4(b)); 当相位差φ₀继续增至0.5π时, 稳定的FPUT现象可传输距离又增至z = 20处(图4(c)). 由此可见, 相位差φ₀可极大程度地影响FPUT现象的稳定性, 通过大量的数值模拟, 我们发现当φ₀从0开始逐渐增加时, FPUT稳定性先逐渐减弱, 当φ₀增加至某临界值φ_0cr, FPUT稳定性减至最弱, 当φ₀继续增加时, FPUT稳定性又逐步增强. 上述结论适用于调制不稳定性带内的其他扰动频率值.

$图 4 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, δ = 0.1　(a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5π\r\nFig. 4. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 1.5, δ = 0.1: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.$

图 4 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 1.5, δ = 0.1　(a), (b) φ₀ = 0.1π; (c), (d) φ₀ = 0.3π; (e), (f) φ₀ = 0.5π

Fig. 4. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 1.5, δ = 0.1: (a), (b) φ₀ = 0.1π; (c), (d) φ₀= 0.3π; (e), (f) φ₀ = 0.5π.

下载: 全尺寸图片幻灯片

图5显示了当扰动振幅分别为δ = 0.1 和δ =0.01时, 临界值φ_0cr随扰动频率的变化关系. 由图5可见, 临界值φ_0cr随扰动频率的增加而快速减小, 且在较大的频率范围内, 临界值φ_0cr随扰动振幅δ的增加而增加. 在此值得注意的是当扰动频率接近截止频率时, 即当Ω → 2时, φ_0cr → 0, 也就是说当扰动频率接近截止频率, 且当扰动信号和泵浦光同相时, FPUT现象的稳定性最差.

$图 5 扰动信号与泵浦光之间的临界相位差与扰动频率的依赖关系(1%噪声)\r\nFig. 5. The dependence of the critical phase difference between the perturbation and the pump on the perturbation frequency (1% noise).$

图 5 扰动信号与泵浦光之间的临界相位差与扰动频率的依赖关系(1%噪声)

Fig. 5. The dependence of the critical phase difference between the perturbation and the pump on the perturbation frequency (1% noise).

下载: 全尺寸图片幻灯片

当扰动频率位于调制不稳定性带外时, 相位差对FPUT现象稳定性的影响与扰动频率位于带内的情况截然不同, 随着相位差φ₀的增加, FPUT现象稳定性呈现单调增强的趋势, 如图6所示, 当φ₀ = 0.1π时(图6(a)), FPUT只可稳定的传输至约z = 5; 当φ₀ = 0.3π时(图6(b)), FPUT可稳定的传输至约z = 15; 当φ₀ = 0.5π时(图6(c)), FPUT可稳定传输距离增至约z = 20.

$图 6 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, δ = 0.25　 (a), (b) φ0 = 0.1π; (c), (d) φ0 = 0.3π; (e), (f) φ0 = 0.5π\r\nFig. 6. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 2.2, δ = 0.25: (a), (b) φ0 = 0.1π; (c), (d) φ0= 0.3π; (e), (f) φ0 = 0.5π.$

图 6 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(1%噪声), 参数为Ω = 2.2, δ = 0.25　 (a), (b) φ₀ = 0.1π; (c), (d) φ₀ = 0.3π; (e), (f) φ₀ = 0.5π

Fig. 6. The evolution of perturbed plane waves with the propagation distance in the temporal and spectral domain (1% noise) with parameters Ω = 2.2, δ = 0.25: (a), (b) φ₀ = 0.1π; (c), (d) φ₀= 0.3π; (e), (f) φ₀ = 0.5π.

下载: 全尺寸图片幻灯片

3.3 噪声对FPUT现象稳定性的影响

噪声在实验中是不可避免的, 同时, 在数值模拟中, 也会因数值方法的不稳定性引起, 故在研究工作中必须考虑噪声对FPUT现象稳定性的影响. 图7展示了当(8)式中噪声系数为0.1时, 即当噪声的振幅为泵浦光振幅的10%时, 扰动平面波随传输距离的时空和相应的频谱演化, 其他参数与图1(噪声系数为0.01)完全相同. 比较图7与图1可看到噪声越大, 所观察到的FPUT现象循环数越少, FPUT现象稳定性越差, 该结论与Wabnitz和Wetzel研究的^[5]结论一致. 物理上这是可以完全理解的, 噪声中有众多不稳定性增益频率成分, 其振幅随着传输距离的增加而增加, 当振幅达到一定量值, 彼此之间以及与一阶扰动频率成分之间均会产生相互作用, 从而破坏FPUT现象的稳定性演化. 当噪声系数增加时, 噪声中所有频率成分的初始振幅都将增加, 这时所有具有非零增益的频率成分的振幅都会随着传输距离的增加而极速增加, 故在较短的传输距离内就能彼此之间以及与一阶及级联的高阶扰动频率成分之间发生强相互作用, 从而快速破坏FPUT现象的稳定性传输.

$图 7 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(10%噪声), 参数为Ω = 1.5, φ0 = 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25\r\nFig. 7. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (10% noise) with Ω = 1.5, φ0 = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.$

图 7 非线性光纤中扰动平面波随传输距离的时空和相应的频谱演化(10%噪声), 参数为Ω = 1.5, φ₀= 0.5π　(a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25

Fig. 7. The evolution of perturbed plane waves in nonlinear fibers with the propagation distance in the temporal and spectral domain (10% noise) with Ω = 1.5, φ₀ = 0.5π: (a), (b) δ = 0.01; (c), (d) δ = 0.1; (e), (f) δ = 0.25.

下载: 全尺寸图片幻灯片

上述扰动振幅及扰动信号与泵浦光之间相位差对FPUT现象的稳定性影响, 不能由调制不稳定性及级联调制不稳定性来解释, 它们只能用来解释一阶及高阶扰动振幅的初始增长, 由于能量守恒, 其振幅并不会无限的增加, 当振幅增加到一定值, 就会出现衰减至初始态的现象, 衰减的物理原因至今尚未清楚, 我们认为扰动振幅对FPUT现象的稳定性影响可能主要与其关联的严格双周期波的稳定性紧密相关. 经典的非线性薛定谔模型方程(1)是可积系统, 理论上讲, 具有非常多的严格双周期波解, 其中一些解在一定条件下可退化到(7)式的初始态, 该些解的稳定性可能均与(7)式中的扰动振幅δ和相位差φ₀紧密相关, 而且与本文上述结论一致.

4. 总结与讨论

FPUT现象的稳定性是影响其实验观测的一个重要特性, 本文结合实验中的实际条件, 通过大量的仿真模拟研究了扰动振幅及扰动信号与泵浦光间相位差对FPUT现象稳定性的影响, 研究发现二者均可以极大程度地影响FPUT现象的稳定性.

当扰动振幅从无穷小量逐渐增加时, FPUT现象的稳定性逐渐增强; 当扰动振幅值达到临界扰动振幅值, FPUT现象的稳定性增至最强; 当扰动振幅继续增加时, FPUT现象的稳定性又逐渐减弱, 该临界扰动振幅值随扰动频率的增加而增加.

扰动信号与泵浦光间相位差对调制不稳定性带内和带外的FPUT现象稳定性影响不同. 当扰动频率位于调制不稳定性带内时, 随着相位差的增加, FPUT现象的稳定性逐步减弱; 当相位差增至临界值, FPUT现象的稳定性减至最弱; 随着相位差的继续增加, FPUT现象的稳定性又开始逐步增强. 临界相位差值随扰动频率的增加而减弱. 当扰动频率位于调制不稳定性带外时, FPUT现象的稳定性随着相位差的增加而增强.

噪声的强度并不影响上述主要结论, 但是噪声的存在将降低FPUT现象的稳定性, 噪声振幅越大, FPUT现象的稳定性越差.

利用上述主要结论, 可以用来指导FPUT现象的实验观测. FPUT现象并不能无限的周期性传输, 它的周期性总是在一定传输距离后被破坏掉, 由上述主要结论, 我们得知在实验中设置的扰动振幅要适当大, 不能受到调制不稳定性的局限性而把扰动振幅总设置在非常小的范围值内, 同时, 扰动信号与泵浦光的相位差要适当, 当扰动频率位于传统调制不稳定性区域内时, 该相位差要尽量避免中间值, 即要么使其为0, 要么为0.5π, 但是当扰动频率位于传统调制不稳定性区域外时, 该相位差要位于0.5π为最佳.

综上, 我们在研究中并不局限于某一特定值的扰动振幅和扰动频率, 其中扰动振幅任意, 扰动频率横跨了调制不稳定性区域内外的频率空间, 故我们的研究结果具有广泛性、可推广性、可为光纤中FPUT现象的理论和实验研究提供有益的帮助.

数据可用性声明

支撑本研究成果的数据集可在科学数据银行https://doi.org/10.57760/sciencedb.j00213.00066中访问获取.

References

[1]	Kim H G, Yang J H, Kim W J, Koo Y H 2016 Nucl. Eng. Technol. 48 1 Google Scholar
[2]	Kurata M 2016 Nucl. Eng. Technol. 48 26 Google Scholar
[3]	Leenaers A, Van den Berghe S, Koonen E, Jacquet P, Jarousse C, Guigon B, Ballagny A, Sannen L 2004 J. Nucl. Mater. 327 121 Google Scholar
[4]	Zinkle S J, Terrani K A, Gehin J C, Ott L J, Snead L L 2014 J. Nucl. Mater. 448 374 Google Scholar
[5]	Terrani K A, Wang D, Ott L J, Montgomery R O 2014 J. Nucl. Mater. 448 512 Google Scholar
[6]	Johnson K D, Raftery A M, Lopes D A, Wallenius J 2016 J. Nucl. Mater. 477 18 Google Scholar
[7]	Watkins J K, Gonzales A, Wagner A R, Sooby E S, Jaques B J 2021 J. Nucl. Mater. 553 153048 Google Scholar
[8]	Wood E S, White J T, Nelson A T 2017 J. Nucl. Mater 484 245 Google Scholar
[9]	Ortega L H, Blamer B J, Evans J A, McDeavitt S M 2016 J. Nucl. Mater. 471 116 Google Scholar
[10]	White J T, Travis A W, Dunwoody J T, Nelson A T 2017 J. Nucl. Mater. 495 463 Google Scholar
[11]	Lopes D A, Uygur S, Johnson K D 2017 J. Nucl. Sci. Technol. 54 405 Google Scholar
[12]	刘续希, 高士森, 喇永孝, 玉栋梁, 柳文波 2024 物理学报 73 148201 Google Scholar Liu X X, Gao S S, La Y X, Yu D L, Liu W B 2024 Acta Phys. Sin. 73 148201 Google Scholar
[13]	刘东昆, 王庆宇, 张田, 周羽, 王翔 2024 物理学报 73 066102 Google Scholar Liu D K, Wang Q Y, Zhang T, Zhou Y, Wang X 2024 Acta Phys. Sin. 73 066102 Google Scholar
[14]	王凯乐, 杨文奎, 史新成, 侯华, 赵宇宏 2023 物理学报 72 076102 Google Scholar Wang K L, Yang W K, Shi X C, Hou H, Zhao Y H 2023 Acta Phys. Sin. 72 076102 Google Scholar
[15]	刘明治, 张瑞杰, 方伟, 章书周, 曲选辉 2012 金属学报 48 1207 Google Scholar Liu M Z, Zhang R J, Fang W, Zhang S Z, Qu X H 2012 Acta. Metall. Sin. 48 1207 Google Scholar
[16]	Kumar V, Fang Z Z, Fife P C 2010 Mater. Sci. Eng. A 528 254 Google Scholar
[17]	Biswas S, Schwen D, Wang H, Okuniewski M, Tomar V 2018 Comput. Mater. Sci. 148 307 Google Scholar
[18]	Du L F, Yang S M, Zhu X W, Jiang J, Hui Q, Du H L 2018 J. Mater. Sci. 53 9567 Google Scholar
[19]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[20]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[21]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[22]	Moelans N, Blanpain B, Wollants P 2008 Phys. Rev. B 78 024113 Google Scholar
[23]	Ahmed K, Yablinsky C A, Schulte A, Allen T, El-Azab A 2013 Modell. Simul. Mater. Sci. Eng. 21 065005 Google Scholar
[24]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[25]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[26]	Biner S B 2017 Programming Phase-Field Modeling (Switzerland: Springer International Publishing) p18
[27]	Holt J B, Almassy M Y 1969 J. Am. Ceram. Soc. 52 631 Google Scholar
[28]	戚晓勇, 柳文波, 何宗倍, 王一帆, 恽迪 2023 金属学报 59 1513 Google Scholar Qi X Y, Liu W B, He Z B, Wang Y F, Yun D 2023 Acta Metall. Sin. 59 1513 Google Scholar
[29]	Bocharov D, Gryaznov D, Zhukovskii Y F, Kotomin E A 2013 J. Nucl. Mater. 435 102 Google Scholar
[30]	Cooper M W D, Gamble K A, Capolungo L, Matthews C, Andersson D A, Beeler B, Stanek C R, Metzger K 2021 J. Nucl. Mater. 555 153129 Google Scholar
[31]	Beeler B, Baskes M, Andersson D, Cooper M W D, Zhang Y F 2019 J. Nucl. Mater. 514 290 Google Scholar
[32]	Cheniour A, Tonks M R, Gong B, Yao T K, He L F, Harp J M, Beeler B, Zhang Y F, Lian J 2020 J. Nucl. Mater 532 152069 Google Scholar
[33]	Chockalingam K, Kouznetsova V G, van der Sluis O, Geers M G D 2016 Comput. Methods Appl. Mech. Eng. 312 492 Google Scholar
[34]	Rahaman M N 1995 Ceramic Processing and Sintering (New York: Marcel Dekker) p446
[35]	Riedel H, Svoboda J 1993 Acta Metall. Mater. 41 1929 Google Scholar
[36]	孙正阳, 杨超, 柳文波 2020 金属学报 56 1295 Google Scholar Sun Z Y, Yang C, Liu W B 2020 Acta Metall. Sin. 56 1295 Google Scholar
[37]	Ahmed K, Allen T, El-Azab A 2016 J. Mater. Sci. 51 1261 Google Scholar
[38]	Yadav V, Vanherpe L, Moelans N 2016 Comput. Mater. Sci. 125 297 Google Scholar

图 1 相场模型示意图 (ρ, ϕ_α, ϕ_β —浓度场变量; η₁—晶粒1的取向场变量; η₂—晶粒2的取向场变量)

Figure 1. Schematic of phase field model (ρ, ϕ_α, ϕ_β —concentration field variable; η₁—orientation field variable of grain 1; η₂—orientation field variable of grain 2.)

DownLoad: Full-Size Img PowerPoint

两个不同相等大圆形晶粒演化的相场模拟　(a) 6 × 10⁴ $\Delta t$ ; (b) 50 × 10⁴ $\Delta t$ ; (c) 200 × 10⁴ $\Delta t$ ; (d) 500 × 10⁴ $\Delta t$

Phase-field simulation of the evolution of two equal-sized circular grains with different phases: (a) 6 × 10⁴ $\Delta t$ ; (b) 50 × 10⁴ $\Delta t$ ; (c) 200 × 10⁴ $\Delta t$ ; (d) 500 × 10⁴ $\Delta t$ .

DownLoad: Full-Size Img PowerPoint

图 3 不同相晶粒的烧结颈对数增长曲线 (l — 颈部长度, t — 时间步)

Figure 3. Logarithmic growth curves of sintering neck of different phase grains (l —neck length, t — time step).

DownLoad: Full-Size Img PowerPoint

演化时间500 × 10⁴ $\Delta t$ ((a)—(c))与1500 × 10⁴ $\Delta t$ ((d)—(f))下的平衡二面角　(a), (d) 2个UN晶粒; (b), (e) 2个U₃Si₂晶粒; (c), (f) 1个UN晶粒和1个U₃Si₂晶粒

Equilibrium dihedral angles at evolution times 500 × 10⁴ $\Delta t$ ((a)–(c)) and 1500 × 10⁴ $\Delta t$ ((d)–(f)): (a), (d) Two UN grains; (b), (e) two U₃Si₂ grains; (c), (f) one UN grain and one U₃Si₂ grain.

DownLoad: Full-Size Img PowerPoint

两个不同相不等大圆形晶粒演化的相场模拟　(a) 2 × 10⁴ $\Delta t$ ; (b) 50 × 10⁴ $\Delta t$ ; (c) 100 × 10⁴ $\Delta t$ ; (d) 500 × 10⁴ $\Delta t$ .

Phase-field simulation of the evolution of two unequal-sized circular grains with different phases: (a) 2 × 10⁴ $\Delta t$ ; (b) 50 × 10⁴ $\Delta t$ ; (c) 100 × 10⁴ $\Delta t$ ; (d) 500 × 10⁴ $\Delta t$ .

DownLoad: Full-Size Img PowerPoint

图 6 晶粒面积与烧结颈尺寸的演化曲线

Figure 6. Evolution curves of grain area and sintering neck sizes.

DownLoad: Full-Size Img PowerPoint

图 7 四种不同相晶粒个数比例下3个晶粒烧结的三叉晶界　(a) 3个UN晶粒; (b) 2个UN晶粒和 1个U₃Si₂晶粒; (c) 1个UN晶粒和2个U₃Si₂晶粒; (d) 3个U₃Si₂晶粒

Figure 7. Trident grain boundaries of the sintering of three grains with four different phase grain number ratios: (a) Three UN grains; (b) two UN grains and one U₃Si₂ grain; (c) one UN grain and two U₃Si₂ grains; (d) three U₃Si₂ grains.

DownLoad: Full-Size Img PowerPoint

图 8 三叉晶界处气孔率的演化曲线

Figure 8. Evolution curves of the porosity at triple grain boundaries.

DownLoad: Full-Size Img PowerPoint

图 9 不同体积分数比的两相多晶组织初始形貌　(a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β

Figure 9. Initial morphology of two-phase polycrystalline structures with different volume fraction ratios: (a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β.

DownLoad: Full-Size Img PowerPoint

图 10 不同体积分数比的两相多晶组织最终形貌　(a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β

Figure 10. Final morphology of two-phase polycrystalline structures with different volume fraction ratios: (a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β.

DownLoad: Full-Size Img PowerPoint

表 1 UN的物理参数^[27–29]

Table 1. Physical parameters of UN at 1823 K^[27–29]

Parameter	Value	Unit	Ref.
$D_\alpha ^{\text{s}}$	7.5 × 10^–12	m²·s^–1	[27]
$D_\alpha ^{{\text{gb}}}$	$0.01D_\alpha ^{\text{s}}$	m²·s^–1	[28]
$\gamma _{\text{s}}^\alpha$	1.6	J·m^–2	[29]
$\gamma _{{\text{gb}}}^\alpha$	0.8	J·m^–2	[29]
δ	6	nm	[29]

DownLoad: CSV

表 2 U₃Si₂的物理参数^[30–32]

Table 2. Physical parameters of U₃Si₂ at 1823 K^[30–32]

Parameter	Value	Unit	Ref.
$D_\beta ^{\text{s}}$	100 $D_\beta ^{{\text{gb}}}$	m²·s^–1
$D_\beta ^{{\text{gb}}}$	6.5923 × 10^–10	m²·s^–1	[30]
$\gamma _{\text{s}}^\beta$	2.0	J·m^–2	[31]
$\gamma _{{\text{gb}}}^\beta$	1.3	J·m^–2	[32]
δ	6	nm	[32]
Note: $D_\alpha ^{\text{s}}$ , $D_\beta ^{\text{s}}$ — surface diffusivity; $D_\alpha ^{{\text{gb}}}$ , $D_\beta ^{{\text{gb}}}$ — grain-boundary diffusivity; $\gamma _{\text{s}}^\alpha$ , $\gamma _{\text{s}}^\beta$ — surface energy; $\gamma _{{\text{gb}}}^\alpha$ , $\gamma _{{\text{gb}}}^\beta$ — grain-boundary energy; δ — diffuse interface width.

DownLoad: CSV

表 3 模拟中的无量纲参数表

Table 3. Non-dimensional parameters used in simulation.

Parameter	Value	Parameter	Value
$\tilde A({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0)$	17	${\tilde \kappa _\eta }$	6.75
$\tilde A({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1)$	11.5	$\tilde {\boldsymbol{M}}({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0)$	6817.5
$\tilde B$	1	$\tilde {\boldsymbol{M}}({\varphi _\alpha } = 0, {\varphi _\beta } = 1)$	6817.5
${\tilde \kappa _\rho }({\varphi _\alpha } = 1, {\varphi _\beta } = 0)$	20.25	$\tilde L$	1
${\tilde \kappa _\rho }({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1)$	14	$\Delta x = \Delta y$	1
${\tilde \kappa _\phi }({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0)$	20.25	$\Delta t$	2 × 10^–5
${\tilde \kappa _\phi }({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1)$	14
Note: $\tilde A$ , $\tilde B$ , ${\tilde \kappa _\rho }$ , ${\tilde \kappa _\phi }$ , ${\tilde \kappa _\eta }$ — non-dimensional parameters of free energy function; $\tilde {\boldsymbol{M}}$ — non-dimensional mobility; $\tilde L$ —non-dimensional Allen-Cahn mobility; ∆x, ∆y — space scale; ∆t — time scale.

DownLoad: CSV

[1]	Kim H G, Yang J H, Kim W J, Koo Y H 2016 Nucl. Eng. Technol. 48 1 Google Scholar
[2]	Kurata M 2016 Nucl. Eng. Technol. 48 26 Google Scholar
[3]	Leenaers A, Van den Berghe S, Koonen E, Jacquet P, Jarousse C, Guigon B, Ballagny A, Sannen L 2004 J. Nucl. Mater. 327 121 Google Scholar
[4]	Zinkle S J, Terrani K A, Gehin J C, Ott L J, Snead L L 2014 J. Nucl. Mater. 448 374 Google Scholar
[5]	Terrani K A, Wang D, Ott L J, Montgomery R O 2014 J. Nucl. Mater. 448 512 Google Scholar
[6]	Johnson K D, Raftery A M, Lopes D A, Wallenius J 2016 J. Nucl. Mater. 477 18 Google Scholar
[7]	Watkins J K, Gonzales A, Wagner A R, Sooby E S, Jaques B J 2021 J. Nucl. Mater. 553 153048 Google Scholar
[8]	Wood E S, White J T, Nelson A T 2017 J. Nucl. Mater 484 245 Google Scholar
[9]	Ortega L H, Blamer B J, Evans J A, McDeavitt S M 2016 J. Nucl. Mater. 471 116 Google Scholar
[10]	White J T, Travis A W, Dunwoody J T, Nelson A T 2017 J. Nucl. Mater. 495 463 Google Scholar
[11]	Lopes D A, Uygur S, Johnson K D 2017 J. Nucl. Sci. Technol. 54 405 Google Scholar
[12]	刘续希, 高士森, 喇永孝, 玉栋梁, 柳文波 2024 物理学报 73 148201 Google Scholar Liu X X, Gao S S, La Y X, Yu D L, Liu W B 2024 Acta Phys. Sin. 73 148201 Google Scholar
[13]	刘东昆, 王庆宇, 张田, 周羽, 王翔 2024 物理学报 73 066102 Google Scholar Liu D K, Wang Q Y, Zhang T, Zhou Y, Wang X 2024 Acta Phys. Sin. 73 066102 Google Scholar
[14]	王凯乐, 杨文奎, 史新成, 侯华, 赵宇宏 2023 物理学报 72 076102 Google Scholar Wang K L, Yang W K, Shi X C, Hou H, Zhao Y H 2023 Acta Phys. Sin. 72 076102 Google Scholar
[15]	刘明治, 张瑞杰, 方伟, 章书周, 曲选辉 2012 金属学报 48 1207 Google Scholar Liu M Z, Zhang R J, Fang W, Zhang S Z, Qu X H 2012 Acta. Metall. Sin. 48 1207 Google Scholar
[16]	Kumar V, Fang Z Z, Fife P C 2010 Mater. Sci. Eng. A 528 254 Google Scholar
[17]	Biswas S, Schwen D, Wang H, Okuniewski M, Tomar V 2018 Comput. Mater. Sci. 148 307 Google Scholar
[18]	Du L F, Yang S M, Zhu X W, Jiang J, Hui Q, Du H L 2018 J. Mater. Sci. 53 9567 Google Scholar
[19]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[20]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[21]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[22]	Moelans N, Blanpain B, Wollants P 2008 Phys. Rev. B 78 024113 Google Scholar
[23]	Ahmed K, Yablinsky C A, Schulte A, Allen T, El-Azab A 2013 Modell. Simul. Mater. Sci. Eng. 21 065005 Google Scholar
[24]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[25]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[26]	Biner S B 2017 Programming Phase-Field Modeling (Switzerland: Springer International Publishing) p18
[27]	Holt J B, Almassy M Y 1969 J. Am. Ceram. Soc. 52 631 Google Scholar
[28]	戚晓勇, 柳文波, 何宗倍, 王一帆, 恽迪 2023 金属学报 59 1513 Google Scholar Qi X Y, Liu W B, He Z B, Wang Y F, Yun D 2023 Acta Metall. Sin. 59 1513 Google Scholar
[29]	Bocharov D, Gryaznov D, Zhukovskii Y F, Kotomin E A 2013 J. Nucl. Mater. 435 102 Google Scholar
[30]	Cooper M W D, Gamble K A, Capolungo L, Matthews C, Andersson D A, Beeler B, Stanek C R, Metzger K 2021 J. Nucl. Mater. 555 153129 Google Scholar
[31]	Beeler B, Baskes M, Andersson D, Cooper M W D, Zhang Y F 2019 J. Nucl. Mater. 514 290 Google Scholar
[32]	Cheniour A, Tonks M R, Gong B, Yao T K, He L F, Harp J M, Beeler B, Zhang Y F, Lian J 2020 J. Nucl. Mater 532 152069 Google Scholar
[33]	Chockalingam K, Kouznetsova V G, van der Sluis O, Geers M G D 2016 Comput. Methods Appl. Mech. Eng. 312 492 Google Scholar
[34]	Rahaman M N 1995 Ceramic Processing and Sintering (New York: Marcel Dekker) p446
[35]	Riedel H, Svoboda J 1993 Acta Metall. Mater. 41 1929 Google Scholar
[36]	孙正阳, 杨超, 柳文波 2020 金属学报 56 1295 Google Scholar Sun Z Y, Yang C, Liu W B 2020 Acta Metall. Sin. 56 1295 Google Scholar
[37]	Ahmed K, Allen T, El-Azab A 2016 J. Mater. Sci. 51 1261 Google Scholar
[38]	Yadav V, Vanherpe L, Moelans N 2016 Comput. Mater. Sci. 125 297 Google Scholar

[1]	Liu Xu-Xi, Gao Shi-Sen, La Yong-Xiao, Yu Dong-Liang, Liu Wen-Bo. Phase-field simulation of high-temperature corrosion of binary Zr-2.5Sn alloy. Acta Physica Sinica, 2024, 73(14): 148201. doi: 10.7498/aps.73.20240393
[2]	Jiang Yan-Bo, Liu Wen-Bo, Sun Zhi-Peng, La Yong-Xiao, Yun Di. Phase-field simulation of void evolution in UO₂ under applied stress. Acta Physica Sinica, 2022, 71(2): 026103. doi: 10.7498/aps.71.20211440
[3]	Song Yan, Jiang Hong-Xiang, Zhao Jiu-Zhou, He Jie, Zhang Li-Li, Li Shi-Xin. Numerical simulations of solidification microstructure evolution process for commercial-purity aluminum alloys inoculated by Al-Ti-B refiner. Acta Physica Sinica, 2021, 70(8): 086402. doi: 10.7498/aps.70.20201431
[4]	Yang Zhao-Xi, Liu Wen-Bo, Zhang Cong-Yu, He Xin-Fu, Sun Zheng-Yang, Jia Li-Xia, Shi Tian-Tian, Yun Di. Phase field simulation of grain boundary segregation and radiation-enhanced segregation in Fe-Cr alloys. Acta Physica Sinica, 2021, 70(11): 116101. doi: 10.7498/aps.70.20201840
[5]	Liu Di, Wang Jing, Wang Jun-Sheng, Huang Hou-Bing. Phase field simulation of misfit strain manipulating domain structure and ferroelectric properties in PbZr_(1–x)Ti_xO₃ thin films. Acta Physica Sinica, 2020, 69(12): 127801. doi: 10.7498/aps.69.20200310
[6]	Hao Gang-Ling, Xu Qiao-Ping, Li Xian-Yu, Wang Wei-Guo. Internal friction characteristics of compacted billet of metal powder in sintering process. Acta Physica Sinica, 2019, 68(12): 126101. doi: 10.7498/aps.68.20190031
[7]	Zhang Jun, Chen Wen-Xiong, Zheng Cheng-Wu, Li Dian-Zhong. Phase-field modeling of ferrite morphology in austenite-to-ferrite transformation with considering anisotropic effects. Acta Physica Sinica, 2017, 66(7): 070701. doi: 10.7498/aps.66.070701
[8]	Duan Pei-Pei, Xing Hui, Chen Zhi, Hao Guan-Hua, Wang Bi-Han, Jin Ke-Xin. Phase-field modeling of free dendritic growth of magnesium based alloy. Acta Physica Sinica, 2015, 64(6): 060201. doi: 10.7498/aps.64.060201
[9]	Du Li-Fei, Zhang Rong, Xing Hui, Zhang Li-Min, Zhang Yang, Liu Lin. Phase-field simulation of solidified microstructure evolution in the presence of lateral constraint. Acta Physica Sinica, 2013, 62(10): 106401. doi: 10.7498/aps.62.106401
[10]	Chen Xi, Lin Zheng-Zhe, Yin Cong, Tang Hao, Hu Yun-Cheng, Ning Xi-Jing. Theoretical prediction of the growth and surface structure of platinum nanoparticles. Acta Physica Sinica, 2012, 61(7): 076801. doi: 10.7498/aps.61.076801
[11]	Pan Shi-Yan, Zhu Ming-Fang. Quantitative phase-field model for dendritic growth with two-sided diffusion. Acta Physica Sinica, 2012, 61(22): 228102. doi: 10.7498/aps.61.228102
[12]	Wei Cheng-Yang, Li Sai-Yi. Effect of temperature gradient on grain growth behavior from phase field simulations. Acta Physica Sinica, 2011, 60(10): 100701. doi: 10.7498/aps.60.100701
[13]	Wang Gang, Xu Dong-Sheng, Yang Rui. Phase field simulation on sideplates formation in Ti-6Al-4V alloy. Acta Physica Sinica, 2009, 58(13): 343-S348. doi: 10.7498/aps.58.343
[14]	Hu Jian-Min, Xin Jiang-Bo, Lü Qiang, Wang Yue-Yuan, Rong Jian-Ying. Powders (Sb2Te3)0.75(1－x)(Bi2Te3)0.25(1－x)(Sb2Se3)x prepared by mechanical alloying and thermoelectric properties of cold-pressed and s. Acta Physica Sinica, 2006, 55(9): 4843-4848. doi: 10.7498/aps.55.4843
[15]	Kuang An-Long, Liu Xing-Chong, Lu Zhong-Lin, Ren Shang-Kun, Liu Cun-Ye, Zhang Feng-Ming, Du You-Wei. Room-temperature ferromagnetism in Mn-doped SnO2 diluted magnetic semiconductor. Acta Physica Sinica, 2005, 54(6): 2934-2937. doi: 10.7498/aps.54.2934
[16]	Li Zheng, He Ye-Qing, Hu Bo-Ping, Wang Zhen-Xi. Thermodynamic effects on the sintered Nd-Fe-B magnets. Acta Physica Sinica, 2005, 54(11): 5400-5404. doi: 10.7498/aps.54.5400
[17]	Duan He, Zheng Yu-Feng, Zhang Xiao-Gang, Sun Yan-Fei, Dong You-Zhong. Hydrothermal synthesis of iron pyrite (FeS22) crystalpowder and the rmal-kinetic study on crystal growth. Acta Physica Sinica, 2005, 54(4): 1659-1664. doi: 10.7498/aps.54.1659
[18]	Huang Rui, Lin Xuan-Ying, Yu Yun-Peng, Lin Kui-Xun, Yao Ruo-He, Huang Wen-Yong, Wei Jun-Hong, Wang Zhao-Kui, Yu Chu-Ying. Control of grain size during low-temperature growth of polycrystalline silicon films. Acta Physica Sinica, 2004, 53(11): 3950-3955. doi: 10.7498/aps.53.3950
[19]	Zhang Jian-Min, Xu Ke-Wei. Investigation of abnormal grain growth andtexture change in Ag and Cu films. Acta Physica Sinica, 2003, 52(1): 145-149. doi: 10.7498/aps.52.145
[20]	Zhang Jian-Min, Xu Ke-Wei. . Acta Physica Sinica, 2002, 51(11): 2562-2566. doi: 10.7498/aps.51.2562

Catalog

Vol. 73, Issue 21 - 2024-11-05

Metrics

Abstract views: 1621
PDF Downloads: 55

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板