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相场法研究Fe-Cu-Mn-Al合金富Cu相析出机制

郭震 赵宇宏 孙远洋 赵宝军 田晓林 侯华

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相场法研究Fe-Cu-Mn-Al合金富Cu相析出机制

郭震, 赵宇宏, 孙远洋, 赵宝军, 田晓林, 侯华

Phase field study of effect of Al on Cu-rich precipitates in Fe-Cu-Mn-Al alloys

Guo Zhen, Zhao Yu-Hong, Sun Yuan-Yang, Zhao Bao-Jun, Tian Xiao-Lin, Hou Hua
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  • 基于Ginzburg-Landau理论采用连续相场法模拟了Fe-15%Cu-3%Mn-xAl(质量分数x = 1%, 3%, 5%)合金在873 K等温时效时纳米富Cu析出相沉淀机制及Al含量对富Cu相析出的阻碍效应. 通过计算成分场变量和结构序参数, 研究了富Cu析出相的形貌、颗粒密度、平均颗粒半径、生长和粗化动力学. 研究结果表明: 在时效早期阶段, 纳米富Cu相通过失稳分解机制析出, 由于原子扩散速率存在差异, 从而形成以富Cu相为核心的核壳结构. 随着时效时间延长, 富Cu相析出物结构由体心立方转变为面心立方. 其中Al和Mn原子在富Cu核外偏析形成Al/Mn簇, 可以将其视为阻碍富Cu析出相形成的缓冲层; 在沉淀过程中, 随着Al含量的增大, Al/Mn金属间相促进了缓冲层的生长, 阻碍富Cu析出相的生长和粗化.
    Low carbon steel plays an important role in many applications due to its high strength. Its high strength comes from the strengthening effect of nano-Cu-rich phase precipitates. In order to effectively adjust the microstructure of Cu-rich phase precipitates and obtain Fe-Cu-based steel with the best properties by adding different alloying elements (Mn, Al), it is necessary to understand the precipitation process of Cu particles. In this paper, based on the Ginzburg-Landau theory, the previous phase field model is modified, and the continuous phase field method is used to simulate the precipitation mechanism of nanometer Cu-rich precipitates and the inhibiting of the effect of Al content on Cu-rich precipitates of Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5% mass fraction) alloy at 873 K isothermal aging. Combining with the free energy derived from thermodynamics database, the microstructure evolution corresponds to the real alloy system. By calculating the composition field variables and structural order parameters, the evolution of phase separation and precipitated phase morphology in aging process are simulated. Moreover, the influence law of morphology, quantity density, average particle radius, growth and coarsening of Cu-rich precipitated phase are discussed. The results show that in the early stage of aging process, the nano-Cu-rich phase precipitates through the spinodal decomposition mechanism, and is randomly distributed in the iron matrix. Furthermore, due to the difference in atomic diffusion rate, the core-shell structure with Cu-rich phase as a core is formed. With the aging time extending, the structure of Cu-rich phase precipitates changes from bcc to fcc. Because of the synergistic effect between Al and Cu, the diffusion of Cu is slowed down. Besides, with the Al and Mn atoms precipitating, Al/Mn clusters are segregated around the Cu-rich precipitates, forming the Al/Mn intermetallic core-shell structure, and gradually wrapping the Cu-rich phase uniformly. During the evolution of the precipitation stage, the Al/Mn clusters are isolated around the Cu-rich precipitation phase, forming a gradually uniform Al/Mn intermetallic phase core shell structure covering the Cu-rich phase, which is to hinder the buffer layer from forming in the precipitation stage of the reservoir. In addition, with the Al content increasing, the Al/Mn intermetallic phase promotes the growth of the buffer layer and hinders the Cu-rich precipitate phase from growing and coarsening.
      通信作者: 赵宇宏, zhaoyuhong@nuc.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 22008224, 52074246, 51774254, 51774253, 51804279, 51801189)、中央引导地方科技发展专项资金项目(批准号: YDZX20191400002796)、山西省科技重大专项(批准号: 20181101014, 20191102008, 20191102007)、山西省平台基地和人才专项(批准号: 201805D211036)、山西省科技成果转化引导专项(批准号: 201804D131039)、山西省青年科技研究基金(批准号: 201801D221152)和装备预研领域基金重点项目(批准号: 61409230407)资助的课题
      Corresponding author: Zhao Yu-Hong, zhaoyuhong@nuc.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 22008224, 52074246, 51774254, 51774253, 51804279, 51801189), the Guiding Local Science and Technology Development Projects by the Central Government, China (Grant No. YDZX20191400002796), the Science and Technology Major Project of Shanxi Province, China (Grant Nos. 20181101014, 20191102008, 20191102007), the Platform and Talent Project of Shanxi Province, China (Grant No. 201805D211036), the Transformation of Scientific and Technological Achievements Special Guide Project of Shanxi Province, China (Grant No. 201804D131039), the Youth Science and Technology Research Foundation of Shanxi Province, China (Grnat No. 201801D221152) and the Key Project of Equipment Pre-research Foundation, China (Grant No. 61409230407)
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    Zhu J M, Zhang T L, Yang Y, Liu C T 2019 Acta Mater. 166 560Google Scholar

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    Han G, Shang C J, Misra R D K, Xie Z J 2019 Physica B 569 68Google Scholar

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    Li B Y, Hu S Y, Li C L, Li Q L, Chen J, Shu G G, Jr C H, Weng Y Q, Xu B, Liu W 2017 Model. Simul. Mater. Sc. 25 6

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    Lv G C, Zhang H, He X F, Yang W, Su Y J 2016 Aip Adv. 6 045004Google Scholar

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    Jiao Z B, Luan J H, Miller M K, Chung Y W, Liu C T 2017 Mater. Today 20 142Google Scholar

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    Shu S P, Wells P B, Almirall N, Odette G R, Morgan D D 2018 Acta Mater. 157 298Google Scholar

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    Odette G R, Liu C L, Wirth B D 1996 MRS Online Proceedings Library Archive 439 457Google Scholar

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    Wen Y R, Hirata A, Zhang Z W, Fujita T, Liu C T, Jiang J H, Chen M W 2013 Acta Mater. 61 2133Google Scholar

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    Miller M K, Wirth B D, Odette G R 2003 Mater. Sci. Eng. A 353 133Google Scholar

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    沈琴, 王晓姣, 赵安宇, 何益锋, 方旭磊, 马佳荣, 刘文庆 2016 金属学报 52 513Google Scholar

    Shen Q, Wang X J, Zhao A Y, He Y F, Fang X L, Ma J R, Liu W Q 2016 Acta Metall. Sin. 52 513Google Scholar

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    Shen Q, Xiong X, Li T, Chen H, Cheng Y M, Liu W Q 2018 Mater. Sci. Eng. A 723 279Google Scholar

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    Vaynman S, Isheim D, Kolli R P, Bhat S P, Seidman D N, Fine M E 2008 Metall. Mater. Trans. A 39 363Google Scholar

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    Sun Y Y, Zhao Y H, Zhao B J, Yang W K, Li X L, Hou H 2019 J. Mater. Sci. 54 11263Google Scholar

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    赵宝军, 赵宇宏, 孙远洋, 杨文奎, 侯华 2019 金属学报 55 593Google Scholar

    Zhao B J, Zhao Y H, Sun Y Y, Yang W K, Hou H 2019 Acta Metall. Sin. 55 593Google Scholar

    [17]

    Wen Z Q, Zhao Y H, Hou H, Wang B, Han P D 2017 Mater. Design 114 398Google Scholar

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    Huang Z W, Zhao Y H, Hou H, Wang Z, Mu Y Q, Niu X F, Han P D 2011 Rare Metal Mat. Eng. 12 2136

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    田晓林, 赵宇宏, 田晋忠, 侯华 2018 物理学报 67 230201Google Scholar

    Tian X L, Zhao Y H, Tian J Z, Hou H 2018 Acta Phys. Sin. 67 230201Google Scholar

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    Chen L Q 2002 Annu. Rev. Mater. Res. 32 113Google Scholar

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    Hou H, Zhao Y H, Zhao Y H 2009 Mater. Sci. Eng. 499 204Google Scholar

    [23]

    Kuang W W, Wang H F, Li X, Zhang J B, Zhou Q, Zhao Y H 2018 Acta Mater. 159 16Google Scholar

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    Zhao Y H, Zhang B, Hou H, Chen W P, Wang M 2019 J. Mater. Sci. Technol. 35 1044Google Scholar

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    Zhang J B, Wang H F, Kuang W W, Zhang Y C, Li S, Zhao Y H, Herlach D M 2018 Acta Mater. 148 86Google Scholar

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    Cahn J W 1961 Acta Metal. 9 795Google Scholar

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    Zhao Y H, Wang S, Zhang B, Yuan Y, Guo Q W, Hou H 2019 J. Solid State Chem. 276 232Google Scholar

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    Koyama T, Hashimoto K, Onodera H 2006 Mater. Trans. 47 2765Google Scholar

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    Koyama T, Onodera H 2005 Mater. Trans. 46 1187Google Scholar

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  • 图 1  时效温度873 K时Fe-15%Cu-3%Mn-1%Al合金沉淀相三维演化相场模拟 (a1)−(a4) t* = 17000; (b1)−(b4) t* = 18500; (c1)−(c4) t* = 20000; (d1)−(d4) t* = 22500

    Fig. 1.  Three-dimensional phase-field simulation of precipitation phase of Fe-15%Cu-3%Mn-1%Al alloy when aged at 873 K: (a1)−(a4) t* = 17000; (b1)−(b4) t* = 18500; (c1)−(c4) t* = 20000; (d1)−(d4) t* = 22500.

    图 2  时效温度873 K时Fe-15%Cu-3%Mn-1%Al合金中富Cu相结构序参数演化曲线

    Fig. 2.  Evolution curves of Cu-rich phase structure order parameter in Fe-15%Cu-3%Mn-1%Al alloy when aged at 873 K.

    图 3  时效温度为873 K时Fe-15%Cu-3%Mn-xAl合金三维富Cu相演化相场模拟 (a1)−(d1) x = 1%; (a2)−(d2) x = 3%; (a3)−(d3) x = 5%; (a1)−(a3) t* = 21000; (b1)−(b3) t* = 22000; (c1)−(c3) t* = 25000

    Fig. 3.  Three dimensional evolution diagrams of Cu rich phase in quaternary alloy Fe-15%Cu-3%Mn-xAl alloy aged at 873 K: (a1)−(d1) x = 1%; (a2)−(d2) x = 3%; (a3)−(d3) x = 5%; (a1)−(a3) t* = 21000; (b1)−(b3) t* = 22000; (c1)−(c3) t* = 25000.

    图 4  时效温度873 K时Fe-15%Cu-3%Mn-xAl(x = 1%, 3%, 5%)合金Gibbs自由能随时间变化曲线

    Fig. 4.  Gibbs free energy curves in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy when aged at 873 K.

    图 5  时效温度873 K时Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金中富Cu析出相颗粒密度随时间变化曲线

    Fig. 5.  Curves of Cu-rich precipitate phase particle density in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy aged at 873 K.

    图 6  时效温度873 K时Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金富Cu析出相平均颗粒半径随时间变化

    Fig. 6.  Average particle radius of Cu-rich precipitates in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy aged at 873 K.

    表 1  相场模型参数表[16]

    Table 1.  Parameters used in phase field simulation[16]

    ParameterValueUnit
    $ {k}_{c}, {k}_{\eta } $${k}_{c}=5.0\times {10}^{-15}, $$ {\mathrm{J} \cdot \mathrm{m}}^{2}/{\mathrm{mol}}^{} $
    ${k}_{c}=1.0\times {10}^{-15} $
    $ {V}_{m} $$ 7.09\times {10}^{-6} $$ {\mathrm{m}}^{3}/{\mathrm{mol}}^{} $
    T$ 873 $$ \mathrm{K} $
    Y$ 214 $$ \mathrm{GPa} $
    $ {L}_{x}\times {L}_{y}\times {L}_{z} $$ 64\times 64\times 64 $$ \mathrm{n}{\mathrm{m}}^{3} $
    W$ 5.0\times {10}^{3} $$ \mathrm{J}/{\mathrm{mol}}^{} $
    $ {D}_{i}^{0, \varphi }\left(\varphi =\alpha, \gamma \right) $${D}_{\mathrm{Cu} }^{0, \alpha }=4.7\times {10}^{-5},$

    ${D}_{\mathrm{Cu} }^{0, \gamma }=4.3\times {10}^{-5} $
    $ {\mathrm{m}}^{2}/{\mathrm{s}}^{} $
    ${D}_{\mathrm{Mn} }^{0, \alpha }=1.49\times {10}^{-4}, $

    ${D}_{\mathrm{Mn} }^{0, \gamma }=1.78\times {10}^{-5} $
    ${D}_{\mathrm{Al} }^{0, \alpha}$[31]$=5.35\times {10}^{-4}, $

    $ {D}_{\mathrm{Al} }^{0, \gamma } $[31]$=2.20\times {10}^{-5} $
    $ {Q}_{i}^{0, \varphi }\left(\varphi =\alpha, \gamma \right) $${Q}_{\mathrm{Cu} }^{0, \alpha }=2.44\times {10}^{5}, $

    ${Q}_{\mathrm{Cu} }^{0, \gamma }=2.80\times {10}^{5} $
    $ \mathrm{J}/{\mathrm{mol}}^{} $
    ${Q}_{\mathrm{Mn} }^{0, \alpha }=2.63\times {10}^{5},$

    $ {Q}_{\mathrm{Mn} }^{0, \gamma }=2.64\times {10}^{5} $
    ${Q}_{\mathrm{Al} }^{0, \alpha }$[31]$=2.71\times {10}^{5}, $

    ${Q}_{\mathrm{Al} }^{0, \gamma } $[31]$=2.67\times {10}^{5} $
    注: $ {k}_{c}, {k}_{\eta } $, 梯度能量系数; $ {V}_{m} $, 摩尔体积; T, 热力学温度; Y, 平均刚度系数; $ {L}_{x}, {L}_{y}, {L}_{z} $, 沿x, y, z轴的模拟区域宽度; W, 双势阱高度; $ {D}_{i}^{0, \varphi } $, 扩散系数; $ {Q}_{i}^{0, \varphi } $, 热扩散激活能.
    下载: 导出CSV
  • [1]

    Zhu J M, Zhang T L, Yang Y, Liu C T 2019 Acta Mater. 166 560Google Scholar

    [2]

    Han G, Shang C J, Misra R D K, Xie Z J 2019 Physica B 569 68Google Scholar

    [3]

    Li B Y, Zhang L, Li C L, Li Q L, Chen J, Shu G G, Weng Y Q, Xu B, Hu S Y, Liu W 2018 J. Nucl. Mater. 507 59Google Scholar

    [4]

    Li B Y, Hu S Y, Li C L, Li Q L, Chen J, Shu G G, Jr C H, Weng Y Q, Xu B, Liu W 2017 Model. Simul. Mater. Sc. 25 6

    [5]

    Lv G C, Zhang H, He X F, Yang W, Su Y J 2016 Aip Adv. 6 045004Google Scholar

    [6]

    Jiao Z B, Luan J H, Miller M K, Chung Y W, Liu C T 2017 Mater. Today 20 142Google Scholar

    [7]

    Shu S P, Wells P B, Almirall N, Odette G R, Morgan D D 2018 Acta Mater. 157 298Google Scholar

    [8]

    Odette G R, Liu C L, Wirth B D 1996 MRS Online Proceedings Library Archive 439 457Google Scholar

    [9]

    Wen Y R, Hirata A, Zhang Z W, Fujita T, Liu C T, Jiang J H, Chen M W 2013 Acta Mater. 61 2133Google Scholar

    [10]

    Miller M K, Wirth B D, Odette G R 2003 Mater. Sci. Eng. A 353 133Google Scholar

    [11]

    Osamura K, Okuda H, Asano K, Furusaka M, Kishida K, Kurosawa F, Uemori R 1994 ISIJ Int. 34 346Google Scholar

    [12]

    沈琴, 王晓姣, 赵安宇, 何益锋, 方旭磊, 马佳荣, 刘文庆 2016 金属学报 52 513Google Scholar

    Shen Q, Wang X J, Zhao A Y, He Y F, Fang X L, Ma J R, Liu W Q 2016 Acta Metall. Sin. 52 513Google Scholar

    [13]

    Shen Q, Xiong X, Li T, Chen H, Cheng Y M, Liu W Q 2018 Mater. Sci. Eng. A 723 279Google Scholar

    [14]

    Vaynman S, Isheim D, Kolli R P, Bhat S P, Seidman D N, Fine M E 2008 Metall. Mater. Trans. A 39 363Google Scholar

    [15]

    Sun Y Y, Zhao Y H, Zhao B J, Yang W K, Li X L, Hou H 2019 J. Mater. Sci. 54 11263Google Scholar

    [16]

    赵宝军, 赵宇宏, 孙远洋, 杨文奎, 侯华 2019 金属学报 55 593Google Scholar

    Zhao B J, Zhao Y H, Sun Y Y, Yang W K, Hou H 2019 Acta Metall. Sin. 55 593Google Scholar

    [17]

    Wen Z Q, Zhao Y H, Hou H, Wang B, Han P D 2017 Mater. Design 114 398Google Scholar

    [18]

    Huang Z W, Zhao Y H, Hou H, Wang Z, Mu Y Q, Niu X F, Han P D 2011 Rare Metal Mat. Eng. 12 2136

    [19]

    田晓林, 赵宇宏, 田晋忠, 侯华 2018 物理学报 67 230201Google Scholar

    Tian X L, Zhao Y H, Tian J Z, Hou H 2018 Acta Phys. Sin. 67 230201Google Scholar

    [20]

    Chen L Q 2002 Annu. Rev. Mater. Res. 32 113Google Scholar

    [21]

    Zhao Y H, Tian X L, Zhao B J, Sun Y Y, Guo H J, Dong M Y, Liu H, Wang X J, Guo Z H, Umar A, Hou H 2018 Sci. Adv. Mater. 10 1793Google Scholar

    [22]

    Hou H, Zhao Y H, Zhao Y H 2009 Mater. Sci. Eng. 499 204Google Scholar

    [23]

    Kuang W W, Wang H F, Li X, Zhang J B, Zhou Q, Zhao Y H 2018 Acta Mater. 159 16Google Scholar

    [24]

    Zhao Y H, Zhang B, Hou H, Chen W P, Wang M 2019 J. Mater. Sci. Technol. 35 1044Google Scholar

    [25]

    Zhang J B, Wang H F, Kuang W W, Zhang Y C, Li S, Zhao Y H, Herlach D M 2018 Acta Mater. 148 86Google Scholar

    [26]

    Cahn J W 1961 Acta Metal. 9 795Google Scholar

    [27]

    Zhao Y H, Wang S, Zhang B, Yuan Y, Guo Q W, Hou H 2019 J. Solid State Chem. 276 232Google Scholar

    [28]

    Koyama T, Hashimoto K, Onodera H 2006 Mater. Trans. 47 2765Google Scholar

    [29]

    Koyama T, Onodera H 2005 Mater. Trans. 46 1187Google Scholar

    [30]

    Dinsdale A T 1991 Calphad 15 317Google Scholar

    [31]

    Bergner D, Khaddour Y 1993 Defect Diffus Forum 6 95

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    [17] 杨 弘, 张清光, 陈 民. 热扰动对过冷熔体中二次枝晶生长影响的相场法模拟. 物理学报, 2005, 54(8): 3740-3744. doi: 10.7498/aps.54.3740
    [18] 李梅娥, 杨根仓, 周尧和. 二元合金高速定向凝固过程的相场法数值模拟. 物理学报, 2005, 54(1): 454-459. doi: 10.7498/aps.54.454
    [19] 龙文元, 蔡启舟, 陈立亮, 魏伯康. 二元合金等温凝固过程的相场模型. 物理学报, 2005, 54(1): 256-262. doi: 10.7498/aps.54.256
    [20] 于艳梅, 杨根仓, 赵达文, 吕衣礼, A. KARMA, C. BECKERMANN. 过冷熔体中枝晶生长的相场法数值模拟. 物理学报, 2001, 50(12): 2423-2428. doi: 10.7498/aps.50.2423
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出版历程
  • 收稿日期:  2020-11-04
  • 修回日期:  2020-12-23
  • 上网日期:  2021-04-14
  • 刊出日期:  2021-04-20

相场法研究Fe-Cu-Mn-Al合金富Cu相析出机制

  • 中北大学材料科学与工程学院, 太原 030051
  • 通信作者: 赵宇宏, zhaoyuhong@nuc.edu.cn
    基金项目: 国家自然科学基金(批准号: 22008224, 52074246, 51774254, 51774253, 51804279, 51801189)、中央引导地方科技发展专项资金项目(批准号: YDZX20191400002796)、山西省科技重大专项(批准号: 20181101014, 20191102008, 20191102007)、山西省平台基地和人才专项(批准号: 201805D211036)、山西省科技成果转化引导专项(批准号: 201804D131039)、山西省青年科技研究基金(批准号: 201801D221152)和装备预研领域基金重点项目(批准号: 61409230407)资助的课题

摘要: 基于Ginzburg-Landau理论采用连续相场法模拟了Fe-15%Cu-3%Mn-xAl(质量分数x = 1%, 3%, 5%)合金在873 K等温时效时纳米富Cu析出相沉淀机制及Al含量对富Cu相析出的阻碍效应. 通过计算成分场变量和结构序参数, 研究了富Cu析出相的形貌、颗粒密度、平均颗粒半径、生长和粗化动力学. 研究结果表明: 在时效早期阶段, 纳米富Cu相通过失稳分解机制析出, 由于原子扩散速率存在差异, 从而形成以富Cu相为核心的核壳结构. 随着时效时间延长, 富Cu相析出物结构由体心立方转变为面心立方. 其中Al和Mn原子在富Cu核外偏析形成Al/Mn簇, 可以将其视为阻碍富Cu析出相形成的缓冲层; 在沉淀过程中, 随着Al含量的增大, Al/Mn金属间相促进了缓冲层的生长, 阻碍富Cu析出相的生长和粗化.

English Abstract

    • 核反应堆压力容器(reactor pressure vessel, RPV)是核电站唯一不可更换设备, 在高能中子辐照下会析出大量纳米富Cu相(Cu-rich precipitate, CRP), 可作为其他相形核中心, 导致RPV钢发生脆化, 影响其使用寿命[1-6]. 研究发现, 通过添加不同合金元素(Mn, Ni, Al等), 可有效调节沉淀相微观结构, 以获得材料的最佳性能[7,8]. Wen等[9]发现, 添加Ni, Mn和Al元素会影响富Cu相析出及有序B2-Ni(Al, Mn)壳的形成, B2壳可缓和应变并阻止富Cu沉淀相与体心立方Fe(bcc-Fe)基体之间的相互扩散, 导致沉淀相粗化率降低. Miller等[10]研究发现, 与Fe-Cu合金中相比, Fe-Cu-Mn合金中析出相数量更多, 尺寸更小. Osamura等[11]研究表明, Fe-Cu合金中添加Mn和Ni元素, 富Cu沉淀物周围形成了富集Ni和Mn的偏析层, 可促进富Cu相的沉淀反应. Shen等[12,13]发现, 在峰值硬度下, Fe-Cu-Ni-Al合金中析出物由富Cu核与NiAl壳结构组成. NiAl壳层可降低界面能及壳层中Cu, Ni和Al原子扩散速率, 阻止富Cu相生长和粗化. 随着时效时间延长, 核壳分解形成新的富Cu相和NiAl相, 这与Wen等[9]和Vaynman等[14]的核壳纳米结构机制相似. 迄今为止, 针对Al, Mn的添加如何影响富Cu相析出的详细机理尚不完全清楚, 有必要进一步研究Fe-Cu-Mn-Al合金中富Cu相析出机制及Al含量影响.

      本工作基于课题组前期工作[15-19], 采用相场法(PFM)[20-25], 耦合相图计算(calculation of phase diagram, CALPHAD)方法导出的热力学数据[26], 建立了Fe-Cu-Mn-Al合金相场模型, 模拟时效过程相分离和沉淀相形态演化, 讨论了Al含量对富Cu析出相的形貌、颗粒密度、平均颗粒半径、生长和粗化的影响规律.

    • 非线性成分守恒场变量的Cahn-Hilliard扩散方程(1)和结构序参数非守恒场变量的Allen-Cahn弛豫方程[27](2)为

      $ \frac{\partial {\mathrm{c}}_{i}\left({{r}},t\right)}{\partial t}=\nabla \cdot \left\{{M}_{i} \cdot \nabla \frac{\partial F}{\partial {c}_{i}\left({{r}},t\right)}\right\}+{\xi }_{{\mathrm{c}}_{i}}\left({{r}},t\right), $

      $ \frac{\partial \eta \left({{r}},t\right)}{\partial t}=-{L}_{\eta }\frac{\partial F}{\partial \eta \left({{r}},t\right)}+{\xi }_{\eta }\left({{r}},t\right), $

      局域成分场变量${\mathrm{c}}_{i}\left({{r}}, t\right)(i=\mathrm{1, 2}, \mathrm{3, 4},$分别代表${\rm{Fe}}, \mathrm{Cu}, \mathrm{Mn}, \mathrm{Al}) $, 其中$ {c}_{1}=1-{c}_{2}-{c}_{3}-{c}_{4} $. 结构序参量$ \eta \left({{r}}, t\right) $表示在空间坐标$ {{r}} $和时间t$ \alpha $相(bcc结构)和$ \gamma $相(面心立方(fcc)结构)的结构转变情况, 通常取$ 0 < \eta < 1 $, $ \eta =0 $表示bcc结构, $ \eta =1 $表示fcc结构; $ {\xi }_{{\mathrm{c}}_{i}}\left({{r}}, t\right) $$ {\xi }_{\eta }\left({{r}}, t\right) $是Gauss噪声项; $ {L}_{\eta } $是表征相结构转变的动力学系数; $ {M}_{i} $是原子扩散迁移率:

      $\begin{split} &{M}_{i}\left(\eta,T\right)\\ =&\;{c}_{0i}\left(1-{c}_{0i}\right)\times \left[\left(1-\eta \right)\frac{{D}_{i}^{\alpha }\left(T\right)}{RT}+\eta \frac{{D}_{i}^{\gamma }\left(T\right)}{RT}\right], \end{split}$

      式中, R为气体摩尔常数, 8.314 J/(mol·K); T是热力学温度; $ {c}_{0 i} $表示合金元素i的初始成分; $ {D}_{i}^{\alpha }\left(T\right) $$ {D}_{i}^{\gamma }\left(T\right) $分别为合金元素i在bcc结构和fcc结构中的互扩散系数:

      $ {D}_{i}^{\varphi }\left(T\right)={D}_{i}^{0,\varphi }\left(T\right)\mathrm{exp}\left(\frac{-{Q}_{i}^{0,\varphi }}{ {R}T}\right), $

      式中, $ \varphi $表示$ \alpha $$ \gamma $相; $ {D}_{i}^{0, \varphi }\left(T\right) $是合金元素i$ \varphi $相中的自扩散系数; $ {Q}_{i}^{0, \varphi } $是合金元素i$ \varphi $相中的热扩散激活能[28].

      微观结构演化驱动力来自于自由能降低, 系统总自由能F[29]

      $\begin{split} F=\;&\int \Big\{\left[1-h\left(\eta \right)\right][{G}^{\alpha }\left({c}_{i}\left({{r}},t\right),T\right)+Y{V}_{m}{\varepsilon }_{0}^{2}\left({c}_{i}\right)]\\ &+h\left(\eta \right){G}^{\gamma }\left({c}_{i}\left({{r}},t\right),T\right) +Wg\left(\eta \right)\\ &+\frac{1}{2}\sum {k}_{c}{\left({\nabla c}_{i}\right)}^{2}+\frac{1}{2}\sum {k}_{\eta }{\left(\nabla \eta \right)}^{2}\Big\}{\rm{d}}V,\\[-14pt] \end{split}$

      式中, $\dfrac{1}{2}\displaystyle\sum {k}_{c}{\left({\nabla c}_{i}\right)}^{2}$$\dfrac{1}{2}\displaystyle\sum {k}_{\eta }{\left(\nabla \eta \right)}^{2}$分别是成分和结构的梯度能函数, $ {k}_{c} $$ {k}_{\eta } $分别是成分和结构的梯度能量系数; $ h\left(\eta \right) $$ g\left(\eta \right) $是无量纲插值函数[29], 其作用是限制结构序参数取值在[0, 1]内, $ h\left(\eta \right)={\eta }^{2}\left(3-2\eta \right) $, $ g\left(\eta \right)=\eta \left(1-\eta \right) $; W是双势阱高度, 通常取正数; Y$ {V}_{m} $分别是平均刚度系数和摩尔体积, $ {\varepsilon }_{0}\left({c}_{i}\right) $是由于晶格错配而引起的本征应变能:

      $ {\varepsilon }_{0}\left({c}_{i}\right)=\sum\limits_{i=2}^{4}{\delta }_{i}\left({c}_{i}-{c}_{0i}\right), $

      式中, $ {\delta }_{i} $是晶格错配系数, $ {\delta }_{i}=\left({a}_{i}-{a}_{1}\right)/{a}_{i} $; $ {a}_{1} $表示基体的晶格常数, $ {a}_{i} $是第i组分的晶格常数. 表1[16]为模拟参数.

      ParameterValueUnit
      $ {k}_{c}, {k}_{\eta } $${k}_{c}=5.0\times {10}^{-15}, $$ {\mathrm{J} \cdot \mathrm{m}}^{2}/{\mathrm{mol}}^{} $
      ${k}_{c}=1.0\times {10}^{-15} $
      $ {V}_{m} $$ 7.09\times {10}^{-6} $$ {\mathrm{m}}^{3}/{\mathrm{mol}}^{} $
      T$ 873 $$ \mathrm{K} $
      Y$ 214 $$ \mathrm{GPa} $
      $ {L}_{x}\times {L}_{y}\times {L}_{z} $$ 64\times 64\times 64 $$ \mathrm{n}{\mathrm{m}}^{3} $
      W$ 5.0\times {10}^{3} $$ \mathrm{J}/{\mathrm{mol}}^{} $
      $ {D}_{i}^{0, \varphi }\left(\varphi =\alpha, \gamma \right) $${D}_{\mathrm{Cu} }^{0, \alpha }=4.7\times {10}^{-5},$

      ${D}_{\mathrm{Cu} }^{0, \gamma }=4.3\times {10}^{-5} $
      $ {\mathrm{m}}^{2}/{\mathrm{s}}^{} $
      ${D}_{\mathrm{Mn} }^{0, \alpha }=1.49\times {10}^{-4}, $

      ${D}_{\mathrm{Mn} }^{0, \gamma }=1.78\times {10}^{-5} $
      ${D}_{\mathrm{Al} }^{0, \alpha}$[31]$=5.35\times {10}^{-4}, $

      $ {D}_{\mathrm{Al} }^{0, \gamma } $[31]$=2.20\times {10}^{-5} $
      $ {Q}_{i}^{0, \varphi }\left(\varphi =\alpha, \gamma \right) $${Q}_{\mathrm{Cu} }^{0, \alpha }=2.44\times {10}^{5}, $

      ${Q}_{\mathrm{Cu} }^{0, \gamma }=2.80\times {10}^{5} $
      $ \mathrm{J}/{\mathrm{mol}}^{} $
      ${Q}_{\mathrm{Mn} }^{0, \alpha }=2.63\times {10}^{5},$

      $ {Q}_{\mathrm{Mn} }^{0, \gamma }=2.64\times {10}^{5} $
      ${Q}_{\mathrm{Al} }^{0, \alpha }$[31]$=2.71\times {10}^{5}, $

      ${Q}_{\mathrm{Al} }^{0, \gamma } $[31]$=2.67\times {10}^{5} $
      注: $ {k}_{c}, {k}_{\eta } $, 梯度能量系数; $ {V}_{m} $, 摩尔体积; T, 热力学温度; Y, 平均刚度系数; $ {L}_{x}, {L}_{y}, {L}_{z} $, 沿x, y, z轴的模拟区域宽度; W, 双势阱高度; $ {D}_{i}^{0, \varphi } $, 扩散系数; $ {Q}_{i}^{0, \varphi } $, 热扩散激活能.

      表 1  相场模型参数表[16]

      Table 1.  Parameters used in phase field simulation[16]

      (5)式中, $ {G}^{\alpha }\left({c}_{i}\left({{r}}, t\right), T\right) $$ {G}^{\gamma }\left({c}_{i}\left({{r}}, t\right), T\right) $分别代表$ \alpha $$ \gamma $相的Gibbs自由能, 是关于$ {c}_{i}\left({{r}}, t\right) $T的函数, 其表达式为

      $ \begin{split} &{G}^{\varphi }\left({c}_{i}\left({{r}},t\right),T\right)\\ =& \sum\limits_{i}{G}_{i}^{\varphi }{c}_{i}+\mathrm{R}T\sum\limits_{i}{c}_{iln}{c}_{i} +\sum\limits_{i}\sum\limits_{j>i}{L}_{i,j}^{\varphi }{c}_{i}{c}_{j} \\ & +\sum\limits_{i}\sum\limits_{j>i}\sum\limits_{k>j}{L}_{i,j,k}^{\varphi }{c}_{i}{c}_{j}{c}_{k},\end{split} $

      其中, $ {G}_{i}^{\varphi } $是纯i元素Gibbs自由能[30]; $ {L}_{i, j}^{\varphi } $$ {L}_{i, j, k}^{\varphi } $是相互作用系数.

      基于相场动力学方程, 将距离、时间、能量分别无量纲化为$ b=L/N $(其中, L为模拟宽度, N为网格数)、$t=\dfrac{{b}^{2}}{{D}_{\mathrm{Cu}}^{\alpha }}{t}^{*}$ (其中$ {t}^{*} $是无量纲时间)、RT形式. 模拟了873 K时, Fe-15%Cu-3%Mn-1%Al合金中富Cu析出相的析出机制以及不同Al含量(x = 1%, 3%, 5%)下富Cu析出相的动态演化规律.

    • 图1为Fe-15%Cu-3%Mn-1%Al合金在873 K时效时的相分离三维原子演化图. 随着时效时间延长, 分别用Fe (图1(a1)(d1))、Cu (图1 (a2)(d2))、Mn (图1 (a3)(d3))、Al (图1 (a4)(d4))来表示富Cu相析出过程, 对应色标在右侧给出. 对比图1(a1)(a4), 初始阶段过饱和固溶体, 高斯噪声相影响导致在$ \mathrm{\alpha } $-Fe基体中产生小的随机成分起伏, 此时富Cu相开始形核, 如图1(a2)中绿色斑点区域, 而Ni与Mn原子尚未看到明显变化. 表明此时体系由$ \alpha $-Fe基体和共格富Cu相组成. $ t^* = 18500 $时, 浓度波逐渐分解, 形成水滴状富Cu沉淀相, 分散在合金中. 同时, Mn, Al开始出现成分起伏, 向富Cu相中心偏聚, 如图1(b1)(b4) 所示. $ t^* = 20000 $时, 基体中球状富Cu相颗粒以Ostwald熟化机制进行粗化, 大颗粒长大、小颗粒消失. 同时, Mn, Al从富Cu区域内扩散至富Cu相界面处, 这主要是由于富Cu相界面处存在较大共格畸变, 畸变大的区域易产生新的无畸变晶粒的核心, 导致Mn, Al在界面处聚集形成以富Cu相为核心, Mn, Al为壳状物的核壳结构, 如图1(a3)(d3)所示, 这与实验结果一致[9].

      图  1  时效温度873 K时Fe-15%Cu-3%Mn-1%Al合金沉淀相三维演化相场模拟 (a1)−(a4) t* = 17000; (b1)−(b4) t* = 18500; (c1)−(c4) t* = 20000; (d1)−(d4) t* = 22500

      Figure 1.  Three-dimensional phase-field simulation of precipitation phase of Fe-15%Cu-3%Mn-1%Al alloy when aged at 873 K: (a1)−(a4) t* = 17000; (b1)−(b4) t* = 18500; (c1)−(c4) t* = 20000; (d1)−(d4) t* = 22500.

      图2为结构序参数随时间变化曲线, 其中$ \eta =0 $表示bcc结构, $ \eta =1 $表示fcc结构. 当$ t^* $ = 25000时, 结构序参数为零且基本未发生变化, 为bcc结构. 当$ t^* $ = 31000时, 富Cu相结构序参数值达到0.2左右, 表明此时富Cu相开始由bcc向fcc转变. 当$ t^* $ = 33000时, 结构序参数值超过0.6, 表明此时富Cu相已基本转变为fcc结构.

      图  2  时效温度873 K时Fe-15%Cu-3%Mn-1%Al合金中富Cu相结构序参数演化曲线

      Figure 2.  Evolution curves of Cu-rich phase structure order parameter in Fe-15%Cu-3%Mn-1%Al alloy when aged at 873 K.

    • 图3为时效温度为873 K时Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金三维富Cu相演化图. 对比图3(a1)(a3), $ t^* $ = 21000, x = 1%时, 纳米富Cu相析出颗粒数量最多, x = 3%次之, x = 5%最少, 表明Al含量增大对抑制富Cu相的生长和粗化. B2结构的Al/Mn相由于具有较小晶格失配, 降低Fe基体/富Cu相界面的界面能和晶格应变能而起缓冲层作用, 导致富Cu相生长缓慢. 因此Cu, Mn和Al原子通过B2结构Al/Mn相的扩散比通过Fe基体的扩散速率慢. 如图3所示, 在相同时效时间内, x = 1%的富Cu析出相的颗粒尺寸始终大于x = 3%和x = 5%.

      图  3  时效温度为873 K时Fe-15%Cu-3%Mn-xAl合金三维富Cu相演化相场模拟 (a1)−(d1) x = 1%; (a2)−(d2) x = 3%; (a3)−(d3) x = 5%; (a1)−(a3) t* = 21000; (b1)−(b3) t* = 22000; (c1)−(c3) t* = 25000

      Figure 3.  Three dimensional evolution diagrams of Cu rich phase in quaternary alloy Fe-15%Cu-3%Mn-xAl alloy aged at 873 K: (a1)−(d1) x = 1%; (a2)−(d2) x = 3%; (a3)−(d3) x = 5%; (a1)−(a3) t* = 21000; (b1)−(b3) t* = 22000; (c1)−(c3) t* = 25000.

    • 时效温度823 K时, Fe-15%Cu-3%Mn-xAl(x = 1%, 3%, 5%)合金中富Cu析出相Gibbs自由能随时间变化曲线如图4所示, Gibbs自由能在初始阶段几乎保持不变, 然后随时效时间延长呈下降趋势. 当Al含量为1%时合金Gibbs自由能高于3%和5%时的情况, 表明随Al含量增大, B2结构Al/Mn壳层通过降低壳层中的Cu, Mn和Al原子的界面能和扩散速率, 富Cu相析出减缓, 析出所需自由能增多, Gibbs自由能降低, 这与图3结果吻合. 在自由能下降阶段存在突起, 由于Ostwald机制, 富Cu析出相中小颗粒析出后分解, 释放能量[3,15], 使Gibbs自由能曲线出现拐点, 形成突起. Al含量越高, Gibbs自由能曲线出现拐点时间越晚, 富Cu析出相中小颗粒分解时间越晚.

      图  4  时效温度873 K时Fe-15%Cu-3%Mn-xAl(x = 1%, 3%, 5%)合金Gibbs自由能随时间变化曲线

      Figure 4.  Gibbs free energy curves in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy when aged at 873 K.

      图5为时效温度873 K时, Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金中富Cu析出相颗粒密度随时间变化曲线. 在时效初期失稳阶段, 颗粒密度迅速增加, 然后在Al含量分别为1%, 3%, 5%条件下, 富Cu相颗粒密度分别在t* = 23800, 25900, 29300开始降低. 这是Al, Mn的加入, 降低的界面能和弹性应变能, 升高的化学成分驱动力造成了析出相所需的临界形核能量降低, 符合Mn和Al可以提高富Cu相的成核速率这一事实[15]. Fe-15%Cu-3%Mn-xAl合金中Al含量越高, 富Cu析出相数量越少, 即Al含量越高, 越促进Mn原子在靠近富Cu析出相界面处偏析, 形成核壳结构, 从而降低了富Cu析出相生长和粗化速率, 这与图3结果一致.

      图  5  时效温度873 K时Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金中富Cu析出相颗粒密度随时间变化曲线

      Figure 5.  Curves of Cu-rich precipitate phase particle density in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy aged at 873 K.

      图6为时效温度为873 K时, Fe-15%Cu-3%Mn-xAl(x = 1%, 3%, 5%)合金中富Cu析出相平均颗粒半径随时间变化曲线. 可以看出, 时效早期沉淀相析出阶段, Al含量分别为1%, 3%, 5%时的 $ \langle r \rangle $保持不变, 且为0. 这是因为孕育期阶段尚未发生相分离, 所以平均粒径为0. 随后生长阶段内, 富Cu相从$ \mathrm{\alpha } $-Fe基体中析出, 发生相分离, 由于合金元素在系统中弥散分布, 并且偏析程度较低, $ \langle r \rangle $开始变大. 其次, 这一阶段持续时间随着Al含量增大而延长表明: Al含量越高, Al与Cu之间存在的协同作用增强, 表现为x = 5%时上升斜率大于x = 1%. 这也可以通过能量变化分析得到验证, 说明Al/Mn相阻碍富Cu相析出. 粗化阶段中由于发生Ostwald粗化, 较大颗粒合并后长大, 较小颗粒分解. 在这一阶段, Al含量增加会抑制富Cu相生长, 较小颗粒消失越慢, 富Cu相的粗化速率越缓慢, $ \langle r \rangle $变化越平稳.

      图  6  时效温度873 K时Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%)合金富Cu析出相平均颗粒半径随时间变化

      Figure 6.  Average particle radius of Cu-rich precipitates in Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%) alloy aged at 873 K.

    • 结合CALPHAD热力学数据, 建立Fe-Cu-Mn-Al合金相场模型研究富Cu相析出机制及Al含量影响, 主要结论如下.

      1)富Cu相通过失稳分解机制析出并随机分布于铁基体中, 具有核-壳分层结构, 随时效时间延长, 富Cu相由bcc转变为fcc结构. Al和Mn原子在富Cu核外偏析形成Al/Mn壳层, 抑制富Cu析出相进一步扩散生长, 可将其视为阻碍富Cu析出相形成的缓冲层, 影响富Cu相析出.

      2)在时效温度823 K下, Fe-15%Cu-3%Mn-xAl (x = 1%, 3%, 5%, 质量分数)合金中, B2-AlMn金属间相的形成阻止富Cu相分离和粗化; 随着Al含量增大, Al/Mn金属间相缓冲层抑制富Cu相进一步扩散生长和粗化的程度增强.

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