﻿ 薄膜中的扩散与应力模型 Models of Diffusion and Stress in Thin Films

Material Sciences
Vol.06 No.06(2016), Article ID:19107,19 pages
10.12677/MS.2016.66053

Models of Diffusion and Stress in Thin Films

Bing Lin, Zhongbao Dai, Mengying Gao, Jiangyong Wang

Department of Physics, Shantou University, Shantou Guangdong

Received: Nov. 4th, 2016; accepted: Nov. 26th, 2016; published: Nov. 30th, 2016

ABSTRACT

Diffusion of atoms in the materials can cause stress, and the produced and/or intrinsic stress may influence the diffusion process, which is more pronounced in the thin films. In this paper, some common diffusion and stress models are summarized and the diffusion equations under stress are deduced based on the thermodynamics. The diffusion profiles with different stress models are then simulated by the finite difference method.

Keywords:Diffusion, Diffusion Coefficient, Diffusion-Induced Stress, Stress-Induced Diffusion

1. 引言

2. 扩散

2.1. 连续模型

1855年，德国生理学家阿道夫·菲克(Adolf Eugen Fick, 1829~1901) [8] 归纳了格雷厄姆(Thomas Graham, 1805~1869)的实验数据 [9] ，并引入扩散系数的概念，将扩散现象用与热学中傅立叶定律(或电学中的欧姆定律)类似的数学式表达，

(2.1)

(2.2)

1896年，英国冶金学家威廉·钱德勒·罗伯茨–奥斯汀爵士(Sir William Chandler Roberts-Austen, 1843~1902)将菲克等人的工作从流体扩展至固体的扩散。他通过实验，得出不同温度和时间下，Au在液体Pb和固体Pb中扩散浓度随深度的剖面图 [10] 。随后，斯凡特·阿伦尼斯(Svante Arrhenius, 1859~1927)给出如下扩散系数与温度的关系式 [11]

(2.3)

(a) 糖放在水中 (b) 溶解：分子扩散 (c) 均匀分布

Figure 1. Diffusion of sucrose molecules in water

Figure 2. Linear (left) and Logarithmic (right) coordinates of impurity (Ge, Cu) diffusion in Al

(2.4)

(2.5)

(2.6)

Figure 3. Gaussian function solution (left) and Error function solution (right) of the diffusion equation

(2.6a)

(2.6b)

(2.7)

(2.8)

Figure 4. Matano plane drifting during diffusion (dash line is the Matano plane)

2.2. 原子模型

1905年，阿尔伯特·爱因斯坦(Albert Einstein, 1879~1955)提出扩散系数、温度、溶质粒子半径与粘度系数之间的关系

(2.9)

(2.10)

1906年，波兰分子统计物理学家玛丽安·斯莫鲁霍夫斯基(Marian von Smoluchowski, 1872~1917)与爱因斯坦建立了布朗运动的统计理论，该理论认为布朗运动实际上代表一种随机涨落现象，涨落可以解释为，原子与布朗颗粒碰撞，并推动其前进 [31] 。该理论在1908年被法国物理学家让·巴蒂斯特·皮兰(Jean Baptiste Perrin)用实验证实 [33] 。

1920年和1921年，奥匈化学家乔治·卡尔·冯·赫维西(Georg Karl von Hevesy, 1885~1966)第一采用同位素标记法分别研究液态Pb [34] 与固态Pb [35] 中的自扩散问题。自扩散与原子跳动频率的关系为

(2.11)

and (2.12)

(2.13)

1951年，人们发现，原子沿晶界扩散的速度比在晶格中的扩散速度快，系数之比甚至达到5个数量级 [38] 。同年，费舍 [39] 和特恩布尔 [40] 分别开始了对晶界扩散的定量研究。这在多晶特别是纳米晶体，薄膜或者多层膜器件中起重要作用。

2.3. 热力学分析

(2.14)

(2.15)

(2.16)

(2.17)

3. 应力

Prussin [48] 最早提出了扩散致应力的概念。他认为，固体中浓度梯度引起的应力，与无受力固体中温度梯度引起的应力类似，都有一个线性应变系数，即

(3.1)

(3.2)

Li [49] 对一些简单形状(薄板，圆柱，球形)进行试验，并给出了(3.1)式中的膨胀系数值，，其中，为摩尔体积。

Larch和Cahn [50] [51] [52] 建立了固体结构中局部扩散通量和由于多相存在或者不均匀分布产生的应力之间的关系，以此为基础的理论预测模型可用于定性，在某些情况下可以定量描述上坡扩散问题。他们给出的厚度为L的薄板应力表达式 [53]

(3.3)

1999年，Gao等人 [55] [56] 研究直接镀在衬底上的金属多晶薄膜的晶界扩散，发现扩散过程中会形成与裂纹类似的楔形晶界。在他们的模型中，假设晶界平行于轴方向，由于质量迁移产生的应力效果类似于在与轴垂直的方向上，某个处插入沿轴方向的位错 [57] 。楔形的插入，晶格收到挤压，引起垂直晶界方向的应力变化，邻近原子沿晶界扩散，其作用相当于产生一个沿晶界方向的正向牵引，如图6所示。

(3.4)

Figure 5. Error function diffusion profiles (left) and corresponding stresses (right) at different times

Figure 6. The geometry of a crack-like grain-boundary wedge

(3.5)

(3.6)

2003年，陈永翀等人基于流体力学的基本概念和方法，推导建立了描述固体互扩散生长的普适方程 [59] ，讨论了应力与扩散的相互关系 [60] ，区别了体积生长和界面生长 [61] ，并给出了固态反应周期层片的理论描述 [62] [63] 。普适方程为：

(3.7)

(3.8)

2005年，Jay Chabraborty [64] 基于Larhe-Cahn理论，针对置换固溶体严格推导建立了非静水应力扩散模型，并以Pd-Cu和Au-Ni二元扩散偶为例数值模拟了非静水压和自应力下所发生的特殊扩散现象。Chabraborty采用误差函数型应力，应力梯度仅局限在薄膜的界面区域，其描述为

(3.9)

2010年，Moskalioviene等人 [65] [66] [67] 根据Onsager理论 [68] 模拟计算了N在渗氮铁中发生应力致间隙扩散导致的深度剖面，很好的与实验数据相吻合。他们采用的应力为 [69]

(3.10)

Figure 7. The variation of the grain boundary stresses with depth as the dislocation density decreases (left) and increases (right)

Figure 8. Different error type stresses (Left: W = 0.05, R = −10, −5, 0, 5, 10; Right: R = 5, W = 0.01, 0.03, 0.05, 0.1, 0.2)

(3.11)

4. 应力对扩散的影响

(4.1)

(4.2)

Figure 9. Sketch of solid under hydrostatic pressure (left) and non-hydrostatic pressure (right). The arrow indicates the direction of pressures and the density of arrow represents the pressure value

(平衡态下，均为常数) (4.3)

(4.4)

(4.5)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

(4.17)

5. 应力作用下扩散的模拟

5.1. Cahn应力模型下的模拟

(5.1)

Larche和Cahn的应力模型为，

(5.2)

(5.3)

(5.4)

(5.5)

(5.6)

(5.7)

(5.8)

5.2. Gao应力模型下的模拟

(5.9)

(5.10)

Figure 10. The evolution of stress at different times (number of time steps: 100, 1000, 5000, 10,000) (left) and the concentration under stress and stress free (right, number of time steps: 10,000)

Figure 11. The calculation dependencies of the dimensionless GB concentration (CB), the normal GB stress (S), and the width of the wedge (W) on the penetration depth along the GB for φ = 0.1 and τ = 1

6. 总结与展望

Models of Diffusion and Stress in Thin Films[J]. 材料科学, 2016, 06(06): 413-431. http://dx.doi.org/10.12677/MS.2016.66053

1. 1. Louthan, M.R., Caskey, G.R., Donovan, J.A.,, Rawl Jr., D.E. (1972) Hydrogen Embrittlement of Metals. Materials Science and Engineering, 10, 357-368. https://doi.org/10.1016/0025-5416(72)90109-7

2. 2. Shih, D.S., Robertson, I.M. and Birnbaum, H.K. (1988) Hydrogen Embrittlement of α Titanium: In Situ TEM Studies. Acta Metallurgica, 36, 111-124. https://doi.org/10.1016/0001-6160(88)90032-6

3. 3. 张俊善. 材料的高温变形与断裂[M]. 北京: 科学出版社, 2007: 489-493.

4. 4. 张在玉, 陈秀华. Cu在CoN和CoSiN薄膜中的扩散研究[J]. 怀化学院学报, 2013, 32(5): 27-32.

5. 5. 王玉中, 赵寿南. 硅扩散应力的理论分析[J]. 华南理工大学学报: 自然科学版, 1995, 23(12): 38-45.

6. 6. Baker, D.R., Verbrugge, M.W. and Bower, A. (2016) Thermodynamics, Stress, and Stefan-Maxwell Dif-fusion in Solids: Application to Small-Strain Materials Used in Commercial Lithium-Ion Batteries. Journal of Solid State Electrochemistry, 20, 163-181. https://doi.org/10.1007/s10008-015-3012-7

7. 7. Diaz, A., Alegre, J. and Cuesta, I.I. (2016) A Review on Diffusion Modelling in Hydrogen Related Failures of Metals. Engineering Failure Analysis, 66, 577-595. https://doi.org/10.1016/j.engfailanal.2016.05.019

8. 8. Fick, A. (1855) Ueber Diffusion. Annalen der Physik, 170, 59-86. https://doi.org/10.1002/andp.18551700105

9. 9. Graham, T. (1950) The Bakerian Lecture: On the Diffusion of Liquids. Philosophical Transactions of the Royal Society of London Series A, 140, 1-46. https://doi.org/10.1098/rstl.1850.0001

10. 10. Roberts-Austen, W.C. (1896) Bakerian Lecture: On the Diffusion in Metals. Philosophical Transactions of the Royal Society of London Series A, 187, 383-415. https://doi.org/10.1098/rsta.1896.0010

11. 11. Arrhenius, S. (1889) Über die Reaktionsgeschwindigkeitbei der Inversion von Rohrzuckerdurch Säuren. Zeitschrift für Physikalische Chemie, 4, 226-248.

12. 12. Peterson, N.L. and Rothman, S.J. (1970) Impurity Diffusion in Aluminum. Physical Review B, 1, 3264-3273. https://doi.org/10.1103/PhysRevB.1.3264

13. 13. Salamon, M. and Mehrer, H. (2005) Interdiffusion, Kirkendall Effect, and Al Self-Diffusion in Iron-Aluminiumalloys. Zeitschrift fur Metallkunde, 96, 4-16. https://doi.org/10.3139/146.018071

14. 14. Shewmon, P. (1989) Diffusion in Solids. The Minerals, Metals & Materials Society, Diffusion in Solids. 2nd Edition, Retroactive Coverage, United States, 246.

15. 15. Crank, J. (1975) The Math-ematics of Diffusion. 2nd Edition, Clarendon Press, Clarendon.

16. 16. 潘金生, 仝健民, 田民波. 材料科学基础[M]. 北京: 清华大学出版社, 1998.

17. 17. 林福民. 数学物理方法简明教程[M]. 北京: 北京大学出版社, 2008.

18. 18. 胡敏. 扩散方程高精度加权差分格式的MATLAB实现[J]. 四川文理学院学报, 2014, 24(5):15-18.

19. 19. 常旭, 石伟. 基于 MATLAB 的渗氮扩散数值模拟程序[J]. 热处理技术与装备, 2015, 36(5): 9-15.

20. 20. Boltzmann, L. (1894) Zur Integration der Diffusionsgleichung bei variabeln Diffusionscoefficienten. Wiedemann’s Annalen, 53, 959-964.

21. 21. Matano, C. (1933) On the Relation between the Diffusion-Coefficients and Concentrations of Solid Metals (The Nickel-Copper System). Japanese Journal of Physics, 8, 109-113.

22. 22. Chen, J., Zhang, C., Wang, J., et al. (2015) Thermodynamic Description, Diffusivities and Atomic Mobilities in Binary Ni-Os System. Calphad, 50, 118-125. https://doi.org/10.1016/j.calphad.2015.06.001

23. 23. Kirkaldy, J.S. (1957) Diffusion in Multicomponent Metallic Systems. Canadian Journal of Physics, 35, 435-440. https://doi.org/10.1139/p57-047

24. 24. 王常珍. 冶金物理化学研究方法[J]. 北京: 冶金工业出版杜, 1982: 307.

25. 25. 杨绮琴, 刘冠昆. 金属在其合金相中扩散系数的测定[J]. 稀有金属, 1992, 16(1): 18-21.

26. 26. 赵长伟, 马沛生, 何明霞. 液相扩散系数测定方法的近期研究进展[J]. 化学工业与工程, 2002, 19(5): 374-379.

27. 27. 吴永炘, 文效忠, 杨志雄, 等. 镀层中基体金属扩散系数的测定[J]. 电镀与涂饰, 1999(4): 008.

28. 28. Brown, R. (1828) A Brief Ac-count of Microscopical Observations Made in the Months of June, July and August 1827, on the Particles Contained in the Pollen of Plants, and on the General Existence of Active Molecules in Organic and Inorganic Bodies. Philosophical Magazine Series 2, 4, 161-173. https://doi.org/10.1080/14786442808674769

29. 29. Mehrer, H. and Stolwijk, N.A. (2009) Heroes and Highlights in the History of Diffusion. Diffusion Fundamentals, 11, 1-32.

30. 30. Narasimhan, T.N. (2009) The Dichotomous History of Diffusion. Physics Today, 62, 48-53.

31. 31. Einstein, A. (1905) Über die von der molekularkinetischen Theorie der Wärmegeforderte Bewegung von in ruhenden Flussigkeitensuspendierten Teilchen. Annalen der Physik, 322, 549-560. https://doi.org/10.1002/andp.19053220806

32. 32. Von Smoluchowski, M. (1906) Zurkinetischentheorie der brownschenmolekularbewegung und der suspensionen. Annalen der Physik, 326, 756-780. https://doi.org/10.1002/andp.19063261405

33. 33. Perrin, J. (1908) La loi de Stokes et le mouvementbrownien. Comptesrendus, 147, 475-476.

34. 34. Groh, J. (1920) Die Selbstdiffusionsgeschwindigkeit des geschmolzenen Bleis. Annalen der Physik, 368, 85-92. https://doi.org/10.1002/andp.19203681705

35. 35. Groh, J. (1921) Die Selbstdiffusion in festem Blei. Annalen der Physik, 370, 216-222. https://doi.org/10.1002/andp.19213701103

36. 36. Kirkendall, T.D., Thomassen, L. and Upthegrove, C. (1939) Rates of Diffusion of Copper and Zinc in Alpha Brass. Transaction of American Institute of Mining, Metallurgical, and Petro-leum Engineers, 133, 186-203.

37. 37. Darken, L.S. (2010) Diffusion, Mobility and Their Interrelation through Free En-ergy in Binary Metallic Systems. Metallurgical and Materials Transactions A, 41, 545-555.

38. 38. Le Claire, A.D. (1951) Grain Boundary Diffusion in Metals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 42, 468-474. https://doi.org/10.1080/14786445108561177

39. 39. Fisher, J.C. (1951) Calculation of Diffusion Penetration Curves for Surface and Grain Boundary Diffusion. Journal of Applied Physics, 22, 74-77. https://doi.org/10.1063/1.1699825

40. 40. Hoffman, R.E. and Turnbull, D. (1951) Lattice and Grain Boundary Self-Diffusion in Silver. Journal of Applied Physics, 22, 634-639. https://doi.org/10.1063/1.1700021

41. 41. Viljoen, E.C., Du Plessis, J., Swart, H.C. and van Wyk, G.N. (1995) Sn Bulk-to-Surface Diffusion in a Cu (111)(Sn) Single Crystal. Surface Science, 342, 1-10. https://doi.org/10.1016/0039-6028(95)00684-2

42. 42. Wang, J.Y., Du Plessis, J., Terblans, J.J. and van Wyk, G.N. (1999) Kinetics near the Discontinuous Surface Transition in the Cu (Ag)(111) Binary Segregating System. Surface Science, 423, 12-18. https://doi.org/10.1016/S0039-6028(98)00819-X

43. 43. Wang, J.Y., Du Plessis, J., Terblans, J.J. and van Wyk, G.N. (1999) The Discontinuous Surface Transition in the Cu (111)(Ag) Binary Segregating System. Surface Science, 419, 197-206. https://doi.org/10.1016/S0039-6028(98)00790-0

44. 44. Laughlin, D. and Cahn, J. (1975) Spinodal Decompo-sition in Age Hardening Copper-Titanium Alloys. Acta Metallurgica, 23, 329-339. https://doi.org/10.1016/0001-6160(75)90125-X

45. 45. 胡赓祥, 等编著. 材料科学基础(第三版) [M]. 上海: 上海交通大学出版社, 2010.

46. 46. Thornton, J. and Hoffman, D. (1989) Stress-Related Effects in Thin Films. Thin Solid Films, 171, 5-31. https://doi.org/10.1016/0040-6090(89)90030-8

47. 47. 邵珊珊. 力及力–电耦合作用下微结构中扩散, 应力和变形分析[D]: [博士学位论文]. 上海: 华东理工大学, 2011.

48. 48. Prussin, S. (1961) Generation and Distribution of Disloca-tions by Solute Diffusion. Journal of Applied Physics, 32, 1876-1881. https://doi.org/10.1063/1.1728256

49. 49. Li, J.C.M. (1978) Physical Chemistry of Some Microstructural Phenomena. Metallurgical Transactions A, 9, 1353- 1380. https://doi.org/10.1007/BF02661808

50. 50. Larche, F. and Cahn, J.W. (1973) A Linear Theory of Thermochemical of Solids under Stress. Acta Metallurgica, 21, 1051-1063. https://doi.org/10.1016/0001-6160(73)90021-7

51. 51. Larch, F. and Cahn, J.W. (1978) A Nonlinear Theory of Thermochemical Equilibrium of Solids under Stress. Acta Metallurgica, 26, 53-60. https://doi.org/10.1016/0001-6160(78)90201-8

52. 52. Larche, F.C. and Cahn, J.W. (1978) Thermochemical Equilibrium of Multiphase Solids under Stress. Acta Metallurgica, 26, 1579-1589. https://doi.org/10.1016/0001-6160(78)90067-6

53. 53. Larche, F.C. and Cahn, J.W. (1982) The Effect of Self-Stress on Diffusion in Solids. Acta Metallurgica, 30, 1835- 1845. https://doi.org/10.1016/0001-6160(82)90023-2

54. 54. 杨小斌, 涂善东. 碳扩散和扩散应力的相互影响分析[J]. 固体力学学报, 2013(S1): 74-78.

55. 55. Gao, H., Zhang, L., Nix, W.D., Thompson, C.V. and Arzt, E. (1999) Crack-Like Grain-Boundary Diffusion Wedges in Thin Metal Films. Acta Materialia, 47, 2865-2878. http://dx.doi.org/10.1016/S1359-6454(99)00178-0

56. 56. Zhang, L. (2000) A Class of Strongly Coupled Elasticity and Diffusion Problems in Thin Metal Films. Stanford University, Stanford.

57. 57. Chateau, J.P., Delafosse, D. and Magnin, T. (2002) Numerical Simulations of Hydrogen-Dislocation Interactions in Fcc Stainless Steels, Part II: Hydrogen Effects on Crack Tip Plasticity at a Stress Corrosion Crack. Acta Materialia, 50, 1523-1538. https://doi.org/10.1016/S1359-6454(02)00009-5

58. 58. Klinger, L. and Rabkin, E. (2011) Theory of the Kirkendall Effect during Grain Boundary Interdiffusion. Acta Materialia, 59, 1389-1399. https://doi.org/10.1016/j.actamat.2010.10.070

59. 59. Chen, Y.C., Zhang, Y.G. and Chen, C.Q. (2004) General Theory of Interdiffusion Growth in Diffusion Couples. Materials Science and Engineering A, 368, 1-9. https://doi.org/10.1016/S0921-5093(03)00480-5

60. 60. 陈永翀. 扩散蠕变理论的基础问题研究[J]. 稀有金属, 2012, 36(2): 171.

61. 61. 陈永翀, 其鲁, 张永刚, 等. 固体互扩散生长的唯象理论[J]. 北京大学学报(自然科学版), 2006, 42(2): 168-174.

62. 62. 陈永翀, 其鲁, 张永刚, 等. 固态反应周期层片型结构分析[J]. 金属学报, 2005, 41(3): 235-241.

63. 63. Chen, Y., Zhang, Y. and Chen, C. (2003) Quantitative Descriptions of Periodic Layer Formation during Solid State Reactions. Materials Science and Engineering A, 362, 135-144. https://doi.org/10.1016/S0921-5093(03)00479-9

64. 64. Chakraborty, J. (2005) Diffusion in Stressed Thin Films. Uni-versitat Stuttgart, Stuttgart.

65. 65. Galdikas, A. and Moskalioviene, T. (2010) Stress Induced Nitrogen Diffusion during Nitriding of Austenitic Stainless Steel. Computational Materials Science, 50, 796-799. https://doi.org/10.1016/j.commatsci.2010.10.018

66. 66. Galdikas, A. and Moskalioviene, T. (2011) Modeling of Stress Induced Nitrogen Diffusion in Nitrided Stainless Steel. Surface and Coatings Technology, 205, 3742-3746. https://doi.org/10.1016/j.surfcoat.2011.01.040

67. 67. Moskalioviene, T. and Galdikas, A. (2012) Stress Induced and Concentration Dependent Diffusion of Nitrogen in Plasma Nitrided Austenitic Stainlesssteel. Vacuum, 86, 1552-1557. https://doi.org/10.1016/j.vacuum.2012.03.026

68. 68. Onsager, L. (1945) Theories and Problems of Liquid Diffusion. Annals of the New York Academy of Sciences, 46, 241- 265. https://doi.org/10.1111/j.1749-6632.1945.tb36170.x

69. 69. Christiansen, T. and Somers, M.A.J. (2006) Avoiding Ghost Stress on Reconstruction of Stress-and Composition- Depth Profiles from Destructive X-Ray Diffraction Depth Profiling. Materials Science and Engineering A, 424, 181- 189. https://doi.org/10.1016/j.msea.2006.03.007

70. 70. Hoeft, D., Latella, B.A. and Short, K.T. (2005) Residual Stress and Cracking in Expanded Austenite Layers. Journal of Physics: Condensed Matter, 17, 3547-3558. https://doi.org/10.1088/0953-8984/17/23/007

71. 71. Christiansen, T.L. and Somers, M.A.J. (2009) Stress and Composition of Carbon Stabilized Expanded Austenite on Stainless Steel. Metallurgical and Materials Transactions A, 40, 1791-1798. https://doi.org/10.1007/s11661-008-9717-9

72. 72. 关振铎, 等编著. 无机材料物理性能[M]. 北京: 清华大学出版社, 1992.

73. 73. Larche, F. and Cahn, J. (1985) The Interaction of Composition and Stress in Crystalline Solids. Acta Met-allurgica, 33, 331-357. https://doi.org/10.1016/0001-6160(85)90077-X

74. 74. Klinger, L. and Rabkin, E. (2011) Grain Boundary Interdiffusion and Stresses in Thin Polycrystalline Films. Journal of Materials Science, 46, 4343-4348. https://doi.org/10.1007/s10853-010-5237-2