Thermal Grooving by Surface Diffusion: a Review of Classical Thermo-Kinetics Approach

Year 2017, , 7 - 16, 30.06.2017

Oncu Akyildiz , Tarik Omer Ogurtani

https://doi.org/10.17350/HJSE19030000042

Cited By: 7

Abstract

I n polycrystalline materials wherever a grain boundary intersects a free surface and whenever the topographic variation associated with the atomic motion is favored by total free energy dissipation, the material surface grooves. In this review, we focused on the grain boundary grooving by surface diffusion which is an active mechanism at moderate temperatures and for grooves small in size. Starting with a description of the classical thermo-kinetics treatment of the process, we briefly reviewed Mullins’ very first modeling effort with a small slope assumption at the groove root and further considerations regarding finite slopes, different grain geometries, and anisotropic surface free energies. We concluded by giving examples of experimental observations in accord with theoretical calculations

Keywords

Thermal grooving, Grain boundary groove, Surface diffusion

References

• 1. Mullins WW, Shewmon PG. The kinetics of grain boundary grooving in copper. Acta Metallurgica, 7, 3 (1959) 163-170. doi: 10.1016/0001-6160(59)90069-0.
• 2. Balluffi RW, Allen SM, Carter WC. Kinetics of Materials. John Wiley & Sons, Inc., Hoboken, NJ, USA, 2005. doi: 10.1002/0471749311.
• 3. Mullins WW. Theory of thermal grooving. Journal of Applied Physics, 28, 3 (1957) 333. doi: 10.1063/1.1722742.
• 4. Sun B, Suo Z, Yang W. A finite element method for simulating interface motion—I. Migration of phase and grain boundaries. Acta Materialia, 45, 5 (1997) 1907-1915. doi: 10.1016/ S1359-6454(96)00323-0.
• 5. Suo Z. Motions of microscopic surfaces in materials. Advances in Applied Mechanics, 33 (1997) 193-294. doi: 10.1016/ S0065-2156(08)70387-9.
• 6. Herring C. The physics of powder metallurgy. (Kinston WE) (p. 143). McGraw-Hill, New York, USA, 1951.
• 7. Gibbs W. The Collected Works of J. Willard Gibbs, Vol. I Thermodynamics. Yale University Press, New Haven, USA, 1948.
• 8. Defay R, Prigogine I. Surface tension and adsorption. (A. Bellemans). John Wiley & Sons., New York, USA, 1966.
• 9. Mullins WW. Solid surface morphologies governed by capillarity. In Metal Surfaces: Structure, Energetics and Kinetics (p. 17), ASM, Cleveland, OH, USA, 1963.
• 10. Robertson WM. Grain-boundary grooving by surface diffusion for finite surface slopes. Journal of Applied Physics, 42, 1 (1971) 463. doi: 10.1063/1.1659625.
• 11. Zhang W, Schneibel JH. Numerical simulation of grainboundary grooving by surface diffusion. Computational Materials Science, 3, 3 (1995) 347-358. doi: 10.1016/0927- 0256(94)00073-L.
• 12. Khenner M, Averbuch A, Israeli M, Nathan M, Glickman EE. Level set modeling of transient electromigration grooving. Computational Materials Science, 20,2 (2001) 235-250. doi: 10.1016/S0927-0256(00)00179-8.
• 13. Hackney SA, Ojard G. Grain boundary grooving at finite grain size. Scripta Metallurgica, 22,11 (1988) 1731-1735. doi: 10.1016/S0036-9748(88)80274-6.
• 14. Khenner M. Numerical simulation of grainboundary grooving by level set method. Journal of Computational Physics, 170, 2 (2001) 764-784. doi: 10.1006/jcph.2001.6760.
• 15. Srolovitz DJ, Safran SA. Capillary instabilities in thin films. I. Energetics. Journal of Applied Physics, 60, 1 (1986) 247. doi: 10.1063/1.337689.
• 16. Huang P, Li Z, Sun J. Finite element analysis for surface diffusion-controlled shape instabilities of plate-like grains. Computational Materials Science, 20,1 (2001) 66-76. doi: 10.1016/S0927-0256(00)00126-9.
• 17. Ogurtani TO, Akyildiz O. Grain boundary grooving and cathode voiding in bamboo-like metallic interconnects by surface drift diffusion under the capillary and electromigration forces. Journal of Applied Physics, 97, 9 (2005) 093520. doi: 10.1063/1.1883305.
• 18. Shenoy VB, Freund LB. A continuum description of the energetics and evolution of stepped surfaces in strained nanostructures. Journal of the Mechanics and Physics of Solids, 50, 9 (2002) 1817-1841. doi: 10.1016/S0022- 5096(02)00015-7.
• 19. Jeong H, Williams, ED. Steps on surfaces: experiment and theory. Surface Science Reports, 34, 6-8 (1999) 171-294. doi: 10.1016/S0167-5729(98)00010-7.
• 20. Mykura H. The variation of the surface tension of nickel with crystallographic orientation. Acta Metallurgica, 9, 6 (1961) 570-576. doi: 10.1016/0001-6160(61)90160-2.
• 21. Bonzel HP, Mullins WW. Smoothing of perturbed vicinal surfaces. Surface Science, 350,1-3 (1996) 285-300. doi: 10.1016/0039-6028(95)01111-0.
• 22. Rabkin E, Klinger L, Semenov VN. Grain boundary grooving at the singular surfaces. Acta Materialia, 48, 7 (2000) 1533- 1540. doi: 10.1016/S1359-6454(99)00432-2.
• 23. Klinger LM, Rabkin E. The effect of stress on grain boundary interdiffusion in a semi-infinite bicrystal. Acta Materialia, 55, 14 (2007) 4689-4698. doi: 10.1016/j. actamat.2007.04.039.
• 24. Rabkin E, Klinger L. The fascination of grain boundary grooves. Advanced Engineering Materials, 3, 5 (2001) 277-282. doi: 10.1002/1527-2648(200105)3:53.0.CO;2-G.
• 25. Zhang W, Sachenko PP, Schneibel JH. Kinetics of thermal grain boundary grooving for changing dihedral angles. Journal of Materials Research, 17, 6 (2002) 1495-1501. doi: 10.1557/JMR.2002.0222.
• 26. Akyildiz O, Oren EE, Ogurtani TO. Mesoscopic nonequilibrium thermodynamics treat-ment of the grain boundary thermal grooving induced by the anisotropic surface drift diffusion, J. Mater. Sci., 46, 18 (2011) 6054-6064.
• 27. Zhang W, Sachenko PP, Gladwell I. Thermal grain boundary grooving with anisotropic surface free energies. Acta Materialia, 52, 1 (2004) 107-116. doi: 10.1016/j. actamat.2003.08.033.
• 28. Xin T, Wong H. A δ-function model of facets. Surface Science, 487, 1-3 (2001) L529-L533. doi: 10.1016/S0039- 6028(01)01158-X.
• 27. Xin T, Wong H. Grain-boundary grooving by surface diffusion with strong surface energy anisotropy. Acta Materialia, 51, 8 (2003) 2305-2317. doi: 10.1016/S1359-6454(03)00039-9.
• 30. Xin T, Wong H. A spike-function model of facets. Materials Science and Engineering A, 364, 1-2 (2004) 287-295. doi: 10.1016/j.msea.2003.08.042.
• 31. Min D, Wong H. Grain-boundary grooving by surface diffusion with asymmetric and strongly anisotropic surface energies. Journal of Applied Physics, 99, 2 (2006a) 023515. doi: 10.1063/1.2159082.
• 32. Du P, Wong H. A delta-function model for axially symmetric crystals. Scripta Materialia, 55,12 (2006) 1171-1174. doi: 10.1016/j.scriptamat.2006.07.042.
• 33. Ogurtani, T. O. Dirichlet extremum problem associated with the asymmetric grain-boundary thermal grooving under the Dirac δ-type anisotropic surface stiffness in bicrystal thin solid films. Journal of Applied Physics, 102, 6 (2007) 063517. doi: 10.1063/1.2781574.
• 34. Ramasubramaniam A, Shenoy VB. On the evolution of faceted grain-boundary grooves by surface diffusion. Acta Materialia, 53 (2005) 2943-2956. doi: 10.1016/j.actamat.2005.03.013.
• 35. Ogurtani TO. Variational formulation of irreversible thermodynamics of surfaces and interfaces with grainboundary triple-junction singularities under the capillary and electromigration forces in anisotropic two-dimensional space. Physical Review B, 73, 23 (2006a) 1-17. doi: 10.1103/PhysRevB.73.235408.
• 36. Verschaffelt JE. The thermomechanics of the superficial layer. I. Generalities; pure substances. Bulletins de l’Academie Royale des Sciences, des Lettres et des Beaux Arts de Belgique, 22 (1936) 373.
• 37. Guggenheim EA. Thermodynamics (3 ed., p. 46). NorthHolland, Amsterdam, Holland, 1959.
• 38. Ogurtani TO, Akyildiz O, Oren EE. Morphological evolution of tilted grain-boundary thermal grooving by surface diffusion in bicrystal thin solid films having strong anisotropic surface Gibbs free energies. Journal of Applied Physics, 104, 1 (2008) 013518. doi: 10.1063/1.2952520.
• 39. Ogurtani TO. Thermal grain-boundary grooving in bicrystal thin solid films having strong anisotropic surface Gibbs free energy represented by the modified cycloid-curtate function. Journal of Crystal Growth, 311, 6 (2009a) 1584- 1593. doi: 10.1016/j.jcrysgro.2009.01.084.
• 40. Sharma SK, Spitz J. Thermal grooving in thin silver films. Journal of Materials Science, 16, 2 (1981) 535-536. doi: 10.1007/BF00738649.
• 41. Tsoga A, Nikolopoulos P. Groove angles and surface mass transport in polycrystalline alumina. Journal of the American Ceramic Society, 77, 4 (1994) 954–960. doi: 10.1111/j.1151-2916.1994.tb07252.x
• 42. Tritscher P, Broadbridge P. Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove. Proceedings of the Royal Society: Mathematical and Physical Sciences (1990-1995), 450, 1940 (1995) 569-587. doi: 10.1098/rspa.1995.0101.
• 43. Schöllhammer J, Chang L, Rabkin E, Baretzky B, Gust W, Mittemeijer E. Measurement of the profile and the dihedral angle of grain boundary grooves by atomic force microscopy. Zeitschrift für Metallkunde, 90 (1999) 687-690.
• 44. Shin W, Seo W, Koumoto K. Grain-boundary grooves and surface diffusion in polycrystalline alumina measured by atomic force microscope. Journal of the European Ceramic Society, 18, 6 (1998) 595-600. doi: 10.1016/S0955- 2219(97)00207-0.
• 45. Lee KY, Case ED. A comparison of theoretical and experimental profiles for thermally-induced grain-boundary grooving. The European Physical Journal Applied Physics, 8, 3 (1999) 197-214. doi: 10.1051/epjap:1999247.
• 46. Rabkin E, Gabelev A, Klinger L, Semenov VN, Bozhko SI. Grain boundary grooving in molybdenum bicrystals. Journal of Materials Science, 41, 16 (2006) 5151-5160. doi: 10.1007/ s10853-006-0438-4.
• 47. Sachenko PP, Schneibel JH, Zhang W. Effect of faceting on the thermal grain-boundary grooving of tungsten.Philosophical Magazine A, 82, 4 (2002) 815-829. doi: 10.1080/01418610208243204.
• 48. Sachenko PP, Schneibel JH, Zhang W. Observations of secondary oscillations in thermal grain boundary grooves. Scripta Materialia, 50, 9 (2004) 1253-1257. doi: 10.1016/j. scriptamat.2004.01.030.
• 49. Sachenko PP, Schneibel JH, Swadener JG, Zhang W. Experimental and simulated grain boundary groove profiles in tungsten. Philosophical Magazine Letters, 80, 9 (2000)627-631. doi: 10.1080/09500830050134345.
• 50. Munoz NE, Gilliss SR, Carter CB. The monitoring of grain-boundary grooves in alumina. Philosophical Magazine Letters, 84, 1 (2003) 21-26. doi: 10.1080/09500830310001614487.
• 51. Ogurtani TO, Oren EE. Irreversible thermodynamics of triple junctions during the intergranular void motion under the electromigration forces. International Journal of Solids and Structures, 42, 13 (2005) 3918-3952. doi: 10.1016/j. ijsolstr.2004.11.013.
• 52. Ogurtani TO. Mesoscopic nonequilibrium thermodynamics of solid surfaces and interfaces with triple junction singularities under the capillary and electromigration forces in anisotropic three-dimensional space. The Journal of chemical physics, 124, 14 (2006b) 144706. doi: 10.1063/1.2185625.