Design & Manufacturing

Korean Society of Die & Mold Engineering (한국금형공학회)
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Redistribution of Vacancy Concentration in Metal Specimens under Stress-induced Diffusion at a High Temperature

고온 환경하 응력 확산에 의한 금속시편내 격자결함 재분포

  • Received : 2018.02.26
  • Accepted : 2018.04.01
  • Published : 2018.04.01
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Abstract

In this study, we calculated the redistribution of vacancy concentration in metal specimens induced by stress-induced diffusion at a high temperature. To deduce the governing equation, we associated the unit volume change equation of strains with a differential equation of vacancy concentration as a function of stress using the stress-strain relationship. In this governing equation, we considered stress as the only chemical potential parameter to stay in the scope of this study, which provided the vacancy concentration equation as of stress gradient in metals. The equation was then mathematically delineated to derive a analytical solution for a transient, one-dimensional diffusion case. With the help of Korhonen's approximation and the boundary conditions, we successfully deduced a general solution from the governing equation. To visualize the feasibility of our solutions, we applied the solution to two different stress-induced cases - a rod with fixed concentrated stresses at both ends and a rod with varying concentrated stresses at both ends. Although it is necessary to legitimatized the model in the future for improvement, our results showed that the model can be used to interpret the location of structural defects, the formation of vacancy, and furthermore the high temperature behavior of metals.

Keywords

Acknowledgement

Supported by : 산업기술평가관리원

References

  1. Korhonen, M.A., Borgesen, P., Tu, K.N. and Li, C.Y., "Effects of microstructure on the formation, shape, and motion of voids during electromigration in passivated copper interconnects", J. Appl. Phys. vol.73, p.3790, 1993. https://doi.org/10.1063/1.354073
  2. M. Hauschildt, M. Gall, S. Thrasher, P. Justison, L. Michaelson, R. Hernandez, H. Kawasaki and P. Ho, "Statistical Analysis of Electromigration Lifetimes and Void Evolution for Cu Interconnects", MRS Symposium Proceedings, vol. 812, p.379, 2004.
  3. Herring, C., "Diffusional viscosity of a polycrystalline solid", J. Appl. Phys., vol. 21, p. 437, 1950. https://doi.org/10.1063/1.1699681
  4. Ashby, M.F., "A first report on deformation -mechanism maps", Acta Metallurgica vol. 20 p. 887, 1972. https://doi.org/10.1016/0001-6160(72)90082-X
  5. Mohamed, F.A. and T.G. Langdon, "Deformation mechanism maps based on grain size", Met. Trans. vol.5 p. 2339, 1974. https://doi.org/10.1007/BF02644014
  6. Rzepka, S., Korhonen, M.A., Weber, E.R. and Li, C.-Y., "Three-Dimensional Finite Element Simulation of Electro and Stress Migration Effects in Interconnect Lines", Mat. Res. Soc. Symp. Proc. vol. 473, p. 329, 1997.
  7. Clement, J.J. and Thompson, C.V., "Modeling Electromigration-Induced Stress Evolution in Confined Metal Lines", J. Appl. Phys., vol. 78, no. 2, pp. 900-904, 1995. https://doi.org/10.1063/1.360281
  8. Clement, J.J., "Reliability Analysis for Encapsulated Interconnect Lines Under DC and Pulsed DC Current Using a Continuum Electromigration Transport Model", J. Appl. Phys., vol. 82, no. 12, pp. 5991-6000, 1997. https://doi.org/10.1063/1.366464
  9. Balluffi, R.W. and Granato, A.V., "Dislocations, Vacancies and Interstitials," in Dislocation in Solids, edited by F.N.R. Nabarro, pp. 1-133, 1979.
  10. Carslaw, H.S. and Jaeger, J.C., "Conduction of Heat in Solids", Clarendon, Oxford, 1947.