Abstract
Mass transport at interfaces is induced by a gradient of chemical potential along the interface; typically, at surfaces, this is caused by a gradient in curvature and, at grain boundaries, by a gradient of normal stress. In addition, interface mass transport in metallic conductors is induced by strong electric fields/currents. On a sufficiently small scale, depending on the temperature, this interface transport dominates bulk diffusion. Continuum equations that specify the interface fluxes in terms of the preceding driving forces and continuity equations that describe the consequences of a divergence of these fluxes are presented; the chemical potential whose gradient is used as a driving force is that in local equilibrium with an element of interface. The equations are subject to boundary conditions at interface junctions that require the total emerging flux to vanish and that require, at junctions that pass flux freely, the chemical potential to be continuous. With the use of several approximations, solutions of the equations are given to describe, in a unified way, basic models of surface morphological evolution, Coble creep and diffusion-based models of sintering, and electromigration. Some of the approximations, not necessarily made simultaneously, are (1) isotropy of interface properties, both within the interface and with regard to the interface orientation; (2) surface slopes everywhere small compared to a reference plane; and (3) steady-state stress in grain boundaries. Limitations and possible extensions of the framework are discussed.
Similar content being viewed by others
References
J. Herring:Appl. Phys., 1950, vol. 21, p. 301.
W.W. Mullins: inMetal Surfaces: Structure, Energetics and Kinetics, ASM, Metals Park, OH, 1963, p. 17.
F.M. d’Heurle:Int. Mater. Rev., 1989, vol. 34, p. 53.
R.W. Siegel:Mater. Sci. Eng. A, 1993, vol. 168 (2) pp. 189–97.
E.D. Williams:Surf. Sci., 1994, vol. 299–300, pp. 502–24.
G. Ehrlich:Surf Sci., 1994, vol. 299–300, pp. 628–42.
W. Selke and P.M. Duxbury:Z. Phys., 1994, vol. B94, p. 215.
P.C. Searson, R. Li, and K. Sieradzki:Phys. Rev. Lett., 1995, Feb.
C. Herring:Physics of Powder Metallurgy, W.E. Kingston, ed., McGraw-Hill, New York, NY, 1951, p. 143.
J.W. Cahn and J.E. Taylor:Acta Metall Mater., 1994, vol. 42, p. 1045.
J. Pan and A.C.F. Cocks:Acta Metall., 1994, vol. 42, pp. 1223–30.
R.J. Asaro and W.A. Tiller:Metall Trans., 1972, vol. 3, pp. 1789–96.
M.A. Grinfeld:Dokl. Akad. Nauk SSSR, 1986, vol. 290, p. 1358, Sov. Phys. Dokl., 1986, vol. 31, p. 831.
D.J. Srolovitz:Acta Metall., 1988, vol. 37, p. 621.
C. Herring: inStructure and Properties of Solid Surfaces, R. Gomer and C.S. Smith, eds., University of Chicago Press, Chicago, IL, 1952, p. 5.
Differential Geometry of Three Dimensions, C.E. Weatherburn, ed., Cambridge University Press, Cambridge, United Kingdom, 1927.
I.A. Blech and C. Herring:Appl. Phys. Lett., 1976, vol. 29, p. 131.
I.A. Blech and E.S. Meieran:J. Appl Phys., 1969, vol. 40, p. 485.
I.A. Blech:J. Apply. Phys., 1976, vol. 42, p. 1203.
W.W. Mullins:Phil. Mag., 1961, vol. 6, p. 1313.
C. Herring:Phys. Rev., 1951, vol. 82, p. 87.
J.C. Heyraud and J.J. Metois:Surf. Sci., 1983, vol. 128, p. 334.
H.P. Bonzel, E. Preuss, and B. Steffen:Surf. Sci., 1984, vol. 145, pp. 20–23.
M. Ozdemir and A. Zangwill:Phys. Rev. B, 1990, vol. 42, p. 5013.
M. Ozdemir and A. Zangwill:Phys. Rev. B, 1992, vol. 45, p. 3718.
H. Spohn:J. Phys. I (France), 1993, vol. 3, p. 69.
W.C. Carter, A.R. Roosen, J.W. Cahn, and J.E. Taylor:Acta Metall. Mater., in press.
H.P. Bonzel, N. Freyer, and E. Preuss:Phys. Rev. Lett., 1986, vol. 57, pp. 1024–27.
C.C. Umback, M.E. Keeffe, and J.M. Blakely:J. Vac. Sci. Technol. A., 1991, vol. 9 (3), p. 1014.
F.A. Nichols and W.W. Mullins:J. Appl. Phys., 1965, vol. 36, p. 1826.
F.A. Nichols and W.W. Mullins:Trans. AIME. 1965, vol. 233, p. 1840.
R.F. Sekerka and T.F. Marinas:Proc. Int. Conf. on Solid-State Phase Transformations, TMS-AIME, Warrendale, PA, 1983, pp. 67–84.
B.D. Coleman, R.S. Falk, and M. Moakher:Phys. D, in press.
A. Piquet et Uzan:Structure et Propriétés des Surfaces des Solides, Colloques Internationaux du Centre National de la Recherche Scientifique No. 187, Paris, 1969.
W. Zhang and J.H. Schneibel:Comput. Mater. Sci, 1995, vol. 3, pp. 347–58.
W.W. Mullins:J. Appl. Phys., 1959, vol. 30, p. 77.
W.W. Mullins:J. Appl. Phys., 1957, vol. 28, p. 333.
N.A. Gjostein: in Metal Surfaces: Structure, Energetics and Kinetics, W.D. Robertson and N.A.Gjostein, eds., ASM, Metals Park, OH, p. 99.
J.M. Blakely: inProg. Mater. Sci., 1963, vol. 10, p. 395.
N.A. Gjostein: inDiffusion, ASM, Metals Park, OH, 1973, pp. 241–74.
H.P. Bonzel: inSurface Physics of Materials, J.M. Blakely, ed., Academic Press, New York, NY, 1975, vol. 2, p. 279.
Landolt-Börnstein, New Series Group IIIc:Crystals and Solid State Physics, vol. 26,Diffusion in Metals and Alloys, 1990, ch. 13.
W.W. Mullins and P.G. Shewmon:Acta Metall., 1959, vol. 7, p. 163.
W.W. Mullins:Acta Metall., 1958, vol. 6, p. 414.
V.Y. Aristov, V.Y. Fradkov, and L.S. Shvindlerman:Phys. Met. Metall., 1978, vol. 45 (5), p. 83.
A. Brokman, R. Kris, W.W. Mullins, and A.J. Vilenkin:Scripta Metall., in press.
H.J. Frost, C.V. Thompson, and D.T. Walton:Acta Metall., 1992, vol. 40, p. 779.
S.A. Hackney and G.C. Ojard:Scripta Metall., 1988, vol. 22, p. 1731.
S.A. Hackney:Scripta Metall., 1988, vol. 22, p. 1273.
F.Y. Génin, W.W. Mullins, and P. Wynblatt:Acta Metall. Mater., 1992, vol. 40, p. 3239.
D.J. Srolovitz and S.A. Safrin:J. Appl. Phys., 1986, vol. 60, pp. 247–54.
D.J. Srolovitz and S.A. Safrin:J. Appl. Phys., 1986, vol. 60, pp. 255–60.
D.A. Smith: private communication.
K.T. Miller, F.F. Lang, and D.B. Marshall:J. Mater. Res., 1990, vol. 5, p. 151.
R. Brandon and F.J. Bradshaw: Royal Aircraft Establishment Technical Report No. 66095, 1966.
A. Pimpinelli:Surf. Sci., 1993, vol. 295, pp. 143–53.
N.C. Bartelt, T.L. Einstein, and E.D. Williams:Surf. Sci., 1994, vol. 312, pp. 411–21.
C. Alfonso, J.M. Bermond, J.C. Heyraud, and J.J. Metois:Surf. Sci., 1992, v. 262, p. 371.
J.G. Amar and F. Family:Phys. Rev. Lett., 1990, vol. 64, p. 543.
L. Golubovic and R. Bruinsma:Phys. Rev. Lett., 1991, vol. 66, p. 321.
J. Villain,J. Phys. I (France), 1991, vol. 1, 19–42.
H. Yan:Phys. Rev. Lett., 1992, v. 68, p. 3048.
C.N. Luse and A. Zangwill,Phys. Rev. B, 1993, vol. 48 (3), 1970.
R. Rosenberg and M. Ohring:J. Appl. Phys., 1971, vol. 42, pp. 5671–79.
F.Y. Génin, W.W. Mullins, and P. Wynblatt:Acta Metall. Mater., 1993, vol. 41, p. 3541.
T.J. Chuang and J.R. Rice:Acta Metall., 1973, vol. 21, p. 1625.
M.D. Thouless:Acta Metall. Mater., 1993, vol. 41, p. 1057.
L.M. Klinger, E.E. Glickman, V.E. Fradkov, W.W. Mullins, and C.L. Bauer,J. Appl. Phys., in press.
H.J. Vogel and L. Ratke:Acta Metall., 1991, vol. 39, pp. 915–23.
A. Brokman, A.J. Vilenkin, and Marchenko: private communication.
R.L. Coble:J. Appl. Phys., 1963, vol. 34, p. 1679.
D.L. Johnson,J. Appl Phys., 1969, vol. 40, p. 192.
U. Smith, N. Kristensen, F. Ericson, and J.E. Schweitz:J. Vac. Sci. Technol., 1991, vol. A9, p. 2527.
F.Y. Génin:Acta Metall. Mater., in press.
F.G. Yost:Scripta Metall., 1989, vol. 23, p. 1323.
J.R. Spingarn and W.D. Nix:Acta Metall., 1978, vol. 26, pp. 1389–98.
P.M. Hazzledine and J.H. Schneibel:Acta Metall., 1993, vol. 41, pp. 1253–62.
T.G. Langdon:Mater. Sci. Eng., 1991, vol. A137, p. 1–11.
W.W. Mullins:Scripta Metall. Mater., 1993, vol. 29, pp. 491–96.
W. Zhang, J.H. Schneibel, and C. Hsueh:Phil. Mag. A., 1994, vol. 70, pp. 1107–18.
R.L. Coble:J. Appl. Phys., 1961, vol. 32, p. 787.
R.L. Eadie, G.C. Weatherly, and K.T. Aust:Acta Metall., 1978, vol. 26, pp. 759–67.
P. Bross and H.E. Exner:Acta Metall., 1979, vol. 27, pp. 1013–20.
F.B. Swinkels and M.F. Ashby:Acta Metall., 1981, v. 29, pp. 259–81.
J. Pan and A.C.F. Cocks:Acta Metall., 1994, vol. 42, p. 1215–222.
J. Svoboda and H. Riedel:Acta Metall. Mater., 1995, vol. 43, pp. 499–506.
A.C.F. Cocks and N.D. Aparicio:Acta Metall. Mater., 1995, vol. 43, pp. 731–41.
J. Pan and A.C.F. Cocks,Acta Metall Mater., vol. 43, p. 1395 (1995).
Z. Li, C.L. Bauer, S. Mahajan, and A.G. Milnes:J. Appl. Phys., 1992, vol. 72.
M. Scherge, C.L. Bauer, and W.W. Mullins:Acta Metall. Mater., in press.
C.L. Bauer and W.W. Mullins:Appl. Phys. Lett., 1992, vol. 61, p. 2987.
M. Scherge, C.L. Bauer, and W.W. Mullins:Proc. 1994 Spring Materials Research Society Symp., Materials Research Society, Pittsburgh, PA, in press.
L. Klinger, E. Glickman, A. Katsman, and L. Levin:Mat. Sci. Eng., 1994, vol. B23, pp. 15–18.
P.R. Besser, M.C. Madden, and P.A. Flinn:J. Appl. Phys., 1992, vol. 72, p. 3792.
W.D. Nix and E. Artz:Metall. Trans. A, 1992, vol. 23A, pp. 2007–13.
H.S. Carslaw and J.C. Yaeger:Conduction of Heat in Solids, Oxford University Press, Oxford, United Kingdom, 1959.
O. Kraft, J. Sanchez, and E. Artzt:Proc. 1992 Spring Meeting of the Materials Research Society, Materials Research Society, Pittsburgh, PA, 1992, vol. 265, p. 119.
S. Shingubara and Y. Nakasaki:Appl. Phys. Lett., 1991, vol. 58, pp. 42–44.
O.D. Kellog:Foundations of Potential Theory, Dover Publishing Co., New York, NY, 1953.
unpublished research.
Z. Suo, W. Wang, and M. Yang:Appl. Phys. Lett., 1994, vol. 64 (15), p. 1944.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mullins, W.W. Mass transport at interfaces in single component systems. Metall Mater Trans A 26, 1917–1929 (1995). https://doi.org/10.1007/BF02670663
Issue Date:
DOI: https://doi.org/10.1007/BF02670663