Thermal grooving by surface diffusion: Mullins revisited and extended to multiple grooves
Author:
P. A. Martin
Journal:
Quart. Appl. Math. 67 (2009), 125-136
MSC (2000):
Primary 35R35
DOI:
https://doi.org/10.1090/S0033-569X-09-01086-4
Published electronically:
January 7, 2009
MathSciNet review:
2497600
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Abstract: We extend the original Mullins theory of surface grooving due to a single interface to multiple interacting grooves formed due to closely spaced flat interfaces. First, we show that Mullins’ analysis for one groove can be simplified by using Fourier cosine transforms instead of Laplace transforms. Second, we solve the corresponding problem for an infinite periodic row of grooves. For both of these problems, symmetry considerations ensure that the interface conditions reduce to boundary conditions. Third, we solve the problem for two interacting grooves. Continuity requirements at the groove roots require sliding at the interfaces or tilting of the groove roots. We adopt the latter model. We find that the groove roots tilt until the surface curvature of the semi-infinite profiles is eliminated.
References
- P. Broadbridge and P. Tritscher, An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces, IMA J. Appl. Math. 53 (1994), no. 3, 249–265. MR 1314257, DOI https://doi.org/10.1093/imamat/53.3.249
- J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta Metall. Mater. 42, 1045–1063 (1994)
- E. Fried and M. E. Gurtin, A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy, Adv. Appl. Mech. 40, 1–177 (2004)
- Morton E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. MR 1402243
- P. A. Martin, Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves, IMA J. Appl. Math. 54 (1995), no. 2, 139–157. MR 1334457, DOI https://doi.org/10.1093/imamat/54.2.139
- W. W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28, 333–339 (1957)
- E. Rabkin, L. Klinger, T. Izyumova, A. Berner, and V. Semonov, Grain boundary grooving with simultaneous grain boundary sliding in Ni-rich NiAl, Acta Mater. 49, 1429–1438 (2001)
- W. M. Robertson, Grain-boundary grooving by surface diffusion for finite surface slopes, J. Appl. Phys. 42, 463–467 (1971)
- P. Tritscher and P. Broadbridge, Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove, Proc. Roy. Soc. A 450, 569–587 (1995)
References
- P. Broadbridge and P. Tritscher, An integrable fourth-order nonlinear evolution equation applied to thermal grooving of metal surfaces, IMA J. Appl. Math. 53, 249–265 (1994) MR 1314257 (95j:35217)
- J. W. Cahn and J. E. Taylor, Surface motion by surface diffusion, Acta Metall. Mater. 42, 1045–1063 (1994)
- E. Fried and M. E. Gurtin, A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy, Adv. Appl. Mech. 40, 1–177 (2004)
- M. E. Gurtin, Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford University Press, 1993 MR 1402243 (97k:73001)
- P. A. Martin, Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves, IMA J. Appl. Math. 54, 139–157 (1995) MR 1334457 (96f:41024)
- W. W. Mullins, Theory of thermal grooving, J. Appl. Phys. 28, 333–339 (1957)
- E. Rabkin, L. Klinger, T. Izyumova, A. Berner, and V. Semonov, Grain boundary grooving with simultaneous grain boundary sliding in Ni-rich NiAl, Acta Mater. 49, 1429–1438 (2001)
- W. M. Robertson, Grain-boundary grooving by surface diffusion for finite surface slopes, J. Appl. Phys. 42, 463–467 (1971)
- P. Tritscher and P. Broadbridge, Grain boundary grooving by surface diffusion: an analytic nonlinear model for a symmetric groove, Proc. Roy. Soc. A 450, 569–587 (1995)
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Additional Information
P. A. Martin
Affiliation:
Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado 80401-1887
Email:
pamartin@mines.edu
Received by editor(s):
July 25, 2007
Published electronically:
January 7, 2009
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.