Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-23T20:38:52.353Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering

Published online by Cambridge University Press:  21 July 2010

JOHN W. BARRETT ,
HARALD GARCKE  and
ROBERT NÜRNBERG
JOHN W. BARRETT
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected], [email protected]
HARALD GARCKE
Affiliation:
NWF I – Mathematik, Universität Regensburg, 93040 Regensburg, Germany email: [email protected]
ROBERT NÜRNBERG
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the coupled surface and grain boundary motion in bi- and tricrystals in three-space dimensions, building on previous work by the authors on the simplified two-dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple-junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical finite-element approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric finite-element approximation we show well posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time-step size. Moreover, the induced discrete equations are linear and easy to solve. A generalisation to anisotropic surface energies is straightforward. Several numerical results in two- and three-space dimensions are presented, including simulations for thermal grooving and sintering. Three-dimensional simulations featuring quadruple junction points, non-standard boundary contact angles and fully anisotropic surface energies are also presented.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 6 , December 2010 , pp. 519 - 556
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barbosa, J. L., do Carmo, M. & Eschenburg, J. (1988) Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Z. 197, 123–138.CrossRefGoogle Scholar
[2]Barrett, J. W., Garcke, H. & Nürnberg, R. (2007a) On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29, 1006–1041.CrossRefGoogle Scholar
[3]Barrett, J. W., Garcke, H. & Nürnberg, R. (2007b) A phase field model for the electromigration of intergranular voids. Interfaces Free Bound. 9, 171–210.Google Scholar
[4]Barrett, J. W., Garcke, H. & Nürnberg, R. (2008a) Numerical approximation of anisotropic geometric evolution equations in the plane. IMA J. Numer. Anal. 28, 292–330.CrossRefGoogle Scholar
[5]Barrett, J. W., Garcke, H. & Nürnberg, R. (2008b) On sharp interface limits of Allen–Cahn/Cahn–Hilliard variational inequalities. Discrete Contin. Dyn. Syst. Ser. S 1, 1–14.Google Scholar
[6]Barrett, J. W., Garcke, H. & Nürnberg, R. (2008c) On the parametric finite element approximation of evolving hypersurfaces in 3. J. Comput. Phys. 227, 4281–4307.CrossRefGoogle Scholar
[7]Barrett, J. W., Garcke, H. & Nürnberg, R. (2008d) A variational formulation of anisotropic geometric evolution equations in higher dimensions. Numer. Math. 109, 1–44.CrossRefGoogle Scholar
[8]Barrett, J. W., Garcke, H. & Nürnberg, R. (2010) Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies. Interfaces Free Bound. 12, 187–234.Google Scholar
[9]Bernoff, A. J., Bertozzi, A. L. & Witelski, T. P. (1998) Axisymmetric surface diffusion: Dynamics and stability of self-similar pinchoff. J. Stat. Phys. 93, 725–776.CrossRefGoogle Scholar
[10]Blackford, J. R. (2007) Sintering and microstructure of ice: A review. J. Phys. D: Appl. Phys. 40, 355–384.CrossRefGoogle Scholar
[11]Brakke, K. A. (1992) The surface evolver. Exp. Math. 1, 141–165.CrossRefGoogle Scholar
[12]Bronsard, L., Garcke, H. & Stoth, B. (1998) A multi-phase Mullins–Sekerka system: Matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem. Proc. R. Soc. Edinburgh Sect. A 128, 481–506.CrossRefGoogle Scholar
[13]Cahn, J. W. & Hoffman, D. W. (1974) A vector thermodynamics for anisotropic surfaces. II. Curved and faceted surfaces. Acta Metall. 22, 1205–1214.CrossRefGoogle Scholar
[14]Ch'ng, H. N. & Pan, J. (2004) Cubic spline elements for modelling microstructural evolution of materials controlled by solid-state diffusion and grain-boundary migration. J. Comput. Phys. 196, 724–750.CrossRefGoogle Scholar
[15]Ch'ng, H. N. & Pan, J. (2005) Modelling microstructural evolution of porous polycrystalline materials and a numerical study of anisotropic sintering. J. Comput. Phys. 204, 430–461.CrossRefGoogle Scholar
[16]Ch'ng, H. N. & Pan, J. (2007) Sintering of particles of different sizes. Acta Mater. 55, 813–824.CrossRefGoogle Scholar
[17]Concus, P. & Finn, R. (1974) On capillary free surfaces in the absence of gravity. Acta Math. 132, 177–198.CrossRefGoogle Scholar
[18]Davi, F. & Gurtin, M. E. (1990) On the motion of a phase interface by surface diffusion. Z. Angew. Math. Phys. 41, 782–811.CrossRefGoogle Scholar
[19]Deckelnick, K., Dziuk, G. & Elliott, C. M. (2003) Error analysis of a semidiscrete numerical scheme for diffusion in axially symmetric surfaces. SIAM J. Numer. Anal. 41, 2161–2179.CrossRefGoogle Scholar
[20]Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S. & Leis, R. (editors), Partial Differential Equations and Calculus of Variations, Vol. 1357, Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 142–155.CrossRefGoogle Scholar
[21]Dziuk, G. (1991) An algorithm for evolutionary surfaces. Numer. Math. 58, 603–611.CrossRefGoogle Scholar
[22]Dziuk, G. (1994) Convergence of a semi-discrete scheme for the curve shortening flow. Math. Models Methods Appl. Sci. 4, 589–606.CrossRefGoogle Scholar
[23]Dziuk, G. & Elliott, C. M. (2007) Surface finite elements for parabolic equations. J. Comput. Math. 25, 385–407.Google Scholar
[24]Finn, R. (1986) Equilibrium capillary surfaces. Grundlehren der Mathematischen Wissenschaften, Vol. 284, Springer-Verlag, New York.Google Scholar
[25]Garcke, H., Nestler, B. & Stoth, B. (1998) On anisotropic order parameter models for multi-phase systems and their sharp interface limits. Physica D 115, 87–108.Google Scholar
[26]Garcke, H., Nestler, B. & Stoth, B. (1999) A multiphase field concept: Numerical simulations of moving phase boundaries and multiple junctions. SIAM J. Appl. Math. 60, 295–315.CrossRefGoogle Scholar
[27]Garcke, H. & Novick-Cohen, A. (2000) A singular limit for a system of degenerate Cahn–Hilliard equations. Adv. Differ. Equ. 5, 401–434.Google Scholar
[28]Garcke, H. & Wieland, S. (2006) Surfactant spreading on thin viscous films: Nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37, 2025–2048.CrossRefGoogle Scholar
[29]Hestenes, M. R. (1975) Pseudoinverses and conjugate gradients. Commun. ACM 18, 40–43.CrossRefGoogle Scholar
[30]Hildebrandt, S. & Tromba, A. (1996) The Parsimonious Universe. Shape and Form in the Natural World, Copernicus, New York.CrossRefGoogle Scholar
[31]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2003) A traveling wave solution for coupled surface and grain boundary motion. Acta Mater. 51, 1981–1989.CrossRefGoogle Scholar
[32]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2004) Coupled surface and grain boundary motion: Nonclassical traveling wave solutions. Adv. Differ. Equ. 9, 299–327.Google Scholar
[33]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2005) A numerical study of grain boundary motion in bicrystals. Acta Mater. 53, 227–235.CrossRefGoogle Scholar
[34]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2006) Numerical analysis of a 3d radially symmetric shrinking grain attached to a free crystal surface. Acta Mater. 54, 2589–2595.CrossRefGoogle Scholar
[35]Kucherenko, S., Pan, J. & Yeomans, J. A. (2000) A combined finite element and finite difference scheme for computer simulation of microstructure evolution and its application to pore-boundary separation during sintering. Comput. Mater. Sci. 18, 76–92.CrossRefGoogle Scholar
[36]Moelans, N., Blanpain, B. & Wollants, P. (2007) A phase field model for grain growth and thermal grooving in thin films with orientation dependent surface energy. Solid State Phenomena 129, 89–94.CrossRefGoogle Scholar
[37]Mullins, W. W. (1958) The effect of thermal grooving on grain boundary motion. Acta Metall. 6, 414–427.CrossRefGoogle Scholar
[38]Novick-Cohen, A. (2000) Triple-junction motion for an Allen–Cahn/Cahn–Hilliard system. Physica D 137, 1–24.Google Scholar
[39]Pan, J. (2003) Modelling sintering at different length scales. Int. Mater. Rev. 48, 69–85.CrossRefGoogle Scholar
[40]Pan, J. & Cocks, A. C. F. (1995) A numerical technique for the analysis of coupled surface and grain-boundary diffusion. Acta Mater. 43, 1395–1406.CrossRefGoogle Scholar
[41]Pan, J., Cocks, A. C. F. & Kucherenko, S. (1997) Finite element formulation of coupled grain-boundary and surface diffusion with grain-boundary migration. Proc. R. Soc. Lond. Ser. A 453, 2161–2184.CrossRefGoogle Scholar
[42]Pan, Z. & Wetton, B. (2008) A numerical method for coupled surface and grain boundary motion. Eur. J. Appl. Math. 19, 311–327.CrossRefGoogle Scholar
[43]Schmidt, A. & Siebert, K. G. (2005) Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Vol. 42, Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin.Google Scholar
[44]Sun, B. & Suo, Z. (1997) A finite element method for simulating interface motion. II. Large shape change due to surface diffusion. Acta Mater. 45, 4953–4962.CrossRefGoogle Scholar
[45]Sun, B., Suo, Z. & Yang, W. (1997) A finite element method for simulating interface motion. I. Migration of phase and grain boundaries. Acta Mater. 45, 1907–1915.CrossRefGoogle Scholar
[46]Taylor, J. E. & Cahn, J. W. (1994) Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Stat. Phys. 77, 183–197.CrossRefGoogle Scholar
[47]Tritscher, P. (1999) Thermal grooving by surface diffusion for an extended bicrystal abutting a half-space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455, 1957–1977.CrossRefGoogle Scholar
[48]Vilenkin, A. J., Kris, R. & Brokman, A. (1997) Breakup and grain growth in thin-film array. J. Appl. Phys. 81, 238–245.CrossRefGoogle Scholar
[49]Wakai, F., Yoshida, M., Shinoda, Y. & Akatsu, T. (2005) Coarsening and grain growth in sintering of two particles of different sizes. Acta Mater. 53, 1361–1371.CrossRefGoogle Scholar
[50]Zhang, H. & Wong, H. (2002a) Coupled grooving and migration of grain boundaries: Regime I. Acta Mater. 50, 1983–1994.CrossRefGoogle Scholar
[51]Zhang, H. & Wong, H. (2002b) Coupled grooving and migration of grain boundaries: Regime II. Acta Mater. 50 (2002), 1995–2012.CrossRefGoogle Scholar
[52]Zhang, W. & Gladwell, I. (2005) Thermal grain boundary grooving with anisotropic surface free energy in three dimensions. J. Cryst. Growth 277, 608–622.CrossRefGoogle Scholar