May  2011, 30(2): 427-454. doi: 10.3934/dcds.2011.30.427

An entropy based theory of the grain boundary character distribution

1. 

Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States

2. 

Fraunhofer Austria Research GmbH, Visual Computing, A-8010 Graz, Austria

3. 

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States

4. 

Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, United States

5. 

Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890

6. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States, United States

Received  October 2010 Revised  November 2010 Published  February 2011

Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains—a texture—appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation.
Citation: Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427
References:
[1]

B. L. Adams, D. Kinderlehrer, I. Livshits, D. Mason, W. W. Mullins, G. S. Rohrer, A. D. Rollett, D. Saylor, S Ta'asan and C. Wu, Extracting grain boundary energy from triple junction measurement, Interface Science, 7 (1999), 321-338. doi: 10.1023/A:1008733728830.

[2]

B. L. Adams, D. Kinderlehrer, W. W. Mullins, A. D. Rollett and S. Ta'asan, Extracting the relative grain boundary free energy and mobility functions from the geometry of microstructures, Scripta Materiala, 38 (1998), 531-536. doi: 10.1016/S1359-6462(97)00530-7.

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

[4]

T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Locally conservative numerical methods for flow in porous media, Comput. Geosci, 6 (2002), 453-481. doi: 10.1023/A:1021295215383.

[5]

T. Arbogast and H. L. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media, Comput. Geosci, 10 (2006), 291-302. doi: 10.1007/s10596-006-9024-8.

[6]

M. Balhoff, A. Mikelić and Mary F. Wheeler, Polynomial filtration laws for low Reynolds number flows through porous media, Transp. Porous Media, 81 (2010), 35-60. doi: 10.1007/s11242-009-9388-z.

[7]

M. T. Balhoff, S. G. Thomas and M. F. Wheeler, Mortar coupling and upscaling of pore-scale models, Comput. Geosci, 12 (2008), 15-27. doi: 10.1007/s10596-007-9058-6.

[8]

K. Barmak, unpublished. unpublished.

[9]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp and S. Ta'asan, Predictive theory for the grain boundary character distribution, in "Proc. Recrystallization and Grain Growth IV,'' (2010).

[10]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp and S. Ta'asan, "Critical Events, Entropy, and the Grain Boundary Character Distribution,'' Center for Nonlinear Analysis 10-CNA-014, Carnegie Mellon University, 2010, to appear in Physical Review B.

[11]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer and S. Ta'asan, Geometric growth and character development in large metastable systems, Rendiconti di Matematica, Serie VII, 29 (2009), 65-81.

[12]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, On a statistical theory of critical events in microstructural evolution, in "Proceedings CMDS 11,'' ENSMP Press, (2007), 185-194.

[13]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, Towards a statistical theory of texture evolution in polycrystals, SIAM Journal Sci. Comp, 30 (2007), 3150-3169. doi: 10.1137/070692352.

[14]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, A new perspective on texture evolution, International Journal on Numerical Analysis and Modeling, 5 (2008), 93-108.

[15]

K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a multiscale approach to grain growth in polycrystals, In "Variational Problems in Materials Science,'' volume 68 of "Progr. Nonlinear Differential Equations Appl,'' Birkhäuser, Basel, (2006), 1-11.

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J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math, 84 (2000), 375-393. doi: 10.1007/s002110050002.

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E. Bouchbinder and J. S. Langer, Nonequilibrium thermodynamics of driven amorphous materials. i. Internal degrees of freedom and volume deformation, Physical Review E, 80 (2009).

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E. Bouchbinder and J. S. Langer, Nonequilibrium thermodynamics of driven amorphous materials. iii. shear-transformation-zone plasticity, Physical Review E, 80 (2009).

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L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379.

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P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,'' Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co, Amsterdam, 1978. doi: 10.1016/S0168-2024(08)70178-4.

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A. Cohen, A stochastic approach to coarsening of cellular networks, Multiscale Model. Simul, 8 (2009/10), 463-480.

[24]

A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, Two-dimensional modelling of soft ferromagnetic films, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci, 457 (2001), 2983-2991.

[25]

Y. Epshteyn and B. Rivière, On the solution of incompressible two-phase flow by a p-version discontinuous Galerkin method, Comm. Numer. Methods Engrg, 22 (2006), 741-751. doi: 10.1002/cnm.846.

[26]

Y. Epshteyn and B. Rivière, Fully implicit discontinuous finite element methods for two-phase flow, Applied Numerical Mathematics, 57 (2007), 383-401. doi: 10.1016/j.apnum.2006.04.004.

[27]

M. Frechet, Sur la distance de deux lois de probabilite, Comptes Rendus de l' Academie des Sciences Serie I-Mathematique, 244 (1957), 689-692.

[28]

C. Gardiner, "Stochastic Methods, 4th Edition,'' Springer-Verlag, 2009.

[29]

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.), 47 (1959), 271-306.

[30]

S. K. Godunov and V. S. Ryaben'kii, "Difference Schemes. An Introduction to the Underlying Theory,'' volume 19 of Studies in Mathematics and its Applications, North-Holland Publishing Co, Amsterdam, 1987. (Translated from the Russian by E. M. Gelbard)

[31]

J. Gruber, H. M. Miller, T. D. Hoffmann, G. S. Rohrer and A. D. Rollett, Misorientation texture development during grain growth. part i: Simulation and experiment, Acta Materialia, 57 (2009), 6102-6112. doi: 10.1016/j.actamat.2009.08.036.

[32]

J. Gruber, A. D. Rollett and G. S. Rohrer, Misorientation texture development during grain growth. part ii: Theory, Acta Materialia, 58 (2010), 14-19. doi: 10.1016/j.actamat.2009.08.032.

[33]

M. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane,'' Oxford, 1993.

[34]

R. Helmig, "Multiphase Flow and Transport Processes in the Subsurface,'' Springer, 1997.

[35]

C. Herring, Surface tension as a motivation for sintering, in "The Physics of Powder Metallurgy'' (Walter E. Kingston, editor), Mcgraw-Hill, New York, (1951), 143-179.

[36]

C. Herring, The use of classical macroscopic concepts in surface energy problems, In "Structure and Properties of Solid Surfaces'' (R. Gomer and C. S. Smith, editors), The University of Chicago Press, Chicago, (1952), 5-81. (Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA)

[37]

E. A. Holm, G. N. Hassold and M. A. Miodownik, On misorientation distribution evolution during anisotropic grain growth, Acta Materialia, 49 (2001), 2981-2991. doi: 10.1016/S1359-6454(01)00207-5.

[38]

A. Iserles, "A First Course in the Numerical Analysis of Differential Equations,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1996.

[39]

R. Jordan, D. Kinderlehrer and F. Otto, Free energy and the fokker-planck equation, Physica D, 107 (1997), 265-271. doi: 10.1016/S0167-2789(97)00093-6.

[40]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[41]

D. Kinderlehrer, J. Lee, I. Livshits, A. Rollett and S. Ta'asan, Mesoscale simulation of grain growth, Recrystalliztion and Grain Growth, pts 1 and 2, 467-470 (2004), 1057-1062.

[42]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Mathematical Models and Methods in Applied Sciences, 11 (2001), 713-729. doi: 10.1142/S0218202501001069.

[43]

D. Kinderlehrer, I. Livshits, G. S. Rohrer, S. Ta'asan and P. Yu, Mesoscale simulation of the evolution of the grain boundary character distribution, Recrystallization and grain growth, pts 1 and 2, 467-470 (2004), 1063-1068.

[44]

D. Kinderlehrer, I. Livshits and S. Ta'asan, A variational approach to modeling and simulation of grain growth, SIAM J. Sci. Comp, 28 (2006), 1694-1715. doi: 10.1137/030601971.

[45]

R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys, 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4.

[46]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,'' Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London, 1959.

[47]

P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math, 7 (1954), 159-193. doi: 10.1002/cpa.3160070112.

[48]

P. D. Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,'' Society for Industrial and Applied Mathematics, Philadelphia, Pa, 1973. (Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.)

[49]

B. Li, J. Lowengrub, A. Rätz and A. Voigt, Geometric evolution laws for thin crystalline films: Modeling and numerics, Commun. Comput. Phys, 6 (2009), 433-482.

[50]

I. M. Lifshitz, E. M. and V. V. Slyozov, The kinetics of precipitation from suprsaturated solid solutions, Journal of Physics and Chemistry of Solids, 19 (1961), 35-50.

[51]

J. S. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Phys. Rev. E (3), 79 (2009), 0311926.

[52]

M. A. Miodownik, P. Smereka, D. J. Srolovitz and E. A. Holm, Scaling of dislocation cell structures: diffusion in orientation space, Proceedings Of The Royal Society A-Mathematical Physical And Engineering Sciences, 457 (2001), 1807-1819. doi: 10.1098/rspa.2001.0794.

[53]

W. W. Mullins, "Solid Surface Morphologies Governed by Capillarity,'' American Society for Metals, Metals Park, Ohio, (1963), 17-66.

[54]

W. W. Mullins, On idealized 2-dimensional grain growth, Scripta Metallurgica, 22 (1988), 1441-1444. doi: 10.1016/S0036-9748(88)80016-4.

[55]

F. Otto, T. Rump and D. Slepčev, Coarsening rates for a droplet model: rigorous upper bounds, SIAM J. Math. Anal, 38 (2006), 503-529 (electronic). doi: 10.1137/050630192.

[56]

G. S. Rohrer, Influence of interface anisotropy on grain growth and coarsening, Annual Review of Materials Research, 35 (2005), 99-126. doi: 10.1146/annurev.matsci.33.041002.094657.

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A. D. Rollett, S.-B. Lee, R. Campman and G. S. Rohrer, Three-dimensional characterization of microstructure by electron back-scatter diffraction, Annual Review of Materials Research, 37 (2007), 627-658. doi: 10.1146/annurev.matsci.37.052506.084401.

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show all references

References:
[1]

B. L. Adams, D. Kinderlehrer, I. Livshits, D. Mason, W. W. Mullins, G. S. Rohrer, A. D. Rollett, D. Saylor, S Ta'asan and C. Wu, Extracting grain boundary energy from triple junction measurement, Interface Science, 7 (1999), 321-338. doi: 10.1023/A:1008733728830.

[2]

B. L. Adams, D. Kinderlehrer, W. W. Mullins, A. D. Rollett and S. Ta'asan, Extracting the relative grain boundary free energy and mobility functions from the geometry of microstructures, Scripta Materiala, 38 (1998), 531-536. doi: 10.1016/S1359-6462(97)00530-7.

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

[4]

T. Arbogast, Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Locally conservative numerical methods for flow in porous media, Comput. Geosci, 6 (2002), 453-481. doi: 10.1023/A:1021295215383.

[5]

T. Arbogast and H. L. Lehr, Homogenization of a Darcy-Stokes system modeling vuggy porous media, Comput. Geosci, 10 (2006), 291-302. doi: 10.1007/s10596-006-9024-8.

[6]

M. Balhoff, A. Mikelić and Mary F. Wheeler, Polynomial filtration laws for low Reynolds number flows through porous media, Transp. Porous Media, 81 (2010), 35-60. doi: 10.1007/s11242-009-9388-z.

[7]

M. T. Balhoff, S. G. Thomas and M. F. Wheeler, Mortar coupling and upscaling of pore-scale models, Comput. Geosci, 12 (2008), 15-27. doi: 10.1007/s10596-007-9058-6.

[8]

K. Barmak, unpublished. unpublished.

[9]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp and S. Ta'asan, Predictive theory for the grain boundary character distribution, in "Proc. Recrystallization and Grain Growth IV,'' (2010).

[10]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp and S. Ta'asan, "Critical Events, Entropy, and the Grain Boundary Character Distribution,'' Center for Nonlinear Analysis 10-CNA-014, Carnegie Mellon University, 2010, to appear in Physical Review B.

[11]

K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer and S. Ta'asan, Geometric growth and character development in large metastable systems, Rendiconti di Matematica, Serie VII, 29 (2009), 65-81.

[12]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, On a statistical theory of critical events in microstructural evolution, in "Proceedings CMDS 11,'' ENSMP Press, (2007), 185-194.

[13]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, Towards a statistical theory of texture evolution in polycrystals, SIAM Journal Sci. Comp, 30 (2007), 3150-3169. doi: 10.1137/070692352.

[14]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, A new perspective on texture evolution, International Journal on Numerical Analysis and Modeling, 5 (2008), 93-108.

[15]

K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a multiscale approach to grain growth in polycrystals, In "Variational Problems in Materials Science,'' volume 68 of "Progr. Nonlinear Differential Equations Appl,'' Birkhäuser, Basel, (2006), 1-11.

[16]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math, 84 (2000), 375-393. doi: 10.1007/s002110050002.

[17]

G. Bertotti, "Hysteresis in Magnetism,'' Academic Press, 1998.

[18]

E. Bouchbinder and J. S. Langer, Nonequilibrium thermodynamics of driven amorphous materials. i. Internal degrees of freedom and volume deformation, Physical Review E, 80 (2009).

[19]

E. Bouchbinder and J. S. Langer, Nonequilibrium thermodynamics of driven amorphous materials. ii. effective-temperature theory, Physical Review E, 80 (2009).

[20]

E. Bouchbinder and J. S. Langer, Nonequilibrium thermodynamics of driven amorphous materials. iii. shear-transformation-zone plasticity, Physical Review E, 80 (2009).

[21]

L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation, Arch. Rational Mech. Anal., 124 (1993), 355-379.

[22]

P. G. Ciarlet, "The Finite Element Method for Elliptic Problems,'' Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co, Amsterdam, 1978. doi: 10.1016/S0168-2024(08)70178-4.

[23]

A. Cohen, A stochastic approach to coarsening of cellular networks, Multiscale Model. Simul, 8 (2009/10), 463-480.

[24]

A. DeSimone, R. V. Kohn, S. Müller, F. Otto and R. Schäfer, Two-dimensional modelling of soft ferromagnetic films, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci, 457 (2001), 2983-2991.

[25]

Y. Epshteyn and B. Rivière, On the solution of incompressible two-phase flow by a p-version discontinuous Galerkin method, Comm. Numer. Methods Engrg, 22 (2006), 741-751. doi: 10.1002/cnm.846.

[26]

Y. Epshteyn and B. Rivière, Fully implicit discontinuous finite element methods for two-phase flow, Applied Numerical Mathematics, 57 (2007), 383-401. doi: 10.1016/j.apnum.2006.04.004.

[27]

M. Frechet, Sur la distance de deux lois de probabilite, Comptes Rendus de l' Academie des Sciences Serie I-Mathematique, 244 (1957), 689-692.

[28]

C. Gardiner, "Stochastic Methods, 4th Edition,'' Springer-Verlag, 2009.

[29]

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.), 47 (1959), 271-306.

[30]

S. K. Godunov and V. S. Ryaben'kii, "Difference Schemes. An Introduction to the Underlying Theory,'' volume 19 of Studies in Mathematics and its Applications, North-Holland Publishing Co, Amsterdam, 1987. (Translated from the Russian by E. M. Gelbard)

[31]

J. Gruber, H. M. Miller, T. D. Hoffmann, G. S. Rohrer and A. D. Rollett, Misorientation texture development during grain growth. part i: Simulation and experiment, Acta Materialia, 57 (2009), 6102-6112. doi: 10.1016/j.actamat.2009.08.036.

[32]

J. Gruber, A. D. Rollett and G. S. Rohrer, Misorientation texture development during grain growth. part ii: Theory, Acta Materialia, 58 (2010), 14-19. doi: 10.1016/j.actamat.2009.08.032.

[33]

M. Gurtin, "Thermomechanics of Evolving Phase Boundaries in the Plane,'' Oxford, 1993.

[34]

R. Helmig, "Multiphase Flow and Transport Processes in the Subsurface,'' Springer, 1997.

[35]

C. Herring, Surface tension as a motivation for sintering, in "The Physics of Powder Metallurgy'' (Walter E. Kingston, editor), Mcgraw-Hill, New York, (1951), 143-179.

[36]

C. Herring, The use of classical macroscopic concepts in surface energy problems, In "Structure and Properties of Solid Surfaces'' (R. Gomer and C. S. Smith, editors), The University of Chicago Press, Chicago, (1952), 5-81. (Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA)

[37]

E. A. Holm, G. N. Hassold and M. A. Miodownik, On misorientation distribution evolution during anisotropic grain growth, Acta Materialia, 49 (2001), 2981-2991. doi: 10.1016/S1359-6454(01)00207-5.

[38]

A. Iserles, "A First Course in the Numerical Analysis of Differential Equations,'' Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1996.

[39]

R. Jordan, D. Kinderlehrer and F. Otto, Free energy and the fokker-planck equation, Physica D, 107 (1997), 265-271. doi: 10.1016/S0167-2789(97)00093-6.

[40]

R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the fokker-planck equation, SIAM J. Math. Analysis, 29 (1998), 1-17. doi: 10.1137/S0036141096303359.

[41]

D. Kinderlehrer, J. Lee, I. Livshits, A. Rollett and S. Ta'asan, Mesoscale simulation of grain growth, Recrystalliztion and Grain Growth, pts 1 and 2, 467-470 (2004), 1057-1062.

[42]

D. Kinderlehrer and C. Liu, Evolution of grain boundaries, Mathematical Models and Methods in Applied Sciences, 11 (2001), 713-729. doi: 10.1142/S0218202501001069.

[43]

D. Kinderlehrer, I. Livshits, G. S. Rohrer, S. Ta'asan and P. Yu, Mesoscale simulation of the evolution of the grain boundary character distribution, Recrystallization and grain growth, pts 1 and 2, 467-470 (2004), 1063-1068.

[44]

D. Kinderlehrer, I. Livshits and S. Ta'asan, A variational approach to modeling and simulation of grain growth, SIAM J. Sci. Comp, 28 (2006), 1694-1715. doi: 10.1137/030601971.

[45]

R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys, 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4.

[46]

L. D. Landau and E. M. Lifshitz, "Fluid Mechanics,'' Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6, Pergamon Press, London, 1959.

[47]

P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math, 7 (1954), 159-193. doi: 10.1002/cpa.3160070112.

[48]

P. D. Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,'' Society for Industrial and Applied Mathematics, Philadelphia, Pa, 1973. (Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11.)

[49]

B. Li, J. Lowengrub, A. Rätz and A. Voigt, Geometric evolution laws for thin crystalline films: Modeling and numerics, Commun. Comput. Phys, 6 (2009), 433-482.

[50]

I. M. Lifshitz, E. M. and V. V. Slyozov, The kinetics of precipitation from suprsaturated solid solutions, Journal of Physics and Chemistry of Solids, 19 (1961), 35-50.

[51]

J. S. Lowengrub, A. Rätz and A. Voigt, Phase-field modeling of the dynamics of multicomponent vesicles: Spinodal decomposition, coarsening, budding, and fission, Phys. Rev. E (3), 79 (2009), 0311926.

[52]

M. A. Miodownik, P. Smereka, D. J. Srolovitz and E. A. Holm, Scaling of dislocation cell structures: diffusion in orientation space, Proceedings Of The Royal Society A-Mathematical Physical And Engineering Sciences, 457 (2001), 1807-1819. doi: 10.1098/rspa.2001.0794.

[53]

W. W. Mullins, "Solid Surface Morphologies Governed by Capillarity,'' American Society for Metals, Metals Park, Ohio, (1963), 17-66.

[54]

W. W. Mullins, On idealized 2-dimensional grain growth, Scripta Metallurgica, 22 (1988), 1441-1444. doi: 10.1016/S0036-9748(88)80016-4.

[55]

F. Otto, T. Rump and D. Slepčev, Coarsening rates for a droplet model: rigorous upper bounds, SIAM J. Math. Anal, 38 (2006), 503-529 (electronic). doi: 10.1137/050630192.

[56]

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