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 & Continuous Dynamical Systems - A, 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. doi: 10.1023/A:1008733728830. Google Scholar

[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. doi: 10.1016/S1359-6462(97)00530-7. Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Lectures in Mathematics ETH Z\, (2008). Google Scholar

[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. doi: 10.1023/A:1021295215383. Google Scholar

[5]

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

[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. doi: 10.1007/s11242-009-9388-z. Google Scholar

[7]

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

[8]

K. Barmak, unpublished., unpublished., (none). Google Scholar

[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, (2010). Google Scholar

[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, (2010). Google Scholar

[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, 29 (2009), 65. Google Scholar

[12]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, On a statistical theory of critical events in microstructural evolution,, in, (2007), 185. Google Scholar

[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. doi: 10.1137/070692352. Google Scholar

[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. Google Scholar

[15]

K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a multiscale approach to grain growth in polycrystals,, In, 68 (2006), 1. Google Scholar

[16]

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

[17]

G. Bertotti, "Hysteresis in Magnetism,'', Academic Press, (1998). Google Scholar

[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). Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[23]

A. Cohen, A stochastic approach to coarsening of cellular networks,, Multiscale Model. Simul, 8 (2009), 463. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cnm.846. Google Scholar

[26]

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

[27]

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

[28]

C. Gardiner, "Stochastic Methods, 4th Edition,'', Springer-Verlag, (2009). Google Scholar

[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. Google Scholar

[30]

S. K. Godunov and V. S. Ryaben'kii, "Difference Schemes. An Introduction to the Underlying Theory,'', volume \textbf{19} of Studies in Mathematics and its Applications, 19 (1987). Google Scholar

[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. doi: 10.1016/j.actamat.2009.08.036. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

C. Herring, Surface tension as a motivation for sintering,, in, (1951), 143. Google Scholar

[36]

C. Herring, The use of classical macroscopic concepts in surface energy problems,, In, (1952), 5. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[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, 467-470 (2004), 467. Google Scholar

[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. doi: 10.1137/030601971. Google Scholar

[45]

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

[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, 6 (1959). Google Scholar

[47]

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

[48]

P. D. Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,'', Society for Industrial and Applied Mathematics, (1973). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. doi: 10.1098/rspa.2001.0794. Google Scholar

[53]

W. W. Mullins, "Solid Surface Morphologies Governed by Capillarity,'', American Society for Metals, (1963), 17. Google Scholar

[54]

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

[55]

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

[56]

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

[57]

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. doi: 10.1146/annurev.matsci.37.052506.084401. Google Scholar

[58]

D. M. Saylor, A. Morawiec and G. S. Rohrer, The relative free energies of grain boundaries in magnesia as a function of five macroscopic parameters,, Acta Materialia, 51 (2003), 3675. doi: 10.1016/S1359-6454(03)00182-4. Google Scholar

[59]

C. S. Smith, Grain shapes and other metallurgical applications of topology,, in, (1952), 65. Google Scholar

[60]

H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods,, J. Comput. Phys, 56 (1984), 363. doi: 10.1016/0021-9991(84)90103-7. Google Scholar

[61]

A. Toselli and O. Widlund, "Domain Decomposition Methods—Algorithms and Theory,", volume \textbf{34} of Springer Series in Computational Mathematics, 34 (2005). Google Scholar

[62]

C. Villani, "Topics in Optimal Transportation,'', volume \textbf{58} of Graduate Studies in Mathematics, 58 (2003). Google Scholar

[63]

J. Von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks,, J. Appl. Phys, 21 (1950), 232. doi: 10.1063/1.1699639. Google Scholar

[64]

C Wagner, Theorie der alterung von niederschlagen durch umlosen (Ostwald-Reifung),, Zeitschrift fur Elektrochemie, 65 (1961), 581. Google Scholar

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. doi: 10.1023/A:1008733728830. Google Scholar

[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. doi: 10.1016/S1359-6462(97)00530-7. Google Scholar

[3]

L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'', Lectures in Mathematics ETH Z\, (2008). Google Scholar

[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. doi: 10.1023/A:1021295215383. Google Scholar

[5]

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

[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. doi: 10.1007/s11242-009-9388-z. Google Scholar

[7]

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

[8]

K. Barmak, unpublished., unpublished., (none). Google Scholar

[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, (2010). Google Scholar

[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, (2010). Google Scholar

[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, 29 (2009), 65. Google Scholar

[12]

K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer and S. Ta'asan, On a statistical theory of critical events in microstructural evolution,, in, (2007), 185. Google Scholar

[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. doi: 10.1137/070692352. Google Scholar

[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. Google Scholar

[15]

K. Barmak, D. Kinderlehrer, I. Livshits and S. Ta'asan, Remarks on a multiscale approach to grain growth in polycrystals,, In, 68 (2006), 1. Google Scholar

[16]

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

[17]

G. Bertotti, "Hysteresis in Magnetism,'', Academic Press, (1998). Google Scholar

[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). Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[22]

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

[23]

A. Cohen, A stochastic approach to coarsening of cellular networks,, Multiscale Model. Simul, 8 (2009), 463. Google Scholar

[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. Google Scholar

[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. doi: 10.1002/cnm.846. Google Scholar

[26]

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

[27]

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

[28]

C. Gardiner, "Stochastic Methods, 4th Edition,'', Springer-Verlag, (2009). Google Scholar

[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. Google Scholar

[30]

S. K. Godunov and V. S. Ryaben'kii, "Difference Schemes. An Introduction to the Underlying Theory,'', volume \textbf{19} of Studies in Mathematics and its Applications, 19 (1987). Google Scholar

[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. doi: 10.1016/j.actamat.2009.08.036. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

C. Herring, Surface tension as a motivation for sintering,, in, (1951), 143. Google Scholar

[36]

C. Herring, The use of classical macroscopic concepts in surface energy problems,, In, (1952), 5. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[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, 467-470 (2004), 467. Google Scholar

[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. doi: 10.1137/030601971. Google Scholar

[45]

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

[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, 6 (1959). Google Scholar

[47]

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

[48]

P. D. Lax, "Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves,'', Society for Industrial and Applied Mathematics, (1973). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. doi: 10.1098/rspa.2001.0794. Google Scholar

[53]

W. W. Mullins, "Solid Surface Morphologies Governed by Capillarity,'', American Society for Metals, (1963), 17. Google Scholar

[54]

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

[55]

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

[56]

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

[57]

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. doi: 10.1146/annurev.matsci.37.052506.084401. Google Scholar

[58]

D. M. Saylor, A. Morawiec and G. S. Rohrer, The relative free energies of grain boundaries in magnesia as a function of five macroscopic parameters,, Acta Materialia, 51 (2003), 3675. doi: 10.1016/S1359-6454(03)00182-4. Google Scholar

[59]

C. S. Smith, Grain shapes and other metallurgical applications of topology,, in, (1952), 65. Google Scholar

[60]

H. B. Stewart and B. Wendroff, Two-phase flow: Models and methods,, J. Comput. Phys, 56 (1984), 363. doi: 10.1016/0021-9991(84)90103-7. Google Scholar

[61]

A. Toselli and O. Widlund, "Domain Decomposition Methods—Algorithms and Theory,", volume \textbf{34} of Springer Series in Computational Mathematics, 34 (2005). Google Scholar

[62]

C. Villani, "Topics in Optimal Transportation,'', volume \textbf{58} of Graduate Studies in Mathematics, 58 (2003). Google Scholar

[63]

J. Von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks,, J. Appl. Phys, 21 (1950), 232. doi: 10.1063/1.1699639. Google Scholar

[64]

C Wagner, Theorie der alterung von niederschlagen durch umlosen (Ostwald-Reifung),, Zeitschrift fur Elektrochemie, 65 (1961), 581. Google Scholar

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