Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-09T01:38:12.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Recent Developments in Material Microstructure: a Theory of Coarsening

Published online by Cambridge University Press:  11 June 2015

K. Barmak ,
E. Eggeling ,
M. Emelianenko ,
Y. Epshteyn ,
D. Kinderlehrer ,
R. Sharp  and
S. Ta’asan
K. Barmak
Affiliation:
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027E-mail address: kb2612@columbia.edu
E. Eggeling
Affiliation:
Fraunhofer Austria Research GmbH, Visual Computing, A-8010 Graz, AustriaE-mail address: eva.eggeling@fraunhofer.at
M. Emelianenko
Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030E-mail address: memelian@gmu.edu
Y. Epshteyn
Affiliation:
Department of Mathematics, The University of Utah, Salt Lake City, UT, 84112E-mail address: epshteyn@math.utah.edu
D. Kinderlehrer
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213E-mail address: davidk@cmu.edu
R. Sharp
Affiliation:
Globys Corporation, Seattle, WA, 98104E-mail address: rsharp@gmail.com
S. Ta’asan
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213E-mail address: shlomo@andrew.cmu.edu
Get access

Abstract

Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The recently discovered grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Here we describe an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution.

Keywords

microstructuregrain boundariestexture
Type
Articles
Information
MRS Online Proceedings Library (OPL) , Volume 1753: Symposium NN – Mathematical and Computational Aspects of Materials Science , 2015 , mrsf14-1753-nn01-01
Copyright
Copyright © Materials Research Society 2015 

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

Adams, B.L., Kinderlehrer, D., Livshits, I., Mason, D., Mullins, W.W., Rohrer, G.S., Rollett, A.D., Saylor, D., Ta’asan, S, and Wu, C.. Extracting grain boundary energy from triple junction measurement. Interface Science, 7:321–338, 1999.CrossRefGoogle Scholar
Adams, BL, Kinderlehrer, D, Mullins, WW, Rollett, AD, and Ta’asan, S. Extracting the relative grain boundary free energy and mobility functions from the geometry of microstructures. Scripta Materiala, 38(4):531–536, Jan 13 1998.CrossRefGoogle Scholar
Bardsley, P.. thesis, university of utah.Google Scholar
Barmak, K., Eggeling, E., Emelianenko, M., Epshteyn, Y., Kinderlehrer, D., Sharp, R., and Ta’asan, S.. Predictive theory for the grain boundary character distribution. In Materials Science Forum, volume 715-716, pages 279–285. Trans Tech Publications, 2012.CrossRefGoogle Scholar
Barmak, K., Eggeling, E., Emelianenko, M., Epshteyn, Y., Kinderlehrer, D., Sharp, R., and Ta’asan, S.. Critical events, entropy, and the grain boundary character distribution. Phys. Rev. B, 83(13):134117, Apr 2011.CrossRefGoogle Scholar
Barmak, K., Eggeling, E., Emelianenko, M., Epshteyn, Y., Kinderlehrer, D., and Ta’asan, S.. Geometric growth and character development in large metastable systems. Rendiconti di Matematica, Serie VII, 29:65–81, 2009.Google Scholar
Barmak, K., Eggeling, E., Kinderlehrer, D., Sharp, R., Ta’asan, S., Rollett, A.D., and Coffey, K.R.. Grain growth and the puzzle of its stagnation in thin films: The curious tale of a tail and an ear. Progress in Materials Science, 58(7):987 – 1055, 2013.CrossRefGoogle Scholar
Barmak, K., Emelianenko, M., Golovaty, D., Kinderlehrer, D., and Ta’asan, S.. On a statistical theory of critical events in microstructural evolution. In Proceedings CMDS 11, pages 185–194. ENSMP Press, 2007.Google Scholar
Barmak, K., Emelianenko, M., Golovaty, D., Kinderlehrer, D., and Ta’asan, S.. Towards a statistical theory of texture evolution in polycrystals. SIAM Journal Sci. Comp., 30(6):3150–3169, 2007.CrossRefGoogle Scholar
Barmak, K., Emelianenko, M., Golovaty, D., Kinderlehrer, D., and Ta’asan, S.. A new perspective on texture evolution. International Journal on Numerical Analysis and Modeling, 5(Sp. Iss. SI):93–108, 2008.Google Scholar
Barmak, Katayun, Eggeling, Eva, Emelianenko, Maria, Epshteyn, Yekaterina, Kinderlehrer, David, Sharp, Richard, and Ta’asan, Shlomo. An entropy based theory of the grain boundary character distribution. Discrete Contin. Dyn. Syst., 30(2):427–454, 2011.Google Scholar
Barmak, Katayun, Eggeling, Eva, Emelianenko, Maria, Epshteyn, Yekaterina, Kinderlehrer, David, Sharp, Richard, and Ta’asan, Shlomo. A theory and challenges for coarsening in microstructure. In Analysis and numerics of partial differential equations, volume 4 of Springer INdAM Ser., pages 193–220. Springer, Milan, 2013.CrossRefGoogle Scholar
Benamou, Jean-David and Brenier, Yann. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84(3):375–393, 2000.CrossRefGoogle Scholar
Bronsard, Lia and Reitich, Fernando. On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rational Mech. Anal., 124(4):355–379, 1993.CrossRefGoogle Scholar
Burke, J.E. and Turnbull, D.. Recrystallization and grain growth. Progress in Metal Physics, 3(C):220–244,IN11–IN12,245–266,IN13–IN14,267–274,IN15,275–292, 1952. cited By (since 1996) 68.CrossRefGoogle Scholar
Gomer, Robert and Smith, Cyril Stanley, editors. Structure and Properties of Solid Surfaces, Chicago, 1952. The University of Chicago Press. Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA.Google Scholar
Gurtin, M.. Thermomechanics of evolving phase boundaries in the plane. Oxford, 1993.Google Scholar
Herring, C.. Surface tension as a motivation for sintering. In Kingston, Walter E., editor, The Physics of Powder Metallurgy, pages 143–179. Mcgraw-Hill, New York, 1951.Google Scholar
Herring, C.. The use of classical macroscopic concepts in surface energy problems. In Gomer and Smith [16], pages 5–81. Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA.Google Scholar
Jordan, Richard, Kinderlehrer, David, and Otto, Felix. The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29(1):1–17, 1998.CrossRefGoogle Scholar
Kinderlehrer, D, Lee, J, Livshits, I, Rollett, A, and Ta’asan, S. Mesoscale simulation of grain growth. Recrystalliztion and grain growth, pts 1 and 2, 467-470(Part 1-2):1057–1062, 2004.Google Scholar
Kinderlehrer, D and Liu, C. Evolution of grain boundaries. Mathematical Models and Methods in Applied Sciences, 11(4):713–729, Jun 2001.CrossRefGoogle Scholar
Kinderlehrer, D, Livshits, I, Rohrer, GS, Ta’asan, S, and Yu, P. Mesoscale simulation of the evolution of the grain boundary character distribution. Recrystallization and grain growth, pts 1 and 2, 467-470(Part 1-2):1063–1068, 2004.Google Scholar
Kinderlehrer, David, Livshits, Irene, and Ta’asan, Shlomo. A variational approach to modeling and simulation of grain growth. SIAM J. Sci. Comp., 28(5):1694–1715, 2006.CrossRefGoogle Scholar
Kinderlehrer, David and Lu, Xin Yang. to appear.Google Scholar
Kohn, Robert V.. Irreversibility and the statistics of grain boundaries. Physics, 4:33, Apr 2011.CrossRefGoogle Scholar
Mullins, W.W.. 2-Dimensional motion of idealized grain growth. Journal Applied Physics, 27(8):900–904, 1956.CrossRefGoogle Scholar
Mullins, W.W.. Solid Surface Morphologies Governed by Capillarity, pages 17–66. American Society for Metals, Metals Park, Ohio, 1963.Google Scholar
Rohrer, GS. Inuence of interface anisotropy on grain growth and coarsening. Annual Review of Materials Research, 35:99–126, 2005.CrossRefGoogle Scholar
Rollett, Anthony D., Lee, S.-B., Campman, R., and Rohrer, G. S.. Three-dimensional characterization of microstructure by electron back-scatter diffraction. Annual Review of Materials Research, 37:627–658, 2007.CrossRefGoogle Scholar
Smith, Cyril Stanley. Grain shapes and other metallurgical applications of topology. In Gomer and Smith [16], pages 65–108. Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA.Google Scholar
von Neumann, John. Discussion remark concerning paper of C. S. Smith “grain shapes and other metallurgical applications of topologyˮ. In Gomer and Smith [16], pages 108–110. Proceedings of a conference arranged by the National Research Council and held in September, 1952, in Lake Geneva, Wisconsin, USA.Google Scholar