Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Ng, F.S.L. (2016)
Publisher: Elsevier
Journal: Acta Materialia
Languages: English
Types: Article
Subjects: Metals and Alloys, Polymers and Plastics, Electronic, Optical and Magnetic Materials, Ceramics and Composites
We develop a statistical-mechanical model of one-dimensional normal grain growth that does not require any drift-velocity parameterization for grain size, such as used in the continuity equation of traditional mean-field theories. The model tracks the population by considering grain sizes in neighbour pairs; the probability of a pair having neighbours of certain sizes is determined by the size-frequency distribution of all pairs. Accordingly, the evolution obeys a partial integro-differential equation (PIDE) over ‘grain size versus neighbour grain size’ space, so that the grain-size distribution is a projection of the PIDE’s solution. This model, which is applicable before as well as after statistically self-similar grain growth has been reached, shows that the traditional continuity equation is invalid outside this state. During statistically self-similar growth, the PIDE correctly predicts the coarsening rate, invariant grain-size distribution and spatial grain-size correlations observed in direct simulations. The PIDE is then reducible to the standard continuity equation, and we derive an explicit expression for the drift velocity. It should be possible to formulate similar parameterization-free models of normal grain growth in two and three dimensions.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] H.V. Atkinson, Theories of normal grain growth in pure single phase systems, Acta Metall. 36 (1988) 469e491.
    • [2] J.P. Platt, W.M. Behr, Grainsize evolution in ductile shear zones: implications for strain localization and the strength of the lithosphere, J. Struct. Geol. 33 (2011) 537e550.
    • [3] W.W. Mullins, The statistical self-similarity hypothesis in grain growth and particle coarsening, J. Appl. Phys. 59 (4) (1986) 1341e1349.
    • [4] J. Lambert, R. Mokso, I. Cantat, P. Cloetens, J.A. Glazier, F. Graner, R. Delannay, Coarsening foams robustly reach a self-similar growth regime, Phys. Rev. Lett. 104 (2010) 248304.
    • [5] M. Castro, R. Cuerno, M.M. García-Hernandez, L. Vazquez, Pattern-wavelength coarsening from topological dynamics in silicon nanofoams, Phys. Rev. Lett. 112 (2014) 094103.
    • [6] C. Castellano, S. Fortunato, V. Loreto, Statistical physics of social dynamics, Rev. Mod. Phys. 81 (2009) 591e646.
    • [7] J. Burke, D. Turnbull, Recrystallization and grain growth, Prog. Metal. Phys. 3 (1952) 220e292.
    • [8] W.W. Mullins, J. Vin~als, Self-similarity and growth kinetics driven by surface free energy reduction, Acta Metall. 37 (1989) 991e997.
    • [9] M.P. Anderson, D.J. Srolovitz, G.S. Grest, P.S. Sahni, Computer simulation of grain growthdI. kinetics, Acta Metall. 32 (1984) 783e791.
    • [10] F. Wakai, N. Enomoto, H. Ogawa, Three-dimensional microstructural evolution in ideal grain growthdgeneral statistics, Acta Mater 48 (2000) 1297e1311.
    • [11] C.E. Krill III, L.-Q. Chen, Computer simulation of 3-D grain growth using a phase-field model, Acta Mater 50 (2002) 3057e3073.
    • [12] M. Elsey, S. Esedoglu, P. Smereka, Large-scale simulation of normal grain growth via diffusion-generated motion, Proc. Roy. Soc. A467 (2011) 381e401.
    • [13] J.A. Glazier, S.P. Gross, J. Stavans, Dynamics of two-dimensional soap froths, Phys. Rev. A 36 (1987) 306e310.
    • [14] M. Hillert, On the theory of normal and abnormal grain growth, Acta Metall. 13 (1965) 227e238.
    • [15] W.W. Mullins, Grain growth of uniform boundaries with scaling, Acta Mater. 46 (1998) 6219e6226.
    • [16] N.P. Louat, On the theory of normal grain growth, Acta Metall. 22 (1974) 721e724.
    • [17] I.M. Lifshitz, V.V. Slyozov, The kinetics of precipitation from supersaturated solid solutions, J. Phys. Chem. Solids 19 (1961) 35e50.
    • [18] C. Wagner, Theorie der Alterung von Niederschla€gen durch Umlo€sen (Ostwald-Reifung), Z. Electrochem 65 (1957) 581e591.
    • [19] D.J. Srolovitz, M.P. Anderson, P.S. Sahni, G.S. Grest, Computer simulation of grain growthdII. grain size distribution, topology, and local dynamics, Acta Metall. 32 (1984) 793e802.
    • [20] P.R. Rios, T.G. Dalpian, V.S. Branda~o, J.A. Castro, A.C.L. Oliveira, Comparison of analytical grain size distributions with three-dimensional computer simulations and experimental data, Scr. Mater. 54 (2006) 1633e1637.
    • [21] O. Hunderi, N. Ryum, The influence of spatial grain size correlation on normal grain growth in one dimension, Acta Mater 44 (1996) 1673e1680.
    • [22] K. Marthinsen, O. Hunderi, N. Ryum, The influence of spatial grain size correlation and topology on normal grain growth in two dimensions, Acta Mater. 44 (1996) 1681e1689.
    • [23] M.W. Nordbakke, N. Ryum, O. Hunderi, Invariant distributions and stationary correlation functions of simulated grain growth process, Acta Mater. 50 (2002) 3661e3670.
    • [24] O. Hunderi, J. Friis, K. Marthinsen, N. Ryum, Grain size correlation during normal grain growth in one dimension, Scr. Mater 55 (2006) 939e942.
    • [25] T.O. Saetre, O. Hunderi, N. Ryum, Modelling grain growth in two dimensions, Acta Metall. 37 (1989) 1381e1387.
    • [26] S.P.A. Gill, A.C.F. Cocks, An investigation of mean-field theories for normal grain growth, Phil. Mag. A75 (1997) 301e313.
    • [27] P.R. Rios, K. Lücke, Comparison of statistical analytical theories of grain growth, Scr. Mater 44 (2001) 2471e2475.
    • [28] D. Zo€llner, P. Streitenberger, Three-dimensional normal grain growth: Monte Carlo Potts model simulation and analytical mean field theory, Scr. Mater 54 (2006) 1697e1702.
    • [29] P. Streitenberger, D. Zo€llner, Effective growth law from three-dimensional grain growth simulations and new analytical grain size distribution, Scr. Mater 55 (2006) 461e464.
    • [30] P. Streitenberger, D. Zo€llner, Triple junction controlled grain growth in twodimensional polycrystals and thin films: self-similar growth laws and grain size distributions, Acta Mater 78 (2014) 114e124.
    • [31] J. von Neumann, Written discussion of grain shapes and other metallurgical applications of topology, in: Metal Interfaces, 1952, pp. 108e110. Am. Soc. Metals, Cleveland.
    • [32] W.W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (8) (1956) 900e904.
    • [33] R.D. MacPherson, D.J. Srolovitz, The von Neumann relation generalized to coarsening of three-dimensional microstructures, Nature 446 (2007) 1053e1055.
    • [34] V.E. Fradkov, A theoretical investigation of two-dimensional grain growth in the 'gas' approximation, Phil. Mag. Lett. 58 (1988) 271e275.
    • [35] V.E. Fradkov, D. Udler, Two-dimensional normal grain growth: topological aspects, Adv. Phys. 43 (1994) 739e789.
    • [36] J. Jeppsson, J. Ågren, M. Hillert, Modified mean field models of normal grain growth, Acta Mater 56 (2008) 5188e5201.
    • [37] M.A. Fortes, V. Ramos, A. Soares, Grain growth in one dimension: the mean field approach, Mat. Sci. Forum 94e96 (1992) 337e344.
    • [38] Q. Du, J.R. Kamm, R.B. Lehoucq, M.L. Parks, A new approach for a nonlocal, nonlinear conservation law, SIAM J. Appl. Math. 72 (2012) 464e487.
    • [39] J.D. Logan, Nonlocal advection equations, Int. J. Sci. Math. Educ. 34 (2003) 271e277.
    • [40] D. Zo€llner, P. Streitenberger, Grain size distributions in normal grain growth, Prakt. Metallogr. 47 (2010) 262e283.
    • [41] O. Hunderi, N. Ryum, The kinetics of normal grain growth, J. Mat. Sci. 15 (1980) 1104e1108.
    • [42] A.D. Rollett, D.J. Srolovitz, M.P. Anderson, Simulation and theory of abnormal grain growthdanisotropic grain boundary energies and mobilities, Acta Metall. 37 (1989) 1227e1240.
    • [43] G.I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics, Cambridge University Press, New York, 1996.
    • [44] J.L. Mason, E.A. Lazar, R.D. MacPherson, D.J. Srolovitz, Geometric and topological properties of the canonical grain-growth microstructure, Phys. Rev. E 92 (2015) 063308.
    • [45] K. Barmak, M. Emelianenko, D. Golovaty, D. Kinderlehrer, S. Ta'asan, Towards a statistical theory of texture evolution in polycrystals, SIAM J. Sci. Comput. 30 (6) (2008) 3150e3169.
    • [46] K. Barmak, E. Eggeling, M. Emelianenko, Y. Epshteyn, D. Kinderlehrer, R. Sharp, S. Ta'asan, Critical events, entropy, and the grain boundary character distribution, Phys. Rev. B 83 (2011) 134117, http://dx.doi.org/10.1103/ PhysRevB.83.134117.
    • [47] E.A. Lazar, R. Pemantle, Coarsening in one dimension: invariant and asymptotic states, Israel J. Math. (2015) (in press). arXiv:1505.07893v2 [mathPR].
  • No related research data.
  • No similar publications.

Share - Bookmark

Cite this article