Grain Growth: Monte Carlo Simulation & Theory

Nanocrystalline Grain Growth

Metals and alloys of nanocrystalline size have quite different mechanical properties compared to conventional materials. This has already been proposed 20 years ago by [Gleiter (1989)]. In recent times it has been proven by experiments ([Koch et al. (2007); Dao et al. (2007)]) that nanocrystalline materials are indeed characterised by high values of yield and fracture strength, hardness and superplastic behaviour at low temperatures implying a size-effect. If we follow for example the Gibbs-Thompson-equation, then the driving force of grain growth is in inverse ratio to the grain size. Therefore, nanocrystalline grains should growth very fast. However, in experiments it has been found that such materials grow very slow even up to relatively high temperatures showing stable grain sizes [Krill et al. (2001)]. Moreover, linear growth kinetics have been observed in clear contradiction to normal parabolic grain growth.
It is important to note that such investigations concerning the stability of nanocrystalline materials during grain growth are of intense technological interest because an increase of the grain size beyoned nm can result in a loss of important materials properties making them unusable in applications.

However, this problem is not new. Already more than 10 years ago a number of investigations have been realised regarding the thermal stability of nanocrystalline materials. An overview by [Malow and Koch (1997)] summarised significant works concerning the stabilization of nanocrystalline grain structures in many materials and the number of factors influencing the grain boundary mobility in nanocrystalline alloys like

grain boundary segregation [Krill et al. (1995); Abe et al. (1992); Eckert et al. (1993)],
solute drag [Knauth et al. (1993)],
pore drag [Höfler et al. (1990)],
second phase (Zener) drag [Boylan et al. (1991)] and
chemical ordering [Bansal et al. (1995); Gao and Fultz (1993, 1994)].

A universal explanation has not been found yet, but possible reasons are currently in discussion. The following list of works is not intended to be exhaustive but represents rather a rough survey (ordered by date of publication):

Michels et al. (1999) have proposed that if grain boundary migration is controlled by solute drag a size-dependence to the retarding force on boundary migration is introduced.
3D molecular dynamics simulations performed by Haslam et al. (2001) have shown that in nanocrystalline materials grain rotations are important.
In the presence of solute atoms grain boundaries are in a metastable thermodynamic equilibrium [Kirchheim (2002)].
Also with a molecular dynamics simulation Zhou et al. (2005) have verified that grain boundary mobility is a function of the size of the system in nano-scale systems, which is important, e.g., for thin films.
Pande and Masumura (2005) discussed that both, grain rotation and finite triple junction mobility, need to be taken into account in the case of nanocrystalline materials. Regarding the finite triple junction mobility they refer to an older paper of [Gottstein and Shvindlerman (1998)].
Grain growth under the condition that triple junctions possess a limited mobility was also modelled by Novikov (2005). He observed that the growth kinetics is linear in an early growth stage and becomes parabolic at later stages concluding that the triple junction drag could explain some features of the growth kinetics in nanocrystalline materials.
In a newer paper [Gottstein and Shvindlerman (2006)] proposed that grain growth in polycrystals can be controlled by grain boundary junction mobilities. The established structure is rather stable, in particular, for ultrafine grained and nanocrystalline materials (see also [Gottstein et al. (2005); Barrales Mora et al. (2008)]).
Farkas et al. (2007) found by molecular dynamics simulation that linear grain growth can be attributed to the fact that mobility of nanosized grain boundaries is sizedependent. The growth process for these very small sizes is accompanied by grain rotation and a decrease in the average grain boundary energy per unit area.

Now, compared to normal grain growth, a field of work with several sophisticated mesoscopic simulation methods, for nanocrystalline grain growth there are until today no extensively tested simulation methods available although it is quite desireable since the mesoscopic simulations on grain growth have the advantage that they recreate grain growth numerically efficiant and simulate large grain ensembles needed for statistical analyses.
Although some years ago Weygand et al. (1998) presented a first approach applying a vertex dynamics model to the 2D simulation of grain growth including a limited mobility of triple points. They observed changed growth kinetics as well as a change in the size distribution.

If we follow theoretical preliminary considerations based on experiments, e.g., [Krill et al. (2001)] and the works of, e.g., [Gottstein and Shvindlerman (2006), Novikov (2005), Farkas et al. (2007)], then the mobility of the grain boundaries is reduced depending on their length. It can be assumed that a limited junction mobility yielding a junction drag can be represented in the Potts method by an effective mobility

,
where a is the average grain boundary junction spacing. The mobilities of grain boundary, triple lines and quadruple points are given by m_gb , m_tj and m_qp, where especially the mobility of the grain boundary can be calculated with the Huang-Humphreys-relation, which yields for high angle grain boundaries m_gb = 1.

Based on this assumption the Monte Carlo Potts model can be modified either by applying the specific mobilities m_gb , m_tj and m_qp directly to the associated grain features or by introducing the effective parameters m_gb:m_tj and m_gb:m_qp.
Detailed information can be found in [Zöllner (2011), Streitenberger and Zöllner (2011)].

[Gleiter (1989)]	H. Gleiter: Nanocrystalline materials. Prog. Mater. Sci., 33, (1989), p. 223-315.
[Koch et al. (2007)]	C.C. Koch, I.A. Ovid’ko, S. Seal and S. Veprek: Structural Nanocrystalline Materials. Cambridge University Press, ISBN 978-0-521-85565-5, (2007).
[Dao et al. (2007)]	M. Dao, L. Lu, R.J. Asaro, J.T.M. De Hosson and E. Ma: Toward a quantitative understanding of mechanical behavior of nanocrystalline metals. Acta Mater., Vol 55, (2007), p. 4041-4065.
[Krill et al. (2001)]	C.E. Krill, L. Helfen, D. Michels, H. Natter and A. Fitch: Size-Dependent Grain-Growth Kinetics Observed in Nanocrystalline Fe. Phys. Rev. Lett., Vol 86, (2001), p. 842-845.
[Malow and Koch (1997)]	T.R. Malow and C.C. Koch: Grain growth in nanocrystalline iron prepared by mechanical attrition. Acta Mater., Vol 45, (1997), p. 2177-2186.
[Krill et al. (1995)]	C.E. Krill, R. Klein, S. Janes and R. Birringer: Thermodynamic stabilization of grain boundaries in nanocrystalline alloys. Mater. Sci. Forum, Vols 179-181, (1995), p.443-448.
[Abe et al. (1992)]	Y.R. Abe, J.C. Holzer and W.L. Johnson: Formation and Stability of Nanocrystalline Nb-Cu Alloys. Mat. Res. Soc. Symp. Proc., Vol 238, (1992), p. 721-726.
[Eckert et al. (1993)]	J. Eckert, J.C. Holzer and W.L. Johnson: Thermal stability and grain growth behavior of mechanically alloyed nanocrystalline Fe-Cu alloys. J. Appl. Phys., Vol 73, (1993), p. 131-141.
[Knauth et al. (1993)]	P. Knauth, A. Charaï and P. Gas: Grain growth of pure nickel and of a Ni-Si solid solution studied by differential scanning calorimetry on nanometer-sized crystals. Scripta Metall. Mater., Vol 28, (1993), p. 325-330.
[Höfler et al. (1990)]	H.J. Höfler and R.S. Averback: Grain growth in nanocrystalline TiO2 and its relation to vickers hardness and fracture toughness. Scripta metall. mater., Vol 24, (1990), p. 2401-2406.
[Boylan et al. (1991)]	K. Boylan, D. Ostrander, U. Erb, G. Palumbo and K.T. Aust: An In-Situ TEM Study of the Thermal Stability of Nanocrystalline Ni-P NiunderlineP. Scripta Metall. Mater., Vol 25, (1991), p. 2711-2716.
[Bansal et al. (1995)]	C. Bansal, Z.Q. Gao and B. Fultz: Grain Growth and Chemical Ordering in (Fe,Mn)3Si. Nanostruct. Mater., Vol 5, (1995), p. 327-336.
[Gao and Fultz (1993)]	Z. Gao and B. Fultz: The thermal stability of nanocrystalline Fe-Si-Nb prepared by mechanical alloying. Nanostruct. Mater., Vol 2, (1993), p. 231-240.
[Gao and Fultz (1994)]	Z. Gao and B. Fultz: Inter-dependence of grain growth, Nb segregation, and chemical ordering in Fe-Si-Nb nanocrystals. Nanostruct. Mater., Vol 4, (1994), p. 939-947.
[Michels et al. (1999)]	A. Michels, C.E. Krill, H. Ehrhardt, R. Birringer and D.T. Wu: Modelling the influence of grain-size-dependent solute drag on the kinetics of grain growth in nanocrystalline materials. Acta Mater., Vol 47, (1999), p. 2143-2152.
[Haslam et al. (2001)]	A.J. Haslam, S.R. Phillpot, D. Wolf, D. Moldovan and H. Gleiter: Mechanisms of grain growth in nanocrystalline fcc metals by molecular-dynamics simulation. Mater. Sci. Eng. A, Vol 318, (2001), p. 293-312.
[Kirchheim (2002)]	R. Kirchheim: Grain coarsening inhibited by solute segregation. Acta Mater., Vol 50, (2002), p. 413-419.
[Zhou et al. (2005)]	L. Zhou, H. Zhang and D.J. Srolovitz: A size effect in grain boundary migration: A molecular dynamics study of bicrystal thin films. Acta Mater., Vol 53, (2005), p. 5273-5279.
[Pande and Masumura (2005)]	C.S. Pande and R.A. Masumura: Grain growth and deformation in nanocrystalline materials. Mater. Sci. Eng. A, Vol 409, (2005), p. 125-130.
[Gottstein and Shvindlerman (1998)]	G. Gottstein and L. S. Shvindlerman: Triple junction dragging and von Neumann-Mullins relation. Scripta Mater., Vol 38, (1998), p. 1541-1547.
[Novikov (2005)]	V.Y. Novikov: On the influence of triple junctions on grain growth kinetics and microstructure evolution in 2D polycrystals. Scripta Mater., Vol 52, (2005), p. 857-861.
[Gottstein and Shvindlerman (2006)]	G. Gottstein and L.S. Shvindlerman: Grain boundary junction engineering. Scripta Mater., Vol 54, (2006), p. 1065-1070.
[Gottstein et al. (2005)]	G. Gottstein, Y. Ma and L.S. Shvindlerman: Triple junction motion and grain microstructure evolution. Acta Mater., Vol 53, (2005), p. 1535-1544.
[Barrales Mora et al. (2008)]	L.A. Barrales Mora, V. Mohles, L.S. Shvindlerman and G. Gottstein: Effect of a finite quadruple junction mobility on grain microstructure evolution: Theory and simulation. Acta Mater., Vol 56, (2008), p. 1151-1164.
[Farkas et al. (2007)]	D. Farkas, S. Mohanty and J. Monk: Linear Grain Growth Kinetics and Rotation in Nanocrystalline Ni. Phys. Rev. Letters, Vol 98, 165502 (2007).
[Weygand et al. (1998)]	D. Weygand, Y. Bréchet, J. Lépinoux: Influence of a reduced mobility of triple points on grain growth in two dimensions. Acta Mater., Vol 46, (1998), p. 6559-6564.
[Zöllner (2011)]	D. Zöllner: A Potts model for junction limited grain growth, Computational Materials Science, Vol. 50, (2011), p. 2712-2719.
[Streitenberger and Zöllner (2010)]	P. Streitenberger and D. Zöllner: Evolution Equations and Size Distributions in Nanocrystalline Grain Growth, Acta Materialia, Vol. 59 (2011), p. 4235-4243.

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