Grain boundary, triple junction and quadruple point mobility controlled normal grain growth

Abstract

Reduction in stored free energy provides the thermodynamic driving force for grain and bubble growth in polycrystals and foams. Evolution of polycrystalline networks exhibit the additional complication that grain growth may be controlled by several kinetic mechanisms through which the decrease in network energy occurs. Polyhedral boundaries, triple junctions (TJs), and quadruple points (QPs) are the geometrically distinct elements of three dimensional networks that follow Plateau’s rules, provided that grain growth is limited by diffusion through, and motion of, cell boundaries. Shvindlerman and co-workers have long recognized the kinetic influences on polycrystalline grain growth of network TJs and QPs. Moreover, the emergence of interesting polycrystalline nanomaterials underscored that TJs can indeed influence grain growth kinetics. Currently there exist few detailed studies concerned either with network distributions of grain size, number of faces per grain, or with ‘grain trajectories’, when grain growth is limited by the motion of its TJs or QPs. By contrast there exist abundant studies of classical grain growth limited by boundary mobility. This study is focused on a topological/geometrical representation of polycrystals to obtain statistical predictions of the grain size and face number distributions, as well as growth ‘trajectories’ during steady-state grain growth. Three limits to grain growth are considered, with grain growth kinetics controlled by boundary, TJ, and QP mobilities.

Keywords:

• grain growth
• grain size distribution
• distribution of number of faces
• triple junctions
• quadruple points

Acknowledgements

The author, PRR, is grateful for his financial support to the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and to the Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ. MEG acknowledges his helpful discussions on continuous and discrete variables held with Dr Semen Köksal, Professor of Mathematics, Florida Institute of Technology, Melbourne, Florida, and gives thanks for the research support provided through the Allen S. Henry Chair, Mechanical and Aerospace Engineering Department.

Notes

No potential conflict of interest was reported by the authors.

1 There is a network restriction imposed on ANHs that they have a minimum -value of . provides a complete series of regular polyhedra with two or more vertices to form network QPs, which allow, theoretically, interconnections within a network among the other trivalent polyhedra. ANHs and their theoretical networks are abstract, fictive constructions, as most ANHs are not constructible objects.

2 Any constructible regular trivalent polyhedron must have an integer number of edges per face, , limited to the set }, corresponding to }. Constructible irregular polyhedra, with different face shapes, and an arbitrary integer number of sides, , on a face (constrained by Smith’s average) often exhibit an average number of edges per face, , which is a repeating decimal. ANHs, by contrast, exhibit extreme symmetry, such that if the average number of edges per face is a repeating decimal, so too is the number of edges on every face, i.e. , which precludes constructibility. Clearly, unless the average number of edges per face is itself an integer – a condition limited to only the four trivalent, regular -hedra, viz., and 12 – the ANH is not constructible. See again Smith’s Equation (6(6) ).

3 The ANH with and is a constructible regular polyhedron with 2 edges per face and 2 vertices, but lacks a flat-face primitive, which would, if it existed, correspond to its Platonic solid. The ANH with faces instead collapses instead to an infinite line segment, as its face curvatures reduce to zero.

4 The free energy density of any crystalline region may be elevated by a number of causes, among them strain energy, composition differences, atomic order, magnetic energy, and boundary curvature, or pressure.

5 Movements of GBs within a polycrystal are geometrically responsible for changes in grain volume, which may result either from direct kinetic migration caused by the curvature of the GBs themselves, or from indirect migration induced kinematically by network forces moving the TJs or QPs.

6 Network forces vanish, in principle, only for the special non-integer ANH that has the ‘critical’ number of flat faces, viz., This ANH forms an abstract, flat face, regular polyhedron with straight edges per face, and a dihedral angle . The critical property of the trivalent -hedron is that it exhibits the most efficient polyhedral partition of space, or minimum surface-to-volume ratio. Thus, the critical ANH stores the minimum face energy for its intrinsic volume, and acts as an ‘attractor’ by setting the minimum free energy ‘floor’ within the network. The present authors showed that local cluster averages of space filling, irregular network grains tend to approximate the critical features of the -hedron. Deviation of these local averages from their critical values increases the local network free energy slightly above its global minimum, so spontaneous growth continues, and the network remains endlessly ‘frustrated’ [8].

7 The non-integer ANH, , has flat-faces, a dihedral angle of , and separates the sub-populations of convex ANH face classes with from concave ones with .

8 The approximation of a ‘zero’ rate of face loss occurs because the continuum solutions for values are not integers.

Additional information

Funding

This work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro/FAPERJ; Conselho Nacional de Desenvolvimento Científico e Tecnológico/CNPq.