If you can look into the seeds of time,
And say which grain will grow and which will not,
Speak then to me, …
– Shakespeare (Macbeth)
Grain growth in polycrystalline solids
Most of the metallic and ceramic materials in use usually consist of many crystallites of differing orientations; such materials are classified as polycrystalline and the crystals of differing orientations that make up the polycrystalline aggregate are known as grains.
In a polycrystalline sample that is held at high enough temperatures, the total number of grains will decrease (with a corresponding increase in the mean size of the surviving grains). This process of known as grain growth.
The shapes and sizes of grains as well as their evolution when the sample is held at high enough temperatures is a problem of both industrial importance and academic interest; it is of industrial importance because many of the material properties of interest are determined by the underlying grain structure; it is of academic interest since grain growth is but one example of a large class of phenomena in which a material system reduces the area of internal interfaces in an effort to minimize its free energy.
There exists an important result for the growth of (idealized) grains in two dimensions (2D), which can be stated as follows:
The rate of change of area of a grain is determined purely by its topology; if it has more than six sides, it would grow; if it has less than six sides, it would shrink; and, if the number of sides are six, such a grain would neither grow nor shrink.
where, is the area of the grain, is the time, and is the number of sides.
This is known as the Neumann-Mullins law following its postulation and derivation by Neumann  and Mullins .
In this post, I want to discuss these two papers [1,2], some of the history behind the contribution of Neumann, and the recent extension of this result by MacPherson and Srolovitz  to three dimensional (3D) systems.
A bit of history: the role of Cyril Stanley Smith
Though the name of C S Smith is not associated with the Neumann-Mullins law, he played a key role in its formulation. Smith, as I understand, delivered the Institute of Metals lecture in 1948 titled Grains, phases and interfaces: An interpretation of microstructure . Unfortunately, I do not have access to this piece at the moment; however, the impact of this work is clear from the Foreword to the proceedings of the seminar called Metal Interfaces held during the thirty-third national metal congress and exposition, Detroit, October 13 to 19, 1951 :
There was a sharp spurt in metallurgical research in this general field [surfaces or interfaces] as a result of a lecture in 1948 by Cyril Stanley Smith which brought new ideas to bear on an old and important topic. The new ideas and data resulting from vigorous and productive research on metal interfaces in the past three years are the basis of this volume.
The contribution of Neumann is but a two page written discussion on the paper of Smith in this volume titled Grain shapes and other metallurgical applications of topology — reminding ourselves of the times when conference proceedings, and the written discussion in them were considered as proper venues for reporting original and far-reaching research findings.
As I see it, the main contribution of C S Smith is in (a) drawing attention to the possible applications of topological ideas to metallurgical problems; (b) setting the grain growth problem as belonging to the same class as cell structures in biological tissues, soap froths, foams, and problems on space filling; and, (c) using models of soap froths as near exact geometric models of grain growth in metals.
By the way, Smith’s paper is probably the only one in metallurgical literature which contains a line drawing based on a painting of Picasso.
It is not clear if Neumann was present at the Detroit meeting in 1951. However, what is clear from Smith’s paper is that he did take active interest in involving mathematicians in this kind of project — one of his acknowledgements in this paper, for example, is to Saunders MacLane — so, it is quite possible that Smith circulated a copy of his paper to Neumann too for comments; further, since Smith also worked in the Manhattan project along with Neumann, and since Neumann gave strong political support to the formation of the field of materials science , it is not out of place to postulate such a strong working relationship between Neumann and Smith on such a problem of metallurgical interest with a strong mathematical component.
For those of you who are interested in learning more about C S Smith, the man and his work, I can do no better than to point to this June 2006 issue of Resonance, which is a special issue honouring C S Smith.
Before I leave Smith behind, I just want to draw your attention to the reply that Smith gave to the discussions section, in which he expressed a hope and a word of caution:
The discussion by Dr. von Neumann is much appreciated, and his conclusions are as remarkable for their simplicity as they are nonobvious on first consideration of the problem. It is greatly to be hoped that he, or some other mathematician, will be able to deduce similar relations in three dimensions and can combine these with the topological requirements to give the equilibrium distribution of bubbles toward which a froth must tend.
… Metals are not soap bubbles, even though soap bubbles are so useful in illustrating the basic principles, as I hope I have demonstrated in the paper.
While it took another fifty years (and another meeting between a mathematician and a materials scientist) for part of Smith’s hope to be fulfilled, his word of caution continues to give directions to research in this area.
The formulations of Neumann and Mullins
the changes of bubble-volume due to diffusion, that occur in a two-dimensional bubble-froth.
And, as Neumann pointed out, to first approximation, this diffusion flow is proportional to the pressure difference, and
The pressure difference of the two adjascent bubbles, at a given point of a wall, is , where is the surface tension of the liquid forming the froth, and is the radius of curvature of the wall at .
Further, since it is a bubble-froth, is a constant throughout the froth. With these given, assuming that the triple junctions in such a forth are at equilibrium (that is, they tend 120 degrees), Neumann obtained the now famous result:
In a two-dimensional bubble-froth the total gas-gain-rate of any bubble is (positively) proportional to , where is the number of sides of the bubble …
The derivation of Mullins is much closer in spirit to what is given in modern text-books (like that of Gottstein and Shvindlerman , for example). Mullins begins his derivation assuming that any boundary segment moves with a velocity which is given by , where is the local curvature, and is the mobility. And, as he notes, the result of Neumann is but a special case in Mullins’ derivation since
… within each cell of a soap froth, the possibility of a rapid mass flow of air maintains a uniform pressure which in turn causes each film to have a constant mean curvature; within a metal grain there is no possibility of a rapid mass flow and its associated uniformity of pressure so that the motion of any portion of a boundary is governed by local conditions only. Thus the problem of grain boundary motion, according to the curvature rule, is a problem in differential geometry.
Thus, the contribution of Mullins is in showing that the Neumann result holds even if the curvature is not a constant on any given domain wall — local curvature is what matters.
3D generalization of MacPherson and Srolovitz
Apparently, there had been many attempts to generalise Neumann-Mullins to 3D; none of these attempts have been successful in obtaining an exact result. However, recently, Srolovitz and MacPherson have not only managed to generalise the results to 3D, but to any higher dimension. The generalization is achived using the following strategy:
- Generalise 2D von Neumann-Mullins to multiply connected domains;
- Volume integrate the result; this is equivalent to considering all possible sections of a 3D structure, and doing a von Neumann-Mullins analysis on each, and putting the results together.
The result of the above manipulations is the introduction of a natural measure of length called mean width (which is a Hadwiger measure); and, using such a measure, Neumann-Mullins can be generalised to N-dimensional domain wall networks and the 2D and 3D networks are just the special cases of such a general result. And, as it turns out, in 3D (and in higher dimensions), curvature driven growth is not purely topological.
Here are some more resources on the 3D generalization.
- The supplementary information to the Nature article which gives the details of the derivation.
- David Kinderlehrer, in a News and Views piece puts the work of MacPherson and Srolovitz in perspective:
A long-standing mathematical model for the growth of grains in two dimensions has been generalized to three and higher dimensions. This will aid our practical understanding of certain crucial properties of materials.
He further notes some of the crucial assumptions made in deriving the 3D result and the future direction of research in the area:
A physical grain network is beset by inhomogeneities and anisotropy, to name just two impediments to ideal growth. Even in the abstract, the effect of such features is unknown: we enter the realm of stochastic analysis through simple combinatorial events such as cell or facet deletion. Such analysis, implemented with automated data acquisition in the laboratory and large-scale simulation at the desk, will be the future direction of the subject.
- The Scientific American commentary.
The Neumann-Mullins result is a classic; not only does it put the problem of grain growth in the same class as that of coarsening in soap froths, namely, curvature driven growth, but also shows the close relationship between problems that are of interest to materials scientists and mathematicians, and how a collaboration between the two can result in startling-ly simple and elegant results. Further, these results also serve as an important benchmark for both experiments and theory. The nearly half-a-century long, hard search for its generalization has brought much progress to the field, and the final result is as satisfactory as the 2D one, and elegant in its own way. However, like all true classics, Neumann and Mullins, even as they maintain their relevance, also lead us in new directions.
In real systems, say, a grain boundary, for example, the boundary energies are anisotropic; the mobilities are not constant; the triple junctions induce drag on the boundary motion; and, all these are experimentally well known. Or, in other words, each of the assumptions made by Neumann, Mullins, MacPherson, and Srolovitz are to be relaxed; and, studies along these lines are pursued, and, I hope I will be able to report on some of them and our own phase field studies in these pages some time soon. In the meanwhile, let Neumann and Mullins engage you.
It is a pleasure to thank Abi, Peter, Shankara, and Prof. Ranganathan for many useful discussions and their comments on some of my earlier writings on Neumann-Mullins and MacPherson-Srolovitz (though, the mistakes, if any, that remain, are all my own).
 J von Neumann, in a written discussion to Grain shapes and other metallurgical applications of topology by C S Smith, in Metal Interfaces, American Society for Metals, Cleveland, Ohio, 1952.
 W W Mullins, Two-dimensional motion of idealized grain boundaries, Journal of Applied Physics, 27, pp. 900-904, 1956.
 R D MacPherson and D J Srolovitz, The von Neumann relation generalized to coarsening of three-dimensional microstructures, Nature 446, pp. 1053-1055, 2007.
 C S Smith, Grains, phases and interfaces — an interpretation of microstructure, Transactions, American Institute of Mining and Metallurgical Engineers, 175, pp. 15-51, 1948.
 R M Brick, Foreword to Metal Interfaces, American Society for Metals, Cleveland, Ohio, 1952.
 R W Cahn, The coming of materials science, Pergamon materials series, Elsevier Science Publishers, 2003.
 G Gottstein and L S Shvindlerman, Grain boundary migration in metals: Thermodynamics, kinetics and applications, CRC Press, 1999.