Grain Growth: Beyond von Neumann-Mullins — Edition 2

Nearly six months ago, I heard Prof. Srolovitz on the topic, and blogged about the talk here; as I noted in that post, unfortunately, I did not have any resources to link to then; instead, I summarised the talk to the extent I understood. In this post, I am happy to link to not just one, but several pieces on the issue:

  1. In the latest issue of Nature, Robert D MacPherson and and David J Srolovitz describe their work. Here is the abstract of their paper:

    Cellular structures or tessellations are ubiquitous in nature. Metals and ceramics commonly consist of space-filling arrays of single-crystal grains separated by a network of grain boundaries, and foams (froths) are networks of gas-filled bubbles separated by liquid walls. Cellular structures also occur in biological tissue, and in magnetic, ferroelectric and complex fluid contexts. In many situations, the cell/grain/bubble walls move under the influence of their surface tension (capillarity), with a velocity proportional to their mean curvature. As a result, the cells evolve and the structure coarsens. Over 50 years ago, von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structure (using the relation between wall velocity and mean curvature, the fact that three domain walls meet at 120° and basic topology). This forms the basis of modern grain growth theory. Here we present an exact and much-sought extension of this result into three (and higher) dimensions. The present results may lead to the development of predictive models for capillarity-driven microstructure evolution in a wide range of industrial and commercial processing scenarios—such as the heat treatment of metals, or even controlling the ‘head’ on a pint of beer.

  2. Here is the supplementary information to the Nature article which gives the details of the derivation.
  3. David Kinderlehrer, in a News and Views piece puts the article 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.

  4. Here is the Scientific American commentary on the piece.

Have fun!

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2 Responses to “Grain Growth: Beyond von Neumann-Mullins — Edition 2”

  1. In Nature this week « Materials, Science of Says:

    […] course the big story, as I blogged elsewhere, is the higher dimensional generalizations of the Neumann-Mullins rule of grain growth by […]

  2. Shapes of crystallites and their evolution « Entertaining Research Says:

    […] As long time readers of this blog might already have noticed, this post is based on a couple that I wrote on this problem: here and here. […]

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