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Going beyond 2D Neumann-Mullins (or, what is popularly known as, solving the beer froth structure)

Mogadalai Gururajan's picture

Introduction

The blogosphere is abuzz with the latest report of the generalisation of the von Neumann-Mullins grain growth relation to 3 (and N) dimensions by MacPherson and Srolovitz (As an interesting aside, almost all the reports say mathematical structure of beer foam structure resolved, or words to that effect --hence, I also decided to join the bandwagon on that one). I heard Prof. Srolovitz describe the work in a seminar nearly six months ago. Based on my notes of the talk, I would like the explain their work in this post. Curvature in the following refers to mean curvature (and not Gaussian).

2D von Neumann-Mullins grain growth relation

Consider a cell structure in 2 dimensions.

  1. Using Euler's polyhedron formula, which states that for a convex polyhedron, V-E+F = 2, where V, E, and F are the vertices, edges and faces of the polyhedron,
  2. assuming isotropic energy (and equal mobility) for all the cell walls, and
  3. assuming that triple junctions are points at which the cell walls make 120 degrees with each other (in other words, achieve equilibrium),

von Neumann showed that six sided cells are stable; the grains with sides greater than six sides grow while those with less than six sides shrink. Neumann made the assumption that the curvature of each of the walls is a constant. Mullins relaxed the curvature condition, and showed that the result holds even if the curvature is not a constant--mean curvature is what matters. Thus, Neumann's results are valid for soap froths, while that of Mullins is valid even for grain boundaries--albeit in two dimensions. This is a purely topological result and can also be derived using Gauss-Bonnet theorem -- see the derivation given by Guenter Gottstein and Lasar S Svindlerman in Grain boundary migration in metals: Thermodynamics, Kinetics, pp. 309-310, CRC Press, New York 1999, for example.

3D (and N) generalization to von Neumann-Mullins relation

Apparently, there had been several attempts to generalize the Neumann-Mullins to 3D in vain, so far. MacPherson and Srolovitz manage to do just that--in the process, they also obtain an N-dimensional generalization. The idea behind the derivation of such a generalized result is as follows:

  1. Generalize Neumann-Mullins to multiply connected domains in 2D; and,
  2. Volume integrate the result.

Obviously, the two step process is nothing but considering all possible 2D sections of the given 3D structure, doing a 2D Neumann-Mullins analysis on each of them, and putting the results of all these analyses together. And, this is also the point where things become a bit (too) mathematical.

In any case, I understand that the net result of the two step process described above is the introduction of a natural measure of length--called mean width--and this measure is a Hadwiger measure. And, the Neumann-Mullins result can be stated in terms of the Hadwiger measures in N dimensions, of which, the 2D and 3D results become a special case. And, the result also shows that in 3- and higher dimensions, the result is not purely topological.

What next?

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 -- Or, in other words, each of the assumptions made by Neumann, Mullins, MacPherson, and Srolovitz are to be relaxed. Thus, this is but a first step in the search for an understanding of grain growth and coarsening studies.

Relevant links

Most of this post is based on this blog post of mine. I have also collected the links to the paper of MacPherson and Srolovitz, the supplementary information to the paper, the News and Views piece on the work by David Kinderlehrer, and the Scientific American news report.

Have fun!

Comments

Pradeep Sharma's picture

Mogadalai,

This is quite interesting. I plan to read some of the papers and links you have suggested. One request for clarification though: Could you expand further on what you mean when you say that in 3D and above, the results are not purely topological?

 

Mogadalai Gururajan's picture

Dear Pradeep,

Thanks for your interest in the papers.

In 2D, the rate of change of area of the domain is dependent only on the number of triple junctions in the boundary:

dA/dt = 2 pi M gamma (n/6 - 1)

A--area of the domain; M--mobility of the domain wall; gamma--surface tension of the domain wall; n--number of triple points. Thus, it does not matter how lengthy each segment of the domain wall is, how big the domain is etc. If n=6, dA/dt =0; if n > 6 (<6), dA/dt is positive (negative) and hence the area increases (decreases). Which is the Neumann-Mullins result.

However, in 3D (and higher dimensions), the rate of change of volume of a given domain depends on two length measures--the mean width of the domain, and the length of the triple lines (edge):

dV/dt = 2 pi M gamma(L/6 - D)

V--volume of the domain; L--length of all the triple lines(edges) in the domain wall; D--Mean width of the domain wall.

Note: The mean width D is the Hadwiger measure, and as the Wiki on Hadwiger measure notes, the name mean width is a misnomer.

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