TY - JOUR
T1 - The von Neumann relation generalized to coarsening of three-dimensional microstructures
AU - MacPherson, Robert D.
AU - Srolovitz, David J.
PY - 2007/4/26
Y1 - 2007/4/26
N2 - 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.
AB - 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.
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U2 - 10.1038/nature05745
DO - 10.1038/nature05745
M3 - 21_Publication in refereed journal
VL - 446
SP - 1053
EP - 1055
JO - Nature
JF - Nature
SN - 0028-0836
IS - 7139
ER -