Von Neumann law

The Von Neumann's law describes the change in the size of two-dimensional cell foam .

Foam cells with number of corners

Due to the pressure differences between neighboring foam cells and the resulting diffusion , the surface area of ​​the individual two-dimensional foam cells also changes over time. In 1952, John von Neumann found the law named after him

${\displaystyle {\dot {A}}=k\cdot (N-6)}$

where indicates the change in the area of ​​a cell with N corners over time. k> 0 is a growth coefficient with unity . It is interesting that the change in area only depends on the number of sides of the foam cell, but not on the neighbors or the area of ​​the cell. An equilibrium is thus established when an area consists entirely of hexagonal cells. ${\displaystyle {\dot {A}}}$ ${\displaystyle m^{2} \over s}$

7-sided cells grow, 5-sided cells shrink

4-corner	${\displaystyle {\dot {A}}=-2k}$	Cell is shrinking
5-corner	${\displaystyle {\dot {A}}=-k}$	Cell is shrinking
6-corner	${\displaystyle {\dot {A}}=0}$	Area constant
7-corner	${\displaystyle {\dot {A}}=k}$	Cell grows
Octagon	${\displaystyle {\dot {A}}=2k}$	Cell grows

Von Neumann's law only applies to two-dimensional foam. A corresponding law for three-dimensional foam was published by Sascha Hilgenfeldt et al. and Robert D. MacPherson & David J. Srolovitz.

literature

• Denis Weaire, Stefan Hutzler: The Physics of Foams . Clarendon Press, Oxford 1999, ISBN 0-19-851097-7 .

Individual evidence

1. ^ J. von Neumann: Metal Interfaces . American Society for Metals, Cleveland 1992.
2. ^ Sascha Hilgenfeldt: An Accurate von Neumann's Law for Three-Dimensional Foams . In: Physical Review Letters . No. 86 , 2001, doi : 10.1103 / PhysRevLett.86.2685 .
3. ^ Robert D. MacPherson, David J. Srolovitz: The von Neumann relation generalized to coarsening of three-dimensional microstructures . In: Nature . tape 446 , no. 7139 , April 26, 2007, p. 1053-1055 , doi : 10.1038 / nature05745 ( nature.com [accessed April 10, 2016]).