Skip to main content Accessibility help

Numerical experiments on two-dimensional foam

  • Thomas Herdtle (a1) and Hassan Aref (a1)


The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.



Hide All
Aboav D. A. 1970 The arrangement of grains in a polycrystal. Metallography 3, 383390.
Aboav D. A. 1980 The arrangement of cells in a net. Metallography 13, 4358.
Aref, H. & Herdtle T. 1990 Fluid networks. In Topological Fluid Mechanics (ed. H. K. Moffatt & A. Tsinober), pp. 745764. Cambridge University Press.
Beenakker C. W. J. 1988 Numerical simulation of a coarsening two-dimensional network Phys. Rev. A 37, 16971702.
Berge B., Simon, A. J. & Libchaber A. 1990 Dynamics of gas bubbles in monolayers Phys. Rev. A 41, 68936900.
Bolton, F. & Weaire D. 1991 The effects of Plateau borders in the two-dimensional soap froth. I. Decoration lemma and diffusion theorem Phil. Mag. B 63, 795809.
Bragg, L. & Nye J. F. 1947 A dynamical model of a crystal structure Proc. R. Soc. Lond. A 190, 474481.
George, A. & Ng E. 1984 SPARSPAK: Waterloo Sparse Matrix Package. Dept. Computer Science, Univ. Waterloo, Res. Rep. CS-84–37, 47 pp.
Glazier J. A. 1989 Dynamics of cellular patterns. Ph.D. thesis, University of Chicago.
Glazier J. A., Gross, S. P. & Stavans J. 1987 Dynamics of two-dimensional soap froths Phys. Rev. A 36, 306312.
Glazier, J. A. & Stavans J. 1989 Nonideal effects in the two-dimensional soap froth Phys. Rev. A 40, 73987401.
GruUnbaum, B. & Shephard G. C. 1987 Tilings and Patterns. W. H. Freeman.
Herdtle, T. & Aref H. 1989 Numerical simulation of two-dimensional foam. Bull. Am. Phys. Soc. 34, 2296 (abstract only).
Herdtle, T. & Aref H. 1991a Relaxation of fractal foam. Phil. Mag. Lett. 64, 335340.
Herdtle, T. & Aref H. 1991b On the geometry of composite bubbles Proc. Roy. Soc. Lond. A 434, 441447.
Isenberg C. 1978 The Science of Soap Films and Soap Bubbles. Somerset: Woodspring Press.
Kermode, J. P. & Weaire D. 1990 2D-FROTH: A program for the investigation of 2-dimensional froths. Comput. Phys. Commun. 60, 75109 (referred to herein as KW).
Knobler C. M. 1990 Seeing phenomena in Flatland: Studies of monolayers by fluorescence microscopy. Sci. 249, 870874.
Kraynik, A. M. & Hansen M. G. 1986 Foam and emulsion rheology: A quasistatic model for large deformations of spatially-periodic cells. J. Rheol. 30, 409439.
Lewis F. T. 1928 The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis. Anat. Rec. 38, 341376.
Lucassen J., Akamatsu, S. & Rondelez F. 1991 Formation, evolution and rheology of two-dimensional foams in spread monolayers at the air-water interface. J. Colloid Interface Sci. 144, 434448.
Maddox J. 1989 Soap bubbles make serious physics. Nature 338, 293.
Neumann von J. 1952 Discussion remark concerning paper of C. S. Smith, ‘Grain shapes and other metallurgical applications of topology’. Metal Interfaces, pp. 108110. Am. Soc. for Metals, Cleveland, Ohio.
Plateau J. A. F. 1873 Statique ExperimeAntale et TheAorique des Liquides Soumis aux Seules Forces MoleAculaires. Gauthier-Villars.
Princen, H. M. & Mason S. G. 1965 The permeability of soap films to gases. J. Colloid Sci. 20, 353375.
Rivier N. 1990 Geometry of random packings and froths. In Physics of Granular Media (ed. D. Bideau & J. Dodds) Nova Science.
Stavans J. 1990 Temporal evolution of two-dimensional drained soap froths Phys. Rev. A 42, 50495051.
Stavans, J. & Glazier J. A. 1989 Soap froth revisited: dynamic scaling in the two-dimensional froth. Phys. Rev. Lett. 62, 13181321.
Sullivan J. M. 1991 Generating and rendering four-dimensional polytopes. Mathematica J. 1 (3), 76.
Weaire D. 1974 Some remarks on the arrangement of grains in a polycrystal. Metallography 7, 157160.
Weaire, D. & Kermode J. P. 1983a The evolution of the structure of a two-dimensional soap froth Phil. Mag. B 47, L29L31.
Weaire, D. & Kermode J. P. 1983b Computer simulation of two-dimensional soap froth I. Method and motivation Phil. Mag. B 48, 245259.
Weaire, D. & Kermode J. P. 1984 Computer simulation of two-dimensional soap froth II. Analysis of results Phil. Mag. B 50, 379395.
Weaire D., Kermode, J. P. & Wejchert J. 1986 On the distribution of cell areas in a Voronoi network Phil. Mag. B 53, L101L105.
Weaire, D. & Lei H. 1990 A note on the statistics of the mature two-dimensional soap froth. Phil. Mag. Lett. 62, 427430.
Weaire, D. & Rivier N. 1984 Soap, cells and statistics – Random patterns in two dimensions. Contemp. Phys. 25, 5999.
Wejchert J., Weaire, D. & Kermode J. P. 1986 Monte Carlo simulation of the evolution of a two-dimensional soap froth Phil. Mag. B 53, 1524.
MathJax is a JavaScript display engine for mathematics. For more information see

Numerical experiments on two-dimensional foam

  • Thomas Herdtle (a1) and Hassan Aref (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed