Equilibrium states and ground state of two-dimensional fluid foams.

We study the equilibrium energies of two-dimensional (2D) noncoarsening fluid foams, which consist of bubbles with fixed areas. The equilibrium states correspond to local minima of the total perimeter. We present a theoretical derivation of energy minima; experiments with ferrofluid foams, which can be either highly distorted, locally relaxed, or globally annealed; and Monte Carlo simulations using the extended large-Q Potts model. For a dry foam with small size variance we develop physical insight and an electrostatic analogy, which enables us to (i) find an approximate value of the global minimum perimeter, accounting for (small) area disorder, the topological distribution, and physical boundary conditions; (ii) conjecture the corresponding pattern and topology: small bubbles sort inward and large bubbles sort outward, topological charges of the same signs "repel" while charges of the opposite signs "attract;" (iii) define local and global markers to determine directly from an image how far a foam is from its ground state; (iv) conjecture that, in a local perimeter minimum at prescribed topology, the pressure distribution and thus the edge curvature are unique. Some results also apply to 3D foams.

[1]  Lionel G. Harrison,et al.  The shortest enclosure of three connected areas in R , 1994 .

[2]  Frank Morgan,et al.  The Isoperimetric Problem on Surfaces , 1999 .

[3]  J. Taylor,et al.  The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces , 1976 .

[4]  J. Bacri,et al.  TWO-DIMENSIONAL MAGNETIC LIQUID FROTH : COARSENING AND TOPOLOGICAL CORRELATIONS , 1997 .

[5]  Frank Morgan,et al.  Mathematicians, Including Undergraduates, Look at Soap Bubbles , 1994 .

[6]  R. Osserman.  The isoperimetric inequality , 1978 .

[7]  D. Weaire,et al.  Soap, cells and statistics – random patterns in two dimensions , 1984 .

[8]  Frank Morgan,et al.  The Hexagonal Honeycomb Conjecture , 1999 .

[9]  J. Zimba,et al.  The standard double soap bubble in R2 uniquely minimizes perimeter , 1993 .

[10]  Y Jiang,et al.  Hysteresis and avalanches in two-dimensional foam rheology simulations. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  M. Vignes-Adler,et al.  DYNAMICS OF 3D REAL FOAM COARSENING , 1998 .

[12]  H. Aref,et al.  The equation of state of a foam , 2000 .

[13]  J. Glazier,et al.  Foams out of stable equilibrium: Cell elongation and side swapping , 1999 .

[14]  Frank Morgan,et al.  Soap bubbles in ${\bf R}^2$ and in surfaces. , 1994 .

[15]  S. Safran.  Domain Growth of Degenerate Phases , 1981 .

[16]  Glazier,et al.  Grain growth in three dimensions depends on grain topology. , 1993, Physical review letters.

[17]  Aste,et al.  From one cell to the whole froth: A dynamical map. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  G. Grest,et al.  Effects of lattice anisotropy and temperature on domain growth in the two-dimensional Potts model. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[19]  S. Alexander,et al.  Amorphous solids: their structure, lattice dynamics and elasticity , 1998 .

[20]  S. Ross.  BUBBLES AND FOAMS—NEW GENERAL LAW , 1969 .

[21]  James A. Glazier,et al.  The kinetics of cellular patterns , 1992 .

[22]  Frank Morgan.  CHAPTER 1 – Geometric Measure Theory , 1995 .

[23]  J. Stavans,et al.  The evolution of cellular structures , 1993 .