2D Fluid Foams

A ``fluid foam'' or ``cellular fluid'' is a material that consists of a collection of cells surrounded by a continuous phase of edges tending to minimize its surface energy. This definition covers a class of systems as diverse as soap foams, emulsions, magnetic garnets, and even grain boundaries in crystals.

The cellular structure of 2D fluid foams is similar to biological tissues. When cells migrate in aggregates, they move from one closely packed configuration to another. When subject to shear, bubbles in a foam rearrange from one metastable configuration to another. One question we'd like to address is will study of these cellular materials help understand biological cells?

For mathematicians, foams may provide an insight to the classic ``isoperimetric problem'': how to determine the minimal perimeter enclosing a cluster of $N$ bubbles with known areas. This problem has attracted much attention recently when Hales proved a two-thousand years old honeycomb conjecture a cluster of 2D bubbles of same area reaches its minimum perimeter when all bubbles are regular hexagons. The conjecture is only true if the cluster has no boundaries, i.e. the bubble cluster is either infinite or has periodic boundary conditions. Besides this result, only the case N=2 ( the double-bubble problem) has been well studied, N=3 (the triple-bubble problem) and larger have been partly studied. We use a physics insight to find the perimeter ``minimizer'' in cases that have thus far out of the reach of rigorous study, including large N, real boundary conditions or area dispersity. We estimate the value of perimeter minimum, and conjecture the corresponding patterns to be candidates of the minimizer. We hope to provide an insight for future rigorous mathematical proofs.

For physicists, foams are an excellent model for studying a class of materials that share a common essence due to the cellular structure and surface energy minimization. The coarsening of foams, gas diffusion across films due to pressure differences resulting in growth of some bubbles and shrinkage and disappearance of others, has been used as a simple model for metallic grain growth since 1950s. That coarsening reaches a scaling state and the properties of the scaling state are also interesting topics related to statistical physics. Drainage of fluid out of foam through the complicated films and plateau borders is of great interest to chemical engineering studies of foam stability.

In their various industrial applications, ranging from food and shaving cream to fire fighting and oil recovery, the most remarkable and technologically relevant features of foams is the range of mechanical properties--from elastic solid to viscous fluid. Foam rheology displays multiple length scales with competing time scales, history dependence and avalanche-like dynamics.

Publications:

• Foam structure:

  1. Foams Out of Stable Equilibrium: Cell Elongation and Side Swapping , Phil. Mag. B 79, 729 (1999).
  2. Equilibrium states and ground state of 2D fluid foams , Phys. Rev. E, 6301, 1402 (2001).
  3. Energy landscape of 2D fluid foams, in "Foams, Emulsions and Their Applications" P. Zitha, J. Banhart and G. Verbist editors (Verlag MIT Publishing, Bremen, Germany, 2000), 321-327.
  4. Equilibrium states and ground state of 2D fluid foams, Phys. Rev. E, 6301, 1402 (2001).

• Foam coarsening:

  1. Grain Growth From Homogeneous Initial Conditions: Anomalous Grain Growth and Special Scaling States, Phys. Rev. E Rapide Comm. 52, R3333 (1995).

• Foam drainage:

  1. Extended Large-Q Potts Model Simulation of Foam Drainage, Phil. Mag. Lett. 74, 119 (1996).
  2. Foam Drainage: Extended Large-Q Potts Model Simulation and a Mean Field Theory, Proceedings of MRS Fall Meeting'96 463, 307 (1997).

• Foam rheology:

  1. Hysteresis and Avalanches in Two Dimensional Foam Rheology Simulations, Phys. Rev. E 59, 5819 (1999).
  2. Ab Initio derivation of mesoscopic stress and stain in foams, in "Foams, Emulsions and Their Applications", P. Zitha, J. Banhart and G. Verbist editors (Verlag MIT Publishing, Bremen, Germany, 2000), 297-304.
  3. A texture tensor to quantify deformations, Granular Matter 5, 64 (2003).
  4. A texture tensor to quantify deformations: the example of two-dimensional flowing foams, Granular Matter 5, 71 (2003).
  5. Yield drag in a two-dimensional foam flow around a circular obstacle: the role of fluid fraction, European Physical Journal E, 23,217-228 (2007).

Some fun pictures:

(colors correspond to bubble topology--number of neighbors. color changes when topological rearrangements occur)

quadrupole

split into a pair of dipoles under shear and relaxed

vortex (?) under shear

Movies:

2D fluid foam under shear

• an ordered foam under boundary shear
• an ordered foam under small bulk shear
• an ordered foam under large bulk shear
• a disordered foam under bulk shear
• a disordered foam under boundary shear

  Foam flow

• foam flow around a large obstacle
• foam flow around a small obstacle

• flowx
• flowy

• a real foam