# Statistical mechanics of two-dimensional shuffled foams: prediction of the correlation between geometry and topology

• Durand, M (Universit Paris Diderot)
• Tuesday 25 February 2014, 11:45-12:05
• Seminar Room 1, Newton Institute.

Foams and Minimal Surfaces

Co-authors: S. Ataei Talebi (Universit Grenoble 1), S. Cox (Aberystwyth University), F. Graner (Universit Paris Diderot), J. Kfer (Universit Lyon 1), C. Quilliet (Universit Grenoble 1)

Two-dimensional foams are characterised by their number of bubbles, \$N_{}\$, area distribution, \$p(A)\$, and number-of-sides distribution, \$p(n)\$. When the liquid fraction is very low (``dry’’ foams), their bubbles are polygonal, with shapes that are locally governed by the laws of Laplace and Plateau. Bubble size distribution and packing (or ``topology”) are crucial in determining extit{e.g.} rheological properties or coarsening rate. When a foam is shuffled (either mechanically or thermally), \$N_{}\$ and \$p(A)\$ remain fixed, but bubbles undergo ``T1’’ neighbour changes, which induce a random exploration of the foam configurations.

We explore the relation between the distributions of bubble number-of-sides (topology) and bubble areas (geometry). We develop a statistical model which takes into account physical ingredients and space-filling constraintes. The model predicts that the mean number of sides of a bubble with area \$A\$ within a foam sample with moderate size dispersity is given by: \$\$ar{n}(A) = 3left(1+dfrac{ qrt{A}}{langle qrt{A} angle} ight),\$\$ where \$langle . angle\$ denotes the average over all bubbles in the foam. The model also relates the extit{topological disorder} \$ Delta n / langle n angle = qrt{langle n2 angle – langle n angle2}/langle n angle\$ to the (known) moments of the size distribution: \$\$left(dfrac{Delta n}{langle n angle} ight)2= rac{ 1 }{4}left(langle A{1/2} angle langle A angle+langle A angle langle A{1/2} angle^{-2} -2 ight).\$\$ Extensive data sets arising from experiments and simulations all collapse surprisingly well on a straight line, even at extremely high values of geometrical disorder.

At the other extreme, when approaching the perfectly regular honeycomb pattern, we identify and quantitatively discuss a crystallisation mechanism whereby topological disorder vanishes.

This talk is part of the Isaac Newton Institute Seminar Series series.