# FMS

## Seminar

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

Durand, M (Université Paris Diderot)
Tuesday 25 February 2014, 11:45-12:05

Seminar Room 1, Newton Institute

#### Abstract

Co-authors: S. Ataei Talebi (Université Grenoble 1), S. Cox (Aberystwyth University), F. Graner (Université Paris Diderot), J. Käfer (Université Lyon 1), C. Quilliet (Université 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 \textit{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: $$\bar{n}(A) = 3\left(1+\dfrac{\sqrt{A}}{\langle \sqrt{A} \rangle} \right),$$ where $\langle . \rangle$ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} $\Delta n / \langle n \rangle =\sqrt{\langle n^2 \rangle - \langle n \rangle^2}/\langle n \rangle$ to the (known) moments of the size distribution: $$\left(\dfrac{\Delta n}{\langle n \rangle}\right)^2=\frac{ 1 }{4}\left(\langle A^{1/2} \rangle \langle A^{-1/2} \rangle+\langle A \rangle \langle A^{1/2} \rangle^{-2} -2 \right).$$ 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.

#### Video

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