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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


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.


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