Statistical mechanics of two-dimensional shuffled foams: prediction of the correlation between geometry and topology
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| Description: |
Durand, M (Université Paris Diderot)
Tuesday 25 February 2014, 11:45-12:05 |
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| Created: | 2014-02-28 17:35 |
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| Collection: | Foams and Minimal Surfaces |
| Publisher: | Isaac Newton Institute |
| Copyright: | Durand, M |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| 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: nˉ(A)=3(1+A−−√⟨A−−√⟩), where ⟨.⟩ denotes the average over all bubbles in the foam. The model also relates the \textit{topological disorder} Δn/⟨n⟩=⟨n2⟩−⟨n⟩2−−−−−−−−−√/⟨n⟩ to the (known) moments of the size distribution: (Δn⟨n⟩)2=14(⟨A1/2⟩⟨A−1/2⟩+⟨A⟩⟨A1/2⟩−2−2). 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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