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SUMMARY:Statistical mechanics of two-dimensional shuffled foams: predictio
 n of the correlation between geometry and topology - Durand\, M (Universit
  Paris Diderot)
DTSTART:20140225T114500Z
DTEND:20140225T120500Z
UID:TALK51073@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:Co-authors: S. Ataei Talebi (Universit Grenoble 1)\, S. Cox (A
 berystwyth University)\, F. Graner (Universit Paris Diderot)\, J. Kfer (Un
 iversit Lyon 1)\, C. Quilliet (Universit Grenoble 1) \n\nTwo-dimensional f
 oams are characterised by their number of bubbles\, $N_{}$\, area distribu
 tion\, $p(A)$\, 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. Bubb
 le size distribution and packing (or ``topology") are crucial in determini
 ng 	extit{e.g.} rheological properties or coarsening rate. When a foam is 
 shuffled (either mechanically or thermally)\, $N_{}$ and $p(A)$ remain fix
 ed\, but bubbles undergo ``T1'' neighbour changes\, which induce a random 
 exploration of the foam configurations. \n\nWe explore the relation betwee
 n the distributions of bubble number-of-sides (topology) and bubble areas 
 (geometry). We develop a statistical model which takes into account physic
 al 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: $$ar{n}(A) = 3left(1+dfrac{ qrt{A}}
 {langle  qrt{A} \nangle} \night)\,$$ where $langle . \nangle$ denotes the 
 average over all bubbles in the foam. The model also relates the 	extit{to
 pological disorder} $ Delta n / langle n \nangle = qrt{langle n^2 \nangle 
 - langle n \nangle^2}/langle n \nangle$ to the (known) moments of the size
  distribution: $$left(dfrac{Delta n}{langle n \nangle}\night)^2=rac{ 1 }{
 4}left(langle A^{1/2} \nangle langle A^{-1/2} \nangle+langle A \nangle lan
 gle A^{1/2} \nangle^{-2} -2 \night).$$ Extensive data sets arising from ex
 periments and simulations all collapse surprisingly well on a straight lin
 e\, even at extremely high values of geometrical disorder. \n\nAt the othe
 r extreme\, when approaching the perfectly regular honeycomb pattern\, we 
 identify and quantitatively discuss a crystallisation mechanism whereby to
 pological disorder vanishes.\n
LOCATION:Seminar Room 1\, Newton Institute
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