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Foams and Emulsions Ref. [10].
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Foams and Emulsions [Ref. [10].
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25
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33744856279
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note
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eff, the geometry of the pattern remains the same.
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26
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F. Elias, C. Flament, J.-C. Bacri, O. Cardoso, and F. Graner, Phys. Rev. E 56, 3310 (1997).
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29
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33744847084
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note
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We use the edge length L(A) of a hexagon as a mean-field approximation. L(A) is independent of the bubble topology and thus constant through a Tl and easy to measure on a picture. We are currently trying to relax ;his approximation to treat both area and topulogical disorder: as discussed in the Appendix, an n-sided bubble has a "reference length" L(A,n) = e(n)√A/n≈3.72√A/n.
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30
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0003745617
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Springer, Berlin
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See, e.g., M. A. Herman and H. Sitter, Molecular Beam Epitaxy, Vol. 7 of Springer Series in Material Science, 2nd ed. (Springer, Berlin, 1996).
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Herman, M.A.1
Sitter, H.2
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31
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33744842791
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note
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If the area disorder were large, the topological disorder would be large, too: e.g., a binary area distribution, where tiny bubbles decorate the vertices of the large-bubble honeycomb lattice; Fig. 2(a) of Ref. [2].
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32
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0032676895
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F. Elias, J.-C. Bacri, F.-rl. de Mouglns, and T. Spengler, Philos. Mag. Lett. 79, 389 (1999).
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Spengler, T.4
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35
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33744869856
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note
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a is roughly constant if the edges remain nearly straight.
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37
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33744840241
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This is usually derived from the Euler theorem [12,14], applied here to a non-simply-convex foam (e.g., on a torus).
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This is usually derived from the Euler theorem [12,14], applied here to a non-simply-convex foam (e.g., on a torus).
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38
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0000195664
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note
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D}. Using the results of J. Avron and D. Levine, Phys. Rev. Lett. 69, 208 (1992), one finds that q is the sum of the Gaussian curvature of the face and the curvature of me edge.
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-
-
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39
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0004231024
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American Society for Metals, Cleveland, quoted, e.g., in [12,13,15].
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J. von Neumann, in Metal Interfaces (American Society for Metals, Cleveland, 1952), pp. 108-110. quoted, e.g., in [12,13,15].
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Von Neumann, J.1
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33744887664
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h=1.
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h=1.
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46
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33744862483
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to be published.
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G. Hruska, D. Leykekhman, D. Pinzon, B. Shoy, ane, J. Foisy, Rocky Mountain J. Math, (to be published).
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Hruska, G.1
Leykekhman, D.2
Pinzon, D.3
Shoy, B.4
Foisy, J.5
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47
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33744856638
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note
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In the rare case where the edge fluid does not wet the box boundaries, e.g., Fig. 1 (b). the edges are tangent to the boundaries. Removing these edges leads to an incomplete foam with edges meeting the boundary perpendicularly.
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48
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0014582879
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with references to P. G. Tait in the 1860s
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S. Ross, Ind. Eng. Chem. 61, 48 (1969) (with references to P. G. Tait in the 1860s);
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Ross, S.1
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49
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33744874812
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note
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i is extremal at λ= 1.
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-
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50
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33744844847
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F. Morgan jpersonal communication.
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F. Morgan jpersonal communication).
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51
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33645051664
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Wiss Springer, Berlin
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L. Fejes Toth, Lagerungen In der Ebene, auf der Kugel und im Raum, Vol. 65 of Die Grundlehren der Math. Wiss (Springer, Berlin, 1972;, p. 84.
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Fejes Toth, L.1
|