PHYSICO-CHEMIE STRUCTURE COARSENING RHEOLOGY DRAINAGE Fluid Foam Physics
Computer simulations Dry 2D foam Minimisation of interfacial energy Wet 2D foam (“bubbly liquid”) Foams in FLATLAND ~30 % LIQUID FRACTION = liquid area / total area
ONE film - Surface tension P2 - pressure Film LengthL P1 R –radius of curvature Note: careful with units! For example in real 2D, is a force, p is force per length etc.. Line tension Line energy Equilibrium (as always 2 points of view possible): 1. Forces must balance or 2. Energy is minimal (under volume constraint) Laplace law 2D Soap films are always arcs of circles!
How do SEVERAL films stick together? p p p 120o p THE STEINER PROBLEM 4-fould vertices are never stable in dry foams! Human beings make “soap film” footpaths
Laplace: Edges are arcs of circles whose radius of curvature r is determined by the pressure difference across the edge SUMMARY:Rules of equilibriumin 2D Plateau (1873): films are arcs of circles three-fold vertices make angles of 120° J. F. Plateau 120°
LOCAL structure « easy » – but GLOBAL structure ? Surface Evolver How to stick MANY bubbles together?
General: Foam minimises internal (interfacial) Energy U and maximises entropy E – minimises FREE ENERGY F How to get there? The T1
Energy Is this foam optimal? E Energy Problem: Large energy barriers E. Temperature cannot provide sufficient energy fluctuations. Need other means of « annealing » (coarsening, rheology, wet foams…) « Structure » Foam structures generally only « locally ideal »(in fact, generally it is impossible to determine the global energy minimum (too complex))
just + Exception 1: Small Clusters = Vaz et al, Journal of Physics-Condensed Matter, 2004
Buckling instability Cox et al, EPJ E, 2003
Exception 2: Periodic structures Final proof of the Honeycomb conjecture: 1999 by HALES (in only 6 months and on only 20 pages…) (S. Hutzler) Answer to: How partition the 2D space into equal-sized cells with minimal perimeter? However: difficult to realise experimentally on large scale - defaults
w z maintains the angles(Plateau’s laws) f(z) “holomorphic” function arcs of circles are mapped onto arcs of circles(Young-Laplace law) f(z) “bilinear” function: Equilibrium foam structure mapped onto equilibrium foam strucure!!! Conformal transformation
Translational symmetry w = (ia)-1log(iaz) A(v) ~ f’(z)~ e2av Drenckhan et al. (2004) , Eur. J. Phys. 25, pp 429 – 438; Mancini, Oguey (2006) Setup: inclined glass plates Experimental result GRAVITY’S RAINBOW
Rotational symmetryf(z) ~ z 1/(1-)A(r) ~ r2 A. > 0, = -1 B. < 0, = 2/3
Sunflower (Y. Couder) repelling drops of ferrofluid (Douady) peacock spiral galaxy foetus shell PHYLLOTAXIS 3 logarithmic spirals f(z) ~ ez Number of each spiral type that cover the plane -> [i j k] consecutive numbers of FIBONACCI SEQUENCE Emulsion (E. Weeks)
Infinite Eukledian space Sphere, rugby ball Torus, Doghnut V – number of vertices E – number of edges C – number of cells EULER’S LAW - Integer depends on geometry of surface covered 2D foam:(Plateau) Two bubbles share one edge n – number of edges = number of neighbours
5-sided cell 7-sided cell 8-sided cell [F. Graner, M. Asipauskas]
Statistics: Measure of Polydispersity (Standard Deviation of bubble area A) Measure of Disorder (Standard Deviation of number of edges n)
some more Statistics: Corellations in n: m(n) – average number of sides of cells which are neighbours of n-sided cells Aboav Law A = 5, B = 8 Aboav-Weaire law in polydisperse foam original papers?
Foams behave just like French administrative divisions... Schliecker 2003
curvature = 1/radius of curvature Make a tour around a vertex and apply Laplace law across each film: Curvature sum rule Original paper?
i Small curvature approx. Make a tour around a bubble Geometric charge For the overall foam (infinitely large) Topological charge <n> = 6 or all edges are counted twice with opposite curvature
example: regular bubbles • Consequences: • n > 6 curved inwards (on average) • n < 6 curved outwards (on average) • if all edges are straight it must be a hexagon!!! curved outwards straight edges curved inwards Constant curvature bubbles n
n A Feltham(Bubble perimeter) L(n) ~ n + no Lewis law(Bubble area) n - 6 A(n) ~ n + no Marchalot et al, EPL 2008 F.T. Lewis, Anat. Records 38, 341 (1928); 50, 235 (1931). F.T. Lewis formulated this law in 1928 whilestudying the skin of a cucumber.
Efficiency parameter : n Interfacial Energy of foam almost independant of topology (Graner et al., Phys. Rev. E, 2000) n Efficiency parameter : Ratio of Linelength of cell to linelength of cell was circular P - Linelength
General foam structures can be well approximated by regular foam bubbles!!! n Regular foam bubbles e(2) ~ 3.78 increases monotonically to e(infinity) ~ 3.71 Total line length of 2D foam i – number of bubbles Shown that this holds by Vaz et al, Phil. Mag. Lett., 2002
Summary dry foam structures in 2D • Films are arcs of circles (Laplace) • Three films meet three-fold in a vertex at 120 degrees (Plateau) • Average number of neighbours • Curvature sum rule • Geometric charge • Aboav-Weaire Law
Wet foams? liquid
Slightly wet foams up to 10 % liquid fraction Decoration Theorem r To obtain the wet foam structure: Take foam structure of an infinitely dry foam and « decorate » its vertices Radius of curvature of gas/liquid interface given by Laplace law: R normally pg – pl << p11-p22 therefore r << R and one can assume r = const. r Theory fails in 3D! Weaire, D. Phil. Mag. Lett. 1999
Example: Dry Wet
Wet foams find more easily a good structure Energy Liquid Fraction
unstable Steiner Problem K. Brakke, Coll. Surf. A, 2005
Experimental realisation of 2D foams Plate-Plate (« Hele-Shaw ») Plate-Pool(« Lisbon ») Free Surface S. Cox, E. Janiaud, Phil. Mag. Lett, 2008
ATTENTION when taking and analysing pictures Digitalcamera Sample Lightdiffuser Base of overhead projector
Example: kissing bubbles Experiment Simulation van der Net et al. Coll and Interfaces A, 2006
Similar systems (Structure and Coarsening) Corals in Brest Langmuir-Blodget Films Ice under crossed polarisers (grain growth) Monolayers of Emulsions Myriam Tissue Magnetic Garnett Films(Bubble Memory), Iglesias et al, Phys. Rev. B, 2002 SuprafrothProzorov, Fidler 2008 (Superconducting [cell walls] vs. normal phase) Ferrofluid « foam »(emulsion), no surfactants! E. Janiaud