By Dublin artist D. Boran

Fluid Foam Physics PHYSICO-CHEMIE STRUCTURE COARSENING RHEOLOGY DRAINAGE Fluid Foam Physics

Plan 2 Tutorials Flatland: the structure of 2D foams The real world: the structure of 3D foams

Wet 2D foam (“bubbly liquid”) Computer simulations Minimisation of interfacial energy Foams in FLATLAND Dry 2D foam Wet 2D foam (“bubbly liquid”) Plateau rules for 2D foams  ~30 % LIQUID FRACTION  = liquid area / total area

ONE film Equilibrium (as always 2 points of view possible): - pressure - Surface tension Film Length L 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? THE STEINER PROBLEM 120o p 4-fould vertices are never stable in dry foams! Human beings make “soap film” footpaths

SUMMARY: Rules of equilibrium in 2D J. F. Plateau SUMMARY: Rules of equilibrium in 2D Plateau (1873):  films are arcs of circles  three-fold vertices make angles of 120° Laplace: Edges are arcs of circles whose radius of curvature r is determined by the pressure difference across the edge 120° Principle rules of 3D foams very similar

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

Is this foam optimal? E Energy « Structure » 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))

Exception 1: Small Clusters + just = 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

SIDE TRACK: On the intelligence of bees Toth Structure, 1964 part of a Kelvin cell (see later) - 0.4 % Energy Toth, F., What the bees know and what they do not know. Bull. Am. Math. Soc., 1964. 70: p. 468-81. But, bees make wet foams sais Weaire! Phelan, Weaire, Nature 1994

Conformal transformation z w f(z) “holomorphic” function maintains the angles (Plateau’s laws) f(z) “bilinear” function: arcs of circles are mapped onto arcs of circles (Young-Laplace law) Equilibrium foam structure mapped onto equilibrium foam strucure!!!

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) Translational symmetry w = (ia)-1log(iaz) A(v) ~ f’(z)~ e2av Setup: inclined glass plates GRAVITY’S RAINBOW Experimental result

Rotational symmetry f(z) ~ z 1/(1-) A(r) ~ r 2 B.  < 0,  = 2/3

PHYLLOTAXIS 3 logarithmic spirals spiral galaxy foetus shell Sunflower (Y. Couder) peacock repelling drops of ferrofluid (Douady) f(z) ~ e z 3 logarithmic spirals Number of each spiral type that cover the plane -> [i j k] consecutive numbers of FIBONACCI SEQUENCE spiral galaxy foetus shell Emulsion (E. Weeks)

EULER’S LAW 2D foam: (Plateau) V – number of vertices E – number of edges C – number of cells EULER’S LAW  - Integer depends on geometry of surface covered Infinite Eukledian space Sphere, rugby ball Torus, Doghnut 2D foam: (Plateau) Two bubbles share one edge n – number of edges = number of neighbours

The 5-7 defect

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: Aboav Law Aboav-Weaire law 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?

Geometric charge Topological charge Make a tour around a bubble 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

Consequences: example: regular bubbles 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

Feltham (Bubble perimeter) 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 : 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!!! 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 Digital camera Sample Light diffuser Base of overhead projector

Example: kissing bubbles Experiment Simulation van der Net et al. Coll and Interfaces A, 2006

Langmuir-Blodget Films 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 Suprafroth Prozorov, Fidler 2008 (Superconducting [cell walls] vs. normal phase) Ferrofluid « foam » (emulsion), no surfactants! E. Janiaud