By Dublin artist D. Boran. 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.
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By Dublin artist D. Boran
Fluid Foam Physics
Dry 2D foam
Minimisation of interfacial energy
Wet 2D foam (“bubbly liquid”)
Foams in FLATLAND
LIQUID FRACTION = liquid area / total area
- Surface tension
R –radius of curvature
Note: careful with units! For example in real 2D, is a force, p is force per length etc..
Equilibrium (as always 2 points of view possible):
1. Forces must balance
2. Energy is minimal
(under volume constraint)
2D Soap films are always arcs of circles!
How do SEVERAL films stick together?
THE STEINER PROBLEM
4-fould vertices are never stable in dry foams!
Human beings make “soap film” footpaths
Edges are arcs of circles whose radius of curvature r is determined by the pressure difference across the edge
SUMMARY:Rules of equilibriumin 2D
films are arcs of circles three-fold vertices make angles of 120°
J. F. Plateau
LOCAL structure « easy » – but GLOBAL structure ?
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?
Is this foam optimal?
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
Vaz et al, Journal of Physics-Condensed Matter, 2004
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…)
Answer to: How partition the 2D space into equal-sized cells with minimal perimeter?
However: difficult to realise experimentally on large scale - defaults
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!!!
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
Rotational symmetryf(z) ~ z 1/(1-)A(r) ~ r2
A. > 0, = -1
B. < 0, = 2/3
Sunflower (Y. Couder)
repelling drops of ferrofluid (Douady)
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
V – number of vertices
E – number of edges
C – number of cells
- Integer depends on geometry of surface covered
Two bubbles share one edge
n – number of edges = number of neighbours
The 5-7 defect
[F. Graner, M. Asipauskas]
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
A = 5, B = 8
in polydisperse foam
Foams behave just like French administrative divisions...
curvature = 1/radius of curvature
Make a tour around a vertex and apply Laplace law across each film:
Curvature sum rule
Small curvature approx.
Make a tour around a bubble
For the overall foam (infinitely large)
<n> = 6 or all edges are counted twice with opposite curvature
example: regular bubbles
Constant curvature bubbles
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)
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
Slightly wet foams up to 10 % liquid fraction
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:
normally pg – pl << p11-p22
therefore r << R and one can assume r = const.
Theory fails in 3D!
Weaire, D. Phil. Mag. Lett. 1999
Wet foams find more easily a good structure
K. Brakke, Coll. Surf. A, 2005
Experimental realisation of 2D foams
Plate-Plate (« Hele-Shaw »)
Plate-Pool(« Lisbon »)
S. Cox, E. Janiaud, Phil. Mag. Lett, 2008
ATTENTION when taking and analysing pictures
Base of overhead projector
Example: kissing bubbles
van der Net et al. Coll and Interfaces A, 2006
Similar systems (Structure and Coarsening)
Corals in Brest
Ice under crossed polarisers (grain growth)
Monolayers of Emulsions
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