1. By Dublin artist D. Boran

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

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

4. 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

5. 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!

6. 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

7. 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

8. LOCAL structure « easy » – but GLOBAL structure ?
Surface Evolver
How to stick MANY bubbles together?

9. General: Foam minimises internal (interfacial) Energy U and maximises entropy E – minimises FREE ENERGY F
How to get there?
The T1

10. 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))

11. Exception 1: Small Clusters+
just =
Vaz et al, Journal of Physics-Condensed Matter, 2004

12. Buckling instability
Cox et al, EPJ E, 2003

13. 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

14. 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., : p
But, bees make wet foams sais Weaire!
Phelan, Weaire, Nature 1994

15. Conformal transformationz 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!!!

16. 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

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

18. 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)

19. 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

20. The 5-7 defect

21. 5-sided cell
7-sided cell
8-sided cell
[F. Graner, M. Asipauskas]

22. Statistics:
Measure of Polydispersity (Standard Deviation of bubble area A)
Measure of Disorder (Standard Deviation of number of edges n)

23. 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?

24. Foams behave just like French administrative divisions...
Schliecker 2003

25. curvature = 1/radius of curvature
Make a tour around a vertex and apply Laplace law across each film:
Curvature sum rule
Original paper?

26. 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

27. 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

28. 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.

29. 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

30. 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

31. 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

32. Wet foams?
liquid

33. 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

34. Example:
Dry
Wet

35. Wet foams find more easily a good structure
Energy
Liquid Fraction

36. unstable
Steiner Problem
K. Brakke, Coll. Surf. A, 2005

37. Experimental realisation of 2D foams
Plate-Plate (« Hele-Shaw »)
Plate-Pool (« Lisbon »)
Free Surface
S. Cox, E. Janiaud, Phil. Mag. Lett, 2008

38. ATTENTION when taking and analysing pictures
Digital camera
Sample
Light diffuser
Base of overhead projector

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

40. 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 (Superconducting [cell walls] vs. normal phase)
Ferrofluid « foam » (emulsion), no surfactants! E. Janiaud