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By Dublin artist D. Boran

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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

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

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:

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