By Dublin artist D. Boran

STRUCTURE COARSENING

DRAINAGERHEOLOGY

Fluid Foam Physics

PHYSICO-CHEMIE

Dry 2D foam

Wet 2D foam (“bubbly liquid”)

Foams in FLATLAND

LIQUID FRACTION = liquid area / total area

~30 %

Computer simulations

Minimisation of interfacial energy

Film Length L

Line tension 2LE 2Line energy

Equilibrium (as always 2 points of view possible):

1. Forces must balance

or

2. Energy is minimal(under volume constraint)

P1

P2

Laplace law

Rppp

221

R – radius of curvature

2D Soap films are always arcs of circles!

ONE film

- Surface tension- pressure

Note: careful with units! For example in real 2D, is a force, p is force per length etc..

120o

Human beings make “soap film” footpaths

THE STEINER PROBLEM

p

p

p

p

How do SEVERAL films stick together?

4-fould vertices are never stable in dry foams!

120°

SUMMARY:Rules of equilibriumin 2D

Plateau (1873):

films are arcs of circles three-fold vertices make angles of 120°

rp

2

Laplace:

Edges are arcs of circles whose radius of curvature r is determined by the pressure difference across the edge

J. F. Plateau

LOCAL structure « easy » – but GLOBAL structure ?

How to stick MANY bubbles together?

Surface Evolver

General: Foam minimises internal (interfacial) Energy U and maximises entropy E – minimises FREE ENERGY F

TSUF

How to get there?

The T1

Foam structures generally only « locally ideal »(in fact, generally it is impossible to determine the global energy minimum (too complex))

Is this foam optimal?

« Structure »

Ene

rgy

En

erg

y

Problem: Large energy barriers E. Temperature cannot provide sufficient energy fluctuations. Need other means of « annealing » (coarsening, rheology, wet foams…)

E

Exception 1: Small Clusters

Vaz et al, Journal of Physics-Condensed Matter, 2004

+

just

=

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)

However: difficult to realise experimentally on large scale - defaults

Answer to: How partition the 2D space into equal-sized cells with minimal perimeter?

Conformal transformationz w

f(z) “holomorphic” function maintains the angles(Plateau’s laws)

f(z) “bilinear” function:

dcba

f

zz

z )(

arcs of circles are mapped onto arcs of circles(Young-Laplace law)

Equilibrium foam structure mapped onto equilibrium foam strucure!!!

Experimental result

Setup: inclined glass plates

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

GRAVITY’S RAINBOW

A. > 0, = -1

B. < 0, = 2/3

Rotational symmetry f(z) ~ z 1/(1-) A(r) ~ r 2

3 logarithmic spirals

PHYLLOTAXIS

spiral galaxyfoetus shell

f(z) ~ e z Sunflower (Y. Couder)

peacock

repelling drops of ferrofluid (Douady)

Emulsion (E. Weeks)

Number of each spiral type that cover the plane -> [i j k] consecutive numbers of FIBONACCI SEQUENCE

CEV

- Integer depends on geometry of surface covered

12

Infinite Eukledian space

Sphere, rugby ball

0 Torus, Doghnut3

2EV 2D foam:

(Plateau)

V – number of vertices

E – number of edges

C – number of cells

EULER’S LAW

CC

E 33

C

2

n

Two bubbles share one edge

06n

n – number of edges = number of neighbours

2

n

C

E

The 5-7 defect

6n

[F. Graner, M. Asipauskas]

5-sided cell

7-sided cell

8-sided cell

Statistics:Measure of Polydispersity (Standard Deviation of bubble area A)

Measure of Disorder (Standard Deviation of number of edges n)

222 nnn

22

2

2

AA

ApAAn

A

some more Statistics:

Corellations in n:m(n) – average number of sides of cells which are neighbours of n-sided cells

n

BAnm )(

n

aanm

n26

6)(

2.1a

Aboav Law

Aboav-Weaire law

original papers?

in polydisperse foam

A = 5, B = 8

Foams behave just like French administrative divisions...

Schliecker 2003

ijij r

1

Make a tour around a vertex and apply Laplace law across each film:

0133221

312312

pppppp

ppp

1p

2p

3p

31r

312312

0312312

Curvature sum rule

curvature = 1/radius of curvature

Original paper?

ijij r

1

Make a tour around a bubble

n

iiiiiln 1,1,3

2 i

3

1, iir

i

1i

2

Small curvature approx.

g

n

iiiii qnl 6

31,1,

Geometric charge

Topological charge

tqn 60

i

igq

For the overall foam (infinitely large)

<n> = 6 or all edges are counted twice with opposite curvature

Consequences:• n > 6 curved inwards (on average)

• n < 6 curved outwards (on average)

• if all edges are straight it must be a hexagon!!!

g

n

iiiii qnl 6

31,1,

curved outwards curved inwardsstraight edges

example: regular bubbles

n

Constant curvature bubbles

n

Lewis law(Bubble area)A(n) ~ n + no

n - 6Marchalot et al, EPL 2008

Feltham(Bubble perimeter)

L(n) ~ n + no

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.

A

n

Interfacial Energy of foam almost independant of topology (Graner et al., Phys. Rev. E, 2000)

PE

21)(A

Pne Ratio of Linelength of cell

to linelength of cell was circular

P - Linelength

54.322

2

R

Re 4

4

L

Le72.3

233

6

2

R

Re

Efficiency parameter :n

Eff

icie

ncy

par

amet

er :

Regular foam bubbles e(2) ~ 3.78 increases monotonically to e(infinity) ~ 3.71

n

iiA

neP 2

1

2

)6(

i – number of bubbles

Total line length of 2D foam

)6(282.0)(3

)6(21

nne

nA

Shown that this holds by Vaz et al, Phil. Mag. Lett., 2002

General foam structures can be well approximated by regular foam bubbles!!!

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

g

n

iiiii qnl 6

31,1,

0312312

6n

n

aanm

n26

6)(

Wet foams?

liquid

Decoration Theorem

Slightly wet foams up to 10 % liquid fraction

Weaire, D. Phil. Mag. Lett. 1999

lp

1gp

rpp lg

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:

Rpp gg

221

normally pg – pl << p11-p2

2

therefore r << R and one can assume r = const.

Theory fails in 3D!

r

2gp

r

Dry Wet

Example:

Wet foams find more easily a good structureEn

erg

yLiq

uid

Fra

cti

on

unstable

K. Brakke, Coll. Surf. A, 2005

Steiner Problem

Experimental realisation of 2D foams

S. Cox, E. Janiaud, Phil. Mag. Lett, 2008

Plate-Plate (« Hele-Shaw »)

Plate-Pool(« Lisbon »)

Free Surface

ATTENTION when taking and analysing pictures

Lightdiffuser

Base of overhead projector

Sample

Digitalcamera

Example: kissing bubbles

Experiment

Simulation

van der Net et al. Coll and Interfaces A, 2006

Similar systems (Structure and Coarsening)

Langmuir-Blodget Films Ice under crossed polarisers (grain

growth)

Myriam

Suprafroth Prozorov, Fidler 2008 (Superconducting [cell walls] vs. normal phase)

Magnetic Garnett Films (Bubble Memory), Iglesias et al, Phys. Rev. B, 2002

Tissue

Ferrofluid « foam » (emulsion), no surfactants! E. Janiaud

Monolayers of Emulsions

Corals in Brest