by dublin artist d. boran. structurecoarsening drainage rheology fluid foam physics physico- chemie
TRANSCRIPT
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 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
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
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
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)(
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
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
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