The Physics of Foams
Simon
Image by M. Boran (Dublin)
Cox
Outline
• Foam structure – rules and description
• Dynamics
Prototypes for many other systems:
metallic grain growth, biological organisms, crystal structure, emulsions,
…
Motivation
Many applications of industrial importance:
•Oil recovery
•Fire-fighting
•Ore separation
•Industrial cleaning
•Vehicle manufacture
•Food products
Dynamic phenomena in Foams
Must first understand the foam’s structure
What is a foam?
• Depends on the length-scale:
• Depends on the liquid content:
hard-spheres, tiling of space, …
How are foams made?
from Weaire & Hutzler, The Physics of Foams (Oxford)
Single bubble
Soap film minimizes its energy = surface area
Least area way to enclose a given volume is a sphere.
Isoperimetric problem
(known to Greeks, proven in 19th century)
Laplace-Young Law
p C
(200 years old)
Mean curvature C of each film is balanced by the pressure difference across it:
Coefficient of proportionality is the surface tension
Soap films have constant mean curvature
Plateau’s Rules
Minimization of area gives geometrical constraints (“observation” = Plateau, proof = Taylor):
• Three (and only three) films meet, at 120°, in a Plateau border
•Plateau borders always meet symmetrically in fours (Maraldi angle).
Tetrahedral and Cubic Frames
For each film, calculate shape that gives surface of zero mean curvature.
Plateau
Bubbles in wire frames
D’Arcy Thompson
Ken Brakke’s Surface Evolver
“The Surface Evolver is software expressly designed for the modeling of soap bubbles, foams, and other liquid surfaces
shaped by minimizing energy subject to various constraints …”
http://www.susqu.edu/brakke/evolver/
“Two-dimensional”
Foams
Lawrence Bragg Cyril Stanley Smith
(crystals) (grain growth)
Easily observable
Plateau & Laplace-Young: in equilibrium, each film is a circular
arc; they meet three-fold at 120°.
Energy proportional to perimeter
Topological changes
• T1: neighbour swapping
• T2: bubble disappearance
(reduces perimeter)
Describing 2D foam structure
• Euler’s Law:
• Second moment of number of edges per bubble:
n
n n
p
n
22
( ) ( ) ( 6 )
6
n
Describing foam structure
Aboav-Weaire Law:
where m(n) is the average number of sides of cells with n-sided neighbours.
Applied (successfully) to many natural and artificial cellular structures.
What is a?
n
n a a
n
m 6 ( )
6 )
(
22D space-filling structure
Honeycomb conjecture Hales
Fejes-Toth
Finite 2D clusters
Find minimal energy cluster for N bubbles.
Proofs for N=2 and 3.
How many possibilities are there for each N?
Morgan et al. Wichiramala
Candidates for N=4 to 23, coloured by topological charge
Work with Graner (Grenoble) and Vaz (Lisbon)
200 bubbles
Honeycomb structure in bulk;
what shape should surface take?
Lotus flowers
Tarnai (Budapest)
Seed heads represented by perimeter minima for bubbles inside a circular constraint?
Also work on fly eyes (Carthew) and sea urchins (Raup, D’Arcy Thompson)
Conformal Foams
Drenckhan et al. (2004) , Eur. J. Phys.f(z) ~ e z
Conformal map f(z) preserves angles (120º)
Bilinear maps preserve arcs of circles
Equilibrium foam structure mapped onto equilibrium foam structure
Logarithmic spiral
Experimental result
Gravity’s Rainbow
Setup
w = (i)-1log(iz) w ~ z1/(1-)
Theoretical prediction
Drenckhan et al. (2004) , Eur. J. Phys.
translational symmetry rotational symmetry
Ordered Foams in 3D
gas - pressure; nozzle diameter
ratio: bubble diameter / tube diameter
(Elias, Hutzler, Drenckhan)
Description of 3D bulk structure
• Topological changes similar, but more possibilities.
•
restricts possible regular structures.
• Second moment:
• Sauter mean radius: (polydisperse)
• Aboav-Weaire Law
39 . ) 13
3 / 1 ( cos 3
2 2 6
12
1
F n
(Euler, Coxeter, Kusner)
2 2
2
( F ) F F
2 3
32 R / R
R
3D space-filling structure
Kelvin’s Bedspring (tetrakaidecahedron)
Polyhedral cells with curved faces packed together to fill space.
What’s the best arrangement? (Kelvin problem)
Euler & Plateau: need structure with average of 13.39 faces and 5.1 edges per face
14 “delicately curved” faces (6 squares, 8 hexagons)
<E>=5.14 See Weaire (ed), The Kelvin Problem (1994)
Weaire-Phelan structure
Kelvin’s candidate structure reigned for 100 years
WP is based on A15 TCP structure/ β-tungsten clathrate
<F>=13.5, <E>=5.111 0.3% lower in surface area
2 pentagonal dodecahedra 6 Goldberg 14-hedra
Swimming pool for 2008 Beijing Olympics (ARUP) Surface Evolver
3D Monodisperse Foams
Quasi-crystals?
Matzke
nergy
Finite 3D clusters
J.M.Sullivan (Berlin)
Find minimal energy cluster for N bubbles.
Must eliminate strange possibilities:
Proof that “obvious” answer is the right one for N=2 bubbles in 3D, but for no greater N.
Finite 3D clusters
DWT
Central bubble from 123 bubble cluster
27 bubbles surround one other
Dynamics
Coarsening Drainage Rheology
Graner, Cloetens (Grenoble)
Coarsening
Von Neumann’s Law - rate of
change of area due to gas diffusion depends only upon number of sides:
) 6 d (
d n k t
A
Gas diffuses across soap films due to pressure differences
between bubbles.
Only in 2D. Also applies to grain growth. T1 s and T2 s
Coarsening
In 3D,
V
2/3G ( F ) ?
dt
d
Stationary bubble has 13.39 faces
Foam Rheology
• Elastic solids at low strain
• Behave as plastic solids as strain increases
• Liquid-like at very high strain
Exploit bubble-scale structure (Plateau’s laws) to predict and model the rheological response of foams.
Energy dissipated through topological changes (even in limit of zero shear-rate).
Properties scale with average bubble area.
2D contraction flow
J.A. Glazier (Indiana)
Shear banding? Localization? cf Lauridsen et al. PRL 2002
Couette Shear (Experiment)
Experiment by G. Debregeas (Paris), PRL ‘01
Much faster than real-time.
Couette Shear Simulations
Quasistatic: Include viscous drag on bounding plates:
Outlook
This apparently complex two-phase material has a well-defined local structure.
This structure allows progress in predicting the dynamic properties of foams
The Voronoi construction provides a useful starting condition (e.g. for simulations and special cases) but neglects the all-important curvature.
