Transcript The Physics of Foams - University of Groningen

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The Physics of Foams
Simon 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
(200 years old)
Mean curvature C of each film is balanced
by the pressure difference across it:
 C  p
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
Plateau
For each film, calculate shape that gives surface of zero mean curvature.
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
(crystals)
Cyril Stanley Smith
(grain growth)
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
(reduces perimeter)
• T2: bubble disappearance
Describing 2D foam structure
• Euler’s Law:
n 6
• Second moment of number of edges per bubble:
2 (n)   p(n)(n  6)
n
2
Describing foam structure
Aboav-Weaire Law:
6a   2 ( n)
m( n )  6  a 
n
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?
2D space-filling structure
Honeycomb conjecture
Hales
Fejes-Toth
Finite 2D clusters
Find minimal energy cluster for N bubbles.
Proofs for N=2 and 3.
Morgan et al.
Wichiramala
How many possibilities are there for each N?
Work with Graner (Grenoble) and Vaz (Lisbon)
Candidates for N=4 to 23, coloured by topological charge
200 bubbles
Honeycomb structure in bulk;
what shape should surface take?
Tarnai (Budapest)
Lotus flowers
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
Conformal map f(z)
preserves angles (120º)
Drenckhan et al. (2004) , Eur. J. Phys.
Bilinear maps preserve
arcs of circles
Equilibrium foam structure mapped onto equilibrium foam structure
f(z) ~ e z
Logarithmic spiral
Gravity’s Rainbow
Drenckhan et al. (2004) , Eur. J. Phys.
Setup
Theoretical prediction
Experimental result
translational symmetry
w = (ia)-1log(iaz)
rotational symmetry
w ~ z1/(1-)
Ordered Foams in 3D
ratio:
bubble diameter / tube diameter
gas - pressure; nozzle diameter
(Elias, Hutzler, Drenckhan)
Description of 3D bulk structure
• Topological changes similar, but more possibilities.
12
2
 2
 13.39
• F 
1
6 n
3 cos (1 / 3)  
(Euler, Coxeter, Kusner)
restricts possible regular structures.
• Second moment:
2 ( F )  F
2
 F
3
2
R

R
/
R
• Sauter mean radius: 32
• Aboav-Weaire Law
2
(polydisperse)
3D space-filling structure
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)
Kelvin’s Bedspring (tetrakaidecahedron)
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
nergy
Matzke
Quasi-crystals?
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.
27 bubbles
surround one
other
Finite 3D clusters
DWT
Central bubble
from 123 bubble cluster
Coarsening
Drainage
Rheology
Graner, Cloetens (Grenoble)
Dynamics
Coarsening
Gas diffuses across soap films
due to pressure differences
between bubbles.
Von Neumann’s Law - rate of
change of area due to gas diffusion
depends only upon number of sides:
dA
 k (n  6)
dt
Only in 2D. Also applies to grain growth.
T1 s and T2 s
Coarsening
In 3D,


d 2/3
V
 G( F ) ?
dt
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.
J.A. Glazier (Indiana)
2D contraction flow
Experiment by G. Debregeas (Paris), PRL ‘01
Couette Shear (Experiment)
Shear banding? Localization?
Much faster
than real-time.
cf Lauridsen et al. PRL 2002
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.