The Physics of Foams

Simon Cox

Imag

e by

M. B

oran

(D

ublin

)

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

pC

(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 surfacesshaped 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

nnpn 22 )6()()(

6n

Describing foam structureAboav-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

naanm

)(66)( 2

2D space-filling structure

Honeycomb conjecture

Hales

Fejes-Toth

Finite 2D clustersFind 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

Wor

k w

ith G

rane

r (G

reno

ble)

and

Vaz

(L

isbo

n)

200 bubblesHoneycomb structure in bulk;

what shape should surface take?

Lotus flowersT

arna

i (B

udap

est)

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 FoamsDrenckhan 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 RainbowSetup

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(cos3

22

6

121

nF

(Euler, Coxeter, Kusner)

222 )( FFF

2332 / RRR

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 structureKelvin’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 dodecahedra6 Goldberg 14-hedra

Swimming pool for 2008 Beijing Olympics (ARUP) Surface Evolver

3D Monodisperse Foams

Quasi-crystals?

Matzke

nergy

Finite 3D clusters

J.M

.Sul

liva

n (B

erli

n)

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

Gra

ner,

Clo

eten

s (G

reno

ble)

Coarsening

Von Neumann’s Law - rate of change of area due to gas diffusion depends only upon number of sides:

)6(d

d nk

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

CoarseningIn 3D, ?)(3/2 FGV

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 flowJ.

A. G

lazi

er (

Indi

ana)

Shear banding? Localization? cf Lauridsen et al. PRL 2002

Couette Shear (Experiment)

Exp

erim

ent b

y G

. Deb

rege

as (

Pari

s), P

RL

‘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.