Analytical Models of Single Bubbles and Foams

Robert Murtagh

Outline of this Presentation1. General introduction to foam What is a foam? Importance of liquid fraction Monodispersity and structure Surface energy and minimisation

2. Z-Cone model for ordered foams Constructing the model How it works Results and comparison with simulation Applications of the model

3. Cone Model for general foams Application of the cone model to the Kelvin foam Contact losses in the Kelvin foam.

4. Conclusions & Outlook

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What is a foam?

• A collection of gas bubbles separated by continuous liquid films.

• A liquid foam is a two-phase system in which a gas is dispersed in a continuous liquid phase.

OR

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Motivation• Structure is concerned with the geometry of soap bubbles

that have been packed together, usually in the bulk of a foam.

Aim: To understand foam structure in static equilibrium, starting from the wet limit, via simple analytical models.

• Bubble assumes lowest possible surface area shape for the volume of gas it contains.

• Using minimal surfaces, we can determine the shape of a bubble, as a function of liquid fraction.

AE

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Importance of liquid fraction• The liquid fraction, ϕ, of a foam is the ratio of the volume of

the liquid phase to the total volume of the foam.

foam

liquid

VV

• High liquid fraction → “Wet” foam

• Most every day foams fall between these limits.

• Extremes are referred to as the dry (0%) and wet limits (approx. 36%).

• Low liquid fraction → “Dry” foam

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The effect of gravityDry foam: Bubbles form polygonal cells governed by Plateau’s Laws

Wet foam: Bubbles are roughly spherical

What about the shape of the bubbles between these two limits?

Top

Bottom

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Structure of monodisperse foam• The term foam structure refers to the particular arrangement

of the bubbles.

• Best known ordered structures arise for monodisperse bubbles.

For many years, the lowest surface area structure for a dry foam was thought to be the Kelvin structure.

In 1994, an even lower surface area to volume ratio structure was discovered by Weaire and Phelan.

Which unit cell, infinitely repeated, partitions space into cells of equal volume such that a minimal amount of surface area separates the cells?

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Surface Evolver• Surface Evolver finds a minimum surface area via

triangulation and minimisation.

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• This method also applies to confined bubbles.

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Structure of monodisperse foam• For very wet foams, the structure resembles a packing of

spheres.• Above critical liquid fraction φc bubbles dissociate from one

another.

φc = 0.26 φc = 0.32

• A random packing of spheres, for example, has φc = 0.36.

• φc is structure dependent.

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Shape of bubbles for intermediate φ?• Bubbles minimise their surface area A subject to their

confinement conditions.

1100

AA

EE

Key Question: How do bubbles interact with each other as they are deformed in a foam?

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How do bubbles interact in a foam?

Morse and Witten, 1993.Lacasse, Grest and Levine, 1996.Durian, 1995.

3D2D & 3D2D

Single bubble deformed under gravity

Single bubble deformed symmetrically between two plates

Soft disk model: harmonic repulsion proportional to overlap

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“Soft” Disk model• Simple dynamic model for

interacting bubbles as overlapping disks.

• Each of the disks experiences a force:

dRR

RkFji

avsd

2

• Model is widely implemented in large-scale sheared foam simulations.

• Can recast as an elastic potential via

2)( ksd

• where ε is excess energy relative to two isolated disks and ξ is a measure of the distance between bubble centres.

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Lacasse et al. in 2D• Harmonicity in 2D was tested

by Lacasse et al.

• Surface Evolver simulations of a single bubble confined and deformed by a number of contacts.

• Contacts are flat regions separated by circular arcs in 2D.

• Fundamentally different from soft disk model due to surface deformability.

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)( 20

R

Perimetersd

0

0

RhR

N.B. Volume of 2D bubble kept constant!

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Morse and Witten: 3D• Situation more complex in

3D.

• Investigated ε for a single bubble pressed against a flat surface by gravity.

• Surface of the bubble is taken to be fully deformable.

)(ln)()( 2 FFMW

• For small contacts, and correspondingly small deformations they found

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Lacasse et al. in 3D

• Investigated response of a bubble in 3D confined by multiple contacts numerically with Surface Evolver.

Z

ZL ZC

1

)1(1)( 2

• Proposed a fit to the data, over a limited range.

Complete analytic solution adduced for two contacts

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How do bubbles interact in a foam?

Morse and Witten, 1993.Lacasse, Grest and Levine, 1996.Durian, 1995.

3D2D & 3D2D

3D: Energy-force relationship for small deformations:

Intermediate deformation: approx. harmonic

Key questions (3D):Energy-compression relationship?Is this relationship linear in Z?Really harmonic in 3D?

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Bubble With 2 ContactsTop down view

Side on view for the case of two contacts

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The “Z-Cone” Approximation

Inspiration:Ziman 1961, Fermi Surface of Copper

2θ

Flat contact

Z21arccos

Surface Evolver FCC (Z = 12): Volume V

Total volume V is redistributed into Z small cones of volume

ZVVcone

Example of one such cone

Area of constant mean curvature

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The “Z-Cone” Approximation

Each of the angular cones is approximated as a circular cone in the model

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How the Z-Cone Model works

)1()( 0 Rhh c

ξ: dimensionless deformation parameter

R0: radius of an undeformed spherical sector (i.e. δ= 0)

Key Concept: Minimal body of revolution bounding a known volume

)0(/ r

Z21arccos

II

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Formally, minimisation of cone surface area A

under constraint of constant volume

using Euler-Lagrange equation

with Lagrange multiplier λ

Mathematics of the model

dzdzdrzrZA h

0

2

2 1)(2/

Cr

rz

z LL

)(1)(2/ 2

22

zrdzdrzr

hZ

L

)tan(3)0()(/

3

0

2

rdzzrZV

h

0atcot

at

zr

hzr

z

z

boundary conditions:

)0(/ r

II

Mathematics of the model)0(/ r Approach:

Minimising the surface area, keeping the volume of the cone constant, using the Euler-Lagrange approach

Dimensionless excess energy

1),(6

12

),()1(1),( 3/2

3/1

22

ZJZZZ

ZKZZ

Z

1)(),(0

A

AZ Elliptic Integrals

Complicated dependence on Z!

Clearly not harmonic!

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One Complication: Elliptic Integrals

The elliptic integrals must be performed numerically.

)0(/ r

Integrals can be used to provide asymptotic analytic expressions for excess energy and deformation in the wet limit.

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Visualising cone deformationII

Excess energy versus deformation

for FCC

There is a very good agreement with Surface Evolver for small deformations.

“Z-Cone Model for the energy of an ordered foam” S. Hutzler, R.P.Murtagh, D.Whyte, S.T.Tobin, D. Weaire (published in Soft Matter 2014).

II

wet limit

Deformation

for Z=12

wet limit

Logarithmic limit

Large deviations from a harmonic potential in both the wet and dry limits

ln2

),(2ZZ

Logarithmic asymptote

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Roughly harmonic for low Z

For intermediate deformation and low Z, there is a region of approximate harmonicity.

Breaks down for larger Z.

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Roughly harmonic for low Z

Guiding line with a slope of 2.2

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Effective Spring Constant

)75.0(21.0 Zkeff

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Energy, Liquid Fraction & Gravity

Dry limit described by ε(φ)=e0 -e1 φ 1/2 )ln(

)()1(18

)(2/1

2

c

c

c

Z

1

43

111

3/1

ZZ

c

c

II

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Beyond the Z-Cone Model

Total of 12 diamond-shaped faces

Two Cone Approach

FCC Kelvin Foam

One Cone Approach

Total of 14 faces; 8 large hexagons and 6 small squares

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Cone Model for Kelvin III

Cone Model for KelvinOne Cone Approach

Two Cone Approach

III

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Key Changes to the Model

)0(/ r

068 VVV sh 0VZVc

Z21arccos

2

1arccos nn

nh

nh

n

n

n

n

22 cot4

sin2arcsin22

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Matching at Cone-Cone Edges

)0(/ r

• The slant height of the two cones rs are equal.

• The ratio of the heights H of the two cones remains in a constant ratio

23864434.0

coscos

s

h

s

h

HH

III

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Matching at Cone-Cone Edges

)0(/ r

• The pressure in each of the cones must be equal.

• The interior gamma angles must sum to a right angle, taking care to account of the different cone types:

232 sh

III

Success of Two Cone Approach

Inset shows relative difference between Evolver result and the cone model

Loss of the square faces occurs somewhat earlier for cone model due to approximations

1468

sh AA

092.0* cone11.0*

III

Contact Losses in Kelvin Foam

• Increasing liquid fraction

III

Contact Loss in Kelvin Foam

092.0* cone

11.0*

III

Contact Loss in Kelvin Foam

The loss of the square faces is associated with a change in the slope of the energy.

Loss of the eight hexagonal faces occurs at the wet limit

1468

sh AA

092.0* cone318.0*, conec

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Contact Loss in Kelvin Foam

)ln(

c

cadd

22

1 ))(ln(

c

bbdd

• Discrepancy suggests that results for bubble-bubble interactions at the wet limit not the same as contact losses away from this limit.

• Investigations into the presence of a pre-emptive instability for this structure is ongoing.

III

Future Work: Curved Contacts

Raa

PRc

1

4

IV

Future Work: Curved Contacts

Z = 2

Large bubbles have a lower excess surface energy than small bubbles!

Smaller (green) bubbles

Larger (purple) bubbles

IV

Future Work: Curved ContactsIV

Future Work: Curved ContactsZ = 6

Large bubbles have a lower excess surface energy than small bubbles!

IV

Conclusions• Analytic models of bubbles provides a convenient way to study the energy

of foams.• Highlight the important role played by logarithmic terms in the bubble-

bubble interaction. • Extending a Durian soft disk model to 3D is neither qualitatively nor

quantitatively justified.• Extend analytical cone model to more general structures.• Loss of square faces associated with increasing liquid fraction was clearly

seen with our model.• Bubble-bubble interaction is more complex than first thought with contact

losses away from the wet limit being not trivial.• In the future, we will further develop the model for other crystal structures

with the goal of describing the energy of a random foam.

Take home message:

Wet foam often considered as overlapping harmonic disks, but there are qualitative differences that we MUST consider, especially when we model wet foams!

AcknowledgementsThis work would not have been possible with out the help of Prof. Stefan Hutzler, Prof. Denis Weaire, David Whyte and the entire Trinity Foams and Complex Systems Group.

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Publications1. S. Hutzler, R. P. Murtagh, D. Whyte, S. T. Tobin and D. Weaire. Z-cone

model for the energy of an ordered foam. Soft Matter, 10, 7103--7108 (2014).

2. D. Whyte, R. P. Murtagh, D. Weaire and S. Hutzler. Applications and extensions of the Z-cone model for the energy of a foam. Colloids and Surfaces A, 473, 55--59 (2015).

3. R. P. Murtagh, D. Whyte, D. Weaire and S. Hutzler. Adaptation of the Z-cone model to the estimation of the energy of a bcc foam. Philosophical Magazine, 95, 4023—4034 (2015).

4. R. P. Murtagh, A. J. Meagher, D. Weaire and S. Hutzler. Evolution of a bubble on a liquid surface containing one or two gas species. (In preparation)