2. Outline of this Presentation
1. 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
3. What is a foam?
• A collection of gas bubbles separated by continuous liquid
films.
3
• A liquid foam is a two-phase system in which a gas is
dispersed in a continuous liquid phase.
OR
I
4. Motivation
4
• 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
5. 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.
5
foam
liquid
V
V
• 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
I
6. The effect of gravity
6
Dry 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
I
7. Structure of monodisperse foam
• The term foam structure refers to the particular
arrangement of the bubbles.
7
• 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?
I
8. Surface Evolver
8
• Surface Evolver finds a minimum surface area via
triangulation and minimisation.
I
• This method also applies to confined bubbles.
9. Structure of monodisperse foam
• For very wet foams, the structure resembles a packing
of spheres.
9
• 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.
I
10. Shape of bubbles for intermediate φ?
10
• Bubbles minimise their surface area A subject to their
confinement conditions.
11
00
A
A
E
E
Key Question: How do bubbles interact with
each other as they are deformed in a foam?
AE
I
11. How do bubbles interact in a foam?
11
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
I
12. “Soft” Disk model
12
• Simple dynamic model for
interacting bubbles as overlapping
disks.
• Each of the disks experiences a
force:
d
RR
R
kF
ji
av
sd
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.
I
13. Lacasse et al. in 2D
13
• 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.
1
2
)( 2
0
R
Perimeter
sd
0
0
R
hR
N.B. Volume of
2D bubble kept
constant!
I
14. Morse and Witten: 3D
14
• 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
I
15. Lacasse et al. in 3D
15
• 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
I
16. How do bubbles interact in a foam?
16
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?
I
17. Bubble With 2 Contacts
Top down viewSide on view for the case of
two contacts
II
18. The “Z-Cone” Approximation
Inspiration:
Ziman 1961,
Fermi Surface
of Copper
2θ
Flat contact
Z
2
1arccos
Surface Evolver FCC
(Z = 12):
Volume V
Total volume V is
redistributed into Z small
cones of volume
Z
VVcone
Example of one such cone
Area of constant
mean curvature
II
20. 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
Z
2
1arccos
II
21. Formally, minimisation of cone surface area A
under constraint of constant volume
using Euler-Lagrange equation
with Lagrange multiplier λ
Mathematics of the model
dz
dz
dr
zrZA
h
0
2
2
1)(2/
C
r
r
z
z
L
L
)(1)(2/ 2
22
zr
dz
dr
zr
h
Z
L
)tan(3
)0(
)(/
3
0
2
r
dzzrZV
h
0atcot
at
zr
hzr
z
z
boundary conditions:
)0(/ r
21
II
22. 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
1
2
),()1(
1),( 3/2
3/1
22
ZJ
Z
Z
Z
ZK
Z
Z
Z
1
)(
),(
0
A
A
Z
Elliptic
Integrals
Complicated dependence on Z!
Clearly not harmonic!
II
23. 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.
II
25. 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
26. Deformation
for Z=12
wet limit
Logarithmic limit
Large deviations
from a harmonic
potential in both the
wet and dry limits
ln2
),(
2
Z
Z
Logarithmic asymptote
II
27. Roughly harmonic for low Z
27
For intermediate deformation and
low Z, there is a region of
approximate harmonicity.
Breaks down for
larger Z.
II
30. 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
4
3
1
1
1
3/1
Z
Z
c
c
II
31. 31
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
III
33. Cone Model for Kelvin
One Cone Approach Two Cone Approach
III
34. Key Changes to the Model
34
)0(/ r
068 VVV sh 0VZVc
Z
2
1arccos
2
1arccos n
n
n
h
n
h
n
n
n
n
22
cot4
sin2
arcsin22
III
35. Matching at Cone-Cone Edges
35
)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
2
3
864434.0
cos
cos
s
h
s
h
H
H
III
36. Matching at Cone-Cone Edges
36
)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:
2
3
2
sh
III
37. 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
1
4
68
sh AA
092.0*
cone11.0*
III
40. 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
1
4
68
sh AA
092.0*
cone318.0*
, conec
III
41. Contact Loss in Kelvin Foam
)ln(
c
c
a
d
d
2
2
1
))(ln(
c
b
b
d
d
• 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
43. Future Work: Curved Contacts
Z = 2
Large bubbles have a
lower excess surface
energy than small
bubbles!
Smaller (green)
bubbles
Larger (purple)
bubbles
IV
45. Future Work: Curved Contacts
Z = 6
Large bubbles have a
lower excess surface
energy than small
bubbles!
IV
46. 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.
47. 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!
48. Acknowledgements
This 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.
49. Publications
49
1. 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)
