1. Surface energy of two-dimensional finite or periodic foams
ICMS, Edinburgh, March 2012
Surface energy of two-dimensional
finite or periodic foams
1. Finite foams – M. Fátima Vaz
2. Periodic foams – Paulo Teixeira

2. ICMS, Edinburgh, March 2012
Problem: N cells of equal or different areas
Identify the arrangement, of the N cells with lowest energy
2D bubbles  minimize the total perimeter as E = P
(Honeycomb = minimum perimeter partion of the plane into regions of equal area – Hales 2001)
- Experiments
- Analytical results
- Surface Evolver
Finite clusters of several types:
A central bubble surrounded by several shells of bubbles with the same or different areas;
Assemblies of finite collections of identical bubbles or bubbles of two- different areas;
Chain clusters which consist of periodic rows of bubbles;
Bubble clusters with defects inside

3. ICMS, Edinburgh, 19-23 March 2012 Experimental procedure
Liquid – glass (Smith (1952), Vaz and Fortes (1997), F. Graner,B. Dollet)‏
Vessel with detergent solution; blow air bubbles under a plate
(a) Bubbles are trapped between a glass plate and a liquid solution, a cluster is obtained‏
(b) By tilting the plate, the liquid fraction increases, the cluster looses rigidity and the bubbles are separated
(c) By levelling the vessel the bubbles assemble into different clusters

4. ICMS, Edinburgh, March 2012
a) A central bubble surrounded by several shells of bubbles with the same or different areas
b) Assemblies of finite collections of identical bubbles or bubbles of two different areas
c) Chain clusters which consist of a periodic single row of bubbles
d) Bubble clusters with defects inside
5-7

5. ICMS, Edinburgh, March 2012
a) A central bubble surrounded by several shells of bubbles with the same or different areas
Vaz, Fortes, Graner, Phil. Mag. Letters, 2002
To propose an approximate equation for the surface energy of 2D free bubble clusters
Based on Graner et al 2001
Ai = area of bubble i
3.722 = perimeter of a regular hexagon of area 1
Equation is exact for the honeycomb - but deviations occur as width of cell area distribution increases
Finite clusters –contribution of the external boundary to the energy

6. ICMS, Edinburgh, 19-23 March 2012 Round cluster:
Large finite cluster – regular hexagonal boundary:

7. ICMS, Edinburgh, March 2012
We compare with exact calculations of the surface energy of symmetrical clusters
Analytically - the clusters are solved by imposing the Plateau laws (circular films meeting at 120° at triple junctions with zero sum of the curvatures) and the areas of the bubbles.
n = number of sides of the central cell
cell areas are 1 and 
- deviations: lower n and lower 
good accuracy for narrow area distribution

8. ICMS, Edinburgh, 19-23 March 2012 Pressure inside planar clusters
Fortes, Morgan, Vaz, Phil. Mag. Letters, 2007
Hexagonal cluster
N= 331 bubbles
p*= average pressure of a cluster
p0= 21/231/4 ≈ = pressure in a half- plane cluster of unit areas = mondisperse cluster

9. ICMS, Edinburgh, March 2012
As the number, N, of unit bubbles become large, the average normalized pressure p* in an energy-minimizing cluster approaches
p0= 2 1/2 3 1/4 ≈
- An equation was derived for the rigorous theoretical upper and lower bounds on the average pressure in terms of N.
- Surface Evolver simulations agree with the estimates.

10. ICMS, Edinburgh, March 2012
Effect of the number of shells on the pressure and energy of two-dimensional free bubble clusters
Vaz, Teixeira, Graner, Cox, Colloids and Surfaces A, 2009
Simulations of two-dimensional hexagonal bubble clusters consisting of:
a central bubble of area  surrounded by s shells or layers of bubbles of unit area.
Monodisperse clusters: a central bubble with area = 10 surrounded by s shells of unit-area bubbles: (a) s = 1, (b) s = 2, (c) s = 3, (d) s = 4, (e) s = 7, (f) s = 10, (g) s = 15 and (h) s = 20.

11. ICMS, Edinburgh, March 2012
Pressure in the central cell, p0, vs the number of shells s, for several 
For  < 10, p0 decreases with increasing s, as in the monodisperse cluster
For  >10, however, p0 becomes an increasing function of s.
The same is true of the average pressure over the entire s-shell cluster
The average pressure in a 20-shell cluster is almost independent of , even for large central bubble areas. It seems safe to conclude that in a large cluster the average pressure does not depend on  and tends to the same limiting value, , as in the monodisperse cluster

12. ICMS, Edinburgh, March 2012
Provides a good account of our results, even for clusters with a large number of shells and large .

13. ICMS, Edinburgh, March 2012
b) Assemblies of finite collections of identical bubbles or bubbles of two different areas
Minimum energy configurations of small bidisperse clusters
Vaz, Cox, Alonso, J. Phys. Condens. Matter, 2004
- Small collections of N bubbles with two different areas = bidisperse clusters
- For experimental simplicity, we restricted to clusters of
N/2 bubbles of area A1 and N/2 bubbles of area A2.
- Experimentally different arrangements were found for each N.
the observed topologies for bidisperse clusters
N = 6 and A1/A2 = 4/3.
Most frequent nº2

14. ICMS, Edinburgh, March 2012
The candidates that appear most frequently in the experiments are expected to be the minimal ones.
We these topologies we computed the energy of each of these clusters using the Surface Evolver.
- Candidates for the minimal energy arrangement.
Lowest energy clusters – Surface Evolver
- N=4, 6, 8, 10

15. ICMS, Edinburgh, March 2012
SE calculated energies are compared with existing approximations.
- Round cluster:
- Regular hexagonal boundary = lower bound for energy
- SE calculations showed good agreement with the analytic predictions
- Some values below the lower bound can be attributted to the small number of bubbles in the cluster

16. ICMS, Edinburgh, March 2012
c) Chain clusters which consist of periodic rows of bubbles
Fortes, Vaz, Cox, Teixeira, Colloids and Surfaces A, 2007
Stability - chain symmetrical clusters show an energy minimum under compression (decreasing the width of the chain clusters)‏
Chain clusters with N rows, each containing n bubbles of unit area and width w, periodic in one direction w
N=4 n=30

17. ICMS, Edinburgh, March 2012
Examples of the buckling instability in chain clusters with N=1, N=2 ,N=4
- Surface Evolver was used to examine chain clusters which are confined in a periodic box
- The width w of the bubbles is decreased until buckling occurs at a critical wb
- Instability occurs when one of the eigenvalues of the Hessian matrix vanishes
- The width of a bubble at the point of buckling wb increases with n for N> 1

18. Simulation of defects in bubble clusters
ICMS, Edinburgh, March 2012
d) Bubble clusters with defects inside
Simulation of defects in bubble clusters
Cox, Teixeira, Vaz, J. Phys. Condens. Matter, 2010
Ex: plastic deformation deals with the interactions between defects
To study a small number of defects in 2D large clusters and assess how the presence of defects affect the energy and pressure of clusters
Our simulations of pairs of defects reveal how the presence of one defect is “felt” by the other defect as a function of their separation
- Isolated defects:
- Dislocations (pair of 5- and 7- sided bubbles)‏
- Disclinations (non- hexagonal bubble)‏
Pairs of defects and interactions

19. ICMS, Edinburgh, March 2012
Surface Evolver simulations - The surface energy E is minimized.
Isolated disclinations
n= 5, 7, 8 and 9
- Isolated dislocation 5/7
- Pair of 5/7 dislocations
opposite Burgers vector

20. ICMS, Edinburgh, 19-23 March 2012 Surface Evolver simulations
Pairs of disclinations of the same strength P (n1 = n2 = 5 or 7) or opposite (n1 = 5 and n2 = 7)‏
P = n-6

21. ICMS, Edinburgh, 19-23 March 2012 Results Pressure in the central cell
Bubble pressure
Energy per unit area (i.e. per bubble),
for all clusters

22. Single disclination ICMS, Edinburgh, 19-23 March 2012 Nematic liquid
Equation derived 
w ( n = 5 disclination) (P = -1)‏
=w (n = 7 disclination) (P = +1)‏
(but the magnitude of w is half of predicted)‏

23. Paired disclination ICMS, Edinburgh, 19-23 March 2012
(for disclinations of opposite sign)‏
n1 = n2 = 5 and n1 = n2 = 7 have similar strain energies, which decrease, with the separation
n1 = 5, n2 =7, increases with d, with a logarithmic fit with M = 2.5 × 10−3
Data converge to the values of a single disclination

24. Single dislocation ICMS, Edinburgh, 19-23 March 2012 Burgers vector
(hexagonal foam  B=1.074)‏
energy decreases with 1/r with =1.88

25. Paired dislocations ICMS, Edinburgh, 19-23 March 2012
Energy increases almost logarithmically with d
Comparasion of data with the same defects embedded in an infinite hexagonal foam
Boundaries do not have a large effect on the interaction between defects
Energy in the hexagonal foam fits ln(kd + 1)‏

26. ICMS, Edinburgh, March 2012
Disclination- affects the energy and the pressure.
The energy of a disclination decreases as the number of shells increases.
The energy and the pressure of a cluster with n = 6 ~ those of cluster with 5/7 dislocation
Analytical approaches for continuous media: defects in foams follow the predicted trends. For example, the energy of two disclinations with opposite strengths a distance d apart is proportional to lnd.
A perfect match between analytical results and simulations is not to be expected. Clusters with n = 5 and 7 are examples of this because these disclinations have the same strength (in absolute value), one would expect that they would have the same energy. However, the two clusters have different boundaries

27. ICMS, Edinburgh, March 2012
Challenge

28. ICMS, Edinburgh, March 2012
Periodic tilings of the plane: what are they?
Minimum perimeter partitions of the plane into equal numbers of regions of two different areas
Fortes, Teixeira, Eur. Phys. J. E, 2001
The minimum-perimeter partition of the plane into regions of equal area is the tiling by regular hexagons - the honeycomb (Hales, 2001)
How do we pave the plane with tiles of two types (sizes) so as to have the shortest boundary?
Consider only 1:1 periodic tilings with at most two cells of each area per repeating unit, and where all cells of the same area are equivalent
Draw all candidate structures and calculate their energies vs area ratio

29. ICMS, Edinburgh, March 2012
Periodic tilings of the plane: what are they?
There are a few others, e.g., 3292, 6262, that are never lowest energy

30. ICMS, Edinburgh, March 2012
Periodic tilings of the plane: what are they?
Results
Range of λ
Minimal tiling
3191
4282
4181
5272
6161
3191 wins if one cell is much smaller than the other
6161 wins if they are about the same size

31. Mixing and sorting of bidisperse 2D bubbles
ICMS, Edinburgh, March 2012
Periodic tilings of the plane: but are they actually realised?
Mixing and sorting of bidisperse 2D bubbles
Teixeira, Graner, Fortes, Eur. Phys. J. E, 2002
We now allow cells to de-mix and consider finite cell clusters. Compare energies of mixed and sorted arrangements
Estimate outer interface energy (all arrangements) and inner interface energy (sorted arrangements)
Inner interface is wall of dislocations (5/7 pairs)

32. ICMS, Edinburgh, March 2012
Periodic tilings of the plane: but are they actually realised?
Results
Work our which arrangement has lowest energy for each (N,λ) pair
- As N increases the weight of the interface decreases and there is sorting

33. ICMS, Edinburgh, March 2012
Periodic tilings of the plane: but are they actually realised?
Results
For infinite N:
Range of λ
Minimal tiling
3191
6+6
4282
4181
5272
Mixed and sorted arrangements alternate
Cells of similar sizes sort!

34. Some outstanding questions in 2D foams
ICMS, Edinburgh, March 2012
Some outstanding questions in 2D foams
We cannot prove that any of these configurations is a minimiser.
We cannot be sure that we did not miss some possible configurations. Do you know a systematic way of generating them?
Can we do a better job estimating boundary energies?
We would like to generalise to
(i) more than two cells of each size per repeating unit;
(ii) different numbers of cells of each size
The number of possible configurations is large. How do we generate/analyse them?
What about non-periodiic configurations?
How do we generalise to wet foams?
What is appropriate reference state for measuring energies of defected clusters?

35. International Centre for Mathematical Sciences Edinburgh
ICMS, Edinburgh, March 2012
Acknowledgements:
International Centre for Mathematical Sciences Edinburgh
British Council trough Treaty of Windsor Programme, grant no. B-20/2010

37. ICMS, Edinburgh, 19-23 March 2012 Disclinations – strain energy
- In a nematic liquid, the energy, w, of a wedge disclination is
where ρ is the distance between the dislocation line and the container wall, a is the molecular dimension and K is the average elastic constant.
The energy per unit length of line, w, for two wedge disclinations of opposite strengths +P and −P a distance d apart is
- In a 2D hexagonal foam, the strain energy density w, i.e. the energy per unit area per unit length, of a disclination cluster of strength P is
where G is the shear modulus and E is the elastic modulus E=4G,
where a0 is the edge length of a hexagonal bubble with area A
(i.e and γ is the film tension)‏
P = n-6

38. ICMS, Edinburgh, 19-23 March 2012 Dislocations - strain energy
The strain energy density w of a dislocation with Burgers vector B in an incompressible foam cluster is
w decreases with the distance r from the core as r−2
The interaction energy of two edge dislocations with opposite signs a distance d apart in the same glide plane can be adapted to
where A is the bubble area.

39. ICMS, Edinburgh, 19-23 March 2012 Results
For isolated defects, we define an excess energy density as
where is obtained in the simulations
and is the energy of a perfect hexagonal 2D cluster
► For paired disclinations, we define an excess energy density as
and is the energy of a joined cluster without defects
where N’ is a factor N’=N+k1s(k2s-d)‏
-depends on the number of rows removed when the clusters are joined which varies with s-d (s=number of shells, d=distance )
- where k1 and k2 are two fitting parameters extracted from the case n1 = n2 = 6;  k1 = 1.1 and k2 = 0.5