some applications plateau’s perfectly foams: tilings ...piteixeira/cftctalk.pdf• two exampes: n...

Introduction: what . . .

Some applications . . .

Plateau’s perfectly . . .

Tilings

Flowers

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Foams: tilings, flowers and defects

Paulo Teixeira

Instituto Superior de Engenharia de Lisboaand Centro de Fısica Teorica e Computacional

Lisbon, Portugal

Collaborators:

Manuel Fortes† and Fatima Vaz (IST, Lisbon)Simon Cox (U. Aberystwyth, UK)Francois Graner (Institut Curie, Paris, France)




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1. Introduction: what are foams?

• A foam consists of pockets, called cells or bubbles, of gas or liquid en-closed in liquid – liquid foams – or solid – solid foams.

• In liquid foams, liquid is distributed over films, Plateau borders (PBs),and nodes.




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• They are held together by the balance of surface tension and air pressure.

• A liquid foam is extremely delicate but incompressible.

• Rheologically it behaves as a Bingham plastic liquid: it is solid up tosome yield stress, then flows like a liquid.

D. Weaire and M. A. Fortes, Adv. Phys. 43, 685 (1994)

• Liquid foams can be wet (up to 30% liquid) or dry (less than 5% liquid).




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2. Some applications of liquid foams

• Beer and sparkling wines

• Whipped cream, ice cream, chocolate mousse

• Household cleaning products

• Toiletries

• Firefighting

• Fractionation and flotation

• Enhanced oil recovery




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3. Plateau’s perfectly dry or ideal foam (1873)

• Films have zero thickness and are endowed with a free energy per unitarea σ: the total foam free energy is:

F = σS

• Films meet three at a time, at 120◦, at triple lines.

• Triple lines meet four at a time, at cos−1(−1/3) ≈ 109.5◦, at vertices.

• Films have constant mean curvature given by the Young-Laplace law:

∆p = 2σ

(1

R1

+1

R2

)• At a vertex, film curvatures sum to zero.




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4. Tilings

• A liquid foam is held together by surface tension.

• Foams are therefore model systems for the solution of area (in 3d) orperimeter (in 2d) minimisation problems.

• Plateau’s laws are necessary conditions for an energy minimum, but donot uniquely determine the equilibrium geometry/topology.

• In 2d, the minimum-perimeter partition of the plane into regions of equalarea is the tiling by regular hexagons – the honeycomb [T. C. Hales,Discrete Comput. Geom. 25, 1 (2001)]:

• It is now natural to look at generalisations to polydisperse foams, possiblyalso of finite size.




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• Just for completeness: what’s it like in 3d? This is the Kelvin problem.

Kelvin solution: cells are identical 14-sided truncated octahedra, or orthictetrakaidecahedra. This was a modelfor the luminiferous aether [W. Thomson,Phil. Mag. 24, 503 (1887)].

Weaire-Phelan solution (A15):cells of two different shapes (butequal volumes) – a dodecahedronand a 14- sided polyhedron. Itbeats Kelvin’s by 0.3% [D. Weaireand R. Phelan, Phil. Mag. Lett.70, 345 (1994)].

• 3d is a lot more difficult and there are no rigorous proofs. We shall keepwell clear of it.




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4.1. How to pave a plane with two types of tile so as tohave the shortest boundary?

• Only 1:1 periodic tilings with at most two cells of each area per repeatingunit, and such that all cells of the same area are equivalent.

• We draw all candidate structures and calculate their energies vs arearatio λ. The minimisers are:




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M. A. Fortes andP. I. C. Teixeira,,Eur. Phys. J. E 6, 133(2001)

Interval of λ Minimal tiling0.645− 1 6161

0.268− 0.645 5272

0.108− 0.268 4181

0.041− 0.108 4282

0− 0.041 3191




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4.2. What is the stable equilibrium state of a 2d bidis-perse, finite foam cluster? Does it stay mixed ordoes it size-sort?

• We now allow the cells to de-mix and consider finite cell clusters. Com-pare energies of mixed and sorted arrangements.

• Estimate outer inteface energy (all arrangements) and inner interfaceenergy (sorted arrangements).

• Inner interface is wall of dislocations(5/7 pairs).




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Work out which arrangementhas lowest energy for each(N, λ).

Mixing and sortingalternate!

P. I. C. Teixeira, F. Graner and M. A. Fortes, Eur. Phys. J. E 9, 161 (2002)




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5. Flowers

• A flower cluster consists of N shells of n petal cells all of unit area,surrounding a central cell of area λ.

• Two exampes: N = 2 and n = 12 (left), and N = 4 and n = 12 (right)

• Investigate stability looking at eigenvalues of Hessian matrix with respectto deformations – use Surface Evolver software.




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• Below a threshold λ, the symmetric clusters become unstable: symmetryis spontaneously broken!

• So the symmetric solution is not always preferred - an ugly, asymmetricstate may have (often very slightly) lower energy.

M. A. Fortes, M. F. Vaz, S. J. Cox and P. I. C. Teixeira,Colloids Surf. A 309, 64 (2007)




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6. DefectsS. J. Cox, P. I. C. Teixeira and M. F. Vaz,

J. Phys.: Condens. Matter 22, 065101 (2010)

6.1. Disclinations

• Very rarely observed in solids, sometimes in foams, often in block copoly-mers and liquid crystals.

• Volterra construction: a disclination of strength P results from cuttingor inserting a slice of angle Pπ/3. The edges of the slice rotate.

• Only wedge disclinations will be considered (2d).

• We want to study both isolated disclinations and interactions betweendisclinations, and compare with analytical results.




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• In foam clusters (n-sided central cell, N cells) they look like this: P = −1(left), P = +1 (right) (P = n− 6).

• All our calculations were performed using Surface Evolver software.




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• Pressure in central cell vs cluster size:

• The pressure of the central cell is strongly correlated with n (decreasesas n increases), and varies only weakly with N .




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• Energy of cluster with central disclination vs cluster size N :

• Fits to numerical data:

E5(N) = 1.86898 + 1.90172N−1/2

E6(N) = 1.86171 + 1.90625N−1/2

E7(N) = 1.86701 + 1.92538N−1/2

• Analytical prediction: E(N) = 1.86 + 1.94N−1/2




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• Excess energy of isolated disclination:

• Lower than theoretical prediction (top line)

w =G

36P 2 , G =

1

2√

3

γ

a0

but agrees there is invariance P → −P .




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• Interaction between disclinations:

• By analogy wih liquid crystals we hypothesise, for two disclinations ofopposite signs:

w = MP 2 ln d

Fit is OK with M = 2.5× 10−3.




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6.2. Dislocations

• Very common in solids and liquid crystals, especially smectics. They areresponsible for plastic deformation.

• Volterra construction: a dislocation results from cutting the material andtranslating the edges relative to each other.

• Only wedge disclinations will be considered (2d).

• We want to study both isolated dislocations and interactions betweendislocations, and compare with analytical results.




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• In foam clusters (n-sided central cell, N cells) they look like this: a 5/7pair.

• All our calculations were performed using Surface Evolver software.




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• Excess energy of a 5/7 dislocation vs distance from core:

• Theoretical prediction is:

w =G

4πB2r−2 , G =

1

2√

3

γ

a0

We get w ∼ r−α, α = 1.88 ± 0.24, but prefactor is about 4 times toolarge.




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• Interaction between two 5/7 dislocations:

• Theoretical prediction is:

w =G

πB2 ln d , G =

1

2√

3

γ

a0

is an underestimate. The dependence is logarithmic, though.




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7. Summary and outlook

• We have calculated the minimum-perimeter periodic partitions of theplane into regions of areas 1 and λ in 1:1 proportion, with no more thantwo regions of each area per unit cell.

• Whether our conclusions would stand if we allowed for larger unit cells,or for aperiodic (e.g., Penrose) tilings, remains an open question.

• However, as λ is varied between 0 and 1, the minimal energy arrangementalternates between mixed (i.e., single-phase) and sorted (i.e., “phase-separated”) arrangements.

• We aim to exend this work to non-1:1 clusters. Investigation of mixingand sorting under shear is also in progress.

• Flower clusters were shown to exhibit a symmetry-breaking transition asthe flower “core” shrinks.

• The behaviour of defects – disclinations and dislocations – in finite foamclusters was shown to follow trends that are similar, but not quite iden-tical, to known analytical results for a continuous medium.




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Acknowledgements

Alliance ProgrammeFCT/British Council Transnational Cooperation SchemeEPSRC UK (EP/D071127/1, EP/D048397/1).




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One last pretty picture. . .

some applications plateau’s perfectly foams: tilings ...piteixeira/cftctalk.pdf• two exampes: n...

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