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

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2 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)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 periodic rows of bubbles; d)Bubble clusters with defects inside

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3 ICMS, Edinburgh, 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

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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

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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 A i = area of bubble i = 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

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6 ICMS, Edinburgh, March Round cluster: - Large finite cluster – regular hexagonal boundary:

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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

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8 ICMS, Edinburgh, March 2012 Fortes, Morgan, Vaz, Phil. Mag. Letters, 2007 Hexagonal cluster N= 331 bubbles p*= average pressure of a cluster p 0 = 2 1/2 3 1/ = pressure in a half- plane cluster of unit areas = mondisperse cluster Pressure inside planar clusters

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9 ICMS, Edinburgh, March As the number, N, of unit bubbles become large, the average normalized pressure p* in an energy-minimizing cluster approaches p 0 = 2 1/2 3 1/ 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.

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10 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. Effect of the number of shells on the pressure and energy of two-dimensional free bubble clusters 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. ICMS, Edinburgh, March 2012 Vaz, Teixeira, Graner, Cox, Colloids and Surfaces A, 2009

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11 - 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 ICMS, Edinburgh, March 2012

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

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13 ICMS, Edinburgh, March 2012 Vaz, Cox, Alonso, J. Phys. Condens. Matter, 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 b) Assemblies of finite collections of identical bubbles or bubbles of two different areas Minimum energy configurations of small bidisperse clusters

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14 ICMS, Edinburgh, March 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

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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

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16 ICMS, Edinburgh, March 2012 c) Chain clusters which consist of periodic rows of bubbles 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 N=4 n=30 Fortes, Vaz, Cox, Teixeira, Colloids and Surfaces A, 2007 w

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17 ICMS, Edinburgh, March 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 w b - Instability occurs when one of the eigenvalues of the Hessian matrix vanishes Examples of the buckling instability in chain clusters with N=1, N=2,N=4 - The width of a bubble at the point of buckling w b increases with n for N> 1

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18 Simulation of defects in bubble clusters - Isolated defects: - Dislocations (pair of 5- and 7- sided bubbles) - Disclinations (non- hexagonal bubble) - Pairs of defects and interactions ICMS, Edinburgh, March 2012 d) Bubble clusters with defects inside Cox, Teixeira, Vaz, J. Phys. Condens. Matter, 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

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19 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 ICMS, Edinburgh, March 2012

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20 Surface Evolver simulations Pairs of disclinations of the same strength P (n 1 = n 2 = 5 or 7) or opposite (n 1 = 5 and n 2 = 7) P = n-6 ICMS, Edinburgh, March 2012

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

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22 Single disclination 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) ICMS, Edinburgh, March 2012

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23 (for disclinations of opposite sign) Paired disclination n 1 = n 2 = 5 and n 1 = n 2 = 7 have similar strain energies, which decrease, with the separation n 1 = 5, n 2 =7, increases with d, with a logarithmic fit with M = 2.5 × 10 3 Data converge to the values of a single disclination ICMS, Edinburgh, March 2012

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

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25 Paired dislocations 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) ICMS, Edinburgh, March 2012

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26 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 ICMS, Edinburgh, March 2012

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27 ICMS, Edinburgh, March 2012 Challenge

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28 Minimum perimeter partitions of the plane into equal numbers of regions of two different areas ICMS, Edinburgh, March 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 Fortes, Teixeira, Eur. Phys. J. E, 2001 Periodic tilings of the plane: what are they?

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

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30 ICMS, Edinburgh, March wins if one cell is much smaller than the other wins if they are about the same size Periodic tilings of the plane: what are they? Range of λMinimal tiling Results

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31 ICMS, Edinburgh, March Estimate outer interface energy (all arrangements) and inner interface energy (sorted arrangements) - Inner interface is wall of dislocations (5/7 pairs) We now allow cells to de-mix and consider finite cell clusters. Compare energies of mixed and sorted arrangements 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

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

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33 ICMS, Edinburgh, March Mixed and sorted arrangements alternate - Cells of similar sizes sort! Results Periodic tilings of the plane: but are they actually realised? Range of λMinimal tiling For infinite N:

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34 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? ICMS, Edinburgh, March 2012

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

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37 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 a 0 is the edge length of a hexagonal bubble with area A (i.e. and γ is the film tension) P = n-6 ICMS, Edinburgh, March 2012

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38 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. 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 Dislocations - strain energy ICMS, Edinburgh, March 2012

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39 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 where is obtained in the simulations and is the energy of a joined cluster without defects where N is a factor N=N+k 1 s(k 2 s-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 k 1 and k 2 are two fitting parameters extracted from the case n 1 = n 2 = 6; k 1 = 1.1 and k 2 = 0.5 Results ICMS, Edinburgh, March 2012

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