Modeling Random Wet 2D Foams with Controlled Polydispersity. Back to the Future?

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1 Modeling Random Wet 2D Foams with Controlled Polydispersity Back to the Future? Andy Kraynik Sandia National Labs (retired) CEAS, University of Manchester University of Erlangen-Nuremberg Simon Cox Aberystwyth University Stephen Neethling Imperial College London

2 Random Foams Kraynik, Reinelt & van Swol (2003) Phys Rev E 67, ; (2004) Phys Rev Lett 93, ; (2005) Colloids Surfaces A Monodisperse Polydisperse Bidisperse Spatially periodic structure 1728 cells Cell volumes vary by three orders of magnitude

3 Outline Motivation Simulating random 2D foams with controlled polydispersity Dry Energy Shear modulus Wet Increasing liquid fraction toward the jamming transition Shear modulus

4 Simulating Random Polydisperse Foam Molecular Dynamics Random Close Packed (RCP) Spheres Laguerre (Weighted Voronoi) Tessellation Surface Evolver Relaxed Foam Tension-Compression Cycles Annealing Relax the lattice to achieve isotropic stress Elastic Recoil

5 Confocal microscopy of Plateau borders in emulsions Eric Weeks, Physics, Emory University and Doug Wise, Physics, Harvard University

6 Plateau Borders in a Wet Kelvin Foam Kelvin Cell f = Surface Tension = s Film Tension = 2s f = 0.01 Plateau borders in liquid foams correspond to struts in solid foams

7 Wet monodisperse foams with 64 cells and isotropic stress f = lost contacts f = f = f =

8 Bubble Overlap f = lost contacts

9 Random Monodisperse Foam f = 0.025

10 Modeling the strut-level geometry of open-cell foams

11 Multiple solutions when the thin film tension is 2s The junction between the Plateau borders and the thin films is not unique computationally and can lead to overlapping interfaces near the cusp when the film tension is 2s.

12 Random Polydisperse 2D Foams Packed Disks Voronoi Polygons Equilibrium

13 Surface Free Energy Density in 3D E = s S F V F = s S V = 3 s R 2 R 3 = 3 R 3 1/3 R 32 s R 3 1/3 = 36 1/3 R R 3 1/3 1 s V 1/3 E = 3 s R 32 = 36 1/3 1 + p s 5.32 = V 1/3 1 + p s V 1/3 Sauter mean radius R 32 = R 3 R 2 polydispersity p = R 32 R 3 1/3 1 0

14 Surface Free Energy Density in 2D E = s L F A F = s L A = 2 s R R 2 = 2 R 2 1/2 R 21 s R 2 1/2 = 2 1/2 R R 2 1/2 1 s A 1/2 E = 2 s R 21 = 2 1/2 1 + p s 3.75 = A 1/2 1 + p s A 1/2 Sauter mean radius R 21 = R 2 R polydispersity p = R 21 R 2 1/2 1 0

15 Shear Modulus 3D 2D

16 Creating Wet Random Polydisperse 2D Foams Weighted-Voronoi (Laguerre) Polygons

17 Weighted-Voronoi Polygons with Primitive Plateau Borders

18 q s 2s cosq

19 Wet Topological Transitions q = 8 o r = 0.02

20 Wet Topological Transitions q = 8 o Neighbor Loss r = 0.05 Neighbor Switch

21 Wet Topological Transition Neighbor Switching

22 Outlook for Random Wet 2D Foams Topology Geometry Rheology Diffusive Coarsening

23 Swelling a foam by adding liquid

24 Transition Between Liquid-Like and Solid-Like Behavior yield stress t y ~ s/r (f-f * ) 2 shear modulus G ~ s/r f(f-f * ) f = volume fraction of dispersed phase f * = onset of jamming s = surface tension R = bubble/drop size Hohler & Cohen-Addad (2005) f * Semi-dilute gas-bubble suspension or liquid-liquid emulsion Concentrated suspension or emulsion Wet Foam Dry Foam Highly Concentrated Emulsion High Internal Phase Emulsion (HIPE)

25 q = 8 o r = 0.02

26 Neighbor Switch r = 0.03

27 r = 0.035

28 Neighbor Loss r = 0.04

29 r = 0.045

30 r = 0.05

31 r = 0.06

32 r = 0.07

33 r = 0.08

34 r = 0.09

35 r = 0.10

36 r = 0.11

37 r = 0.12

38 r = 0.13

39 r = 0.14

40 r = 0.145

41 r = 0.15

42 Beyond Jamming r = 0.155

43 r = 0.16

44 r = 0.18

45 r = 0.20

46 r = 0.25

47 r = 0.30

48 r = 0.35

49 r = 0.39

50 r = 0.39

51 r = 0.05 r = 0.39

52 Physics of Flocculated (Adhesive, Sticky) Emulsions Aveyard et al. Langmuir 18, , (2002) P = 4πR 1 + q3 6 + E = 4πsR 1 q3 3 +

53

54

55 The surface area of a foam cell is about 10% greater than an equal-volume sphere = S (36 V 2 ) -1/3 =

56 Shear Modulus: Theory B.V. Derjaguin (1933) Kolloid-Z. 64, 1. Affine deformation of random soap films G = 4/15 E = E D. Stamenovic (1991) J. Coll. Int. Sci. 145, 255. Deformation of idealized foam vertex G = 1/6 E = E Kraynik & Reinelt (1996) J. Coll. Int. Sci. 181, 511. Surface Evolver calculations for ordered foams Kelvin Bergman (T) Weaire-Phelan Williams Friauf-Laves (C15)

57 Shear Modulus: Measurements for foams and emulsions H.M. Princen & A.D. Kiss (1986) J Coll Int Sci Static shear modulus of highly concentrated liquid-liquid emulsions G ~ s R 1 32 f 1/3 (f - f c ) R 32 = S R 3 / S R 2 dry limit (f=1) G = 0.51 s R 32 1 T.G. Mason, J. Bibette & D.A. Weitz (1995) PRL Elasticity of compressed emulsions G ~ s R 1 f (f - f c ) A. Saint-Jalmes & D.J. Durian (1999) J Rheology Vanishing elasticity of wet foams

58 Surface Evolver simulation of random wet foam f = 0.025