1. gwnow@amu.edu.pl The filling up tetrahedral nodes in the monodisperse foams and emulsions with Reuleaux-like tetrahedra Department of Physical Chemistry Faculty of Chemistry, UAM, Poznań Waldemar Nowicki, Grażyna Nowicka

2. Model: The three phase fluid system: A, B and C phase A and B fluids form droplets/bubbles dispersed into liquid C The volume of the dispersion medium C is so low that the dispersion is a system of space-filling polyhedra organized into a network.

3. The aim of the study: Are 3D patterns stable in three-phase bidisperse cellular fluids? Can these patterns be formed spontaneously? Do the transition states associated with local energy minima?

4. Plateau’s laws: Films meet at triple edges at 2/3  (120°) Edges meet at tetrahedral vertices at arccos(1/3) (109.5°) Laplace’s law: The curvature of a film separating two bubbles balances the pressure difference between them Plateau’s laws: Films meet at triple edges at 2/3  (120°) Edges meet at tetrahedral vertices at arccos(1/3) (109.5°) Laplace’s law: The curvature of a film separating two bubbles balances the pressure difference between them 2-phase cellular fluids (foams)

5. The energy and structure of cellular fluid are dominated by interfacial tension.  The structure can be found by the interfacial energy minimization. The energy and structure of cellular fluid are dominated by interfacial tension.  The structure can be found by the interfacial energy minimization. 3-phase cellular fluids

6. Monodisperse foams Arystotle – tetrahedra fill the space (On the Heavens ) Kelvin – the best partition – slightly curved 14-sided polyhedra (tetrakaidecahedra ). Thomson W. (Lord Kelvin), On the division of space with minimum partitional area, Phil. Mag., 24, 503 (1887) Weaire-Phelan – two kinds of cells of equal volume: dodecahedra, and 14- sided polyhedra with two opposite hexagonal faces and 12 pentagonal faces (0.3% in area better than Kelvin's partition) Weaire D., Phelan R., A counterexample to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett., 69, 107 (1994) Experiment – the light tomography of foams Thomas P.D., Darton R.C., Whalley P.B., Liquid foam structure analysis by visible light tomography, Chem. Eng. J., 187 (1995) 187-192 Garcia-Gonzales R., Monnreau C., Thovert J.-F., Adler P.M., Vignes-Adler W., Conductivity of real foams, Colloid Surf. A, 151 (1999) 497-503 Monodisperse foams Arystotle – tetrahedra fill the space (On the Heavens ) Kelvin – the best partition – slightly curved 14-sided polyhedra (tetrakaidecahedra ). Thomson W. (Lord Kelvin), On the division of space with minimum partitional area, Phil. Mag., 24, 503 (1887) Weaire-Phelan – two kinds of cells of equal volume: dodecahedra, and 14- sided polyhedra with two opposite hexagonal faces and 12 pentagonal faces (0.3% in area better than Kelvin's partition) Weaire D., Phelan R., A counterexample to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett., 69, 107 (1994) Experiment – the light tomography of foams Thomas P.D., Darton R.C., Whalley P.B., Liquid foam structure analysis by visible light tomography, Chem. Eng. J., 187 (1995) 187-192 Garcia-Gonzales R., Monnreau C., Thovert J.-F., Adler P.M., Vignes-Adler W., Conductivity of real foams, Colloid Surf. A, 151 (1999) 497-503

7. 2D bidisperse cellular fluids SURUZ 2003

8. Surface Evolver by Keneth Brakke (Susquehanna University)

9. 3 dimensional bi-disperse cellular fluids

10. tetrahedron (3 4 3 –6 ) Interfacial energy vs. curvature radius

11. tetrahedron (3 4 3 –6 ) Interfacial energy vs. curvature radius

12. sphere (1 1 ) Interfacial energy vs. curvature radius

13. lens (1 2 1 –1 ) Interfacial energy vs. curvature radius

14. trihedron (2 3 2 –3 ) Interfacial energy vs. curvature radius

15. Minimum curvature radius vs. relative interfacial tension

16. The mixing energy – the change in the interfacial energy which accompanies the transfer of A cell from the A-C network to the B-C network

17. tetrahedron (3 4 3 –6 ) Mixing energy vs. volume fraction

18. R=R min tetrahedron (3 4 3 –6 ) Mixing energy vs. volume fraction

19. sphere (1 1 ) Mixing energy vs. volume fraction

20. R=R min lens (1 2 1 –1 ) Mixing energy vs. volume fraction

21. R=R min trihedron (2 3 2 –3 ) Mixing energy vs. volume fraction

22. 5.10 13.39  1 1  1 2 1 –1  2 3 2 –3  3 4 3 –6 Mixing energy vs. relative interfacial tension

23. 5.10 13.39  1 1  1 2 1 –1  2 3 2 –3  3 4 3 –6  0.1

24. Small cells introduced to the monodisperse network produce the stable highly- organized patterns at any  values. At  =1 patterns cannot be formed spontaneously. For small  values patterns are able to self- organize.

25. Thank you for your attention ???