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

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.

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?

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)

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

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) Garcia-Gonzales R., Monnreau C., Thovert J.-F., Adler P.M., Vignes-Adler W., Conductivity of real foams, Colloid Surf. A, 151 (1999) 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) Garcia-Gonzales R., Monnreau C., Thovert J.-F., Adler P.M., Vignes-Adler W., Conductivity of real foams, Colloid Surf. A, 151 (1999)

2D bidisperse cellular fluids SURUZ 2003

Surface Evolver by Keneth Brakke (Susquehanna University)

3 dimensional bi-disperse cellular fluids

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

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

sphere (1 1 ) Interfacial energy vs. curvature radius

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

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

Minimum curvature radius vs. relative interfacial tension

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

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

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

sphere (1 1 ) Mixing energy vs. volume fraction

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

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

 1 1  –1  –3  –6 Mixing energy vs. relative interfacial tension

 1 1  –1  –3  –6  0.1

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.

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