SPONTANEOUS TOPOLOGICAL TRANSITIONS IN BIDISPERSE CELLULAR FLUIDS Rafał Olejniczak and Waldemar Nowicki Department of Physical Chemistry, Faculty of Chemistry, A. Mickiewicz University, Poznań, Poland

Model 3 phase fluid system 3 phase fluid system Cells A, B immersed in liquid C Cells A, B immersed in liquid C All fluids are immiscible and incompressible All fluids are immiscible and incompressible

Model 3 phase fluid system 3 phase fluid system Cells A, B immersed in liquid C Cells A, B immersed in liquid C All fluids are immiscible and incompressible All fluids are immiscible and incompressible

Plateau’s laws Films meet at triple edges at 2/3 π (120°) Films meet at triple edges at 2/3 π (120°) Edges meet at tetrahedral nodes at arccos 1/3 (109°3’) - tetrahedral angle Edges meet at tetrahedral nodes at arccos 1/3 (109°3’) - tetrahedral angle

Laplace’s law The average curvature of a film separating two bubbles is determined by pressure difference between them The average curvature of a film separating two bubbles is determined by pressure difference between them

Explanation of the title

For low surface tension value and many bubbles TESSELLATION TESSELLATION A regular tiling of polygons (2D), polyhedra (3D)

Aristotle Similar cells Similar cells Tetrahedra fill up the space Tetrahedra fill up the space „ On the Heavens ”

Kelvin Similar cells Similar cells Minimum surface area Minimum surface area The best block: 14-sided polyhedron (tetraidecahedron) The best block: 14-sided polyhedron (tetraidecahedron) Thomson W. (Lord Kelvin), On the division of space with minimum partitional area, Phil. Mag., 24, 503 (1887)

Weaire and Phelan Two kinds of equal-volume cells: dodecahedron & 14- sided tetrakaidecahedron Two kinds of equal-volume cells: dodecahedron & 14- sided tetrakaidecahedron Unit cell structured from 8 cells Unit cell structured from 8 cells 0,3% in area better partition than Kelvin’s partition 0,3% in area better partition than Kelvin’s partition Weaire D., Phelan R., A counterexample to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett., 69, 107 (1994)

Recapitulation TESSELLATION Aristotle Kelvin Weaire and Phelan

Results – effect of μ Film curvature radius vs μ for 1 0 Film curvature radius vs μ for 1 0 Concave → convex shape Concave → convex shape R/V 1/3 = for sphere

Results – dependence of topology on μ Value of μ determines the shape of cells Value of μ determines the shape of cells ¤ at low μ, 2 0 form two pentahedra ¤ at low μ, 2 0 form two pentahedra ¤ at μ ¤ at μ ≥ 0,3 a trihedron and a trigonal prism are formed 2 0 object

Results – effect of μ Dependencies of different interfacial energy components on μ for 2 0 object Dependencies of different interfacial energy components on μ for 2 0 object

Results – effect of μ Dependencies of the Dependencies of the E T on μ for:

Results – spontaneous processes

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Conclusions At low μ - final product is 1 0 ; At high μ - no strictly defined final product is observed; Multiplets located at nodes can increase their multiplicities by the association of X 1 objects; Some geometrical structures are not fully adaptable to the node symmetry (e.g. 5 0 are energetically forbidden);

Kelvin partition in Surface Evolver

Weaire and Phelan partition in Surface Evolver

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