SPONTANEOUS SPONTANEOUS TOPOLOGICAL TRANSITIONS TOPOLOGICAL TRANSITIONS
IN BIDISPERSE CELLULAR IN BIDISPERSE CELLULAR FLUIDSFLUIDS
Rafał ORafał Olejniczaklejniczak and Waldemar and Waldemar NNowickiowicki
Department of Physical Chemistry, Faculty of Chemistry, A. Mickiewicz University, Poznań,
Poland
ModelModel
3 phase fluid system3 phase fluid system Cells A, B immersed in liquid CCells A, B immersed in liquid C All fluids are immiscible and All fluids are immiscible and
incompressible incompressible
Plateau’s lawsPlateau’s laws
Films meet at triple edges at 2/3 Films meet at triple edges at 2/3 ππ (120(120°°)) Edges meet at tetrahedral nodes at arccos Edges meet at tetrahedral nodes at arccos
1/3 (1091/3 (109°°3’) - tetrahedral angle3’) - tetrahedral angle
Laplace’s lawLaplace’s law
The average curvature of a film separating The average curvature of a film separating two bubbles is determined by pressure two bubbles is determined by pressure difference between themdifference between them
For low surface tension value For low surface tension value and many bubblesand many bubbles
TESSELLATIONTESSELLATION
A regular tiling of polygons (A regular tiling of polygons (2D2D), polyhedra (), polyhedra (3D3D))
AristotleAristotle
Similar cellsSimilar cells Tetrahedra fill up the Tetrahedra fill up the
spacespace
„ „ On the Heavens On the Heavens ””
KelvinKelvin
Similar cellsSimilar cells Minimum surface areaMinimum 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 Weaire and PhelanPhelan
Two kinds of equal-volume cells: dodecahedron & Two kinds of equal-volume cells: dodecahedron & 14-sided tetrakaidecahedron14-sided tetrakaidecahedron
Unit cell structured from 8 cellsUnit cell structured from 8 cells 0,3% in area better partition than Kelvin’s partition0,3% 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)
Results – effect of Results – effect of μμ
Film curvature Film curvature radius vs radius vs μμ for for 1100
Concave Concave → → convex convex shapeshape
R/V 1/3= 0.620 for sphere 31/ /VR
1E-3 0,1 1 10-20
-10
0
10
R/V1/3=0,620
R/V
1/3
Results – effect of Results – effect of μμ
Dependencies of Dependencies of different interfacial different interfacial energy energy components on components on μμ for 2for 20 0 objectobject
0 3 6 9
0
20
40
E
EACB
EACA
EBCB
ET
Results – effect of Results – effect of μμ
Dependencies of Dependencies of thethe
EETT on on μμ for: for:
2233
2222
2211
2200 0,1 1 100,1
1
10
ET
23
22
21
20
ConclusionsConclusions At low μ - final product is 10;
At high μ - no strictly defined final product is observed;
Multiplets located at nodes can increase their multiplicities by the association of X1 objects;
Some geometrical structures are not fully
adaptable to the node symmetry (e.g. 50 are energetically forbidden);