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

Explanation of the Explanation of the titletitle

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

Results – spontaneous Results – spontaneous processesprocesses

Results – spontaneous Results – spontaneous processesprocesses

+

Results – spontaneous Results – spontaneous processesprocesses

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);

Thank you four your Thank you four your attentionattention