gwnow@amu.edu.pl

The filling up tetrahedral nodes in the monodisperse foams and emulsions with Reuleaux-like tetrahedra

The filling up tetrahedral nodes in the monodisperse foams and emulsions with Reuleaux-like tetrahedra

Department of Physical ChemistryFaculty of Chemistry, UAM, Poznań

Waldemar Nowicki, Grażyna NowickaWaldemar 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)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 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) 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

2Dbidispersecellularfluids

SURUZ2003

Surface Evolver by Keneth Brakke (Susquehanna University)

3 dimensional bi-disperse cellular fluids

tetrahedron (343–6)

22-n12n SSE

2

4 4RS

22

4 sin42

3 RrS

3

4tan

43

tanarctan4

Rr

2arcsin2

Interfacial energy vs. curvature radius

tetrahedron (343–6) Interfacial energy vs. curvature radius

1

2

1

2

sphere (11) Interfacial energy vs. curvature radius

lens (121–1) Interfacial energy vs. curvature radius

trihedron (232–3) Interfacial energy vs. curvature radius

Minimum curvature radius vs. relative interfacial tension

1

2

1

2

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

tetrahedron (343–6) Mixing energy vs. volume fraction

ref

refN

E

EE

11

3

R

mixB,222

3

R

mixA,2ref

V

V

A

SN

V

V

A

SWE

22WSEK

ENEE KN

R=Rmin

tetrahedron (343–6) Mixing energy vs. volume fraction

1

2

1

2

sphere (11)

Mixing energy vs. volume fraction

R=Rmin

lens (121–1) Mixing energy vs. volume fraction

R=Rmin

trihedron (232–3) Mixing energy vs. volume fraction

5.1013.39 11 121–1 232–3 343–6

Mixing energy vs. relative interfacial tension

1

2

1

2

5.1013.39 11 121–1 232–3 343–6

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.

Thank youfor your attention

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