gwnow@amu.edu.pl

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 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

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.

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

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 sin423 RrS

3

4tan

43tanarctan4

Rr

2arcsin2

Interfacial energy vs. curvature radius

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

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

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

refNE

EE

11

3

R

mixB,222

3

R

mixA,2ref

VV

ASN

VV

ASWE

22WSEK

ENEE KN

R=Rmin

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

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

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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