Języki
Strony
Prawny
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
???
Top Related