KELVIN FOAM

In general the foam density reduces as the amount of blowing agent is increased, with a lower limit set by foam stability. It is possible to model the factors which affect the final density Mahapatro and co-workers (206) used a regular Kelvin foam model to analyse the expansion of PE foams. The foam has uniform sized cells, each with eight hexagonal faces and four square... 

Micromechanics theories for closed cell foams are less well advanced for than those for open cell foams. The elastic moduli of the closed-cell Kelvin foam were obtained by Finite Element Analysis (FEA) by Kraynik and co-workers (a. 14), and the high strain compressive response predicted by Mills and Zhu (a. 15). The Young s moduli predicted by the Kraynik model, which assumes the cell faces remain flat, lie above the experimental data (Figure 7), while those predicted by the Mills and Zhu model, which assumes that inplane compressive stresses will buckle faces, lie beneath the data. The experimental data is closer to the Mills and Zhu model at low densities, but closer to the Kraynik theory at high foam densities. 

Young s modulus for LDPE foams versus relative density, compared with predictions for the Kelvin foam (a.l5)... 

Mills and Zhu (a. 15) used a Kelvin foam model, in which face tensions restrain the bending of cell edges... 

A Kelvin foam model with planar cell faces was used (a. 17) to predict the thermal expansion coefficient of LDPE foams as a function of density. The expansion of the heated gas is resisted by biaxial elastic stresses in the cell faces. However SEM shows that the cell faces are slightly wrinkled or buckled as a result of processing. This decreases the bulk modulus of the... 

The density of chemicaUy-blown LDPE foam was altered by varying the amount of blowing agent, degree of crosslinking of the polymer, and the foam expansion temperature. A theory was proposed for the equilibrium density, based on the gas pressures in a Kelvin foam structure, and a rubber-elastic analysis of the biaxial stretching of the cell faces. 20 refs. 

Micro-mechanics models for foam deformation are simplifications of the real structure. Figure 4.23 shows a repeating element of the Kelvin foam cell of Fig. 4.22, prior to deformation. The flat surface at the front is a mirror symmetry plane through the polymer structure, as is the hidden flat surface... 

Figure 4.23 Section of Kelvin foam model, R — 0.027, at compressive strains of 0, 20 and 50%, with contours of principal stress MPa. The slender beam approximation is shown on the left-hand figure, with the applied load (Mills, N. J. to be published).

Figure 4.24 Compressive stress-strain curve for PU foam of density 31 kgm, compressed in-plane —, and through thickness —, compared with the Kelvin foam prediction for compression along [III] direction —...

Micellar Discrete cubic (I1J2) bcc packing Im3m (3D) 72.74 76 78 716 ., etc. Bilayer lines the faces of the Kelvin foam... 

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

For a long time it was thought that soap foams, grains in metals and so on were icosahedra. It took Lord Kelvin (of the degree K) to get it right. 

Simone and Gibson (a. 16) predicted the effect of wrinkled cell faces (in aluminium closed cell foams), on the Young s modulus, by FEA of a modified Kelvin... 

The faces in low density LDPE foams are partly buckled or wrinkled, as a result of processing (a.l7). This affects both the bulk modulus and the Young s modulus. The foam bulk modulus Kp is predicted, using the Kelvin closed cell foam model, to be ... 

Support for this postulation came from work done on the shape of the ideal foam cell [32-40]. Ross and co-worker [34,35] proposed three minimal geometric structures, i.e. those which will subdivide space with minimum parti-tional area. These were the pentagonal dodecahedron, the minimal tet-rakaidecahedron, originally suggested by Thomson (Lord Kelvin), and the P-tetrakaidecahedron (Fig. 4). 

The pentagonal dodecahedron, however, is not entirely space-filling, i.e. a close-packed array of such figures has a number of interstitial voids. On the other hand, Kelvin s tetrakaidecahedron and the P-tetrakaidecahedron are. The latter requires 4% more surface area, so a system of such figures would spontaneously rearrange to the more stable array of Kelvin cells. Thus, it would seem that Kelvin s tetrakaidecahedron is the ideal candidate nevertheless, this is not observed in real systems Pentagonal faces are shown on foam cells. These... 

However, Weaire et al. [41] have recently shown that it is possible to produce monodisperse dry foams containing Kelvin polyhedra. Upon addition of liquid, a structural rearrangement occurs at 0.87, to a system made up of mainly pentagonal faces. Thus, it would seem that a transition from pentagonal dodecahedral to tetrakaidecahedral packing may take place on reduction of foam liquid content. 

The theoretical analysis for two-dimensional foams and emulsions has recently been expanded to three dimensions [38], with Kelvin s minimal tet-rakaidecahedron as the unit cell. The system is subjected to a uniaxial extensional strain. As the elastic limit, or yield point, is approached, the cell shape tends towards a rhombic dodecahedron however, at the yield point, the shrinking quadrilateral faces of the polyhedron have finite (albeit small) area. 

There are two types of foams closed cell foams and open cell (or reticulated) foams. In open foams, air or other fluids are free to circulate. These are used for filters and as skeletons. They are often made by collapsing the walls of closed cell foams. Closed cell foams are much stiffer and stronger than open cell foams because compression is partially resisted by increased air pressure inside the cells. Figure 19.1 shows that the geometry of open and closed cell foams can modeled by Kelvin tetrakaidecahedra. 

Later observations of Irish scientists [74-75] have shown that in the surface layers of a foam or in a foam in narrow cylinders or in the narrow space between two plates, Kelvin s polyhedra are observed as well as fragments of clathrate structures, consisting of 6 tetrakaidecahedra and 2 dodecahedra [75]. 

Shear stress for three-dimensional foams using the Kelvin s tetrakaidecahedron model is given in [29], The value of Young s modulus (modulus of extension) was calculated to be... 

The physicochemical properties of foam and foam films have attracted scientific interest as far back as a hundred years ago though some investigations of soap foams were carried out in the seventeen century. Some foam forming recipes must have been known even earlier. The foundations of the research on foam films and foams have been laid by such prominent scientists as Hook, Newton, Kelvin and Gibbs. Hook s and Newton s works contain original observations on black spots in soap films. 

At high shear rates in some systems, the onions become large and very monodisperse in size, and they then order into a macrocrystalline packing. At rest, it is clear that the onions are not spherical, but polyhedral, because they must fill space. In the perfectly ordered macrocrystalline state, the typical shape of the space-filling onions appears to be that of the Kelvin tetrakaidecahedron, which is a model structure for liquid foams (see Section 9.5.1). These well-defined MLVs might be important as encapsulants in the pharmaceutical or cosmetics industries (Roux and Diat 1992). 

It was repeatedly proposed to use Kelvin s tetrakaidecahedron (that is, minimal truncated octahedron) [381, 407, 479] with eight hexagonal and six quadrangular faces as the polyhedral model of a foam cell and of a cell of any three-dimensional biological tissue. Note, however, that it was statistically shown [195] that Kelvin s tetrakaidecahedron is encountered in biological tissues among other tetrakaidecahedral cells only in 10% of the cases. 

FIGURE 15.17 IGFs without grains, (a and b) Models of the Kelvin cell and a Weaire-Phelan foam used to describe soap bubbles (c-e) how these relate to the structure of TJs and QJs. 

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