Foam cells shape

Urethane foam, cell shape,compressive modulus,density. 

Cell Structure. A complete knowledge of the cell stmcture of a cellular polymer requires a definition of its cell sizes, cell shapes, and location of each cell in the foam. 

CellgeometTy is governed predominantly by the final foam density and the external forces exerted on the cellular stmcture prior to its stabilization in the expanded state. In a foam prepared without such external forces, the cells tend to be spherical or ellipsoidal at gas volumes less than 70—80% of the total volume, and they tend toward the shape of packed regular dodecahedra at greater gas volumes. These shapes have been shown to be consistent with surface chemistry arguments (144,146,147). Photographs of actual foam cells (Fig. 2) show a broad range of variations in shape. 

Density and polymer composition have a large effect on compressive strength and modulus (Fig. 3). The dependence of compressive properties on cell size has been discussed (22). The cell shape or geometry has also been shown important in determining the compressive properties (22,59,60,153,154). In fact, the foam cell stmcture is controlled in some cases to optimize certain physical properties of rigid cellular polymers. 

Those stmctural variables most important to the tensile properties are polymer composition, density, and cell shape. Variation with use temperature has also been characterized (157). Flexural strength and modulus of rigid foams both increase with increasing density in the same manner as the compressive and tensile properties. More specific data on particular foams are available from manufacturers Hterature and in References 22,59,60,131 and 156. Shear strength and modulus of rigid foams depend on the polymer composition and state, density, and cell shape. The shear properties increase with increasing density and with decreasing temperature (157). 

Mechanical Properties and Structural Performance. As a result of the manufacturing process, some cellular plastics have an elongated cell shape and thus exhibit anisotropy in mechanical, thermal, and expansion properties (35,36). Efforts are underway to develop manufacturing techniques that reduce such anisotropy and its effects. In general, higher strengths occur for the paraHel-to-rise direction than in the perpendicular-to-rise orientation. Properties of these materials show variabiUty due to specimen form and position in the bulk material and to uncertainty in the axes with respect to direction of foam rise. Expanded and molded bead products exhibit Httie anisotropy. 

Fig. 26.2. The microstructure of wood. Woods ore foams of relative densities between 0.07 and 0.5, with cell walls which ore fibre-reinforced. The properties ore very anisotropic, partly because of the cell shape and partly because the cell-wall fibres ore aligned near the axial direction.

The transverse modulus is lower partly because the cell wall is less stiff in this direction, but partly because the foam structure is intrinsically anisotropic because of the cell shape. When wood is loaded across the grain, the cell walls bend (Fig. 26.5b,c). It behaves like a foam (Chapter 25) for which... 

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 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. 

The last topic in evaluating the suitability of reticulated foam as the scaffold of a composite is somewhat qualitative. It is known that hepatic cells do not function when cultured on a flat plate. At least part of the reason for this is the deformation of the cells developed during the spreading process. It seems likely that if a cell is sufficiently deactivated by a flat surface, the effect will be as severe as when culturing on a convex surface such as the outside shape of a hollow fiber. A reticulated foam, however, presents a cell with several opportunities for a more natural attachment. The dodecahedron structure of each foam cell would appear to be a more natural scaffold for attachment. This perhaps explains the claimed superiority of a scaffold based on a reticulated foam of Gion et al. - over the HepatAssist hollow fiber device. Part of the research program that we will propose is that the effects of conformational aspects of an efficient scaffold will be quantified. 

The volume and shape of Plateau borders depend on the expansion ratio of the foam. In a spherical monodisperse foam with close packing of bubbles all air/liquid interfaces are spherical and the liquid volume which belongs to one cell can be derived from the difference between the volumes of the corresponding polyhedron (for example, a dodecahedron) and the inscribed in it sphere, having in mind the co-ordination number of the foam cell. 

The shape of foam films and border profiles in large interval of foam expansion ratio from 10 to 1500 has been experimentally studied in [83], A regular pentagonal dodecahedron made up of transparent organic glass with an elastic rubber balloon inside it which took the shape of a sphere at inflation (Fig. 1.10) was used as a model of foam cell. 

The edges of this dodecahedron sized a - 8.5 cm. When the volume of the rubber balloon at inflation became bigger than the volume of the sphere inscribed in the dodecahedron, the balloon was deformed by the dedecahedron faces and took a shape close to the respective shape of a bubble in a monodisperse dodecahedral foam with a definite expansion ratio. The expansion ratio of the foam was determined by the volume of liquid (surfactant solution or black ink in the presence of sodium dodecylsulphate) poured into the dodecahedron. An electric bulb fixed in the centre of the balloon was used to take pictures of the model of the foam cell obtained. The film shape and the projection of the borders and vertexes on the dodecahedron face are clearly seen in Fig. 1.10. 

Shear strength and modulus of rigid foams depend on the polymer composition and state, density, and cell shape. Shear properties increase with increasing density and decreasing temperature [30]. 

Cell Shape of Foamed Polymers 7,1 Cell Shape Models... 

As with cell shapes of a real foam, cell sizes in this material can also be characterized only by nominal (effective) values. The actual effective values depend, first, on the observation method (whether direct — macroscopic, or indirect — adsorption, volumetric, picnometric, etc.). Secondly, they depend on the particular simplified model of the structure and cell shape and thirdly on the method of processing the measured data. 

The volumetric weight and the ratio between the number of open and closed cells are the fundamental morphological parameters of foamed plastics. Nevertheless, it has been reported that even for the same volumetric weight and number of open (or closed) cells the strength and thermophysical parameters may be widely different in plastic foams made of the same polymer grade. The differences in cell shapes and sizes are responsible for this fact. 

Fig. 23 a and b. Cell shape variation (a) in nonreticulated and (b) in reticulated flexible polyurethane foams under compressive and tensile loads (1) initial state, (2) compression, (3) tension... 

As pointed out in Section 7, the plastio foam cells have a very complex shape, most frequently that of irregular polygons, which makes the use of the above formulas too cumbersome nevertheless, the order of the specific surface value given by the equations is correct. 

Denser foams behave differently in compression tests. The brittle failure of samples involves cracking along the inclined and longitudinal planes (Fig. 29 d). When compressive load is applied to a material of high apparent density, such as polyurethane foam (Fig. 29 e), the load will go on increasing continuously even after has been attained. At the same time, the cross-section of the sample will increase and the sample assume a barrel-like shape. The foam cells will begin to crumple at the same time. 

Foam cell size and shape, foam cell size distribution (if aerosol-type product)... 

In rigid urethane foams, the cell shapes are elliptical like eggs, and, therefore, the compressive strengths in the direction perpendicular to foam rise is smaller than the direction parallel to foam rise. Therefore, if urethane foams are required to have the same compressive strength as pyranyl foams in the direction perpendicular to foam rise (i.e., compressive strength in the direction vertical to the panel substrate) urethane foams must have foam densities greater than those of pyranyl foams. Adhesion of pyranyl foams to various substrates, e.g., steel, phosphated steel, stainless steel and aluminum, as well as paper and wood, is very good. 

The foam expansion ratio can be characterised by the liquid volume fraction in the foam, which is the sum of the volume fractions of the films, plateau borders and vertexes. Alternatively, the foam density can be used as a measure of the foam expansion ratio. The reduced pressure in the foam plateau border can be measured using a capillary manometer [4], while the bubble size and shape distribution in a foam can be determined by microphotography of the foam. Information about the liquid distribution between films and plateau borders is obtained from the data on the border radius of curvature, the film thickness, and the film-to-plateau border number ratio obtained in an elementary foam cell. 

Actual foam contains bubbles whose shape is intermediate between spheres and polyhedra. Such foam is said to be cellular [214, 280]. The distinction between the cellular and polyhedral kinds of foam is rather conventional and is determined by very low moisture contents (of the order of some tenth of per cent). Nevertheless, the polyhedral model of foam cells is used rather frequently [38,125,244,438,480],... 

Dispersity. Foam cells usually have the shape of rounded polyhedra. Therefore, it is convenient to choose the radius a of the volume-equivalent bubble, that is, of a spherical bubble of the same volume as the cell as the single linear dimension characterizing the interior scale of foam. Foam consisting of cells of the same size is said to be monodisperse. This kind of foam is extremely rare. Usually, there exists a spectrum of radii ai,..., a in this case, the foam dispersity characterizes the mean linear dimension of the cell [214] ... 

Preliminary remarks. Models of the foam cell. The polyhedral shape of foam cells is the limit shape as the foam multiplicity grows infinitely. At the same time, this is a rather convenient structural model for actual foam with finite multiplicity. A polyhedron constructed of liquid films must satisfy the following two rules, stated by Plateau [9, 379, 407] ... 

Foam cells have the shape of equal pentagonal dodecahedra with rounded edges and vertices. 

A complicated many-layer structure of foam cells is formed when gas bubbles are sparged into solutions of surfactants. According to [429], each bubble has a two-sided envelope which is a layer of the solvent containing hydrophilic polar parts of surfactant molecules (see Figure 7.2). Nonpolar hydrophobic parts of molecules on the inner surface of the envelope are oriented toward the bubble, and on the outer surface, outward the envelope. Between two cells, each of which is a capsule with envelope, there is a lamella, that is, an interlayer of a complicated structure. In the middle of the lamella, there is a liquid layer that is a continuous phase. On each of two surfaces of this layer, there is a monolayer of the surfactant. The hydrophobic parts ( tails ) of surfactants molecules in each monolayer and the tails of the envelope form two direct plate micellae [413], which separate the envelope and the continuous liquid film at the center. Thus, gas bubbles in foam are separated at least by five distinct layers. The multilayer structure of a foam lamella is well seen in photographs (e.g., see [429], p. 54). This fact is also confirmed by the ladder-type shape of the disjoining pressure... 

Effective viscosity of viscoelastic foam. Foam exhibits viscous properties even if it is in the solid-shaped state. The point is that fluid elements of a foam cell also contribute to the shear deformation. The rate of energy dissipation in these elements depends on the frequency of the applied action, or, which... 

Matzke, E. B., The three-dimensional shape of bubbles in foam-an analysis of the role of surface forces in three-dimensional cell shape determination, Amer. J. Botany, Vol. 33, No. 1, pp. 58-80, 1946. 

The phenomena just mentioned lead to formation of a polyhedral foam the shape of the air cells approximates polyhedra. For cells of equal volume, the shape would be about that of a regular dodecahedron (a body bounded by 12 regular pentagons), and the edge q then equals about 0.8/-, where r is the radius of a sphere of equal volume. Actually, the structure is less regular, because of polydispersity. Moreover, close packing of true dodecahedrons is not possible. In a two-dimensional foam, say, one layer of bubbles... 

The edges of foam cells are the Gibbs-Plateau borders (channels) filled with dispersion medium (see Chapter VII, 1). It was shown by Plateau that only three films may be joined by one border, and that the films must meet at 120°. The surface of Gibbs- Plateau border has a complex concave shape dictated by the condition requiring that the sum of two main curvatures remains constant. The capillary pressure under a concave surface is the reason for the lowered pressure in the Gibbs-Plateau border. 

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