Cellular foam models

Foams usually possess a finite low-frequency elastic modulus, along with static and dynamic yield stresses. These and other aspects of foam flow and rheology can be captured qualitatively and even semiquantitatively by cellular foam models. 

The earliest analysis of the deformation properties of a liquid foam is that of Princen (1983), who modeled two-dimensional foams and dense emulsions at rest by an array of regular hexagons (see Fig. 9-32a). While Princen s model was limited to hexagons with a particular orientation relative to the imposed shearing flow, this restriction was lifted in the work of Khan and Armstrong (1986) and Kraynik and Hansen (1986). Polydisperse cell sizes have also been considered (Weaire et al. 1986 Khan and Armstrong 1987 Weaire and Fu 1988 Kraynik et al. 1991 Okuzono et al. 1993), as well as wet foams with 

The stress tensor for a two-dimensional hexagonal foam can be computed either from a work-energy argument or by averaging the stresses in the microstructure over a representative volume (or unit cell) of the foam. Both approaches give the following for the shear stress cT 2 and first normal stress difference N  

Doi and Ohta (1991) derived very similar expressions for cri2, A/i, N2 for the elastic stresses of emulsions in step shearing strains. Larson (1997) has shown that Eqs. (9-57a) and (9-57b) can be derived as an approximation from a general phenomenological film model for affinely stretching, constant-tension interfaces thus cti2, N, and N2 can all be represented by the simple tensor expression 

Nevertheless, some of the predictions of simple regular foam models are relevant to real foams. One such property is the linear modulus Go, which is the slope of the stress-strain curve at zero strain. From Eq. (9-57a), we obtain 

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Emulsions with a high volume fraction of droplets (0 > 0.64) and foams show solidlike properties such as a yield stress and a low-frequency plateau value of G. The magnitudes of the yield stress and elastic modulus can be estimated using simple cellular foam models. These and related models show that at low shear rates where the shear stress is close to the yield value, the flow occurs by way of intermittent bubble-reorganization events. The dissipative processes that occur during foam and emulsion flows are still under active investigation. 

Most published literature analyzed the elastic modulus of silica aerogels by drawing inspiration from the cellular solids models. For example, Ashby and Gibson (1997) describe the open cellular foam model compressive modulus to follow power law dependence on the relative density as shown in Eq. (5.1) where C and /i are geometric constants that depend on the topological features and microstructure undergoing cell wall bending as the dominant deformation. 

An experimental and theoretical study of the degassing of an LDPE high-density foam is presented. Measurements of the mass, dimensions, and density as a function of storage time are reported. A geometrical model is described to represent the basic mass transport and volume relaxation processes in a cellular system. Model predictions were compared with experimental results. 12 refs. 

Cellular Structure Models and Calculation of Mechanical Properties of Foamed Polymers... 

Here p and p are the liquid and gas densities, respectively, g is the vector of the gravitational acceleration, and AP is the capillary rarefaction given by (7.1.10) and (7.1.15). The kinetic coefficient H was called the coefficient of hydroconductivity and calculated for polyhedral foam models [245, 246]. Generally speaking, the variable H is a tensor, but usually the isotropic approximation is used, where this parameter is a scalar. Various expressions for the coefficient H were proposed and made more precise in [125, 214, 245]. Thus, different approaches used to calculate the coefficient of hydroconductivity were analyzed in [488]. For example, the structure of spherical and cellular foam was studied under the assumption that liquid flows through a porous layer according... 

The approaches discussed so far suggest that silica aerogels have complexly networked structures that the final properties depend not only on the topological features but also on the process parameters. For certain, most of the authors used density as the single most important physical property in their analysis. Hence, the models and proposed theories could only provide an estimate of the real values. Nevertheless, both the cellular foam and parametric models revealed a certain... 

Barber, A.D. A model for a cellular foam, PhD Thesis, University of Nottingham, 1973,... 

Experimental and theoretical studies have led to significant gains in knowledge on the structure (both cellular structure and polymer morphology) of foams and on the physical mechanisms that control the different properties of foams. Models have been developed to predict the Young and bulk modulus, creep and gas diffusion coefficient, thermal expansion, and thermal conduc-... 

Properties of peroxide cross-linked polyethylene foams manufactured by a nitrogen solution process, were examined for thermal conductivity, cellular structure and matrix polymer morphology. Theoretical models were used to determine the relative contributions of each heat transfer mechanism to the total thermal conductivity. Thermal radiation was found to contribute some 22-34% of the total and this was related to the foam s mean cell structure and the presence of any carbon black filler. There was no clear trend of thermal conductivity with density, but mainly by cell size. 27 refs. 

A change in composition or a commercial food product s formulation is most likely to affect its cellular stmcture, especially if formed by extrusion or puffing. Thus, studying the effect of stmcture or composition in isolation may not be an easy task. However, there are ways to investigate their effects. For example, freezing at different rates usually produces ice crystals of different sizes, which upon dehydration can produce foams with almost identical composition but different cellular stmcture. Freeze-dried model foams, based on food gums with and without additives can be used to study the effect of the cell wall material in foams that have a similar stmcture (see, e.g., Nussinovitch et al. 2000, 2001). Whether this kind of study will generate wide interest, however, is uncertain. 

Assuming that structural data are available, and that a property has been correctly measured, the next problem is to establish a relationship. Fundamental models are preferred by engineers because tlrey are based on basic principles of physics and the physical chemistry of the described phenomenon. Once it is realized that foods are essentially composite hierarchical structures, we can borrow models and theories developed for nonfood systems and apply them. A good example is the adaptation of mechanical principles for the description of cellular solids, (Gibson and Ashby 1988) to the properties of solid food foams (Attenburrow et al. 1989 Warburton et al. 1990). Examples are provided in Chapter 10. 

For a few decades now cellular and porous systems have been classified in morphological terms by simulating the real systems by one or another imaginary, and always simplified, geometrical or stereometrical scheme using an artificially ordered-structure model. Such classifications have always been based on the concept that in any cellular or porous system it is possible to isolate a structural element (cell or pore). However, the diversity of pore and cell types even in small-sized real foamed systems does, in most cases, not permit a definition by only one single geometrical structural parameter, as for other types of solids (type and volume of elementary cell, interplanar or interatomic distances, etc.)... 

The macrostructure of a real dispersed material, including plastic foams, cannot be visualized without recurring to a geometrical model hence, the concepts of the material structure are always conventional, being valid only on the basis of the GSE model used. Nevertheless, such concepts are necessary both for a quantitative description of a macrostructure and for a comparative analysis of the morphological parameters of the different cellular plastics. 

The cellular anisotropy of plastic foams may be evaluated by e.g. an anisotropy coefficient q which is equal to the ratio between average cell dimensions along the major symmetry axes of the respective model. An isotropic material Is characterized by only one anisotropy coefficient (q = 1), a transversally anisotropic material by two(qj = q and q ) and an orthotropic medium by three coefficients (q q q ). 

Fig. 30a and b. Rod model of a cellular structure (a) and tension-failure diagram (b) of this model for rigid foamed polymers... 

Eq. (77) may be valid for other rigid foams as well, since the model underlying the calculation procedure encompasses morphologies of a large enough range of real cellular polymers. 

Fig. 4 The lipid influx/efflux rheostat model maintains lipid uptake and export mechanisms in a balance. ATP synthase is regulated by apoA-I or apoE leading to enhanced conversion of ATP to ADP. The absence of apoA-I would lead to enhanced sinking in phagocytosis since actin can bind ATP, polymerize, and form F-actin which is essential for type 11 phagocytosis. Hence apoA-I could lead to increased influx. On the other hand, apoA-I binds to ABCAl leading to enhanced lipid efflux. Dysfunction of this equilibrium may lead to severe disturbances of cellular lipid traffic. This is obvious in Tangier disease patients where ABCAl is inoperative and apoA-/-dependent cholesterol is absent. Cholesterol influx, however, is enhanced due to apoA-Z-dependent stimulation of ATP synthase B leading to cholesteryl ester formation and enhanced foam cell formation...

The unique cellular morphologies of foams play a key role in determining their deformation mechanisms [51. They also allow the development of very simple alternative equations based on the mechanical models of beam theory (a branch of civil engineering) combined with scaling concepts, to estimate both the thermoelastic properties and the strengths of foams. Such simple relationships have been developed for foams manifesting elastomeric, elastic-plastic and elastic-brittle responses to mechanical defonnation. While much of this work has focused on the responses of foams to compressive defonnation because of the special importance of this deformation mode in many applications of foams, the responses of foams to tensile and shear deformation have also been considered within this theoretical framework. 

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],... 

Using a continuum rather than cellular model implies that the growing foam is regarded as a fluid with a continuously distributed source of flow. Hence, the flow rate is also a function of position the (hydrodynamic) pressure gradient is resulted from inertia, gravity and stress mechanisms operating in the fluid. 

Kinetic studies have been made on the thermal decomposition of a poly(oxypropylene)triol-toluene di-isocyanate copolymer foam. Following a diffusion rate-controlled step, the cellular structure collapses to a viscous liquid and degradation then occurs on a random scission basis. Products of degradation of A-monosubstituted and A A-disubstituted polyurethanes have been analysed by direct pyrolysis in the ion source of a mass spectrometer. The mono-substituted polymers depolymerize quantitatively to di-isocyanates and diols, whereas the disubstituted materials decompose selectively to secondary amines, olefins, and carbon dioxide. The behaviour of the monosubstituted polymers has been confirmed in an i.r. study of the degradation of model compounds. A study of the thermal degradation in vacuum of polyurethanes prepared from butanediol, methylene bis(4-phenylisocyanate), and hexanedioic acid-ethylene glycol-propylene glycol polyesters has been reported and reaction mechanisms proposed. ... 

In the cellular model (Gibson, 1988), the porous solid is defined as a material built up of struts or plates, which form the edges and the faces of the cells. The most important structural characteristic of the cellular models is their relative density, which can be expressed as a function of the cell edge length, I and the size of the cross section, t. For an open cell foam made of cubic polyhedra, p/ps is proportional to (r//). Then, the elastic and the mechanical features have been calculated respectively E oc and Kic oc (p/ps). ... 

The model generated several sets of random patterns of cellular domains, with cells of varying sizes that mirrored those of the actual foam. However, due to... 

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