Shear modulus of a foam

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

The shear modulus of a foam is a proportionality coefficient between the applied stress and the shear deformation A/, i.e. 

Although aH these models provide a description of the rheological behavior of very dry foams, they do not adequately describe the behavior of foams that have more fluid in them. The shear modulus of wet foams must ultimately go to zero as the volume fraction of the bubbles decreases. The foam only attains a solid-like behavior when the bubbles are packed at a sufficiently large volume fraction that they begin to deform. In fact, it is the additional energy of the bubbles caused by their deformation that must lead to the development of a shear modulus. However, exactly how this modulus develops, and its dependence on the volume fraction of gas, is not fuHy understood. 

Recently, Kinra and Ker 137) published data of the shear modulus of syntactic foams consisting of hollow glass spheres in a poly(methyl methacrylate) matrix. The glass spheres had a mean radius of 45 pm and a wall thickness of 1.2 pm. Reliable values are known for the shear modulus of the polymer G0, the shear modulus of glass Gs, and Poisson s ratio of the polymer G0 = 1120 MPa, Gs = 2800 MPa, and v0 = 0.35. Using these values, the upper curve 1 of Fig. 24 was calculated by Nielsen for the modulus of the foam as a function of the volume fraction of hollow spheres. These calculated values are, however, too high compared with the experimental values reported by Kinra and Ker. 

Fig. 24. Relative shear modulus of a PMMA/hollow-glass microspheres syntactic foam 67 1) values calculated with a uniform shell thickness of 1.2 pm and Gh/G0 = 1.01 2) values calculated with a shell thickness of 0.6 pm and GH/G = 0.5. Filled circles are the experimental data of Kinra and Ker at 1 MHz1371...

Another important rheological property of dry foams and highly concentrated emulsions is G, the shear modulus. Princen and Kiss [57] demonstrated that this property was dependent on < >, the volume fraction of the system. Previously, Stamenovic et al. [58] and, much earlier, Derjaguin and coworker [59], had derived an expression for the shear modulus of foams of volume fraction very close to unity. The value was found to depend on the surface tension of the liquid phase (in foams), for the particular case of (Jja 1. However, Princen demonstrated that the values of G obtained were overestimated by a factor of two. This error was attributed to the model used by Stamenovic and coworker, which failed to maintain the equilibrium condition that three films always meet at angles of 120° during deformation. 

The compressibility of a foam is determined by (i) the ability of the gas to compress and (ii) its wetting power, which is determined by the properties of the foaming solution [4]. As with any disperse system, a foam may acquire the properties of a solid body - that is, it can maintain its shape and it possesses a shear modulus (see below). 

Princen [35] used a two-dimensional hexagonal package model to derive an expression for the shear modulus and yield stress of a foam, taking into account the foam expansion ratio and the contact angles. 

The first expression for the shear modulus of random dry foams (and emulsions) was derived by Deijaguin (91). It is based on the assumption that the foam is a eolleetion of randomly oriented films of eonstant tension 2o and negligible thiekness, and that eaeh film responds affinely to the applied shear strain, as would an imaginary surfaee element in a eontinuum. Evaluating the eontribution to the shear stress of a film of given orientation and averaging over all orientations then leads to... 

Fig. 13. Shear modulus as a function of temperature for PU and PS foams with density, process, and orientation as parameters...

Foams and emulsions soften as their packing fraction decreases—their shear modulus gets smaller and ultimately vanishes continuously at the critical packing fraction Hence, the collective response of a foam or emulsion can be much softer than its individual contacts. 

Rheology. The rheology of foam is striking it simultaneously shares the hallmark rheological properties of soHds, Hquids, and gases. Like an ordinary soHd, foams have a finite shear modulus and respond elastically to a small shear stress. However, if the appHed stress is increased beyond the yield stress, the foam flows like a viscous Hquid. In addition, because they contain a large volume fraction of gas, foams are quite compressible, like gases. Thus foams defy classification as soHd, Hquid, or vapor, and their mechanical response to external forces can be very complex. 

H.M. Princen and A.D. Kiss Rheology of Foams and Highly Concentrated Emulsions III. Static Shear Modulus. I. Colloid Interface Sci. 112, 427 (1986). 

In a subsequent theoretical study, Stamenovic [60] obtained an expression for the shear modulus independent of foam geometry or deformation model. The value of G was reported to depend only on the capillary pressure, which is the difference between the gas pressure in the foam cells and the external pressure, again for the case of <)> ca 1. Budiansky et al. [61] employed a foam model consisting of 3D dodecahedral cells, and found that the ratio of shear modulus to capillary pressure was close to that obtained by Princen, but within the experimental limits given by Stamenovic and Wilson. 

The foam retains some of the properties of the phases that is formed from. For example, its compressibility is determined mainly by the ability of the gas to compress, and its wetting power by the properties of the foaming solution. At the same time, being a disperse system, the foam acquires the properties of a solid body maintains its shape, possesses a shear modulus, etc. 

Experimental determination of the shear modulus by the technique of rotating cylinder has give values of G/pg = 0.23-0.7 for a foam from 0.02% saponin solution [30]. The author explained the value scattering with the non-homogeneity of the foam. For a foam from 0.1% aqueous saponin solution, G/pa = 8.1 which is 15 times higher than the theoretical one and obviously results from the structuring in the adsorption layers. 

Princen HM, Kiss AD (1986) Rheology of foams and highly concentrated emulsions. III. Static shear modulus. J Colloid Interface Sci 112 427-437 Ramsaywak PC, Labb6 G, Siemann S et al. (2004) Molecular cloning, expression, purilication, and characterization of fructose 1,6-bisphosphate aldolase from Mycobacterium tuberculosis -a novel Class II A tetramer. Protein Expres Purif 37 220-228 Richard JP (1993) Mechanism for the formation of methylglyoxal from triosephosphates. Biochem Soc 121 549-553... 

Pri ncen, H.H. Kiss, A.D. Rheology of foams and highly concentrated emulsions. III. Static shear modulus. J. Colloid Interface Sci., 112 427. 1986. 

As pointed out by Reinelt and Kraynik (54), however, the idealized vertex does not adequately represent an equilibrium structure. Similar reservations apply to the work of Budiansky and Kimmel (95), who considered the behavior of an isolated foam cell in the form of a rectangular pentagonal dodecahedron and obtained a shear modulus between the two above values. 

For concentrated emulsions and foams, Princen [182, 183] proposed a stress-strain theory based on a two-dimensional cell model. Consider a steady state shearing of such a system. Initially, at small values of strain, the stress increases linearly as in elastic body. As the strain increases, the stress reaches a yield point, and then at higher deformation it catastrophically drops to negative values. The reason for the latter behavior is the creation of unstable cell structure that relaxes by recoil. For real emulsions the shear modulus and yield stress are expected to follow the expressions ... 

Sandwich panels constructed of a cellular or honeycomb core and laminated outer skins are treated as I beams to calculate the moment and modulus. The laminate skin is the I beam flange, and the core is the beam shear weh in calculations. Under a bending load, the flexural stiffiiess of a sandwich panel is proportional to the cube of its thickness [4]. For these reasons, modified I beams constructed as a foam core-outer skin sandwich... 

In these expressions, G is shear modulus, a and 6 are respectively the inner and outer radii of the microballoon, v is Poisson s ratio, (pm is the maximum packing factor that can be achieved, and 2 is the volume fraction of microballoons. The maximum packing factor, (pm, is about 62.5% (18). The subscript H, S, F, and 1 indicate the hollow sphere, the solid sphere, the foam, and the polymer matrix, respectively. The shear modulus can be converted to Yoimg s modulus by... 

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