Yield stress of a foam

Bikerman [7] has proposed a formula for the yield stress of a foam, accounting for the angle between the plate (determining the shear stress) and the films... 

The yield stress of a foam depends to a considerable extent on the character of foam interaction with the tube walls or the cylindrical surface of the viscometer, used in the study of its rheological properties. At low flow rates and smooth tube walls the maximum shear stress of the foam layers contacting the wall can be less than the shear stress of the foam matrix (shear of bubble layers). Hence, the foam flow will occur as a movement of a continuous medium in a cylinder covered with a thin lubricating layer of thickness 2-10 pm [9,16], In this case t0 is ca. 1 Pa, that is, much less than its theoretical value. 

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. 

H.M. Princen Rheology of Foams and Highly Concentrated Emulsions I. Elastic Properties and Yield Stress of a Cyhndrical Model System. J. Colloid Interface Sci. 91, 160 (1983). 

Princen HM. Rheology of foams and highly concentrated emulsions. I. Elastic properties and yield stress of a cylindrical model system. J Colloid Interface Sci 1983 91 160-175. 

Variation of initial compressive yield stress of ESI/LDPE foams with temperature, compared with an EVA and a... 

In reality, the microstmcture of LDPE foams remains very similar as the density inaeases from 18 to 100 kg m, the main changes being in the cell face thickness. The fraction of polymer in the cell faces is greater than 70%, and the initial compressive yield stress of LDPE varies approximately with the 1.5th power of the density (a.15). This does not mean that the model behind Equation (7) is appropriate. 

Harden s (27) market survey of the growth of polyolefin foams production and sales shows that 114 x 10 kg of PE was used to make PE foam in 2001. The growth rate for the next 6 years was predicted as 5-6% per year, due to recovery in the US economy and to penetration of the automotive sector. In North America, 50% of the demand was for uncrosslinked foam, 24% for crosslinked PE foams, 15% for EPP, 6% for PP foams, 3% for EVA foams and 2% for polyethylene bead (EPE) foam. As protective packaging is the largest PE foam use sector, PE foam competes with a number of other packaging materials. Substitution of bead foam products (EPP, EPE, ARCEL copolymer) by extruded non-crosslinked PE foams, produced by the metallocene process was expected on the grounds of reduced costs. Compared with EPS foams the polyolefin foams have a lower yield stress for a given density. Compared with PU foams, the upper use temperature of polyolefin foams tends to be lower. Eor both these reasons, these foams are likely to coexist. 

To determine rheological parameters such as the yield stress and effective viscosity of a foam, commercial rheometers are available rotational and conlinuous-lfow-tubc viscometry are most commonly employed (See also Rheology). However, obtaining reproducible results independent of the sample geometry is a diflicull goal which arguably has not been achieved in most of the experiments reported in the scientific lileralure... 

Foams have a wide variety of applications that exploit Iheir different physical properties. The luw density, or high volume fraction of gas. enable foams to float on top of other fluids and to till large volumes with relatively little fluid material. These features are of particular importance in iheir use lor lire lighting. The very high internal surface area of foams makes them useful in many separation processes. The unique rheology of foams also results in a wide variety of uses, as a foam can behave as a solid, while slill being able to flow once its yield stress is exceeded. Foams are used in food, oil recovery, detergents, textiles, and cosmetics. 

Reference to Table 14 will show the effect of increasing levels of APES on the compressive properties of an anhydride cured epoxide/silica microballoon foam, the APES being added on the resin content. The notation w/r (wt% resin) has been used in the tables. Both the yield stress and strain to failure increased steadily with increased silane content, with a corresponding increase in compressive modulus. At the 5 wt% level there was no real increase in yield stress but a marked increase in strain to failure, resulting in a lower modulus. However, at the 4% level the compressive strength was more than double that of the nonsilane control. 

If the liquid laminae of a foam system can be converted to impermeable solid membranes, the film viscosity can be regarded as having become infinite, and the resulting solid foam will be permanent. Likewise, if the laminae are composed of a gingham plastic or a thixotrope, the foam will be permanently stable for bubbles whose buoyancy does not permit exceeding the yield stress. For other non-newtonian fluids, however, and for all newtonian ones, no matter how viscous, the viscosity can only delay but never prevent foam disappearance. The popular theory, held since the days of Plateau, that foam life is proportional to surface viscosity and inversely proportional to interfacial tension, is not correct, according to Biker-man (op. cit., p. 161), who points out that it is contradicted by experiment. 

Calculations of the elastic constants and yield stress of disordered two-dimensional foams as a function of the gas volume fraction have been reported by Weaire et al. [26-28]. 

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. 

Equation (7.24) indicates that, if the foam is required to have a certain compressive yield stress, yet minimum density, the yield stress of the polymer in the bulk state must be high. Polystyrene has a yield stress at high strain rates of 120MPa whereas polypropylene has a yield stress of 60 MPa. Consequently, an EPS helmet will have a lower density than an EPP helmet designed to meet the same impact tests. Therefore, EPS is optimal for helmets that offer single-impact protection. 

Ventilation of helmets has become a selling point. Helmets are advertised on the number or size of the ventilation openings. Such openings mean that the foam density must be increased, to increase the compressive yield stress of the remaining material. Aerodynamics has been applied to increase the air flow through such ventilation holes. However, given the lack of research on heat transfer from the head, the benefit of some of the styling features has not been established... 

As the temperature increases, the polymer chains become more ductile, leading to reductions in the moduli and yield strength for both the pure SMP and foam. This behavior is particularly pronounced at temperatures higher than the Tg. Figure 3.11 shows the stress-strain responses at 79 °C, just above the Tg region. There is approximately a 97% reduction in the yield strength of the foam eompared to room temperature and that of the pure SMP behaves like a mbber, obscuring the yield point. When the temperature is increased to 121 °C, even the foam tends to... 

This basic model seems contrived, in that no direct experimental justification is given for it by Pelton and Goddard [47], Indeed these workers argue [47] that groups in this treatment are a mathematical convenience to facilitate the derivation and not a physically observable cluster of bubbles. However, in a later paper, which uses a similar model, Pelton [48] claims to observe formation of secondary bubbles in the foam column. It is claimed that they expand in size by coalescence until the buoyancy force exceeds the yield stress in the foam whereupon they rise rapidly to the top of the foam column and rupture. 

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. 

One simple rheological model that is often used to describe the behavior of foams is that of a Bingham plastic. This appHes for flows over length scales sufficiently large that the foam can be reasonably considered as a continuous medium. The Bingham plastic model combines the properties of a yield stress like that of a soHd with the viscous flow of a Hquid. In simple Newtonian fluids, the shear stress T is proportional to the strain rate y, with the constant of proportionaHty being the fluid viscosity. In Bingham plastics, by contrast, the relation between stress and strain rate is r = where is... 

Consistent with this model, foams exhibit plug flow when forced through a channel or pipe. In the center of the channel the foam flows as a soHd plug, with a constant velocity. AH the shear flow occurs near the waHs, where the yield stress has been exceeded and the foam behaves like a viscous Hquid. At the waH, foams can exhibit waH sHp such that bubbles adjacent to the waH have nonzero velocity. The amount of waH sHp present has a significant influence on the overaH flow rate obtained for a given pressure gradient. 

Cellular materials can collapse by another mechanism. If the cell-wall material is plastic (as many polymers are) then the foam as a whole shows plastic behaviour. The stress-strain curve still looks like Fig. 25.9, but now the plateau is caused by plastic collapse. Plastic collapse occurs when the moment exerted on the cell walls exceeds its fully plastic moment, creating plastic hinges as shown in Fig. 25.12. Then the collapse stress (7 1 of the foam is related to the yield strength Gy of the wall by... 

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