Yield Stress and Elastic Modulus

We have seen that the rheological properties of weakly flocculated gels can be predicted at least qualitatively using reasonable particle-particle interaction potentials derived from van der Waals and polymer depletion forces. Can a similar approach succeed in predicting the mechanical properties of strongly flocculated gels  

Sensitivity to the shape of W D) differentiates weakly from strongly interacting particles. For the former, the precise shape of the potential is not important we saw in Section 7.2.4 that even a simple square-well potential is an adequate approximation. But insensitivity to the shape of the potential can only be expected when the particles are only weakly bound by that potential, so that rapid, thermally driven changes in particle-particle separation average out the details of the shape of the potential. For strongly flocculated gels, the particle-particle separations remain trapped near the minimum in the potential well, and the shape of the well near this minimum matters much more. 

Nevertheless, if one assumes a static, rather than a thermodynamic, equilibrium, one can attempt to estimate the dependence of the yield stress Oy and the modulus G on the shape and depth of the interparticle potential. Imagine that a gel is subjected to a shear strain Y that homogeneously displaces particles from their positions of static equilibrium. Pairs of particles are pulled apart by this strain, and separations between particle centers of mass should increase roughly by an amount yrQ, where ro = 2a + Dq is the separation between centers of mass in the absence of strain. Hence, the imposition of a strain y increases the gap between particle surfaces from Dq to 

A force F — —W D) with D = y 2a + Dq) Dq is produced by this increased separation between the particles, where W is the derivative of W with respect to D. This force would restore the original interparticle spacing if the shearing stress were removed. 

The macroscopic stress cr is this force times the number of interparticle bonds that cross a unit area of the sample this latter factor should scale as 0 /a (Russel et al. 1989). As long as the local applied force increases with increased strain, cr increases with increasing strain, and the gel maintains its mechanical stability. But once the strain reaches the point -that the slope W of the potential is a maximum (see Fig. 7-23), any further strain produces a decreasing force, and the interparticle structure breaks apart. This corresponds to the point of yield. Thus, the yield strain yy is given by the condition that the second derivative W of W D) is zero that is, W Dy) = 0, where Dy = lyyU + (yy + 1)Dq is the value of D for which W = 0. Very roughly, we might expect that W is a maximum (W — 0) when separation D = Dy % on the order of twice Dq, the value of D at static equilibrium. This would imply that the yield strain yy is roughly hence, for particles 100 nm in 

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A. Flores, F. J. Balta Calleja, G. E. Attenburrow, and D. C. Bassett, Microhardness Studies of Chain-extended PE III. Correlation with Yield Stress and Elastic Modulus, Polymer, 41, 5431 (2000). 

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. 

Fig. 27.2 provides a comparison of different plastics and metals with respect to yield stress and elastic modulus. The plotted values for the various types of plastic are median values of hundreds of grades while the plotted values for the metals correspond to Aluminum 6061-0, Titanium 6-4, tempered AISI... 

Figure 27.2 Comparison of yield stress and elastic modulus.

Tempel (1961) surmised that fat crystal network structure consisted of an assembly of chains, each chain being an assembly of a linear array of closely aligned particles. The chains were considered to be branched and interlinked to form a three-dimensional network, with liquid fat filling the voids. Unfortunately, this theory did not successfully account for the nonlinear dependence of rheological phenomena such as yield stress and elastic modulus on the proportion of solid fat. Fat crystal networks are now viewed as being composed of haphazardly interlinked aggregates (Tempel, 1979). 

Since polyethylene mainly crosslinks, the effect of irradiation is to generally enhance the physical properties. For example, for HOPE an increase in both the yield stress and secant modulus at 0.5% strain is observed [52], The effect on the properties above the melting point have been extensively studied [82] and there is a direct relationship of the elastic modulus measured at 160 °C to the dose. For irradi-... 

Oy, E are the yield strength and elastic modulus respectively K is the stress intensity factor "k, P are constants... 

Table 5.2 lists polymers and their tendency toward crystallinity. Yield stress and strength, and hardness increase with an increase in crystallinity as does elastic modulus and stiffness. Physical factors that increase crystallinity, such as slower cooling and annealing, also tend to increase the stiffness, hardness, and modulus of a polymeric material. Thus polymers with at least some degree of crystallinity are denser, stiffer, and stronger than amorphous polymers. However, the amorphous region contributes to the toughness and flexibility of polymers. 

The simplest model assumes ideal elastic behavior (Figure 7.12A). At a stress below the yield stress (Fy), the sample behaves perfectly elastically. In this region, a modulus of elasticity can be determined. At the yield stress, the sample flows. It continues to flow until the stress is lowered again to below the yield stress value. Therefore, both the elastic modulus and yield stress describe the behavior of a plastic material. They can be determined easily by compression testing. The continuous network of fat crystals in a fat bears the stress below the yield stress and therefore contributes solid or elastic properties to the material (Narine and Marangoni, 1999a). 

On the basis of what has been discussed, we are in the position to provide a unified understanding and approach to the composite elastic modulus, yield stress, and stress-strain curve of polymers dispersed with particles in uniaxial compression. The interaction between filler particles is treated by a mean field analysis, and the system as a whole is macroscopically homogeneous. Effective Young s modulus (JE0) of the composite is given by [44]... 

Systematic studies about their relative mechanical performance are rare. The pioneering work of Fujiyama who compared y-quinacridone with quinacridonequinone shows in an incontestable way the non-equivalence of different highly efficient -promoters [21]. For an optimal concentration, the -nucleated resins differed in their notched impact strength, yield stress, elongation at break and elastic modulus. Our own investigations confirmed this feature. 

Compression tests, as well as tension tests, were carried out on the samples of varying AN content. All samples exhibited a compressive yield stress, and with further increase of strain, a subsequent drop in stress, occurred before the stress again started to rise. From the test data, the elastic modulus and the yield strength were determined as a function of nitrile content and the results are shown in Fig. 29. 

March (1964) approximated an elastoplastic indentation to the expansion of a spherical cavity under hydrostatic pressure in an infinite elastic-plastic medium. In this simplified model, the indentation pressure Pm is related to the yield stress Y and elastic modulus, E, through ... 

Both destructive and nondestructive measurements can be done on an Instron Material Tester. In this system, the sample is loaded in a test cell, and the compression or tension force is measured when the upper part of the cell is moved over a given distance (time). Within the elastic limit of the gel, the elastic modulus E (or gel strength) is obtained from the initial slope of the nondestructive stress/strain curve additional deformation results in the breakage of the sample, giving the characteristic parameters—yield stress and breaking strain. 

Like any disperse system, foams produce non-Newtonian systems, and to characterize their rheological properties information must be obtained on the elasticity modulus (the modulus of compressibility and expansion), the shear modulus, yield stress and effective viscosity, and elastic recovery. 

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