A number of food emulsions exhibit rheological behavior known as plasticity (e.g., margarine, butter, and certain spreads) (Sherman 1968a,c, 1970; Tung and Paulson 1995). A plastic material has elastic properties below a certain applied stress, known as the yield stress, but flows like a fluid when this stress is exceeded.
FIGURE 8.8 Hysteresis curve for liquids whose viscosity depends on the length of time they are sheared.
FIGURE 8.9 Rheological behavior of ideal and nonideal plastics. 18.104.22.168. Ideal Plastics
The ideal plastic material is referred to as a Bingham plastic, after the scientist who first proposed this type of rheological behavior (Sherman 1970). Two equations are needed to describe the rheological behavior of a Bingham plastic, one below the yield stress and one above it:
where G is the shear modulus, n is the viscosity, and T0 is the yield stress. The rheological properties of an ideal plastic are shown in Figure 8.9.
Foods that exhibit plastic behavior usually consist of a network of aggregated molecules or particles dispersed in a liquid matrix (Clark 1987, Edwards et al. 1987, Tung and Paulson 1995). For example, margarine and butter consist of a network of tiny fat crystals dispersed in a liquid oil phase (Moran 1994). Below a certain applied stress, there is a small deformation of the sample, but the weak bonds between the crystals are not disrupted. When the critical yield stress is exceeded, the weak bonds are broken and the crystals slide past one another, leading to flow of the sample. Once the force is removed, the flow stops. A similar type of behavior can be observed in emulsions containing three-dimensional networks of aggregated droplets.
Above the yield stress, the fluid flow may exhibit non-Newtonian behavior similar to that described earlier for liquids (e.g., pseudoplastic, dilatant, thixotropic, or rheopectic). The material may also exhibit nonideal elastic behavior below the yield stress (e.g., the yield point may not be sharply defined; instead, the stress may increase dramatically, but not instantaneously, as the shear rate is increased) (Figure 8.9). This would occur if the material did not all begin to flow at a particular stress, but there was a gradual breakdown of the network structure over a range of stresses (Sherman 1968a,c).
Many food emulsions are not pure liquids or pure solids, but have rheological properties that are partly viscous and partly elastic (Sherman 1968a,c, 1970; Dickinson 1992). Plastic materials exhibit elastic behavior below a certain value of the applied stress and viscous
behavior above this value. In contrast, viscoelastic materials exhibit both viscous and elastic behavior simultaneously. In an ideal elastic solid, all the mechanical energy applied to the material is stored in the deformed bonds and is returned to mechanical energy once the force is removed (i.e., there is no loss of mechanical energy). On the other hand, in an ideal liquid, all of the mechanical energy applied to the material is dissipated due to friction (i.e., the mechanical energy is converted to heat). In a viscoelastic material, part of the energy is stored as mechanical energy within the material, and part of the energy is dissipated. For this reason, when a force is applied to a viscoelastic material, it does not instantaneously adopt its new dimensions, nor does it instantaneously return to its undeformed state when the force is removed (as an ideal elastic material would). In addition, the material may even remain permanently deformed once the force is removed. The rheological properties of a viscoelastic material are characterized by a complex elastic modulus (E*) which is comprised of an elastic and a viscous contribution:
Here, E is known as the storage modulus and E" as the loss modulus.
Two types of experimental tests are commonly used to characterize the rheological properties of viscoelastic materials: one based on transient measurements and the other on dynamic measurements (Whorlow 1992). Both types of tests can be carried out by the application of simple shear, simple compression, or bulk compression to the material being analyzed. Simple shear tests are the most commonly used to analyze food emulsions, and therefore only these will be considered here. Nevertheless, the same basic principles are also relevant to compression tests.
22.214.171.124. Transient Tests
In a transient experiment, a constant stress is applied to a material and the resulting strain is measured as a function of time or vice versa.
Creep. In a creep experiment, a constant stress is applied to a material and the change in its dimensions with time are monitored, which results in a strain versus time curve (Sherman 1968c, 1970). The data are usually expressed in terms of a parameter called the compliance (J), which is equal to the ratio of the strain to the applied stress (and is therefore the reciprocal of the modulus). The compliance is proportional to the strain, but it is a better parameter to use to characterize the rheological properties of the material because it takes into account the magnitude of the applied stress. The time dependence of the compliance of a material can also be measured when the stress is removed, which is referred to as a creep recovery experiment. A typical compliance versus time curve for a viscoelastic material is shown in Figure 8.10 (Sherman 1968a). This curve can be divided into three regions:
1. A region of instantaneous elastic deformation in which the bonds between the particles are stretched elastically. In this region, the material acts like an elastic solid with a compliance J0 given by the ratio of the strain to the applied stress.
2. A region of retarded elastic compliance in which some bonds are breaking and some are reforming. In this region, the material has viscoelastic properties and its compliance is given by JR = JM [1 - exp(-t/TM)], where JM and tm are the mean compliance and retardation time.
3. A region of Newtonian compliance (J) when the bonds are disrupted and do not reform so that the material only flows: JN = t/nN.
FIGURE 8.10 A typical creep versus time curve for a viscoelastic material, such as ice cream.
The total creep compliance of the system is therefore given by:
J(t) = Jo + Jr(t) + JN(t) = Jo + Jm[1 - exp(-t / TM)] + t / nn (8-9)
This type of material is usually referred to as a viscoelastic liquid, because it continues to flow for as long as the stress is applied. Some materials exhibit a different type of behavior and are referred to as viscoelastic solids. When a constant stress is applied to a viscoelastic solid, the creep compliance increases up to a finite equilibrium value (J) at long times rather than continuously increasing. When the force is removed, the compliance returns to zero, unlike a viscoelastic liquid, which does not return to its initial shape.
Stress Relaxation. Instead of applying a constant force and measuring the change in the strain with time, it is also possible to apply a constant strain and measure the change in the stress acting on the material with time. This type of experiment is referred to as a stress relaxation. The same type of information can be obtained from creep and stress relaxation experiments, and the method used largely depends on the type of rheological instrument available.
In a dynamic experiment, a sinusoidal stress is applied to a material and the resulting sinusoidal strain is measured or vice versa (Tung and Paulson 1995, Liu and Masliyah 1996). In this section, only the case where a stress is applied to the sample and the resultant strain is measured is considered. The applied stress is characterized by its maximum amplitude (t0) and its angular frequency (ra). The resulting strain has the same frequency as the applied stress, but its phase is different because of relaxation mechanisms associated with the material (Whorlow 1992). Information about the viscoelastic properties of the material can therefore be obtained by measuring the maximum amplitude (y0) and phase shift (5) of the strain (Figure 8.11). The amplitude of the applied stress used in this type of test is usually so small that the material is in the linear viscoelastic region (i.e., the stress is proportional to the strain), and the properties of the material are not affected by the experiment (van Vliet 1995, Liu and Masliyah 1996). If the applied stress varies sinusoidally with time, then (Whorlow 1992):
and the resulting harmonic strain is
The compliance of the material is therefore given by:
J(t) = — = — (cos 8 cos rot + sin 8 sin rot) (8.12) T 0 T 0
where J (= Yo cos 8/t0) is known as the storage compliance, which is the in-phase component of the compliance, and J' (= Yo sin 8/t0) is known as the loss compliance, which is the 90° out-of-phase component of the compliance. The in-phase component of the compliance is determined by the elastic properties of the material, whereas the 90° out-of-phase component is determined by the viscous properties. This is because the stress is proportional to the strain (t ^ y) for elastic materials, whereas it is proportional to the rate of strain (t ^ dy/dt) for viscous materials (Macosko 1994).
The dynamic rheological properties of a material can therefore be characterized by measuring the frequency dependence of the applied stress and the resulting strain and then plotting a graph of J and J" versus frequency. Alternatively, the data are often presented in terms of the magnitude of the complex compliance (J* = J - iJ") and the phase angle:
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