Nonideality may manifest itself in a number of different ways; for example, the viscosity of a liquid may depend on the shear rate and/or the time over which the shear stress is applied, or the fluid may exhibit some elastic as well as viscous properties (Macosko 1994, Tung and Paulson 1995). Plastic and viscoelastic materials, which have some elastic characteristics, are considered in later sections.
Shear-Rate-Dependent Nonideal Liquids. In an ideal liquid, the viscosity is independent of shear rate and the length of time the liquid is sheared (i.e., the ratio of the shear stress to the shear rate does not depend on shear rate or time) (Figure 8.5). In practice, many food emulsions have viscosities which depend on the shear rate and the length of time the emulsion is sheared (Dickinson 1992). In this section, emulsions in which the viscosity depends on shear rate but is independent of the shearing time are examined (Dickinson 1992). In the following section, emulsions in which the viscosity depends on both the shear rate and shearing time are examined.
Rate of Strain (a)
Rate of Strain (b)
FIGURE 8.5 Comparison of the viscosity of ideal and nonideal liquids. (a) Shear stress versus shear rate. (b) Viscosity versus shear rate.
The viscosity of an emulsion may either increase or decrease as the shear rate is increased, rather than staying constant, as for a Newtonian liquid (Figure 8.5). In these systems, the viscosity at a particular shear rate is referred to as the apparent viscosity. The dependence of the apparent viscosity on shear rate means that it is crucial to stipulate the shear rate used to carry out the measurements when reporting data. The choice of shear rate when measuring the apparent viscosity of a nonideal liquid is a particularly important consideration when carrying out rheological measurements in a laboratory which are supposed to mimic some process which occurs in a food naturally (e.g., flow through a pipe, creaming of an emulsion droplet, or mastication). The test in the laboratory should use a shear rate that is as close as possible to that which the food experiences in practice.
The two most common types of shear-rate-dependent nonideal liquids are:
1. Pseudoplastic fluids. Pseudoplastic flow is the most common type of nonideal behavior exhibited by food emulsions. It manifests itself as a decrease in the apparent viscosity of a fluid as the shear rate is increased and is therefore often referred to as shear thinning (Figure 8.5). Pseudoplasticity may occur for a variety of reasons in food emulsions (e.g., the spatial distribution of the particles may be altered by the shear field, nonspherical particles may become aligned with the flow field, solvent molecules bound to the particles may be removed, or flocs may be deformed and disrupted) (Hunter 1993, Mewis and Macosko 1994).
2. Dilatant fluids. Dilatant behavior is much less common than pseudoplastic behavior. It manifests itself as an increase in the apparent viscosity as the shear rate is increased and is therefore often referred to as shear thickening (Figure 8.5). Dilantancy is often observed in concentrated emulsions or suspensions where the particles are packed tightly together (Hunter 1989). At intermediate shear rates, the particles form two-dimensional "sheets" which slide over each other relatively easily, but at higher shear rates, these sheets are disrupted and so the viscosity increases (Pal 1996). Shear thickening may also occur when the particles in an emulsion become flocculated because of an increased collision frequency (Section 22.214.171.124); therefore, this process usually leads to time-dependent behavior and so will be considered in the following section.
Liquids that exhibit pseudoplastic behavior often have a viscosity versus shear rate profile similar to that shown in Figure 8.6. The viscosity decreases from a constant value at low shear rates (n) to another constant value at high shear rates (nJ. A number of mathematical equations have been developed to describe the rheological behavior of shear-rate-dependent nonideal liquids. The major difference is the range of shear rates over which they are
applicable. If measurements are carried out across the whole shear rate range (which often extends many orders of magnitude), then the viscosity can often be described by the Meter equation (Hunter 1989):
where T is the shear stress where the viscosity is midway between the low and high shear rate limits and n is the power index. The rheological properties of this type of system can therefore be characterized by four parameters: n0, Ti, and n.
If measurements are only carried out at shear rates that are sufficiently less than the high shear rate plateau (Figure 8.6), then the rheology can be described by the Ellis equation (Hunter 1993):
If measurements are carried out at intermediate shear rates (i.e., above the low shear plateau and below the high shear plateau), the rheology can often be described by a simple power-law model (Hunter 1993):
The constants A and B are usually referred to as the consistency index and the power index, respectively (Dickinson 1992). For an ideal liquid, B = 1; for an emulsion which exhibits shear thinning, B < 1; and for an emulsion which exhibits shear thickening, B > 1. Equations 8.5 are easy to use since they only contain two unknown parameters, which can simply be obtained from a plot of log (t) versus log (dy/dt). Nevertheless, these equations should only be used after it has been proven experimentally that the relationship between log (T) and log (dy/dt) is linear over the shear rates used.
Time-Dependent Nonideal Liquids. The apparent viscosity of the fluids described in the previous section depended on the shear rate, but not on the length of time that the shear was applied. There are many food emulsions whose apparent viscosity either increases or decreases with time during the application of shear. In some cases, this change is reversible and the fluid will recover its original rheological characteristics if it is allowed to stand at rest for a sufficiently long period. In other cases, the change brought about by shearing the sample is irreversible, and the sample will not recover its original characteristics.
An appreciation of the time dependency of the flow properties of food emulsions is of great practical importance in the food industry. The duration of pumping or mixing operations, for instance, must be carefully controlled so that the food sample has an apparent viscosity which is suitable for the next processing operation. If a food is mixed or pumped for too long, it may become too thick or too runny and thus lose its desirable rheological properties.
The dependence of the rheology of a liquid on time is often associated with some kind of relaxation process (Hunter 1993, Mewis and Macosko 1994). When an external force is applied to a system that is initially at equilibrium, the material takes a certain length of time to reach the new equilibrium conditions, which is characterized by a relaxation time (xRl. When the measurement time is of the same order as the relaxation time, it is possible to observe changes in the properties of the system with time. Thus the rheological properties of an emulsion depend on the time scale of the experiment. Time-dependent nonideal fluids are classified in two different categories:
1. Thixotropic behavior. A thixotropic fluid is one in which the apparent viscosity decreases with time when the fluid is subjected to a constant shear rate (Figure 8.7). Emulsions which exhibit this type of behavior often contain particles (droplets, crystals, or biopolymers) which are aggregated by weak forces. Shearing of the material causes the aggregated particles to be progressively deformed and disrupted, which decreases the resistance to flow and therefore causes a reduction in viscosity over time. If the relaxation time associated with the disruption of the flocs is shorter than the measurement time, then the viscosity will be observed to tend to a constant final value. This value may correspond to the point where the rate of structure disruption is equal to the rate of structure reformation or where there is no more structure to be broken down. In pseudoplastic liquids, the breakdown of the aggregated particles occurs so rapidly that the system almost immediately attains its new equilibrium position, and so it appears as though the viscosity is independent of time.
2. Rheopectic. In some food emulsions, the apparent viscosity of the fluid increases with time when it is subjected to a constant shear rate (Figure 8.7). One of the most common reasons for this type of behavior is that shearing increases both the frequency and efficiency of collisions between droplets, which leads to enhanced aggregation (Section 7.4.1) and consequently an increase in apparent viscosity over time.
In some fluids, the time-dependent rheological properties are irreversible (i.e., once the shear force is removed, the system may not fully regain its initial rheological properties). Liquids that experience this type of permanent change are called rheodestructive. This type of behavior might occur when flocs are disrupted by an intense shear stress and are unable to reform when the shear stress is removed. Otherwise, the structure and rheological properties of a material may return to their original values. In this case, the recovery time is often an important characteristic of the material.
The rheological properties of time-dependent nonideal liquids can be characterized by measuring the change in their viscosity over time. From these measurements, one can obtain a relaxation time for the structural rearrangements that occur in the emulsion. Nevertheless,
this type of experiment is often inconvenient if one also wants to obtain information about the dependence of the viscosity on shear rate. One would have to establish the relaxation time for the structural rearrangements at each shear rate, and then ensure that the samples were sheared for a time that was long enough for them to reach their steady-state rheology. If the relaxation time of a sample is relatively long, this type of measurement would be time consuming and laborious to carry out. Instead, it is often more convenient to measure the viscosity of a fluid when the shear rate is increased from zero to a certain value and then decreased back to zero again (Figure 8.8). When there is a significant structural relaxation in a system, the upward curve is different from the downward curve and one obtains a hysteresis loop. The area within the loop depends on the degree of relaxation that occurs and the rate at which the shear rate is altered. The slower the shear rate is altered, the more time the system has to reach its equilibrium value, and therefore the smaller the area within the hysteresis loop. By carrying out measurements as a function of the rate at which the shear rate is increased, it is possible to obtain information about the relaxation time.
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