where T is a critical shear stress that is related to the size of the droplets (t = kT/pr3), and P is a dimensionless constant with a value of about 0.431 (Hunter 1989). The value of Ti is a characteristic of a particular system which describes the relative importance of the translational Brownian motion and hydrodynamic shear forces. When T << T, Brown-ian motion dominates and the particles have a random distribution, but when T >> T, the shear forces dominate and the particles become organized into "strings" or "layers" along the lines of the shear field, which causes less energy dissipation. This equation indicates that the viscosity decreases from a constant value at low shear stresses (n0) to another constant value at high shear stresses (nJ. The viscosity can decrease by as much as 30% from its low shear rate value, with the actual amount depending on the dispersed-phase volume fraction (Hunter 1989). The shear rate at which the viscosity starts to decrease from its n0 value is highly dependent on the particle size. For large particles, T is often so low that it is not possible to observe any shear thinning behavior, but for smaller particles, shear thinning behavior may be observed at the shear rates typically used in a rheological experiment.
The Dougherty-Krieger equation can still be used to describe the dependence of the suspension viscosity on dispersed-phase volume fraction, but the value of used in the equation is shear rate dependent. This is because droplets can pack more efficiently at higher shear rates, and therefore §c increases with shear rate (Hunter 1989).
8.4.6. Suspensions of Nonflocculated Particles with Repulsive Interactions
Most food emulsions contain droplets which have various types of colloidal interaction acting between them (e.g., van der Waals, electrostatic, steric, hydrophobic, depletion, etc.) (Chapter 3). The precise nature of these interactions has a dramatic influence on the rheol-ogy of particulate suspensions. For example, two emulsions with the same droplet concentration could have rheological properties ranging from a low-viscosity Newtonian liquid to a highly viscoelastic material, depending on the nature of the colloidal interactions. In this section, suspensions that contain isolated spherical particles with repulsive interactions are considered. In the following section, the rheological properties of flocculated emulsions are considered.
The major types of droplet repulsion in most food emulsions are due to electrostatic and steric interactions. These repulsive interactions prevent the droplets from coming into close contact when they collide with each other and therefore increase the effective volume fraction of the droplets (Tadros 1994, Mewis and Macosko 1994):
where 8 is equal to half the distance of closest separation between the two droplets.
For steric stabilization, 8 is approximately equal to the thickness of the adsorbed layer. For electrostatically stabilized systems, it is related to the Debye length (k-1) and can be described by the following equation at low shear stresses (Mewis and Macosko 1994):
where a = 4n £oeR¥or2K exp(2rK)/kT, e0 is the dielectric permittivity of a vacuum, eR is the relative dielectric permittivity of the continuous phase, ¥0 is the electrical potential at the droplet surface, r is the radius, K-1 is the Debye length, k is Boltzmann's constant, and Tis the absolute temperature. For electrically charged oil droplets, the distance of closest contact therefore decreases as the surface charge decreases or as the ionic strength of the aqueous phase increases.
It is convenient to categorize droplets with repulsive interactions as being either "hard" particles or "soft" particles (Liu and Masliyah 1996). A hard particle is incompressible, and so its effective size is independent of shear rate or droplet concentration. On the other hand, a soft particle is compressible, and so its effective size may be reduced at high shear rates or droplet concentrations. Sterically stabilized droplets with dense interfacial layers are usually considered to act like hard particles because the layer is relatively incompressible, whereas electrostatically stabilized droplets or sterically stabilized droplets with open interfacial layers are usually considered to act like soft particles because the layer is compressible.
The viscosity of an emulsion that contains droplets with repulsive interactions can be related to the dispersed-phase volume fraction by replacing the value of < in the Dougherty-Krieger equation (Equation 8.33) with the effective volume fraction (<eff). A plot of viscosity versus <eff then falls on the same curve as for an emulsion that contains droplets with no longrange colloidal interactions. In emulsions that contain "soft" particles, it is also necessary to replace the value of <c with <ceff, to take into account that the particles may be compressed at higher volume fractions and can therefore pack more efficiently. As a consequence, the viscosity of an emulsion that contains soft particles is lower than one that contains hard particles at the same effective volume fraction.
The influence of repulsive interactions on the rheology of emulsions depends on the magnitude of 8 relative to the size of the particles. For relatively large particles (i.e., 8 << r), this effect is negligible, but for small droplets or droplets with thick layers around them (i.e., 8 ~ r), this effect can significantly increase the viscosity of a suspension (Tadros 1994).
The rheological properties of electrostatically stabilized systems that contain small emulsion droplets are particularly sensitive to pH and salt concentration. The viscosity would decrease initially with increasing salt concentration because screening of the charges decreases 8. Above a certain salt concentration, the interactions between the droplets would become attractive, rather than repulsive, and therefore flocculation will occur, causing an increase in emulsion viscosity with salt. The rheological properties of electrostatically stabilized emulsions are therefore particularly sensitive to the pH, salt concentration, and type of ions present.
8.4.7. Concentrated Suspensions of Flocculated Particles
In concentrated emulsions, the flocs are close enough together to interact with each other, through hydrodynamic interactions, colloidal interactions, or entanglement. The viscosity of flocculated emulsions can be described by the Dougherty-Krieger equation by assuming that the flocs are "soft" particles with the actual droplet volume fraction replaced by the effective droplet volume fraction given by Equation 8.27. At a given actual droplet volume fraction, the emulsion viscosity increases as the size of the flocs increases or the packing of the flocs becomes more open (lower D).
At sufficiently high droplet concentrations, flocculation may lead to the formation of a three-dimensional network of aggregated droplets (Sherman 1970, Goodwin and Ottewill
1991, Pal 1996). The more open the structure of the droplets within the flocs, the lower the value of the actual droplet volume fraction where the network is formed. Network formation causes a suspension of particles to exhibit plastic and/or viscoelastic characteristics (Pal 1996). The network of aggregated droplets acts like a solid at low shear stresses because the applied forces are not sufficient to overcome the forces holding the droplets together. Once a critical shear stress is exceeded, the bonds between the droplets are disrupted and so the droplets can flow past one another. If some of the bonds are capable of reforming during the shearing process, then the emulsion will exhibit viscoelastic behavior (Sherman 1968a). At higher shear stresses, the rate of bond disruption greatly exceeds that of bond formation and the emulsion acts like a liquid. Consequently, a suspension that contains a three-dimensional network of aggregated particles often has a yield stress, below which it acts like an elastic solid and above which it acts like a liquid. Above the yield stress, the suspension often exhibits strong shear thinning behavior as more and more flocs are deformed and disrupted. The magnitude of the yield stress depends on the strength of the attractive forces holding the particles together: the greater the attractive forces, the greater the yield stress (Pal 1996). The rheology of the system is also sensitive to the structural organization of the droplets (e.g., whether they are loosely or densely packed and the number of bonds per droplet) (Bremer
8.4.8. Emulsions with Semisolid Continuous Phases
A number of food emulsions consist of droplets dispersed in a continuous phase which is either partly crystalline or gelled (Sherman 1970, Dickinson and Stainsby 1982, Dickinson 1992, Moran 1994). Butter and margarine consist of water droplets suspended in a liquid oil phase which contains a three-dimensional network of aggregated fat crystals. Many meat products, desserts, and sauces consist of oil droplets suspended in an aqueous phase of aggregated biopolymer molecules. The rheological properties of these systems are usually dominated by the properties of the continuous phase, and therefore their properties can be described using theories developed for networks of aggregated fat crystals or biopolymer molecules (Clark 1987). Nevertheless, in some systems the presence of the emulsion droplets does play a significant role in determining the overall rheological behavior (see below).
It is important that spreadable products, such as butter, margarine, and low-fat spreads, retain their shape when they are removed from the refrigerator, but spread easily when a knife is applied (Sherman 1970, Moran 1994). These products must therefore be designed so that they exhibit plastic properties (i.e., they have a yield stress below which they are elastic and above which they are viscous). The plastic behavior of this type of product is usually attributed to the presence of a three-dimensional network of aggregated fat crystals. Low shear stresses are not sufficiently large to disrupt the bonds which hold the aggregated crystals together, and so the product exhibits solid-like behavior. Above the yield stress, the applied shear stress is sufficiently large to cause the bonds to be disrupted, so that the fat crystals flow over each other and the product exhibits viscous-like behavior. After the stress is removed, the bonds between the fat crystals reform over time, and therefore the product regains its elastic behavior. The creation of a product with the desired rheological characteristics involves careful selection and blending of various food oils, as well as control of the cooling and shearing conditions used during the manufacture of the product. To the author's knowledge, little systematic research has been carried out to establish the influence of the characteristics of the water droplets on the rheological prop erties of these products (e.g., particle size distribution, emulsifier type, and dispersed-phase volume fraction).
A great deal of research has recently been carried out to determine the influence of oil droplets on the rheology of filled gels (Jost et al. 1986; Aguilera and Kessler 1989; Xiong et al. 1991; Xiong and Kinsella 1991; Yost and Kinsella 1993; McClements et al. 1993c; Dickinson and Hong 1995a,b, 1996; Dickinson and Yamamoto 1996). These filled gels are created by heating oil-in-water emulsions which contain a significant amount of whey protein in the continuous phase above a temperature where the proteins unfold and form a three-dimensional network of aggregated molecules. The oil droplets can act as either structure promoters or structure breakers, depending on the nature of their interaction with the gel network. When the droplets are stabilized by dairy proteins, the attractive interactions between the adsorbed proteins and those in the network reinforce the network and increase the gel strength. Conversely, when the droplets are stabilized by small-molecule surfactants (which do not interact strongly with the protein network), the presence of the droplets tends to weaken the network and decrease the gel strength. Quite complex rheological behavior can be observed in emulsions that contain mixtures of proteins and small-molecule surfactants (Dickinson and Hong 1995a,b; Dickinson and Yamamoto 1996). The shear modulus of filled gels that contain protein-stabilized oil droplets increases dramatically when a small amount of surfactant is added to the system, but decreases at higher surfactant values. The incorporation of surfactants into filled gels may therefore prove to be an effective means of controlling their rheological properties.
The influence of the emulsion droplets also depends on their size relative to the pore size of the gel network (McClements et al. 1993c, Yost and Kinsella 1993). If the droplets are larger than the pore size, they tend to disrupt the network and decrease the gel strength, but if they are smaller than the pore size, they are easily accommodated into the network without disrupting it.
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