Food emulsions exhibit a wide range of different rheological properties, ranging from low-viscosity liquids to fairly rigid solids. The rheological behavior of a particular food depends on the type and concentration of ingredients it contains, as well as the processing and storage conditions it has experienced. In this section, the relationship between the rheological properties of emulsions and their composition and microstructure is discussed. We begin by considering the rheology of dilute suspensions of noninteracting rigid spheres, because the theory describing the properties of this type of system is well established (Hiemenz 1986, Hunter 1986, Mewis and Macosko 1994, Tadros 1994). Nevertheless, many food emulsions are concentrated and contain nonrigid, nonspherical, and/or interacting droplets (Dickinson 1992). The theoretical understanding of these types of systems is less well developed, although some progress has been made, which will be reviewed.

8.4.1. Dilute Suspensions of Rigid Spherical Particles

The viscosity of a liquid increases upon the addition of rigid spherical particles because the particles disturb the normal flow of the fluid, causing greater energy dissipation due to friction (Hunter 1986, Mewis and Macosko 1994). Einstein derived an equation to relate the viscosity of a suspension of rigid spheres to its composition:

where n0 is the viscosity of the liquid surrounding the droplets and ^ is the dispersed-phase volume fraction. This equation assumes that the liquid is Newtonian, the particles are rigid and spherical, that there are no particle-particle interactions, that there is no slip at the particle-fluid interface, and that Brownian motion effects are unimportant. The Einstein equation predicts that the viscosity of a dilute suspension of spherical particles increases linearly with particle volume fraction and is independent of particle size and shear rate. The Einstein equation gives excellent agreement with experimental measurements for suspensions that conform to the above criteria, often up to particle concentrations of about 5%.

It is convenient to define a parameter known as the intrinsic viscosity of a suspension, [n] (Dickinson and Stainsby 1982):

For rigid spherical particles, the intrinsic viscosity tends to 2.5 as the volume fraction tends to zero. For nonspherical particles or for particles that swell due to the adsorption of solvent, the intrinsic viscosity is larger than 2.5 (Hiemenz 1986), whereas it may be smaller for fluid particles (Sherman 1968a,c; Dickinson and Stainsby 1982). For these systems, measurements of [n] can provide valuable information about their shape or degree of solvation.

8.4.2. Dilute Suspensions of Fluid Spherical Particles

Food emulsions usually contain fluid, rather than solid, particles. In the presence of a flow field, the liquid within a droplet is caused to circulate because it is dragged along by the liquid (continuous phase) that flows past the droplet (Sherman 1968a,c; Dickinson and Stainsby 1982). Consequently, the difference in velocity between the materials on either side of the droplet surface is less than for a solid particle, which means that less energy is lost due to friction and therefore the viscosity of the suspension is lower. The greater the viscosity of the fluid within a droplet, the more it acts like a rigid sphere, and therefore the higher the viscosity of the suspension.

The viscosity of a suspension of noninteracting spherical droplets is given by (Tadros where ndr0p is the viscosity of the liquid in the droplets. For droplets containing relatively high-viscosity liquids (ndrop/no >> 1), the intrinsic viscosity tends to 2.5, and therefore this equation tends to that derived by Einstein (Equation 8.19). For droplets that contain relatively low-viscosity fluids (ndrop/n0 << 1), such as air bubbles, the intrinsic viscosity tends to unity, and so the suspension viscosity is given by n = n0(1 + $). One would therefore expect the viscosity of oil-in-water or water-in-oil emulsions to be somewhere between these two extremes. In practice, the droplets in most food emulsions are coated by a layer of emulsifier molecules that forms a viscoelastic membrane. This membrane retards the transmittance of the tangential stress from the continuous phase into the droplet and therefore hinders the flow of the fluid within the droplet (Pal et al. 1992, Tadros 1994). For this reason, most food emulsions contain droplets which act like rigid spheres, and so their viscosities at low concentrations can be described by the Einstein equation.

At sufficiently high flow rates, the hydrodynamic forces can become so large that they overcome the interfacial forces holding the droplets together and cause the droplets to become deformed and eventually disrupted (Chapter 6). The shear rates required to cause droplet disruption are usually so high that the flow profile is turbulent rather than laminar, and so it is not possible to make viscosity measurements. Nevertheless, a knowledge of the viscosity of fluids at high shear rates is important for engineers who design mixers and homogenizers.

8.4.3. Dilute Suspensions of Rigid Nonspherical Particles

Many of the particles in food emulsions may have nonspherical shapes (e.g., flocculated droplets, partially crystalline droplets, fat crystals, ice crystals, or biopolymer molecules) (Dickinson 1992). Consequently, it is important to appreciate the effects of particle shape on suspension viscosity. The shape of many particles can be approximated as prolate spheroids (rod-like) or oblate spheroids (disk-like). A spheroid is characterized by its axis ratio rp = a/ b, where a is the major axis and b is the minor axis (Figure 8.15). For a sphere, a = b; for a prolate spheroid, a > b; and for an oblate spheroid, a < b. The flow profile of a fluid around a nonspherical particle causes a greater degree of energy dissipation than that around a spherical particle, which leads to an increase in viscosity (Hunter 1986, Hiemenz 1986, Mewis and Macosko 1994). The magnitude of this effect depends on the rotation and orientation of the spherical particle. For example, the viscosity of a rod-like particle is much lower when it is aligned parallel to the fluid flow, rather than perpendicular, because the parallel orientation offers less resistance to flow.

The orientation of a spheroid particle in a flow field is governed by a balance between the hydrodynamic forces that act upon it and its rotational Brownian motion (Mewis and Macosko 1994). The hydrodynamic forces favor the alignment of the particle along the direction of the flow field, because this reduces the energy dissipation. On the other hand, the alignment of the particles is opposed by their rotational Brownian motion, which favors the complete randomization of their orientations. The relative importance of the hydrodynamic and Brown-ian forces is expressed in terms of a dimensionless number, known as the Peclet number (Pe). For simple shear flow (Mewis and Macosko 1994):

where « is the shear rate and DR is the rotational Brownian diffusion coefficient, which depends on particle shape:

for rigid spheres

32 nnb3

for circular disks

Flow field

Flow field

FIGURE 8.16 At low shear rates, the particles rotate freely in all directions, but as the shear rate increases, they become more and more aligned with the flow field. This causes a reduction in the viscosity with increasing shear rate (i.e., pseudoplasticity).

FIGURE 8.16 At low shear rates, the particles rotate freely in all directions, but as the shear rate increases, they become more and more aligned with the flow field. This causes a reduction in the viscosity with increasing shear rate (i.e., pseudoplasticity).

Dr = 8nnr3 (ln 2rp ~ 0.5) for long thin rods (8.26)

When the Peclet number is much less than unity (Pe << 1), the rotational Brownian motion dominates, and the particles tend to rotate freely in the liquid. This type of behavior is observed when the particles are small, the shear rate is low, and/or the viscosity of the surrounding fluid is low. When the Peclet number is much greater than unity (Pe >> 1), the hydrodynamic forces dominate, and the particles become aligned with the flow field (Figure 8.16). This type of behavior is observed when the particles are large, the shear rate is high, and/or the viscosity of the surrounding liquid is high.

The viscosity of a suspension of nonspherical particles therefore depends on the shear rate. At low shear rates (i.e., Pe << 1), the viscosity has a constant high value. As the shear rate is increased, the hydrodynamic forces become more important, and so the particles become oriented with the flow field, which causes a reduction in the viscosity. At high shear rates (i.e., Pe >> 1), the hydrodynamic forces dominate and the particles remain aligned with the shear field, and therefore the viscosity has a constant low value (Figure 8.16). Thus suspensions of nonspherical particles exhibit shear thinning behavior. The shear rate at which the viscosity starts to decrease depends on the size and shape of the particles, as well as the viscosity of the surrounding liquid. Mathematical formulae similar to Equation 8.3 have been developed to calculate the influence of shear rate on the viscosity of suspensions of nonspherical particles, but these usually have to be solved numerically. Nevertheless, explicit expressions are available for systems that contain very small or very large particles.

When the attractive forces between the droplets dominate the repulsive forces, and are sufficiently greater than the thermal energy of the system, then droplets can aggregate into a primary or secondary minimum (Chapter 3). The rheological properties of many food emulsions are dominated by the fact that the droplets are flocculated, and so it is important to understand the factors which determine the rheological characteristics of these systems. It is often convenient to categorize systems as being either strongly flocculated (wattractive > 20 kT) or weakly flocculated (1 kT < wattractive < 20 kT), depending on the strength of the attraction between the droplets (Liu and Masliyah 1996).

A dilute suspension of flocculated droplets consists of flocs which are so far apart that they do not interact with each other through colloidal or hydrodynamic forces. This type of suspension has a higher viscosity than a suspension that contains the same concentration of isolated particles because the particles in the flocs trap some of the continuous phase and therefore have a higher effective volume fraction than the actual volume fraction (Liu and Masliyah 1996). In addition, the flocs may rotate in solution because of their rotational Brownian motion, sweeping out an additional amount of the continuous phase and thus increasing their effective volume fraction even more.

Suspensions of flocculated particles tend to exhibit pronounced shear thinning behavior (Figure 8.17). At low shear rates, the hydrodynamic forces are not large enough to disrupt the bonds holding the particles together, and so the flocs act like particles with a fixed size and shape, resulting in a constant viscosity. As the shear rate is increased, the hydrodynamic forces become large enough to cause flocs to become deformed and eventually disrupted. The deformation of the flocs results in their becoming elongated and aligned with the shear field, which results in a reduction in the viscosity. The disruption of the flocs decreases their effective volume fraction and therefore also contributes to a decrease in the suspension viscosity. The viscosity reaches a constant value at high shear rates, either because all of the flocs are completely disrupted so that only individual droplets remain or because the number of flocculated droplets remains constant since the rate of floc formation is equal to that of floc disruption (Campanella et al. 1995).

Depending on the nature of the interdroplet pair potential (Chapter 3), it is also possible to observe shear thickening due to particle flocculation under the influence of the shear field (de Vries 1963). Some emulsions contain droplets which are not flocculated under quiescent conditions because there is a sufficiently high energy barrier to prevent the droplets from falling into a primary minimum. However, when a shear stress is applied to the emulsions, the frequency of collisions and the impact force between the droplets increase, which can

FIGURE 8.17 An emulsion that contains flocculated droplets exhibits shear thinning behavior because the flocs are deformed and disrupted in the shear field.

Flow field

Flow field

Shear rate cause the droplets to gain sufficient energy to "jump" over the energy barrier and become flocculated, therefore leading to shear thickening.

Quite complicated behavior can therefore be observed in some emulsions (Pal et al. 1992, Liu and Masliyah 1996). For example, an emulsion that contains droplets which are weakly flocculated in a secondary minimum exhibits shear thinning at fairly low shear rates, but shows shear thickening when the shear rate exceeds some critical level where the droplets have sufficient energy to "jump" over the energy barrier and fall into the primary minimum. The value of this critical shear rate increases as the height of the energy barrier increases. A knowledge of the interdroplet pair potential is therefore extremely useful for understanding and predicting the rheological behavior of food emulsions.

The size, shape, and structure of flocs largely determine the rheological behavior of dilute suspensions of flocculated particles (Dickinson and Stainsby 1982, Liu and Masliyah 1996). Flocs formed by the aggregation of emulsion droplets often have structures that are fractal (Chapter 7). The effective volume fraction (0eff) of a fractal floc is related to the size of the floc and the fractal dimension by the following expression (Bremer 1992):

where r is the droplet radius and R is the floc radius. The viscosity of a dilute emulsion containing fractal flocs can therefore be established by substituting this expression into the Einstein equation:*

If it is assumed that the flocs are approximately spherical, then [n] = 2.5. This equation offers a useful insight into the relationship between the rheology and microstructure of flocculated emulsions.

Flocs with fairly open structures (i.e., lower D) have higher viscosities than those with compact structures because they have higher effective volume fractions (Equation 8.28). As mentioned earlier, the viscosity decreases with increasing shear rate partly because of disruption of the flocs (i.e., a decrease in R). The shear stress at which the viscosity decreases depends on the magnitude of the forces holding the droplets together within a floc. The greater the strength of the forces, the larger the shear rate required to deform and disrupt the flocs. Thus the dependence of the viscosity of an emulsion on shear stress can be used to provide valuable information about the strength of the bonds holding the droplets together (Sherman 1970, Dickinson and Stainsby 1982, Hunter 1989).

8.4.5. Concentrated Suspensions of Nonflocculated Particles in the Absence of Colloidal Interactions

When the concentration of particles in a suspension exceeds a few percent, the particles begin to interact with each other through a combination of hydrodynamic and colloidal interactions, and this alters the viscosity of the system (Hunter 1986, Mewis and Macosko 1994, Tadros 1994). In this section, we examine the viscosity of concentrated suspensions in the absence

* Alternatively, the expression for the effective volume fraction can be placed in the Dougherty-Krieger equation developed for more concentrated emulsions (Barnes 1994).

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