The Rheology of Colloidal and Noncolloidal Food Dispersions


  • D. B. Genovese,

    1. Authors Genovese and Lozano are with PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, 8000 Bahía Blanca, Argentina. Author Rao is with Dept. of Food Science and Technology, Cornell Univ., Geneva, NY 14456-0462, Direct inquiries to author Genovese (E-mail:
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  • J. E. Lozano,

    1. Authors Genovese and Lozano are with PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, 8000 Bahía Blanca, Argentina. Author Rao is with Dept. of Food Science and Technology, Cornell Univ., Geneva, NY 14456-0462, Direct inquiries to author Genovese (E-mail:
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  • M. A. Rao

    1. Authors Genovese and Lozano are with PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, 8000 Bahía Blanca, Argentina. Author Rao is with Dept. of Food Science and Technology, Cornell Univ., Geneva, NY 14456-0462, Direct inquiries to author Genovese (E-mail:
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ABSTRACT:  Rheological data on a food together with data on its composition and structure or microstructure should lead to understanding the interrelationships between them. A number of foods are dispersions of solids in liquids, liquids in liquids, or gas in liquids. The dispersed particles may be colloidal in nature with dimensions < 10 μm, or larger noncolloidal particles (> 10 μm). For both colloidal and noncolloidal dispersions (either in dilute or concentrated regimes), several theoretical equations exist that provide insights into the role of key rheological parameters, such as particle volume fraction and size, interparticle forces, and fractal dimension on their viscosity, yield stress, and modulus. When theoretical models cannot be easily applied to foods with complex structures, structural analysis and structure-based models provide insight into the role of solids loading and interparticle bonding on rheological behavior. In this review, recent studies on colloidal and noncolloidal food dispersions in which theoretical models as well as structural analysis were employed are discussed.


Processed foods are edible structures that are created as a result of the responses of proteins, mono and polysaccharides, and lipids in aqueous media to different processing methods such as thermal processing, homogenization, and other physical treatments. Most, if not all, of those responses are physical in nature. The measured rheological responses occur at the macroscopic level. However, they are affected by the changes and properties at the molecular and microscopic level. A major challenge is to establish links between the macroscopic rheological properties with changes at the molecular and microscopic level (Rao 2006). Rheological data on a food together with data on its composition and structure or microstructure should lead to understanding the interrelationships between them. In turn, such knowledge should lead to the improvement in food quality through creation of foods with desirable structures and rheological behavior.

Many foods are dispersions of either solids in a liquid medium, usually an aqueous solution, or of liquid droplets in another liquid, named emulsions.

Three kinds of forces coexist to various degrees in flowing dispersions: hydrodynamic, Brownian, and colloidal forces. Hydrodynamic forces arise from the relative motion of particles to the surrounding fluid. The Brownian force is the ever-present thermal randomizing force. Colloidal forces are potential forces and are elastic in nature (Zhou and others 2001; Qin and Zaman 2003). The relative magnitude of these forces and, therefore, the bulk rheology depend on the particle size. Brownian motion and interparticle forces quickly equilibrate for sub-nanometer-size dispersions, while hydrodynamic forces dominate for particles larger than approximately 10 μm. For particles in the intermediate range the flow behavior is determined by a combination of hydrodynamic forces, Brownian motion, and interparticle forces (Russel 1980; Qin and Zaman 2003). Consequently, dispersions may be classified according to the size and nature of the particles in them (Table 1).

Table 1—.  Classification of dispersions
 Particle size/natureForces governing viscosity
  1. aFood dispersions covered in this review.

Polymers approximately 1 nmMolecular/colloidalDilute: intermolecular-Brownian
Concentrated: Entanglements and Reptation
1 nm-10 μmaMicroscopic/colloidalLow inline image: Hydrodynamic-interparticle-Brownian
High inline image: Hydrodynamic
10 μm-100 μmaMicroscopic/noncolloidalHydrodynamic
> 100 μmMacroscopic/noncolloidalHydrodynamic

Colloidal dispersions may be defined as polyphasic or heterogeneous systems where the dispersed phase is subdivided into discrete units (particles/droplets) that are large compared to simple molecules, but small enough so that interfacial and inertial forces are significant in governing system properties (Sennet and Olivier 1965). The size range of colloidal (or Brownian) particles is not defined rigidly, but typically considered to be 1 nm to 10 μm (Russel and others 1989; McClements 1999; Qin and Zaman 2003), as for example cloudy fruit juices, milk, soy protein suspensions, and mayonnaise (a colloidal emulsion). Brownian motion promotes collisions between pairs of colloidal particles, while interparticle forces determine if 2 colliding particles aggregate or not.

The particle arrangements in colloidal systems depend on the volume occupied by the particles in relation to the total volume, that is, the particle volume fraction, φ. At low φ, the mean distance between particles is large compared to particle radius. Thus, particles are able to move freely throughout the medium driven by Brownian forces. This regime (φ→ 0) is considered as the dilute limit. As φ increases, hydrodynamic interactions and the probability of collision between particles become important. As a consequence, the Brownian motion of each particle is hindered by the presence of the other ones. In this region, the suspension is considered to be concentrated (Quemada and Berli 2002).

Colloidal suspensions can be either dispersed or flocculated, depending on the magnitude of particle–particle interaction energy and particle concentration. Two limit cases are usually considered for the mechanism of aggregation: perikinetic aggregation, where the particle encounters are due to Brownian motion, and orthokinetic aggregation, where the collisions are caused by the gradient velocity field. This last mechanism dominates for particles of micrometer size and larger (Berli and others 1999b).

In weakly flocculated dispersions, particles form a volume-spanning network. For aggregation resulting from long-range attractions, deformation of the network changes the relative positions of the particles, thereby increasing the total potential energy of the system and producing an interparticle force tending to restore the equilibrium structure. The nonequilibrium nature of the structure gives a complex, history-dependent rheological behavior (Russel and others 1989). A colloidal gel is a special state of strongly flocculated systems where a continuous network of particles (often < 0.1 μm) is formed by aggregation, with the resulting suspension having a very high viscosity and a finite shear modulus.

Several foods contain microscopic particles that are >10 μm and up to approximately 100 μm (for example, chocolate, fruit purees and sauces, starch and vegetable pastes); some of them may be flocs or aggregates of colloidal particles. For such large particles, Brownian motion and interparticle forces are negligible compared to hydrodynamic forces. However, nonhydrodynamic parameters such as particle shape, particle size and size distribution, particle deformability, and liquid polarity could affect the structure and the resulting flow behavior (Tsai and Zammouri 1988). Such dispersions have been called simply “noncolloidal” or non-Brownian.

Other foods contain macroscopic particles (>100 μm), for example, vegetable or cream soups, fruits in syrups or yogurts, dressings or sauces with seeds, and pasta or meat in sauces. Gondret and Petit (1997) referred to dispersions of glass spheres of diameter 45 to 450 μm as macroscopic. Those food dispersions are usually considered to be liquid/particle mixtures, and sometimes the liquid itself is a colloidal dispersion (Martínez-Padilla 2005). In this case, when the particle size has atomic dimensions, the system consists of a molecular or ionic dispersion, although strictly speaking it is a 1-phase system (Sennet and Olivier 1965).

It should be emphasized that most foods are complex polydisperse systems that may contain particles from a broad size range (that is, both microscopic and macroscopic particles) immersed (or even imbibed) in an aqueous (or lipid) medium, which in turn may be a molecular dispersion itself. The rheological behavior of such systems has been covered by several groups, including Rao and coworkers (Vitali and Rao 1984; Tanglertpaibul and Rao 1987; Yoo and Rao 1996).

In addition to solid–liquid and liquid–liquid systems, liquid food foams are biphasic systems where a gas bubble phase is dispersed in a continuous liquid phase, with the gas–liquid interface being under tension. Their complex rheological behavior is influenced by many factors, such as air phase volume, liquid phase viscosity, interfacial tension and viscosity, bubble size, size distribution, and shape (Herzhaft 1999; Vernon-Carter and others 2001; Thakur and others 2003). There are also solid food foams (typically cooked products such as breads or cakes), which consist of a discontinuous air phase dispersed in a continuous solid phase. Solid foams are elastic or plastic materials. Their mechanical (rheological) behavior is beyond the scope of this work, but we may mention that it is primarily dependent on the physical properties of the solid phase and the bulk density of the material (Pernell and others 2002).

The present review is focused on microscopic colloidal and noncolloidal food dispersions. The objective is to discuss the role of structure on their rheological properties in terms of theoretical models and structural analysis. Measurement of rheological properties has been well covered in texts such as those of Steffe (1996) and Rao (1999); therefore, they are not covered here.

Theoretical Rheological Models and Their Application

Theoretical models are derived from fundamental concepts, and they provide valuable guidelines on understanding the role of structure. They indicate the factors that influence a rheological parameter. As mentioned, the viscosity of colloidal dispersions is affected by interparticle forces and Brownian motion at low shear rates, while hydrodynamic forces dominate at high shear rates (Berli and others 1999a). On the other hand, the viscosity of noncolloidal dispersions is governed by hydrodynamic forces within all the range of shear rates.

Viscosity models for noncolloidal dispersions

The hydrodynamic disturbance of the flow field induced by solid particles in liquid media leads to an increase in the energy dissipation and an increase in viscosity (Zhou and others 2001). The relative viscosity of dispersions of solid particles in a liquid medium is defined as


where η is the viscosity of the dispersion, and ηs is the viscosity of the continuous phase. In general, ηr of dispersions of rigid, noninteracting spherical particles (hard-sphere systems) depends on the volume fraction, φ, and the intrinsic viscosity, [η], of the dispersed solids (Eq. 2 and 3). At low particle concentrations (in the dilute regime), it is described by the well-known Einstein's equation


Theoretically, [η] depends on particle shape, being 2.5 for rigid spheres (Barnes 2000).

One relationship derived for concentrated dispersions is the widely used Krieger–Dougherty equation (Krieger and Dougherty 1959)


where φm is the maximum packing fraction of solids. Although the theoretical φm value of monodisperse spheres is 0.74 (in a face-centered cubic array), experimental observations have shown that loose random packing is close to 0.60, and that dense random packing is close to 0.64 (Quemada and Berli 2002; Servais and others 2002; Qin and Zaman 2003).

Effect of particle shape and particle size distribution When the particles are nonspherical there is an extra energy dissipation and consequently an increase in the viscosity. In dilute dispersions this increase is reflected by the intrinsic viscosity in Einstein equation (Eq. 2). Simple formulas has been derived by Barnes (2000) as [η]= 0.07q5/3 for rod-like (prolate) particles, and [η]= 0.3q for disk-like (oblate) particles, where q is the axial ratio. The effect of particle shape on concentrated suspensions was studied by Kitano and others (1981), who proposed a modification of the Krieger–Dougherty relationship (Eq. 3) for suspensions of nonspherical particles


where β is an adjustable parameter whose value decreases as the aspect ratio (L/D) of the suspended particles increases; for example, when L/D = 1 (smooth spheres) β=φm, and when 6 < L/D < 8 (rough crystals) β= 0.44 (Metzner 1985; Rao 1999).

Equation 3 is valid for monomodal spherical particles. For suspensions of polydispersed spherical particles in Newtonian fluids, φm is higher since small particles may occupy the space between the larger particles. Under flow conditions, the small particles act as a lubricant for the flow of the larger particles, thereby reducing the overall viscosity (Servais and others 2002; Martínez-Padilla 2005). In other words, for a given particle concentration (φ), the viscosity decreases at increasing polydispersity (particle size distribution width). Farris (1968) assembled many monomodal distributions and showed that the overall resulting viscosity was the product of the relative viscosity associated with each discrete unimodal size distribution, ηrii), assuming no interactions between the particles of different class sizes


Effect of particle deformability At high concentrations, deformable particles can accommodate each other at rest and squeeze past each other during flow, increasing φm and reducing [η] in Eq. 3, resulting in a lower viscosity (Barnes 2000). Often, the data necessary to apply a theoretical model are not easy to obtain. For example, in a starch dispersion of a specific concentration, c, φ changes during heating of the dispersion; it has a low value in the ungelatinized state and it increases continuously to a maximum value. After the maximum value has been attained, on further heating, the volume fraction of a native starch often decreases substantially while that of a cross-linked starch remains nearly the same (Yang and Rao 1998; Rao 1999; Tattiyakul and Rao 2000). Because of their deformable nature, it is difficult to determine volume fraction of starch dispersions accurately and it is preferable to work with starch granule mass fraction, (cQ), where c is dry starch concentration, w/w, and Q is the mass of hydrated starch granules per unit weight of dry starch.

Yang and Rao (1998) and Liao and others (1999) obtained complex viscosity, η*, versus temperature data of a starch dispersion at several oscillatory frequencies (not shown here). Because the profiles were similar in shape, by choosing an arbitrary reference frequency (ωr), all the η*-temperature curves at the different frequencies were reduced to a single master curve of reduced complex viscosity η*R as in Figure 1.

Figure 1—.

Change of reduced complex viscosity of a cornstarch dispersion during heating, due to granule swelling and rupture. Master curve of experimental data at different oscillatory frequencies and heating rates. Reprinted from Yang and Rao (1998), with permission from Blackwell Publishing.

The shape of the curve in Figure 1 reflects the aforementioned changes in the volume fraction of starch granules. Initially at low temperatures, the granules are in the raw state and the volume fraction is low. As the granules are heated they swell due to water absorption (segment A-B-C), the volume fraction increases and reaches a maximum value (C). With further heating, the granules rupture and disintegrate, resulting in a gradual decrease in the volume fraction of the granules in the dispersion (segment C-D). The rupture of granules also results in the release of amylose that contributes to the viscosity of the continuous phase of the starch dispersion. Thus, in Figure 1, the segment C-D is not a mirror image of the segment ABC. The leached amylose and the granule remnants contribute to a viscosity at point D that is higher than that at point A.

Viscosity models for colloidal dispersions

In stable colloidal dispersions, repulsive forces keep neighboring particles away from each other. Therefore, for flow to occur particles must be forced to move against the force fields of the other particles, demanding an extra energy (Berli and others 1999a). The predictive models for the viscosity of colloidal dispersions are divided into 2 categories based on their scaling technique. One is the separation of contributions method, in which the contributions from individual factors are separated from each other. The other is the effective volume fraction method, in which all the contributions from different factors are lumped into 1 factor, the effective maximum packing fraction (Qin and Zaman 2003).

In the separation of contributions method, the relative viscosity of a colloidal dispersion has been modeled (Ogawa and others 1997; Quemada and Berli 2002) as the sum of a “hard-sphere” contribution (ηhsr) and a “colloidal forces” contribution (ηcfr)


The term ηhsr is considered to be the relative viscosity of an ideal dispersion of hard spheres or, more generally, the relative viscosity of a noncolloidal dispersion of rigid particles. Thus, it can be calculated by selecting the appropriate expression from Eq. 2 to 5.

The term ηcfr involves the increase in viscosity due to interparticle colloidal forces. They can be classified as attractive (including van der Waals, electrostatic attractive, hydrophobic, bridging, and depletion) and repulsive (including electrostatic repulsive, steric, hydration, and structural) forces (McClements 1999; Zhou and others 2001). The total or net interaction potential (U) between pairs of particles as a function of the interparticle distance (r) may be predicted by the extended DLVO theory.

Charged particles in an electrolyte present an arrangement of charges in the interface called the electrical double layer (EDL). The distortion of the EDL by the shear field leads to an increase in the viscosity due to increased energy dissipation. This effect was first considered by Smoluchowski and is called the primary electroviscous effect (Hidalgo-Álvarez and others 1996; Rubio-Hernández and others 2004). For dilute dispersions of spherical particles, it appears as a correction p, the primary electroviscous coefficient, to the Einstein equation (Eq. 2), such that combined with Eq. 6 gives


The coefficient p is a function of the potential in the slipping plane or ζ–potential (a measure of the particle surface potential), and the relative size of the particle radius (a) with respect to the thickness of the EDL, calculated as the Debye length (κ−1). Several theoretical expressions have been derived for p (see, for example, Hidalgo-Álvarez and others 1996; Rubio-Hernández and others 2004).

However, Eq. 7 only considers the contribution of electrostatic repulsive forces to the viscosity. Genovese and Lozano (2006) proposed a more general, semiempirical expression for ηcfr in terms of the maximum net repulsive potential between pairs of particles (UMax), also known as the energy barrier or activation energy


where α is a dimensionless proportionality constant, kB is Boltzmann's constant, and T is the absolute temperature. The value of UMax predicts the stability of colloidal dispersions, and is obtained at the maximum of the U(r) curve.

Figure 2 shows the 3 components of Eq. 6 as a function of φ for cloudy apple juice from 10 to 50° Brix (Genovese and Lozano 2006). Values of ηcfr were obtained by difference between empirical ηr values and theoretical ηhsr values from Eq. 2. Values of UMax were obtained from the balance between van der Waals, electrostatic, and hydration interparticle forces. By applying Eq. 8, the value α= 0.483 was obtained.

Figure 2—.

Cloudy apple juice relative viscosity compared with particle volume fraction: experimental (•), and predicted with Eq. 6 (dashed line); theoretical hard-sphere contribution, Eq. 2 (full line); colloidal-forces contribution: semi-empirical values (○), and fitted with Eq. 8 (dashed-dotted line). Reprinted from Genovese and Lozano (2006), with permission from Elsevier.

Figure 3 shows ηcfr (φ) of cloudy apple juice at different pHs and ionic strengths (Benítez and others 2007). Values of UMax were obtained combining the extended DLVO theory and turbidity experimental data. A unique value of α= 0.267 was obtained from Eq. 8 for all samples.For concentrated colloidal suspensions, Ogawa and others (1997) derived the following expression based on the theory of activation processes:


where c1 and c2 (theoretically π/6) are numerical constants, d is the particle diameter, and σp (approximately 0 for low shears) is the particle stress (as in Eq. 32 also).

Figure 3—.

Effect of particle volume fraction on the colloidal forces contribution to relative viscosity, for different liquid medium conditions. Experimental data (symbols) fitted with Eq. 8 (full lines). Reprinted from Benítez and others (2007), with permission from Elsevier.

In the effective volume fraction method, it is considered that repulsive forces keep particles apart from one another, thus increasing their effective radius from a to aeff. Consequently, the effective volume fraction is given as


In this context, the viscosity of a colloidal dispersion can be simply obtained by replacing φ by φeff in a hard-sphere viscosity equation, like Eq. 3 (Quemada and Berli 2002)


By allowing aeff to be shear-dependent, Buscall (1991, 1994) found an approximate theoretical expression


where k approximately ½ and Keff) = 0.016 + 0.52φeff are phenomenological factors.

Aggregating colloidal dispersions and microgel suspensions The aggregation of colloidal particles leads to the formation of fractal structures, that is, highly branched aggregates. A fractal dimension indicates the degree to which an image or object outline deviates from smoothness and regularity. The term fractal was coined by Mandelbrot (1982), who introduced dimensions “between” the conventional Euclidean dimensions of 1, 2, and 3 in order to describe structures that are not Euclidean lines, surfaces, or solids.

Berli and others (1999) proposed a viscosity equation for an aggregating soy suspension by using the modified Krieger–Dougherty equation (Eq. 3)


where ηm is the viscosity of the medium filling the space between the aggregates (instead of ηs), Df is the fractal dimension of the aggregates, and N is the averaged number of particles in a cluster given by the expression


where t0 is the initial time at which N≈ 1, and inline image the aggregation rate constant. Combining Eq. 13 and 14 it is possible to measure the degree of aggregation of a suspension (and to estimate the value of Df) from viscosity-time data.

Microgel suspensions are particles composed of a central zone of cross-linked polymer and an external layer of polymer chains. Berli and Quemada (2000) proposed the following model for these systems


where σc is the critical shear stress, χ is a rheological index, ahs is the equivalent hard-sphere radius, which involves the core radius plus the hydrodynamic thickness of the polymer layer, and φ0 and φ are the maximum packing fractions for σ→ 0 and σ→∞, respectively.

Modulus of colloidal gels of fractal flocs

Shih and others (1990) developed a scaling relationship to explain the elastic properties of colloidal gels by considering the structure of the gel network to be a collection of close packed fractal flocs of colloidal particles. At high particle concentrations, the links between flocs have lower elastic constant than the flocs themselves. This regime should be applicable to gels that are well above the gelation threshold (Shih and others 1990), where the elastic modulus, G′, is related to the particle volume fraction (φ) by the following relationship:


where D is the Euclidean dimension of the network—usually 3.

Russel and others (1989) pointed out that the fractal dimension reflects the internal structure of the flocs and depends on the mode of aggregation. In rapid flocculation, Df= 1.75 for cluster–cluster aggregation, and Df= 2.5 for particle–particle aggregation. On the other hand, slow flocculation allows particles to penetrate further into the floc and rearrange its configuration before sticking, thus increasing Df to approximately 2.0 and 3.0, respectively.

Fractal dimension of food dispersionsGenovese and Rao (2003a) studied dispersions of 2 starches that were substantially different from each other: a cross-linked waxy maize (CLWM) and tapioca, a tuber starch, with 19.3% amylose. The volume fraction occupied by the starch granules was calculated as the granule mass fraction, that is, φ=cQ. From plots (not shown here) of elastic modulus (G′) compared with oscillatory frequency (ω) of the starch dispersions the plateau equilibrium values, G0, were determined (Shih and others 1990). The equilibrium G0 values obtained for CLWM and tapioca starch dispersions were plotted against φ (Figure 4). The double logarithmic plot resulted in reasonable straight lines (R2≥ 0.95) for both starches, fulfilling the power law relationship inferred by Eq. 18. From the slope of the lines, the fractal dimensions of the 2 types of starch granules were calculated to be Df= 2.81 for CLWM and Df= 2.79 for tapioca, respectively. This was interpreted as both starches having granules with highly convoluted surfaces. The higher G0 values of CLWM starch dispersions were attributed to the higher rigidity of their granules (Genovese and Rao 2003a). Studies on fractal dimension of a number of foods, some of them based on rheological data, have been reported and the values obtained, listed in Table 2, illustrate typical magnitudes.

Figure 4—.

Elastic plateau modulus of 2 starch dispersions as a function of their granules volume fraction. Experimental data (symbols) fitted with Eq. 17. Reprinted from Genovese and Rao (2003a), with permission from AACC International.

Table 2—.  Fractal dimension of structural elements in a few foods based on rheological data (Rao 2006)
Network of particlesFractal dimension,DfReference
Palm oil or lard fat2.82–2.88Marangoni and Rousseau (1998)
Cocoa butter2.37Narine and Marangoni (1999)
Salatrim®2.90Narine and Marangoni (1999)
Milk fat and/canola oil blends1.97–1.99Marangoni and Hartel (1998)
Whey protein isolate + CaCl2 gels2.3–2.6Hongsprabhas and others (1999)
Soy protein isolate gels, pH 3.8 and 0.2M NaCl2.3.Renkema and van Vliet (2004)
Starch gels2.79–2.81Genovese and Rao (2003a)
Egg white protein gel, pH 3.71.9–2.1Ould-Eleya and others 2004

Yield stress models

In general, the magnitude of the yield stress increases with increasing particle volume fraction, decreasing particle size, and increasing magnitude of interparticle forces (Poslinski and others 1988). It has been claimed that yield stress will not occur in a hard-sphere system until the maximum packing fraction is reached (Zhou and others 2001). Most investigations have established and/or quantified yield stresses in suspensions of interacting (colloidal or surface active) particles. Just a few works (Husband and others 1993; Marquez and others 2006) were found on yield stress determination of highly concentrated noncolloidal suspensions. In calcium carbonate filled polyisobutylene suspensions (27 μm in diameter, 0%-57% volume fractions), Husband and others (1993) found a proportional dependence of the yield stress with particle volume fraction. Unfortunately, the reasons/mechanisms of that yielding behavior were not clear, and no theoretical models were proposed in those works.

Colloidal dispersions Volume fraction (φ) and particle size (d) affect the density of the interparticle links and the microstructure, which in turn govern the yield stress behavior of concentrated colloidal dispersions (Zhou and others 2001). Consequently, most theoretical yield stress models are mainly functions of φ, d, and particle–particle interactions. A representative selection of the many models found in the literature is presented next.

Michaels and Bolger (1962) derived the following expression for suspensions of aggregated flocs, where the aggregates form tenuous networks


where σ0 is the apparent yield stress, F is a dimensionless orientation function, H is the interfloc adhesive force, d is the diameter of the individual flocs, and φf is the minimum floc concentration required to form a continuous aggregate network.

By considering the colloidal interparticle forces, Poslinski and others (1988) derived an expression for the yield stress of solid spheres dispersed in a polymeric matrix


where Z is the total number of nearest neighbors of each sphere in a particular packing configuration, A is the Hamaker constant (a measure of the van der Waals attractive forces), ɛ is the dielectric constant of the continuous medium, ɛ0 the permittivity of vacuum, and ψ0 is the surface potential of the spheres. The 2nd term in Eq. 20 accounts for the electrostatic repulsive forces.

Scales and others (1998) proposed a general interactive model for the shear yield stress of flocculated suspensions. For monodisperse spherical particles, the model reduces to an expression similar to Eq. 20:


where M(φ) is the mean coordination number. The terms in the bracket account for the van der Waals and EDL forces acting between pairs of particles, or the strength of 1 particle bond.

For polydisperse spheres, the previous model is quite complex and may be consulted in the work of Scales and others (1998). A simpler model has been suggested for calculation of the yield stress of a mixture suspension (Zhou and others 2001):


where φi and σ0i are the volume fraction and yield stress of the ith component of the suspension, respectively. Unlike hard-sphere dispersions in which polydispersity reduces the viscosity (Eq. 5), a suspension with a broad particle size distribution exhibits higher yield stress than a narrow size distributed suspension (Zhou and others 2001).

From Eq. 12, Buscall (1991, 1994) derived Eq. 23, in which the yield stress arises when the effective particle radius is high enough to produce a dense packing of particles, thus


where am is the maximum radius that the particle can take because of the spatial constraints of the other particles, and the other terms are the same as in Eq. 12.

Based on Eq. 9, Ogawa and others (1997) proposed a simpler model, indicating that when the effect of repulsive interaction is strong, there is an apparent yield stress given by


It should be noted that Eq. 20 and 21 only consider DLVO (van der Waals and repulsive electrostatic) interparticle forces, while Eq. 23 and 24 are functions of the total interaction potential, U, which may include also non-DLVO (steric, hydration, depletion, and so on) interactions.

From analyses of experimental data and theoretical analysis from several works, Zhou and others (2001) suggested the following general expression:


where constant B is related with bond strength depending on material properties and system surface chemistry condition.

The power-law exponent in Eq. 25, v, has been correlated with the fractal dimension, which is believed to be associated with the interconnection and space-filling ability of the network microstructure. To describe this behavior, de Rooij and others (1994) developed a microrheological model in terms of fractal microstructures (Eq. 26):


This model describes the suspension in terms of particle chains where interparticle bonds can be soft or rigid, similar to the study of Shih and others (1990) (Eq. 18).

Finally, it is worth mentioning the model of Berli and Quemada (2000) for microgel suspensions (Eq. 15 to 17), where the yield stress is simply given by


which is only valid if φ> φ0 approximately φm, that is, when particles are densely packed.

The differences among Eq. 19 to 27 suggest that there is not a unique or general theoretical model to describe the yield stress of concentrated colloidal dispersions, but instead the appropriate model should be chosen/derived for each particular system.

Foams Foams behave as elastic solids under small deformations, and flow like viscous liquids above a certain yield stress (Pernell and others 2000; Kampf and others 2003). Based on Eq. 25 and other models for foams (Höhler and others 1999; Princen and Kiss 1989; Davis and others 2004), Raharitsifa and others (2006) proposed a generalized expression for the yield stress of foams as a function of their structural properties:


where φ is the air volume fraction, parameter B′ is expected to include the effect of interfacial tension, b is a fitting parameter, and D32 is the Sauter mean bubble diameter or surface average diameter. For spherical bubbles


where di is the diameter of each bubble, and i varies from 1 to the number of bubbles observed. Raharitsifa and others (2006) applied Eq. (28) to experimental data (Figure 5) on apple juice foams prepared with different concentrations of 2 different foaming agents (methylcellulose and egg white), resulting in B′=18.9, v=10.9, and b=1.37 (R2=0.975).

Figure 5—.

Yield stress of apple juice foams as function of air volume fraction and bubble mean diameter for different foaming agents: (•) methylcellulose, and (○) egg white. Experimental data (symbols) fitted with Eq. 28 (mesh). From Raharitsifa and others (2006).

Structural Models and Analyses

Structural models

These models are derived from consideration of the structure and often kinetics of changes in it. They are used, together with experimental data, to estimate values of parameters that help characterize the rheological behavior of a food sample. One such model is that of Cross (1965) (Eq. 30) that has been used extensively to characterize the flow behavior of polymer dispersions and other shear-thinning fluids.


For the shear rate, inline image, where ηa= (η0)/2, the Cross time constant inline image. Generally, inline image gives an order of magnitude of the critical shear rate marking the end of the zero shear rate Newtonian plateau or the onset of the shear-thinning region (Rao 1999).

The Casson model (Eq. 31) is another structure-based model (Casson 1959) that, although developed for characterizing printing inks originally, has been used to characterize chocolate and other food dispersions that exhibit yield stress.


From a σ0.5 compared with inline image plot, the Casson yield stress is calculated as the square of the intercept, σ0c= (K0c)2 and the Casson plastic viscosity as the square of the slope, ηCa= (Kc)2.

Structural analyses

While application of structure-based models to rheological data does provide useful information, structure-based analysis can provide valuable insight into the role of the structure of a dispersed system. Bodenstab and others (2003) estimated the contributions to flow shear stress of soy milk by suspended particles and the suspending fluid.


where σs is the shear stress caused by the viscous forces generated by the suspending fluid (continuous phase), and σp is the shear stress caused by interaction between suspended particles. The former is temperature dependent and in the latter the temperature dependency can be neglected. In the direct interparticle interactions, Coulomb's mechanical friction forces, hydrogen bonds, electrostatic attraction, and hydrophobic attraction may be important. The viscous contribution was estimated from the expression


where the subscripts 1 and 2 refer to the values of stress and viscosity of the dispersion at temperatures 1 and 2, respectively. Typical values of temperature that were used were 10 °C and 25 °C, and 20 °C and 40 °C (Bodenstab and others 2003). Considerable care had to be used to obtain reliable samples of the continuous phase of the soy milk without solid residues, a concern with many other food dispersions such as fruit and vegetable products (Rao 1987). Another concern is that significant differences in structure of the sample may exist at the 2 different temperatures used. Nevertheless, for the soy milks, interparticle interactions were found to be significant at particle concentrations above about 20 g/100 g.

In the kinetic or structural approach to rheology of dispersions (Michaels and Bolger 1962), the basic flow units are assumed to be small clusters or flocs that at low shear rates give the dispersion a finite yield stress. The clusters associate randomly to form weakly bonded aggregates and tenuous networks, giving rise to plastic and structural properties. Based on the work of Michaels and Bolger (1962), from an energy balance at the point of maximum deformation (yield point) in the vane test, the contributions of different structural components to the total yield stress, σ0s, may be estimated (Genovese and Rao 2003b)


The stress to break the bonds between the flocs may be calculated as the difference between the static, σ0s, and the dynamic, σ0d, yield stresses of the samples with undisrupted and disrupted structure, respectively.


where σb is the stress required to break the bonds between the flocs, σv is the stress dissipated due to purely viscous drag, and σn is the stress required to break the aggregate network. Given that inline image is very small in most dispersions, one can estimate the 2 contributions σb and σn. As 1 example of structural analysis of processed foods, the contribution of bonding to the static yield stress of products that were homogenized, such as mayonnaise, ranged between 53% and 65%, while that of finished, nonhomogenized products such as apple sauce was about 20% (Genovese and Rao 2005). In another study (Tárrega and others 2006), the influence of starch concentration and λ-carrageenan on the contribution of bonding in skim milk-based dispersions was examined.

A texture map can be created by plotting yield stress values against the corresponding values of the deformation. The texture map of 3 heated starch dispersions based on static and dynamic yield stresses compared with deformation is shown in Figure 6 (Genovese and Rao 2003b). Unlike a traditional texture map, a map based on static and dynamic yield stresses indicates the behavior of a food with both undisrupted and disrupted structure, and should be valuable in evaluation of product quality.

Figure 6—.

Texture map of cross-linked waxy maize (cwm), amioca (ami), and tapioca (tap) starch dispersions with undisrupted (u) and disrupted (d) structures. From Genovese and Rao (2003b).


Some food dispersions (for example, cloudy apple juice) contain colloidal particles with dimensions < 10 μm, and others (for example, tomato concentrates, orange juice) contain larger, noncolloidal particles with dimensions > 10 μm. Theoretical models have been derived assuming that the particles are rigid to predict viscosity, yield stress, and modulus of both types of dispersions. These models provide valuable guidelines with respect to the role of key rheological parameters such as volume fraction, size, and fractal dimension of the particles, as well as interparticle forces. However, due to the complex, including nonrigid, nature of particles in most food dispersions, those models have been modified to describe viscosity data. Also due to the complex structure of foods, structural analyses have been developed that are based on experimental rheological data on dispersions and they provide insight into the nature of food microstructure.

Finally, it is worth to mention that foods are subjected to extensional shear in the mouth and in some unit operations (such as fiber spinning and dough mixing). Therefore, extensional viscosity and its measurement are also of interest. Briefly, for low-viscosity fluids, techniques based on creating a stagnation point such as flow between opposing jets and filament stretching have been developed. For high-viscosity and semisolid foods, biaxial (compression-extension) deformation technique has been used (Rao 1999). Further, to better understand how food structure deforms under elongational flow at the microscale, equipment was developed to visualize food structure behavior under compression–extension deformation by combining confocal microscopy and compression apparatus (Nicolas and Paques 2003). One can anticipate that more rheo-optical studies will be conducted, which should provide both visual and rheological data on foods at the microscale.


Authors Genovese and Lozano are grateful for financial support from CONICET. Author Rao acknowledges support from USDA-NRI competitive grants.


particle radius, m


maximum particle radius, m


Hamaker constant, J


constant in Eq. 28, dimensionless

B, B'

constants in Eq. 25 and 28, N and N.mb-2, respectively


dry starch concentration, dimensionless

c1, c2

constants in Eq. 9, dimensionless


particle diameter, m


Euclidean dimension, dimensionless


fractal dimension, dimensionless


Sauter mean diameter of bubbles, m


orientation function, dimensionless


elastic modulus, Pa


plateau modulus, Pa


interfloc adhesive force, N


phenomenological factor in Eq. 12, dimensionless

inline image

aggregation rate constant, s−1


Boltzmann constant, J/°K


phenomenological factor in Eq. 12, dimensionless


square root of Casson plastic viscosity, (Pa.s)1/2


square root of Casson yield stress, Pa1/2


Cross exponent, dimensionless


mean coordination number, dimensionless


averaged number of particles in a cluster, dimensionless


primary electroviscous effect coefficient, dimensionless


axial ratio, dimensionless


swelling factor, dimensionless


distance between pairs of particles, m

t, t0

time, initial time, s


absolute temperature, °K


total interaction potential between pairs of particles, J


energy barrier between pairs of particles, J


constant in Eq. 25 and 28, dimensionless


total number of nearest neighbors of each sphere, dimensionless

Greek Letters

constant in Eq. 8, dimensionless


Cross time constant, s


constant in Eq. 4, dimensionless

inline image

shear rate, s−1

inline image

critical shear rate, s−1


dielectric constant of the continuous medium, dimensionless


permittivity of vacuum, F/m


zeta potential, V


dispersion viscosity, Pa.s


apparent viscosity, Pa.s


filling medium viscosity, Pa.s


relative viscosity, dimensionless


solvent viscosity, Pa.s


zero-shear viscosity (apparent viscosity as inline image), Pa.s


infinite-shear viscosity (apparent viscosity as inline image), Pa.s


intrinsic viscosity, dimensionless


reciprocal Debye length, m−1


shear stress, Pa


shear stress required to break the bonds between flocs, Pa


critical shear stress, Pa


stress required to break the aggregate network, Pa


shear stress caused by particle interaction, Pa


shear stress caused by flow of the suspending fluid, Pa


shear stress dissipated due to purely viscous drag, Pa


yield stress, Pa


dynamic yield stress, Pa


static yield stress, Pa


volume fraction of the dispersed phase, dimensionless


minimum floc volume fraction, dimensionless


maximum packing fraction, dimensionless

φ0, φ

φm for σ→ 0 and σ→∞, respectively, dimensionless


rheological index, dimensionless


surface potential, V


colloidal forces