Open
Published Online: 08 January 2019
Accepted: December 2018
Journal of Rheology 63, 179 (2019); https://doi.org/10.1122/1.5062837
more...View Affiliations

Protein-stabilized emulsions can be seen as mixtures of unadsorbed proteins and protein-stabilized droplets. To identify the contributions of these two components to the overall viscosity of sodium caseinate o/w emulsions, the rheological behavior of pure suspensions of proteins and droplets was characterized, and their properties were used to model the behavior of their mixtures. These materials are conveniently studied in the framework developed for soft colloids. Here, the use of viscosity models for the two types of pure suspensions facilitates the development of a semiempirical model that relates the viscosity of protein-stabilized emulsions to their composition.
Despite their complexity, food products can be conveniently studied from the perspective of colloid science [11. Mezzenga, R., P. Schurtenberger, A. Burbidge, and M. Michel, “Understanding foods as soft materials,” Nat. Mater. 4, 729–740 (2005). https://doi.org/10.1038/nmat1496]. In the last three decades, research in the field of food colloids has led to major advances in understanding their structure over a wide range of lengthscales [22. Dickinson, E., “Food colloids research: Historical perspective and outlook,” Adv. Colloid Interface Sci. 165, 7–13 (2011). https://doi.org/10.1016/j.cis.2010.05.007], which has proved key to developing a good control of their flavor and texture properties [33. Vilgis, T. A., “Soft matter food physics—The physics of food and cooking,” Rep. Prog. Phys. 78, 124602 (2015). https://doi.org/10.1088/0034-4885/78/12/124602].
Many food products such as mayonnaise, ice cream, and yoghurt involve protein-stabilized emulsions either during their fabrication or as the final product. Proteins have particularly favorable properties as emulsifiers because of their ability to strongly adsorb at oil/water interfaces and to stabilize oil droplets by steric and electrostatic repulsion. However, proteins do not completely adsorb at the interface, leaving a residual fraction of protein suspended in the continuous phase after emulsification [4,54. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h5. Srinivasan, M., H. Singh, and P. A. Munro, “Adsorption behaviour of sodium and calcium caseinates in oil-in-water emulsions,” Int. Dairy J. 9, 337–341 (1999). https://doi.org/10.1016/S0958-6946(99)00084-9]. Protein-stabilized emulsions are thus mixtures of protein-stabilized droplets and suspended proteins, as illustrated in Fig. 1. Understanding the contributions of these two components to the properties of the final emulsion remains a challenge.
When considered separately, the droplets in protein-stabilized emulsions can be considered as colloidal particles with some degree of softness. It is thus possible to compare the rheological properties of protein-stabilized emulsions to other types of soft particle suspension and to model their behavior. From a theoretical point of view, particles, colloidal or not, can be described as soft if they have the ability to change the size and shape at high concentration [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007]. Such a definition covers a striking variety of systems, including gel microparticles [7,87. Adams, S., W. J. Frith, and J. R. Stokes, “Influence of particle modulus on the rheological properties of agar microgel suspensions,” J. Rheol. 48, 1195–1213 (2004). https://doi.org/10.1122/1.17737828. Shewan, H. M., and J. R. Stokes, “Viscosity of soft spherical micro-hydrogel suspensions,” J. Colloid Interface Sci. 442, 75–81 (2015). https://doi.org/10.1016/j.jcis.2014.11.064], microgels [9,109. Cloitre, M., R. Borrega, F. Monti, and L. Leibler, “Structure and flow of polyelectrolyte microgels: From suspensions to glasses,” C. R. Phys. 4, 221–230 (2003). https://doi.org/10.1016/S1631-0705(03)00046-X10. Tan, B. H., K. C. Tam, Y. C. Lam, and C. B. Tan, “Microstructure and rheological properties of ph-responsive coreshell particles,” Polymer 46, 10066–10076 (2005). https://doi.org/10.1016/j.polymer.2005.08.007], star polymers [11,1211. Roovers, J., “Concentration-dependence of the relative viscosity of star polymers,” Macromolecules 27, 5359–5364 (1994). https://doi.org/10.1021/ma00097a01512. Winkler, R. G., D. A. Fedosov, and G. Gompper, “Dynamical and rheological properties of soft colloid suspensions,” Curr. Opin. Colloid Interface Sci. 19, 594–610 (2014). https://doi.org/10.1016/j.cocis.2014.09.005], and block copolymer micelles [1313. Leermakers, F. A. M., J. C. Eriksson, and H. Lyklema, Chapter 4—Association colloids and their equilibrium modelling, in Fundamentals of Interface and Colloid Science, edited by J. Lyklema (Elsevier, Amsterdam, 2005), Vol. 5, pp. 4.1–4.123.]. These systems have been the focus of many studies in the last two decades; however, one major challenge to comparing the behavior of such diverse materials is the availability of a well defined volume fraction ϕ for the suspensions.
To overcome the challenge of defining the volume fraction of soft colloids, a common approach is to use an effective volume fraction ϕ eff proportional to the concentration c, ϕ eff = k 0 × c, where k 0 is a constant indicating the voluminosity of the soft particle of interest, usually determined in the dilute ( ϕ < 1 %) or semidilute regime ( ϕ < 15 %). Such a definition for ϕ eff does not take into account the deformation or shrinking of the particle at high concentrations, high values of ( ϕ eff > 1) can thus be reached. k 0 can be estimated using osmometry [1414. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-8], light scattering [1515. Vlassopoulos, D., G. Fytas, S. Pispas, and N. Hadjichristidis, “Spherical polymeric brushes viewed as soft colloidal particles: Zero-shear viscosity,” Phys. B Condens. Matter 296, 184–189 (2001). https://doi.org/10.1016/S0921-4526(00)00798-5], or viscosimetry [10,11,1610. Tan, B. H., K. C. Tam, Y. C. Lam, and C. B. Tan, “Microstructure and rheological properties of ph-responsive coreshell particles,” Polymer 46, 10066–10076 (2005). https://doi.org/10.1016/j.polymer.2005.08.00711. Roovers, J., “Concentration-dependence of the relative viscosity of star polymers,” Macromolecules 27, 5359–5364 (1994). https://doi.org/10.1021/ma00097a01516. Boulet, M., M. Britten, and F. Lamarche, “Voluminosity of some food proteins in aqueous diepersions at various ph and ionic strengths,” Food Hydrocoll. 12, 433–441 (1998). https://doi.org/10.1016/S0268-005X(98)00009-5]. In this study, k 0 was estimated, for each individual component of the emulsions, by modeling the relative zero-shear viscosity η 0 / η s behavior of the pure suspensions in the semidilute regime with the Batchelor equation for hard spheres [1717. Batchelor, G. K., “The effect of Brownian motion on the bulk stress in a suspension of spherical particles,” J. Fluid Mech. 83, 97–117 (1977). https://doi.org/10.1017/s0022112077001062]
η 0 η s = 1 + 2.5 ϕ eff + 6.2 ϕ eff 2 . (1)
Sodium caseinate is used here to stabilize emulsions as a case-study, because of its outstanding properties as a surface-active agent and stabilizer and because it is widely used in industry. Sodium caseinate is produced by replacing the calcium in native milk casein micelles with sodium to increase its solubility [1818. Dalgleish, D. G., and A. J. R. Law, “Sodium caseinates—Composition and properties of different preparations,” J. Soc. Dairy Technol. 41, 1–4 (1988). https://doi.org/10.1111/j.1471-0307.1988.tb00571.x], a process which also leads to the disruption of the micelles. It has been established that sodium caseinate is not present as a monomer in suspension but rather in the form of small aggregates [1919. Lucey, J. A., M. Srinivasan, H. Singh, and P. A. Munro, “Characterization of commercial and experimental sodium caseinates by multiangle laser light scattering and size-exclusion chromatography,” J. Agric. Food Chem. 48, 1610–1616 (2000). https://doi.org/10.1021/jf990769z]. The exact nature of the interactions in play in the formation of these aggregates is not well-known but they have been characterized as elongated and their size estimated to be around 20 nm [14,19,2014. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-819. Lucey, J. A., M. Srinivasan, H. Singh, and P. A. Munro, “Characterization of commercial and experimental sodium caseinates by multiangle laser light scattering and size-exclusion chromatography,” J. Agric. Food Chem. 48, 1610–1616 (2000). https://doi.org/10.1021/jf990769z20. Huppertz, T., I. Gazi, H. Luyten, H. Nieuwenhuijse, A. Alting, and E. Schokker, “Hydration of casein micelles and caseinates: Implications for casein micelle structure,” Int. Dairy J. 74, 1–11 (2017). https://doi.org/10.1016/j.idairyj.2017.03.006]. Some larger aggregates can also form in the presence of residual traces of calcium or oil from the original milk; however, these only represent a small fraction of the protein [1818. Dalgleish, D. G., and A. J. R. Law, “Sodium caseinates—Composition and properties of different preparations,” J. Soc. Dairy Technol. 41, 1–4 (1988). https://doi.org/10.1111/j.1471-0307.1988.tb00571.x]. The viscosity behavior of sodium caseinate as a function of concentration shows similarities with hard-sphere suspensions at relatively small concentrations, but at higher concentrations, over c > 130 g l 1, the viscosity continues to increase with a power law rather than diverging [14,2114. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-821. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003] as would be expected for a hard-sphere suspension [2222. Faroughi, S. A., and C. Huber, “A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number,” Rheol. Acta 54, 85–108 (2014). https://doi.org/10.1007/s00397-014-0825-8].
Proteins can be considered as irreversibly adsorbed [2323. Norde, W., J. Buijs, and H. Lyklema, Chapter 3—Adsorption of globular proteins, in Fundamentals of Interface and Colloid Science, edited by J. Lyklema (Elsevier, Amsterdam, 2005), Vol. 5, pp. 3.1–3.59.] because multiple segments can adsorb, and a saturated monolayer can be reached at the oil–water interface [2424. Spyropoulos, F., E. A. K. Heuer, T. B. Mills, and S. Bakalis, Protein-stabilised emulsions and rheological aspects of structure and mouthfeel, in Practical Food Rheology an Interpretive Approach, edited by I. T. Norton, F. Spyropoulos, and P. Cox (Wiley-Blackwell, Oxford, 2010), Serial 9, pp. 193–2018.] at low concentrations (around 10 × 10 4 wt. % for caseins [2525. Graham, D. E., and M. C. Phillips, “Proteins at liquid interfaces: II. Adsorption isotherms,” J. Colloid Interface Sci. 70, 415–426 (1979). https://doi.org/10.1016/0021-9797(79)90049-3]). The desorption that may occur upon changes in concentration of the emulsion is thus considered to be negligible.
In a different context, caseinate-stabilized emulsions are known to build 3D networks that display viscoelastic properties, and these structures can be used to prevent creaming of the droplets in industrial applications [26,2726. Dickinson, E., “Structure formation in casein-based gels, foams, and emulsions,” Colloids Surf. A Physicochem. Eng. Aspects 288, 3–11 (2006). https://doi.org/10.1016/j.colsurfa.2006.01.01227. Dickinson, E., “Emulsion gels: The structuring of soft solids with protein-stabilized oil droplets,” Food Hydrocoll. 28, 224–241 (2012). https://doi.org/10.1016/j.foodhyd.2011.12.017]. However, this gelation is in general induced by the destabilization of sodium caseinate, by means of a change of pH or enzymatic reactions. Here, no such structure formation occurs as the proteins, and consequently the droplets to which they are adsorbed, are electrostatically stabilized at pH above 5.8 [2828. Chen, J. S., E. Dickinson, and M. Edwards, “Rheology of acid-induced sodium caseinate stabilized emulsion gels,” J. Texture Stud. 30, 377–396 (1999). https://doi.org/10.1111/j.1745-4603.1999.tb00226.x]. The emulsions thus behave as liquids in the range of concentrations studied here.
In this study, the rheology of protein-stabilized emulsions is examined within the framework of soft colloidal particles. Modeling proteins in this way ignores protein-specific elements, such as surface hydration, conformation changes, association, and surface charge distribution [29,3029. Sarangapani, P. S., S. D. Hudson, K. B. Migler, and J. A. Pathak, “The limitations of an exclusively colloidal view of protein solution hydrodynamics and rheology,” Biophys. J. 105, 2418–2426 (2013). https://doi.org/10.1016/j.bpj.2013.10.01230. Sarangapani, P. S., S. D. Hudson, R. L. Jones, J. F. Douglas, and J. A. Pathak, “Critical examination of the colloidal particle model of globular proteins,” Biophys. J. 108, 724–737 (2015). https://doi.org/10.1016/j.bpj.2014.11.3483], but it provides a convenient theoretical framework to separate and discuss the contributions of both sodium caseinate and the droplets to the viscosity of emulsions. Similarly, protein-stabilized droplets can be seen as comprising an oil core and a soft protein shell [3131. Bressy, L., P. Hebraud, V. Schmitt, and J. Bibette, “Rheology of emulsions stabilized by solid interfaces,” Langmuir 19, 598–604 (2003). https://doi.org/10.1021/la0264466], allowing for a unifying approach for both components of the emulsions. The terminology used for soft colloids is also applied here [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007], and samples containing only unadsorbed proteins (or protein-stabilized droplets) are termed protein suspensions (or droplet suspensions, respectively) to reflect the theoretical approach adopted for these systems.
The aim of this study is to present a predictive model of the viscosity of protein-stabilized emulsions that takes into account the presence and behavior of both the protein-stabilized droplets and the unadsorbed protein. A first step is to characterize separately the flow behavior and viscosity of suspensions of purified protein-stabilized droplets and of protein suspensions over a wide range of concentrations. This also allows a critical assessment of the soft colloidal approach. These components are then combined to form mixtures of well-characterized composition, and their viscosity is compared to a semiempirical model. Because they are well dispersed, most of the suspensions and emulsions display a Newtonian behavior at low shear, with shear thinning at higher shear-rates. In this context, we model the concentration dependence of zero-shear viscosity and the shear thinning behavior separately to confirm the apparent colloidal nature of the components of the emulsions and protein suspensions.
A. Preparation of protein suspensions
Because of its excellent ability to stabilize emulsions, sodium caseinate was used in this study. Sodium caseinate (Excellion S grade, spray-dried, 90.4 wt. % dry protein, 5.5 wt. % moisture, 4.1 wt. % ash—including 1.4 wt. % sodium and 1.0 wt. % calcium) was kindly provided by Friesland Campina DMV (Netherlands). It was further purified by first suspending it in deionized water, at 5%–9% (w/w), and then by mixing thoroughly with a magnetic stirrer for 16 h. After complete dispersion, a turbid suspension was obtained, which was centrifuged at 40 000 ×g (Evolution RC, Sorvall with rotor SA 600, Sorvall and clear 50 ml tubes, Beckmann) for 4 h at 21  °C. Subsequently, the supernatant, made of residual fat contamination, and the sediment were separated from the suspension, which was now clearer. The solution was then filtered using a 50 ml stirred ultrafiltration cell (Micon, Millipore) with a 0.45 μm membrane (Sartolon Polyamid, Sartorius). In order to avoid spoilage of the protein solution, 0.05% of ProClin 50 (Sigma Aldrich), a commercial 2-methyl-4-isothiazolin-3-one solution, was added. The suspension at 5 % (w/w) was then diluted to the required concentration. Concentrated suspensions of sodium caseinate were prepared by evaporating a stock solution of sodium caseinate at 5 % (w/w), prepared following the previous protocol, using a rotary evaporator (Rotavapor R-210, Buchi). Mild conditions were used to avoid changing the structure of the proteins: the water bath was set at 40  °C and a vacuum of 45 mbar was used to evaporate water. The concentration of all the suspensions after purification was estimated by refractometry, using a refractometer RM 50 (Mettler Toledo), light-emitting diode at 589.3 nm, and a refractive index increment of d n / d c = ( 0.1888 ± 00033 ) nl g 1 [3232. Zhao, H., P. H. Brown, and P. Schuck, “On the distribution of protein refractive index increments,” Biophys. J. 100, 2309–2317 (2011). https://doi.org/10.1016/j.bpj.2011.03.004].
Size analysis by flow field fractionation (kindly performed by PostNova Analytics Ltd) showed that the resulting suspensions of sodium caseinate were composed of small aggregates of a hydrodynamic radius of 11 nm at 96 %, while the remaining 4 % formed larger aggregates with a wide range of sizes (hydrodynamic radii from 40 to 120 nm) as shown in Fig. 2.
B. Preparation of emulsions
Nanosized caseinate-stabilized droplets were prepared in two steps. First, the preemulsion was produced by mixing 45 mg ml 1 sodium caseinate solution (prepared as detailed previously) with glyceryl trioctanoate ( ρ = 0.956 g ml 1, Sigma Aldrich) at a weight ratio 4:1 using a rotor stator system (L4R, Silverson). This preemulsion was then stored at 4 °C for 4 h to reduce the amount of foam. It was then passed through a high-pressure homogeniser (Microfluidizer, Microfluidics) with an input pressure of 5 bar, equivalent to a pressure of 1000 bar in the microchamber, three times consecutively. After three passes, a stationary regime was reached where the size of droplets could not be reduced any further.
The size of the droplets produced by this emulsification protocol was measured using dynamic light scattering (DLS) (Zetasizer Nano ZS, Malvern Panalytical) and static light scattering (SLS) (Mastersizer, Malvern), with the refractive index of glyceryl-trioctanoate n oil = 1.458. DLS measures the hydrodynamic radius and gave R h , 43 = ( 110 ± 49 ) nm, while SLS measures the optical cross section and gave R opt , 43 = ( 65 ± 31 ) nm, where both sizes are given by the volume average radius R 43 and its standard deviation. The size distributions can be found in the supplementary material in Fig. S1 [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.]. The difference in the two sizes arises from the influence of the layer of adsorbed proteins on the hydrodynamic radius, while the optical cross section only takes into account the scattering by the oil core of the droplet. Because this study is focused on the rheological properties of the emulsion, the use of the hydrodynamic size is preferred here. This protocol for emulsification produced droplets of radius around 110 nm as measured by DLS (Zetasizer, Malvern) and 65 nm by SLS (Mastersizer, Malvern).
Because not all the protein content was adsorbed at the interface, an additional centrifugation step was required to separate the droplets from the continuous phase of protein suspension. This separation was performed by spinning the emulsion at 235 000 ×g with an ultracentrifuge (Discovery SE, Sorvall, with fixed-angle rotor 45Ti, Beckmann Coulter) for 16 h at 21 °C. The concentrated droplets then formed a solid layer at the top of the subnatant that could be carefully removed with a spatula. The subnatant containing proteins and some residual droplets was discarded. The drying of a small fraction of the concentrated droplet layer and the weighing of its dry content yielded a concentration of the droplet paste of ( 0.519 ± 0008 ) g ml 1, so the concentration in droplets of all the suspensions was derived from the dilution parameters. Only one centrifugation step was employed to separate the droplets from the proteins, as it was felt that further steps may lead to protein desorption and coalescence. The pure nanosized droplets were then redispersed at the required concentration, in the range 0.008–0.39 g ml 1 in deionized water for 1–30 min with a magnetic stirrer.
C. Preparation of mixtures
To prepare emulsions with a controlled concentration of proteins in suspension, the concentrated droplets were resuspended in a protein suspension at the desired concentration using a magnetic stirrer and a stirring plate for 5 min to 2 h.
D. Viscosity measurements
Rotational rheology measurements were performed using a stress-controlled MCR 502 rheometer (Anton Paar) and a Couette geometry (smooth bob and smooth cup, 17 ml radius) at 25 °C. For each sample, three measurements are performed and averaged to obtain the flow curve. The values of viscosity on the plateau at low shear are averaged to determine the zero-shear viscosity. Viscosity measurements were performed at different concentrations for protein suspensions, protein-stabilized droplet suspensions, and mixtures.
In order to study the rheological behavior of protein-stabilized emulsions, the approach used here is to separate the original emulsion into its two components, namely, unadsorbed protein assemblies and protein-coated droplets, and to characterize the suspensions of each of these components. Despite their intrinsic complexity due to their biological natures, random coil proteins such as sodium caseinate can conveniently be considered as colloidal suspensions, as we demonstrate in the discussion below.
A. Viscosity of suspensions in the semidilute regime: Determination of volume fraction
The weight concentration (in g ml 1) is a sufficient parameter to describe the composition in the case of one suspension, but only the use of the volume fraction of the suspended particles allows meaningful comparisons between protein assemblies and droplets. In the framework of soft colloids, the effective volume fraction ϕ eff of a colloidal suspension can be determined by modeling the viscosity in the semidilute regime with a hard-sphere model.
The relative zero-shear viscosities of semidilute samples are displayed in Fig. 3 as a function of the mass concentration of protein or droplets (viscosity data at the full range of concentration can be found in Fig. S2 in the supplementary material [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.]). As can be seen, protein suspensions reach a higher viscosity at a lower weight fraction than droplet suspensions. This is because the protein is highly hydrated and swollen and so occupies a greater volume per unit mass than do the droplets, where the main contributor to the occupied volume is the oil core.
The viscosity behavior of each type of suspension in the semidilute regime can be described by a theoretical model such as Batchelor’s equation [1717. Batchelor, G. K., “The effect of Brownian motion on the bulk stress in a suspension of spherical particles,” J. Fluid Mech. 83, 97–117 (1977). https://doi.org/10.1017/s0022112077001062], Eq. (1), as a function of the volume fraction ϕ. This involves assuming that the particles in the suspension of interest do not have specific interparticle interactions or liquid interfaces in this regime and can be accurately described as hard spheres.
In addition, as a first approximation, the effective volume fraction ϕ eff of soft particles in suspension is assumed to be proportional to the weight concentration c
ϕ eff = k 0 × c , (2)
where k 0 is a constant expressed in ml 1 g. This equation is combined with Eq. (1) in order to obtain an expression for the viscosity as a function of the concentration. When applied to experimental viscosity values for suspensions of protein or droplets at concentrations in the semidilute regime, such an expression allows estimation of k 0. The effective volume fraction ϕ eff of the suspensions can then be calculated using Eq. (2).
When fitted to the viscosity data for pure sodium caseinate and pure droplets, as described above, Eq. (1) gives satisfactory fits as shown in Fig. 3. The resulting values for k 0 are, for protein suspensions, k 0 , prot = ( 8.53 ± 23 ) ml 1 g, and for droplet suspensions, k 0 , drop = ( 2.16 ± 13 ) ml 1 g.
The protein result is in reasonable agreement with previous results, where determinations of the volume fraction using the intrinsic viscosity gave ϕ eff , prot = 6.4 c [2121. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003] and ϕ eff , prot = 6.5 ± 5 c [2020. Huppertz, T., I. Gazi, H. Luyten, H. Nieuwenhuijse, A. Alting, and E. Schokker, “Hydration of casein micelles and caseinates: Implications for casein micelle structure,” Int. Dairy J. 74, 1–11 (2017). https://doi.org/10.1016/j.idairyj.2017.03.006], while osmometry measurements (at a higher temperature) gave ϕ eff , prot = 4.47 c [1414. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-8]. For droplet suspensions, k 0 , drop corresponds to the voluminosity of the whole droplets. If these were purely made of a hard oil core, their voluminosity would be 1 / ρ oil = 1.05 ml 1 g. The higher value observed can be attributed to the layer of adsorbed proteins at the surface of the droplets. This is an indication that the nanosized droplets can be modeled as core–shell particles.
These results make it possible to calculate the effective volume fractions ϕ eff of both types of suspensions, which is a necessary step to allowing their comparison. It is however important to keep in mind that ϕ eff is an estimate of the volume fraction using the hard sphere-assumption, which is likely to break down as the concentration is increased, where deswelling, deformation, and interpenetration of the particles may occur [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007].
B. Modeling the viscosity behaviors of colloidal suspensions
In order to identify the contributions of the components to the viscosity of the mixture, it is important to characterize the viscosity behaviors of the pure suspensions of caseinate-stabilized nanosized droplets and of sodium caseinate. This is achieved by modeling the volume fraction dependence of the viscosity with equations for hard and soft colloidal particles.
1. Suspensions of protein-stabilized droplets
The viscosity of protein-stabilized droplet suspensions is displayed in Fig. 4. A sharp divergence is observed at high volume fraction, and this behavior is typical of hard-sphere suspensions [3333. de Kruif, C. G., E. M. F. van Iersel, A. Vrij, and W. B. Russel, “Hard sphere colloidal dispersions: Viscosity as a function of shear rate and volume fraction,” J. Chem. Phys. 83, 4717–4725 (1985). https://doi.org/10.1063/1.448997]. It is thus appropriate to use one of the relationships derived for such systems to model the viscosity behavior of droplet suspensions.
Among the multiple models for the viscosity of hard-sphere suspensions that have been proposed over time, the theoretical model developed by Quemada [3434. Quemada, D., “Rheology of concentrated disperse systems and minimum energy dissipation principle,” Rheol. Acta 16, 82–94 (1977). https://doi.org/10.1007/BF01516932] is used in this work
η 0 η s = 1 ϕ ϕ m 2 , (3)
where the parameter ϕ m is the maximum volume fraction at which the viscosity of the suspension diverges
lim ϕ ϕ m η 0 η s = . (4)
The Quemada model fits remarkably well to the experimental data of the relative viscosity η 0 / η s of suspensions of droplets when ϕ is approximated by the effective volume fraction ϕ eff. The value for the maximum volume fraction is found to be ϕ m = 0.79 ± 2. Despite the similarity in viscosity behavior between the droplet suspensions and hard-sphere suspensions, the maximum volume fraction found here is considerably higher than the theoretical value of ϕ m = ϕ rcp = 0.64 for randomly close-packed hard spheres.
A possible explanation for this discrepancy is the polydispersity of the droplet suspension. Indeed, random close-packing is highly affected by the size distribution of the particles, as smaller particles can occupy the gaps between larger particles [3535. Farris, R. J., “Prediction of the viscosity of multimodal suspensions from unimodal viscosity data,” Trans. Soc. Rheol. 12, 281–301 (1968). https://doi.org/10.1122/1.549109]. In a recent study, Shewan and Stokes modeled the viscosity of hard-sphere suspensions using a maximum volume fraction predicted by a numerical model developed by Farr and Groot [36,3736. Shewan, H. M., and J. R. Stokes, “Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution,” J. Nonnewton. Fluid Mech. 222, 72–81 (2015). https://doi.org/10.1016/j.jnnfm.2014.09.00237. Farr, R. S., and R. D. Groot, “Close packing density of polydisperse hard spheres,” J. Chem. Phys. 131, 244104 (2009). https://doi.org/10.1063/1.3276799], which allows the maximum volume fraction of multiple hard-sphere suspensions to be predicted from their size distribution.
Here, the same approach is used with the size distributions of the protein-stabilized droplets obtained from both the Mastersizer and the Zetasizer. The numerically estimated random close-packing volume fraction ϕ rcp is close for both size distributions, and its value is ϕ rcp = 0.68. Although this is a higher maximum volume fraction than for a monodisperse hard-sphere suspension, it is still considerably lower than the experimental value, ϕ m = 0.79. Such a high random close-packing fraction can be achieved numerically only if a fraction of much smaller droplets is added to the distribution obtained by light scattering. The hypothesis of the presence of small droplets, undetectable by light scattering without previous fractionation, is supported by the observation of such droplets upon fractionation of a very similar emulsion in a previous study [3838. Dalgleish, D. G., S. J. West, and F. R. Hallett, “The characterization of small emulsion droplets made from milk proteins and triglyceride oil,” Colloids Surf. A Physicochem. Eng. Aspects 123, 145–153 (1997). https://doi.org/10.1016/S0927-7757(97)03783-7].
It is also possible that other mechanisms than the polydispersity come into play at high volume fractions of droplets. Although it would be hard to quantify, it is likely that the soft layer of adsorbed proteins may undergo some changes at high volume fraction, such as deswelling or interpenetration.
2. Protein suspensions
Sodium caseinate is known to aggregate in solution to form clusters or micelles [14,20,2114. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-820. Huppertz, T., I. Gazi, H. Luyten, H. Nieuwenhuijse, A. Alting, and E. Schokker, “Hydration of casein micelles and caseinates: Implications for casein micelle structure,” Int. Dairy J. 74, 1–11 (2017). https://doi.org/10.1016/j.idairyj.2017.03.00621. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003]. These differ from protein-stabilized droplets because of their swollen structure and likely dynamic nature. The viscosity behavior of the suspensions they form is displayed in Fig. 5.
TABLE I. Parameters for the modified Quemada model for soft colloids, Eq. (5), applied to sodium caseinate suspensions.
Parameter Value Standard error
ϕ m 0.93 0.02
n 6.1 0.4
At high concentrations, the viscosity does not diverge as quickly as for the suspensions of droplets. This result is in agreement with previous studies on sodium caseinate, in which suspensions at higher concentrations were studied [14,21,3914. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-821. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.00339. Loveday, S. M., M. A. Rao, L. K. Creamer, and H. Singh, “Rheological behavior of high-concentration sodium caseinate dispersions,” J. Food Sci. 75, N30–N35 (2010). https://doi.org/10.1111/j.1750-3841.2009.01493.x]. In these works, it was shown that the viscosity does not diverge but follows a power law η 0 / η s ( ϕ eff , prot ) 12.
The behavior displayed by sodium caseinate resembles that of core–shell microgels [1010. Tan, B. H., K. C. Tam, Y. C. Lam, and C. B. Tan, “Microstructure and rheological properties of ph-responsive coreshell particles,” Polymer 46, 10066–10076 (2005). https://doi.org/10.1016/j.polymer.2005.08.007] and soft spherical brushes [1515. Vlassopoulos, D., G. Fytas, S. Pispas, and N. Hadjichristidis, “Spherical polymeric brushes viewed as soft colloidal particles: Zero-shear viscosity,” Phys. B Condens. Matter 296, 184–189 (2001). https://doi.org/10.1016/S0921-4526(00)00798-5]; hence, a soft colloid framework (as reviewed, e.g., in [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007]) seems suitable for the study of these suspensions.
A general feature of the viscosity behavior of soft colloidal suspensions is the oblique asymptote at high concentrations. This behavior is believed to arise because, as the concentration increases, the effective volume occupied by each particle decreases, by deswelling or interpenetration. Thus, the strong viscosity divergence of hard-sphere suspensions is absent for soft colloids. To describe the behavior of such suspensions, a model is thus required that takes into account this distinctive limit at high concentrations while retaining the hard-sphere behavior at lower concentrations.
A semiempirical modification that fulfills the above criteria is the substitution of the maximum volume fraction ϕ m by a ϕ-dependent parameter ϕ m that takes the form ϕ m = ϕ m n + ϕ n 1 / n.
As a result, a modified version of Eq. (3) can be derived that takes into account the softness of the particles via a concentration-dependent maximum volume fraction ϕ m . This semiempirical viscosity model is expressed as
η 0 η s = 1 ϕ ϕ m 2 , (5)
where
ϕ m = ϕ m 1 + ϕ ϕ m n 1 / n .
The addition of the exponent n as a parameter expresses the discrepancy from the hard-sphere model. The smaller the n is, the lower the volume fraction ϕ at which ϕ m diverges from ϕ m, and the less sharp the divergence in viscosity.
The model in Eq. (5) was applied to fit the experimental data displayed in Fig. 5, and the resulting fitting parameters are listed in Table I.
The use of this approach gives a good fit of the viscosity behavior of sodium caseinate in the range of concentrations used here. In addition, this semiempirical model also satisfactorily describes the viscosity of sodium caseinate suspensions at higher concentration from [14,2114. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-821. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003]. It is worth noting that the inflection of viscosity is slightly sharper for the model than for the experimental data.
Using this model, ϕ m = 0.93 ± 002, while for monodisperse hard spheres, the expected value would be of 0.64. While it is not clear that ϕ m in our model should be consistent with the packing of hard spheres, such high values have also been obtained for other soft colloidal systems, such as microgels [40,4140. Borrega, R., M. Cloitre, I. Betremieux, B. Ernst, and L. Leibler, “Concentration dependence of the low-shear viscosity of polyelectrolyte micro-networks: From hard spheres to soft microgels,” Europhys. Lett. 47, 729–735 (1999). https://doi.org/10.1209/epl/i1999-00451-141. Pellet, C., and M. Cloitre, “The glass and jamming transitions of soft polyelectrolyte microgel suspensions,” Soft Matter 12, 3710–3720 (2016). https://doi.org/10.1039/c5sm03001c] or star polymers [4242. Erwin, B. M., M. Cloitre, M. Gauthier, and D. Vlassopoulos, “Dynamics and rheology of colloidal star polymers,” Soft Matter 6, 2825–2833 (2010). https://doi.org/10.1039/b926526k]. This is mainly because the effective volume fraction ϕ eff is an ambiguous parameter for soft colloids that does not necessarily describe the actual volume fraction occupied by the particles [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007]. It is indeed likely that osmotic deswelling of the charged protein aggregates occurs at high concentrations, leading to a high effective volume fraction.
The power law toward which the relative viscosity η 0 / η s described by Eq. (5) tends at high concentration (i.e., ϕ > ϕ m) can be calculated by developing ϕ m . Indeed, at high concentration, ϕ m converges toward ϕ × [ 1 + ( 1 / n ) × ( ϕ m / ϕ ) n ] ( ϕ eff ) n, so η 0 / η s converges toward [ 1 + n ( ϕ eff / ϕ m ) n ] 2 (detailed calculations are provided in the supplementary material [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.]). Using the value in Table I, the relative viscosity of sodium caseinate suspensions is found to follow the power law η 0 / η s ( ϕ eff , prot ) 12.2 ± 8. This value is in good agreement with the literature where η 0 / η s ( ϕ eff , prot ) 12 in the concentrated regime [14,21,3914. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-821. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.00339. Loveday, S. M., M. A. Rao, L. K. Creamer, and H. Singh, “Rheological behavior of high-concentration sodium caseinate dispersions,” J. Food Sci. 75, N30–N35 (2010). https://doi.org/10.1111/j.1750-3841.2009.01493.x].
It is interesting to note that Eq. (5) provides a good model for the behavior of particle suspensions and emulsions whose particles have a wide range of softness, as will be detailed elsewhere. Within this context, the concentration behavior of sodium caseinate suspensions seems to indicate that they can also be regarded as suspensions of soft particles. This interpretation of the behavior can be further tested by considering the shear-rate dependent response of both the emulsions and sodium caseinate suspensions.
C. Shear thinning behavior of protein and droplet suspensions
Over most of the concentration range studied here, the protein suspensions display Newtonian behavior. However, at high concentration of protein, shear thinning is observed at high-shear-rates (flow curves in the supplementary material [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.]). By comparison, the droplet suspensions display shear thinning at a much broader range of concentrations. This behavior is common in colloidal suspensions [33,4333. de Kruif, C. G., E. M. F. van Iersel, A. Vrij, and W. B. Russel, “Hard sphere colloidal dispersions: Viscosity as a function of shear rate and volume fraction,” J. Chem. Phys. 83, 4717–4725 (1985). https://doi.org/10.1063/1.44899743. Helgeson, M. E., N. J. Wagner, and D. Vlassopoulos, “Viscoelasticity and shear melting of colloidal star polymer glasses,” J. Rheol. 51, 297–316 (2007). https://doi.org/10.1122/1.2433935], as well as polymer and surfactant solutions and arises from a variety of mechanisms [44,4544. Cross, M. M., “Rheology of non-Newtonian fluids—A new flow equation for pseudoplastic systems,” J. Colloid Sci. 20, 417–437 (1965). https://doi.org/10.1016/0095-8522(65)90022-X45. Cross, M. M., “Kinetic interpretation of non-Newtonian flow,” J. Colloid Interface Sci. 33, 30–35 (1970). https://doi.org/10.1016/0021-9797(70)90068-8]. In nonaggregated suspensions of Brownian particles, shear thinning arises from the competition between Brownian motion (which increases the effective diameter of the particles) and the hydrodynamic forces arising from shear. Shear thinning then occurs over a range where the two types of forces balance, as characterized by the dimensionless reduced shear stress ( σ r) being of order unity. σ r is given by
σ r = σ R 3 k T , (6)
where σ is the shear stress, R is the radius of the colloidal particle, k is the Boltzmann constant, and T is the temperature of the suspension (here T = 298 K).
In such suspensions, the flow curve can be described using the following equation for the viscosity as a function of shear stress [46–4846. Woods, M. E., and I. M. Krieger, “Rheological studies on dispersions of uniform colloidal spheres: I. Aqueous dispersions in steady shear flow,” J. Colloid Interface Sci. 34, 91–99 (1970). https://doi.org/10.1016/0021-9797(70)90262-647. Krieger, I. M., Rheology of monodisperse latices,” Adv. Colloid Interface Sci. 3, 111–136 (1972). https://doi.org/10.1016/0001-8686(72)80001-048. Frith, W. J., J. Mewis, and T. A. Strivens, “Rheology of concentrated suspensions—Experimental investigations,” Powder Technol. 51, 27–34 (1987). https://doi.org/10.1016/0032-5910(87)80037-2]:
η η s = η + η 0 η 1 + σ r / σ r , c m , (7)
where η 0 is the zero-shear viscosity, η is the high-shear limit of the viscosity, m is an exponent that describes the sharpness of the change in regime between η 0 and η , and σ r , c is the reduced critical shear stress.
Because shear thinning arises from the competition between Brownian motion and the applied external flow, the use of a dimensionless stress that takes into account the size of the colloidal particles allows meaningful comparisons between the different suspensions [46,4846. Woods, M. E., and I. M. Krieger, “Rheological studies on dispersions of uniform colloidal spheres: I. Aqueous dispersions in steady shear flow,” J. Colloid Interface Sci. 34, 91–99 (1970). https://doi.org/10.1016/0021-9797(70)90262-648. Frith, W. J., J. Mewis, and T. A. Strivens, “Rheology of concentrated suspensions—Experimental investigations,” Powder Technol. 51, 27–34 (1987). https://doi.org/10.1016/0032-5910(87)80037-2]. Here, we use this approach to compare the flow behavior of the protein and droplet suspensions and to test further the hypothesis that the protein suspensions can be considered to behave as though they are suspensions of soft particles.
Fitting the flow curves with Eq. (7) allows for the extraction of the critical stress σ c. The behavior of this parameter as a function of the zero-shear relative viscosity (as a proxy for concentration) is shown in Fig. 6(a). The corresponding values of σ r , c are calculated using R drop R h , drop = 110 nm and R prot R h , prot = 11 nm and are displayed in Fig. 6(b).
As can be observed, the protein suspensions require a much higher stress to produce a decrease in viscosity than do the droplet suspensions, as σ c is more than 2 orders of magnitudes higher. However, this difference is largely absent when the reduced critical shear stress is used, indicating that the main difference between both systems is the size of the particles and that there are no differences in interparticle interactions at high concentrations, notably no further extensive aggregation of sodium caseinate.
Shear thinning is thus another aspect of the rheology of sodium caseinate that shows an apparent colloidal behavior rather than polymeric behavior. This result reinforces the relevance of the soft colloidal framework as an approach for studying the viscosity of sodium caseinate and sodium caseinate-stabilized droplets.
After having studied separately the components of protein-stabilized emulsions, the next logical step is to investigate mixtures of both with well-characterized compositions by combining purified droplets and protein suspensions. In addition, the soft colloidal framework developed above provides a basis for the development of a predictive approach to the viscosity of mixtures of proteins and droplets, as formed upon emulsification of oil in a sodium caseinate suspension. These topics are the subject of the current section.
These mixtures are composed of water and of two types of colloidal particles (droplets and protein aggregates); hence, they are conveniently represented as a ternary mixture, as displayed in Fig. 7. This representation is limited by the high volume fractions reached by proteins in suspension; hence, some data points lie outside of the diagram. The two-dimensional space of composition for the mixtures can be described by the total effective volume fraction ϕ eff , tot = ϕ eff , prot + ϕ eff , drop and the ratio of their different components χ prot
χ prot = ϕ eff , prot ϕ eff , prot + ϕ eff , drop . (8)
χ prot describes the relative fraction of protein in the emulsion compared to the droplets: χ prot = 1 for samples containing only proteins, χ prot = 0 for samples containing only protein-stabilized droplets, and χ prot = 0.5 for mixtures containing an equal volume fraction of proteins and protein-stabilized droplets.
The viscosity of the mixtures containing both proteins and protein-stabilized droplets was measured as for the pure suspensions. The values can be compared with the pure suspensions using the total volume fraction for the mixtures ϕ eff , tot and are displayed in Fig. 8. The mixtures all display viscosities between those of the pure droplets and of the pure proteins at a given volume fraction, their exact value depending on their compositional index χ prot.
Notably, no phase separation is observed in the emulsion samples on the timescale of the experiments. Despite sodium caseinate-stabilized emulsions being notoriously prone to depletion induced-flocculation [49–5149. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.493950. Dickinson, E., “Flocculation of protein-stabilized oil-in-water emulsions,” Colloids Surf. B Biointerfaces 81, 130–140 (2010). https://doi.org/10.1016/j.colsurfb.2010.06.03351. Dickinson, E., “Caseins in emulsions: Interfacial properties and interactions,” Int. Dairy J. 9, 305–312 (1999). https://doi.org/10.1016/S0958-6946(99)00079-5] caused by the presence of unadsorbed sodium caseinate [44. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h], the emulsions prepared here present a rheological behavior that differs from those of flocculated samples. Indeed, most of the mixtures of this study are Newtonian fluids. In addition, the comparison of the non-Newtonian behavior of the most concentrated samples prepared here, as can be found in the supplementary material (Fig. S4) [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.], with the flow curves of emulsions in which depletion flocculation is occurring in [4949. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.4939] shows clear differences. Concentrated mixture samples display a clear zero-shear viscosity plateau and present shear thinning at shear stresses one order of magnitude higher than equivalent flocculated samples. This is an unusual result as sodium caseinate-stabilized emulsions are notoriously prone to depletion induced-flocculation caused by the presence of unadsorbed sodium caseinate [4,31,49–514. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h31. Bressy, L., P. Hebraud, V. Schmitt, and J. Bibette, “Rheology of emulsions stabilized by solid interfaces,” Langmuir 19, 598–604 (2003). https://doi.org/10.1021/la026446649. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.493950. Dickinson, E., “Flocculation of protein-stabilized oil-in-water emulsions,” Colloids Surf. B Biointerfaces 81, 130–140 (2010). https://doi.org/10.1016/j.colsurfb.2010.06.03351. Dickinson, E., “Caseins in emulsions: Interfacial properties and interactions,” Int. Dairy J. 9, 305–312 (1999). https://doi.org/10.1016/S0958-6946(99)00079-5]. Presumably, this unusual behavior is due to the small size of the droplets, which are only one order of magnitude larger than the naturally occurring caseinate structures.
The weakness of the depletion interaction probably arises from a combination of the small size of the droplets and the reduced calcium concentration in the samples. Indeed, depletion flocculation has been exploited to obtain monodisperse emulsions in a similar system, in which caseinate-stabilized droplets of radius 180 nm were separated from the emulsion by the depletion-induced creaming of larger droplets [3131. Bressy, L., P. Hebraud, V. Schmitt, and J. Bibette, “Rheology of emulsions stabilized by solid interfaces,” Langmuir 19, 598–604 (2003). https://doi.org/10.1021/la0264466], following a depletion crystallization scheme [5252. Bibette, J., “Depletion interactions and fractionated crystallization for polydisperse emulsion purification,” J. Colloid Interface Sci. 147, 474–478 (1991). https://doi.org/10.1016/0021-9797(91)90181-7]. This shows that droplets of radius below 180 nm like those studied here are less prone to depletion-induced-flocculation. In addition, the purification protocol used here to clean the protein may change the concentration of calcium, a common impurity in commercial sodium caseinate that has been shown to play an important role in the stability of caseinate-stabilized emulsions [5353. Dickinson, E., S. J. Radford, and M. Golding, “Stability and rheology of emulsions containing sodium caseinate: Combined effects of ionic calcium and non-ionic surfactant,” Food Hydrocoll. 17, 211–220 (2003). https://doi.org/10.1016/S0268-005x(02)00055-3].
The understanding and models developed for the suspensions of proteins and droplets in Sec. III can thus be used to develop a semiempirical model to describe the viscosity of mixtures.
A. Semiempirical predictive model
Models have been developed previously to predict the viscosity of suspensions of multimodal particles, for example, in [5454. Mendoza, C. I., “A simple semiempirical model for the effective viscosity of multicomponent suspensions,” Rheol. Acta 56, 487–499 (2017). https://doi.org/10.1007/s00397-017-1011-6] or [5555. Mwasame, P. M., N. J. Wagner, and A. N. Beris, “Modeling the effects of polydispersity on the viscosity of noncolloidal hard sphere suspensions,” J. Rheol. 60, 225–240 (2016). https://doi.org/10.1122/1.4938048], the latter was then extended for mixtures of components of different viscosity behaviors in [5656. Mwasame, P. M., N. J. Wagner, and A. N. Beris, “Modeling the viscosity of polydisperse suspensions: Improvements in prediction of limiting behavior,” Phys. Fluids 28, 061701 (2016). https://doi.org/10.1063/1.4953407]. However, these models are mathematically complex and do not describe accurately our experimental results.
Instead, a simple and useful approach is to consider that each component of the mixture is independent from the other, as in the early model for multimodal suspensions described in [3535. Farris, R. J., “Prediction of the viscosity of multimodal suspensions from unimodal viscosity data,” Trans. Soc. Rheol. 12, 281–301 (1968). https://doi.org/10.1122/1.549109]. In this case, the protein suspension acts as a viscous suspending medium for the droplets, whose viscosity behavior was previously characterized and modeled by Eq. (3). Because the viscosity behavior of the protein suspension is also known, it can be combined with the droplet behavior to determine the viscosity of the mixture. This approach is illustrated in Fig. 9.
1. Development of the model
Considering the suspending medium alone first, it is useful to consider the protein content of the aqueous phase residing in the interstices between the droplets, ϕ prot i
ϕ prot i = V prot V prot + V water = ϕ prot ϕ prot + ϕ water = ϕ prot 1 ϕ droplet , (9)
where it is assumed that ϕ prot ϕ eff , prot = k 0 , prot × c prot and ϕ droplet ϕ eff , drop = k 0 , drop × c drop according to Eq. (2), with k 0 , prot and k 0 , drop determined previously using the Batchelor equation fitted to the viscosities of semidilute suspensions of pure proteins and pure droplets.
The study of the pure suspensions of protein-stabilized droplets and proteins makes it possible to model the viscosity behavior of both suspensions
The relative viscosity of a suspension of protein-stabilized droplets η r , drop ( ϕ ) is described by Eq. (3) with the parameter ϕ m = 0.79 ± 2 (Quemada model for hard spheres [3434. Quemada, D., “Rheology of concentrated disperse systems and minimum energy dissipation principle,” Rheol. Acta 16, 82–94 (1977). https://doi.org/10.1007/BF01516932]).
The relative viscosity of a suspension of sodium caseinate η r , prot ( ϕ ) is described by Eq. (5) with the parameters listed in Table I (modified Quemada model) and using ϕ prot i as described above.
These elements are then combined to predict the relative viscosity of the mixture η r , mix p, in the absence of specific interactions between the droplets and the proteins, thus,
η r , mix p ( ϕ eff , prot , ϕ eff , drop ) = η r , prot ϕ prot i × η r , drop ϕ eff , drop . (10)
2. Application of the model
The values of the relative viscosity calculated for each mixture using Eq. (10) are compared to the experimentally measured relative viscosity η r , mix m in Fig. 10. Details of the estimated viscosity of the continuous phase of the mixture can be found in the supplementary material (Fig. S5) [5858. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases.].
Despite the simplicity of this model, it provides a reasonably accurate prediction of the viscosity of protein-stabilized emulsions. This result seems to indicate that there are no specific interactions between the proteins and the droplets, neither at a molecular scale between unadsorbed and adsorbed proteins nor at a larger length scale where depletion interactions could occur. This is likely to be related to the small size of the droplets in this specific system, and increasing the droplet size may result in a decreased accuracy of this simple model.
The small inaccuracies in the predicted viscosities probably lie in the slightly imperfect fit of Eqs. (3) and (5). First, at moderate viscosity ( η r < 10), the slight discrepancy between predicted and measured viscosity of the samples with a high χ prot is probably a reflection of the modest underestimation of the viscosity of protein suspensions for 2 < ϕ eff < 10 by Eq. (5).
At higher concentrations, the effective volume fraction approximation may break down. Indeed, as observed previously for pure suspensions, ϕ eff can reach high values and may not correspond exactly to the volume fraction actually occupied by the particles, especially in the case of ϕ eff , prot. A natural consequence is that the relationship ϕ eff , prot + ϕ eff , drop + ϕ eff , water = 1 may not hold, leading to an overestimation of ϕ prot i when calculated by Eq. (9). It should be noted that the lack of unifying definition of the volume fraction for soft colloids is a particularly relevant challenge when dealing with mixtures. An approach to address this problem could be to take the viscosity behavior of one of the two components as a reference and map the volume fraction of the other component to follow this reference viscosity [5656. Mwasame, P. M., N. J. Wagner, and A. N. Beris, “Modeling the viscosity of polydisperse suspensions: Improvements in prediction of limiting behavior,” Phys. Fluids 28, 061701 (2016). https://doi.org/10.1063/1.4953407], but it would considerably increase the complexity of the model.
Finally, another possible source of discrepancy is the assumption that the proteins in the interstices will reach the same random close-packing fraction as for proteins in bulk ϕ rcp , prot. However, at high droplet volume fraction, there are geometrical arguments to support the hypothesis of a different random close-packing volume fraction due to excluded volume effects. Therefore, this assumption may lead to a decreased accuracy of the model at high concentrations.
To summarize, in this section, we have shown that the preliminary study of the individual components of a mixture allows the subsequent prediction of the viscosity of mixtures of these components with reasonable accuracy, providing that the composition of the mixtures is known.
3. Reversal of the model: Estimation of the composition of emulsions
A common challenge when formulating protein-stabilized emulsions is to estimate the amount of protein adsorbed at the interface as opposed to the protein suspended in the aqueous phase. Here, we suggest that reversing the semiempirical model developed in Secs. IV A 1 and IV A 2 allows estimation of the amount of proteins in suspension after emulsification with a simple viscosity measurement, which can be performed online in advanced industrial processing lines. The calculation process is illustrated in Fig. 11.
To assess the accuracy of the suggested method, a case in point is the emulsion used to prepare the sodium caseinate droplets in this study after microfluidization. It is composed of 20 wt. % oil and 4.0 wt. % sodium caseinate, and its relative viscosity was measured to be η r , mix m = 10.
The first step is to calculate the contribution of the oil droplets to the viscosity of the mixture, in order to isolate the protein contribution. A 20 wt. % content in oil corresponds to ϕ eff , drop = 0.40, so η r , drop = 1 ϕ eff , drop / ϕ m 2 = 4.1.
It is then possible, using Eq. (10), to calculate the viscosity of the continuous phase η r , prot ( ϕ prot i ) = η r , mix m / η r , drop = 2.4, assumed to arise from the presence of unadsorbed proteins. In order to estimate the volume fraction of proteins in the interstices ϕ prot i, the equation below has to be solved
1 + ϕ eff , prot , m ϕ prot i n 1 / n = 1 1 η r , prot . (11)
Finally, numerically solving Eq. (11) with the values for n and ϕ m from Table I gives ϕ prot i = 0.33 ± 006. This result corresponds to a volume fraction of unadsorbed proteins in the overall emulsion ϕ eff , prot = ϕ prot i ( 1 ϕ eff , drop ) = 0.20 ± 004 or expressed as a concentration in the emulsion c = ( 23 ± 5 ) mg ml 1. This has to be compared with the initial concentration of 45 mg ml 1 in proteins before emulsification. Thus, only half of the amount of proteins adsorb at the interface, while the other half is still in suspension. The validity of this result is difficult to assess experimentally because the presence of small oil droplets in the subnatant after ultracentrifugation of the emulsion prevents the use of optical techniques for the determination of the protein concentration, such as absorption or refractometry.
This result can be converted into a surface coverage to be compared with studies on sodium caseinate-stabilized emulsions using micron-sized droplets. It is estimated that 1 l of emulsion contains 20 wt. % of oil, and with a droplet size of R opt , c = 65 nm presents a surface area of oil of 9.6 × 10 3 m 2. Furthermore, the reversal of the semipredictive model yielded that 22 ± 5 g of sodium caseinate is adsorbed at the interfaces. Thus, the surface coverage found here is ( 2.3 ± 05 ) mg m 2. This result is in good correspondence with studies on similar emulsions at larger droplet sizes, in which a surface coverage of around 3.0 mg m 2 was found upon emulsification with a protein excess [4,5,494. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h5. Srinivasan, M., H. Singh, and P. A. Munro, “Adsorption behaviour of sodium and calcium caseinates in oil-in-water emulsions,” Int. Dairy J. 9, 337–341 (1999). https://doi.org/10.1016/S0958-6946(99)00084-949. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.4939]. This plausible surface coverage thus provides a validation for the use of the measurement of the viscosity as a tool to estimate the amount of unadsorbed proteins present in the emulsions studied here.
The semiempirical model for the viscosity of emulsions developed in this study, once calibrated, can thus be used not only as a predictive tool for mixtures of droplets and proteins of known composition but also as a method to estimate the amount of adsorbed proteins without the need for further separation of the components. This extension of the model to industrial applications would however require further validation, as some practical complications may emerge from the time-dependence of these systems over longer times. Indeed, variations in the aggregation state of the protein, Ostwald ripening or depletion-induced-flocculation, may occur. The reliability of the calibration also relies on the repeatability of the emulsification, which may be reduced by the variations of composition of different batches of sodium caseinate.
Previous studies have attempted to compare the rheological properties of sodium caseinate to those of a suspension of hard spheres and found that agreement at high concentrations is poor [14,2114. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-821. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003]. As a result, it was concluded that a colloidal model is inadequate to describe the observed behavior. Here, we argue that this is mainly due to the choice of hard spheres as colloidal reference. We have shown that using the framework developed for soft colloidal particles, such as microgels and block copolymer micelles [66. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007], helps toward a better description of the viscosity behavior of the protein dispersions. Although this approach neglects the additional layer of complexity due to the biological nature of the sodium caseinate, such as inhomogeneous charge distribution and dynamic aggregation [29,3029. Sarangapani, P. S., S. D. Hudson, K. B. Migler, and J. A. Pathak, “The limitations of an exclusively colloidal view of protein solution hydrodynamics and rheology,” Biophys. J. 105, 2418–2426 (2013). https://doi.org/10.1016/j.bpj.2013.10.01230. Sarangapani, P. S., S. D. Hudson, R. L. Jones, J. F. Douglas, and J. A. Pathak, “Critical examination of the colloidal particle model of globular proteins,” Biophys. J. 108, 724–737 (2015). https://doi.org/10.1016/j.bpj.2014.11.3483], it gives a satisfactory model that can be used to build a better description of protein-stabilized emulsions. Interestingly, the soft colloidal approach can also be successfully applied to the rheology of noncolloidal food particles, such as fruit purees [5757. Leverrier, C., G. Almeida, P. Menut, and G. Cuvelier, “Design of model apple cells suspensions: Rheological properties and impact of the continuous phase,” Food Biophys. 12, 383–396 (2017). https://doi.org/10.1007/s11483-017-9494-3].
In addition, a protocol was developed for preparing pure suspensions of protein-stabilized droplets rather than emulsions containing unadsorbed proteins. The viscosity behavior of the nanosized droplets appeared to be very similar to the hard-sphere model. The main discrepancy is the high effective volume fraction at which the viscosity diverges, which may be due to the size distribution of droplets or arise from the softness of the layer of adsorbed proteins.
Finally, examining protein-stabilized emulsions as ternary mixtures of water, unadsorbed proteins and droplets has allowed us to develop a semiempirical model for their viscosity. The contributions of each component to the overall viscosity of the emulsions being quantified by the analysis of the properties of the pure suspensions of droplets or proteins. The model can also be reversed to estimate the composition, after emulsification, of a protein-stabilized emulsion given its viscosity. It should be noted, however, that the droplet size is likely to be critical to the success of the model, as it is expected that flocculation of droplets will occur for larger droplets [31,4,49–5131. Bressy, L., P. Hebraud, V. Schmitt, and J. Bibette, “Rheology of emulsions stabilized by solid interfaces,” Langmuir 19, 598–604 (2003). https://doi.org/10.1021/la02644664. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h49. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.493950. Dickinson, E., “Flocculation of protein-stabilized oil-in-water emulsions,” Colloids Surf. B Biointerfaces 81, 130–140 (2010). https://doi.org/10.1016/j.colsurfb.2010.06.03351. Dickinson, E., “Caseins in emulsions: Interfacial properties and interactions,” Int. Dairy J. 9, 305–312 (1999). https://doi.org/10.1016/S0958-6946(99)00079-5]. This is due to the depletion interaction generated by the proteins in the mixture, which is not taken into account in the present model. For this reason, it would be interesting to explore further the influence of the droplet size on the viscosity behavior of emulsions. In addition, increasing the droplet size would change the hardness of the droplets by decreasing the internal pressure as well as the influence of the soft layer of proteins, adding further complexity to the system.
This project forms part of the Marie Curie European Training Network COLLDENSE that has received funding from the European Unions Horizon 2020 research and innovation program Marie Skłodowska-Curie Actions (Grant Agreement No. 642774). The authors wish to acknowledge DMV for graciously providing the sodium caseinate sample used in this study and PostNova Analytics Ltd. for graciously performing the field flow fractionation measurement of sodium caseinate.
  1. 1. Mezzenga, R., P. Schurtenberger, A. Burbidge, and M. Michel, “Understanding foods as soft materials,” Nat. Mater. 4, 729–740 (2005). https://doi.org/10.1038/nmat1496, Google ScholarCrossref
  2. 2. Dickinson, E., “Food colloids research: Historical perspective and outlook,” Adv. Colloid Interface Sci. 165, 7–13 (2011). https://doi.org/10.1016/j.cis.2010.05.007, Google ScholarCrossref
  3. 3. Vilgis, T. A., “Soft matter food physics—The physics of food and cooking,” Rep. Prog. Phys. 78, 124602 (2015). https://doi.org/10.1088/0034-4885/78/12/124602, Google ScholarCrossref
  4. 4. Srinivasan, M., H. Singh, and P. A. Munro, “Sodium caseinate-stabilized emulsions: Factors affecting coverage and composition of surface proteins,” J. Agric. Food Chem. 44, 3807–3811 (1996). https://doi.org/10.1021/jf960135h, Google ScholarCrossref
  5. 5. Srinivasan, M., H. Singh, and P. A. Munro, “Adsorption behaviour of sodium and calcium caseinates in oil-in-water emulsions,” Int. Dairy J. 9, 337–341 (1999). https://doi.org/10.1016/S0958-6946(99)00084-9, Google ScholarCrossref
  6. 6. Vlassopoulos, D., and M. Cloitre, “Tunable rheology of dense soft deformable colloids,” Curr. Opin. Colloid Interface Sci. 19, 561–574 (2014). https://doi.org/10.1016/j.cocis.2014.09.007, Google ScholarCrossref
  7. 7. Adams, S., W. J. Frith, and J. R. Stokes, “Influence of particle modulus on the rheological properties of agar microgel suspensions,” J. Rheol. 48, 1195–1213 (2004). https://doi.org/10.1122/1.1773782, Google ScholarScitation, ISI
  8. 8. Shewan, H. M., and J. R. Stokes, “Viscosity of soft spherical micro-hydrogel suspensions,” J. Colloid Interface Sci. 442, 75–81 (2015). https://doi.org/10.1016/j.jcis.2014.11.064, Google ScholarCrossref
  9. 9. Cloitre, M., R. Borrega, F. Monti, and L. Leibler, “Structure and flow of polyelectrolyte microgels: From suspensions to glasses,” C. R. Phys. 4, 221–230 (2003). https://doi.org/10.1016/S1631-0705(03)00046-X, Google ScholarCrossref
  10. 10. Tan, B. H., K. C. Tam, Y. C. Lam, and C. B. Tan, “Microstructure and rheological properties of ph-responsive coreshell particles,” Polymer 46, 10066–10076 (2005). https://doi.org/10.1016/j.polymer.2005.08.007, Google ScholarCrossref
  11. 11. Roovers, J., “Concentration-dependence of the relative viscosity of star polymers,” Macromolecules 27, 5359–5364 (1994). https://doi.org/10.1021/ma00097a015, Google ScholarCrossref
  12. 12. Winkler, R. G., D. A. Fedosov, and G. Gompper, “Dynamical and rheological properties of soft colloid suspensions,” Curr. Opin. Colloid Interface Sci. 19, 594–610 (2014). https://doi.org/10.1016/j.cocis.2014.09.005, Google ScholarCrossref
  13. 13. Leermakers, F. A. M., J. C. Eriksson, and H. Lyklema, Chapter 4—Association colloids and their equilibrium modelling, in Fundamentals of Interface and Colloid Science, edited by J. Lyklema (Elsevier, Amsterdam, 2005), Vol. 5, pp. 4.1–4.123. Google Scholar
  14. 14. Farrer, D., and A. Lips, “On the self-assembly of sodium caseinate,” Int. Dairy J. 9, 281–286 (1999). https://doi.org/10.1016/S0958-6946(99)00075-8, Google ScholarCrossref
  15. 15. Vlassopoulos, D., G. Fytas, S. Pispas, and N. Hadjichristidis, “Spherical polymeric brushes viewed as soft colloidal particles: Zero-shear viscosity,” Phys. B Condens. Matter 296, 184–189 (2001). https://doi.org/10.1016/S0921-4526(00)00798-5, Google ScholarCrossref
  16. 16. Boulet, M., M. Britten, and F. Lamarche, “Voluminosity of some food proteins in aqueous diepersions at various ph and ionic strengths,” Food Hydrocoll. 12, 433–441 (1998). https://doi.org/10.1016/S0268-005X(98)00009-5, Google ScholarCrossref
  17. 17. Batchelor, G. K., “The effect of Brownian motion on the bulk stress in a suspension of spherical particles,” J. Fluid Mech. 83, 97–117 (1977). https://doi.org/10.1017/s0022112077001062, Google ScholarCrossref
  18. 18. Dalgleish, D. G., and A. J. R. Law, “Sodium caseinates—Composition and properties of different preparations,” J. Soc. Dairy Technol. 41, 1–4 (1988). https://doi.org/10.1111/j.1471-0307.1988.tb00571.x, Google ScholarCrossref
  19. 19. Lucey, J. A., M. Srinivasan, H. Singh, and P. A. Munro, “Characterization of commercial and experimental sodium caseinates by multiangle laser light scattering and size-exclusion chromatography,” J. Agric. Food Chem. 48, 1610–1616 (2000). https://doi.org/10.1021/jf990769z, Google ScholarCrossref
  20. 20. Huppertz, T., I. Gazi, H. Luyten, H. Nieuwenhuijse, A. Alting, and E. Schokker, “Hydration of casein micelles and caseinates: Implications for casein micelle structure,” Int. Dairy J. 74, 1–11 (2017). https://doi.org/10.1016/j.idairyj.2017.03.006, Google ScholarCrossref
  21. 21. Pitkowski, A., D. Durand, and T. Nicolai, “Structure and dynamical mechanical properties of suspensions of sodium caseinate,” J. Colloid Interface Sci. 326, 96–102 (2008). https://doi.org/10.1016/j.jcis.2008.07.003, Google ScholarCrossref
  22. 22. Faroughi, S. A., and C. Huber, “A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number,” Rheol. Acta 54, 85–108 (2014). https://doi.org/10.1007/s00397-014-0825-8, Google ScholarCrossref
  23. 23. Norde, W., J. Buijs, and H. Lyklema, Chapter 3—Adsorption of globular proteins, in Fundamentals of Interface and Colloid Science, edited by J. Lyklema (Elsevier, Amsterdam, 2005), Vol. 5, pp. 3.1–3.59. Google Scholar
  24. 24. Spyropoulos, F., E. A. K. Heuer, T. B. Mills, and S. Bakalis, Protein-stabilised emulsions and rheological aspects of structure and mouthfeel, in Practical Food Rheology an Interpretive Approach, edited by I. T. Norton, F. Spyropoulos, and P. Cox (Wiley-Blackwell, Oxford, 2010), Serial 9, pp. 193–2018. Google Scholar
  25. 25. Graham, D. E., and M. C. Phillips, “Proteins at liquid interfaces: II. Adsorption isotherms,” J. Colloid Interface Sci. 70, 415–426 (1979). https://doi.org/10.1016/0021-9797(79)90049-3, Google ScholarCrossref
  26. 26. Dickinson, E., “Structure formation in casein-based gels, foams, and emulsions,” Colloids Surf. A Physicochem. Eng. Aspects 288, 3–11 (2006). https://doi.org/10.1016/j.colsurfa.2006.01.012, Google ScholarCrossref
  27. 27. Dickinson, E., “Emulsion gels: The structuring of soft solids with protein-stabilized oil droplets,” Food Hydrocoll. 28, 224–241 (2012). https://doi.org/10.1016/j.foodhyd.2011.12.017, Google ScholarCrossref
  28. 28. Chen, J. S., E. Dickinson, and M. Edwards, “Rheology of acid-induced sodium caseinate stabilized emulsion gels,” J. Texture Stud. 30, 377–396 (1999). https://doi.org/10.1111/j.1745-4603.1999.tb00226.x, Google ScholarCrossref
  29. 29. Sarangapani, P. S., S. D. Hudson, K. B. Migler, and J. A. Pathak, “The limitations of an exclusively colloidal view of protein solution hydrodynamics and rheology,” Biophys. J. 105, 2418–2426 (2013). https://doi.org/10.1016/j.bpj.2013.10.012, Google ScholarCrossref
  30. 30. Sarangapani, P. S., S. D. Hudson, R. L. Jones, J. F. Douglas, and J. A. Pathak, “Critical examination of the colloidal particle model of globular proteins,” Biophys. J. 108, 724–737 (2015). https://doi.org/10.1016/j.bpj.2014.11.3483, Google ScholarCrossref
  31. 31. Bressy, L., P. Hebraud, V. Schmitt, and J. Bibette, “Rheology of emulsions stabilized by solid interfaces,” Langmuir 19, 598–604 (2003). https://doi.org/10.1021/la0264466, Google ScholarCrossref
  32. 32. Zhao, H., P. H. Brown, and P. Schuck, “On the distribution of protein refractive index increments,” Biophys. J. 100, 2309–2317 (2011). https://doi.org/10.1016/j.bpj.2011.03.004, Google ScholarCrossref
  33. 33. de Kruif, C. G., E. M. F. van Iersel, A. Vrij, and W. B. Russel, “Hard sphere colloidal dispersions: Viscosity as a function of shear rate and volume fraction,” J. Chem. Phys. 83, 4717–4725 (1985). https://doi.org/10.1063/1.448997, Google ScholarCrossref, ISI
  34. 34. Quemada, D., “Rheology of concentrated disperse systems and minimum energy dissipation principle,” Rheol. Acta 16, 82–94 (1977). https://doi.org/10.1007/BF01516932, Google ScholarCrossref
  35. 35. Farris, R. J., “Prediction of the viscosity of multimodal suspensions from unimodal viscosity data,” Trans. Soc. Rheol. 12, 281–301 (1968). https://doi.org/10.1122/1.549109, Google ScholarScitation
  36. 36. Shewan, H. M., and J. R. Stokes, “Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution,” J. Nonnewton. Fluid Mech. 222, 72–81 (2015). https://doi.org/10.1016/j.jnnfm.2014.09.002, Google ScholarCrossref
  37. 37. Farr, R. S., and R. D. Groot, “Close packing density of polydisperse hard spheres,” J. Chem. Phys. 131, 244104 (2009). https://doi.org/10.1063/1.3276799, Google ScholarCrossref, ISI
  38. 38. Dalgleish, D. G., S. J. West, and F. R. Hallett, “The characterization of small emulsion droplets made from milk proteins and triglyceride oil,” Colloids Surf. A Physicochem. Eng. Aspects 123, 145–153 (1997). https://doi.org/10.1016/S0927-7757(97)03783-7, Google ScholarCrossref
  39. 39. Loveday, S. M., M. A. Rao, L. K. Creamer, and H. Singh, “Rheological behavior of high-concentration sodium caseinate dispersions,” J. Food Sci. 75, N30–N35 (2010). https://doi.org/10.1111/j.1750-3841.2009.01493.x, Google ScholarCrossref
  40. 40. Borrega, R., M. Cloitre, I. Betremieux, B. Ernst, and L. Leibler, “Concentration dependence of the low-shear viscosity of polyelectrolyte micro-networks: From hard spheres to soft microgels,” Europhys. Lett. 47, 729–735 (1999). https://doi.org/10.1209/epl/i1999-00451-1, Google ScholarCrossref
  41. 41. Pellet, C., and M. Cloitre, “The glass and jamming transitions of soft polyelectrolyte microgel suspensions,” Soft Matter 12, 3710–3720 (2016). https://doi.org/10.1039/c5sm03001c, Google ScholarCrossref
  42. 42. Erwin, B. M., M. Cloitre, M. Gauthier, and D. Vlassopoulos, “Dynamics and rheology of colloidal star polymers,” Soft Matter 6, 2825–2833 (2010). https://doi.org/10.1039/b926526k, Google ScholarCrossref
  43. 43. Helgeson, M. E., N. J. Wagner, and D. Vlassopoulos, “Viscoelasticity and shear melting of colloidal star polymer glasses,” J. Rheol. 51, 297–316 (2007). https://doi.org/10.1122/1.2433935, Google ScholarScitation, ISI
  44. 44. Cross, M. M., “Rheology of non-Newtonian fluids—A new flow equation for pseudoplastic systems,” J. Colloid Sci. 20, 417–437 (1965). https://doi.org/10.1016/0095-8522(65)90022-X, Google ScholarCrossref
  45. 45. Cross, M. M., “Kinetic interpretation of non-Newtonian flow,” J. Colloid Interface Sci. 33, 30–35 (1970). https://doi.org/10.1016/0021-9797(70)90068-8, Google ScholarCrossref
  46. 46. Woods, M. E., and I. M. Krieger, “Rheological studies on dispersions of uniform colloidal spheres: I. Aqueous dispersions in steady shear flow,” J. Colloid Interface Sci. 34, 91–99 (1970). https://doi.org/10.1016/0021-9797(70)90262-6, Google ScholarCrossref
  47. 47. Krieger, I. M., Rheology of monodisperse latices,” Adv. Colloid Interface Sci. 3, 111–136 (1972). https://doi.org/10.1016/0001-8686(72)80001-0, Google ScholarCrossref
  48. 48. Frith, W. J., J. Mewis, and T. A. Strivens, “Rheology of concentrated suspensions—Experimental investigations,” Powder Technol. 51, 27–34 (1987). https://doi.org/10.1016/0032-5910(87)80037-2, Google ScholarCrossref
  49. 49. Dickinson, E., and M. Golding, “Rheology of sodium caseinate stabilized oil-in-water emulsions,” J. Colloid Interface Sci. 191, 166–176 (1997). https://doi.org/10.1006/jcis.1997.4939, Google ScholarCrossref
  50. 50. Dickinson, E., “Flocculation of protein-stabilized oil-in-water emulsions,” Colloids Surf. B Biointerfaces 81, 130–140 (2010). https://doi.org/10.1016/j.colsurfb.2010.06.033, Google ScholarCrossref
  51. 51. Dickinson, E., “Caseins in emulsions: Interfacial properties and interactions,” Int. Dairy J. 9, 305–312 (1999). https://doi.org/10.1016/S0958-6946(99)00079-5, Google ScholarCrossref
  52. 52. Bibette, J., “Depletion interactions and fractionated crystallization for polydisperse emulsion purification,” J. Colloid Interface Sci. 147, 474–478 (1991). https://doi.org/10.1016/0021-9797(91)90181-7, Google ScholarCrossref
  53. 53. Dickinson, E., S. J. Radford, and M. Golding, “Stability and rheology of emulsions containing sodium caseinate: Combined effects of ionic calcium and non-ionic surfactant,” Food Hydrocoll. 17, 211–220 (2003). https://doi.org/10.1016/S0268-005x(02)00055-3, Google ScholarCrossref
  54. 54. Mendoza, C. I., “A simple semiempirical model for the effective viscosity of multicomponent suspensions,” Rheol. Acta 56, 487–499 (2017). https://doi.org/10.1007/s00397-017-1011-6, Google ScholarCrossref
  55. 55. Mwasame, P. M., N. J. Wagner, and A. N. Beris, “Modeling the effects of polydispersity on the viscosity of noncolloidal hard sphere suspensions,” J. Rheol. 60, 225–240 (2016). https://doi.org/10.1122/1.4938048, Google ScholarScitation, ISI
  56. 56. Mwasame, P. M., N. J. Wagner, and A. N. Beris, “Modeling the viscosity of polydisperse suspensions: Improvements in prediction of limiting behavior,” Phys. Fluids 28, 061701 (2016). https://doi.org/10.1063/1.4953407, Google ScholarCrossref, ISI
  57. 57. Leverrier, C., G. Almeida, P. Menut, and G. Cuvelier, “Design of model apple cells suspensions: Rheological properties and impact of the continuous phase,” Food Biophys. 12, 383–396 (2017). https://doi.org/10.1007/s11483-017-9494-3, Google ScholarCrossref
  58. 58. See supplementary material at https://doi.org/10.1122/1.5062837 for information on the size distribution of the droplets, the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behavior of Eq. (5), flow curves of the shear thinning samples, and the contributions to the viscosity of mixtures by the dispersed and continuous phases. Google Scholar
  59. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).