Journal of Colloid and Interface Science 287 (2005) 468–475 www.elsevier.com/locate/jcis
Dynamic mechanical properties of suspensions of micellar casein particles Maud Panouillé, Lazhar Benyahia, Dominique Durand, Taco Nicolai ∗ Polymères, Colloïdes, Interfaces, UMR CNRS, Université du Maine, 72085 Le Mans cedex 9, France Received 8 November 2004; accepted 2 February 2005 Available online 25 March 2005
Abstract Small micellar casein particles, so-called submicelles, were obtained by removing colloidal calcium phosphate from native casein by adding sodium polyphosphate. Aqueous submicelle suspensions were characterized using light scattering and rheology as a function of concentration and temperature. The casein submicelles behave like soft spheres that jam at a critical concentration (Cc ) of about 100 g L−1 . The viscosity does not diverge at Cc , but increases sharply, similarly to that of multiarm star polymers. Cc increases weakly with increasing temperature, which leads to a strong decrease of the viscosity close to and above Cc . Concentrated submicelle suspensions show strong shear-thinning above a critical shear rate and the shear stress becomes independent of the shear rate. The critical shear rates at different temperatures and concentrations are inversely proportional to the zero-shear viscosity. At much higher shear rates, the shear stress fluctuates strongly in time indicating inhomogeneous flow. The frequency dependence of casein submicelle suspensions is characterized by elastic behavior at high frequencies (concentrations) and viscous behavior at low frequencies (concentrations). 2005 Elsevier Inc. All rights reserved. Keywords: Casein; Submicelles; Rheology; Soft spheres; Jamming
1. Introduction Casein is the main protein component of milk. It exists in biological conditions as a supramolecular assembly, called “casein micelle” for historical reasons, composed of four types of individual casein molecules linked together by colloidal calcium phosphate (CCP). It is a roughly spherical particle with mean diameter 160 nm and is very stable [1]. Its stability is provided by a steric stabilizing layer of hydrophilic hairs on the surface of the micelle [2]. The casein micelles dissociate after removal of CCP and form smaller subunits, so-called submicelles. Acidification [3], chelating agents such as EDTA, phosphate, or citrate [4–6], or dialysis against Ca2+ -free buffers may be used to dissociate native casein into submicelles [7,8]. The characteristics of the submicelles depend on their environment [9–11]. * Corresponding author. Fax: +33 2 43 83 35 58.
E-mail addresses:
[email protected] (D. Durand),
[email protected] (T. Nicolai). 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2005.02.007
Radii between 5 and 10 nm and molar masses between 100 and 500 kg mol−1 have been reported in the literature [6,12–18]. Results of neutron scattering studies give a specific volume of the submicelles, v, between 4 × 10−3 and 5.5 × 10−3 L g−1 [13]. A casein submicelle is in fact a small micelle composed of a mixture of individual casein molecules called caseinate and is most likely not present as such in native casein. There are four main types of casein with molar masses between 20 and 25 kg mol−1 . These polypeptides are charged and contain hydrophilic and hydrophobic blocks [19]. Two of these caseins, β-casein and κ-casein, have a hydrophilic and a hydrophobic extremity and form by themselves spherical micelles in aqueous solution with a radius of about 10 nm [16,20]. The aim of this work is to study the rheological behavior of casein submicelle suspensions formed after removal of CCP over a wide range of concentrations and to investigate the influence of ageing and temperature.
M. Panouillé et al. / Journal of Colloid and Interface Science 287 (2005) 468–475
2. Experimental 2.1. Materials The casein used for this study is native phosphocaseinate (NPCP), prepared according to the method described by Pierre et al. and Schuck et al. [21,22] and supplied by INRA– LRTL, Rennes. The chemical composition of the powder is reported in Table 1. A mixture of sodium polyphosphates was used that contained approximately equal quantities of ortho-, pyro-, tri-, and tetraphosphate (between 10 and 15% each), while the remaining phosphate consisted of larger oligomers.
bination with a Malvern goniometer and a Spectra Physics laser operating with vertically polarized light with wavelength λ = 532 nm. Data were collected at scattering angles θ between 10◦ and 140◦ , which correspond to scattering wave vectors 2.7 × 10−3 < q < 3.0 × 10−2 nm−1 with q = (4πn/λ) sin(θ/2), n being the refractive index. The temperature was controlled by a thermostat bath and was set at 20 ◦ C. The relative scattering intensity Ir was determined by subtracting the solvent scattering from the total scattering intensity and dividing by the scattering intensity of toluene. In general Ir can be written as [26,27] Ir = KCMw S(q),
2.2. Preparation of casein submicelle suspensions Casein powder was dissolved in aqueous solutions prepared with Millipore water (18 M). For the lower concentrations (up to 80 g L−1 casein) the powder was directly dissolved in a 20 g L−1 (w/w) sodium polyphosphate solution (Joha, BK Giulini Chemie) by stirring for several hours and the pH was adjusted to 6. For higher concentrations, the viscosity of casein suspensions renders this method impossible. Therefore, the powder was dissolved in Millipore water at 170 g L−1 by stirring for several days and the pH was adjusted to 6. The casein suspensions were mixed at 40 ◦ C with a concentrated polyphosphate solution at pH 6 to obtain the desired casein concentration and a polyphosphate concentration of 20 g L−1 . 20 g L−1 polyphosphate is sufficient for complete dissociation of the casein micelles even at the highest casein concentration [23]. In this way homogeneous casein submicelle suspensions could be prepared up to about 160 g L−1 . At higher concentrations the suspensions become extremely viscous at 40 ◦ C and ageing effects interfere if the suspensions are prepared at higher temperatures. The casein concentration was determined by measuring the absorption at 280 nm using an extinction coefficient of 0.81, which is in accordance with the data found in the literature [24,25].
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(1)
where Mw is the weight average molar mass, K is an optical constant that depends on the refractive index increment and the experimental setup, and S(q) is the q-dependent static structure factor. We have used 0.189 for the refractive index increment of casein and we have used a toluene standard with Rayleigh ratio 2.79×10−5 cm−1 . In highly diluted solutions when interactions between the particles can be neglected, S(q) = 1 for q → 0, and extrapolation of Ir /KC to q = 0 yields Mw . At finite concentrations this procedure gives an apparent molar mass, Ma , that is inversely proportional to the osmotic compressibility. 2.4. Rheology The viscosity of suspensions with η < 10 Pa s was measured with a Contraves Low-Shear 40, using a Couette geometry with an inner diameter of 12 mm and an outer diameter of 12.5 mm. Steady shear rate and oscillation measurements were made with a TA Instruments Rheolyst AR1000 stress-imposed rheometer with a cone (40 mm, 0.58◦ )–plate geometry. The temperature was controlled by a Peltier system and paraffin oil was added to prevent water evaporation.
2.3. Light scattering measurements Static and dynamic light scattering measurements were made using an ALV-5000 multibit multi-τ correlator in comTable 1 Chemical composition of the native phosphocaseinate powder Total solids (TS) Total nitrogen matter (TNM) Noncasein nitrogen (NCN) Nonprotein nitrogen (NPN) Ash
g kg−1 920 831 43 3 80
Ca P Na Cl
%TS 3.2 1.7 0.0 0.2
3. Results 3.1. Light scattering Static and dynamic light scattering experiments were done on casein submicelle suspensions over a range of concentrations at a temperature of 20 ◦ C. Even after centrifugation and filtration, the solutions contained a very small weight fraction of large particles with Rhz ≈ 100 nm, which we assume to be residual fat globules [28]. The scattering intensity of the submicelles was obtained by correcting the total intensity for the contribution of these large particles. The q-dependence of Ir for the submicelles was found to be negligible for all concentrations and the apparent molar mass of the submicelles was calculated as Ma = Ir /KC, where we have assumed that all the casein forms submicelles.
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Fig. 1. Concentration dependence of the apparent molar mass of casein submicelles. The solid line represents Eq. (2) with Mw = 4.5 × 105 g mol−1 and φ = 5.2 × 10−3 C.
Fig. 1 shows the concentration dependence of Ma up to C = 60 g L−1 . At higher concentrations the contribution of the fat globules to the scattering dominates, so that Ma can no longer be determined accurately. For hard spheres the concentration dependence of Ma can be described by the socalled Carnahan–Starling equation [29], (1 − φ)4 , Ma = Mw (2) 1 + 4φ + 4φ 2 − 4φ 3 + φ 4 where φ is the volume fraction of the spheres, which is proportional to the concentration: φ = vC, with v the specific volume of the spheres. The concentration dependence of Ma for the casein submicelles can be well described by Eq. (2), but we need to replace φ by an effective thermodynamic volume fraction, φt , because the submicelles are soft and charged. The solid line in Fig. 1 represents Eq. (2) with φ = φt = 5.2 × 10−3 C. The effective thermodynamic specific volume of the submicelles at 20 ◦ C is thus vt = 5.2 × 10−3 L g−1 . The weight average molar mass of the submicelles obtained from extrapolation to infinite dilution is 4.5 × 105 g mol−1 . The z-average hydrodynamic radius (Rhz ) was obtained from dynamic light scattering as described elsewhere [28], and was found to be 11 nm and only weakly dependent on C up to 60 g L−1 . These results are consistent with those found in the literature about submicelles [6,12–18]. An average specific volume of the micelles 3 N /3M , where N is Avocan be calculated as vh = 4πRhz a w a gadro’s number. Using Rhz and Mw deduced from light scattering we find vh = 6 × 10−3 L g−1 . This value is consistent with vt , but is not very precise, because a small variation of Rh results in a large variation of vh . In addition, submicelles are polydisperse, and light scattering measurements give strong weight to larger particles.
Fig. 2. Concentration dependence of the zero-shear viscosity at 20 ◦ C of suspensions of casein submicelles (open circles) in 20 g L−1 polyphosphate or sodium caseinate in pure water (filled squares), in 20 g L−1 polyphosphate (filled circles), or in a phosphate—NaCl buffer (from Farrer and Lips [42]) (filled triangles). The solid line represents Eq. (3) with Cc = 103 g L−1 . The insert shows the initial concentration dependence of the reduced viscosity of submicelle suspensions. The solid line represents a linear least-squares fit.
3.2. Viscosity We determined the viscosity, η, as a function of the shear rate over a wide range of submicelle concentrations. We will first discuss the viscosity obtained in the linear regime at low shear rates, η0 , and discuss the shear rate dependence in the following section. The zero-shear viscosity of casein submicelle solutions is plotted in Fig. 2 as a function of the casein concentration at 20 ◦ C. Fig. 2 shows that the zero-shear viscosity increases dramatically when C approaches 100 g L−1 . The intrinsic viscosity, [η], and the so-called Huggins coefficient, KH , are calculated using the initial concentration dependence of the reduced viscosity (ηred = (η0 − ηs )/(ηs C)); ηred = [η](1 + KH [η]C): where ηs is the viscosity of the solvent. We find that at 20 ◦ C [η] = 9.5 × 10−3 L g−1 and KH = 6.6; see insert of Fig. 2. For hard spheres [η] = 2.5v [30] and if we apply this relation to the submicelles we find an effective viscoelastic specific volume vη = 4 × 10−3 L g−1 . However, the Huggins coefficient is much larger than that expected for hard spheres with this specific volume (KH = 1), which shows that the interactions between the charged submicelles are stronger than those between equivalent hard spheres. The viscosity of noninteracting hard spheres increases with increasing concentration and diverges at a critical volume fraction close to that of random close packing: φc ≈ 0.6 [31]. It has been suggested that the strong rise of the viscosity can be described by the following equation [32]: η0 = ηs (1 − C/Cc )−2 .
(3)
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(a)
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(b)
Fig. 3. (a) Shear rate dependence of the viscosity of casein suspensions at different concentrations indicated in the figure at 20 ◦ C. The solid lines represent fits to Eq. (4). (b) Master curve of the data obtained by reducing the viscosity by the zero-shear viscosity and the shear rate by its critical value. The solid line represents Eq. (4).
This equation was found to give a good description of the viscosity also for soft spheres, but in that case one has to consider an effective volume fraction [33]. For casein submicelles it works quite well up to C = 100 g L−1 using Cc = 103 g L−1 . Using vη we find that the critical effective volume fraction is φc = 0.41, which is significantly smaller than that for non-interacting spheres with the same intrinsic viscosity. We believe that this reflects the repulsive electrostatic interactions, which also lead to a larger Huggins coefficient. If we use vh or vt to calculate Cc we find values of φc closer to 0.6. The results discussed here were obtained in 20 g L−1 polyphosphate. Using instead 10 g L−1 polyphosphate yields similar results except that the viscosity increases at a slightly lower concentration (Cc = 98 g L−1 ). The viscosity does not diverge at Cc , as expected for hard spheres, but continues to increase very strongly, as was found earlier for soft spheres [33], and multiarm star polymers [34,35]. The reason is that even if the micelles are close-packed the system can relax stress through hopping or restructuring of the micelles, albeit very slowly. 3.3. Flow measurements The shear viscosity is plotted in Fig. 3a as a function of shear rate for a number of submicelle concentrations. At low casein concentrations the steady shear viscosity does not depend on the shear rate over the range investigated, i.e., at least up to γ˙ = 103 s−1 . For C > 70 g L−1 we observe an increasing effect of shear thinning, which is generally observed for dense particle suspensions. The results at different concentrations can be superimposed by plotting η0 /η as a function of γ˙ /γ˙c ; see Fig. 3b. The shear rate dependence of
Fig. 4. Relation between the zero-shear viscosity obtained from flow (circles) and oscillatory (squares) measurements and the critical shear rate and frequency, respectively, for casein submicelle suspensions. The solid line has slope −1.
η can be described by the following empirical equation: η0 . η= (4) 1 + (γ˙ /γ˙c ) Fig. 4 shows that, within the experimental error, γ˙c is inversely proportional to the zero shear rate viscosity: γ˙c−1 = 2.5 × 10−3 η0 . γ˙c−1 is comparable to the time it takes for the submicelles to diffuse over a distance equal to their diameter, tD = (4πRh3 η0 )/(kT ), which gives about 4 × 10−3 η0 , close to experimental results. Clearly, we cannot reach the
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Fig. 5. Example of the time dependence of the shear stress during flow at different shear rates for a casein suspension at C = 141 g L−1 and 10 ◦ C.
high shear rate limit determined by tD in the neat solvent (10−6 s), for these small particles. The observation that η ∝ γ˙ −1 for γ˙ γ˙c implies that the shear stress σ = η · γ˙ becomes independent of the shear rate at high shear rates. However, at shear rates much higher than γ˙c the flow becomes irregular and the shear stress fluctuates in time, especially for highly viscous samples. An example of this behavior is shown in Fig. 5, where we have plotted the shear stress as a function of time at different shear rates. Over a range of shear rates below and close to γ˙c the shear stress is constant in time and almost independent of the shear rate. However, at much higher shear rates the shear stress is lower and evolves irregularly with time. In addition, we observed an important normal stress with erratic time dependence.
Fig. 6. Master curve of the complex shear viscosity reduced by the zeroshear viscosity as a function of the frequency reduced by its critical value for casein submicelle suspensions at different concentrations indicated in the figure at 20 ◦ C. The solid line represents |η∗ | = η0 /(1 + ω/ωc ).
3.4. Oscillation measurements Oscillatory shear measurements at radial frequencies (ω) between 5 and 0.05 rad s−1 were made at 20 ◦ C in the linear regime. We have calculated the complex viscosity, |η∗ |, from the shear modulus for casein concentrations ranging from 78 to 157 g L−1 and obtained a master curve by plotting |η∗ |/η0 as a function of ω/ωc ; see Fig. 6. We have chosen ωc = γ˙c for the reference concentration. The frequency dependence of |η∗ | is different from the shear rate dependence of η and it cannot be described by Eq. (4). It follows that the so-called Cox–Merz rule is not observed for this system. However, the dependence of ωc on the zero-shear viscosity is the same as obtained in the flow experiments, see Fig. 4. The frequency-dependent storage (G ) and loss (G ) moduli obtained at different casein concentrations superimpose if plotted as a function of ω/ωc , as shown in Fig. 7. The master curve shows a crossover from elastic behavior for ω ωc to liquidlike behavior for ω ωc . The large crossover implies a broad distribution of relaxation times. Apparently, the high-frequency elastic modulus does not
Fig. 7. Master curve of the storage (filled symbols) and loss (open symbols) shear moduli as a function of the reduced frequency obtained for casein suspensions at different concentrations at 20 ◦ C. The dashed lines indicate the limiting liquidlike behavior at low reduced frequencies.
vary significantly with the casein concentration in the relatively narrow range investigated, because vertical shifts were not necessary. 3.5. Temperature dependence Light scattering experiments show that Mw increases by about 25% between 20 to 40 ◦ C and that Rh increases by about 10%. However, vt and vh remain the same within experimental error.
M. Panouillé et al. / Journal of Colloid and Interface Science 287 (2005) 468–475
Fig. 8. Concentration dependence of the zero-shear viscosity of submicelle suspensions at different temperatures. The solid lines represent fits to Eq. (3). The insert shows a master curve of the same data obtained by normalizing the viscosity by the solvent viscosity and the concentration by its critical value.
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Fig. 10. Time dependence of the zero-shear viscosity of casein submicelle suspensions at 20 ◦ C.
has the same dependence on η0 . In oscillatory shear measurements the same frequency dependence was observed at different temperatures after normalization with ωc . It appears that increasing the temperature at constant casein concentration is equivalent to decreasing the casein concentration at constant temperature. The crucial parameter that determines the rheological properties is C/Cc , which can be modified by varying either the casein concentration or Cc through the temperature. The decrease of the effective volume of the submicelles with increasing temperature is rather weak in the range between 10 and 60 ◦ C, but the effect on the viscosity is still important for dense suspensions, for which the viscosity depends very strongly on the effective volume fraction. 3.6. Ageing effects
Fig. 9. Shear rate dependence of the viscosity of casein suspensions at different temperatures at C = 141 g L−1 . The solid lines represent fits to Eq. (4).
Fig. 8 shows the effect of the temperature on the concentration dependence of η. At each temperature the concentration dependence can be described by Eq. (3), but the critical concentration increases from 99 g L−1 at 10 ◦ C to 115 g L−1 at 60 ◦ C. The nonlinear rheological properties at different temperatures are similar to those at different concentrations. Fig. 9 shows the effect of the shear rate on the viscosity of a dense suspension (141 g L−1 ) at different temperatures. The results at different temperatures can be superimposed using the same normalization as at different concentrations and γ˙c
Fig. 10 shows the effect of ageing on the zero-shear-rate viscosity of the casein submicelle suspensions at 20 ◦ C. Initially the viscosity decreases, but at longer times it increases again, because aggregates are formed that increase in size and number [28]. At some point the aggregation leads to gelation and the viscosity diverges. The gelation process accelerates at higher casein concentrations and at higher temperatures. The results shown in this paper were obtained approximately 5 h after preparation of the suspensions. It is clear that the results especially at higher concentrations depend on the age of the samples. Light scattering measurements for C < 60 g L−1 at 20 ◦ C show that Mw and Rhz do not change significantly during the first few days, but that the repulsive interaction decreases. This decrease of the interaction between the submicelles explains the decrease of the viscosity. A detailed study of the aggregation and gelation process will be presented elsewhere [36].
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4. Discussion The rheological properties of dense casein submicelle suspensions are similar to those observed for soft or hard spheres [33,37–39]. The viscosity of suspensions of spheres increases dramatically with increasing concentration when the effective volume fraction approaches that of close packing. For monodisperse, noninteracting hard spheres the specific volume is the same whether it is obtained from Rh and Mw (vh ), the concentration dependence of Ma (vt ), or the intrinsic viscosity (vη ). However, for casein submicelles that are polydisperse, charged, and soft these values are different (vh = 6 × 10−3 L g−1 , vt = 5.2 × 10−3 L g−1 , and vη = 4.0 × 10−3 L g−1 ), because polydispersity and interaction influence these parameters differently. Varying the external conditions influences the effective specific volume and thus Cc by modifying the composition of the submicelles and/or the strength of the (electrostatic) interactions. We have observed small variations of Cc at different temperatures and ionic strengths. However, even a small variation of Cc has dramatic effects on the viscoelastic properties for dense suspensions and one should be careful not to misinterpret strong variation of viscoelasticity in terms of large structural variation. In contrast to the case of hard spheres, we do not observe complete jamming of the submicelles above Cc , but rather a viscosity that increases steeply, similarly to that observed for multiarm star polymers [40]. At high frequencies dense suspensions behave as solids, but there is a transition to liquidlike behavior at lower frequencies. We may speculate that stress relaxation occurs by a hopping process of the micelles as was suggested for polymeric micelles [41], but the effects of polydispersity and electrostatic interactions will make it difficult to develop a quantitative theory for the present system. The relaxation process is characterized by a broad distribution of relaxation times and slows down rapidly with increasing effective volume fraction, i.e., increasing concentration or decreasing temperature. Over the relatively small concentration and temperature range where the relaxation process can be observed, the elastic modulus varies only weakly. The strong concentration and temperature dependence of the viscosity is thus essentially caused by the strong variation of the relaxation time. The high-frequency storage modulus, Gel , of the suspensions is about 3 × 103 Pa. If we consider only entropy and treat each submicelle as a spring with an elastic energy of kT , then Gel = kT n, with n the number density of submicelles. Using n = CNa /Mw , we obtain an elastic modulus between 5 × 102 and 103 Pa for C between 80 and 160 g L−1 at 20 ◦ C. Comparing these estimates with experimentally observed elastic moduli, we find that entropic elasticity of the micelles is not enough and that entropic elasticity of individual casein chains and/or enthalpy contributes. In the flow measurements we observed that the shear stress increases linearly with increasing shear rate until the increase weakens at γ˙c and σ reaches a constant value of
about 5 × 102 Pa independent of the temperature. This value was about the same at different concentrations over the very small range where it could be measured (110–150 g L−1 ). Interestingly, the maximum shear stress corresponds to an applied energy of approximately kT per submicelle. At higher stresses the flow is no longer controlled by spontaneous Brownian motion, because for γ˙ > γ˙c the stress is build up faster than the spontaneous relaxation of the system. At much higher shear rates the stress at constant shear rate fluctuates in an erratic manner, probably because the flow becomes inhomogeneous. It is interesting to compare the present results on casein submicelles with those obtained on micelles formed by caseinate, which is a natural mixture of casein molecules without the CCP that stabilizes the native casein complex. Caseinate forms mixed micelles with the same size as the submicelles [15–17]. We have repeated a number of experiments on a commercial sodium caseinate sample (DMV) prepared by precipitation at low pH and obtained results that are very similar to the ones shown here for submicelles formed by adding polyphosphate to native casein micelles; see Fig. 2. The concentration dependence of the viscosity at 20 ◦ C can again be described by Eq. (3) up to about 10 Pa s, but with a somewhat higher critical concentration of about 120 g L−1 in pure water or in 20 g L−1 polyphosphate. However, for C > Cc the increase of the viscosity is sharper for submicelles that for caseinate particles. This suggests that caseinate particles are more deformable spheres or that repulsion between them is smaller than between submicelles. These results are in agreement with those obtained by Farrer and Lips for sodium caseinate in a phosphate–NaCl buffer at pH 6.8 (see Fig. 2) [42]. It seems that the ionic strength and pH between 6 and 6.8 are not very important for the viscosity of caseinate suspensions. The decrease of viscosity with increasing temperature has also been reported for concentrated sodium caseinate suspensions [42,43]. Carr et al. observed a strong reversible increase of shear moduli G and G between 50 and 5 ◦ C for a sodium caseinate concentration of 140 g L−1 and misinterpreted this as reversible cold gelation [44]. The intrinsic viscosity of native casein micelles is found to be 10.4 × 10−3 g L−1 [45] or to ranging from 8 to 11 × 10−3 L g−1 [46], which is comparable with the intrinsic viscosity of submicelles. However, the viscosity of concentrated casein micelles suspensions is much lower than that for submicelles or caseinate (results not shown). The effect of ageing shows that the submicelles are not in thermodynamic equilibrium, but represent a metastable state. Subtle rearrangements decrease the strength of the interactions without significantly varying the size or the molar mass of the submicelles. At long times these rearrangements destabilize the submicelles and they start to form aggregates. The aggregation process leads to sedimention of large flocs at low casein concentrations (C < 20 g L−1 ) or to gelation at higher concentrations [28,36].
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5. Summary Casein submicelles are charged micellar particles formed by removal of colloidal calcium phosphate. They are polydisperse spherical particles with a mean radius of 11 nm. In aqueous solution they behave like soft spheres with an effective specific volume of about 5 × 10−3 L g−1 . The viscosity of submicelle suspensions increases strongly at a critical concentration of about 100 g L−1 that is weakly dependent on the temperature and the ionic strength. At high volume fractions the shear modulus shows elastic behavior at high frequencies and a transition to liquid behavior at low frequencies. The high-frequency elastic modulus is only very weakly dependent on the concentration in the range 110–150 g L−1 or the temperature in the range 10–60 ◦ C, but the relaxation time increases strongly with increasing concentration or decreasing temperature. The systems show shear thinning at shear rates above a critical value that is close to the transition frequency observed in oscillatory shear measurements. The shear and the normal stress increase linearly with increasing shear rate up to a critical shear rate above which they become independent of the shear rate. At much higher shear rates the stress fluctuates with time, indicating inhomogeneous flow that varies in time. The submicelles are not in thermodynamic equilibrium, but evolve with time. The initial effect is a relatively slow decrease of the viscosity, which is followed by a rapid aggregation and gelation process. Ageing of the suspensions is quicker at higher temperatures and casein concentrations.
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