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Dynamic Rheological Properties of Highly Concentrated Protein-Stabilized Emulsions Y. Hemar and D. S. Horne* Hannah Research Institute, Ayr, KA6 5HL, Scotland, U.K. Received June 29, 1999. In Final Form: December 13, 1999 We have measured the dynamic rheological properties of highly concentrated emulsions of soya oil in water, stabilized with sodium caseinate, as a function of oil phase volume fraction (φ g 0.70). Experiments were carried out in oscillatory mode on a controlled stress rheometer, as a function of applied stress and frequency. At low stresses these emulsions behave as a gel whose elastic properties, as a function of φ, follow the empirical model of Princen (J. Colloid Interface Sci. 1986, 112, 427) and are controlled by the Laplace pressure of the oil droplets. At higher stresses, the frequency spectra show a Maxwellian behavior with a single relaxation time. The frequency sweeps can be superimposed at each φ value onto a single master curve scaled against their Maxwellian relaxation time. Strain sweeps, derived from the frequency spectra at these higher stresses are also superimposable using as scaling factor the ratio, reference stress to the applied stress. A frequency-dependent yield stress is also observed, the response varying with the oil volume fraction.
Introduction Emulsions, dispersions of oil droplets stabilized in water by surfactant or protein, have been the subject of extensive study because of their important industrial applications in the fields of food, pharmaceuticals, paints, cosmetics, etc. The role of the surfactant emulsifier is to prevent or inhibit recoalescence of the emulsion droplets during emulsion preparation, by adsorbing to the droplet surface and creating a short-ranged interfacial repulsion between the droplets.1 In the case of food emulsions this surfactant is frequently, but not exclusively, protein, often derived from milk or eggs. In food emulsions, the volume fraction of oil, φ, can range from a few percent, as in homogenized milks, to greater than 80% in a mayonnaise. The magnitude of the volume fraction has profound effects on the viscoelastic properties of the emulsion. In the low volume fraction range, homogenized milks are freely flowing liquids with Newtonian viscosities, where the oil droplets can be treated by hard-sphere models,2 while mayonnaise, a highly concentrated suspension of oil droplets, exhibits solidlike characteristics. These allow the mayonnaise to maintain its shape under the relatively small stresses of gravity but permit it to yield and flow on the application of a little more effort when the sample is subjected to the spreading shear of a knife. A reasonable phenomenological description of this viscoelastic behavior is to speak of a yield stress needed to exceed the elastic response of the concentrated emulsion and induce flow. Indeed in the majority of rheological tests carried out to characterize mayonnaise and salad dressings, measurement of flow properties has been predominant, determining flow and consistency indices,3-5 and fitting the data to HerschelBuckley, Casson, and power law models.4-6 (1) Becher, P. Emulsions, Theory and Practice; Reinhold: New York, 1965. (2) Dickinson, E. J. Dairy Sci. 1997, 91, 2607. (3) Barbosa-Canovas, G. V.; Peleg, M. J. Texture Stud. 1983, 14, 214. (4) Paredes, M. D. C.; Rao, M. A.; Bourne, M. C. J. Texture Stud. 1988, 19, 247. (5) Paredes, M. D. C.; Rao, M. A.; Bourne, M. C. J. Texture Stud. 1988, 20, 235. (6) Bistany, K. L.; Kokini, J. L. J. Texture Stud. 1983, 14, 113.
Theoretical studies have, however, concentrated on the elastic behavior of highly concentrated emulsions or foams, which share many of their complex rheological characteristics.7-14 These theories attempt to relate the macroscopic properties of the emulsions and foams to their microscopic structure, to the physical characteristics of their constituents and interfaces, and to their transport properties on the scale of the dispersed phase. While they have provided a rational basis for describing the salient features of the rheology of highly concentrated emulsions and have reinforced the need for systematic rheological measurements, these theories have not been the subject of extensive experimental comparison. Simplistically, the mechanical properties of these materials originate from the jamming of the particles.14 When the particles are concentrated above the random close packing limit, they cannot move freely and are trapped by their neighbors. This is the origin of their solidlike behavior. But as these particles are soft, the system is able to flow when subjected to sufficient stress. The simplest microscopic model to describe the elasticity and yield behavior of such an emulsion is that of Princen.7,9 Developed originally as a two-dimensional geometric model of hexagonally close-packed drops, and later extended by analogy to three dimensions, Princen7 predicted that for a monodisperse emulsion system with droplet radius R, the elasticity and yield stress of the droplets were controlled by the ratio of surface tension (F) to droplet diameter (2R), effectively the Laplace pressure of the undeformed oil droplet, F/2R. In a further development, Princen and Kiss9 were able to relate the static shear modulus, G0, to the oil volume fraction, φ, through the relation
G0 ) 1.77
F 1/3 φ (φ - φ0) R32
(1)
where R32 is the Sauter mean droplet radius and φ0 is the maximum packing fraction of close-packed undistorted droplets. The main purposes of this paper are to quantify and test the predictions of the Princen model for a highly concentrated model mayonnaise-type emulsion of soya oil
10.1021/la9908440 CCC: $19.00 © 2000 American Chemical Society Published on Web 02/12/2000
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stabilized by sodium caseinate and also to describe some extensions of our measurements into the nonlinear oscillatory flow region where we have observed some interesting and unexpected pseudo-Maxwellian scaling behavior. A preliminary report on the testing of the Princen theory has appeared previously.15 Materials and Methods Soya bean oil was purchased from Sigma Chemical Co. (Poole, Dorset, U.K.). Sodium caseinate was prepared from fresh milk from the Hannah Research Institute herd by isoelectric precipitation, followed by resuspension and neutralization. The product solution was then freeze-dried for storage. The emulsion was prepared by mixing 100 mL of soya bean oil and 400 mL of distilled water containing 10 g of sodium caseinate using a laboratory emulsifier/mixer (Silverson Machines Ltd., Chesham, Bucks., U.K.) for 3 h at its maximum speed. A deliberately coarse emulsion was prepared because we wished to avoid producing gels on concentration which would be too stiff to flow under the stresses available to us on our rheometer. To reduce the polydispersity of our emulsion, we stood the above preparation overnight in a separating funnel, allowing a cream layer to form. The bottom and top portions of this cream layer were discarded and we retained the intermediate portion. Droplet size distributions in the original emulsion and this central cream layer were determined using a Malvern Mastersizer X, equipped with version 1.2b software (Malvern Instruments, Malvern, U.K.). A highly concentrated parent emulsion was obtained from the retained cream layer by centrifugation on a Sorvall RC-5B centrifuge, equipped with an SS34 rotor, at 5000 rpm for 5 min. The oil volume fraction of this emulsion was determined by freezedrying a portion to extract the water, weighing the sample before and after water loss. To prepare the required range of volume fractions, yet keeping interfacial tension and size distribution rigorously constant, we made use of the fact that these highly concentrated emulsions have an “osmotic pressure”16 as a result of the tendency of the deformed droplets to regain their spherical shape. Thus dilution was achieved by adding some continuous aqueous phase on top of the parent emulsion and gently stirring to leave a uniform emulsion of lower φ but identical droplet size and interfacial tension. As a further precaution, the added solution contained 5 mg of sodium caseinate per gram of water, to maintain the protein coating at saturation coverage. The rheological measurements were performed using a controlled stress Bohlin CVO rheometer equipped with stainless steel cone and plate geometry. Cone diameter was 40 mm and cone angle 4°. All measurements were performed in oscillation mode at a temperature of 20 ( 0.1 °C.
Results and Discussion Emulsion Properties. Figure 1 shows an optical micrograph obtained from a highly concentrated parent emulsion (measured φ ) 0.89). No special preparations were required for this micrograph, the emulsion simply being trapped between two glass slides. The micrograph reveals each droplet (the dark areas) as a separate and distinct entity surrounded by a lighter aqueous phase. The distortions of droplet size and shape brought about by packing to this high internal phase volume are (7) Princen, H. M. J. Colloid Interface Sci. 1983, 91, 160. (8) Khan, S. A.; Armstrong, R. C. J. Non-Newtonian Fluid Mech. 1986, 22, 1. (9) Princen, H. M.; Kiss, A. D. J. Colloid Interface Sci. 1986, 112, 427. (10) Budinsky, B.; Kimmel, E. J. Appl. Mech. 1991, 58, 289. (11) Stamenovich, D. J. Colloid Interface Sci. 1991, 145, 255. (12) Reinelt, D.; Kraynik, A. M. J. Colloid Interface Sci. 1993, 156, 460. (13) Hemar, Y.; Hocquart, R.; Lequeux, F. J. Phys. II 1995, 5, 1567. (14) Derec, C.; Ajdari, A.; Lequeux, F. Faraday Discuss. 1999, 112, 195. (15) Hemar, Y.; Horne, D. S. In Food Emulsions and Foams: Interfaces, Interactions and Stability; Dickinson, E., Rodriguez Patino, J. M., Eds.; Royal Society of Chemistry: Cambridge, U.K., 1999; p 319. (16) Princen, H. M. J. Colloid Interface Sci. 1985, 105, 150.
Figure 1. Optical micrograph revealing the microstructure of the parent emulsion at φ ) 0.89.
Figure 2. Particle size distributions, p(d), obtained by light scattering using a Malvern Mastersizer, as a function of particle diameter before (open symbols) and after (closed symbols) creaming. The creamed sample is from the intermediate portion of the creamed layer produced on overnight standing. This intermediate portion is then subjected to centrifugation to obtain the highly concentrated parent emulsion.
particularly apparent in the mid-size-range droplets where hexagonal or pentagonal structures are clearly evident. Droplet size distributions, plotted semilogarithmically as volume fraction versus particle size (Figure 2), showed a typical bimodal shape. Allowing the emulsion to cream overnight reduced the content of smaller droplets and pushed the main peak in the normalized distribution to a higher value. Linear Rheological Measurements. By definition, the linear viscoelasticity region occurs over that region of strain where the complex modulus is independent of that strain. Typical behavior for the shear moduli (both G′ and G′′) of our highly concentrated emulsions is shown in Figure 3 for three different volume fractions. Measurements were carried out at a frequency of 1 Hz, and applied stress was ramped from a low minimum to produce the measured strain. At low strains, the storage modulus, G′, was always much greater than the loss modulus, G′′, confirming the dominant role of the elastic properties in these emulsions in this region of strain. If we determine the linear viscoelastic region solely on the basis of the storage modulus, then we see that this
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Figure 3. Strain dependence of the storage modulus, G′ (solid symbols), and loss modulus, G′′ (open symbols), for three volume fractions, 0.89 (9), 0.75 (b), and 0.70 (4), at an oscillation frequency of 1 Hz.
increased strongly with oil volume fraction and remained strain independent below γ ≈ 0.02 for φ ) 0.89, γ ≈ 0.006 for φ ) 0.75, and γ ≈ 0.002 for φ ) 0.70. In a separate series of experiments carried out at frequency 0.1 Hz (data not shown), critical strains at these volume fractions were found to be virtually identical. Clearly this yield point was also a sensitive function of volume fraction. Beyond these critical strains, there was a noticeable drop in the storage modulus while at the higher volume fractions the loss modulus began to rise perceptibly, indicating the transition to nonlinear behavior and plastic flow. Beyond the value of the last strain point plotted for each data set in Figure 3, a macroscopic slippage occurred at the sample/ rheometer interface and the measured moduli became erratic and noisy. Confining ourselves to very low stresses where measured strain always remained below the critical yield strain determined previously, we carried out frequency sweep measurements on our samples. The rheological spectra, measured over 3 decades of frequency, are shown in Figure 4, for the same three volume fractions used previously. At these stresses, G′(ω) for the two higher volume fractions was essentially independent of frequency, ω. At the lowest volume fraction depicted, however, G′(ω) dropped off with frequency as the frequency was reduced, while remaining independent of frequency at the other end of the spectrum. This emulsion, even at the lowest stresses applied, was flowing over the long time scales to which these frequencies are equivalent. In all cases the loss modulus G′′(ω) was at least a factor of 10 smaller than the storage modulus, confirming once more the gellike nature of these emulsions under low stress. The loss moduli do, however, clearly increase with frequency, if somewhat noisily. Broadly similar behavior has been observed in other types of concentrated emulsions both in frequency sweep and stress sweep experiments. Slight differences in detail are apparent, however. Mason et al.17 observed minima (17) Mason, T. G.; Lacasse, M.-D.; Grest, G. S.; Levine, D.; Bibette, J.; Weitz, D. A. Phys. Rev. E 1997, 56, 3.
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Figure 4. Frequency dependence of the storage modulus, G′ (solid symbols), and loss modulus, G′′ (open symbols), for three volume fractions, 0.89 (9), 0.75 (b), and 0.70 (4), obtained at a developed strain always less than the critical yield strains determined in the prior stress sweep experiments.
in their plots of G′′(ω) vs frequency for concentrated monodisperse emulsions stabilized by surfactant, while Ebert et al.18 found G′′(ω) independent of frequency, possibly as a result of polydispersity in their emulsions. Further measurements were made, with similar results, at other volume fractions diluted from the parent emulsion. The elastic modulus measured at 1 Hz is shown in Figure 5 as a function of oil phase volume fraction. Since the frequency spectra are independent of frequency, we take these values as equivalent to the static shear modulus of the Princen formula. The solid line shows the fit of these data to the Princen equation (eq 1). We find the calculated value for φ0, the close-packing limit, to be equal to 0.70, slightly smaller than the value of 0.712 given by Princen and Kiss.9 Possible explanations for this difference lie in the differing polydispersities of the preparations or in the differing nature of the stabilizing films of our highly concentrated emulsions. Our fit also gives F/R32 ∼ 2400 Pa for the Laplace pressure of our droplets. Assuming that the surface tension of our caseinate film is approximately 24 mN‚m-1, similar to that of β-casein or RS1-casein,19 the extracted droplet size, R32, is about 10 µm. This value is close to the maximum value in the main peak of our droplet size distribution function, confirming the applicability of the Princen model to explaining the elastic behavior of our highly concentrated emulsions. Nonlinear Viscoelasticity. In characterizing the linear viscoelastic behavior, we took great care to minimize the stresses applied to our emulsions. To investigate the nonlinear viscoelastic region, we have measured the frequency spectra of our highly concentrated emulsions as a function of applied stress. Results are shown in Figure 6 for an oil phase volume fraction of 0.85. For these measurements independent frequency sweeps from low to high were carried out, with a fresh sample being loaded for each of the applied stresses. These spectra show (18) Ebert, G.; Platz, G.; Rehage, H. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1158. (19) Castle, J.; Dickinson, E.; Murray, B. S.; Stainsby, G. ACS Symp. Ser. 1987, 343, 118.
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Figure 5. Static storage modulus, taken as the G′ value at 1 Hz in the frequency spectra, as a function of volume fraction for two series of emulsion dilutions from the parent emulsion (b, O). These data points are used to produce the fit to the Princen formula (solid line). The third set of data points (9) are values of G′ in the linear viscoelastic region obtained in the stress sweep experiments (also made at 1 Hz).
Maxwellian-like behavior; the curves are fitted to a Maxwell model with a single relaxation time according to the equations
G′ ) G∞ G′′ ) G∞
(ωτ)2 1 + (ωτ)2 ωτ 1 + (ωτ)2
(2a) (2b)
where τ is the relaxation time and ω is the applied frequency. Similar Maxwellian behavior was observed by Ravey et al.20 for their fluorocarbon gel emulsions at φ ) 0.95. The first point to note in the plots of Figure 6 is that the Maxwellian relaxation time decreased with the applied stress. Even for low stresses, these highly concentrated emulsions will flow but with an extremely long relaxation time and a consequently often negligible rate of flow. Next, no matter what the applied stress, the elastic modulus at high frequencies reached the same plateau value. This was the same value as that measured in the linear viscoelasticity, low stress experiments, which we could now recognize as being in a region of infinite (or at least very long) relaxation times, indicating that droplet elasticity in the flow regime continued to be governed by the Laplace pressure of the droplets. Within the error of our determinations, the viscous modulus, G′′, always reached the same maximum value when reciprocal frequency equaled the relaxation time of the Maxwellian function (eq 2b). Though the origin of this viscous component is not fully understood, it also remained apparently unaffected by the stresses required to induce flow in these highly concentrated emulsion systems. Taken together, these two observations imply that the applied (20) Ravey, J. C.; Ste´be´, M. J.; Sauvage, S. Colloids Surf., A 1994, 91, 237.
Figure 6. Frequency dependence of elastic moduli, G′ (closed symbols) and G′′ (open symbols), for an emulsion with volume fraction, φ ) 0.85, measured at different applied stresses, 44 Pa (9), 48 Pa (b), and 55 Pa (2). The curves drawn through the points obtained for each stress are fits of the data to a Maxwell model (eq 2) with a single relaxation time.
stress and consequent flow are not manifestly changing the droplet size distribution. Decreases in size due to droplet shear, or increases due to droplet coalescence, would change the Laplace pressure, and this would manifest itself as shifts in the shear modulus values in response to the different shearing forces. Similar plots (data not shown) were obtained for the parent and other diluted emulsions. Lowering the volume fraction reduces the plateau elastic modulus in line with the predictions of the Princen formalism7,9 and the results shown in Figure 5. The effect of the oil phase volume fraction on the gross shifts in the mechanical spectra at these higher applied stresses is most easily observed in the plots of Figure 7 where we have plotted the Maxwellian relaxation time, τ, as a function of applied stress in double log format. Up to a volume fraction, φ ) 0.85, the lines are steeply parallel, demonstrating a high power law dependence of relaxation time on applied stress. Only the data points for the parent emulsion (φ ) 0.89) and its first dilution (data not plotted) show an anomalous deviation from this abrupt, almost all-or-nothing behavior, with unexpectedly short relaxation times at low applied stresses. Indeed, this sample was flowing under stresses where no flow was observed in more dilute emulsions, possibly pointing to the existence of another mechanism at these higher phase volumes. Despite this postulated change, the mechanical spectra show superposition behavior. Defining a frequency shift factor, an, as an ) τ(σ)/τ(σ0) where σ0 is the lowest stress where measurements were performed at each volume fraction, we obtained the master curves of Figure 8 to describe the viscoelastic responses at each emulsion concentration. Each master curve consists of data obtained with at least five different applied stresses, shifted only on the horizontal frequency scale. The shifts are not arbitrary but are determined by the viscoelastic response measured at frequency, 1/τ. As can be seen in Figure 7, this shift factor is not small but encompasses approximately 2 orders of magnitude, producing an effective
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Figure 7. Effect of oil phase volume fraction on the behavior of relaxation times, derived from Maxwell model fits to frequency spectra, as a function of applied stress. Oil phase volume fractions were 0.75 (1), 0.80 (2), 0.85 (b), and 0.89 (9).
widening of the mechanical frequency spectrum to almost 5 logarithmic decades. On the low side of the crossover frequency, the data are consistent with classical Maxwellian slopes of +1 for the viscous component and +2 for the elastic component, at all volume fractions. At higher frequencies the elastic moduli tend to the plateau values referred to previously, whereas the loss moduli show an enriched response and indications, in the minima and slight rise, of a second family of relaxation phenomena at high frequency. The Maxwellian crossover point for G′ and G′′ occurs at approximately the same value of reduced frequency independent of volume fraction. Moreover, the two clusters of relaxation times seem to be separated by a constant distance on the reduced frequency scale, independent of oil phase volume fraction, and possibly correspond to the R and β transitions of bulk glassy systems or to those depicted in classical mode coupling theories.21,22 Our observed behavior in these concentrated emulsions is also close to that depicted by Derec et al.14 for their model of the liquid-phase system beyond their defined jamming transition point at finite applied strain. For each point in Figure 6, the stress applied to the sample produces a particular response dependent on the frequency of the oscillation. When the elastic and viscous moduli were plotted as a function of this developed strain amplitude, again in log-log format, the plots of Figure 9 were obtained for our four representative volume fractions. These can be made to produce a master curve (Figure 10) at each volume fraction by multiplying each strain by a reducing factor (σ0/σ) where σ is the applied stress and σ0 is the reference stress, arbitrarily chosen as the highest value employed for each group. We note immediately the power law behavior with G′ ∝ γ -2 and G′′ ∝ γ -1. We are unable to formulate a fundamental interpretation of this scaling behavior of the strain amplitude with applied stress but draw attention to the following points related to the phenomenon. First, it allows the measurements to be extended and oscillatory flow behavior to be demonstrated in the regions of strain amplitude well beyond the range (21) Sollich, P. Phys. Rev. E 1998, 58, 738. (22) He´braud, P.; Lequeux, F. Phys. Rev. Lett. 1998, 81, 2934.
Figure 8. Master curves produced by the superposition of frequency spectra of the type plotted in Figure 6, expressed as a function of reduced frequency, ωan. The frequency shift factor is defined by the ratio of the relaxation time at stress, σ, to that at an arbitrarily selected value, σ0, in the series, i.e., an ) τ(σ)/ τ(σ0). Oil phase volume fractions are indicated for each box.
where macroscopic slippage was normally encountered. For this reason, we believe our measurements to be largely free of macroscopic wall-slip and fracture effects. Second
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Figure 9. Double log plots of elastic and viscous moduli plotted as a function of strain developed following the application of the indicated stresses at a range of frequencies. Oil phase volume fractions, again, are indicated for each box.
Figure 10. Master plots produced by superposition of the strain sweeps of Figure 9. The shift factor is defined as the ratio of applied stress to the highest stress employed at each volume fraction.
the largest shifts are required at the highest oil phase volume fractions (φ g 0.87), that of the parent emulsion and its first dilution, where we recall anomalous relaxation effects were encountered. Here, at the lowest stresses employed, either the system is not behaving as elastically as it should, meaning that the results should be scaled
vertically, or the developed strain is not as large as it should be. We dismiss the former suggestion because, when the data for φ ) 0.89 is plotted against frequency, the elastic modulus values lie along the plateau line and its approach and do not stand as outliers. This suggests that at the highest concentrations, flow behavior is
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σ)
F 1/3 φ Y(φ) R32
(3)
where Y(φ) was an empirically determined function fitted to their data set. This arbitrary function took the form a + b log(1 - φ). Using the value of F/R32 ) 2400 Pa determined using our elasticity data and fitting only the high-frequency data points, we obtained the fitted line, shown solid in Figure 11, with best fit values of a ) 0.065 and b ) 0.115. These values lie within 1% of those determined by Princen and Kiss23 for their surfactantstabilized emulsions using shear viscosity procedures. We would thus conclude that whether we determine yield stress by the crossover of shear modulus components, as we have done here, or by varying shear rate measurements of viscosity, we are looking at the same behavior in these highly concentrated emulsions. They represent two sides of the same coin. Concluding Remarks
Figure 11. Frequency dependence of apparent yield stress as a function of volume fraction. Plotted yield stress values were observed at frequency 4 Hz (9), 0.5 Hz (b) and e0.1 Hz (2) from data, such as that exemplified in Figure 7. Yield values plotted as open squares (0) were obtained as crossover points in directly measured stress sweeps at 1 Hz, of which Figure 3 provides examples. The solid line is the fit of eq 3, with the Princen empirical function for yield stress, to the points obtained at 4 Hz only.
characterized by a lower yield stress, a shorter relaxation time, and smaller strain movements, when compared to that encountered in samples in the oil volume fraction range 0.7-0.85. Nevertheless, we emphasize that at all volume fractions our highly concentrated emulsions showed a Maxwellian response to applied stress when flow was observed. The data of Figure 7 can be interpreted as defining an apparent yield stress with the measured Maxwellian relaxation time, τ, defining the operative frequency. Using such a definition we can view the frequency dependence of the yield behavior as a function of oil phase volume fraction (Figure 11). Thus we see that yield stress is not such a strong function of droplet volume fraction as gel elasticity, only increasing by just over an order of magnitude over the volume fraction range of our highly concentrated emulsions, compared to the 3 orders observed for the elasticity. Moreover, our lowest volume fraction emulsion, near the estimated close-packing limit, is showing a finite yield stress compared to the almost zero elasticity measured. Over most of the volume fraction range, yield stress is largely independent of frequency. Divergence from this behavior sets in again at the high volume fraction end of our range, systematically diminishing at the lower frequencies. From whatever viewpoint we choose to examine these results, yield stress or relaxation time, there is a systematic change in the behavior at the highest volume fractions, but one which does not affect the scaling or Maxwellian behavior we also observe. Any mechanism proposed will have to reconcile these two, perhaps conflicting, observations. Princen and Kiss23 expressed the dependence of the yield stress on volume fraction through a relation of the form (23) Princen, H. M.; Kiss, A. D. J. Colloid Interface Sci. 1989, 128, 176.
Despite being composed of fluids of relatively low individual viscosities, these highly concentrated caseinatestabilized soya oil-in-water emulsions possess a striking shear rigidity, characteristic of a Hookean solid. This elasticity develops and grows by around 3 orders of magnitude only when the emulsion droplets are concentrated beyond the spherical close-packing limit, which we determine to be around φ ≈ 0.70 for our polydisperse emulsion. It is created because the droplets have been compressed, packed together, and deformed to create flat facets where neighboring droplets touch. Shear deformations of this structure create additional droplet surface area, as modeled by Princen,7,9,23 and give rise to the emulsion’s elastic modulus. As we have shown here, with an excellent fit to Princen’s formula, this elasticity is a function of the Laplace pressure, F/2R23, at contact. Importantly, no significant effect related to the stabilizing protein film itself is evident. Our emulsion droplets are of such a size that the effects of film thickness are negligible, but no effects ascribable to the nature of the film or to film interactions between opposing droplets can be discerned. As also predicted by the Princen model, our emulsions show a volume fraction dependent yield stress. They flow when the applied stress exceeds a critical value where sufficient deformation is produced to permit the droplets to slide past each other. This yield stress is a sensitive function of the volume fraction difference above the closepacking limit but is dependent on the frequency or rate of application of that force. The behavior we observe in our food emulsions is similar, qualitatively and quantitatively, to that observed by others in surfactant-stabilized systems, again confirming that any effects, which might be related to the different types of emulsifiers, are secondary. Indeed, because we have carried out our measurements at a range of applied stresses, we have observed behavior through the transition from linear to nonlinear viscoelasticity behavior. We have demonstrated that the gel-like properties found by Ebert et al.18 and the Maxwellian response observed by Ravey et al.20 can be induced in the same emulsion and that their occurrence depends only on the stress applied. In the nonlinear viscoelastic regime, the flowing emulsions exhibit a Maxwellian behavior characterized by a stress-dependent relaxation time. The frequency spectra scale with this relaxation time and the high-frequency moduli components reach constant values, irrespective of the applied stress, the same values measured at (appar-
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ently) subcritical stresses. Thus, though some critical barrier must be exceeded to induce flow, it is not one where bonds have to be broken or droplets sheared and size distributions altered. Rather, it is droplet deformation, a function of internal Laplace pressure, which has to be brought about to cause the structure to slip and flow. Acknowledgment. The authors thank Mrs. C. M. Davidson for technical assistance and Dr. K. Hendry for
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the optical micrography. This research was carried out as part of the EU Framework IV Project CT96-1216 “Structure, rheology and physical stability of particle systems containing proteins and lipids” and their financial support is gratefully acknowledged. Core funding for the Hannah Research Institute is provided by the Scottish Executive Rural Affairs Department. LA9908440
