Scaling and fractal analysis of viscoelastic properties of heat-induced protein gels

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1 Food Hydrocolloids 18 (2004) Scaling and fractal analysis of viscoelastic properties of heat-induced protein gels M.M. Ould Eleya, S. Ko, S. Gunasekaran* Food and Bioprocess Engineering Laboratory, Department of Biological Systems Engineering, University of Wisconsin-Madison, Agricultural Engineering Building 220, 460 Henry Mall, Madison, WI 53706, USA Accepted 23 May 2003 Abstract The scaling of viscoelastic properties of heat-induced gels prepared from egg white was investigated at five ph values (ph 3 11) and various protein concentrations (5 20%) using small-deformation oscillatory rheological measurements. Protein gels were formed by heating samples at 80 8C for 1 h followed by cooling to 25 8C. Storage modulus, G 0 and critical strain, g 0 of the gels exhibited ph-dependent powerlaw relationships with protein concentration. Based on the power-law exponent values, fractal dimension, d f of heat-induced protein gels was estimated using scaling models from the literature. Low d f values ( ) were obtained for gels prepared at acidic and neutral phs (3 7) whereas higher d f values ( ) were obtained for gels formed under basic ph conditions. These d f values lied well within the range of fractal dimension values ( ) reported for protein gels. However, they slightly differed from d f for diffusion-limited and reactionlimited cluster cluster aggregation processes, which made it difficult to justify an assumption regarding the nature of the aggregation process of these protein systems. q 2003 Elsevier Ltd. All rights reserved. Keywords: Scaling model; Fractal dimension; Rheology; Gel; Protein 1. Introduction The ability of globular proteins to form heat-induced gels is one of their most important functional properties. On heating, protein molecules undergo conformational changes characterized by a partial unfolding of the globular structure and resulting in the exposure of reactive groups such as the buried hydrophobic regions and the sulfhydryl groups. The unfolded or denatured molecules may aggregate irreversibly via non-covalent and covalent bonds. The non-covalent bonds are due to van der Waals attractive forces, electrostatic interactions, hydrogen bonding, and hydrophobic interactions. The covalent bonds are intermolecular disulphide bonds formed through sulfhydryl-disulphide interchange reactions (Kinsella & Whitehead, 1989; Mine, 1996; Ziegler & Foegeding, 1990). Numerous studies have shown that protein aggregation results in the formation of disordered structures (aggregates or flocs) of protein particles that behave as stochastic mass-fractals on a length * Corresponding author. Tel.: þ ; fax: þ address: guna@facstaff.wisc.edu (S. Gunasekaran). scale larger than the size of primary particles (Gimel, Durand, & Nicolai, 1994; Hagiwara, Kumagai, & Matsunaga, 1997; Hagiwara, Kumagai, & Nakamura, 1998; Ikeda, Foegeding, & Hagiwara, 1999; Marangoni, Barbut, McGauley, Marcone, & Narine, 2000; Vétier, Desobry Banon, Ould Eleya, & Hardy, 1997). When protein concentration is large enough, the fractal aggregates grow until they occupy the total liquid volume, at which moment a three-dimensional continuous network or gel is formed, a structure that has many interesting practical properties such as elasticity and water-holding capacity. The control of protein gels for various industrial applications requires a better understanding of the relationship between the structure of aggregates and the macroscopic properties of the gel. Fractal dimension of protein aggregates has been determined using various experimental techniques such as rheology, microscopy, light scattering, small-angle neutron scattering, and gel permeability measurements (Bremer, Bijsterbosch, Schrijvers, & van Vliet, 1990; Bremer, van Vliet, & Walstra, 1989; Gimel et al., 1994; Hagiwara et al., 1997, 1998; Ikeda et al., 1999; Marangoni et al., 2000; Verheul, Roefs, Mellema, & de X/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi: /s x(03)

2 316 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) Nomenclature b empirical parameter that measures rate of increase of G 0 with time C protein concentration (%, w/v) d Euclidean dimension ( ¼ 3) d f fractal dimensionality G 0 storage modulus G 00 loss modulus G 0 0 long-time storage modulus in the linear region G 0 1 storage modulus at infinite time n scaling exponent in power-law relationship of G 0 and C m scaling exponent in power-law relationship of g 0 and C t time T temperature x fractal dimensionality of the backbones ð1 # x, d f Þ a microscopic elastic constant in the model of Wu and Morbidelli ð0 # a # 1Þ b exponent in model of Wu and Morbidelli which depends on a and x d phase angle critical strain; limit of linearity g 0 Kruif, 1998; Vétier et al., 1997; Vreeker, Hoekstra, den Boer, & Agterof, 1992). While static light scattering technique is widely accepted as the best technique to investigate the internal structure of fractal aggregates, it has limitations particularly for concentrated systems. Microscopic image analyses are the most direct method for evaluating fractal dimension. However, it is often difficult to obtain an image of an intact protein gel due to possible damage to gel structure during sample preparation (Ikeda et al., 1999). Rheological measurement have the advantage that they are easy to conduct, and can be applied to highly concentrated systems. To determine the structure of colloidal gels from its rheological properties, a scaling model or rheological fractal model is needed. Various scaling models have been developed and applied to several protein and colloidal gel systems during the past few decades (Bremer et al., 1990, 1989; Mellema, van Vliet, & van Opheudsen, 2000, 2002; Shih, Shih, Kim, Liu, & Aksay, 1990; Wu & Morbidelli, 2001). These models relate the rheological properties of the gel (i.e. storage modulus G 0 ; critical strain g 0 ; and yield stress s y ) to the aggregate size ðjþ in the gel ðj, f 1=ðd2d fþ Þ and, thus, to the aggregate fractal dimension d f ; and volume fraction f of particles in the system. Bremer et al. (1990, 1989) derived two models the so-called straight strands (or type 1) and curved strands (or type 2) models that relate the gel elasticity to the fractal dimension of aggregates. For type 1 gel model, the strands composing the gel network are stretched under applied stress while for the type 2 gel model the strands in the gel network are bent under applied stress. To measure the fractal dimensionality d f of the gel using these models, one needs to combine rheological measurements with alternative experimental methods (e.g. microscopy, gel permeability) and/or make an assumption on the nature of the strands composing the network (e.g. straight, curved). A similar but more detailed model has been recently developed by Mellema et al. (2000, 2002). These authors derived three scaling laws for the elastic modulus G 0 ; critical strain g 0 ; and yield stress s y of the gel and incorporated, in these relationships, two scaling exponents describe the number of deformable links in a strand and the dominant type of microscopic deformation. This model allows to estimate the fractal dimension of the gel from the above scaling laws only if one assumes that the value of the exponent n in the scaling law relating the yield stress and particle volume fraction ðs y, f n=ðd2d fþ Þ is known. One of the advantages of this model is that if fractal dimension d f is known, e.g. from other experimental measurements such as microscopy, it is possible to identify the type of gel structure based on the values of scaling exponents. According to this model, five types of gel structure (i.e. random, curved, hinged, straight, rigid) can be distinguished depending on the nature of strands (e.g. randomly curved, flexible) and of the microscopic deformation (i.e. bending, stretching). This model has been applied to experimental data on rennetinduced casein gels and revealed that such gels contain straight strands with a large number of deformable links. In contrast to the scaling models mentioned earlier, the model initially developed by Shih et al. (1990) and recently extended by Wu and Morbidelli (2001) allows an estimation of the fractal dimension d f solely based on the rheological properties of the gel. These models are presented in detail in Section 2. It should be mentioned that fractal models can only be applied for a limited range of particle volume fractions. The volume fraction of particles in the system should be well above that of the gelation threshold and generally smaller than 0.3. At larger volume fractions, the fractal floc will have a diameter ðj ¼ af 1=ðd2d fþ Þ only a few times that of the primary particle, which makes the use of fractal arguments unjustified. In addition, fractal models implicitly assume that the critical gel concentration ðf c Þ is equal to zero, which is unrealistic. Alternative models, based on branching theory or percolation theory, exist in the literature and have been reported to be more broadly applicable than fractal models. In a recent study on various b-lactoglobulin gels, Kavanagh, Clark, and Ross-Murphy (2000) showed that a model based on the branching (cascade) theory gave a better description of the storage modulus concentration relationships than the fractal model developed by Bremer et al.

3 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) (1989). The cascade model fitted well the modulus concentration data for various b-lactoglobulin gels over a wide range of concentrations and allowed estimating critical gel concentration. Unlike fractal models, the cascade model does not, however, contain structural parameters (i.e. fractal dimension) that can be used to characterize the microstructure of the gel. Models based on the percolation theory have been shown to provide good descriptions for the mechanical properties of gels near the percolation threshold. Recently, Van der Linden and Sagis (2001) showed the modulus concentration dependence for various protein and colloidal gels can be well described using a percolation model ðg 0, ðf 2 f c Þ t Þ; with constant scaling exponent t ðt ¼ 1:79 ^ 0:25Þ and a critical percolation threshold that varied with the gel type. Percolation models, however, only rigorously hold close to the percolation boundary (e.g. gel point). We used egg white as a model of globular protein system and investigated its gelation over a range of ph values (3 11) and protein concentrations (5 20% w/v). Egg white, known also as egg albumen, is a protein system extracted from egg and consisting of different fractions with ovalbumin constituting the major protein component (,55% of total egg albumen) and the main gelling agent. Our approach consisted of determining experimental relationships between rheological parameters (G 0 ; g 0 )of heat-induced protein gels and concentration and of applying fractal models to these experimental data. Our objective was to evaluate fractal dimension of aggregates in heat-induced protein gels. We were particularly interested in determining how ph affects fractal dimension and power-law behavior of protein gels since it is well known that the microstructure and physical properties of such gels are highly phdependent. 2. Fractal models 2.1. Model of Shih et al. Shih et al. (1990) developed a scaling model relating G 0 and g 0 to particle volume fraction f for a colloidal gel far from its gelation threshold. Depending on strength of interand intra-floc links, two regimes were defined: strong-link regime (inter-floc links are stronger than intra-floc links) and weak-link regime (inter-floc links are weaker than intrafloc links). In the strong-link regime: G 0, f ðdþxþ=ðd2d fþ g 0, f 2ð1þxÞ=ðd2d fþ In the weak-link regime: G 0, f 1=ðd2d fþ g 0, f 1=ðd2d fþ ð1þ ð2þ ð3þ ð4þ where d is the Euclidean dimension, d f is the fractal dimension of the flocs and x is the fractal dimension of the floc backbone ð1 # x, d f Þ: 2.2. Model of Wu and Morbidelli Based on the model of Shih et al. (1990), Wu and Morbidelli (2001) developed a new model that relates G 0 and g 0 of the gel to volume fraction of primary particles. An appropriate effective microscopic elastic constant a (where a [ ½0; 1Š) was introduced to account for elastic contributions of both inter- and intra-floc links. It indicates the relative importance of these two contributions and allows identifying different gelation regimes prevailing in the system. G 0, f b=ðd2d fþ ð5þ g 0, f ðd2b21þ=ðd2d fþ b ¼ðd 2 2Þþð2 þ xþð1 2 aþ where x is the fractal dimension of the floc backbone ð1 # x, d f Þ as in the Shih et al. model. 3. Material and methods 3.1. Preparation of protein solutions A commercial spray-dried egg white protein powder (Oskaloosa Food Products Corp., Oskaloosa, IA) was used for the experiments. Protein suspensions at various concentrations (5 20% w/v) were prepared by dispersing appropriate amounts of spray-dried egg white in distilled water. Protein solutions were then held overnight at 4 8C for complete hydration, and their ph was adjusted at room temperature to the five following ph values: 3, 5, 7, 9, and 11 using 1N of HCl or NaOH. Since we did not use buffer solutions we checked the variability of ph as a result of thermal treatment by measuring the ph of a protein solution before and after heating at 80 8C for 1 h. Results indicated that only a slight variation in ph (less than 0.5 unit) was observed for some samples Dynamic small-strain rheological measurements A dynamic controlled-stress rheometer (Bohlin CVO, Bohlin Rheologi Inc., Cranbury, NJ) was used with a 20-mm parallel-plate geometry and a 1-mm gap between the plates. Samples of protein solutions (5 20% w/v) were placed between the two parallel plates of the rheometer. Samples were covered with a thin layer of paraffin oil to prevent sample dehydration during experiment. Gels were formed in situ by allowing samples to cure at 80 8C for 1 h. Storage moduli ðg 0 Þ; loss moduli ðg 00 Þ; and phase angles ðdþ of samples were measured as a function of time at 1 Hz and a maximum target strain of Strain value was checked over time and variations were less than 5% of target strain. Samples were ð6þ ð7þ

4 318 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) Results and discussion 4.1. Time dependence of G 0 Fig. 1. Gelation profile of egg white sample at ph 7 and different protein concentrations (test conditions: T ¼ 80 8C; frequency ¼ 1Hz; strain ¼ 1%). cooled to room temperature (25 8C), kept at this temperature for 30 min, and then subjected to a stress sweep experiment at a constant frequency of 1 Hz. The strain values of the samples were measured and critical strain g 0 values were determined from the G 0 -strain profiles of the gels Determination of fractal dimension The volume fraction of particles ðfþ in the gels was assumed to be proportional to the protein concentration ðcþ: Fractal dimension values of egg white gels were evaluated using values of slopes of log G 0 versus log C and of log g 0 versus log C and based on the models of Shih et al. (Eqs. (2) and (3)) and Wu and Morbidelli (Eqs. (6) and (7)). Fig. 1 shows typical gel formation kinetics of egg white at ph 7 for various protein concentrations. Similar profiles (results not shown) were also obtained for egg white samples at the other ph values (ph 3, 5, 9, and 11). On heating egg white solutions at 80 8C, G 0 rapidly increased with time and then tended to reach a plateau. These patterns were similar to those found in identical experiments on other protein gels (Ikeda et al., 1999; Ould Eleya & Gunasekaran, 2002; Rueb & Zukoski, 1997). As shown in Fig. 1, the growth of the storage modulus with time for all protein concentrations could be approximated as a firstorder process (solid curves) with: G 0 ðtþ ¼G 0 1½1 2 expð2btþš where t is the time, b an empirical parameter that measures the rate of increase of G 0 ; and G 0 1 the value of G 0 at infinite time. Such a first-order reaction kinetics have previously been observed for gelation of protein and colloidal systems (Ikeda et al., 1999; Rueb & Zukoski, 1997) Strain dependence of G 0 Changes in G 0 with strain for egg white gels, prepared at ph 7 and at various protein concentrations are presented in Fig. 2. For all protein concentrations, G 0 remained almost constant and then suddenly decreased as strain increases. This sudden decrease in G 0 indicated breaking of bonds within the gel network and a transition from a linear to a non-linear behavior. Similar patterns (results not shown) ð8þ Fig. 2. Storage modulus strain profile of egg white gels at ph 7 and different protein concentrations (%). The inset shows how critical strain or the limit of linearity was determined (test conditions: T ¼ 25 8C; frequency ¼ 1 Hz).

5 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) were also obtained for egg white gels at the other ph values (3, 5, 9, and 11). As previously reported (Rueb & Zukoski, 1997; Shih et al., 1990), the strain amplitude at which G 0 just begins to decrease by 5% from its maximum value was determined and taken as a measure of the limiting or critical strain g 0 of the gel. This is illustrated in the inset of Fig Scaling of the viscoelastic properties of protein gels The values of G 0 at the end of the heating period at 80 8C (Fig. 1), and maximal G 0 values in the linear region at 25 8C (Fig. 2) were determined for the five ph values, and plotted as a function of protein concentration (Fig. 3). This figure shows that, for all ph values studied, both G 0 at 80 8C and G 0 at 25 8C exhibited a power-law behavior or a scaling relationship with protein concentration that can be fitted to the form: G 0, C n ; where n is the power-law exponent. As expected, n values are positive for all gels. At the same ph, the scaling exponent n of G 0 versus C at 80 8C was almost the same as that of G 0 versus C at 25 8C, suggesting that cooling samples from 80 to 25 8C did not induce important structural changes in the gel network. However, an increase in the storage modulus of the gel during the cooling period was observed for all samples, revealing a strengthening of the gel. Similar increase in G 0 of protein gels with decreasing temperature has previously been observed for various systems. Such an increase is generally attributed to strengthening of attractive forces such as van der Waals Fig. 3. Double-logarithmic plots of the storage modulus G 0 of egg white gels as a function of protein concentration at various ph values and at 80 8C (open symbols) and 25 8C (filled symbols) (test conditions: frequency ¼ 1 Hz; strain ¼ 1%).

6 320 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) Table 1 Rheological data and evaluated structural parameters of egg white gels at various ph values ph Power-law exponents Model of Shih et al. (1990) Model of Wu and Morbidelli (2001) Regime n a m b d f c x c d f d b d a d at x ¼ 1:0 a d at x ¼ 1: ^ Transition gel ^ Transition gel ^ Transition gel ^ Transition gel ^ Transition gel a Power-law exponent relating G 0 to concentration: G 0, C n : b Power-law exponent relating g 0 to concentration: g 0, C m : c Values of fractal dimensions d f and x based on the model of Shih et al. (1990). d Values of d f ; b; and a parameters based on the model of Wu and Morbidelli (2001). interactions and hydrogen bonding between protein particles within the gel network (Aguilera, 1995; Ould Eleya & Turgeon, 2000). The n values summarized in Table 1 clearly show that it is strongly ph-dependent and ranged from 3.3 to 6.9 as ph is varied from 3 to 11. These values are close to those found for other protein gels (Fernandes, 1994; Hagiwara et al., 1997, 1998; Ikeda et al., 1999; Kavanagh et al., 2000; Ould Eleya & Gunasekaran, 2002; Verheul et al., 1998), where exponent n ranged from 2 to 7 depending on the experimental conditions. Fig. 4 shows the changes in the maximum linear strain or critical strain g 0 of egg white gels at 25 8C as a function of protein concentration. For all the ph values investigated, the maximum linear strain or limiting strain of egg white gels showed a power-law relationship with protein concentration, i.e. g 0, C m ; with m the power-law exponent. Negative m values were obtained for all gels (Table 1). As for the exponent n ðg 0, C n Þ; m was also strongly sensitive to the gel ph. The lowest value of the exponent m was obtained for gels prepared at ph 11.0 ðm ¼ 23:3Þ whereas the highest value was observed for gels prepared at ph 5 ðm ¼ 20:7Þ: These m values agreed well with power-law exponent values reported for other globular protein gels. Wu and Morbidelli (2001) reported that m values covered a wide range of both negative and positive values between and 5.3, depending on the nature of protein gels and how they were prepared. The lowest m value (2 3.4) was obtained for b-lactoglobulin transparent gels prepared at ph 7 without salts, whereas, the highest m value (5.3) was obtained for bovine serum albumin turbid gels prepared in acetate buffer at ph 5.1, and in presence of 0.1 M of NaCl (6 7) Fractal analysis The fractal dimension d f of protein aggregates in egg white gels at 25 8C were estimated from power-law exponents n and m using scaling models of Shih et al. (1990) and Wu and Morbidelli (2001). These d f values are given in Table 1. It can be seen from data in this table that for all ph values, the estimated m values are negative, thus indicating that the gel system can be considered to be in the strong-link regime according to the model of Shih et al. (1990). Therefore, using Eqs. (2) and (3), one can easily estimate d f and x: Fractal dimension d f values, ranging from 1.9 to 2.4, and x values varying between 20.4 and 0.8, are obtained for egg white gels prepared in the ph range 3 11 (Table 1). These d f values agreed well with those of other proteins gels, where fractal dimension values between 1.5 and 2.8 have been reported (Bremer et al., 1990, 1989; Hagiwara et al., 1997, 1998; Ikeda et al., 1999; Marangoni et al., 2000; Verheul et al., 1998; Vreeker et al., 1992). The strong-link model of Shih et al. (1990) gave, however, unrealistically negative values for x; particularly at ph 5, where a value of 20.4 was obtained (Table 1) which suggested the inapplicability of such model in this case. Reasonable values for fractal dimension x of the aggregate backbone in colloidal gels are comprised between 1 and d f ; and usually lie in the range (17). Recently, Wu and Morbidelli (2001) reproduced experimental data of Hagiwara et al. (1997) and Ikeda et al. (1999) and applied the strong-link model of Shih et al. on these data. They found negative x values and concluded that the failure of the application of the strong-link model ða ¼ 0Þ to these data was because the inter- and intra-floc links in such gel systems are comparable in strength ð0, a, 1Þ: Wu and Morbidelli (2001) evaluated the fractal dimension and a values for such gel systems using their model and found that indeed a values were between 0.5 and 0.7, which suggested that these gels are in the transition regime. We followed a similar approach. Fractal dimension d f values were evaluated for different egg white gels using the model of Wu and Morbidelli (see Eqs. (6) (8)) and results are given in Table 1. The d f values are identical to those estimated using the strong-link model of Shih et al. (1990). Similarly, Wu and Morbidelli (2001) applied their model to several experimental data that they reproduced from the literature and compared d f values calculated using their model to those estimated using the model of Shih et al. (1990). In most cases, they found that the two models gave the same d f

7 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) Fig. 4. Double-logarithmic plots of critical strain g 0 of egg white gels as function of protein concentration at various ph values (test conditions: T ¼ 25 8C; frequency ¼ 1 Hz). values. However, unlike the Shih et al. model that suggested that all our gels were in the strong-link regime (e.g. negative m values), the Wu and Morbidelli model showed that all our gels are in the transition regime. The model of Wu and Morbidelli (2001) allows to derive two additional parameters, b and a: The parameter a indicates the relative importance of the elastic contributions of both inter- and intra-floc links, and can be determined based on the values of b and x (see Eq. (7)). We estimated parameter a values for the two x values (1 and 1.3) that are commonly considered to be a good approximation of the fractal dimension of the backbone of colloidal aggregates. In both cases (Table 1), the a values were between 0 and 1, which indicated that, over the ph region investigated, egg white gels are in the transition regime. However, it should be mentioned that at ph 11 a (0.05) is very close to zero especially for x ¼ 1; which suggest that at the gel tends to the strong-link regime. According to Wu and Morbidelli (2001), the parameter a allows identifying the different gelation regimes (i.e. weak-link, transition, strong-link) prevailing in the system. Strong-link and weak-link regimes, as in the theory of Shih et al. (1990), are found for a ¼ 0 and a ¼ 1; respectively, whereas transition regimes are obtained for a values in the range 0, a, 1: The effects of ph on the fractal dimension d f value of protein aggregates in egg white heat-induced gels are shown in Table 1. ph-dependent fractal dimensions were found with d f values in the range of for gels prepared in

8 322 M.M. Ould Eleya et al. / Food Hydrocolloids 18 (2004) the ph region 3 7 and in the range of for protein gels formed under basic ph conditions (ph 9 and 11). As the case for other globular proteins such as BSA and b- lactoglobulin, this ph dependence of fractal dimension of egg white arises from the nature of the gel structure formed at different ph values. It is well known that at low ionic strength, ovalbumin, the main component of egg white, forms transparent or slightly turbid gels below and its isoelectric point (pi 4.5) and highly turbid particulate gels around its pi. The pi of egg white protein system is higher than that of ovalbumin since egg white contains other proteins such as ovatransferrin (pi 6.1) and lysozyme (pi 10.7) with higher pis. These d f values differ from fractal dimensions estimated using three-dimensional computer simulations. Results of three-dimensional simulations revealed that fractal aggregates formed under a rapid or a diffusion-limited process exhibited a fractal dimension of 1.8 whereas, those formed under a slow or a reaction-limited process showed a fractal dimensionality of 2.1 (Meakin & Jullien, 1988). This has also been confirmed experimentally for various aggregating colloidal particles of latex, silica, and gold (Aubert & Cannell, 1986; Lin et al., 1989; Weitz & Oliveira, 1984). Fractal aggregates prepared from biological macromolecules such as proteins exhibited in general higher d f values than those predicted by simulations due the nature of protein aggregates and the presence of some aggregate restructuring (Aubert & Cannell, 1986; Lin et al., 1990). This could explain why our d f values were slightly higher than those of model aggregate systems. Further, this also suggests that one should be cautious in drawing conclusions regarding the gelation process, e.g. diffusion-limited versus reactionlimited, based solely on the value of fractal dimension. 5. Conclusions The scaling of rheological properties and fractal analysis of heat-induced protein gels, prepared using egg white powder, were investigated for a range of concentrations and ph values (3 11). Both storage modulus, G 0 and critical strain, g 0 of protein gels exhibited ph-dependent power-law relationships with protein concentration. Fractal dimension, d f of aggregates in heat-induced protein gels were estimated based on these power-law exponent values using scaling models of Shih et al. (1990) and Wu and Morbidelli (2001). The two scaling models gave identical d f values. The Shih et al. model gave, however, unrealistically negative values for x; particularly at ph 5 suggesting the inapplicability of the strong-link regime in this case. The use of Wu and Morbidelli on our experimental data showed that the parameter a took values between 0.05 and 0.5, revealing that in these cases inter- and intra-floc links are comparable and thus all our gels can be considered to be in the transition regime. The fractal dimension d f was found to be strongly ph-dependent. Low d f values ( ) were obtained for gels prepared at acidic and neutral phs (3, 5, 7) whereas higher d f values ( ) were found for gels formed under basic ph conditions. These d f values lied well within the range of fractal dimension values ( ) of protein gels reported in the literature. They slightly differed, however, from fractal dimension values determined numerically or experimentally for diffusion-limited and reaction-limited cluster cluster aggregation processes, which rendered us cautious in making assumptions regarding the gelation process. 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