A Model Relating Structure of Colloidal Gels to Their Elastic Properties

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A Model Relating Structure of Colloidal Gels to Their Elastic Properties Hua Wu and Massimo Morbidelli* Laboratorium fuer Technische Chemie ETH Zentrum, CAB C40, Universitaetstrasse 6 CH-8092 Zurich, Switzerland Received August 4, 2000. In Final Form: November 2, 2000 A scaling model relating the microscopic structure parameters of colloidal gels to their macroscopic elastic properties is proposed. This model allows one to estimate the fractal dimension in any gelation regime, purely based on rheological properties (storage modulus and limit of linearity), without resorting to other types of measurements. In particular, an appropriate effective microscopic elastic constant is introduced to account for the mutual elastic contributions of both inter- and intrafloc links. This leads to a new parameter, R ∈ [0,1], which indicates the relative importance of these two contributions. This parameter can be estimated from the rheological data, and its value determines the prevailing gel regime. The model is applied to 12 gelation systems reported in the literature and identifies three gelation regimes: strong-link (R ) 0), weak-link (R ) 1), and transition (0 < R < 1). For the first two regimes, the new model correctly reproduces the results from the model developed by Shih et al.1 However, for the transition regime, corresponding to R ∈ (0.4,0.7), the new model is the only one providing a physically sound interpretation of the experimental results.

I. Introduction Colloidal systems are common in various fields, such as physics, chemistry, biology, geology, and so forth, where small particles under certain conditions can come together to form large complex structures, often referred to as aggregates, clusters, or flocs. When the concentration of the small particles is large, the aggregating system may form a gelsa structure that gives new properties to the system, for example, scattering, conductivity, and elasticity. There is a significant interest in understanding the structure of gels in order to control it for engineering applications. The structures of gels are highly disordered, but there is much experimental evidence that in certain length scales they are often self-similar and can be described in terms of fractal geometry.2 Thus, the most suitable method to quantify the structure of gels is fractal analysis. There are several experimental techniques available to analyze fractal structures in aggregates or gels. These measure physical quantities related to the distribution of mass in space and can be based on rheology, microscopy, scattering, and so forth. Scattering techniques are probably the most reliable, but they are mostly used in dilute systems with particle volume fractions substantially smaller than 1%. At larger particle volume fractions, the techniques based on rheological measurements using a rheometer are better suited to characterize the structure of gels.1-8 * To whom correspondence should be addressed. E-mail: [email protected]. Tel: 0041-1-6323034. (1) Shih, W. H.; Shih, W. Y.; Kim, S. I.; Lin, J.; Aksay, I. A. Phys. Rev. A 1990, 42, 4772. (2) Mandelbrot, B. The Fractal Geometry of Nature; W. H. Freeman: New York, 1982. (3) Hagiwara, T.; Kumagai, H.; Matsunaga, T. J. Agric. Food Chem. 1997, 45, 3807. (4) Hagiwara, T.; Kumagai, H.; Nakamura, K. Food Hydrocolloids 1998, 12, 29. (5) Ikeda, S.; Foegeding, E. A.; Hagiwara, T. Langmuir 1999, 15, 8584. (6) Marangoni, A. G.; Rousseau, D. J. Am. Oil Chem. Soc. 1996, 73, 991. (7) Narine, S. S.; Marangoni, A. G. Food Res. Int. 1999, 32, 227. (8) Gisler, T.; Ball, R. C.; Weitz, D. A. Phys. Rev. Lett. 1999, 82, 1064.

To determine the structure of a gel from rheological measurements, one needs a model, the so-called scaling theory that relates the structure of gels to the rheological properties. Brown and Ball9-11 were the first to develop a scaling theory to relate the structure of gels to their elastic properties. Shih et al.1 extended the results of Brown and Ball, including the work of Buscall et al.12 and Kantor and Webman,13 and developed a scaling model by defining two separate regimes: the strong-link regime at low particle concentrations and the weak-link regime at high particle concentrations. In this model, it is assumed that the structure of gels is constituted by fractal flocs, which during gelation aggregate with each other. The elastic properties of a floc are dominated by its effective backbone, which can be approximated as a linear chain of springs. In the strong-link regime, where the interfloc links are stronger than the intrafloc links, the macroscopic elasticity of the gel is given by that of intralinks. In this case, it is possible to derive two relationships relating the fractal dimension of the flocs (df) and the fractal dimension of the backbones (x) to the storage modulus (G′) and the limit of linearity (γ0):

G′ ∝ φA

(I-1)

γ0 ∝ φ B

(I-2)

where φ is the volume fraction and the exponents, A and B, have the form

A ) (d + x)/(d - df)

(I-3)

B ) -(1 + x)/(d - df)

(I-4)

(9) Brown, W. D.; Ball, R. C. J. Phys. A 1985, 18, L517. (10) Brown, W. D. Ph.D. Dissertation, University of Cambridge, Cambridge, U.K., 1987. (11) Ball, R. C. Physica D 1989, 38, 13. (12) Buscall, R.; Mills, P. D. A.; Goodwin, J. W.; Lawson, D. W. J. Chem. Soc., Faraday Trans. 1 1988, 84, 4249. (13) Kantor, Y.; Webman, I. Phys. Rev. Lett. 1984, 52, 1891.

10.1021/la001121f CCC: $20.00 © 2001 American Chemical Society Published on Web 01/26/2001

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where d is the Euclidean dimension of the system. Thus, from experimentally measured values of G′ and γ0 as a function of φ, one can estimate A and B and then from eqs I-3 and I-4 the values of both df and x. In the weak-link regime, where the flocs are more rigid than the interfloc links, the elasticity of the interfloc links determines the elasticity of the gel. In this case, Shih et al.1 derived the same expressions, eqs I-1 and I-2, for G′ and γ0, respectively, but with different expressions for the exponents:

A ) B ) 1/(d - df)

(I-5)

Comparing eq I-5 to eq I-4, one can see that in the weaklink regime the limit of linearity increases with increasing volume fraction, whereas in the strong-link regime it decreases. Both situations have been verified by experiments using various gelation systems.1,3-5,6,7 It is evident that the strong- and weak-link regimes described by Shih et al.1 represent only two extreme situations. The transition from one to the other with the change in particle concentration must be continuous, leading to intermediate situations where both inter- and intrafloc links contribute to the gel’s overall elasticity. Therefore, we need further developments of the scaling model that can deal with these more general situations. To this aim, Mellema et al.14 have recently derived a general relationship between G′ and φ, which has again the form of eq I-1 but with the exponent A defined as follows:

A ) a/(d - df)

(I-6)

where the parameter a ) 2 + δ + 1, with δ and  being two scaling exponents, defines the type of gel structure and ranges from 1 to (3 + x), corresponding to eqs I-5 and I-3, respectively. Thus, it covers all the situations from the weak- to the strong-link regime. This equation provides only one relation between the two structure parameters (a and df) and the gel elastic properties. Therefore, this has to be coupled with another technique (e.g., confocal laser scanning microscopy) to estimate the fractal dimension (df). In this work, based again on the physical model of Shih et al.,1 we develop a new scaling model that relates the structure parameters of the gel to its elastic properties and is valid not only for the strong- and the weak-link regimes but also for the intermediate regimes. The derived model is supported by various experimental evidences reported in the literature. II. Theory Let us consider the structure of a colloidal gel (network) as a collection of flocs. The flocs are fractal objects with average size ξ and fractal dimension df. If the particle volume fraction inside the flocs is assumed to be equal to the overall volume fraction φ, then the scaling relation between ξ and φ is given by15

ξ ∝ φ1(df-d)

(1)

where d is the Euclidean dimension of the system. It is well-known that the size of the flocs that constitute a gel is in the range from one to hundreds of microns.5,7 Thus, (14) Mellema, M.; van Vliet, T.; van Opheusden, J. H. J. In Proceedings of Second International Symposium on Food Rheology and Structure; ETH: Zurich, 2000. (15) Dietler, G.; Aubert, C.; Cannel, D. S.; Wiltzinus, P. Phys. Rev. Lett. 1986, 57, 3117.

the flocs can be considered as microscopic elements that built the macroscopic gel. On the basis of the above considerations, let us classify a macroscopic gel system into two levels of structure: intramicrostructure characterizing structure within a floc and intermicrostructure characterizing the collection of flocs. Thus, the macroscopic elasticity of a gel must be related to both intra- and intermicroscopic elasticity. To describe such a relation, we define an effective microscopic elastic constant, Keff, which includes effects of both intraand intermicroscopic elasticity. When the two elasticities are comparable, we can regard the overall microscopic elasticity as given by the two elasticities in series as follows:

1 1 1 ) + Keff Kξ Kl

(2)

where Kξ and Kl are the intra- and intermicroscopic elastic constants, respectively. It is evident that Keff approaches Kξ when Kl . Kξ and Kl when Kξ . Kl. The macroscopic elastic constant of the gel, K, can be written in terms of Keff as

(

1 1 Ld-2 ξd-2 ∝ ) ξd-2 + K Keff Kξ Kl

)

(3)

where L is the size of the macroscopic gel. This expression can be rewritten as follows:

K∝

(Lξ )

d-2

Kξ

1 + (Kξ/Kl)

(4)

To best utilize this relation, it is convenient to convert the denominator of the last term in eq 4 into a power law relationship. For this, we note that in a given range of Kξ/Kl values the following relation applies:

()

Kl 1 = 1 + Kξ/Kl Kξ

R

(5)

where R is a constant in the range [0, 1], which depends on the given range of Kξ/Kl. Substituting eq 5 into eq 4, we have

K∝

() ( ) L ξ

d-2

Kξ

Kl Kξ

R

(6)

Note from eq 5 that when Kl . Kξ, R ) 0 and we have K ∝ Kξ, which corresponds to the strong-link model of Shih et al.1 in which the gel elasticity is dominated by the intramicroscopic elasticity Kξ. On the other hand, when Kl , Kξ we see from eq 5 that R ) 1 and then from eq 6 that K ∝ Kl, which corresponds to the weak-link regime. Intermediate regimes are obtained for values of a in the range 0 < R < 1. The next step is to find the scaling relations between the intra- and intermicroscopic elastic constants, Kξ and Kl, and the microstructure parameters, ξ and df. The elastic constant, Kl, refers to the interaction among the flocs, and therefore it is expected to be independent of the microstructure parameters of the flocs. On the other hand, to compute the elastic constant, Kξ, we can follow Shih et al.1 and consider the elastic backbone as a linear chain of springs. Then, Kξ can be computed using the KantorWebman relation13 as follows:

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Kξ ∝

1 Ns Rc2

Wu and Morbidelli

(7)

where Ns is the number of springs in the chain and Rc is the radius of gyration of the projection of the nodes of the chain, which exhibits the following linear scaling relations:1

Ns ∝ ξx

()

Fξ ) Keff (∆L)ξ ) Kξ

( )

Kl R ∆L 1 ∆L ∝ Kξ L/ξ ξ(2+x)(1-R)-1 L

(16)

Assuming that a singly connected bond breaks at a given critical force (e.g., Fξ ) 1), we obtain the limit of linearity, γ0, as follows:

γ0 )

(8)

∆L ∝ ξ(2+x)(1-R)-1 ) ξβ-d+1 ∝ φB L

(17)

where

and

Rc2 ∝ ξ2

(9)

The exponent x represents the backbone fractal dimension or tortuosity of the network, whose value for a colloidal gel is in the range of 1-1.3. With eqs 8 and 9, eq 7 becomes

Kξ ∝ 1/ξ2+x

(10)

Substituting eq 10 in eq 6 and considering that Kl is independent of ξ, we have that

K ∝ 1/ξβ

(11)

β ) (d - 2) + (2 + x)(1 - R)

(12)

where

is a constant, which for a three-dimensional system (d ) 3) ranges between 1, for R ) 1, and 3 + x, for R ) 0. Finally, substituting eq 1 in eq 11 we obtain the relation between the macroscopic elastic constant and the particle volume fraction in terms of the microstructure parameters:

K ∝ φA

(13)

A ) β/(d - df)

(14)

where

Note that with the above procedure we have obtained a relation for the exponent A similar to that of Mellema et al.,14 where the coefficient a is replaced by β, but the expression of a is different from eq 12, although both range between 1 and (3 + x). As mentioned above, from the K values obtained experimentally as a function of the particle volume fraction φ, one can estimate the exponent, A, and therefore a relation between the fractal dimension, df, and β. To estimate both parameters, we need a second relation, involving a different rheological property. In the following, we extend the approach above to derive a scaling model for the limit of linearity, γ0, following an approach originally proposed by Shih et al.1 The limit of linearity is defined as the situation where the weakest bonds break and the linear elastic behavior vanishes. These are generally the singly connected bonds. Suppose that we impose a macroscopic deformation on the system. ∆L, the corresponding deformation of a floc, is given by

(∆L)ξ ∝

∆L L/ξ

(15)

Considering the effective microscopic elastic constant Keff defined by eq 2 and using eqs 5, 10, and 15, we can compute the elastic force acting on a floc:

B ) (d - β - 1)/(d - df)

(18)

Thus, from the experimental values of γ0 as a function of the particle volume fraction φ, one can estimate the exponent B, which together with the value of A in eq 13 allows us to determine the values of both β and df. It is worth mentioning in this context that care must be taken when determining experimentally the limit of linearity, as discussed for example by Hagiwara et al.3 and Buscall et al.16 Equation 18 reproduces correctly the limiting conditions of eqs I-4 and I-5 derived by Shih et al.1 In particular, in the case of strong-link gels R ) 0 and from eqs 12 and 18 we get B ) -(1 + x)/(d - df), whereas for weak-link gels R ) 1 and then B ) 1/(d - df). III. Results and Discussion To validate the rheological model derived above for the storage modulus and the limit of linearity, we use various experimental data in the literature.1,3-5,17 The data are classified in three classes: weak-link gels, strong-link gels, and transition gels. For each set of data, the results of the analysis reported in the original papers following the Shih et al. model are first summarized; the rheological model developed above is then used, and the obtained results are compared and discussed. 3.1. Weak-Link Regime. Hagiwara et al.4 recently investigated the microstructure of heat-induced bovine serum albumin (BSA) protein gels by measuring their macroscopic elastic properties. In particular, they measured both the elasticity (E) and the limit of linearity (γ0) of the gels as a function of the BSA concentration (Cp). The results showed that the γ0 value increases with the BSA concentration, which, based on the Shih et al. model, indicates a weak-link gel. Accordingly, the fractal dimension can be obtained directly from the slope of the E versus Cp plot in a double logarithmic plane using eq I-5. Note that in this procedure the experimental results of the limit of linearity are used only qualitatively, that is, to identify the weak-link gel regime. They are not used to estimate the fractal dimension. In particular, three gels have been studied by Hagiwara et al.,4 which are denoted by numbers 1, 2, and 3 in Table 1. System 1 is a 50 mol/m3 acetate buffered BSA solution (pH 5.1, with 100 mol/m3 NaCl), system 2 is a 50 mol/m3 HEPES buffered BSA solution (pH 7.0, with 30 mol/m3 CaCl2), and system 3 is a 50 mol/m3 HEPES buffered BSA solution (pH 7.0, with 5 mol/m3 CaCl2). In the original paper, the values of the slopes, A, of the log-log plot of E versus Cp for the three systems are reported as shown in column 3 of Table 1. For the log-log plot of γ0 versus (16) Buscall, R.; McGowan, I. J.; Mills, P. D. A.; Stewart, R. F.; Sutton, D.; White, L. R.; Yates, G. E. J. Non-Newtonian Fluid Mech. 1987, 24, 183. (17) Vreeker, R.; Hockstra, L. L.; den Boer, D. C.; Agterof, W. G. M. Food Hydrocolloids 1992, 6, 423.

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Table 1. Experimental Rheological Data of Various Gelation Systems and Evaluated Microscopic Structural Parameters gel no.

colloidal system

Aa

Bb

dfc

1 2 3 4 5 6 7 8 9 10 11 12

BSA BSA BSA WPI Catapal Dispal BSA β-LG WPI WPI WPI β-LG

5.65 5.54 2.58 4.10 4.1 4.2 4.28 5.00 5.38 2.72 3.37 3.24

5.35 5.01 2.91 3.60 -2.1 -2.3 -2.34 -3.43 -0.839 -0.159 -0.86 0.357

2.82 2.82 2.61 2.74 2.0 1.95 2.00 2.14 2.2 1.5 1.8 2.69

xc

1.1 1.4 1.3 1.3 1.3 1.0 1.0

dfd

βd

Rd at x ) 1.3

note

reference

2.82 2.81 2.64 2.76 2.0 1.95 1.97 1.73 2.56 2.22 2.20 2.44

1.03 1.05 0.94 1.06 4.1 4.4 4.41 6.37 2.37 2.12 2.69 1.80

0.99 0.98 1.02 0.98 -0.06 -0.03 -0.03 -0.63 0.58 0.66 0.49 0.76

weak-link gel weak-link gel weak-link gel weak-link gel strong-link gel strong-link gel strong-link gel uncertain data transition gel transition gel transition gel transition gel

Hagiwara et al. (ref 4) Hagiwara et al. (ref 4) Hagiwara et al. (ref 4) Vreeker et al. (ref 17) Shih et al. (ref 1) Shih et al. (ref 1) Hagiwara et al. (ref 3) Hagiwara et al. (ref 3) Ikeda et al. (ref 5) Ikeda et al. (ref 5) Ikeda et al. (ref 5) Hagiwara et al. (ref 3)

a Slope value from log-log plot of the storage modulus vs colloids concentration. b Slope value from log-log plot of the limit of linearity vs colloids concentration. c Values reported by the original authors, based on the model of Shih et al. (ref 1). d Values predicted based on the present model.

Cp, only raw data have been reported. These are reproduced in parts a, b, and c of Figure 1 for the above three systems, respectively. From the slopes, the corresponding B values have been estimated and reported in column 4 of Table 1. As mentioned above, only the values of parameter A have been used in the original paper to estimate the fractal dimension, leading to the values reported in column 5 of Table 1. On the other hand, it can be observed that the estimated values of A and B (columns 3 and 4) are rather similar, thus supporting the assumption of the weak-link gel regime. Let us now use the estimated A and B values in the frame of the rheological model developed above. Using eqs 14 and 18, we obtain the values of df and β, reported in columns 7 and 8, respectively. In addition, using the β values from eq 12 and assuming a reasonable value of x ) 1.3 we obtain the values for the R parameter shown in column 9. These are very close to one, thus indicating a weak-link gel regime. The consequence is that the fractal dimension values estimated by the Shih et al. model (column 5) and by the rheological model developed above (column 7) are indeed quite close. A similar conclusion is obtained for the gelation system (number 4 in Table 1) of whey protein at pH 5.4 with a protein weight fraction larger than 3%, studied by Vreeker et al.17 The values of the exponents A and B, estimated by the authors from the slope of log-log plots of the storage modulus and the yield stress versus protein weight fraction, are 4.1 and 3.6, respectively. Assuming that the yield stress is proportional to the limit of linearity, from eqs 14 and 18, we obtain df ) 2.74 and β ) 1.06. Because the R value calculated from such a β value, assuming x ) 1.3, is very close to one (i.e., 0.98), it is expected that the fractal dimension evaluated from the present model must be very close to that from the weak-link model of Shih et al.,1 which is in fact equal to 2.76. In the same paper, the fractal dimension was also measured by light scattering, for whey protein aggregates at much lower concentration, that is, 0.1% instead of more than 3%. The obtained df value of 2.2 is significantly lower than the value of 2.74 obtained in our analysis. We believe that this is due to the increase of fractal dimension with particle volume fraction as discussed for example by Mellema et al.18 and Dickinson19 using suitable simulations. For the same rheological data, a smaller fractal dimension, equal to 2.3, which is close to that from light scattering, was estimated by Vreeker et al.17 using the (18) Mellema, M.; van Opheusden, J. H. J.; van Vliet, T. J. Chem. Phys. 1999, 111, 6129. (19) Dickinson, E. J. Colloid Interface Sci. 2000, 225, 2.

strong-link model. However, such an argument is not consistent with the B value, which clearly indicates that the system is actually in the weak-link regime. 3.2. Strong-Link Regime. Shih et al.1 investigated experimentally two gels constituted of aqueous suspensions of two boehmite alumina powders, Catapal and Dispal, with volume fractions greater than 3%. From the measured storage modulus and limit of linearity as a function of the particle volume fraction, they estimated the exponents A and B to be 4.1 and -2.1 for Catapal gels and 4.2 and -2.3 for Dispal gels. Because both systems exhibit negative B values, they were classified as stronglink gels. In this case, the expressions of A and B from the Shih et al. model (eqs I-3 and I-4) contain two unknown parameters, df and x. This is a close problem, and the fractal dimensions of both the aggregates (df) and the backbone (x) can be estimated as reported in columns 5 and 6 of Table 1. Now, let us use the values of A and B obtained by Shih et al. and apply the new expressions of A and B [eqs 14 and 18] to estimate the values of df and β, reported in columns 7 and 8 of Table 1. From the obtained β value, using eq 12 and assuming the fractal dimension of backbones x ) 1.3, we estimate values of R very close to zero, that is, R ) -0.06 and -0.03 for Catapal and Dispal, respectively. This confirms that the two gels are in the strong-link regime. Consequently, the df values estimated through this analysis coincide with those originally estimated by Shih et al.1 To further verify the proposed model in the strong-link regime, we introduce two other gels studied by Hagiwara et al.,3 which are heat-induced gels of BSA and β-lactoglobulin proteins without addition of salts, corresponding to systems 7 and 8 in Table 1. In the original paper, the values of the limit of linearity as a function of protein concentration reported in the log-log plane exhibit a negative slope, thus indicating a strong-link regime according to the Shih et al. model. However, in this case the slope, that is, the value of parameter B, was not estimated. Only the slope of the log-log plot of elasticity versus protein concentration, that is, the value of parameter A, was in fact used to estimate df, while taking for x an arbitrary value between 1 and 1.3. To apply the proposed model, we need the B value. For this, the experimental data from the original paper of Hagiwara et al. have been reproduced in parts a and b of Figure 2 for systems 7 and 8, respectively, and from these the B values, -2.34 and -3.43, have been estimated. Note that for system 8 the value of γ0 at Cp ) 155 kg/m3 in Figure 2b has been ignored because with it the estimated

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Wu and Morbidelli

Figure 2. Values of limit of linearity, γ0, of heat-induced protein gels as a function of protein concentration, Cp: (a) bovine serum albumin protein gels and (b) β-lactoglobulin protein gels. The data are from Hagiwara et al. (ref 3).

Figure 1. Values of limit of linearity, γ0, of heat-induced BSA protein gels as a function of protein concentration, Cp, generated from (a) a 50 mol/m3 acetate buffered BSA solution (pH 5.1, with 100 mol/m3 NaCl), (b) a 50 mol/m3 HEPES buffered BSA solution (pH 7.0, with 30 mol/m3 CaCl2), and (c) a 50 mol/m3 HEPES buffered BSA solution (pH 7.0, with 5 mol/m3 CaCl2). The data are from Hagiwara et al. (ref 4).

B value would be too small, leading to an unrealistic value of the fractal dimension larger than 3. From the above B values together with the A values reported by Hagiwara et al., we have estimated df and β, as shown in Table 1. The df value for system 7 is very close to that reported by Hagiwara et al., whereas for system 8 the two values differ significantly. This is probably due to some scattering in the experimental data for system 8, particularly evident in Figure 2b. Assuming in fact the strong-link regime (i.e., R ) 0), the obtained β values in eq 12 would lead to x )

1.41 and 3.37 for systems 7 and 8, respectively. These are rather large values for x, which for system 8 exceed the range of [1, 1.3] calculated by Shih et al. for clustercluster aggregation systems. On the other hand, assuming x ) 1.3 one would estimate R ) -0.03 and -0.63, which for system 7 indicates that the system is in the stronglink regime, while it confirms the inconsistency of the data for system 8. 3.3. Transition Regime. Ikeda et al.5 investigated the effect of ionic strength on the fractal structure of heatinduced whey protein isolate (WPI) gels. For each given ionic strength or NaCl concentration, they measured both the storage modulus, G′, and the fracture strain, Y0, as a function of the WPI weight fraction, φ. The results obtained at NaCl concentrations equal to 25, 100, and 500 mol/m3 are reproduced in Figure 3. It is seen that both G′ and Y0 satisfy the power law relations with respect to φ, with values for the exponents A and B given in Table 1 (systems 9, 10, and 11). It can be seen that in all three cases, the estimated B values are negative, thus indicating a stronglink regime in the Shih et al. theory. However, using the corresponding relations (eqs I-3 and I-4) to estimate df and x, one obtains unrealistically negative values for x, that is, -0.723, -0.876, and -0.315 for systems 9, 10, and 11, respectively. To overcome this problem, Ikeda et al.,5 similarly to Hagiwara et al.,3 ignored the B value, chose arbitrarily the x value between 1.0 and 1.3, and used only the A value and eq I-3 to estimate df.

Elastic Properties of Colloidal Gels

Figure 3. Values of (a) storage modulus, G′, and (b) fracture strain, Y0, as a function of protein weight fraction, φ, for heatinduced whey protein isolate gels at three levels of ionic strength: (b) 25, (9) 100, and (2) 500 mol/m3 NaCl. The data are from Ikeda et al. (ref 5).

Figure 4. Values of the fractal dimension, df, of heat-induced whey protein isolate gels as a function of the ionic strength (NaCl concentration), predicted by (O) this model and (b) the Shih et al. strong-link model (ref 1).

The fractal dimensions, df, estimated by Ikeda et al.5 are shown in Figure 4 (filled circles) as a function of NaCl concentration. It is seen that for NaCl concentrations smaller than 100 mol/m3, the fractal dimension decreases

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as the NaCl concentration increases. This is reasonable, because for increasing ionic strengths the screening of the electrostatic repulsive forces increases, thus making the van der Waals attractive force more important, which moves the aggregation mechanism from reaction-limited to diffusion-limited, and reduces the fractal dimension. However, after reaching a minimum value at 100 mol/m3 NaCl concentration the fractal dimension starts to increase with the NaCl concentration, a behavior difficult to justify. From these results, Ikeda et al. questioned the applicability of the Shih et al. model to these systems. Now, let us apply the developed rheological model to the experimental results obtained by Ikeda et al.5 By use of the values of A and B and eqs 14 and 18, the values of df and β have been estimated and reported in Table 1. The values of the fractal dimension are compared in Figure 4 with those estimated by Ikeda et al. Note that only the experimental results of G′ and Y0 as a function of φ for the NaCl concentrations equal to 25, 100, and 500 mol/m3 have been reported in the original paper, so we cannot estimate the df values using the present model for the remaining experimental points in Figure 4. Nevertheless, some interesting conclusions can be drawn already from these three points. In particular, the df value estimated by Ikeda et al. at 100 mol/m3 NaCl is smaller than that at 500 mol/m3, whereas it is slightly larger when using the new model, thus eliminating the appearance of a minimum. Moreover, the trend of the obtained results indicates that for the NaCl concentration smaller than 100 mol/m3, the fractal dimension decreases rapidly as the NaCl concentration increases, and then it approaches an asymptotic value (about 2.2) at very high NaCl concentrations, which is now a realistic behavior. It should be noted that all the values of the fractal dimension estimated by the present model are higher than those reported by Ikeda et al. For example, at 25 mol/m3 NaCl the fractal dimension reported by Ikeda et al. is 2.2, whereas that based on the present model is 2.56. For protein gelation systems, it is not uncommon to find in the literature4,7 fractal dimension values substantially larger than 2.2. On the other hand, it is true that computer simulations20 indicated that aggregates formed in the reaction-limited regime are characterized by fractal dimensions equal to about 2.0-2.2, but these have been confirmed only by experiments under dilute conditions.17,21 As the particle volume fraction increases, it is reasonable that the fractal dimension also increases, as it must eventually approach 3 when the volume fraction becomes 1. In general, the fractal model can be applied only for particle volume fractions smaller than 20%. At larger volume fractions, the number of particles per cluster would in fact be so small that the use of a fractal model, and therefore the power law dependences given by eqs 13 and 17, can no longer be justified. For this, more advanced theoretical models would be needed. As a self-consistency test for the developed model, using the β values obtained above in eq 12 and assuming again x ) 1.3, we can calculate the R values, reported in column 9 of Table 1. For all three gels, R takes values between 0.5 and 0.7, thus indicating that in these cases inter- and intrafloc links are comparable, and then we are in the transition regime. This justifies the failure of the application of the strong-link model (i.e., R ) 0) to these systems, as found by Ikeda et al.5 (20) Meakin, P. Annu. Rev. Phys. Chem. 1988, 39, 237. (21) Gimel, J.-C.; Durand, D.; Nicolai, T. Macromolecules 1994, 27, 583.

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Figure 5. Values of limit of linearity, γ0, of heat-induced β-lactoglobulin protein gels with addition of 30 mol/m3 CaCl2 as a function of protein concentration, Cp. The data are from Hagiwara et al. (ref 3).

The last gel considered, number 12 in Table 1, studied by Hagiwara et al.,3 should also be classified in the transition regime. This is a heat-induced β-lactoglobulin protein gel with addition of 30 mol/m3 CaCl2. In the original paper, the value is reported for A but not for B. Therefore, we have reproduced their experimental data of the limit of linearity as a function of the protein concentration in Figure 5 and estimated from their slope the value B ) 0.357. Because B is positive, Hagiwara et al. used the weak-link model to estimate the fractal dimension and obtained df ) 2.69. Using the new model with the same values of A and B, we obtain a different fractal dimension, df ) 2.44, and β ) 1.80. This contradiction is only apparent and can be resolved by estimating using eq 12 and assuming x ) 1.3 the parameter R ) 0.76. This indicates that this gel belongs to the transition regime, and therefore the results based on the weak-link model are not correct. This is further supported by the observation that the estimated values of A and B are substantially different, in obvious disagreement with eq I-5 from the weak-link regime. IV. Conclusions A new scaling model based on the physical model of Shih et al.1 has been developed to relate the microscopic structure parameters of colloidal gels to their macroscopic elastic properties. In particular, we refer to the storage modulus and the limit of linearity as a function of particle

Wu and Morbidelli

volume fraction, which are both achievable through rheological measurements. The model, by introducing an appropriate effective microscopic elastic constant, accounts for the elastic contributions of both inter- and intrafloc links. It contains a new parameter, R ∈ [0, 1], which indicates the relative importance of these two contributions and allows one to identify the different types of gelation regimes prevailing in the system. For extreme values of this parameter, the new model correctly reproduces the two limiting regimes described by Shih et al.,1 that is, the strong-link regime for R ) 0 and the weak-link regime for R ) 1. Transition regimes are obtained for values of R in the range 0 < R < 1. The use of the developed model is twofold. (1) From rheological data relating to storage modulus and limit of linearity, one can estimate directly the fractal dimension, df, and an auxiliary parameter, β. Therefore, one can estimate the fractal dimension in any gelation regime, purely based on rheological properties, without resorting to other types of measurements as in previous approaches. (2) In addition, using the estimated value of the auxiliary parameter, β ) (d - 2) + (2 + x)(1 - R), and assuming the backbone fractal dimension x in the range [1, 1.3], one can compute the corresponding R value and identify the gelation regime prevailing in the system. The developed rheological model has been applied to several experimental studies reported in the literature, and the results are summarized in Table 1. These systems can be well classified into the three (strong-link, weaklink, and transition) regimes. For values of x in the range [1, 1.3], the estimated R values are in the range [0.98, 1.06], that is, very close to unity, for the weak-link regime and in the range [-0.136, 0.06], that is, very close to zero, for the strong-link regime. In both situations, the present and the Shih et al. models give the same values of the fractal dimension. For the transition regime corresponding to R ∈ (0.4,0.7), however, only the developed model provides physically consistent results. Finally, from all the results summarized in Table 1 one may conclude that the fractal dimension increases as the gel changes from the strong-link type (df ) 1.7-2.0) to the transition type (df ) 2.2-2.5) to the weak-link type (df ) 2.6-2.8). This is consistent with the fact that the transition from strong-link to weak-link gel occurs as the particle volume fraction increases. Acknowledgment. Financial support from NSF and useful discussions with Dr. Jan Sefcik are gratefully acknowledged. LA001121F