Langmuir 2001, 17, 5821-5824
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Isotropic Force Percolation in Protein Gels Erik van der Linden* and Leonard M. C. Sagis Food Physics Group, Department of Agrotechnology and Nutritional Sciences, Bomenweg 2, 6703 HD Wageningen, The Netherlands Received May 10, 2001. In Final Form: June 28, 2001 We have determined elasticity data of various ovalbumin gels as a function of concentration of ovalbumin and analyzed these data in terms of a percolation model. In addition, we have analyzed elasticity data of a multitude of other protein gels. We find that all data can be described by one percolation model. The critical exponent is 1.79 ( 0.25, in accordance with isotropic percolation. The value of the critical percolation threshold varies by an order of magnitude. The findings suggest that the effects of external fields on gelation can be captured by the effects of these external fields on the critical percolation threshold.
Introduction Generic understanding on the elasticity of gels finds applicability in a wide range of industrial areas.1 During the past few decades, a considerable amount of work has been undertaken to arrive at a generic description of gel elasticity. Much of this effort has been based on percolation theories2 and on the application of fractal theory.3 The work based on percolation theories recognizes the existence of a gelation threshold concentration, depending on the strength of interaction between the monomers that form the network4 and on their morphology.4,5 The elasticity, G′, is in this case written as a power law according to G′∼ (c - cp)t, where c is the concentration of monomers in the system, cp is the critical percolation threshold concentration, and t is a scaling exponent, which depends on the type of Hamiltonian that describes the network elasticity. In contrast, the fractal models assume cp ) 0 and predict a power law behavior written as G′ ∼ cw, where the exponent w is related to the fractal dimension of so-called flocs that form the network.6-8 The fractal dimension depends on the interaction between the monomers. Recently, it was reported that the fractal models could not be used in finding a generic explanation for the temperature and concentration dependence of the elasticity of alumina particle gels.8 This lack of general applicability also followed from work by Grant and Russel on silica particle gels.9 In their case, the use of a fractal model implied a structural change as a function of temperature which was contradicted by the experimental observation of the structure, using light scattering. In contrast, when using a percolation model, the temperature dependence could be incorporated in cp. As a consequence of this percolation approach, both the prefactor in the power law (characterizing the elasticity of the interparticle (1) Keller, A. Faraday Discuss. 1995, 101, 1-49. (2) Stauffer, D.; Coniglio, A.; Adam, M. Adv. Polym. Sci. 1982, 44, 103-158. (3) Shih, W. H.; Shih, W. Y.; Kim, S.; Liu, J.; Aksay, I. A. Phys. Rev. A 1990, 42, 4772-4779. (4) Safran, S. A.; Webman, I.; Grest, G. S. Phys. Rev. A 1985, 32, 506-511. (5) Haan, S. W.; Zwanzig, R. J. Phys. A: Math. Gen. 1977, 10, 15471555. (6) Bremer, L. G. B.; Bijsterbosch, B. H.; Schrijvers, R.; van Vliet, T.; Walstra, P. Colloids Surf. 1990, 51, 159-170. (7) Buscall, R.; Mills, P. D. A.; Goodwin, J. W.; Lawson, D. W. J. Chem. Soc., Faraday Trans. 1 1988, 84, 4249-4260. (8) Yanez, J. A.; Laarz, E.; Bergstro¨m, L. J. Colloid Interface Sci. 1999, 209, 162-172. (9) Grant, M. C.; Russel, W. B. Phys. Rev. E 1993, 47, 2606-2614.
bonds) and the scaling exponent t (characterizing the structure) turned out to be independent of temperature, which is in accordance with experiment. Hence, the percolation model seems to offer a higher level of generality than fractal models. To further test the general applicability of percolation models, we have determined elasticity of ovalbumin gels as a function of pH and protein concentration. On the basis of data obtained from the literature, we also examined whether the percolation model is valid for other protein systems. Protein gels are excellent candidates for this study since the critical percolation threshold can differ by orders of magnitude, depending on the type of protein, solvent characteristics, and temperature. These differences are independent of the total protein concentration, thereby allowing a distinction between threshold concentration and total concentration. Furthermore, a large variety of self-assembling structures is possible, ranging from thin fibers (e.g., actin,10 soy protein,11 and β-lactoglobulin far from its isoelectric point12) to particle gels (e.g., whey protein near its isoelectric point13). This large variety of structures in principle allows testing the predictions of the dependence of the scaling exponent on the mesostructure.14-16 Percolation Models One of the first theoretical models for the elasticity of a gel, based on the percolation concept, was developed by De Gennes.17 He proposed a critical exponent t equal to 11/ for dimensionality d ) 3.17 This scaling relation is 6 based on an analogy between the electrical conductivity and the elastic modulus, which is rigorous for a lattice model of gelation with an isotropic force between nearest neighbors on the lattice. De Gennes17 remarks that the validity of this analogy may be extended to the situation where the connection between the sites that form the network is not through single bonds but through flexible chains. (10) MacKintosh, F. C.; Kas, J.; Janmey, P. A. Phys. Rev. Lett. 1995, 75, 4425-4428. (11) Nakamura, T.; Utsumi, S.; Mori, T. J. Agric. Food Chem. 1984, 32, 349-352. (12) Aymard, P.; Nicolai, T.; Durand, D. Macromolecules 1999, 32, 2542-2552. (13) Verheul, M. Aggregation and Gelation of Whey Proteins; Ph.D. Thesis, Twente University, The Netherlands, 1998. (14) Pandey, R. B.; Stauffer, D. Phys. Rev. Lett. 1983, 51, 527-529. (15) Mitescu, C. D.; Musolf, M. J. J. Phys., Lett. 1983, 44, L679L683. (16) Kantor, Y.; Webman, I. Phys. Rev. Lett. 1984, 52, 1891-1894. (17) De Gennes, P. G. J. Phys. Lett. 1976, 37, L-1-L-2.
10.1021/la010705u CCC: $20.00 © 2001 American Chemical Society Published on Web 08/24/2001
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More recent theoretical work,15 also using an isotropic force between nearest neighbors, resulted in a somewhat higher value for t, that is, t ) 2.06 ( 0.16. This value turned out to be in accordance with subsequent numerical work,14 giving t ) 2 ( 0.2, and seems in agreement with the Alexander-Orbach conjecture.18 If one considers not an isotropic force between neighboring sites but a central force, the numerical work of Feng and Sen19 strongly suggests that the presence of this type of force leads to a different universality class, with a value of t ) 4.4 ( 0.6 in the case of a threedimensional face-centered cubic lattice. In addition to the two models above (i.e., the isotropic force model and the central force model), Kantor and Webman16 have considered a model based on two contributions to the elastic energy, that is, a central force term and a bending energy term. This approach would apply to macroscopically inhomogeneous materials, consisting of rigid and soft regions. Their approach also leads to a universality class that is different from the isotropic force problem, with a lower bound of a critical exponent τ (equivalent to t) of 2.85 and an upper bound of 3.55. In the case of a homogeneous system where percolation theory does not apply, the exponent becomes 3.2 Experimental values for the exponent t have been reported for a variety of gels.8,9,20-22 Some gels, like gelatin,23 exhibit values for t in the range of 11/6,8,17,20,21 but others deviate from this result considerably.9,22 Experimental Section Ovalbumin was purchased from Sigma (Ref. A5503), grade V, with a purity of at least 98%. The protein was dissolved in an aqueous solution of set ionic strength (NaCl) and was stirred for half an hour. The pH was adjusted afterward by adding 0.1-5 M HCl, and the solution was stirred for another 1.5 h. The ovalbumin solutions were centrifuged at 20.000g for 30 min (temperature, (4 °C) to remove the insoluble materials. After centrifugation, the clear solutions were filtered through a 0.45 µm filter (FP030/2, Schleicher and Schuell). The final concentration of the ovalbumin solution was determined by a biuret assay. From this stock solution, protein solutions of different concentrations were prepared. Concentrations ovalbumin are expressed as % (w/w). The elastic modulus G′ was determined with a Bohlin VOR strain controlled rheometer, with a concentric cylinder geometry. The solutions were heated at 80 °C for 1 h and subsequently cooled to 20 °C for 2 h. After the cooling step, a strain sweep (frequency, 1 Hz; temperature, 20 °C; strain, 0.000 206-0.206) was performed. The value for G′ was obtained in the linear regime, where G′ is independent of the applied strain.
Results We first analyze our own data obtained for the ovalbumin systems. We assume the general scaling G′∼ (c cp)t, where t and cp still have to be determined. We first describe in detail the fitting procedure for the system at pH ) 2, 30 mM. Figure 1a shows the plot of (G′)1/t versus c, for t ranging between 1.7 and 4.5. These values for t are the lower and upper bounds as predicted by the various theories discussed above. (18) Alexander, S.; Orbach, R. J. Phys. Lett. 1982, 43, L-625L-631. (19) Feng, S.; Sen, P. N. Phys. Rev. Lett. 1984, 52, 216-219. (20) Takigawa, T.; Takahashi, M.; Urayama, K.; Masuda, T. Chem. Phys. Lett. 1992, 195, 509-512. (21) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1979, 41, 197-218. (22) Kanai, H.; Navarrete, R. C.; Macosko, C. W.; Scriven, L. E. Rheol. Acta 1992, 31, 333-344. (23) De Gennes, P. G. Scaling properties in polymer physics; Cornell University Press: Ithaca, NY, 1979.
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In the fitting procedure, we make use of the physical fact that extrapolation of (G′)1/t to zero should yield the same cp for all t > 0. The scaling assumption has some further implications. Namely, when t is close to the correct value, the points will be on a straight line. When t is larger than the correct value, the fit through the points will bend away from the straight line and will lie below it. In that case, the slope of that fit at the intercept with the horizontal axis will be infinite. When t is smaller than the correct value, the fit through the points will also bend away from the straight line but will now lie above it. In that case, the slope of the fit at the intercept with the horizontal axis will be zero. So, in the fitting procedure we use the fact that the curvature of the fit will change if different values for t are chosen, while the intercept cp will have to remain the same. Using the arguments above, we determine cp from Figure 1a. We use fits through the points that are closest to a straight line in determining an average intercept. We incorporated linear fits for t ) 1.7 and t ) 1.8 in Figure 1a. From Figure 1b, which enlarges part of Figure 1a, the bending away from the straight line (fit for t ) 1.7) is better visualized. We note that the linear fit for t ) 1.7 yields a R2 value of 0.9817, which is indeed larger than the R2 values for the linear fits through the points indicated by +, b, and *, that is, 0.9541, 0.9574, and 0.9614, respectively. We thus determine cp to be between 4% and 4.5%. Inserting cp ) 4% while fitting the data according to G′∼ (c - cp)t yields t ) 1.85, while inserting cp ) 4.5% yields t ) 1.57. We thus conclude that the above procedure yields t ) 1.7 ( 0.2 with cp ) 4.3 ( 0.3. The procedure uses graphic deduction of cp as a first step and determination of the value for t as the second step. The procedure yields an accuracy in t of about 10% due to the accuracy in graphic deduction of cp of about 10%. In Table 1, we have summarized the results for our ovalbumin system under different conditions. The accuracy in graphic deduction of cp is about 10% for any system. In case it is assumed that cp ) 0, that is, using a fractal model, one may determine an exponent, w, defined by G′ ∼ cw. This exponent w is given for comparison. We also have analyzed data on elastic moduli for some other protein gels and one non-protein gel. We have used the same procedure as for ovalbumin. The data, on casein, soy-glycinin, soy β-conglycinin, and Catapal (non-protein), were constructed from refs 6, 24, and 3, respectively. The results for these systems are also summarized in Table 1. Again, the accuracy in cp is about 10%. For the protein systems we analyzed, we find an average value of the percolation exponent t ) 1.79 ( 0.25, while cp varies by an order of magnitude. The value for t is in accordance with predictions of the so-called isotropic percolation model (i.e., t ) 2.06 ( 0.1615), which assumes an isotropic force between nearest neighbors on the lattice. Thus, our results suggest that the isotropic percolation model applies to the elasticity of the protein systems studied. We note that on the basis of the value of the exponent, the systems in Table 1 apparently exhibit a homogeneous structure, with one parameter describing the elastic energy of that structure. This in contrast to systems studied by Grant et al.,9 Kanai et al.,22 and Trappe and Weitz25 who report on systems in a two-phase region9 and systems that are presheared,22,25 respectively. These systems are expected to be nonhomogeneous and, according to Kantor and Webman, should belong to a different universality class, (24) Renkema, M.; Knabben, J.; Vliet, T. v. The effect of pH on heatinduced gel formation by soy proteins. Second International Symposium on Food Rheology and Structure, Zurich, 2000. (25) Trappe, V.; Weitz, D. A. Phys. Rev. Lett. 2000, 85, 449-452.
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Figure 1. (a) (G′(/N/m2))1/t as a function of concentration, c (in w/w %), for t ) 1.7 (-), 1.8 (s), 1.9 ()), 2 (9), 2.5 (2), 3 (×), 3.5 (*), 4 (b), and 4.5 (+) (ovalbumin, pH ) 2, 30 mM). The lines drawn are linear fits through the data taking t ) 1.7, with a R2 value of 0.9817, and t ) 1.8 with a R2 value of 0.9806. (b) Enlargement of (a), better visualizing the bending away from the straight line which is the linear fit through the data for t ) 1.7. The linear fit for t ) 1.7 yields a R2 value of 0.9817, which is larger than the R2 values for the linear fits through the points indicated by +, b, and *, i.e., 0.9541, 0.9574, and 0.9614, respectively. Table 1. Values for cp (in w/w %) and the Exponent t, as well as the Values of the Exponent w, Referring to the Case Where cp Is Assumed To Be Zeroa system
pH
NaCl (mM)
w
cp (w/w %)
t
ovalbumine ovalbumine ovalbumine ovalbumine ovalbumine casein (type 1) casein (type 2) soy glycinin soy glycinin soy β-conglycinin soy β-conglycinin Catapal
3 3 2 3.6 3.6 4.6 4.6 3.8 7.6 3.8 7.6 5.5
30 0 30 0 30 140 140 200 200 200 200 160
2.6 5.8 4.0 3.8 2.8 2.56 4.6 2.98 3.52 3.18 4.64 4.1
2.5 6.5 4.3 3.5 2.0 5 12 1.5 1.5 1 1 2.5
1.61 1.56 1.71 1.76 1.80 1.93 2.0 1.56 2.04 1.82 1.9 1.7
a
Partly, the data are constructed from the literature (see text).
characterized by a critical exponent with a lower bound of 2.85 and an upper bound of 3.55. Indeed, Grant et al.9 report 3 ( 0.5, Kanai et al.22 report 1-4, depending on the initial particle concentration and the strength of the preshearing effect, and Trappe and Weitz report 4.1. Comparison between the values for the exponents w and t in Table 1 for the various systems (including the non-protein system, Catapal3) shows that the exponent w varies considerably with the type of system, while t
remains around 1.8. In establishing a generic description, it therefore appears justified to use a percolation model. Our results imply that systematic efforts to predict elastic moduli of protein systems as a function of concentration may be restricted to making predictions on cp. In general, cp will depend on the type of protein and on parameters that affect molecular interactions such as pH and salt concentration, as is apparent from Table 1. For example, a comparison of cp for ovalbumin at pH ) 3 at 0 and 30 mM NaCl shows that cp decreases upon increasing salt concentration, that is, upon increasing proteinprotein attraction. Such a decrease in cp upon increasing attraction between the particles has also been reported for microemulsions.4,26 Notably, in the area of microemulsions the concept of using cp as a function of experimental parameters has been successfully applied to explain conductivity, Kerr effect, and light scattering data in a self-consistent way.26 In particular, the effect of an external electric field on cp has been incorporated successfully.27 Recently, in a similar fashion, to explain viscoelastic behavior of systems containing weakly attractive carbon black particles, the interplay of interaction and critical (26) Hilfiker, R.; Eicke, H.-F. J. Chem. Soc., Faraday Trans. 1 1987, 83, 1621-1629. (27) Eicke, H.-F.; Hilfiker, R.; Thomas, H. Chem. Phys. Lett. 1986, 125, 295-298.
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percolation concentration has been used by Trappe and Weitz.25 In their work, a change in the attractive interaction is also found to shift the percolation threshold while the viscoelastic properties of the network also remain scale invariant. The above suggests a general applicability of the socalled isotropic force percolation model for the protein systems studied.15 It also suggests that a systematic study of cp as a function of preparation conditions and/or protein type (i.e., charge distribution and consequent anisotropic interaction, leading to an anisotropic aggregate morphology) would be fruitful in describing the resulting gel properties of protein systems. Such a route would allow
van der Linden and Sagis
one also to systematically incorporate effects by external fields, such as for example gravitational and shear fields, and effects due to mixing with polysaccharides and other proteins. Acknowledgment. EvdL gratefully acknowledges stimulating discussions with Dr. Rolf Hilfiker on fractal and percolation theories. Mireille Weijers is gratefully acknowledged for performing the rheology measuments on ovalbumin. LA010705U
