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Estimation of Fractal Dimension in Colloidal Gels Marco Lattuada, Hua Wu, Anwar Hasmy,† and Massimo Morbidelli* Swiss Federal Institute of Technology Zurich, Institute for Chemical- and Bioengineering, ETH-Ho¨ nggerberg/HCI, CH-8093 Zurich, Switzerland, and Laboratorio de Fı´sica Estadı´stica de Sistemas Desordenados, Centro de Fı´sica, IVIC, Apartado 21827, Caracas 1020A, Venezuela Received January 9, 2003. In Final Form: May 5, 2003 Gels are complex structures that can be described in terms of the fractal dimension, df, of the clusters that constitute them. Classical techniques, based on the structure factor obtained through scattering measurements, provide erroneous values of df, which differ from the values estimated from the corresponding particle-density correlation function. The source of this error is identified in this work, and a procedure to get the correct value of df from experimental scattering data is indicated.
Introduction Colloidal systems, when aggregating at relatively large particle volume fractions, may form a continuous threedimensional network, usually referred to as a gel. Although the structure of a gel is highly disordered and complex, as a first approximation it can be described as a connection of clusters regarded as fractal objects1 and therefore characterized only by one parameter, i.e., the fractal dimension, df. Light, X-ray, and neutron scattering are still the most widely used techniques to determine the fractal dimension in various systems such as colloidal gels, aerogels, rock, clay, polymer, and micellar solutions,2-8 even though other alternative experimental methods have been proposed, like microscopy,9 rheology,10 permeability measurements.11 However, the application of scattering techniques to concentrated systems is still challenging, because of high turbidity and multiple scattering. In these cases, one often needs to use a very small sample thickness3,5-7 or to apply some special techniques to eliminate multiple scattering.12 Recently, Lattuada et al.13 have shown experimentally that multiple scattering affects only the magnitude of the scattered light intensity but not the slope of the log-log plot of the intensity curve. Since df is estimated from this slope, it can be concluded that multiple scattering does not affect the estimated df value. * To whom correspondence should be addressed. Tel: 0041-16323034. E-mail:
[email protected]. † IVIC. (1) Poon, W. C. K.; Haw, M. D. Adv. Colloid Interface Sci. 1997, 73, 71. (2) Ehrburger-Dolle, F.; Hindermann-Bischoff, M.; Livet, F.; Bley, F.; Rochas, C.; Geissler, E. Langmuir 2001, 17, 329. (3) Pignon, F.; Piau, J.-M.; Magnin, A. Phys. Rev. Lett. 1996, 76, 4857. (4) Petekidis, G.; Galloway, L. A.; Egelhaaf, S. U.; Cates, M. E.; Poon, W. C. K. Langmuir 2002, 18, 4248. (5) Sorensen, C. M.; Hageman, W. B.; Rush, T. J.; Huang, H.; Oh, C. Phys. Rev. Lett. 1998, 80, 1782. (6) Radlinski, A. P.; Radlinska, E. Z.; Agamalian, M.; Wignall, G. D.; Lindner, P.; Randl, G. O. Phys. Rev. Lett. 1999, 82, 3078. (7) Broseta, D.; Barre, L.; Vizika, O. Phys. Rev. Lett. 2001, 86, 5313. (8) Hasmy, A.; Foret, M.; Pelous, J.; Jullien, R. Phys. Rev. B 1993, 48, 9345. (9) Cai, J.; Lu, N.; Sorensen, C. M. Langmuir 1993, 9, 2861. (10) Wu, H.; Morbidelli, M. Langmuir 2001, 17, 1030. (11) Mellema, M.; Heesakkers, J. W. M.; van Opheusden, J. H. J.; van Vliet, T. Langmuir 2000, 16, 6847. (12) Nicolai, T.; Urban, C.; Schurtenberger, P. J. Colloid Interface Sci. 2001, 240, 419. (13) Lattuada, M.; Wu, H.; Morbidelli, M. Phys. Rev. E 2001, 64, 061404-1.
However, besides these practical problems, some evidence has been reported that question whether the fractal dimension obtained from the scattering structure factor S(q), which is the only experimentally measurable quantity, is really representative of the cluster structure. Hasmy et al.14,15 have recently performed Monte Carlo (MC) simulations of colloidal gelation inside cubic boxes, under diffusion-limited cluster aggregation (DLCA) conditions. They obtained the particle-density correlation function of gels and from this estimated the “true” df value of the clusters that constitute the gels. The df values obtained as a function of the particle volume fraction are shown by the closed squares in Figure 1a. However, when computing the structure factor through Fourier transformation of the particle-density correlation function and estimating the df value from the slope of the log-log plot of the structure factor, values of df substantially smaller than the “true” ones are obtained. In addition, the so obtained values, as shown (open squares) in Figure 1a, decrease as the particle volume fraction increases. It should be pointed out that since the gel blobs are not rigorous fractal objects, we actually refer here to effective fractal dimensions. The fractal dimension estimated from the particle density correlation function was considered by Hasmy et al.14,15 as “true”, because the particle density correlation function gives the most direct information about the microstructure of a system. It is also reasonable on physical grounds that the df value increases with the particle volume fraction, since it must eventually approach 3 when the particle volume fraction becomes close to the closest packing threshold (∼0.72). To verify if the same phenomenon occurs for gels formed in reaction-limited conditions (RLCA), we have performed MC simulations under RLCA conditions at several particle volume fractions and have determined the df values of the generated gels from both the structure factor and the particle density correlation function. The results are shown in Figure 1b, where it can be seen that the situation is similar to that reported in the case of DLCA gels, i.e. the df value determined from the particle density correlation function increases with the particle volume fraction, while the one determined from the structure factor decreases as the particle volume fraction increases. In the same figure are also shown two df values of RLCA gels (closed circles), determined experimentally from the slope of the (14) Hasmy, A.; Jullien, R. J. Non-Cryst. Solids 1995, 186, 342. (15) Hasmy, A. J. Sol-Gel Sci. Technol. 1999, 15, 137.
10.1021/la034043f CCC: $25.00 © 2003 American Chemical Society Published on Web 06/17/2003
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elsewhere.14 We give here a short outline of its main features. The algorithm starts by placing in a cubic box of given size, L, a given number of spherical particles, N, of radius, Rp, according to a chosen particle volume fraction. At each step a cluster (or particle) is randomly chosen with a probability proportional to its diffusion coefficient, which, according to the Stokes-Einstein formula, is inversely proportional to its size. The cluster size and mass are related through the usual fractal scaling. Under DLCA conditions, every time two clusters collide, a new cluster is formed with mass equal to the sum of those of the colliding clusters, while under RLCA conditions we consider that the formation of a new cluster occurs with a probability of 0.001. Periodic boundary conditions are applied, so that when a cluster exits the box from one side, it reenters it immediately from the opposite one. The calculation ends when all the particles are connected to form a single cluster. The program also detects the gel time, which is defined as the time when the size of a formed cluster spans the entire box. Once a simulation is finished, the density correlation function is calculated by counting the number of pairs of particles dN located at a distance between r and r+dr and dividing this number by the volume of the spherical shell with radius r:
g(r) )
1 dN N24r2 dr
(1)
The result is then averaged over several simulations of the same system, so as to achieve a good statistical description. Results and Discussion
Figure 1. Values of the fractal dimension df in colloidal gels from Monte Carlo simulations as a function of the particle volume fraction φ, estimated from the structure factor S(q) or from the particle-density correlation function g(r). (a) DLCA simulations (Hasmy and Julien4). (b) RLCA simulations (this work), compared with experimental data (Lattuada et al.3) estimated on the basis of the structure factor S(q), determined from light scattering.
static structure factor using fluorinated polymer MFA latexes under RLCA conditions.13 It is seen that the obtained df values are in good agreement with those estimated from the structure factor of the gels generated by MC simulations and decrease with the particle volume fraction. From all the above, questions arise about the applicability of scattering techniques to the estimation of df in gels. In this work we explain why the df value estimated from the static structure factor is smaller than the “true” one and decreases with increasing particle volume fraction. For this, we analyze in detail the application of the Fourier transform to the particle density correlation function for getting the structure factor. Since there is no direct experimental measurement for the correlation function, this has to be done using three-dimensional off-lattice MC data. The Monte Carlo Algorithm The MC off-lattice cluster-cluster aggregation algorithm used to generate the data has been described
Figure 2a shows the particle density correlation function (open circles), g(r), in a gel formed at particle volume fraction φ ) 0.02, obtained from MC simulations under DLCA conditions, where r is the normalized (by particle diameter) distance between two particles. The g(r) profile can be roughly divided into three regimes: nonfractal (r < 3), fractal (3 < r < ξ), and intercluster and homogeneity regimes, where ξ, corresponding to the location of the minimum of g(r), characterizes the mean size of clusters in the gel and scales with φ as follows:
ξ ∼ φ-1/(3-df)
(2)
Similarly, Figure 2b shows the g(r) profile in the case of a MC-generated RLCA gel at φ ) 0.02, where again the same three regimes observed in the case of DLCA gels are present. Therefore, the same conclusions that are drawn in the following for DLCA gels can be extended also to RLCA gels. The nonfractal regime is due to the short-range shell region.16,17 In the intercluster and homogeneity regime, the local minimum characterizes the interconnections between clusters, and g(r) approaches unity as r increases, corresponding to uniform conditions. In the fractal regime, g(r) scales following the power law
g(r) ∼ rdf-3 for 3 < r < ξ
(3)
Thus, from the slope of the log-log plot of g(r) in this regime, one can estimate the “true” df value, which in the case of Figure 2a is df ) 2.1. This is done by fitting with (16) Dimon, P.; Sinha, S. K.; Weitz, D. A.; Safinya, C. R.; Smith, G. S.; Varady, W. A.; Lindsay, H. M. Phys. Rev. Lett. 1986, 57, 595. (17) Nicolai, T.; Durand, D.; Gimel, J.-C. in Light ScatteringPrinciples and Development; Brown, W., Ed.; Clarendon Press: Oxford, 1996.
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Figure 3. Structure factor S(q) calculated through the Fourier transform (3) of g(r). Curve 1: transform of the original g(r). Curve 2: transform only of the portion of g(r) g 1. Curve 3: transform of g(r) after having substituted the nonfractal regime with the scaling (2) as shown in Figure 2. (a) φ ) 0.02; (b) φ ) 0.04.
Figure 2. Particle density correlation function g(r) obtained from Monte Carlo simulations. (a) DLCA conditions, φ ) 0.02 and box size L ) 57.7 (Hasmy and Julien4). (b) RLCA conditions, φ )0.02 and box size L ) 86.6.
eq 3 the portion of the g(r) curve that exhibits the most linear behavior in double logarithmic coordinates. In particular, we have considered the portion between r ) 3.0 and r ) 3.7-4.0, depending upon the particle volume fraction. The structure factor of the gel, S(q), is the Fourier transform of g(r) and can be expressed as follows:18
S(q) ) 1 +
sin(qr) 4πr2 dr qr
∫0∞[g(r) - 1]
6φ π
(4)
where q is the scattering wave vector. Figure 3a shows the S(q) curve (curve 1) obtained from g(r) in Figure 2a. The maximum of S(q) at low q corresponds to the onset of the intercluster regime, i.e., the minimum in g(r), and S(q)f0 as qf0, since in the MC data the contribution to the scattered intensity due to the density fluctuations (always present in the experimental data) is neglected. Actually, if we ignore the intercluster regime, i.e., make the transformation only of the part of g(r) g 1 before the local minimum, we obtain another S(q) curve, which is curve 2 in Figure 3a. This curve can be considered as the structure factor of an isolated aggregate having the same characteristics as the clusters constituting the gel in Figure 2a. It is seen that such treatment changes the S(q) curve (18) Feigin, L. A.; Svergun, D. I. Structure Analysis by Small-Angle X-ray and Neutron Scattering; Plenum Press: New York, 1987.
only in the range of low q values before the maximum, while curves 1 and 2 after the maximum are identical. As mentioned above, the linear part of the S(q) curve after the maximum corresponds to the fractal regime. In principle, the slope of this part should be equal to -df. However, the obtained df ) 1.7, as already found by Hasmy et al.,14,15 is substantially smaller than the “true” one (df ) 2.1) obtained from the g(r) in Figure 3a. This indicates that when we perform the Fourier transform given by eq 4, the information in the fractal regime is “contaminated” by the other regimes of g(r). This is obviously not related to the intercluster and homogeneity regime, because variations of this regime, as demonstrated by curves 1 and 2 in Figure 3a, do not change the S(q) curve after the maximum. Then, let us investigate the effect of the nonfractal regime of g(r), which corresponds to the damped oscillations in the region of large q values of the S(q) curve. To do this, we substitute the MC data in the nonfractal regime with those based on the fractal scaling given by eq 3, which is equivalent to extending the fractal regime in Figure 3 to r ) 1. With such modified g(r), we obtain from eq 4 the new structure factor, given by curve 3 in Figure 3a. The shape of curve 3 is similar to that of curve 1, but more importantly, the slope of its linear part after the maximum now gives the value of df ) 2.1, which equals the “true” one. This result clearly indicates that the smaller df value estimated from the structure factor is due to the effect of the short range, nonfractal regime. In Figure 3b the argument above is repeated using a larger particle volume fraction (φ ) 0.04). It can be seen that the value of df ) 1.5 from S(q) (curve 1) is smaller than that estimated from the corresponding curve in Figure 3a, i.e., it decreases with the particle volume fraction. Again, if we substitute the MC data in the nonfractal regime of the g(r) curve with those based on the fractal scaling (3), the df value estimated from the new S(q) (curve 3) equals the one estimated from g(r), which, as shown later in the context of Figure 4, is given
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Figure 4. Comparison of the particle density correlation functions in gels, obtained from Monte Carlo simulations under DLCA conditions, for three values of the particle volume fraction.
by df ) 2.3. Moreover, it is seen from Figure 3a,b that the difference between the df values from S(q) and g(r) increases as the particle volume fraction increases, which implies that the effect of the nonfractal regime increases. This can be justified by considering that the mean cluster size in the gel decreases with φ, thus reducing the size of the fractal regime while that of the nonfractal regime remains unchanged. This clearly appears in Figure 4, where the g(r) curves for gels at three different φ values are compared. In the case of φ ) 0.06, the fractal regime becomes so small that the estimation of the “true” df value from the g(r) curve becomes difficult (in this case, caution must be taken even in applying the concept of fractal objects). On the other hand, it can be shown that when the particle volume fraction decreases below a certain value, the effect of the nonfractal regime becomes negligible, since the fractal regime becomes substantially larger than the nonfractal one, and then the same df value is estimated from S(q) and g(r). To further investigate the effects of the short range, nonfractal regime on the shape of the structure factor, we consider the Fourier transform of only this nonfractal regime, i.e., the portion of g(r) with r < 2:
Snf(q) ) 1 +
sin(qr) 4πr2 dr qr
∫02[g(r) - 1]
6φ π
(5)
The obtained Snf(q) is shown in Figure 5, together with the structure factors of gels at φ ) 0.02, 0.04, and 0.06. It is clearly seen that Snf(q) is responsible for the oscillations at large q values. The first peak appears at about q ) 2, and then the oscillations decay as q increases. It is not surprising that these features are the same for all particle volume fractions, since, as shown in Figure 4, the nonfractal regime of g(r) is practically independent of φ. Since the S(q) curves for each φ value contain the Snf(q) curve, it can be deduced from Figure 5 that the closer the first peak of Snf(q) is to the first peak of S(q), the stronger is the effect of Snf(q) in decreasing the slope of S(q) in the region between the two maxima. This implies that, since as φ increases the two maxima get closer, the estimated df value decreases. It should be pointed out that the difference in the df from g(r) and S(q) was also investigated by Gonzalez et
Figure 5. Fourier transform Snf(q) of the nonfractal regime of g(r), as given by eq 5, compared with the structure factors at various particle volume fractions, for DLCA gels.
al.,19,20 using three-dimensional on-lattice MC data, instead of the off-lattice MC data used in this work. Since the on-lattice MC model does not predict the short range (nonfractal) regime, it is clear from the analysis above that, using these data, the df values predicted from the g(r) or from the corresponding S(q) curve should be the same. We have verified that this is indeed the case by a very careful interpolation of the calculated g(r) reported by Gonzalez et al.20 in their Figure 4, before the Fourier transformation to compute the S(q). The fact that Gonzalez et al. found different df values, and actually larger values when using the S(q) curve, is most likely to be attributed to a improper Fourier transformation of the original g(r) curve, which appears in fact somewhat scattered. In this context, it is confirmed that off-lattice MC data are more representative of experimental data than on-lattice ones. In summary, the value of the gel fractal dimension estimated from S(q) is smaller than the “true” one estimated from g(r) due to the effect of the short range correlation (nonfractal) regime. For very diluted gels, where the cluster size is very large since these are fully developed fractal objects, this effect can be neglected. However, for more concentrated gels, since the mean cluster size decreases with the particle volume fraction, the effect of the short-range correlation becomes significant and the df value estimated from S(q) decreases with φ, as shown in Figure 1a,b. Accordingly, since the scattering techniques allow one to measure S(q), the correspondingly estimated df values are corrupted by the effect of the shortrange correlation. A possible methodology to remove such a problem is as follows. From the results in Figure 4, it can be inferred that the shape of g(r) in the nonfractal regime depends only on the type of aggregation process (e.g. DLCA or RLCA) but not on the particle volume fraction, as previously observed by Dimon et al.16 Accordingly, this regime can be modeled, for instance, using the MC simulations. With this computed nonfractal regime as a constraint, the remaining regimes (fractal, intercluster, and homogeneity) can be recovered by inverse Fourier transform of the gel structure factor measured by scattering techniques. Finally, the “true” df value can be estimated from the fractal regime of the so reconstructed g(r) function. (19) Gonza`lez, A. E.; Ramı´rez-Santiago, G. J. Colloids Interface Sci. 1996, 182, 254. (20) Gonza`lez, A. E.; Lach-Hab, M.; Blaisten-Barojas, E. J. Sol-Gel Sci. Technol. 1999, 15, 119.
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Conclusions In this work, we have investigated the “unusual” problem, encountered both with Monte Carlo simulations and with light scattering experiments, that the fractal dimension df in colloidal gels estimated from the structure factor S(q) decreases as the particle volume fraction φ increases. For this, the density correlation function, g(r), of DLCA gels has been first generated using the Monte Carlo off-lattice cluster-cluster aggregation algorithm. Then the g(r) function has been analyzed to determine the effect that each of its parts has on the shape of the S(q) curve obtained from the Fourier transform of g(r). It has been shown that the structure factor S(q) leads to df values smaller than the “true” ones, i.e. those computed directly from the correlation function g(r), because of the effect of the short range, nonfractal regime of g(r). In other words, the nonfractal regime of g(r) induces a perturbation in the scattering behavior of the gel, which leads to a reduction in the slope of the linear part of S(q) in the log-log plane and consequently to a reduced apparent
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fractal dimension. This effect becomes more significant at high volume fractions, because the fractal regime of g(r) reduces with increasing the particle volume fraction while the nonfractal regime of g(r) remains unchanged. It is to be noted that this effect is so strong that, although the real fractal dimension computed from g(r) increases with the particle volume fraction, the fractal dimension estimated from S(q) instead decreases with the particle volume fraction. In conclusion, to estimate correctly the fractal dimension in gels, a suitable model of the nonfractal regime of the density correlation function should be proposed and used as a constraint to recover its fractal regime by inverse Fourier transform of the experimental scattering data. Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant No. 2000-061883). LA034043F
