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 d f, 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 d f from experimental scattering data is indicated. Introduction Colloidal systems, when aggregating at relatively large particle volume fractions, may form a continuous three- dimensional 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 objects 1 and therefore characterized only by one parameter, i.e., the fractal dimension, d f . 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 measure- ments. 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 thickness 3,5-7 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 d f is estimated from this slope, it can be concluded that multiple scattering does not affect the estimated d f value. However, besides these practical problems, some evi- dence has been reported that question whether the fractal dimension obtained from the scattering structure factor S(q), which is the only experimentally measurable quan- tity, 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) con- ditions. They obtained the particle-density correlation function of gels and from this estimated the “true” d f value of the clusters that constitute the gels. The d f 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 trans- formation of the particle-density correlation function and estimating the d f value from the slope of the log-log plot of the structure factor, values of d f 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 d f 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 d f 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 d f 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 d f values of RLCA gels (closed circles), determined experimentally from the slope of the * To whom correspondence should be addressed. Tel: 0041-1- 6323034. E-mail: morbidelli@tech.chem.ethz.ch. † 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. 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