Test of the Physical Interpretation of the Structural Coefficient for Colloidal Clusters M. Tirado-Miranda, † A. Schmitt, † J. Callejas-Ferna ´ ndez, † and A. Ferna ´ ndez-Barbero* ,‡ Biocolloid and Fluid Physics Group, Department of Applied Physics, University of Granada, 18071 Granada, Spain, and Complex Fluid Physics Group, Department of Applied Physics, University of Almerı ´a, 04120 Almerı ´a, Spain Received April 1, 2000. In Final Form: June 7, 2000 I. Introduction It is well established that random mesoscopic aggrega- tion leads to clusters with internal structure described by a fractal geometry. Branched clusters grow under pure diffusive conditions, exhibiting a fractal dimension around 1.75. Although clusters seem to be good fractals, discussion is actually held on the unicity of the fractal dimension for describing the whole aggregate morphology. 1-4 The fractal dimension, d f , links the number of primary particles per cluster, n, to the aggregate radius of gyration, R g , according to the relationship n ) k 0 (R g /R 0 ) d f , 5 where R 0 is the monomer radius. The structural coefficient, k 0 , has proven to contain additional information on cluster morphology. In fact, for fractal aggregates with identical d f , R g , and R 0 , but different structural coefficient, the lower the k 0 value, the smaller the number of primary particles contained in a cluster. Larger distance should then exist among them, being the structural coefficient related to that distance. Hence, both structural coefficient and fractal dimension parametrize how fractal objects fill space. Oh and Sorensen 6 found dependence of the structural coefficient on the monomer overlapping in a cluster: k 0 ) k 0 (1) δ d F , where δ ) 2R 0 /l, with l being the center-to- center distance. Thus, k 0 should increase as overlap increases. This description coming from simple arguments, agrees with simulations and experiments from stereoviews of three-dimensional (3D) aggregates for which high k 0 values were found. 1 In the present paper, that correlation is extended for systems in which an extra separation is considered between monomers. The relationship presents a similar form, with δ ) 2R 0 /(2R 0 + S), where S is the surface-to- surface distance. An experimental test is performed using surfactant-covered particles aggregating under diffusive conditions. Thus, monomers into the clusters are forced to be located at a certain distance. The mean diffusion coefficient, assessed by dynamic light scattering (DLS), was employed to monitor the aggregation processes. Fractal dimensions were measured by static light scat- tering (SLS). The structural coefficient, k 0 , was determined for bare and surfactant-covered particles. Results not only support a physical interpretation of the structural coef- ficient in terms of a separation among monomers, but also allow that separation to be calculated. II. Theoretical Background The average diffusion coefficient for aggregating col- loidal particles is experimentally assessed from light scattering as 5 The cluster structure factor, S(qR g ), accounts for the spatial distribution of individual particles within the clusters (q is the scattering vector and R g is the radius of gyration for a cluster formed by n monomers). The diffusion coefficient, D(n), describes the system hydrodynamics as a function of the particle size and the cluster size distribution, N(n,t), containing information on the ag- gregation mechanism. The cutoff size, n c , is the upper limit of particles per aggregate, which rises as clusters grow up. The cluster structure factor, S(qR g ), is related to the mean light scattered intensity according to I(q,t) ∼ S(qR g ). The structure characteristic length scale is determined by the cluster size, R g . For fractal clusters, the structure factor shows a long-time asymptotic power law behavior, S(qR g ) ∼ (qR g ) -d F , valid in the range qR 0 e 1 e qR g . The size and structure of the aggregates determine cluster motion. 7,8 The diffusion coefficient depends on the translational and rotational diffusion coefficients, D t (n) and D r (n), respectively. The total diffusion coefficient is then written as D(n) ) D t (n) + D r (n), where coupling effects have been neglected. Assuming fractal structure for clusters, the average translational diffusion coefficient, 〈D t (n)〉, is expressed as a function of the number average mean cluster size, 〈n〉, according to 〈D t (n)〉 ) B〈n〉 -1/d f . 6 B ) D 0 k 0 1/d f is a constant containing the cluster fractal dimension and the structural coefficient, k 0 , and the diffusion coefficient for free monomeric particles, D 0 . For certain experimental conditions (described below), the rotational contribution is negligible and the overall diffusion coefficient is identified only to the translational coefficient. The cluster size distribution, N(n,t), is usually described for diluted systems by Smoluchowski’s equation. The physical information on the aggregation mechanism is contained into the aggregation kernels, k ij , which param- etrize the rate at which i-mers bind to j-mers. Diffusion-limited cluster aggregation is modeled by a Brownian kernel 9 but no analytical solutions are known for the Smoluchowski equation. However, constant kernel has proven to be a good approximation. 10 For monomeric initial conditions the cluster size distribution for constant kernel (the kernel is size independent, i.e., k ij ) k 11 ) cte) † University of Granada. ‡ University of Almerı ´a. * To whom correspondence should be addressed. (1) Samson, R. J.; Mulholland, G. W.; Gentry, J. W. Langmuir 1987, 3, 272. (2) Wu, M. K.; Friedlander, S. K. J. Colloid Interface Sci. 1993, 159, 246. (3) Cai, J.; Lu, N.; Sorensen, C. M. J. Colloid Interface Sci. 1995, 171, 470. (4) Sorensen, C. M.; Roberts, G. J. Colloid Interface Sci. 1997, 186, 447. (5) Jullien, R.; Botet, R. Aggregation and Fractal Aggregates; World Scientific Publishing Co.: Singapore, 1987. (6) Oh C.; Sorensen, C. M. J. Colloid Interface Sci. 1997, 193, 17. (7) Dietler, G.; Aubert, C.; Cannell, D. S.; Wiltzius, P. Phys. Rev. Lett. 1986, 57, 3117. (8) Hurd, A. J.; Flower, W. L. J. Colloid Interface Sci. 1988, 122, 178. (9) Broide M. L.; Cohen R. J. Phys. Rev. Lett. 1990, 64, 2026. (10) Ferna ´ ndez-Barbero, A.; Schmitt, A.; Cabrerizo-Vı ´lchez, M. A.; Martı ´nez-Garcı ´a R. Physica A 1996, 230, 53. D h (t) ) ∑ n)1 n c N(n,t)n 2 S[qR g (n)] D(n) ∑ n)1 n c N(n,t)n 2 S[qR g (n)] (1) 7541 Langmuir 2000, 16, 7541-7544 10.1021/la0005107 CCC: $19.00 © 2000 American Chemical Society Published on Web 08/18/2000