Fractal Geometry of Colloidal Aggregates

Abstract

Measurement of the fractal dimension, D, of colloidal aggregates of small silica particles is reported. We observe power-law decay of the structure factor [S(k) -k o] by both light and x-ray scattering showing that the aggregates are fractal. D is found to be 2.12+0.05, which is intermediate between recent numerical results for the kinetic models of diffusionlimited aggregation (D = 2.5) and cluster aggregation (D = 1.75), but is rather close to the value for lattice animals (D = 2.0), which are equilibrium structures. PACS numbers: 64.60.Cn, 05.40.+j, 61.10.Fr Understanding aggregation has been a primary goal in the field of colloidal physics for many years. 'In addition to its importance in commercial processes, aggregation is a prototypical example of a complicated random process which may display such features as self-similarity, scaling, and universality. 2 These features have been revealed by computer simulations.It has been shown recently, for example, that the two most popular models, diffusion-limited aggregation3 4 (DLA) and cluster-cluster aggregation 6 (CA), produce ramified structures that are self-similar in that the two-point density-density correlation function p2(r) is of a power-law form, p, (r) -r ", for values of r intermediate between the lattice constant or monomer size a and the cluster size R. Structures described by Eq. (1) are self-similar and are known as fractals; their essential geometric properties are independent of length scale. In ddimensional space, they are characterized by a fractal or Hausdorff-Besicovitch dimension D related to A by D = d -A. An immediate consequence of Eq.(1) is that the radius of gyration of a cluster RG is related to the number of particles it contains N, by N, -RD (2) A uniform object has D = d, while more open structures in which the density decreases with distance from the center have D & d. Numerical simulations have shown D to be -Sd/6 for DLA in d dimensions for both lattice (2~d~6) and nonlattice (d=2, 3) diffusion, independent of sticking coefficient s (0. 1» s~1). Cluster aggregation in which many particles diffuse and stick together to form clusters which also diffuse and stick yields self-similar aggregates having D =1.45 and 1.75 in two and three dimensions, respectively.

Light Scattering Study of the Formation and Structure of Partially Hydrolyzed Poly(acrylamide)/Calcium(II) Complexes

“…The RLCA has been observed in several colloid systems and modeled by computer simulation. [17][18][19][20] In general, the fractal dimension d f is defined as M ∼ R df , where M is the molar mass and R is the cluster size. 16 In RLCA, d f ∼ 1.55 and ∼ 2, respectively, in 2-dimensional and hierarchical 3-dimensional simulations.…”

Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media

“…An example that perhaps best illustrates the utility of this correspondence is the superconducting network, where we found that all the physics is given by the unscreened perimeter: ~s = du [Eq. (17)]. …”

Application of the Cluster–Cluster Aggregation model to an open system

“…The Cluster-Cluster Aggregation (CCA) model bridges the study of colloid aggregation by computer simulation and laboratory experiment [3][4][5][6][7][8][9][10]. Two distinct and limiting regimes of irreversible colloid aggregation have been identified by computer simulation with the CCA model [11,12].…”