Aerogelation Process Simulation by a Cluster-Cluster Aggregation Algorithm

A ballistically-limited cluster-cluster aggregation (BLCA) model was developed to simulate aerogelation processes. In the model, the clusters move along linear paths, in random directions, in a finite box. When two aggregates contact each other, they are combined irreversibly to form a larger aggregate. As expected, the simulations show that the aggregation time is much shorter than that obtained with diffusion-limited cluster-cluster aggregation (DLCA) models. The minimum concentration, c_g, required for gel formation scales as L^D–3, where L is the length of the sides of the box and D is the fractal dimension of the aggregates (D 1.95). For a concentration c larger than c_g, the mean free path of the aggregating clusters, , scales as c^–1.1. The pair correlation function g(r) and its Fourier transform S(q) were determined for the single large aggregates formed at the end of the simulations. These functions indicate that there is a characteristic length which scales as c^1/(D–3). As observed previously for the DLCA model, there is a discrepancy between the fractal dimensions obtained from g(r) and S(q).     

The diffusion-limited reaction A+B→0 on a fractal substrate

We show in this paper how the segregation of reactants in the diffusion-limited reaction A+B0 on a fractal substrate arises. For spectral dimensions d_s 2 we obtain segregation controlled by the source and/or the intrinsic lifetime of the particles.     

Scaling properties of diffusion-limited reactions on fractal and euclidean geometries

We review our scaling results for the diffusion-limited reactions A + A 0 and A+B0 on Euclidean and fractal geometries. These scaling results embody the anomalies that are observed in these reactions in low dimensions; we collect these observations under a single phenomenological umbrella. Although we are not able to fix all the exponents in our scaling expressions from first principles, we establish bounds that bracket the observed numerical results.     

格点巢分形上的渗流模型

该文研究了一类格点分形图 (格点巢分形 )上的渗流模型 ,证明了该模型没有临界现象 ,进一步给出一个指数衰减律 .同时 ,指出一般有限分岔图上的渗流模型没有临界现象     

Pattern formation by growing droplets: The touch-and-stop model of growth

We investigate a novel model of pattern formation phenomena. In this model spherical droplets are nucleated on a substrate and grow at constant velocity; when two droplets touch each other they stop their growth. We examine the heterogeneous process in which the droplet formation is initiated on randomly distributed centers of nucleation and the homogeneous process in which droplets are nucleated spontaneously at constant rate. For the former process, we find that in arbitrary dimensiond the system reaches a jamming state where further growth becomes impossible. For the latter process, we observe the appearance of fractal structures. We develop mean-field theories that predict that the fraction of uncovered material (t) approaches to the jamming limit as (t)–()exp(Ct ^d ) for the heterogeneous process and as a power law for the homogeneous process. Exact solutions in one dimension are obtained and numerical simulations ford=1–3 are performed and compared with mean-field predictions.     

Different types of self-avoiding walks on deterministic fractals

Normal and indefinitely-growing (IG) self-avoiding walks (SAWs) are exactly enumerated on several deterministic fractals (the Manderbrot-Given curve with and without dangling bonds, and the 3-simplex). On then th fractal generation, of linear sizeL, the average number of steps behaves asymptotically as N=AL ^D _saw+B. In contrast to SAWs on regular lattices, on these factals IGSAWs and normal SAWs have the same fractal dimensionD _saw. However, they have different amplitudes (A) and correction terms (B).     

Self-similar sets in complete metric spaces

We develop a theory for Hausdorff dimension and measure of self-similar sets in complete metric spaces. This theory differs significantly from the well-known one for Euclidean spaces. The open set condition no longer implies equality of Hausdorff and similarity dimension of self-similar sets and that has nonzero Hausdorff measure in this dimension. We investigate the relationship between such properties in the general case.

     

Phonons and fractons in sol-gel alumina: Raman study

We report Raman scattering from the boehmite,γ-, δ- andα-phases of the alumina gel. Samples are characterized by transmission and scanning electron microscopy, X-ray diffraction and density measurements. The main Raman line in the boehmite phase is red-shifted as well as asymmetrically broadened with respect to that in the crystalline boehmite, signifying the nanocrystalline nature of the gel. Raman signatures are absent in theγ- andδ-phases due to the disorder in cation vacancies. We also show that low frequency Raman scattering from the boehmite phase resembles that from a fractal network, characterized in terms of fraction dimension . Taking Hausdorff dimension D of the boehmite gel to be 2.5 (or 3.0), the value of is 1.33±0.02 (or 1.44±0.02), which is close to the theoretically predicted value of 4/3.