Roberto A. Garza-López

Invariance relations for random walks on hexagonal lattices

Abstract We consider the problem of random walks on finite, N =(2 k ×2 k ) hexagonal lattices with a single, deep trap, and subject to periodic boundary conditions. An exact expression is obtained for calculating the invariance relation linking the set M of n th nearest-neighbor sites surrounding the trapping site, viz., (2M−3)N−{2M+6+3M[ ln (M/6)/ ln (2)]}. This result may be used to obtain approximate values of the overall mean walklength 〈 n 〉. The results are compared with exact numerical results, with the predictions of the asymptotic expression of Montroll and Weiss, and linked to current studies in nanotube chemistry.

Tortuosity factor for permeant flow through a fractal solid

The theory of finite Markov processes is used to calculate a normalized tortuosity for a porous solid with a well-defined internal structure characterized by an interconnected network of serial and parallel channels. The model introduced is based on the Menger sponge, a symmetric fractal set of dimension D=ln 20/ln 3. Numerically exact values of the mean walklength 〈n〉 for a permeant diffusing through this system are calculated both in the presence and absence of a uniform gradient (bias or external field) acting on the permeant. The ratio of walklengths is then used to define unambiguously a normalized tortuosity for the medium. Different assumptions on the initial spatial distribution of the permeant are investigated and the study is designed so that the effects of one or multiple exit sites are quantified. As expected, the tortuosity factor is dependent on system size, and quantitative results are presented for the first- and second-generation Menger sponge. Our calculations document that for a given s...

‘‘Order–disorder’’ phenomena in a diffusion‐reaction model of interacting dipoles on a surface

We study the reaction efficiency of a surficial process in which a diffusing, tumbling dipole A reacts (eventually and irreversibly) with a stationary target dipole B. In contrast to earlier studies of such irreversible diffusion‐reaction events (A+B→C), we consider the situation where at each and every site of the space accessible to the diffusing coreactant A, there is also embedded a fixed dipole. To quantify the influence on the reaction efficiency of (angle‐averaged, dipole–dipole) potential interactions between the tumbling dipole A and the ensemble of stationary dipoles, we design a lattice‐statistical model to describe this problem and use both analytical methods and numerical techniques rooted in the theory of finite Markov processes to work out its consequences. Specifically, we define the reaction space to be an n×n=N square‐planar lattice with the target dipole occupying the centrosymmetric site in that space and determine the mean number of steps required before the irreversible event, A+B→C,...

Pattern development in cellular automata triggered by site-specific reactive processes

Abstract The (discretized) evolution of regular and fractal patterns, initiated by a site-specific event, is studied via simulation on a simple model. Consistent with recent theories of diffusion on percolation networks, our results show that the development of fractal patterns is distinctly slower than for regular (euclidean) patterns. The possible relevance of our results to the evolution of symmetry-breaking instabilities is brought out.

Diffusion–reaction processes involving interacting dipoles on Euclidean vs. fractal architectures

Abstract The relative importance of nearest-neighbor and non-nearest-neighbor excursions of a dipolar species interacting with a fixed target molecule in the presence of a constellation of N fixed dipoles is explored. Particular attention is paid to the topology of the reaction space, and we contrast the results obtained for reactive processes confined to the d =2 surface of a Cartesian shell vs. those obtained for a fractal solid, here the D =2.7268 Menger sponge. The relevance of the results obtained to experimental studies on the lifetime of reactive species diffusing in/on the surface/interior of a zeolite is brought out.

Transition behavior on lattices with ladders

Abstract We consider a d =2-dimensional, planar lattice of coordination number v =4 separated into distinct regions by a two/three-leg ladder. Random walks on this composite inhomogeneous lattice are studied as a function of the coupling of the ladder to the host lattice and with respect to the location of the trapping site. We find that the efficiency of trapping at sites on the host lattice decreases, whereas the efficiency of trapping at ladder sites increases, both dramatically, as the coupling of the ladder to the lattice decreases.

Diffusion of a dimer through a structured medium

Abstract We study the random motion of a dimer diffusing through a compartmentalized system, the spatial dimensions of which are comparable to the size of the diffusing species. By solving numerically the stochastic master equation, the transit time across the medium is determined. Two different motions of the dimer are considered. First, the dimer is assumed to execute purely translational displacements (only); second, a “tumbling” motion is studied by allowing the dimer to pivot about either end as it translates through the medium. For two choices of initial condition, the ratio of the transit time for tumbling versus translation is found to be ≈2.4 for the two segmented geometries considered here.